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author:
- 'I. E. Kalcheva [^1]'
- 'M. G. Hoare'
- 'J. S. Urquhart'
- 'S. Kurtz'
- 'S. L. Lumsden'
- |
\
C. R. Purcell
- 'A. A. Zijlstra'
bibliography:
- 'Breport.bib'
date: 'Received Month Day, 2018; accepted Month Day, 2018'
subtitle: 'III. A catalogue of northern ultra-compact H [II]{} regions'
title: 'The Coordinated Radio and Infrared Survey for High-Mass Star Formation'
---
=1
Introduction
============
The puzzle of the birth and early life of stars exceeding 8$ $ is not yet fully assembled. Some of the obstacles towards building a complete evolutionary sequence for these massive stars include their rarity due to their brief lifetime and the rapid evolution of each observable stage. The main sequence is reached while the young star is still embedded within a dense core and as a result the early phases of its development are hidden behind a heavy veil of dust. A well-founded distinction between global and individual properties of sources in each evolutionary stage is hampered by the strong influence of other objects within the multiple systems where massive stars typically form.
It is a vital task for modern astronomy to overcome these challenges. Massive stars affect not only their immediate surroundings, but also shape their parent galaxy. Their formation controls phase changes in the interstellar medium (ISM) via the profuse emission of ionising UV photons [@molinari:2014]. Processes associated with their evolution, such as winds, outflows, expanding regions and supernovae, stir the ISM and enrich it with heavy elements [@yorke:2007]. This makes their understanding a stepping stone towards a more detailed picture of the Milky Way, as well as the extent to which galaxy formation and evolution in general is driven by stellar populations.
After a massive star has formed, it ionises a pocket of hydrogen gas which remains confined in its vicinity while expanding – thus forming an region. regions are highly convenient tracers of massive star formation, as they are clearly visible across the Galactic plane in the cm-regime [@bania:2009]. Distinguished as a separate observational class by [@churchwell:1989a], ultra-compact (UCHII) regions link the accretion phase when a massive protostar is formed, and the development of a more diffuse and less obscured region. UCHII regions are defined as embedded photoionised regions $\lesssim$ 0.1 pc in diameter, with emission measures $\gtrsim$ 10$^{7}$ $\rm ~pc ~cm^{-6}$ and electron densities $n_{\rm e}$ $\gtrsim$ 10$^{4}$ $\rm ~cm^{-3}$ [@churchwell:1989a]. They are the most luminous objects in the Milky Way in the far-IR, and are observable in the radio part of the spectrum if their luminosities are equivalent or higher than a B0.5 main-sequence star. The Lyman continuum ionising flux corresponding to zero-age main-sequence stars with spectral class from B2 to O5 is in the range $10^{44}$ – $10^{49}$ $\rm ~photons ~s^{-1}$. Estimating the distances to UCHII regions, together with their density distributions, luminosities, morphologies, kinematics, and relationship to the parent molecular clouds is essential. These properties can be used to help understand not only the effect of UCHII regions on their environment, but also test the existing evolutionary models of massive star formation and the structure of the Milky Way [@hoare:2007].
The bounds of current understanding of massive star formation are widened by the modern family of Galactic plane surveys, covering the dust (from hot to cold), the molecular and the ionised gas. These include the GLIMPSE programme [@churchwell:2009; @benjamin:2003] and its companion MIPSGAL survey (mid-IR) [@carey:2009], the UKIDSS GPS survey (near-IR) [@lucas:2008], the BU-FCRAO Galactic Ring Survey (CO) [@jackson:2006], the ATLASGAL survey (sub-mm) [@schuller:2009], the VGPS survey () [@stil:2006], the CORNISH survey (radio) [@hoare:2012; @purcell:2013]. These legacy surveys provide resolution and sensitivity apposite to the detection and discerning of sources occupying angular scales down to $\sim$ 1$''$. At the same time, they cover wide areas on the sky and overcome the high extinction of the plane. In this way, a multi-wavelength treasure trove of unbiased, high-resolution and statistically representative data are available to aid the studies of the earliest phases of massive star formation.
The CORNISH survey[^2], the first Galactic plane survey that is comparable in resolution and coverage with the GLIMPSE data, maps the compact ionised gas within the ISM. At present, the CORNISH catalogue of the northern Galactic plane, imaged with the VLA, is the most uniformly sensitive, homogeneous and complete list of northern compact radio sources at 5 GHz. The CORNISH team identified 240 ultra-compact region candidates. The sample provides the largest unbiased and uniform collection of these objects to date.
Previous radio UCHII samples comprise predominantly IR-targeted surveys based on IRAS point sources with far-IR colours similar to well-known UCHIIs. [@churchwell:1989a; @churchwell:1989b] selected a sample of 75 UCHIIs (out of $\sim$ 1600 candidates in the Galaxy) to observe at 6 cm and 2 cm with the VLA using this method and classified them morphologically. Similarly, [@miralles:1994] selected and observed 12 sources at 6 and 2 cm with the VLA. [@garay:1993] also based their selection on strong IRAS point sources associated with compact regions and produced multi-frequency observations with the VLA (resolved and morphologically classified). [@kurtz:1994] performed radio-continuum observations on 59 UCHIIs, again IRAS-selected. [@depree:2005] located and resolved a hundred objects within the massive star forming regions W49A and Sgr B2 from VLA radio continuum and radio recombination line emission observations (and revisited the [@churchwell:1989a] morphological classification).
The RMS survey [@urquhart:2007; @urquhart:2009; @lumsden:2013] marked a new era of massive star formation studies. Colour-selected sources from MSX [@price:2001] and 2MASS [@skrutskie:2006] were followed up by arcsecond-resolution IR, as well as mm and radio observations. These, together with archival data, were used to identify for the first time a Galaxy-wide sample of $\sim$ 2000 candidate massive young stellar objects (MYSOs) and regions in approximately equal numbers [@urquhart:2012].
The biggest disadvantage of IR-selection in UCHII studies is the discrimination against the most deeply-embedded sources. The issue is resolved by unbiased radio surveys. The first larger-scale unbiased survey at 1.4 GHz (inner Galaxy, VLA B and A/B configuration) was conducted by [@zoo:1990] and was followed by (VLA C configuration) 5 GHz observations [@becker:1994] covering about a fourth of the GLIMPSE region. The survey [now contained within a larger collection of re-reduced archival radio data known as MAGPIS, see @helfand:2006] is useful for the study of extended thermal sources such as evolved regions, bubbles, etc. However, UCHII regions are unresolved or marginally resolved, and even in some instances missed altogether due to insufficient sensitivity. The CORNISH survey covers the entire GLIMPSE region and its noise level of 0.4 mJy ensures the detection of virtually all UCHIIs around a B0.5V star or earlier within the covered area [@hoare:2012].
This work explores the sample of northern ultra-compact regions from the CORNISH survey, the majority of which are also conveniently available within the related surveys, to study the properties of this deeply embedded phase. The sample selection procedure is presented (§\[id\]) and the nature of the identified sources is verified through their observational properties (§\[validation\]) and spectral indices (§\[spind\]). Candidate short-timescale variable sources are presented in §\[variability\]. The methodology of obtaining the distance information and the computed distances are presented in §\[distances\]. The derived physical properties are discussed in §\[physprop\]. Results from performing automated polygon-based aperture photometry on UKIDSS and GLIMPSE infrared associations are discussed in §\[IR\] and presented in an extended source catalogue table. The spectral energy distributions of the sample sources from near-IR to sub-mm wavelengths were explored and utilized via SED fitting to obtain the UCHII bolometric luminosities (§\[lbol\]). In §\[comparison\_methods\], different UCHII search methods in blind surveys are compared. The present work is summarised in §\[summary\].
Identification of the CORNISH UCHII sample {#id}
==========================================
The CORNISH catalogue comprises 3062 sources above a 7$\sigma$ detection limit. Above this limit, less than one spurious source is expected [@purcell:2013]. The 240 UCHII regions were selected from this high-reliability catalogue. All UCHIIs were visually identified, following criteria similar to the RMS survey, where millimetre, infrared and radio data were used for source classification [see @lumsden:2013]. It should be noted that the CORNISH team also identified 48 diffuse regions (as judged by comparison to the MAGPIS and GLIMPSE surveys), which are a part of the larger sample of CORNISH regions.
The full UCHII radio sample has counterparts in GLIMPSE, in all four bands (namely, IRAC 3.6, 4.5, 5.8, and 8.0 $\upmu$m), with excellent positional accuracy in both surveys. This was utilised for the source identification. In the case of UCHIIs, there is overall a good agreement between the mid-IR and the 5 GHz source morphology, which ensures that the same source was captured by both surveys. A particularly good check for this are the 8 $\upmu$m images. They show the morphology produced by a combination of warm Lyman-$\alpha$ heated dust inside the ionised zone [@hoare:1991] and polyaromatic hydrocarbon (PAH) emission from just outside the ionisation front [@watson:2008]. This can be seen in Fig. \[fig:iding\]. Comparison between both wavelengths is therefore useful for the distinction of adjacent unrelated sources and over-resolved emission [see @purcell:2013]. It can also reveal the most heavily obscured objects (those deeply embedded in infrared-dark clouds (IRDCs) or hidden behind dust lanes), as shown in Fig. \[fig:big\_difference1\].
MYSOs, unlike UCHIIs, do not have strong 8.0 $\upmu$m PAH emission, which is consistent with the lack of a strong UV continuum [e.g. @gibb:2004]. They are also generally undetected at 5 GHz, even though there are a few known MYSOs observed at radio wavelengths, with radio luminosities ($S_{\nu}D^{2}$) always below $\sim$ 30 mJy $\rm kpc^{2}$ (discussed in @hoare:2007, @lumsden:2013, and seen from the recent sample by @purser:2016). Sources above this limit are thus regions or planetary nebulae (PNe).
This leaves PNe as possible contaminants of the selected sample. Unlike PNe, UCHIIs are found within molecular clouds, often in close proximity to IR clusters and dust lanes, which aids the visual classification. A lower but significant fraction of sources are found near other radio sources. About 33% of the CORNISH UCHIIs are situated in a radio cluster (within 12$''$ of another source), with 30% in a sky region containing more than seven detections of 7$\sigma$ sources. The outlines of 24% of the UCHIIs overlap one or more 7$\sigma$ sources (see §\[validation\]). regions are expected to be strong sources in 1 mm continuum (which maps the cool dust), whereas planetary nebulae are not. BOLOCAM 1.1 mm images [see @rosolowsky:2010] centred at the radio source position were visually inspected in conjunction with the IR images to verify that the UCHII sample is not contaminated by PNe.
It is easier to sift out other classes of sources such as radio stars and radio galaxies. Radio stars can be distinguished by their lack of mid- and far-IR emission, whereas radio galaxies have no infrared counterparts.
Radio properties of the CORNISH UCHIIs {#validation}
======================================
![Observational properties of the candidate sample – the confinement to the Galactic plane (top panel), small angular sizes (middle panel), and total radio fluxes (bottom panel) are consistent with UCHII regions. The dot-dashed line in the bottom panel marks the 7$\sigma$ (2.8 mJy) sensitivity limit of CORNISH.[]{data-label="fig:lat"}](hist_b "fig:"){width="0.95\columnwidth"}\
![Observational properties of the candidate sample – the confinement to the Galactic plane (top panel), small angular sizes (middle panel), and total radio fluxes (bottom panel) are consistent with UCHII regions. The dot-dashed line in the bottom panel marks the 7$\sigma$ (2.8 mJy) sensitivity limit of CORNISH.[]{data-label="fig:lat"}](hist_angsize "fig:"){width="\columnwidth"}\
![Observational properties of the candidate sample – the confinement to the Galactic plane (top panel), small angular sizes (middle panel), and total radio fluxes (bottom panel) are consistent with UCHII regions. The dot-dashed line in the bottom panel marks the 7$\sigma$ (2.8 mJy) sensitivity limit of CORNISH.[]{data-label="fig:lat"}](hist_flux "fig:"){width="\columnwidth"}
The distribution of the Galactic latitudes, angular sizes and integrated fluxes of the sample of candidate UCHIIs are shown in Fig. \[fig:lat\]. The sources are closely confined to the Galactic plane, as expected for very young massive star forming regions. Using the CORNISH survey, [@urquhart:2013] fit a scale-height of 20.7 $\pm$ 1.7 pc for compact and ultra-compact regions.
The majority of the ultra-compact regions have angular sizes below 5$''$, with the histogram peaking towards unresolved sources. As discussed in [@purcell:2013], sources begin to suffer from over-resolution above angular sizes of 14$''$. Most sources have integrated fluxes above 10 mJy, and the brightest source has a flux density of 12.6 Jy. The integrated flux histogram shows a clear downturn towards the lowest values, that is, the UCHII sample is close to complete.
![Ratio of integrated to peak flux vs. angular size of the 239 CORNISH UCHIIs. Fluxes estimated from Gaussian fits and from polygon apertures are shown as black circles and grey diamonds, respectively.[]{data-label="fig:resunres"}](int_peak){width="\columnwidth"}
The CORNISH beam size is 1.5$''$ and all sources with $\theta <$ 1.8$''$ are marked as unresolved in the catalogue table (detailed checks for the entire CORNISH catalogue are discussed in @purcell:2013). Within the CORNISH UCHII candidate sample, the flux was measured by fitting a Gaussian in 90 out of the 239 cases (angular size range 1.5$''$ to 6.2$''$), and for the remaining 149 sources, a hand-drawn polygon was used instead (angular size range from 1.8$''$ to 23.4$''$). The integrated and peak fluxes were compared as a function of angular size of each source, as shown in Fig. \[fig:resunres\]. Naturally, those sources whose fluxes were measured from a Gaussian fit show a clear trend of the flux ratio with increasing angular size, whereas the remaining sources with manually drawn contours show more variation.
Lower-resolution radio counterparts {#lowres}
-----------------------------------
![Comparison of the MAGPIS 6 cm and CORNISH integrated fluxes (black circles) with plotted line of equality, plotted against the corresponding CORNISH angular size.[]{data-label="fig:fluxes"}](fc_fm){width="\columnwidth"}
The Multi-Array Galactic Plane Imaging Survey (MAGPIS) [@helfand:2006] is useful for the study of evolved regions and other extended, optically thin thermal emitters. However, it is not well-suited to explore dense, thermal sources, as those are unresolved or only marginally resolved. Catalogues at 20 cm (VLA B,C,D configuration) and at 6 cm (VLA C configuration) are available [@white:2005]. The 6 cm catalogue covers $\sim$ 23% of the northern-GLIMPSE region, and at the survey resolution most of the detected CORNISH counterparts are unresolved – no morphological information is available.
The benefits of a comparison between CORNISH and MAGPIS are explained in detail in [@purcell:2013] (see Figs. 20, 21). In brief, for extended sources in the CORNISH sample, the measured flux density could be a lower limit in cases where some of the extended emission was filtered out due to gaps in $\textit{uv}$ coverage, and the measured angular sizes could be underestimated by as much as 50% for non-Gaussian sources in interferometric measurements [see e.g. @panagia:1978]. To test for such instances, a comparison to the MAGPIS 6 cm data is useful, as the survey configuration allowed the recovery of more diffuse emission at the cost of lower resolution.
It is important to note that there are instances of repeats in the combined 6cm-20cm MAGPIS catalogue. This is due to the fact that sources matching more than once are listed multiple times, for example a 6 cm source listed with each 20 cm counterpart and vice versa [@white:2005]. Since the structure of the MAGPIS catalogue leads to repeats even within a small cross-matching radius, to obtain a sufficiently reliable cross-match with the MAGPIS catalogue table, the matched sample had to be limited to only 47 associations. Visual identification of the associations was therefore preferred. The CORNISH team visually identified 216 20 cm matches (out of which 162 have both a 6 cm and 20 cm MAGPIS detection).
The fluxes of the 20 cm and 6 cm MAGPIS UCHII associations were measured independently of the catalogue table values via automated aperture photometry scripts. Radio outlines from the CORNISH database $-$ hand-drawn polygons in the case of extended radio sources, and Gaussian outlines in the case of compact radio sources, were utilised. In order to use these outlines as apertures for the lower-resolution MAPGIS data, they were expanded accordingly. The necessary ‘padding’ value (i.e. the required radial expansion) was determined after multiple runs with different aperture sizes. Based on the curve of growth, a padding value of 4$''$ (i.e. total expansion of 8$''$) was chosen for the flux measurement.
Fig. \[fig:fluxes\] shows that the majority of the flux ratio values occupy the range between $\sim$ 0.2 and 1.1, which indicates that some flux was not recovered at the higher resolution (around 65% of the flux was detected for these sources). The slightly negative slope of the flux ratio is due to worsening over-resolution with increasing angular size. Sources overlapping with one or more 5$\sigma$ or 7$\sigma$ CORNISH neighbours were excluded, as these sources are unresolved and merged in the MAGPIS 6 cm images and their flux measurements are unreliable. The outliers above the equality line, that is, with CORNISH fluxes significantly higher than their MAGPIS 6 cm counterpart, are investigated for potential short-timescale variability in §\[variability\].
The reliability of the measured 216 20-cm and 162 6-cm flux values was judged on the basis of their median brightness level of the sky. For all sources with abnormal (i.e. outside the range of the majority of sources) median level in the sky-annuli, upper flux limits are included in the presented flux table (Table \[table:fluxes\_M\] in Appendix \[appendixD\]). Sources with consistent sky values that are also not overlapping with a 5$\sigma$ or 7$\sigma$ source are here considered to be the highly-reliable MAGPIS flux measurement subset. These were used to compute the spectral indices of the CORNISH UCHIIs – see §\[spind\].
A few MAGPIS UCHII associations were picked at random out of the highly-reliable subset and their fluxes were also measured with `CASA`, by Gaussian fits in unresolved cases or computing the flux for the extended source region otherwise[^3]. Table \[table:casa\] shows a comparison of the flux values in the published MAGPIS catalogue table with our remeasured fluxes for these sources. Clearly, neither the photometry performed with the automated script or with `CASA` reproduced the catalogued 6 cm MAGPIS values to a reasonable degree, with a much better agreement between our two photometric measurements. This also appears to be true when comparing the results for extended 20 cm sources. This is why the fluxes of all available lower-resolution counterparts were remeasured in this work. Our results obtained for unresolved 20 cm sources appear to be overall in better agreement with the MAGPIS catalogue.
---------------------------------- ---------------------- -------------------- ------------------- ---------------- -------------------- ------------------- -----------------
CORNISH name $F_{\rm C}$ $F_{\rm M6cm}$ $F_{\rm M6cm}$ $F_{\rm M6cm}$ $F_{\rm M20cm}$ $F_{\rm M20cm}$ $F_{\rm M20cm}$
(cat.) (ap.) ($\texttt{CASA}$) (cat.) (ap.) ($\texttt{CASA}$) (cat.)
G011.1104$-$00.3985 305.37 $\pm$ 28.55 303.63 $\pm$ 9.12 320.3 112.37 326.73 $\pm$ 3.86 370.91 187.21
G012.1988$-$00.0345 62.71 $\pm$ 5.92 102.95 $\pm$ 1.47 110.18 59.82 42.52 $\pm$ 1.34 48.00 62.05
G016.3913$-$00.1383$^{\ddagger}$ 124.27 $\pm$ 15.43 43.18 $\pm$ 3.61 42.89 20.34 57.52 $\pm$ 5.18 66.71 58.23
G018.7106+00.0002 107.46 $\pm$ 10.62 177.74 $\pm$ 1.71 188.78 102.96 36.60 $\pm$ 0.94 41.97 35.39
G021.8751+00.0075 566.73 $\pm$ 54.14 1064.73 $\pm$ 8.85 1073.00 264.32 685.90 $\pm$ 7.84 719.53 599.63
G037.8731$-$00.3996 2561.21 $\pm$ 234.04 5189.35 $\pm$23.38 5219.50 1518.5 1769.56 $\pm$ 6.57 1800.10 1279.40
---------------------------------- ---------------------- -------------------- ------------------- ---------------- -------------------- ------------------- -----------------
\[table:casa\]
5GHz–1.4GHz spectral indices {#spind}
----------------------------
![Spectral indices between 1.4 GHz and 5 GHz against brightness temperature of the corresponding CORNISH source at 5 GHz. The model lines signify different $T_{\rm e}$. All CORNISH sources with an overlapping 5$\sigma$ or 7$\sigma$ source were excluded, as they are not resolved in MAGPIS.[]{data-label="fig:spind_tb"}](alpha){width="\columnwidth"}
The spectral indices ($\alpha$) were computed from the relation $$\alpha =\frac{\mathrm{ln}\left({S}_{\rm 5GHz}/{S}_{\rm 1.4GHz}\right)}{\mathrm{ln}\left(5/1.4\right)} ~,
\label{eq:sp}$$ where ${S}_{\rm 5GHz}$ and ${S}_{\rm 1.4GHz}$ are the integrated fluxes at 6 cm and 20 cm, respectively. The uncertainty was found from $$\Delta \alpha =\frac{\sqrt{{\left({\sigma }_{\rm 5GHz}/{{S}_{\rm 5GHz}}\right)}^{2}+{\left({\sigma }_{\rm 1.4GHz}/{{S}_{\rm 1.4GHz}}\right)}^{2}}}{\mathrm{ln}\left({5}/{1.4}\right)}.
\label{eq:sperr}$$
The indices were calculated for the CORNISH 6 cm flux and using the newly measured 20 cm fluxes. As discussed in the previous section, for the purpose of obtaining reliable spectral indices, all overlapping sources were excluded, thus making it possible to compute 93 spectral indices. This should be sufficiently representative of the sample as a whole, as the selection is in this way based on image quality alone.
The brightness temperatures of the sources in the UCHII sample at 6 cm were also calculated. This was done for all CORNISH UCHII regions (see Fig. \[fig:tb\]), using the equation $${T}_{\rm b}=\frac{1.36{\lambda }^{2}{S}_{\nu}}{{\theta }^{2}} ~,
\label{eq:Tb}$$ where the flux density ${S}_{\nu}$ is in mJy, the brightness temperature ${T}_{\rm b}$ is in K, the wavelength $\lambda$ is in cm and the HPBW (half-power beam width) $\theta$ for a Gaussian beam is in arcseconds. For all unresolved sources, this provides a lower limit. ${T}_{\rm b}$ serves as a measure of optical thickness – increasing with larger optical depth and reaching the electron temperature ($T_{\rm e}$) of the ionised region in the optically thick limit [@siodmiak:2001]. The optical depths $\tau_{\nu}$ at 5 GHz were found from $${T}_{\rm b}={T}_{\rm e}\left(1-{e}^{-\tau_{\nu}}\right) ~,
\label{eq:tau}$$ for ${T}_{\rm e}$ = $10^{4}$ K (Eqn. 4 in @siodmiak:2001). We note that the optical depth results (Fig. \[fig:optdepth\]) are lower limits in the case of unresolved sources (due to the angular size dependence of ${T}_{\rm b}$), as well as for the most extended sources (due to the possible underestimation of the 5 GHz flux, see Fig. \[fig:fluxes\]).
The spectral index analysis was performed by plotting the UCHII spectral indices (Fig. \[fig:spind\_tb\]) as a function of their corresponding 6 cm brightness temperatures and comparing the result to the theoretical model developed by [@bojicic:2011] (see Eqn. 6 in their paper), who followed up on the work by [@siodmiak:2001]. The focus of these works were planetary nebulae, however the model is general and should be applicable to all ionised nebulae. It assumes a uniform nebula characterized by its electron temperature and optical thickness at a corresponding reference frequency. The individual spectral index values (along with the measured fluxes at 6 cm and 20 cm) can be found in Table \[table:fluxes\_M\].
The majority of UCHII candidates were found to be within the expected theoretical limits for free-free emission ($-0.1 \lesssim \alpha \lesssim 2$) $-$ that is, between sources with optically thin or optically thick radio continuum emission at both frequencies. None of the sources have a spectral index above 1.5. However, $\sim$ 18% of the sources are found below the lower limit (taking errors into account), which is inconsistent with thermal emission. The reason for the apparently non-thermal spectral indices is likely to be the difference in the VLA configuration at 1.4 GHz and 5 GHz (i.e. the $uv$ coverage at higher resolution resulting in filtering out of some of the flux, as seen in Fig. \[fig:fluxes\]). If the 6 cm fluxes were $\sim$ 1.6 times higher (as high as the measured 6 cm MAPGIS fluxes, Fig. \[fig:fluxes\]), all but two sources (G010.6297$-$00.3380 and G023.8985+00.0647) would be shifted within the thermal bounds. Despite this, we choose to use the CORNISH 6 cm data over the MAGPIS 6 cm images, as the latter show significantly greater image-to-image variations in median sky levels and thus provide fewer 6 cm frames that are viable for the spectral index calculation (only 71/162 MAGPIS 6 cm sources when overlapping sources are excluded). The 20 cm images appear overall of higher quality than the MAGPIS 6 cm data. Therefore, by utilising the CORNISH 6 cm images and the (reduced sample of 93) MAGPIS 20 cm images, we have limited our spectral index analysis to the best available data. Due to the time difference between the 20 cm observations and (both sets of) the 6 cm observations, source variability, as discussed in §\[variability\], is another possible explanation for some instances of non-thermal (appearing) UCHIIs. Unfortunately, it is not possible at this time to quantify this effect without more 20 cm observations.
The location of each UCHII in Fig. \[fig:spind\_tb\] is determined by both its optical thickness and its electron temperature. The model lines in the diagram correspond to applying Eqn. 6 of [@bojicic:2011] for electron temperatures $T_{\rm e}$ = 5 $\times \ 10^{3}$ K, $T_{\rm e}$ = $\rm 1 \ \times \ 10^{4}$ K and $T_{\rm e}$ = 2 $\times \ 10^{4}$ K. For reference, the mean value for the electron temperature of Galactic regions has been estimated to be 8000 K [@quireza:2006]. At lower brightness temperatures, the UCHIIs scatter around the optically thin limit, where we find the majority of the sample. The much larger associated uncertainties of the sources with lower $T_{\rm b}$ (i.e. of lower opacity at 6 cm) should be noted in this case. With increasing $T_{\rm b}$, $\alpha$ increases in agreement with the model, corresponding to $5 \times 10^{3} \rm \ K \lesssim $ $T_{\rm e}$ $\lesssim 2 \times 10^{4} \rm \ K$, revealing sources that appear optically thick. Lower electron temperatures would be a better fit to the sample if we take into account the possible under-estimation of the 6 cm flux.
The presented spectral index results are a validation of the UCHII region nature of our sample.
Evidence for short-timescale UCHII variability? {#variability}
-----------------------------------------------
All sources with available MAGPIS and CORNISH 6 cm images were revisited to look for significant flux changes between the two epochs that cannot be attributed to image quality or other individual reasons. Such sources could be variable over short timescales comparable with the difference in time between the MAGPIS 6 cm survey (obs. 1989–1991) and the CORNISH survey (obs. 2006–2008). The flux ratio lower limits were taken into account to quantify this. The higher flux at the earlier epoch (due to the detection of more extended emission) prevents a reliable investigation of instances of intrinsic flux decrease, particularly for sources larger than 5$''$. A further hindrance is that any hypothetical UCHIIs that are completely invisible at the later epoch due to a significant flux decrease over time cannot be reliably differentiated from more extended region phases due to the lower resolution at the earlier epoch.
Due to these limitations imposed by the different VLA configurations used at the two epochs, in the context of this work, we use the term variability to refer to increase in flux in the $\sim$ 15 years separating the two surveys. The sources which appear to be intrinsically variable all have a flux increase greater than $\sim$ 50%. These are listed in Table \[table:variables\].
CORNISH name $F_{\rm C}/F_{\rm M,6cm}$
--------------------- ---------------------------
G011.0328+00.0274 2.42 $\pm$ 0.61
G011.9786$-$00.0973 3.56 $\pm$ 1.83
G014.5988+00.0198 2.92 $\pm$ 0.80
G016.3913$-$00.1383 2.88 $\pm$ 0.15
G023.4553$-$00.2010 17.42 $\pm$ 1.28
G025.7157+00.0487 4.33 $\pm$ 0.49
G030.7579+00.2042 73.85 $\pm$ 4.49
G030.7661$-$00.0348 7.04 $\pm$ 0.56
G037.7347$-$00.1128 22.90 $\pm$ 1.09
: Ratio of CORNISH to MAGPIS 5 GHz fluxes for candidate variable UCHII regions (observed 15 years apart). We note that in some cases, the source was not detected in MAGPIS, so the upper flux limit was used instead for the comparison.
\[table:variables\]
#### {#section .unnumbered}
The 6-cm variables appear to have several properties in common:
1. They exist in relative isolation (no overlapping sources, no other radio sources within $\gtrsim$ 1$'$, no busy complexes). An exception to this is G030.7661$-$00.0348, which is likely experiencing substantially different effects than the rest in the bustling environment of W43;
2. They are all near-IR dark (apart from G030.7661$-$00.0348);
3. Most are particularly compact ($\lesssim$ 0.1 pc), with the exception of G016.3913$-$00.1383.
4. Their 5GHz–1.4GHz spectral indices are not anomalous with respect to the rest of the sample.
The potentially variable sources comprise $\sim$ 5% of the CORNISH UCHII sample. In four cases (G011.9786$-$00.0973, G014.5988+00.0198, G023.4553$-$00.2010, and G030.7661$-$00.0348), the CORNISH source was not detected at both 6 and 20 cm at the earlier observational epoch. Three sources (G025.7157$+$00.0487, G030.7579+00.2042, and G037.7347$-$00.1128) all have only a 20 cm detection at the earlier epoch (and there is a dim 20 cm counterpart for G011.0328+00.0274). The extended G016.3913$-$00.1383 has 6 and 20 cm counterparts, but these appear smaller and dimmer at the earlier epoch. Although there are some suggestive correlations, this group of candidate variable sources is clearly statistically insufficient to establish any common pattern linking the (presence or lack of) emission at 6 and 20 cm at the same epoch. Unfortunately, the 6 cm variability cannot be linked in any way to variability at 20 cm, due to the lack of other available 20 cm data of comparable quality for the 6-cm variables.
Time-variable radio flux densities of several ultra-compact and hyper-compact (HC) regions have been reported previously (@acord:1998, @hernandez:2004, @rodriguez:2007, @gomez:2008, @madrid:2008). The flux changes have been associated with morphological changes across observational epochs, on timescales of a few years. The UC and HC regions have been caught expanding [@acord:1998]. [@madrid:2008] discuss contracting UC and HC regions; however their sources are unresolved. Variability over observable timescales could be caused by factors that are either external or internal to the forming star. The former could be the result of chaotic motions of the material surrounding the ionising star. In this scenario, optically thick gas (e.g. clumps in the stellar wind) would occasionally block the outgoing radiation, shielding the outer ionised gas layers and thus neutralising them [@peters:2010a]. In the latter scenario, the forming star itself is undergoing changes [@hernandez:2004] $-$ surface temperature fluctuations affect the UV flux and thus the region size.
Theoretical studies have reproduced this behaviour, referred to as flickering. The three-dimensional collapse simulations of massive star formation of [@peters:2010a; @peters:2010b; @peters:2010c] and [@klassen:2012a] include feedback by ionising radiation and show time variability leading to changes in region appearance and flux comparable to observations. The [@peters:2010a] model produces flickering on scales of $\sim$ 10 years. In this model, accretion has not ceased prior to the UCHII stage. The infalling neutral flow becomes ionised when in close proximity to the star. The region is gravitationally trapped early on, which is followed by a fluctuation between trapped and extended states, and thus changes in flux, size and morphology are seen over time. [@madrid:2011] performed statistical analysis of simulated radio-continuum observations separated by 10 year steps, using the [@peters:2010a; @peters:2010b; @peters:2010c] models to form regions. They found that 7% of the simulated HC and UC regions have a detectable flux increase (larger than 10%) and 3% have a detectable flux decrease, but expect only $\sim$ 0.3% of their regions to have a flux increase of over 50%. The observations discussed in this work show that $\sim$ 5% of the CORNISH UCHII regions have become brighter by 50% or more over a comparable time scale ($\sim$ 15 years), based on the 6 cm data. In practice, any similar statistic of observed sources with flux increase $\lesssim$ 10% would be unreliable, given the associated flux uncertainties. In any case, assuming that all of our candidate variable sources truly undergo intrinsic changes, variable HC and UC regions could be significantly more common and their brightness could fluctuate more than predicted. If this is the case, invoking ongoing accretion cannot account for the observed dramatic change in flux. The [@peters:2010a] model does not include radiation pressure, magnetic fields, winds and outflows, all of which are components of the star formation process and might be related to variability.
Distances
=========
\[analysis\]
A crucial step towards characterising UCHII regions is to determine the distance to each source. One can then convert measured parameters (e.g. fluxes and angular sizes) into physical quantities (e.g. luminosities and physical sizes). Accurately derived distances to UCHII regions are used to test the current models of the face-on Galactic structure [see e.g. @urquhart:2013]. As the heavy obscuration hinders the use of any optical distance determination techniques, the distances to most UCHII regions are kinematically derived. The kinematic distance is found by fitting the radial velocity of the source to a Galactic rotation curve (e.g. @brand:1993 [@reid:2009]), using radio or mm spectral line data. Errors in the calculated distance arise when the source radial velocity differs from the one assumed by the model (e.g. velocity errors of about 10$\kms$ due to the velocities departing from circular rotation as a result of streaming motions) [@bania:2009].
Distance estimates for the outer Galaxy are relatively straightforward. However, a major obstacle arises when one seeks the kinematic distance for objects within the Solar circle. At Galactic radii smaller than that of the Sun, two possible distances exist for each radial velocity. These distances, known as near and far, are situated at equal intervals from the tangent point distance. The kinematic distance ambiguity (KDA) is not present only for the tangent point velocity, which is the maximum radial velocity. Different methods exist to assign the correct kinematic distance to the sources of interest – for example, Emission/Absorption (E/A), Self-Absorption (SA), or using absorption lines from other molecules, for example formaldehyde ($\rm H_{2}CO$) (as discussed by @bania:2009).
These methods are particularly effective for bright UCHII regions, whose free-free continuum emission is substantially stronger than the Galactic emission, thus resulting in unambiguous absorption spectra. As the maximum radial velocity along the line of sight is always the tangent velocity, lack of absorption between the source and tangent velocity reveals that the source is located at the near distance. Otherwise, the far distance is assigned (or the tangent distance, in the cases when the source velocity equals the tangent velocity). In this work, the E/A method was adopted [see e.g. @urquhart:2012]. This method makes use of CO emission line data and absorption to obtain the near and far kinematic distances and to attempt to resolve the KDA.
Distances from ATLASGAL and RMS
-------------------------------
The work by [@urquhart:2013] presents an unbiased and complete sample of 170 molecular clumps with 213 embedded compact and ultra-compact regions over the common GLIMPSE, ATLASGAL and CORNISH survey region. Table 3 from [@urquhart:2013] contains distances to all CHII- and UCHII-hosting clumps and Table 4 contains Lyman continuum fluxes (${F}_{\rm Ly}$) and source physical sizes. Out of the associated 213 regions, 203 also belong to the CORNISH UCHII sample (ten were classified as more extended). Of the remaining 36 UCHII candidates without ATLASGAL distances, eight were found to have a distance estimate in the Red MSX source survey (RMS) database[^4] [@lumsden:2013]. These are the distances for G010.6297$-$00.3380, G030.6881$-$00.0718, G032.0297+00.0491, G035.0524$-$00.5177, G038.5493+00.1646, G048.6099+00.0270, G060.8842$-$00.1286, and G061.7207+00.8630.
Distance estimates for remaining sources using CO data
------------------------------------------------------
[@anderson:2009] describe a large-scale study of the molecular properties of regions of different sizes and morphologies using fully sampled CO maps. This is the BU-FCRAO Galactic Ring Survey [@jackson:2006], which uses $^{13}$CO $J=1\longrightarrow0$ emission. This has advantages over the commonly used $^{12}$CO isotopologue, as $^{13}$CO is $\sim$ 50 times less abundant and thus provides a smaller optical depth, and consequently smaller line widths and better separation of velocity components along the line of sight.
CO data-cubes from the Galactic Ring Survey[^5] were used to obtain radial velocities for the remaining sources without ATLASGAL or RMS distances. The positional accuracy of the GRS is $\sim$ 2.3$''$, which is equivalent to 1/10 of the spacing between grid points on the map. Data cubes for the available sources were obtained, and the radial velocities were measured for the sources coinciding with the CORNISH coordinates. The emission line structure of the data is very complex, often with multiple emission lines. This reflects the complexity of the ($l, b, v_{\rm LSR}$) structure of the molecular gas in the line of sight of the region. The sources were located within the 15$'$ data cubes by going manually through the cube channels and then mapping the cube once the source was found at or close to the precise CORNISH coordinates. A source’s velocity was taken to be equal to the velocity of the most prominent emission line in the map at the exact source position. The final CO source velocity results were compared to Table 3 in [@anderson:2009] for the sources in common. The code by [@reid:2009] was used to obtain near and far kinematic distances corresponding to each radial velocity estimate (see §\[kda\]).
--------------------- ------- ---------------- --------
G024.4698+00.4954 103 C24.47+0.49 102.67
G024.4721+00.4877 102.8 C24.47+0.49 102.67
G024.4736+00.4950 102.6 C24.47+0.49 102.67
G024.8497+00.0881 109.3 C24.81+0.10 108.31
G030.7661$-$00.0348 96.1 C30.78$-$0.03 94.76
G030.7661$-$00.0348 96.1 U30.84$-$0.11b 96.89
G031.2420$-$00.1106 21.1 U31.24$-$0.11a 21.07
G034.2544+00.1460 57.7 U34.26+0.15 57.1
G034.2571+00.1466 57.7 U34.26+0.15 57.1
G037.9723$-$00.0965 54.7 C38.05$-$0.04 54.1
G049.4640$-$00.3511 59.5 U49.49$-$0.37 60.08
G049.4891$-$00.3763 60.9 U49.49$-$0.37 60.08
G050.3157+00.6747 26.5 U50.32+0.68 26.31
--------------------- ------- ---------------- --------
: Comparison between velocities (in $\kms$) measured in this work and by [@anderson:2009], Table 3 for the sources in common. As the peak channel was used to calculate the velocities, the associated errors are given by the channel width. This equals 0.21 $\rm km~s^{-1}$ for each BU-FCRAO Galactic Ring Survey cube. In practice this error is outweighed by the error due to peculiar motions, which is $\sim$ 10 $\kms$.
\[table:comparison\]
Resolving the KDA {#kda}
-----------------
The Emission/Absorption (E/A) method was implemented as a standard way to choose between the calculated near and far distances [see e.g. @bania:2009; @urquhart:2012]. The method has proven to be very successful for KDA resolution [@bania:2009].
CORNISH name KDA Strong abs. $d$ (kpc)
--------------------- ----- ------------- ----------------
G024.4698+00.4954 n 5.5 $\pm$ 0.3
G024.4721+00.4877 n 5.5 $\pm$ 0.3
G024.4736+00.4950 n 5.5 $\pm$ 0.3
G024.8497+00.0881 n 5.8 $\pm$ 0.3
G026.0083+00.1369 f 13.8 $\pm$ 0.5
G026.8304$-$00.2067 f 11.9 $\pm$ 0.3
G029.7704+00.2189 f 9.8 $\pm$ 0.3
G030.7579+00.2042 t 7.2 $\pm$ 0.6
G030.7661$-$00.0348 t 7.2 $\pm$ 0.6
G031.2420$-$00.1106 f 12.7 $\pm$ 0.4
G034.2544+00.1460 n 3.6 $\pm$ 0.4
G034.2571+00.1466 n 3.6 $\pm$ 0.4
G035.4570$-$00.1791 f 9.7 $\pm$ 0.4
G037.7562+00.5605 f 12.2 $\pm$ 0.5
G037.9723$-$00.0965 f 9.7 $\pm$ 0.4
G045.5431$-$00.0073 t 5.9 $\pm$ 0.9
G049.4640$-$00.3511 t 5.5 $\pm$ 2.2
G049.4891$-$00.3763 t 5.5 $\pm$ 2.2
G050.3157+00.6747 f 8.6 $\pm$ 0.5
G061.4763+00.0892 t 4.0 $\pm$ 1.5
G061.4770+00.0891 t 4.0 $\pm$ 1.5
: KDA-resolved UCHII distances. The distances and errors were derived with the [@reid:2009] code and the E/A method was used to resolve the KDA. The sources that are found at the near, far, and tangent distance, are labelled with n, f, and t, correspondingly. The third column indicates whether the spectra used for the KDA resolution were of good quality (i.e. mostly in absorption, marked with ) or poor (i.e. mostly in emission, indicated with ).
\[table:kda\_dist\]
Out of the 28 CORNISH UCHIIs with missing CO distances, CO and VGPS data were found for 21 (CO data were not available within the range 10$^{\circ}\leq l \leq 17\degree)$. In the VGPS spectra, the brightness temperature at the source location is overall $\lesssim$ 100 K, as expected for optically thin gas. Therefore, a background source of higher temperature should be seen in absorption. The presence of a region would provide a sufficiently strong continuum to detect a line in absorption. It should be noted, however, that the VGPS synthesized beam size is $\sim$ 45$''$ (FWHM) at 21 cm (see @stil:2006 for the VGPS survey paper). The region may not be detected at all if the source size is not comparable to the survey beam size [@urquhart:2012]. The extent of the effects of this on our distance estimates are hard to quantify. Moreover, neighbouring CORNISH UCHIIs found within the beam cannot be distinguished. This is the case, for instance, for G024.4721+00.4877, G024.4736+00.4950, and G024.4698+00.4954, whose spectra are practically identical. The latter issue is alleviated by the fact that spatially close UCHIIs are likely to belong to the same star-forming region, and hence to be situated at the same distance.
To assign a near or a far distance, the data were plotted, together with a line marking the assigned CO radial velocity and its associated error ($\pm$ 10$\kms$ due to peculiar motions) and the calculated tangent velocity was included as well (see Appendix \[appendixB\]). All sources with CO velocity within 10$\kms$ of the tangent point velocity were assigned the tangent point distance, in order to limit wrong assignments to the near distance (as suggested by @bania:2009). Additional uncertainty arises due to half of the spectra being dominated by emission instead of absorption at the UCHII source position. The presence or lack of convincingly strong absorption is indicated in Table \[table:kda\_dist\] for each region. An example of a reliable spectrum is that coinciding with G049.4891$-$00.3763, as it clearly shows very strong absorption lines. Emission-dominated spectra appear similar to the one coincident with G029.7704+00.2189 (Appendix \[appendixB\]).
All 21 UCHIIs with available GRS CO and VGPS data were assigned a distance, and four more distances were adopted from the paper by [@cesaroni:2015]. These are the distances for G010.3204$-$00.2328, G011.1712$-$00.0662, G014.1741+00.0245, and G016.3913$-$00.1383. It should be noted that for these four sources, as for all sources without data to resolve the ambiguity, [@cesaroni:2015] assigned the far distance. Placing sources without a distance solution at the near distance is preferable. Only three sources (G010.8519$-$00.4407, G011.9786$-$00.0973, and G014.1046+00.0918) with no known distance and no available emission and absorption data remain in our sample. From the group of 21 UCHIIs, six were assigned the near distance, eight – the far, and the remaining seven sources were assigned the tangent point distance.
Derived physical properties {#physprop}
===========================
[0.5]{}
{width="0.95\columnwidth"} \[fig:dist\]
[0.5]{}
{width="0.95\columnwidth"} \[fig:galdist\]
[0.5]{}
{width="0.95\columnwidth"} \[fig:phsize\]
[0.5]{}
{width="0.95\columnwidth"} \[fig:ly\]
[0.5]{}
{width="0.95\columnwidth"} \[fig:tb\]
[0.5]{}
{width="0.95\columnwidth"} \[fig:optdepth\]
[0.5]{}
{width="0.95\columnwidth"} \[fig:em\]
[0.5]{}
{width="0.95\columnwidth"} \[fig:ne\]
A summary of the physical properties of the sample is presented in Fig. \[fig:allparameters\]. The distributions of heliocentric and galactocentric distances, physical sizes, optical depths, brightness temperatures, Lyman continuum fluxes, emission measures, and electron densities are shown. The computed values corresponding to each CORNISH UCHII can be found in Appendix \[appendixE\].
Fig. \[fig:dist\] shows the distribution of the UCHII regions with heliocentric distance. The five peaks are at 2, 4, 11, 13, and 17 kpc. As noted by [@urquhart:2013] (who also find five similarly situated peaks at 2, 5, 10, 12, and 16 kpc), they likely correspond, in turn, to the near side of the Sagittarius arm, the end of the bar and Scutum-Centaurus arm, the far sides of the Sagittarius and Perseus arms, and the Norma arm. [@urquhart:2013] studied the associated clumps to the CORNISH compact regions and found that the most prominent peak in the heliocentric distance histogram at $\sim$ 11 kpc corresponds to the W49A complex (as this peak is not seen for clumps).
The galactocentric distances were also estimated (Fig. \[fig:galdist\]). The galactocentric distance distribution depends only on the choice of Galactic rotation curve [@urquhart:2011]. The peaks are located at $\sim$ 4.5, 6, and 7.5 kpc, similarly to the findings of [@urquhart:2011] (peaks at $\sim$ 4, 6, and 8 kpc) for the young massive star sample in the RMS survey. [@urquhart:2011] identify the peak at approximately 4 kpc to be at the intersection of the Long Bar and the Scutum-Centaurus arm, also coinciding with the W43 complex. The 6 kpc peak is coincident with the Sagittarius arm, and the 7.5 kpc peak corresponds to the Perseus arm (the bin is dominated by the W49A complex – the most active star-forming region in the Galaxy).
The distribution of the physical sizes is shown in Fig. \[fig:phsize\]. All sources with available deconvolved angular sizes [see @purcell:2013] are presented in the histogram in blue. The physical sizes of 36 UCHIIs could not be obtained because the corresponding sources were unresolved and hence their true sizes are unknown (and three more sources do not have a computed distance). Upper limits on the physical sizes were computed for these unresolved sources with available distances (grey region in the histogram). Most sources ($\sim$ 66%) are larger than 0.05 pc and smaller than 0.2 pc in diameter, and the distribution peaks at 0.1 pc, which is consistent with the typical sizes of UCHII regions. About 12.5% are with sizes between 0.01 pc and 0.05 pc, and the remaining $\sim$ 21.5% sources are between 0.2 pc and 0.9 pc. The investigated properties of these sources are in accord with the rest of the sample and therefore likely exhibit the natural variation in size one would expect for a continuum of spectral subtypes and ages, and for different ambient densities.
Lyman continuum fluxes from [@urquhart:2013] were available for 203 out of the 239 sources in the CORNISH sample. The Lyman continuum fluxes ${N}_{i}$ for 33 of the remaining sources were computed from $$\left[\frac{{N}_{i}}{\rm ~photons~ {s}^{-1}}\right]=9\times{10}^{43}\left[\frac{{S}_{\nu }}{\rm ~mJy}\right]\left[\frac{{d}^{2}}{\rm ~kpc}\right]\left[\frac{{\nu }^{0.1}}{5\rm ~GHz}\right] ~,
\label{eq:Ni}$$ where ${S}_{\nu}$ is the flux at a frequency $\nu$ and $d$ is the distance (Eqn. 6 from @urquhart:2013). The associated error on ${N}_{i}$ is 10%, which is dominated by the distance error. As for the emission measures and electron densities, a dust-free, optically thin emitter is assumed here. The Lyman continuum flux distribution (Fig. \[fig:ly\]) of the CORNISH UCHII sample peaks between 47.2 < log($N_{\rm~ i}$) < 48.5 $\rm~photons~s^{-1}$, which is also consistent with the Lyman continuum ionising flux from zero-age main-sequence stars with spectral class from B0 to O7. The computed Lyman continuum fluxes should be treated as lower limits.
The brightness temperature distribution at 6 cm and the corresponding optical depth distribution, as calculated in §\[spind\] (Eqns. \[eq:Tb\] and \[eq:tau\]), are shown in Figs. \[fig:tb\] and \[fig:optdepth\], respectively. The optical depth histogram, as well as Fig. \[fig:spind\_tb\], confirm that assuming optically thin sources at 5 GHz is justified for the majority of UCHIIs in our sample.
The emission measures ($EM$, Fig. \[fig:em\]) and the electron densities ($n_{\rm e}$, Fig. \[fig:ne\]) of the UCHII regions were computed from the equations
$$\left[\frac{EM}{\rm~ pc \ {cm}^{-6}}\right]=1.7\times{10}^{7}\left[\frac{{S}_{\nu }}{\rm~Jy}\right]{\left[\frac{\nu}{\rm~ GHz}\right]}^{0.1}{\left[\frac{{T}_{\rm e}}{\rm~ K}\right]^{0.35}}{\left[\frac{\theta_{S}}{''}\right]^{-2}} ~
\label{eq:EM}$$
and $$\begin{split}
\left[\frac{n_{\rm e}}{\rm~ {cm}^{-3}}\right]=2.3\times{10}^{6}\left[\frac{{S}_{\nu }}{\rm~Jy}\right]^{0.5}{\left[\frac{\nu}{\rm~ GHz}\right]}^{0.05}{\left[\frac{{T}_{\rm e}}{\rm~ K}\right]^{0.175}}{\left[\frac{\theta_{S}}{''}\right]}^{-1.5}
\\
\times {\left[\frac{d}{\rm~ pc}\right]}^{-0.5} ~,
\label{eq:ne}
\end{split}$$ where ${T}_{\rm e}$ = $10^{4}$ K is the electron temperature and $\theta_{S}$ is the source angular size. Both equations were adopted from [@sanchez:2013], who followed the formalism of [@mezger:1967] and [@rubin:1968]. It is assumed that the cm continuum flux is emitted from homogeneous optically thin regions. As can be seen from the optical depth results shown in Fig. \[fig:optdepth\], this is a good description for the sample. The typical uncertainties on the flux density and angular diameter imply an uncertainty on the emission measure of 30% and uncertainty on the electron density of 20%. Taking the 5$\sigma$ flux sensitivity of CORNISH (2 mJy, see @hoare:2012) and the 1.5$''$ resolution, we estimate the CORNISH sensitivity to log(*EM*) to be $\sim$ 5.5 $\rm pc \ cm^{-6}$.
The computed UCHII emission measures and electron densities are generally consistent with the results by [@kurtz:1994]. No sources in the sample have computed electron densities and emission measures that would exceed $10^{5} \ \rm cm^{-3}$ and $10^{8} \ \rm pc \ cm^{-6}$, respectively, even when taking into account the associated uncertainties. Hyper-compact regions have electron densities in excess of $10^{6} \ \rm cm^{-3}$ and emission measures in excess of $10^{10} \ \rm pc \ cm^{-6}$ [@hoare:2007]. Thus, the CORNISH survey has identified only UCHIIs; no HCHIIs are reported. This is not surprising, because the high density of HCHIIs implies high turnover frequencies, $\sim$ 30 GHz. For an optically thick free-free spectrum with $S_\nu \propto \nu^2$, the flux density at the CORNISH observing frequency of 5 GHz will be of the order of $40\times$ lower than the flux density near the turnover frequency.
The computed optical depths at 5 GHz were used to quantify how much of the 5 GHz flux density would be missed in (the few) potential cases of optically thick regions (as well as how much this effect varies from source to source). In 80% of all the regions observed at 5 GHz, the difference between the measured and the theoretical unattenuated flux is below 10%. The bright UCHII G049.4905$-$00.3688 has the highest computed difference ( $\sim$ 56.4%). The distribution of the flux difference due to attenuation tapers off above differences greater than $\sim$ 20%, indicating that there is most likely no significant fraction of sources that have been missed altogether. The same should be true even if the electron temperature varies from region to region (within the expected physical bounds). It should also be noted that the computed Lyman continuum flux in this case is not significantly underestimated due to optically thick free-free emission, but could still be affected by loss of ionising photons (e.g. via dust absorption), or for radio flux that was not recovered.
Infrared properties {#IR}
===================
Associations with mid-infrared data {#glimpse}
-----------------------------------
![Colour-colour diagram of the CORNISH UCHIIs (excluding all sources with one or more saturated GLIMPSE images or otherwise unreliable GLIMPSE fluxes). The arrow shows the reddening vector, based on the extinction law of [@indebetouw:2005]. The plot is in the Vega magnitude system.[]{data-label="fig:colour"}](mag_mag_glimpse){width="\columnwidth"}
As discussed by [@watson:2008], the emission detected in the vicinity of a hot star is dominated by a different emission process in each of the four mid-IR GLIMPSE-IRAC bands. In particular, the presence of PAH emission is based on 8 $\upmu$m, 5.8 $\upmu$m, and lack of 4.5 $\upmu$m emission. Similar to larger IR bubbles, the mid-IR images of UCHII regions also show the presence of 8 $\upmu$m shells, dominated by strong PAH features in the IRAC bands at 3.6, 5.8, and 8 $\upmu$m. The inner surface of each shell is located at a distance from the ionising star that is equal to the PAH destruction radius. The 3.6 $\upmu$m band is dominated by stars, with contributions from the diffuse PAH feature at 3.3 $\upmu$m and perhaps from scattered starlight. The 4.5 $\upmu$m band exhibits no PAH features; the brightest contributors are stars and the diffuse emission is due to the shared contribution of lines from regions and from shocked molecular gas. The diffuse emission in the 5.8 $\upmu$m band is dominated by the 6.2 $\upmu$m PAH feature, apart from the immediate vicinity of O stars (where PAHs are destroyed). The diffuse emission in the 8 $\upmu$m band is dominated by the 7.7 and 8.6 $\upmu$m PAH features, or by thermal emission of dust heated by the hot stars and Lyman-$\alpha$ photons.
The CORNISH survey was designed to cover the GLIMPSE region of the Galactic plane [@hoare:2012], ensuring that all sources have mid-infrared counterparts to the radio continuum sources. Photometry of all GLIMPSE UCHII sources in the four IRAC bands was performed. Selecting the correct size of the IR source would not have been straightforward without knowledge of the position and size of the ultra-compact radio source, as the IR environment is more complex than in the radio view due to the different contributions to the emission. It is also difficult to disentangle individual sources in busy neighbourhoods.
In order to use the hand-drawn polygons (in the case of extended radio sources), and Gaussian outlines (in the case of compact radio sources) as apertures for the IR data, they were expanded accordingly. The exact padding value necessary for each of the four GLIMPSE bands was chosen after measuring (for each band) the counts at different aperture sizes and examining where the curve of growth begins to plateau before starting to increase again with the inclusion of unrelated sources. This was done for a small sub-sample of sources with a range of sizes representative of the UCHII sample, in all four bands. The padding radii used for the 3.6, 4.5, 5.8, and 8 $\upmu$m images were 2$''$, 2.4$''$, 3.4$''$, and 4.3$''$, respectively. The measured fluxes are included in Table \[glimpse\_long\] in Appendix \[appendixC\]. Median absolute deviation from the median (MADFM) background estimation was utilised (for all photometry in this work), as it is insensitive to the presence of outliers and is a reliable estimate of the noise [@purcell:2013][^6]. From all 956 GLIMPSE images (all four bands), 36 8.0 $\upmu$m images were found to be saturated at the source location after visual inspection, as well as two 5.8 $\upmu$m images and one 4.5 $\upmu$m image. These, together with non-detections, were excluded from the final results. The photometric results for G031.2801+00.0632 were also excluded, as diagnostic diagrams showed them to be dominated by a neighbouring YSO (seen in all GLIMPSE bands) rather than the UCHII region. This left 180 sources with 3.6 $\upmu$m fluxes, 191 – with 4.5 $\upmu$m fluxes, 190 – with 5.8 $\upmu$m fluxes, and 184 – with 8.0 $\upmu$m fluxes.
A colour-colour plot is shown in Fig. \[fig:colour\]. Only sources for which it was possible to compute both the \[3.6\]$-$\[4.5\] and \[5.8\]$-$\[8.0\] colours were included (and no upper or lower limits for the remaining sources), to avoid overcrowding the plot. This was not found to affect the exhibited trends for the mid-IR colours. From the 174 sources with reliable flux values in all bands, $\sim$ 85% occupy the zone 1.5 $<$ \[5.8\]$-$\[8.0\] $<$ 2. The \[3.6\]$-$\[4.5\] colour ranges between 0.1 and 2.1. This is similar to the results reported by [@fuente:2009] for 19 ultra-compact regions. They find that about 75% of the UCHIIs are grouped around \[5.8\]$-$\[8.0\] $\simeq$ 1.7 and 0.5 $\lesssim$ \[3.6\]$-$\[4.5\] $\lesssim$ 2.0.
Associations with near-infrared data and extinctions {#ukidss}
----------------------------------------------------
![Near-IR colour-colour diagram of near-IR nebulae associated with the CORNISH UCHIIs. The stellar contamination has been removed. Lower limits for sources visible only in H and K are also shown. The arrow shows the extinction vector, calculated using the extinction law from [@stead:2009]. The predicted intrinsic colours of an ionised nebula are shown with a black triangle. The observed data and the intrinsic colours are shown in the AB magnitude system.[]{data-label="fig:colour_nearir"}](jh_hk_ukidss_nebulae){width="\columnwidth"}
![Visual extinctions of the UCHIIs, computed from their near-IR fluxes from four methods – $F_{\rm H}/F_{5 \rm GHz}$, $F_{\rm K}/F_{5 \rm GHz}$, $J-H$, and $H-K$. The top panel compares the H- and K-band derived $A_{\rm V}$. In the bottom panel, the comparison is shown for the $J-H$ against the $H-K$ derived $A_{\rm V}$. Lower limits on the extinction are shown in the cases where the UCHII is seen only in H and K (grey arrows). Lines of equality are also plotted.[]{data-label="fig:Av"}](Av_h_k "fig:"){width="\columnwidth"} ![Visual extinctions of the UCHIIs, computed from their near-IR fluxes from four methods – $F_{\rm H}/F_{5 \rm GHz}$, $F_{\rm K}/F_{5 \rm GHz}$, $J-H$, and $H-K$. The top panel compares the H- and K-band derived $A_{\rm V}$. In the bottom panel, the comparison is shown for the $J-H$ against the $H-K$ derived $A_{\rm V}$. Lower limits on the extinction are shown in the cases where the UCHII is seen only in H and K (grey arrows). Lines of equality are also plotted.[]{data-label="fig:Av"}](Av_jh_hk "fig:"){width="\columnwidth"}
--------------------- ------- ----------------- --------------- ------------ ----------------- ------------------
G029.9559$-$00.0168 K/6cm 2.33 $\pm$ 0.07 G29.96$-$0.02 K/radio 2.14 $\pm$ 0.08 [@watson:1997]
Br$\gamma$ 2.20 $\pm$ 0.25
Br$\gamma$ 2.16 $\pm$ 0.07 [@moore:2005]
G043.8894$-$00.7840 K/6cm 2.83 $\pm$ 0.12 G43.89$-$0.78 Br$\gamma$ 3.32 $\pm$ 0.21 [@moore:2005]
G045.4545+00.0591 K/6cm 2.21 $\pm$ 0.11 G45.45+0.06 K/6cm 2.5 [@feldt:1998]
G049.4905$-$00.3688 K/6cm 2.33 $\pm$ 0.07 W51d K/6cm 2.6 $\pm$ 0.3 [@goldader:1994]
Br$\gamma$ 1.59 $\pm$ 0.07 [@moore:2005]
--------------------- ------- ----------------- --------------- ------------ ----------------- ------------------
\[table:av\_comparison\]
The visual extinction in the line of sight to UCHIIs can be estimated from their J, H, and K fluxes, assuming the diffuse emission is due to purely nebular gas emission. Several methods can be utilised to achieve this, such as the use of near-IR colours, or ratios of the near-IR flux to the radio flux [@willner:1972]. When using the colour-dependent methods, provided sufficiently reliable magnitude measurements of the embedded sources, one’s choice comes down to a compromise between scattering effects and infrared excess. Dust excess (typically pronounced in the K-band) causes the $H-K$ colours to appear redder, and scattered light (due to dust grains) at shorter wavelengths results in bluer $J-H$ colours [@porter:1998]. It is useful to compare the results from the different methods and in this way weigh the severity of systematic issues while providing an extinction range for the studied sources.
The UKIDSS Galactic plane survey [@lucas:2008] covered the northern and equatorial Galactic plane at $|b|<5\degree $ in the J (1.17$-$1.33 $\upmu$m), H (1.49$-$1.78 $\upmu$m), K (2.03$-$2.37 $\upmu$m) bands and provides an opportunity to investigate the near-IR properties of the CORNISH UCHII sample, such as fluxes and detection statistics. Only a point-source UKIDSS catalogue is available at present. Therefore automated photometry was performed in the same manner as described in §\[glimpse\] on all CORNISH sources with available UKIDSS data. In total, 230 sources had available UKIDSS images in the J band, 228 – in H, 227 – in K. The visual inspection revealed that out of all sources, 83 have a visible nebula in K, out of which 31 also have an H-band nebula. Out of these, 14 nebulae are visible in J[^7]. Aperture photometry (with median background subtraction) of all contaminant bright stars found within the expanded polygon aperture used for the automated J, H, and K flux measurements was also performed. This was done for all images with a visible near-IR nebula coinciding with an UCHII. The measured stellar fluxes were subtracted from the total photometric fluxes in order to obtain the nebular fluxes. The near-IR UCHII fluxes and corresponding AB magnitudes are presented in Table \[ukidss\_long\] in Appendix \[appendixC\]. Figure \[fig:colour\_nearir\] shows a diagram of the $J-H$ against $H-K$ nebular colours. The mean colour of the nebulae visible in all three bands is 2.1 for $J-H$, and 1.1 for $H-K$.
Figure \[fig:Av\] presents a comparison between the computed visual extinctions for the UCHII sample obtained from four methods: $F_{\rm H}/F_{5 \rm GHz}$ and $F_{\rm K}/F_{5 \rm GHz}$ (top panel); $J-H$ and $H-K$ (bottom panel). The empirically-derived $R_{V}$-dependent extinction law $A_{\lambda}/A_{V}$ from [@cardelli:1989] (Eqns. 1-3b) was used to convert from near-IR to visual extinction. The (standard for the ISM) optical total-to-selective extinction ratio $R_{V}$ = 3.1 was assumed. This has been found to reach values of 5-6 towards dense clouds [e.g. @cardelli:1989] but using $R_{V}$ = 5 did not affect the results within error.
The near-IR extinction was computed from the difference between the measured and expected near-IR magnitudes. For the flux-ratio methods, the expected H- and K-band fluxes were obtained by utilising the intrinsic ratios between IR and radio flux found by [@willner:1972], $F_{\rm H}/F_{\rm 5GHz}$ = 0.26 and $F_{\rm K}/F_{\rm 5GHz}$ = 0.3. Using the Willner ratios, a value of 0.68 for $H-K$ was computed in this work. In order to obtain the $J-H$ extinction, $F_{\rm J}/F_{\rm 5GHz}$ = 0.43 was computed, using Equation 1 and Table 2 in [@brussaard:1962], taking into account the significant Paschen-$\beta$ line contribution to the J band [@hummer:1987]. This resulted in a value of $-$0.1 for $J-H$. These predicted intrinsic $J-H$ and $H-K$ colours of ionised nebulae agree well with those from near-IR photometry of planetary nebulae (with 2MASS data) studied by [@larios:2005]: $H-K$ = 0.65 and $J-H$ $\sim$ $-$0.1. [@weidmann:2013] also found comparable values, using PNe in the VVV survey[^8]: $H-K$ = 0.62 and $J-H$ = 0.0. This comparison is in the Vega system, taking into account the 2MASS (Vega system) and UKIDSS (AB system) magnitude offsets for each band. Our computed intrinsic AB colours, $J-H$ = $-$0.54 and $H-K$ = 0.15, were converted to the Vega system using Table 7 in [@hewett:2006]: the AB offsets for the $J$, $H$, and $K$ bands are 0.938, 1.379, and 1.9, respectively.
The presented results from the four different extinction methods are consistent with previous estimates of visual extinctions towards UCHIIs, which are $\sim$ 0–50 mags [see e.g. @hanson:2002]. [@moore:2005] calculated the extinction for a number of compact and ultra-compact regions, using observed hydrogen recombination lines. A comparison between extinction results from this work and from literature is presented in Table \[table:av\_comparison\], showing good agreement.
As can be seen in Fig. \[fig:Av\], the offsets between the extinctions obtained from the different methods are clearly systematic. Values derived using the K band are $\sim$ 10 magnitudes lower (i.e. brighter) than those using H (and J). An addition of $\sim$ 10 magnitudes to the K band would bring the $F_{\rm H}/F_{5 \rm GHz}$ and $F_{\rm K}/F_{5 \rm GHz}$ methods to agreement and eliminate most of the unphysical negative values. Such an addition would mimic eliminating the expected boost to the K band from the contribution of very hot dust in the vicinity of the ionising star. However, a $\sim$ 10-magnitude addition to $K$ in the $A_{\rm V,(J-H)}$ vs. $A_{\rm V,(H-K)}$ diagram to exclude potential hot dust contribution would actually result in a $\sim$ 20-magnitude systematic offset between these two methods. Such a discrepancy likely stems from the general nature of each pair of utilised methods and highlights the need for in-depth investigation, preferably on a case-to-case basis (beyond the scope of this work).
Two of the UCHIIs with computed extinction have coinciding $XMM-Newton$ hard x-ray sources. The extinction was computed independently from x-ray spectral fitting (which also revealed that the sources have $k_{\rm B}T \ > \ 2$ keV, i.e. $T \ > \ 10^{7}$ K). The $H-K$ $-$ derived visual extinction for G030.7661$-$00.0348 is $\sim$ 35 mags, whereas the x-ray$-$fitted hydrogen column density translates to $A_{\rm V}$ $\approx$ 67 mags (towards the W43 star cluster as a whole). G025.3824$-$00.1812 has $H-K$ $-$ derived $A_{\rm V}$ $\approx$ 12, and the value derived from the x-ray spectral fit is $\sim$ 11 mags[^9].
Spectral energy distributions and bolometric luminosities {#lbol}
=========================================================
![Average SED for all UCHIIs with available multi-wavelength data, normalised to the sample median distance of 9.8 kpc. The plot includes the $J$, $H$, and $K$ UKIDSS fluxes and the 3.6, 4.5, 5.8, and 8.0 $\upmu$m GLIMPSE fluxes from this work, together with fluxes from MSX (21 $\upmu$m), WISE (22 $\upmu$m), HiGAL (70, 160, 250, 350, and 500 $\upmu$m), and ATLASGAL (870 $\upmu$m) (see table A.1 in @cesaroni:2015).[]{data-label="fig:seds"}](avg_sed){width="\columnwidth"}
The computed UKIDSS and GLIMPSE fluxes were combined with multi-wavelength data from MSX (21 $\upmu$m), WISE (22 $\upmu$m), HiGAL (70, 160, 250, 350, and 500 $\upmu$m), and ATLASGAL (870 $\upmu$m). These data were available for 177 UCHIIs and the SEDs were reconstructed for these sources. The majority of the SEDs have very reasonable shapes and exhibit the same average shape. There are a few SEDs with irregularities, typically the flux at 350 $\upmu$m and 22 $\upmu$m. There are many SEDs (90/177) where the 4.5 $\upmu$m flux is low (comparable to the 3.6 $\upmu$m flux), which is most likely caused by the gap in PAH emission at this wavelength. The average SED for the sample (when all sources are placed at the median distance of 9.8 kpc) is presented in Fig. \[fig:seds\].
![Bolometric luminosity distribution for the UCHIIs with fitted SEDs.[]{data-label="fig:lbol_hist"}](hist_Lbol){width="\columnwidth"}
![5 GHz radio luminosity against bolometric luminosity for the UCHIIs with fitted SEDs, shown with circles. A sample of confirmed ionised MYSO jets by [@purser:2017] is also included (indicated by diamonds). The dash-dotted line marks the expected radio luminosity from Lyman continuum emission. The dotted line shows the empirical relation extrapolated for low-mass YSO jets [@anglada:1995]. The angular size of all sources is colour-coded and emphasized through the marker sizes (the jet sample marker sizes are exaggerated, as their resolution is higher).[]{data-label="fig:lbol"}](Lbol){width="\columnwidth"}
The bolometric luminosities of the CORNISH UCHIIs were calculated from fitting the SEDs of the sources with available data across multiple wavelengths. The SED Fitter from [@robitaille:2007] was utilised, following the procedure prescribed by [@mottram:2011]. The complete sample was split into subsamples according to their distance (as computed in this work) to limit the distance range when running the fits. The fitted bolometric luminosities were then converted to the final bolometric luminosity values by replacing the automatically fitted distances with the distances presented in this work. The results are shown in Fig. \[fig:lbol\_hist\]. These are in good agreement with Fig. 19 in [@urquhart:2013], which summarises the RMS bolometric luminosities of 135 associated clumps. We note that the models used in the SED fitting are tailored to YSOs and do not take into account additional sources of dust heating in the ionised region, such as Lyman-$\alpha$ [@hoare:1991]. However, the SEDs of UCHIIs and YSOs are sufficiently similar up to radio wavelengths (which are not included in the models) and the fits are useful for computing the bolometric luminosities.
Possible contamination by MYSO jet emission was investigated with a plot of radio luminosity at 5 GHz against bolometric luminosity of the UCHII regions and comparing to a sample of confirmed MYSO jets by [@purser:2017]. This is presented in Fig. \[fig:lbol\]. The UCHII regions are brighter, as expected, and the two populations clearly separate. The predicted optically-thin 5 GHz radio luminosities corresponding to a range of bolometric luminosities, from the stellar models by [@thompson:1984] for $L_{\rm bol} \ \leq \ 10^{3} \ \rm L_{\odot}$ and from the models summarised in Table 1 of @davies:2011 for $L_{\rm bol} \ > \ 10^{3} \ \rm L_{\odot}$, are shown with a dash-dotted line in the plot. The radio continuum flux can be inferred from the Lyman continuum flux, as the latter determines the amount of ionised material and thus the number of free electrons participating in the thermal bremsstrahlung process.
The majority of CORNISH UCHIIs have radio luminosity that is between 1% and 100% of the theoretical value at the corresponding region bolometric luminosity (i.e. below the model line). A mixture of UCHII angular sizes are found at equal fractions of the model luminosity, ruling out angular size as the culprit behind the large variation in observed luminosity. It is to be expected that a large portion of UCHIIs would be dimmer than predicted, as the stellar models do not account for the portion of Lyman continuum flux that is absorbed by dust.
A significant number of sources (about a third) can be found above the line, with radio luminosities up to ten times or more than those predicted. The presence of regions in the ‘forbidden’ area above the model line is attributed to a Lyman excess. There are different explanations for such regions, as discussed, for example, by [@sanchez:2013], [@cesaroni:2015; @cesaroni:2016], and in the references therein. One explanation is the assumption of spherical symmetry, whereas in reality Lyman photons could be leaking in directions tracing lower gas density, which could be away from the line of sight. It is unlikely for the contribution to be from overlapping 5$\sigma$ or 7$\sigma$ UCHIIs, as no systematic trends were found in their distribution in the plot. [@cesaroni:2016] used molecular tracers to look for outflows and accretion shocks in the vicinity of 200 CORNISH regions, and found no evidence to support any outflow-related phenomenon. Instead, they found that Lyman-excess sources are more associated with infall than non-excess sources, and propose ongoing accretion and accretion shocks as an explanation, but their HCO$^{+}$ measurements are only sensitive to large-scale infall ($\lesssim$ 1 pc), which is not direct evidence of accretion.
Two sources, G065.2462+00.3505 and G011.0328+00.0274, were found to have computed bolometric luminosities below the lower limit for region formation – that is, for a B3 type star with $L_{\rm bol} \ \sim \ 10^{3} \ \rm L_{\odot}$ ([@boehm:1981], [@meynet:2003]). In the case of G011.0328+00.0274, this is likely due to an individual issue with the distance determination or the bolometric flux determination, as the radio-to-bolometric flux ratio is not extreme. G065.2462+00.3505, originally in the UCHII catalogue, was reclassified as a radio star. Although this unresolved radio source is embedded within an IR nebula $\sim$ 50$''$ in diameter, there is only a near-IR stellar counterpart but no trace of a compact counterpart at 8 $\upmu$m or 70 $\upmu$m.
It should be noted that 6 sources (G011.9786$-$00.0973, G014.5988+00.0198, G018.6654+00.0294, G031.1590+00.0465, G036.4062+00.0221, and G043.7960$-$00.1286) could not be shown in the plot. No bolometric luminosity was computed for them as the available supplementary data was not sufficient to build a good portion of their SEDs for a fit. Their $S_{\rm 5 GHz}D^{2}$ values range from $\sim$ 34 to $\sim$ 828 $\rm mJy \ kpc^{2}$, and would thus still separate from the MYSO sample in the plot.
Comparison of UCHII search methods in blind surveys {#comparison_methods}
===================================================
We argue that high resolution blind radio surveys are the most reliable way to obtain the UCHII population census of the Milky Way. With the CORNISH sample of genuine UCHII regions, our total estimate is $\sim$ 750 UCHIIs in the Galaxy. This was obtained by scaling up the sample size (239 sources) by a geometric correction factor of $\sim$ 3.1 – the ratio between the detected number of RMS UCHII regions in the total RMS and CORNISH area – 500 and 160, respectively (see @urquhart:2008 [@lumsden:2013]). Colour-selected RMS sources with detectable radio emission were classified as UCHIIs due to their mid-IR morphologies. Practically the same scaling factor is obtained when taking the total RMS sample of 900 regions and the 297 RMS regions in the CORNISH area. The RMS survey encompasses $10\degree<l<350\degree$ and $|b|<5\degree$ and provides the best current map of the non-uniform distribution of massive star formation throughout the Galaxy. UCHII regions exhibit the same Galactic scale-height, $\sim$ 0.6, in CORNISH and RMS. Since the RMS UCHII counts in the total and in the CORNISH area were obtained in the same manner, their ratio should not be greatly affected by the systematic limitations of the IR-selection. We note that UCHIIs near the Galactic centre are missed in our total estimate, to avoid assumptions for the UCHII number density in this region not covered by RMS.
The sections below highlight limitations of the other UCHII search methods employed in Galactic plane surveys.
Sub-mm – ATLASGAL
-----------------
[@urquhart:2013] used ATLASGAL-CORNISH associations to compute the surface density of UCHII regions as a function of Galactocentric distance. They estimated that the Galactic UCHII population comprises $\sim$ 400 sources around B0 and earlier type stars, out of which only $\sim$ 45 around O6 or earlier type stars are detectable. However, the depth of ATLASGAL falls short of detecting all UCHII regions within the common ATLASGAL-CORNISH area – about 30 UCHII regions were missed as a result. The deeper SCUBA-2 survey should provide a higher detection certainty with the same search method.
Mid-IR – RMS
------------
The whole CORNISH region is covered in RMS; despite this, about half of the CORNISH UCHIIs have RMS counterparts. There are $\sim$ 40 further associations with the more diffuse CORNISH sample (48 regions). Many of the UCHIIs ‘missing’ from RMS are, in fact, detected within big complexes but not listed as individual objects. [@urquhart:2013] discussed the larger total number of RMS versus ATLASGAL regions – the RMS sample contains extended regions identified from their mid-IR morphology, with radio continuum emission lower than the CORNISH surface brightness sensitivity. The RMS sample thus includes fewer individually listed UCHII regions, together with a number of more extended regions.
Previous radio surveys of the Galactic plane
--------------------------------------------
[@giveon:2008] found 494 MSX matches to the [@white:2005] catalogue within the area shared with CORNISH ($\sim$ 23%). [@giveon:2005] believe their sample is dominated by UCHII regions. As discussed in §\[lowres\], through visual inspection we only found 162 CORNISH UCHII regions in common with the lower-resolution 6 cm data. A catalogue table cross-match found even fewer sources in common – 111, with 20 cross-matches to 37 CORNISH diffuse regions within the shared area. Thus we are finding a factor of 3-4 fewer UCHIIs than implied by [@giveon:2005; @giveon:2008] – the vast majority of their sample are not genuine ultra-compact regions but rather represent more extended region phases.
Summary
=======
The CORNISH UCHII sample is the largest complete and unbiased high-resolution collection of ultra-compact regions to date. Within the mapped region ($10\degree<l<65\degree, |b|<1\degree$), 239 UCHIIs have been confirmed from 240 candidates visually identified at 5 GHz radio-continuum emission. In this work, we explored the observational properties, spectral indices, physical characteristics and spectral energy distributions of this early stage of massive star formation. In summary:
1. The selection procedure for the CORNISH UCHIIs is robust and the nature of the sample as a whole was reliably identified.
2. The majority ($\sim$ 82%) of UCHIIs have spectral indices that are consistent with the expected theoretical limits for thermal free-free emission. The instances of non-thermal spectral indices could be naturally resulting from the difference in VLA configuration between the higher- and lower-frequency datasets, or are the result of combining 6 cm and 20 cm fluxes of variable thermal sources from two epochs.
3. We conclude that at least 5% of UCHIIs have exhibited a significant flux increase (by $\sim$ 50% or more) between two observational epochs separated by $\sim$ 15 years.
4. Distances were computed for 21 UCHIIs which had no literature distance (or their KDA had not been previously resolved) prior to this work. The derived physical properties of the UCHII sample agree well with theoretical expectations.
5. We have presented results of extended source photometry of UCHII regions in the mid- and near-IR. The GLIMPSE and UKIDSS colours of the sample follow the expected trends set by results obtained from earlier, smaller samples. We expect the mid-IR results to be particularly reliable, as they combine the precise knowledge of position, radio size, and shape provided by the CORNISH survey with the good correspondence (in the vast majority of cases) to the mid-IR counterparts. The results of the extended near-IR photometry (particularly the J and H bands) should be used with much more care, due to the difficulty in accurate subtraction of the stellar contamination in the busy, diffuse environments of star forming regions seen at these wavelengths.
6. Extinctions towards the UCHII regions were computed using the intrinsic H- and K-band to radio flux ratio from [@willner:1972], as well as from the $J-H$ and $H-K$ nebular colours.
7. The average spectral energy distribution of the UCHII sample (from gathering available multi-wavelength data and combining them with the new near- and mid-IR results) is in excellent agreement with the expected shape (see e.g. @faison:1998, Fig. 1 in @hoare:2012), with a peak between 70 and 160 $\upmu$m. Bolometric luminosities were computed by fitting the individual SEDs. In a plot of radio luminosity against bolometric luminosity, the CORNISH UCHII sample is clearly a separate population to confirmed MYSO jets. About a third of the UCHIIs exhibit a Lyman excess.
8. High resolution blind radio surveys are the best way to definitively find the UCHII population of the Galaxy. Radio selection provides a more reliable statistic than infrared and mm selection. We found a factor of 3-4 fewer genuine ultra-compact regions than in previous lower resolution radio areal surveys, which, in conjunction with up-to-date models (see @davies:2011), goes towards alleviating the *lifetime problem* posed by [@churchwell:1989a].
IEK acknowledges the support of the Science and Technology Facilities Council of the United Kingdom (STFC) through the award of a studentship. This publication has made use of data from the CORNISH survey database (<http://cornish.leeds.ac.uk/public/index.php>) and the RMS survey database (<http://rms.leeds.ac.uk/cgi-bin/public/RMS_DATABASE.cgi>). This work has also made use of the SIMBAD database (CDS, Strasbourg, France).
KDA resolution – plots {#appendixB}
======================
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Radio fluxes and spectral indices {#appendixD}
=================================
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G010.3009$-$00.1477 & 631.39 $\pm$ 59.3 & 1169.19 $\pm$ 9.41 & 456.88 $\pm$ 74.73 & 0.25 $\pm$ 0.15\
G010.3204$-$00.2328 & 32.43 $\pm$ 5 & <28.78 & 48.94 $\pm$ 2.56 & $-$0.32 $\pm$ 0.13\
G010.3204$-$00.2586 & 18.2 $\pm$ 2.26 & 47.09 $\pm$ 3.28 & 38.2 $\pm$ 1.97 & $-$0.58 $\pm$ 0.11\
G010.4724+00.0275 & 38.43 $\pm$ 4.38 & 92.82 $\pm$ 1.35 & 20.69 $\pm$ 3.36 & 0.49 $\pm$ 0.16\
G010.4736+00.0274 & 19.3 $\pm$ 2.39 & 94.83 $\pm$ 1.01 & 17.68 $\pm$ 3.02 & 0.07 $\pm$ 0.17\
G010.6218$-$00.3848 & 37.06 $\pm$ 4.04 & 1238.9 $\pm$ 2.96 & 167.89 $\pm$ 4.76 & $-$1.19 $\pm$ 0.09\
G010.6223$-$00.3788 & 483.33 $\pm$ 49.87 & 1039.47 $\pm$ 7.42 & 518.5 $\pm$ 15.91 & $-$0.06 $\pm$ 0.08\
G010.6234$-$00.3837 & 1952.22 $\pm$ 176.18 & 3921.35 $\pm$ 6.97 & 747.85 $\pm$ 14.52 & 0.75 $\pm$ 0.07\
G010.6240$-$00.3813 & 71.65 $\pm$ 7.24 & 492.7 $\pm$ 2.53 & 237.88 $\pm$ 8.45 & $-$0.94 $\pm$ 0.08\
G010.6297$-$00.3380 & 26.39 $\pm$ 4.29 & 44.91 $\pm$ 2.27 & 74.02 $\pm$ 1.42 & $-$0.81 $\pm$ 0.13\
G010.8519$-$00.4407 & 11.09 $\pm$ 1.46 & & 7.92 $\pm$ 0.53 & 0.26 $\pm$ 0.12\
G010.9584+00.0221 & 195.97 $\pm$ 18.33 & 326.65 $\pm$ 1.39 & 43.71 $\pm$ 1.19 & 1.18 $\pm$ 0.08\
G010.9656+00.0089 & 51.75 $\pm$ 6.1 & 206.89 $\pm$ 2.11 & 141.83 $\pm$ 9.47 & $-$0.79 $\pm$ 0.11\
G011.0328+00.0274 & 5.69 $\pm$ 1.06 & <4.14 & 4.06 $\pm$ 1.07 & 0.26 $\pm$ 0.25\
G011.1104$-$00.3985 & 305.37 $\pm$ 28.55 & 303.63 $\pm$ 9.12 & 326.73 $\pm$ 3.86 & $-$0.05 $\pm$ 0.07\
G011.1712$-$00.0662 & 102.17 $\pm$ 12.73 & 131.49 $\pm$ 5.72 & 77.43 $\pm$ 4.17 & 0.22 $\pm$ 0.11\
G011.9032$-$00.1407 & 25.57 $\pm$ 3.3 & 83.07 $\pm$ 1.68 & 30.46 $\pm$ 0.89 & $-$0.14 $\pm$ 0.1\
G011.9039$-$00.1411 & 16.81 $\pm$ 1.88 & 69.77 $\pm$ 1.08 & 25.13 $\pm$ 0.56 & $-$0.32 $\pm$ 0.09\
G011.9368$-$00.6158 & 1155.9 $\pm$ 105.38 & & 690.47 $\pm$ 5.18 & 0.4 $\pm$ 0.07\
G011.9446$-$00.0369 & 943.58 $\pm$ 98.5 & 2036.45 $\pm$ 24.35 & 1116.9 $\pm$ 77 & $-$0.13 $\pm$ 0.1\
G011.9786$-$00.0973 & 4.46 $\pm$ 1.08 & <6.82 & <2.64 & >0.41\
G012.1988$-$00.0345 & 62.71 $\pm$ 5.92 & 102.95 $\pm$ 1.47 & 42.52 $\pm$ 1.34 & 0.31 $\pm$ 0.08\
G012.2081$-$00.1019 & 207.87 $\pm$ 19.73 & 309.84 $\pm$ 3.25 & 115.09 $\pm$ 5.64 & 0.46 $\pm$ 0.08\
G012.4294$-$00.0479 & 45.17 $\pm$ 4.35 & 103.16 $\pm$ 4.93 & 56.7 $\pm$ 3.79 & $-$0.18 $\pm$ 0.09\
G012.8050$-$00.2007 & 12616.4 $\pm$ 1120.83 & 26528.17 $\pm$ 88.83 & 6884.66 $\pm$ 10.35 & 0.48 $\pm$ 0.07\
G012.8131$-$00.1976 & 1500.39 $\pm$ 147.3 & 593.9 $\pm$ 103.73 & 731.91 $\pm$ 4.37 & 0.56 $\pm$ 0.08\
G012.9995$-$00.3583 & 20.14 $\pm$ 3.7 & 6.68 $\pm$ 0.62 & 4.28 $\pm$ 0.34 & 1.22 $\pm$ 0.16\
G013.2099$-$00.1428 & 946.76 $\pm$ 87.46 & 1864.65 $\pm$ 10.02 & 818.4 $\pm$ 9.95 & 0.11 $\pm$ 0.07\
G013.3850+00.0684 & 603.94 $\pm$ 60.83 & 1223.45 $\pm$ 21.43 & 953.46 $\pm$ 9.06 & $-$0.36 $\pm$ 0.08\
G013.8726+00.2818 & 1447.55 $\pm$ 129.84 & 4276.31 $\pm$ 22.89 & 2422 $\pm$ 16.98 & $-$0.4 $\pm$ 0.07\
G014.1046+00.0918 & 24.61 $\pm$ 3.11 & 83.46 $\pm$ 4.89 & 66.7 $\pm$ 1.56 & $-$0.78 $\pm$ 0.1\
G014.1741+00.0245 & 47.73 $\pm$ 7.42 & 139.51 $\pm$ 2.76 & 86.1 $\pm$ 1.69 & $-$0.46 $\pm$ 0.12\
G014.2460$-$00.0728 & 51.26 $\pm$ 6.18 & 87.89 $\pm$ 3.65 & 63.45 $\pm$ 1.29 & $-$0.17 $\pm$ 0.1\
G014.5988+00.0198 & 4.39 $\pm$ 1.09 & <3.45 & 7.14 $\pm$ 1.92 & $-$0.38 $\pm$ 0.29\
G014.7785$-$00.3328 & 18.25 $\pm$ 2.47 & 39.47 $\pm$ 1.08 & <12.66 & >0.29\
G016.1448+00.0088 & 14.76 $\pm$ 1.55 & 27.8 $\pm$ 1.15 & 6.87 $\pm$ 0.52 & 0.6 $\pm$ 0.1\
G016.3913$-$00.1383 & 124.27 $\pm$ 15.43 & 43.18 $\pm$ 3.61 & 57.52 $\pm$ 5.18 & 0.61 $\pm$ 0.12\
G016.9445$-$00.0738 & 519.34 $\pm$ 47.78 & 901.84 $\pm$ 5.5 & 242.3 $\pm$ 4.44 & 0.6 $\pm$ 0.07\
G017.0299$-$00.0696 & 5.38 $\pm$ 1.06 & <7.01 & <4.75 & >0.10\
G017.1141$-$00.1124 & 17.21 $\pm$ 2.19 & 34.3 $\pm$ 1.93 & 21.07 $\pm$ 2.01 & $-$0.16 $\pm$ 0.13\
G017.5549+00.1654 & 7.13 $\pm$ 1.22 & 12.15 $\pm$ 1.65 & 7.9 $\pm$ 1.28 & $-$0.08 $\pm$ 0.19\
G017.9850+00.1266 & 10.42 $\pm$ 2.07 & 18.55 $\pm$ 2.7 & 10.11 $\pm$ 1.11 & 0.02 $\pm$ 0.18\
G018.1460$-$00.2839 & 856.18 $\pm$ 82.85 & 546.59 $\pm$ 33.33 & 1084.54 $\pm$ 342.92 & $-$0.19 $\pm$ 0.26\
G018.3024$-$00.3910 & 1277.88 $\pm$ 114.83 & 2032.05 $\pm$ 23.59 & 1372.98 $\pm$ 26.86 & $-$0.06 $\pm$ 0.07\
G018.4433$-$00.0056 & 81.31 $\pm$ 7.3 & 116.98 $\pm$ 2.35 & 56.96 $\pm$ 4.46 & 0.28 $\pm$ 0.09\
G018.4614$-$00.0038 & 342.12 $\pm$ 31.5 & 548.25 $\pm$ 2.21 & 117.05 $\pm$ 4.72 & 0.84 $\pm$ 0.08\
G018.6654+00.0294 & 5.65 $\pm$ 0.85 & 7.57 $\pm$ 1.45 & 3.7 $\pm$ 0.42 & 0.33 $\pm$ 0.15\
G018.7106+00.0002 & 107.46 $\pm$ 10.62 & 177.74 $\pm$ 1.71 & 36.6 $\pm$ 0.94 & 0.85 $\pm$ 0.08\
G018.7612+00.2630 & 51.38 $\pm$ 4.67 & 31.23 $\pm$ 2.24 & 21.99 $\pm$ 0.61 & 0.67 $\pm$ 0.07\
G018.8250$-$00.4675 & 11.41 $\pm$ 2.17 & & <7.83 & >0.30\
G018.8338$-$00.3002 & 131.38 $\pm$ 13.35 & 219.21 $\pm$ 3.86 & 103.4 $\pm$ 1.73 & 0.19 $\pm$ 0.08\
G019.0035+00.1280 & 6.41 $\pm$ 1.52 & <9.77 & 5.06 $\pm$ 0.56 & 0.19 $\pm$ 0.21\
G019.0754$-$00.2874 & 380.69 $\pm$ 37.06 & 605.36 $\pm$ 7.04 & 472.33 $\pm$ 27.38 & $-$0.17 $\pm$ 0.09\
G019.0767$-$00.2882 & 129.48 $\pm$ 13.15 & 231.03 $\pm$ 1.95 & 123.2 $\pm$ 5.79 & 0.04 $\pm$ 0.09\
G019.4752+00.1728 & 37.22 $\pm$ 4.56 & 58.72 $\pm$ 2.55 & 41.58 $\pm$ 0.89 & $-$0.09 $\pm$ 0.1\
G019.6087$-$00.2351 & 2900.88 $\pm$ 260.93 & 5170.32 $\pm$ 17.38 & 2170.95 $\pm$ 7.04 & 0.23 $\pm$ 0.07\
G019.6090$-$00.2313 & 259.95 $\pm$ 26.87 & 1079.53 $\pm$ 3.43 & 512.96 $\pm$ 2.52 & $-$0.53 $\pm$ 0.08\
G019.7281$-$00.1135 & 26.23 $\pm$ 3.09 & 27.19 $\pm$ 1.41 & 29 $\pm$ 0.61 & $-$0.08 $\pm$ 0.09\
G019.7549$-$00.1282 & 36.52 $\pm$ 3.29 & 46.08 $\pm$ 0.73 & 8.91 $\pm$ 0.3 & 1.11 $\pm$ 0.08\
G020.0720$-$00.1421 & 210.13 $\pm$ 21.54 & 416.03 $\pm$ 3.51 & 211.18 $\pm$ 7.05 & 0 $\pm$ 0.08\
G020.0797$-$00.1337 & 14.09 $\pm$ 1.91 & 89.89 $\pm$ 1.23 & 45.64 $\pm$ 0.77 & $-$0.92 $\pm$ 0.11\
G020.0809$-$00.1362 & 498.19 $\pm$ 45.06 & 857.54 $\pm$ 1.3 & 135.56 $\pm$ 0.9 & 1.02 $\pm$ 0.07\
G020.3633$-$00.0136 & 55.11 $\pm$ 5.93 & 98.78 $\pm$ 1.33 & 53.41 $\pm$ 0.62 & 0.02 $\pm$ 0.09\
G020.7619$-$00.0646 & 10.03 $\pm$ 2.08 & 20.82 $\pm$ 6.97 & 13.62 $\pm$ 2.06 & $-$0.24 $\pm$ 0.2\
G020.9636$-$00.0744 & 11.28 $\pm$ 1.84 & 16.75 $\pm$ 1.8 & 14.24 $\pm$ 0.57 & $-$0.18 $\pm$ 0.13\
G021.3571$-$00.1766 & 24.93 $\pm$ 2.34 & 44.61 $\pm$ 2.46 & 16.13 $\pm$ 0.49 & 0.34 $\pm$ 0.08\
G021.3855$-$00.2541 & 113.91 $\pm$ 11.24 & 207.13 $\pm$ 4.78 & 46.24 $\pm$ 0.76 & 0.71 $\pm$ 0.08\
G021.6034$-$00.1685 & 19.84 $\pm$ 3.45 & 46.78 $\pm$ 1.32 & 29.12 $\pm$ 0.81 & $-$0.3 $\pm$ 0.14\
G021.8751+00.0075 & 566.73 $\pm$ 54.14 & 1064.73 $\pm$ 8.85 & 685.9 $\pm$ 7.84 & $-$0.15 $\pm$ 0.08\
G023.1974$-$00.0006 & 10.01 $\pm$ 1.54 & 18.65 $\pm$ 1.24 & 13.27 $\pm$ 0.73 & $-$0.22 $\pm$ 0.13\
G023.2654+00.0765 & 88.57 $\pm$ 9.87 & 149.89 $\pm$ 4.39 & 90.95 $\pm$ 0.96 & $-$0.02 $\pm$ 0.09\
G023.4553$-$00.2010 & 14.39 $\pm$ 1.56 & <3.16 & <6.18 & >0.66\
G023.4835+00.0964 & 8.23 $\pm$ 1.39 & <7.96 & 7.35 $\pm$ 0.55 & 0.09 $\pm$ 0.15\
G023.7110+00.1705 & 208.5 $\pm$ 20.34 & 548.53 $\pm$ 1.38 & 279.31 $\pm$ 33.22 & $-$0.23 $\pm$ 0.12\
G023.8618$-$00.1250 & 39.16 $\pm$ 6.29 & 38.86 $\pm$ 2.59 & 48.71 $\pm$ 1.63 & $-$0.17 $\pm$ 0.13\
G023.8985+00.0647 & 43.42 $\pm$ 5.45 & 115.83 $\pm$ 5.25 & 102.69 $\pm$ 1.07 & $-$0.68 $\pm$ 0.1\
G023.9564+00.1493 & 1161.18 $\pm$ 104.78 & 2658.26 $\pm$ 8.5 & 1319.94 $\pm$ 13.31 & $-$0.1 $\pm$ 0.07\
G024.1839+00.1199 & 3.79 $\pm$ 0.79 & <3.44 & 3.39 $\pm$ 0.39 & 0.09 $\pm$ 0.19\
G024.4698+00.4954 & 29.5 $\pm$ 4.53 & & <276.17 & >$-$1.78\
G024.4721+00.4877 & 55.19 $\pm$ 7.73 & & 257.36 $\pm$ 20.99 & $-$1.21 $\pm$ 0.13\
G024.4736+00.4950 & 99.43 $\pm$ 12.24 & & <404.40 & >$-$1.10\
G024.4921$-$00.0386 & 140.12 $\pm$ 13.5 & 332.61 $\pm$ 2.12 & 173.02 $\pm$ 1.82 & $-$0.17 $\pm$ 0.08\
G024.5065$-$00.2224 & 205.57 $\pm$ 19.72 & 458.99 $\pm$ 9.78 & 266.55 $\pm$ 34.34 & $-$0.2 $\pm$ 0.13\
G024.8497+00.0881 & 19.93 $\pm$ 3.82 & 20.98 $\pm$ 6.23 & 91.44 $\pm$ 18.7 & $-$1.2 $\pm$ 0.22\
G025.3809$-$00.1815 & 460.83 $\pm$ 42.66 & 1799.06 $\pm$ 7.23 & 992.44 $\pm$ 112.85 & $-$0.6 $\pm$ 0.12\
G025.3824$-$00.1812 & 200.13 $\pm$ 20.03 & 929.75 $\pm$ 2.71 & 419.49 $\pm$ 42.51 & $-$0.58 $\pm$ 0.11\
G025.3948+00.0332 & 296.86 $\pm$ 27.46 & 549.92 $\pm$ 3.95 & 303.63 $\pm$ 6.08 & $-$0.02 $\pm$ 0.07\
G025.3970+00.5614 & 121.17 $\pm$ 11.53 & & 63.72 $\pm$ 0.5 & 0.5 $\pm$ 0.08\
G025.3981$-$00.1411 & 2132.24 $\pm$ 194.31 & 4590.88 $\pm$ 10.91 & 1837.09 $\pm$ 10.65 & 0.12 $\pm$ 0.07\
G025.3983+00.5617 & 51.93 $\pm$ 4.82 & & 71.21 $\pm$ 0.59 & $-$0.25 $\pm$ 0.07\
G025.3991$-$00.1366 & 29.38 $\pm$ 6.41 & & 197.73 $\pm$ 1.92 & $-$1.5 $\pm$ 0.17\
G025.7157+00.0487 & 20.79 $\pm$ 2.96 & <6.82 & 18.63 $\pm$ 0.91 & 0.09 $\pm$ 0.12\
G025.8011$-$00.1568 & 31.95 $\pm$ 2.96 & 56.84 $\pm$ 1.25 & 17.07 $\pm$ 0.53 & 0.49 $\pm$ 0.08\
G026.0083+00.1369 & 6.58 $\pm$ 1.03 & 8.13 $\pm$ 2 & 2.77 $\pm$ 0.39 & 0.68 $\pm$ 0.17\
G026.0916$-$00.0565 & 11.59 $\pm$ 2.13 & 110.38 $\pm$ 4.56 & 80.36 $\pm$ 6.89 & $-$1.52 $\pm$ 0.16\
G026.1094$-$00.0937 & 4.72 $\pm$ 0.84 & <7.73 & <8.57 & >$-$0.47\
G026.5976$-$00.0236 & 69.92 $\pm$ 7.45 & 145.65 $\pm$ 2.86 & 67.39 $\pm$ 1.9 & 0.03 $\pm$ 0.09\
G026.6089$-$00.2121 & 201.41 $\pm$ 21.67 & 380.66 $\pm$ 6.04 & 207.49 $\pm$ 1.26 & $-$0.02 $\pm$ 0.08\
G026.8304$-$00.2067 & 12.31 $\pm$ 1.51 & 19.08 $\pm$ 2.28 & 12.92 $\pm$ 0.33 & $-$0.04 $\pm$ 0.1\
G027.1859$-$00.0816 & 19.78 $\pm$ 2.17 & 30.5 $\pm$ 1.63 & 27.48 $\pm$ 0.24 & $-$0.26 $\pm$ 0.09\
G027.2800+00.1447 & 428.04 $\pm$ 42.07 & 769.86 $\pm$ 3.98 & 390.46 $\pm$ 23.49 & 0.07 $\pm$ 0.09\
G027.3644$-$00.1657 & 60.14 $\pm$ 6.13 & 98.6 $\pm$ 1.54 & 39.44 $\pm$ 0.85 & 0.33 $\pm$ 0.08\
G027.5637+00.0845 & 162.53 $\pm$ 18.81 & 149.05 $\pm$ 7.96 & 153.85 $\pm$ 2.94 & 0.04 $\pm$ 0.09\
G027.9352+00.2056 & 4.44 $\pm$ 1.04 & 20.34 $\pm$ 1.24 & 8.16 $\pm$ 0.66 & $-$0.48 $\pm$ 0.19\
G028.1985$-$00.0503 & 136.26 $\pm$ 12.94 & 395.1 $\pm$ 2.08 & 130.88 $\pm$ 1.17 & 0.03 $\pm$ 0.07\
G028.2003$-$00.0494 & 161.66 $\pm$ 15.59 & 395.27 $\pm$ 2.07 & 91.59 $\pm$ 1.24 & 0.45 $\pm$ 0.08\
G028.2879$-$00.3641 & 552.77 $\pm$ 51.9 & 939.64 $\pm$ 4.66 & 381.7 $\pm$ 1.56 & 0.29 $\pm$ 0.07\
G028.4518+00.0027 & 33.75 $\pm$ 3.22 & 73.57 $\pm$ 2 & 50.21 $\pm$ 2.52 & $-$0.31 $\pm$ 0.08\
G028.5816+00.1447 & 40.03 $\pm$ 7.76 & 65.99 $\pm$ 2.2 & 44.2 $\pm$ 0.81 & $-$0.08 $\pm$ 0.15\
G028.6082+00.0185 & 210.15 $\pm$ 20.28 & 379.49 $\pm$ 3.61 & 171.16 $\pm$ 1.93 & 0.16 $\pm$ 0.08\
G028.6523+00.0273 & 228.85 $\pm$ 22.02 & 439.25 $\pm$ 2.75 & 255.07 $\pm$ 3.97 & $-$0.09 $\pm$ 0.08\
G028.6869+00.1770 & 102.98 $\pm$ 11.05 & 145.12 $\pm$ 3.36 & 123.81 $\pm$ 1.14 & $-$0.14 $\pm$ 0.08\
G029.7704+00.2189 & 49.98 $\pm$ 8.84 & 36.37 $\pm$ 5.17 & 82.1 $\pm$ 1.63 & $-$0.39 $\pm$ 0.14\
G029.9559$-$00.0168 & 3116.2 $\pm$ 296.94 & 5643.42 $\pm$ 9.45 & 1935.82 $\pm$ 55.93 & 0.37 $\pm$ 0.08\
G030.0096$-$00.2734 & 4.54 $\pm$ 0.94 & 5.1 $\pm$ 1.31 & <1.32 & >0.97\
G030.2527+00.0540 & 96.79 $\pm$ 10.2 & 158.43 $\pm$ 5.97 & 127.15 $\pm$ 3.02 & $-$0.21 $\pm$ 0.08\
G030.5313+00.0205 & 85.54 $\pm$ 9.65 & 488.53 $\pm$ 2.75 & 243.43 $\pm$ 4.33 & $-$0.82 $\pm$ 0.09\
G030.5353+00.0204 & 710.36 $\pm$ 66.36 & 1385.62 $\pm$ 4.56 & 557.91 $\pm$ 19.1 & 0.19 $\pm$ 0.08\
G030.5887$-$00.0428 & 92.37 $\pm$ 8.33 & 99.57 $\pm$ 2.56 & 5.93 $\pm$ 0.99 & 2.16 $\pm$ 0.15\
G030.6881$-$00.0718 & 466.99 $\pm$ 45.69 & 992.36 $\pm$ 9.39 & 656.15 $\pm$ 75.69 & $-$0.27 $\pm$ 0.12\
G030.7197$-$00.0829 & 969.33 $\pm$ 96.01 & 873.67 $\pm$ 4.51 & 430.79 $\pm$ 15.79 & 0.64 $\pm$ 0.08\
G030.7532$-$00.0511 & 301.66 $\pm$ 29.35 & 165.02 $\pm$ 4.59 & <551.85 & >$-$0.47\
G030.7579+00.2042 & 26.23 $\pm$ 6.12 & <4.78 & 38.8 $\pm$ 1.62 & $-$0.31 $\pm$ 0.19\
G030.7661$-$00.0348 & 87.53 $\pm$ 14.67 & <19.92 & <262.05 & >$-$0.86\
G030.8662+00.1143 & 325.47 $\pm$ 32.96 & 476.58 $\pm$ 3.15 & 133.59 $\pm$ 2.15 & 0.7 $\pm$ 0.08\
G030.9581+00.0869 & 25.79 $\pm$ 4.68 & 18.13 $\pm$ 3.13 & 45.36 $\pm$ 0.8 & $-$0.44 $\pm$ 0.14\
G031.0495+00.4697 & 13.64 $\pm$ 1.49 & & 9.29 $\pm$ 2.44 & 0.3 $\pm$ 0.22\
G031.0595+00.0922 & 11.7 $\pm$ 1.51 & 16.49 $\pm$ 3.31 & 22.06 $\pm$ 3.74 & $-$0.5 $\pm$ 0.17\
G031.1590+00.0465 & 7.04 $\pm$ 1.26 & 8.65 $\pm$ 1.42 & 10.45 $\pm$ 0.87 & $-$0.31 $\pm$ 0.15\
G031.1596+00.0448 & 23.83 $\pm$ 2.28 & 19.34 $\pm$ 1.29 & 17.06 $\pm$ 0.95 & 0.26 $\pm$ 0.09\
G031.2420$-$00.1106 & 296.24 $\pm$ 27.05 & 800.06 $\pm$ 4.23 & 476.69 $\pm$ 2.51 & $-$0.37 $\pm$ 0.07\
G031.2435$-$00.1103 & 353.06 $\pm$ 32.36 & 601.08 $\pm$ 2.7 & 279 $\pm$ 1.02 & 0.18 $\pm$ 0.07\
G031.2448$-$00.1132 & 37.39 $\pm$ 4.2 & 46.07 $\pm$ 1.94 & 56.56 $\pm$ 1.27 & $-$0.33 $\pm$ 0.09\
G031.2801+00.0632 & 268.86 $\pm$ 25.67 & 593.67 $\pm$ 9.48 & 325.13 $\pm$ 3.15 & $-$0.15 $\pm$ 0.08\
G031.3959$-$00.2570 & 80.96 $\pm$ 10.38 & 195.56 $\pm$ 9.22 & 170.16 $\pm$ 22.76 & $-$0.58 $\pm$ 0.15\
G031.4130+00.3065 & 954.8 $\pm$ 87.99 & 1798.53 $\pm$ 14.61 & 927.3 $\pm$ 4.15 & 0.02 $\pm$ 0.07\
G031.5815+00.0744 & 14.48 $\pm$ 1.89 & 20.26 $\pm$ 3.03 & 17.83 $\pm$ 1.09 & $-$0.16 $\pm$ 0.11\
G032.0297+00.0491 & 26.68 $\pm$ 3.82 & 34.34 $\pm$ 2.6 & 29.25 $\pm$ 1.06 & $-$0.07 $\pm$ 0.12\
G032.1502+00.1329 & 533.63 $\pm$ 59.66 & 1207.3 $\pm$ 16.24 & 631.48 $\pm$ 9.29 & $-$0.13 $\pm$ 0.09\
G032.2730$-$00.2258 & 309.28 $\pm$ 34.38 & 735.13 $\pm$ 7.64 & 307.37 $\pm$ 3.27 & 0 $\pm$ 0.09\
G032.4727+00.2036 & 97.38 $\pm$ 9.67 & 180.96 $\pm$ 2.05 & 48.75 $\pm$ 0.59 & 0.54 $\pm$ 0.08\
G032.7398+00.1940 & 3.39 $\pm$ 0.78 & 3.23 $\pm$ 0.9 & 2.45 $\pm$ 0.73 & 0.26 $\pm$ 0.3\
G032.7492$-$00.0643 & 13.12 $\pm$ 1.67 & 12.82 $\pm$ 2.5 & 14.42 $\pm$ 1 & $-$0.07 $\pm$ 0.11\
G032.7966+00.1909 & 3123.37 $\pm$ 281.38 & 3841.71 $\pm$ 10.04 & 1445.26 $\pm$ 6.98 & 0.61 $\pm$ 0.07\
G032.7982+00.1937 & 17.22 $\pm$ 2 & & 13.18 $\pm$ 2.22 & 0.21 $\pm$ 0.16\
G032.9273+00.6060 & 285.57 $\pm$ 31.27 & & 260.63 $\pm$ 4.14 & 0.07 $\pm$ 0.09\
G033.1328$-$00.0923 & 378.59 $\pm$ 34.75 & 949.93 $\pm$ 5.99 & 192.26 $\pm$ 1.36 & 0.53 $\pm$ 0.07\
G033.4163$-$00.0036 & 75.16 $\pm$ 9.16 & 173.65 $\pm$ 11.75 & 108.73 $\pm$ 4.87 & $-$0.29 $\pm$ 0.1\
G033.8100$-$00.1864 & 107.63 $\pm$ 10.51 & 152.81 $\pm$ 1.64 & 59.67 $\pm$ 1.2 & 0.46 $\pm$ 0.08\
G033.9145+00.1105 & 842.22 $\pm$ 88.66 & 1661.36 $\pm$ 15.8 & 803.4 $\pm$ 7.5 & 0.04 $\pm$ 0.08\
G034.0901+00.4365 & 9.62 $\pm$ 1.9 & & 35.22 $\pm$ 4.34 & $-$1.02 $\pm$ 0.18\
G034.1978$-$00.5912 & 10.54 $\pm$ 2.42 & & 13.14 $\pm$ 1.13 & $-$0.17 $\pm$ 0.19\
G034.2544+00.1460 & 352.37 $\pm$ 39.13 & 1625.53 $\pm$ 42.08 & <1031.69 & >$-$0.84\
G034.2571+00.1466 & 47.8 $\pm$ 8.94 & 937.04 $\pm$ 59.69 & <678.32 & >$-$2.08\
G034.2572+00.1535 & 1762.63 $\pm$ 163.28 & 2771.66 $\pm$ 11.59 & 655.95 $\pm$ 4 & 0.78 $\pm$ 0.07\
G034.4032+00.2277 & 8.92 $\pm$ 1.24 & 14.97 $\pm$ 1.18 & 4.08 $\pm$ 0.74 & 0.61 $\pm$ 0.18\
G034.5920+00.2434 & 20.23 $\pm$ 2.4 & 35.02 $\pm$ 2.23 & 18.19 $\pm$ 0.89 & 0.08 $\pm$ 0.1\
G035.0242+00.3502 & 11.44 $\pm$ 1.23 & 22.25 $\pm$ 0.6 & 2.78 $\pm$ 0.63 & 1.11 $\pm$ 0.2\
G035.0524$-$00.5177 & 67.75 $\pm$ 7.56 & & 125.86 $\pm$ 1.7 & $-$0.49 $\pm$ 0.09\
G035.4570$-$00.1791 & 7.52 $\pm$ 1.07 & 12.75 $\pm$ 0.89 & 6.43 $\pm$ 0.7 & 0.12 $\pm$ 0.14\
G035.4669+00.1394 & 317.6 $\pm$ 29.36 & 573.76 $\pm$ 5.67 & 249.89 $\pm$ 1.4 & 0.19 $\pm$ 0.07\
G035.5781$-$00.0305 & 187.75 $\pm$ 18.44 & 252.49 $\pm$ 2.34 & 68.91 $\pm$ 1.39 & 0.79 $\pm$ 0.08\
G036.4057+00.0226 & 22.34 $\pm$ 2.21 & 58.2 $\pm$ 2.01 & 20.9 $\pm$ 0.78 & 0.05 $\pm$ 0.08\
G036.4062+00.0221 & 9.31 $\pm$ 1.15 & 52.18 $\pm$ 1.76 & 18.82 $\pm$ 0.7 & $-$0.55 $\pm$ 0.1\
G037.5457$-$00.1120 & 406.46 $\pm$ 41.28 & 913.02 $\pm$ 20.15 & 454.75 $\pm$ 8.67 & $-$0.09 $\pm$ 0.08\
G037.7347$-$00.1128 & 16.02 $\pm$ 1.63 & <2.28 & 9.54 $\pm$ 1.04 & 0.41 $\pm$ 0.12\
G037.7562+00.5605 & 35.68 $\pm$ 8.12 & & 43.73 $\pm$ 2.4 & $-$0.16 $\pm$ 0.18\
G037.8197+00.4140 & 25.99 $\pm$ 4.33 & 81.53 $\pm$ 3.1 & 36.66 $\pm$ 1.22 & $-$0.27 $\pm$ 0.13\
G037.8209+00.4125 & 20.24 $\pm$ 2.72 & 51.89 $\pm$ 2.14 & 29.07 $\pm$ 0.9 & $-$0.28 $\pm$ 0.11\
G037.8683$-$00.6008 & 210.28 $\pm$ 21.72 & & 237.48 $\pm$ 1.86 & $-$0.1 $\pm$ 0.08\
G037.8731$-$00.3996 & 2561.21 $\pm$ 234.04 & 5189.35 $\pm$ 23.38 & 1769.56 $\pm$ 6.57 & 0.29 $\pm$ 0.07\
G037.9723$-$00.0965 & 20.89 $\pm$ 2.52 & 26.37 $\pm$ 2.96 & 19.41 $\pm$ 0.77 & 0.06 $\pm$ 0.1\
G038.5493+00.1646 & 88.3 $\pm$ 11.26 & 248.94 $\pm$ 4.7 & 138.65 $\pm$ 1.73 & $-$0.35 $\pm$ 0.1\
G038.6465$-$00.2260 & 11.52 $\pm$ 1.98 & 35.13 $\pm$ 1.76 & 19.09 $\pm$ 0.91 & $-$0.4 $\pm$ 0.14\
G038.6529+00.0875 & 7.84 $\pm$ 1.04 & 36.01 $\pm$ 1.54 & 12.96 $\pm$ 0.57 & $-$0.39 $\pm$ 0.11\
G038.6934$-$00.4524 & 19.88 $\pm$ 2.44 & & 21.86 $\pm$ 0.88 & $-$0.07 $\pm$ 0.1\
G038.8756+00.3080 & 311.31 $\pm$ 29.87 & 599.65 $\pm$ 2.25 & 215.24 $\pm$ 1.37 & 0.29 $\pm$ 0.08\
G039.1956+00.2255 & 62.27 $\pm$ 6.41 & 100.08 $\pm$ 1.72 & 27.91 $\pm$ 0.74 & 0.63 $\pm$ 0.08\
G039.8824$-$00.3460 & 276.87 $\pm$ 26.38 & 636.36 $\pm$ 3.18 & 236.6 $\pm$ 1.31 & 0.12 $\pm$ 0.07\
G040.4251+00.7002 & 11.1 $\pm$ 2.42 & & 25.95 $\pm$ 0.99 & $-$0.67 $\pm$ 0.17\
G041.7419+00.0973 & 227.4 $\pm$ 23.95 & 584.48 $\pm$ 8.18 & 289.36 $\pm$ 2.11 & $-$0.19 $\pm$ 0.08\
G042.1090$-$00.4469 & 14.8 $\pm$ 1.53 & & 19.72 $\pm$ 0.54 & $-$0.23 $\pm$ 0.08\
G042.4345$-$00.2605 & 83.65 $\pm$ 9.25 & & 127.2 $\pm$ 8.07 & $-$0.33 $\pm$ 0.1\
G043.1460+00.0139 & 694.6 $\pm$ 69.66 & & 924.55 $\pm$ 4.97 & $-$0.22 $\pm$ 0.08\
G043.1489+00.0130 & 44.13 $\pm$ 6.07 & & 266.28 $\pm$ 5.85 & $-$1.41 $\pm$ 0.11\
G043.1520+00.0115 & 306.63 $\pm$ 34.78 & & 462.01 $\pm$ 49.65 & $-$0.32 $\pm$ 0.12\
G043.1651$-$00.0283 & 2714.29 $\pm$ 262.82 & & 2011.44 $\pm$ 89.31 & 0.24 $\pm$ 0.08\
G043.1652+00.0129 & 160.07 $\pm$ 17.72 & & 120.87 $\pm$ 37.07 & 0.22 $\pm$ 0.26\
G043.1657+00.0116 & 98.25 $\pm$ 10.44 & & 374.41 $\pm$ 30.64 & $-$1.05 $\pm$ 0.11\
G043.1665+00.0106 & 1365.68 $\pm$ 125.16 & & 743.01 $\pm$ 51.46 & 0.48 $\pm$ 0.09\
G043.1674+00.0128 & 74.54 $\pm$ 9.28 & & 199.37 $\pm$ 29.72 & $-$0.77 $\pm$ 0.15\
G043.1677+00.0196 & 115.47 $\pm$ 17.88 & & 478.89 $\pm$ 39.85 & $-$1.12 $\pm$ 0.14\
G043.1684+00.0087 & 185.25 $\pm$ 21.98 & & 633.19 $\pm$ 67.95 & $-$0.97 $\pm$ 0.13\
G043.1684+00.0124 & 63.6 $\pm$ 6.86 & & 185.7 $\pm$ 47.2 & $-$0.84 $\pm$ 0.22\
G043.1699+00.0115 & 43.79 $\pm$ 7.45 & & 248.48 $\pm$ 68.98 & $-$1.36 $\pm$ 0.26\
G043.1701+00.0078 & 1108.08 $\pm$ 103.11 & & 1863.27 $\pm$ 630.08 & $-$0.41 $\pm$ 0.28\
G043.1706$-$00.0003 & 170.72 $\pm$ 16.13 & & 349.99 $\pm$ 77.45 & $-$0.56 $\pm$ 0.19\
G043.1716+00.0001 & 51.74 $\pm$ 11.17 & & 466.43 $\pm$ 137.69 & $-$1.73 $\pm$ 0.29\
G043.1720+00.0080 & 98.59 $\pm$ 12.52 & & <724.09 & >$-$1.57\
G043.1763+00.0248 & 159.42 $\pm$ 21.7 & & 610.27 $\pm$ 27.29 & $-$1.05 $\pm$ 0.11\
G043.1778$-$00.5181 & 181.65 $\pm$ 23.04 & & 206.16 $\pm$ 3.51 & $-$0.1 $\pm$ 0.1\
G043.2371$-$00.0453 & 178.78 $\pm$ 20.41 & & 383.53 $\pm$ 3.41 & $-$0.6 $\pm$ 0.09\
G043.3064$-$00.2114 & 20.08 $\pm$ 3.25 & & <7.44 & >0.78\
G043.7954$-$00.1274 & 24.58 $\pm$ 2.97 & & 14.63 $\pm$ 0.92 & 0.41 $\pm$ 0.11\
G043.7960$-$00.1286 & 9.78 $\pm$ 1.94 & & 19.24 $\pm$ 0.84 & $-$0.53 $\pm$ 0.16\
G043.8894$-$00.7840 & 528.18 $\pm$ 48.82 & & 367.2 $\pm$ 2.91 & 0.29 $\pm$ 0.07\
G044.3103+00.0410 & 5.47 $\pm$ 0.94 & & 4.47 $\pm$ 0.65 & 0.16 $\pm$ 0.18\
G044.4228+00.5377 & 4.27 $\pm$ 0.87 & & 4.27 $\pm$ 1.1 & 0 $\pm$ 0.26\
G045.0694+00.1323 & 46.17 $\pm$ 4.44 & & 45.05 $\pm$ 0.51 & 0.02 $\pm$ 0.08\
G045.0712+00.1321 & 146.67 $\pm$ 14.65 & & 38.72 $\pm$ 0.55 & 1.05 $\pm$ 0.08\
G045.1223+00.1321 & 2984.27 $\pm$ 274.33 & & 1420.06 $\pm$ 4.98 & 0.58 $\pm$ 0.07\
G045.1242+00.1356 & 62.55 $\pm$ 11.86 & & 220.37 $\pm$ 57.06 & $-$0.99 $\pm$ 0.25\
G045.4545+00.0591 & 1029.45 $\pm$ 98.24 & & 1447.75 $\pm$ 233.21 & $-$0.27 $\pm$ 0.15\
G045.4559+00.0613 & 51.59 $\pm$ 7.58 & & 493.27 $\pm$ 104.52 & $-$1.77 $\pm$ 0.2\
G045.4656+00.0452 & 62.26 $\pm$ 5.79 & & 25.31 $\pm$ 1.66 & 0.71 $\pm$ 0.09\
G045.5431$-$00.0073 & 49.16 $\pm$ 7.8 & & 79.92 $\pm$ 2.24 & $-$0.38 $\pm$ 0.13\
Extended IR source fluxes – GLIMPSE and UKIDSS {#appendixC}
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G010.3009$-$00.1477 & 191.81 $\pm$ 80.80 & 316.14 $\pm$ 99.64 & 1982.12 $\pm$ 712.14 & 5117.70 $\pm$ 2547.03 & 10.69 $\pm$ 0.45 & 10.15 $\pm$ 0.34 & 8.16 $\pm$ 0.39 & 7.13 $\pm$ 0.54\
G010.3204$-$00.2328 & 107.48 $\pm$ 3.13 & 159.13 $\pm$ 4.36 & 812.79 $\pm$ 22.51 & 1646.41 $\pm$ 53.34 & 11.32 $\pm$ 0.03 & 10.90 $\pm$ 0.03 & 9.13 $\pm$ 0.03 & 8.36 $\pm$ 0.03\
G010.3204$-$00.2586 & 8.68 $\pm$ 2.98 & 57.50 $\pm$ 4.64 & 213.39 $\pm$ 41.53 & 495.73 $\pm$ 136.43 & 14.05 $\pm$ 0.37 & 12.00 $\pm$ 0.09 & 10.58 $\pm$ 0.21 & 9.66 $\pm$ 0.30\
G010.6218$-$00.3848 & 5.80 $\pm$ 3.48 & 26.34 $\pm$ 4.77 & & & 14.49 $\pm$ 0.65 & 12.85 $\pm$ 0.20 & &\
G010.6223$-$00.3788 & & 53.54 $\pm$ 23.54 & & & & 12.08 $\pm$ 0.48 & &\
G010.6234$-$00.3837 & 36.98 $\pm$ 15.23 & 147.58 $\pm$ 21.27 & 498.96 $\pm$ 259.22 & 1128.85 $\pm$ 700.78 & 12.48 $\pm$ 0.44 & 10.98 $\pm$ 0.16 & 9.65 $\pm$ 0.56 & 8.77 $\pm$ 0.67\
G010.6240$-$00.3813 & & & 117.75 $\pm$ 61.52 & 522.44 $\pm$ 159.40 & & & 11.22 $\pm$ 0.57 & 9.60 $\pm$ 0.33\
G010.6297$-$00.3380 & 106.73 $\pm$ 3.35 & 162.38 $\pm$ 3.85 & 1325.45 $\pm$ 29.06 & 3592.56 $\pm$ 83.78 & 11.33 $\pm$ 0.03 & 10.87 $\pm$ 0.03 & 8.59 $\pm$ 0.02 & 7.51 $\pm$ 0.03\
G010.9584+00.0221 & 25.54 $\pm$ 2.71 & 85.09 $\pm$ 2.70 & 250.68 $\pm$ 22.46 & 522.95 $\pm$ 80.51 & 12.88 $\pm$ 0.11 & 11.58 $\pm$ 0.03 & 10.40 $\pm$ 0.10 & 9.60 $\pm$ 0.17\
G010.9656+00.0089 & 47.99 $\pm$ 7.31 & 65.33 $\pm$ 7.72 & 378.14 $\pm$ 59.23 & 1163.15 $\pm$ 175.39 & 12.20 $\pm$ 0.16 & 11.86 $\pm$ 0.13 & 9.96 $\pm$ 0.17 & 8.74 $\pm$ 0.16\
G011.0328+00.0274 & 15.93 $\pm$ 1.73 & 23.95 $\pm$ 1.64 & 201.82 $\pm$ 5.16 & 532.55 $\pm$ 10.22 & 13.39 $\pm$ 0.12 & 12.95 $\pm$ 0.07 & 10.64 $\pm$ 0.03 & 9.58 $\pm$ 0.02\
G011.1104$-$00.3985 & 255.78 $\pm$ 14.95 & 391.37 $\pm$ 19.24 & 2148.49 $\pm$ 197.93 & 5957.55 $\pm$ 585.03 & 10.38 $\pm$ 0.06 & 9.92 $\pm$ 0.05 & 8.07 $\pm$ 0.10 & 6.96 $\pm$ 0.11\
G011.1712$-$00.0662 & 119.78 $\pm$ 12.64 & 114.47 $\pm$ 11.19 & 691.14 $\pm$ 30.49 & 1766.80 $\pm$ 48.46 & 11.20 $\pm$ 0.11 & 11.25 $\pm$ 0.11 & 9.30 $\pm$ 0.05 & 8.28 $\pm$ 0.03\
G011.9032$-$00.1407 & & 40.18 $\pm$ 4.12 & 118.80 $\pm$ 12.85 & 248.42 $\pm$ 30.12 & & 12.39 $\pm$ 0.11 & 11.21 $\pm$ 0.12 & 10.41 $\pm$ 0.13\
G011.9368$-$00.6158 & 256.25 $\pm$ 8.38 & 609.17 $\pm$ 14.57 & 3641.50 $\pm$ 77.35 & 10277.65 $\pm$ 103.72 & 10.38 $\pm$ 0.04 & 9.44 $\pm$ 0.03 & 7.50 $\pm$ 0.02 & 6.37 $\pm$ 0.01\
G011.9446$-$00.0369 & 820.14 $\pm$ 101.43 & 1048.06 $\pm$ 82.95 & 4647.68 $\pm$ 513.01 & 13816.60 $\pm$ 1300.61 & 9.12 $\pm$ 0.13 & 8.85 $\pm$ 0.09 & 7.23 $\pm$ 0.12 & 6.05 $\pm$ 0.10\
G012.1988$-$00.0345 & 29.57 $\pm$ 2.53 & 177.31 $\pm$ 2.66 & 512.85 $\pm$ 7.35 & 811.23 $\pm$ 12.64 & 12.72 $\pm$ 0.09 & 10.78 $\pm$ 0.02 & 9.63 $\pm$ 0.02 & 9.13 $\pm$ 0.02\
G012.2081$-$00.1019 & 60.29 $\pm$ 8.03 & 53.71 $\pm$ 7.44 & 218.63 $\pm$ 46.01 & 623.32 $\pm$ 129.21 & 11.95 $\pm$ 0.14 & 12.07 $\pm$ 0.15 & 10.55 $\pm$ 0.23 & 9.41 $\pm$ 0.22\
G012.4294$-$00.0479 & & 15.77 $\pm$ 5.48 & 149.58 $\pm$ 59.20 & 498.99 $\pm$ 180.38 & & 13.41 $\pm$ 0.37 & 10.96 $\pm$ 0.43 & 9.65 $\pm$ 0.39\
G012.4317$-$01.1112 & 122.36 $\pm$ 8.64 & 235.54 $\pm$ 8.96 & 1969.68 $\pm$ 83.09 & 5774.60 $\pm$ 264.67 & 11.18 $\pm$ 0.08 & 10.47 $\pm$ 0.04 & 8.16 $\pm$ 0.05 & 7.00 $\pm$ 0.05\
G012.8050$-$00.2007 & 914.06 $\pm$ 90.47 & 2585.81 $\pm$ 124.41 & 13458.14 $\pm$ 1259.96 & 40822.39 $\pm$ 3200.87 & 9.00 $\pm$ 0.11 & 7.87 $\pm$ 0.05 & 6.08 $\pm$ 0.10 & 4.87 $\pm$ 0.08\
G012.8131$-$00.1976 & 101.42 $\pm$ 14.56 & 380.61 $\pm$ 21.59 & 2485.64 $\pm$ 279.36 & 7721.11 $\pm$ 700.72 & 11.38 $\pm$ 0.16 & 9.95 $\pm$ 0.06 & 7.91 $\pm$ 0.12 & 6.68 $\pm$ 0.10\
G012.9995$-$00.3583 & 16.35 $\pm$ 0.98 & 60.61 $\pm$ 1.36 & 320.47 $\pm$ 5.36 & 802.16 $\pm$ 9.71 & 13.37 $\pm$ 0.06 & 11.94 $\pm$ 0.02 & 10.14 $\pm$ 0.02 & 9.14 $\pm$ 0.01\
G013.2099$-$00.1428 & 44.47 $\pm$ 18.40 & 193.80 $\pm$ 25.50 & 1385.79 $\pm$ 201.71 & 3876.50 $\pm$ 669.53 & 12.28 $\pm$ 0.44 & 10.68 $\pm$ 0.14 & 8.55 $\pm$ 0.16 & 7.43 $\pm$ 0.19\
G013.3850+00.0684 & 410.91 $\pm$ 42.92 & 521.85 $\pm$ 41.26 & 3639.65 $\pm$ 245.02 & 9936.61 $\pm$ 498.90 & 9.87 $\pm$ 0.11 & 9.61 $\pm$ 0.09 & 7.50 $\pm$ 0.07 & 6.41 $\pm$ 0.05\
G013.8726+00.2818 & & 900.18 $\pm$ 419.46 & 6252.28 $\pm$ 2905.30 & 21687.65 $\pm$ 9209.99 & & 9.01 $\pm$ 0.50 & 6.91 $\pm$ 0.50 & 5.56 $\pm$ 0.46\
G014.1741+00.0245 & 39.69 $\pm$ 11.61 & 54.71 $\pm$ 9.87 & 436.16 $\pm$ 59.02 & 1309.20 $\pm$ 160.70 & 12.40 $\pm$ 0.32 & 12.05 $\pm$ 0.19 & 9.80 $\pm$ 0.15 & 8.61 $\pm$ 0.13\
G014.2460$-$00.0728 & & 15.58 $\pm$ 5.25 & 134.93 $\pm$ 33.36 & 429.36 $\pm$ 80.03 & & 13.42 $\pm$ 0.36 & 11.07 $\pm$ 0.27 & 9.82 $\pm$ 0.20\
G014.5988+00.0198 & 17.98 $\pm$ 1.94 & 65.09 $\pm$ 2.38 & 186.92 $\pm$ 8.35 & 338.53 $\pm$ 29.88 & 13.26 $\pm$ 0.12 & 11.87 $\pm$ 0.04 & 10.72 $\pm$ 0.05 & 10.08 $\pm$ 0.10\
G014.7785$-$00.3328 & 8.86 $\pm$ 1.07 & 24.07 $\pm$ 1.04 & 96.88 $\pm$ 4.95 & 291.01 $\pm$ 12.93 & 14.03 $\pm$ 0.13 & 12.95 $\pm$ 0.05 & 11.43 $\pm$ 0.06 & 10.24 $\pm$ 0.05\
G016.1448+00.0088 & 16.55 $\pm$ 2.29 & 29.86 $\pm$ 1.99 & 235.40 $\pm$ 7.96 & 635.54 $\pm$ 14.93 & 13.35 $\pm$ 0.15 & 12.71 $\pm$ 0.07 & 10.47 $\pm$ 0.04 & 9.39 $\pm$ 0.03\
G016.3913$-$00.1383 & 78.31 $\pm$ 13.08 & 83.00 $\pm$ 11.12 & 384.31 $\pm$ 53.40 & 947.84 $\pm$ 97.22 & 11.67 $\pm$ 0.18 & 11.60 $\pm$ 0.14 & 9.94 $\pm$ 0.15 & 8.96 $\pm$ 0.11\
G016.9445$-$00.0738 & 91.62 $\pm$ 5.62 & 188.82 $\pm$ 6.22 & 1034.45 $\pm$ 34.01 & 2645.22 $\pm$ 91.93 & 11.50 $\pm$ 0.07 & 10.71 $\pm$ 0.04 & 8.86 $\pm$ 0.04 & 7.84 $\pm$ 0.04\
G017.0299$-$00.0696 & 28.70 $\pm$ 2.85 & 38.88 $\pm$ 2.66 & 97.05 $\pm$ 20.13 & 182.44 $\pm$ 71.63 & 12.76 $\pm$ 0.11 & 12.43 $\pm$ 0.07 & 11.43 $\pm$ 0.22 & 10.75 $\pm$ 0.42\
G017.1141$-$00.1124 & 63.88 $\pm$ 4.15 & 80.79 $\pm$ 3.55 & 637.16 $\pm$ 14.37 & 1692.88 $\pm$ 26.65 & 11.89 $\pm$ 0.07 & 11.63 $\pm$ 0.05 & 9.39 $\pm$ 0.02 & 8.33 $\pm$ 0.02\
G017.5549+00.1654 & 21.75 $\pm$ 2.53 & 26.90 $\pm$ 2.59 & 182.66 $\pm$ 35.80 & 485.29 $\pm$ 139.22 & 13.06 $\pm$ 0.13 & 12.83 $\pm$ 0.10 & 10.75 $\pm$ 0.21 & 9.68 $\pm$ 0.31\
G017.9850+00.1266 & 104.15 $\pm$ 4.12 & 81.18 $\pm$ 3.40 & 390.02 $\pm$ 11.80 & 999.52 $\pm$ 12.53 & 11.36 $\pm$ 0.04 & 11.63 $\pm$ 0.05 & 9.92 $\pm$ 0.03 & 8.90 $\pm$ 0.01\
G018.1460$-$00.2839 & 1395.50 $\pm$ 429.02 & 1693.76 $\pm$ 402.28 & 7930.30 $\pm$ 2666.90 & 24389.19 $\pm$ 7411.47 & 8.54 $\pm$ 0.33 & 8.33 $\pm$ 0.26 & 6.65 $\pm$ 0.36 & 5.43 $\pm$ 0.33\
G018.3024$-$00.3910 & 1898.17 $\pm$ 57.13 & 2404.38 $\pm$ 59.05 & 14926.67 $\pm$ 460.53 & 39597.30 $\pm$ 856.07 & 8.20 $\pm$ 0.03 & 7.95 $\pm$ 0.03 & 5.97 $\pm$ 0.03 & 4.91 $\pm$ 0.02\
G018.4433$-$00.0056 & 9.92 $\pm$ 2.22 & 19.13 $\pm$ 2.68 & 126.12 $\pm$ 11.92 & 393.09 $\pm$ 27.58 & 13.91 $\pm$ 0.24 & 13.20 $\pm$ 0.15 & 11.15 $\pm$ 0.10 & 9.91 $\pm$ 0.08\
G018.4614$-$00.0038 & 31.09 $\pm$ 3.54 & 111.95 $\pm$ 3.01 & 358.18 $\pm$ 12.63 & 958.09 $\pm$ 23.80 & 12.67 $\pm$ 0.12 & 11.28 $\pm$ 0.03 & 10.01 $\pm$ 0.04 & 8.95 $\pm$ 0.03\
G018.6654+00.0294 & 7.52 $\pm$ 2.38 & 16.02 $\pm$ 2.92 & 89.40 $\pm$ 7.87 & 232.82 $\pm$ 20.87 & 14.21 $\pm$ 0.34 & 13.39 $\pm$ 0.20 & 11.52 $\pm$ 0.10 & 10.48 $\pm$ 0.10\
G018.7106+00.0002 & 49.17 $\pm$ 1.95 & 74.35 $\pm$ 1.91 & 285.55 $\pm$ 7.83 & 585.66 $\pm$ 19.86 & 12.17 $\pm$ 0.04 & 11.72 $\pm$ 0.03 & 10.26 $\pm$ 0.03 & 9.48 $\pm$ 0.04\
G018.7612+00.2630 & 28.92 $\pm$ 6.59 & 45.43 $\pm$ 5.73 & 343.32 $\pm$ 27.21 & 903.15 $\pm$ 85.56 & 12.75 $\pm$ 0.25 & 12.26 $\pm$ 0.14 & 10.06 $\pm$ 0.09 & 9.01 $\pm$ 0.10\
G018.8250$-$00.4675 & 53.07 $\pm$ 4.44 & 69.51 $\pm$ 2.92 & 601.06 $\pm$ 30.77 & 1433.97 $\pm$ 100.38 & 12.09 $\pm$ 0.09 & 11.79 $\pm$ 0.05 & 9.45 $\pm$ 0.06 & 8.51 $\pm$ 0.08\
G019.0035+00.1280 & 13.53 $\pm$ 2.68 & 22.32 $\pm$ 3.12 & 106.10 $\pm$ 12.84 & 280.81 $\pm$ 17.05 & 13.57 $\pm$ 0.21 & 13.03 $\pm$ 0.15 & 11.34 $\pm$ 0.13 & 10.28 $\pm$ 0.07\
G019.0754$-$00.2874 & 251.49 $\pm$ 74.31 & 375.62 $\pm$ 77.71 & 2487.01 $\pm$ 690.27 & 6920.80 $\pm$ 2186.79 & 10.40 $\pm$ 0.32 & 9.96 $\pm$ 0.22 & 7.91 $\pm$ 0.30 & 6.80 $\pm$ 0.34\
G019.0767$-$00.2882 & 68.24 $\pm$ 6.47 & 100.86 $\pm$ 7.81 & 946.54 $\pm$ 90.19 & 3167.07 $\pm$ 296.65 & 11.81 $\pm$ 0.10 & 11.39 $\pm$ 0.08 & 8.96 $\pm$ 0.10 & 7.65 $\pm$ 0.10\
G019.4752+00.1728 & 12.48 $\pm$ 3.03 & 27.45 $\pm$ 4.42 & 175.62 $\pm$ 27.95 & 483.39 $\pm$ 37.65 & 13.66 $\pm$ 0.26 & 12.80 $\pm$ 0.17 & 10.79 $\pm$ 0.17 & 9.69 $\pm$ 0.08\
G019.6087$-$00.2351 & 760.05 $\pm$ 51.30 & 1310.98 $\pm$ 48.99 & 6499.63 $\pm$ 311.51 & 17921.26 $\pm$ 869.50 & 9.20 $\pm$ 0.07 & 8.61 $\pm$ 0.04 & 6.87 $\pm$ 0.05 & 5.77 $\pm$ 0.05\
G019.6090$-$00.2313 & 56.47 $\pm$ 15.69 & 151.16 $\pm$ 15.49 & 652.04 $\pm$ 159.76 & 2260.90 $\pm$ 397.04 & 12.02 $\pm$ 0.30 & 10.95 $\pm$ 0.11 & 9.36 $\pm$ 0.26 & 8.01 $\pm$ 0.19\
G019.7281$-$00.1135 & 34.63 $\pm$ 2.95 & 61.13 $\pm$ 2.29 & 546.84 $\pm$ 11.05 & 1412.71 $\pm$ 34.49 & 12.55 $\pm$ 0.09 & 11.93 $\pm$ 0.04 & 9.56 $\pm$ 0.02 & 8.52 $\pm$ 0.03\
G020.0720$-$00.1421 & 115.94 $\pm$ 11.86 & 203.74 $\pm$ 11.73 & 846.74 $\pm$ 48.86 & 2385.06 $\pm$ 95.67 & 11.24 $\pm$ 0.11 & 10.63 $\pm$ 0.06 & 9.08 $\pm$ 0.06 & 7.96 $\pm$ 0.04\
G020.0797$-$00.1337 & & 10.10 $\pm$ 2.36 & 83.23 $\pm$ 9.37 & 330.94 $\pm$ 12.02 & & 13.89 $\pm$ 0.25 & 11.60 $\pm$ 0.12 & 10.10 $\pm$ 0.04\
G020.0809$-$00.1362 & 22.21 $\pm$ 8.21 & 86.98 $\pm$ 9.52 & 281.08 $\pm$ 47.21 & 742.47 $\pm$ 22.29 & 13.03 $\pm$ 0.40 & 11.55 $\pm$ 0.12 & 10.28 $\pm$ 0.18 & 9.22 $\pm$ 0.03\
G020.3633$-$00.0136 & 4.54 $\pm$ 2.59 & 17.40 $\pm$ 2.32 & 86.49 $\pm$ 5.55 & 251.89 $\pm$ 8.53 & 14.76 $\pm$ 0.62 & 13.30 $\pm$ 0.14 & 11.56 $\pm$ 0.07 & 10.40 $\pm$ 0.04\
G020.9636$-$00.0744 & 10.17 $\pm$ 2.93 & 20.53 $\pm$ 2.78 & 140.53 $\pm$ 10.63 & 390.77 $\pm$ 32.59 & 13.88 $\pm$ 0.31 & 13.12 $\pm$ 0.15 & 11.03 $\pm$ 0.08 & 9.92 $\pm$ 0.09\
G021.3855$-$00.2541 & 53.14 $\pm$ 2.75 & 108.23 $\pm$ 2.37 & 504.32 $\pm$ 12.87 & 1600.15 $\pm$ 17.22 & 12.09 $\pm$ 0.06 & 11.31 $\pm$ 0.02 & 9.64 $\pm$ 0.03 & 8.39 $\pm$ 0.01\
G021.6034$-$00.1685 & 27.22 $\pm$ 3.30 & 34.99 $\pm$ 2.71 & 282.22 $\pm$ 9.96 & 754.83 $\pm$ 19.23 & 12.81 $\pm$ 0.13 & 12.54 $\pm$ 0.08 & 10.27 $\pm$ 0.04 & 9.21 $\pm$ 0.03\
G021.8751+00.0075 & 211.88 $\pm$ 43.15 & 314.86 $\pm$ 47.90 & 2194.29 $\pm$ 227.29 & 6067.15 $\pm$ 537.18 & 10.58 $\pm$ 0.22 & 10.15 $\pm$ 0.16 & 8.05 $\pm$ 0.11 & 6.94 $\pm$ 0.10\
G023.1974$-$00.0006 & 70.96 $\pm$ 3.14 & 90.61 $\pm$ 2.80 & 695.26 $\pm$ 18.21 & 1881.53 $\pm$ 59.25 & 11.77 $\pm$ 0.05 & 11.51 $\pm$ 0.03 & 9.29 $\pm$ 0.03 & 8.21 $\pm$ 0.03\
G023.2654+00.0765 & 68.81 $\pm$ 5.87 & 117.37 $\pm$ 5.71 & 853.94 $\pm$ 48.71 & 2236.12 $\pm$ 95.03 & 11.81 $\pm$ 0.09 & 11.23 $\pm$ 0.05 & 9.07 $\pm$ 0.06 & 8.03 $\pm$ 0.05\
G023.4835+00.0964 & 42.91 $\pm$ 3.19 & 55.98 $\pm$ 3.67 & 178.26 $\pm$ 27.42 & 298.54 $\pm$ 64.71 & 12.32 $\pm$ 0.08 & 12.03 $\pm$ 0.07 & 10.77 $\pm$ 0.17 & 10.21 $\pm$ 0.23\
G023.7110+00.1705 & 227.87 $\pm$ 13.88 & 324.14 $\pm$ 17.80 & 1738.13 $\pm$ 150.25 & 5138.15 $\pm$ 396.72 & 10.51 $\pm$ 0.07 & 10.12 $\pm$ 0.06 & 8.30 $\pm$ 0.09 & 7.12 $\pm$ 0.08\
G023.8618$-$00.1250 & 37.45 $\pm$ 5.17 & 56.44 $\pm$ 5.20 & 417.85 $\pm$ 55.07 & 976.60 $\pm$ 175.64 & 12.47 $\pm$ 0.15 & 12.02 $\pm$ 0.10 & 9.85 $\pm$ 0.14 & 8.93 $\pm$ 0.19\
G023.8985+00.0647 & 54.81 $\pm$ 9.43 & 89.85 $\pm$ 10.68 & 607.22 $\pm$ 67.15 & 1417.84 $\pm$ 154.64 & 12.05 $\pm$ 0.19 & 11.52 $\pm$ 0.13 & 9.44 $\pm$ 0.12 & 8.52 $\pm$ 0.12\
G023.9564+00.1493 & 1775.59 $\pm$ 57.21 & 2305.34 $\pm$ 61.92 & 13569.53 $\pm$ 491.26 & 37519.12 $\pm$ 1406.80 & 8.28 $\pm$ 0.03 & 7.99 $\pm$ 0.03 & 6.07 $\pm$ 0.04 & 4.96 $\pm$ 0.04\
G024.1839+00.1199 & 30.76 $\pm$ 1.32 & 61.75 $\pm$ 1.22 & 268.60 $\pm$ 6.60 & 667.40 $\pm$ 16.03 & 12.68 $\pm$ 0.05 & 11.92 $\pm$ 0.02 & 10.33 $\pm$ 0.03 & 9.34 $\pm$ 0.03\
G024.4698+00.4954 & & & 524.66 $\pm$ 297.18 & & & & 9.60 $\pm$ 0.61 &\
G024.4721+00.4877 & 150.30 $\pm$ 28.44 & 188.65 $\pm$ 26.00 & 941.01 $\pm$ 273.80 & 4025.91 $\pm$ 1574.29 & 10.96 $\pm$ 0.20 & 10.71 $\pm$ 0.15 & 8.97 $\pm$ 0.31 & 7.39 $\pm$ 0.42\
G024.4736+00.4950 & 106.84 $\pm$ 45.20 & 139.24 $\pm$ 55.13 & 961.10 $\pm$ 548.22 & 3734.72 $\pm$ 2386.14 & 11.33 $\pm$ 0.46 & 11.04 $\pm$ 0.43 & 8.94 $\pm$ 0.62 & 7.47 $\pm$ 0.69\
G024.5065$-$00.2224 & 290.25 $\pm$ 20.68 & 405.36 $\pm$ 22.98 & 2601.48 $\pm$ 241.89 & 7109.98 $\pm$ 767.93 & 10.24 $\pm$ 0.08 & 9.88 $\pm$ 0.06 & 7.86 $\pm$ 0.10 & 6.77 $\pm$ 0.12\
G024.8497+00.0881 & 77.97 $\pm$ 8.87 & 110.89 $\pm$ 14.29 & 541.36 $\pm$ 58.58 & 1424.67 $\pm$ 240.40 & 11.67 $\pm$ 0.12 & 11.29 $\pm$ 0.14 & 9.57 $\pm$ 0.12 & 8.52 $\pm$ 0.18\
G025.3948+00.0332 & 72.83 $\pm$ 6.46 & 260.66 $\pm$ 7.16 & 1129.20 $\pm$ 37.97 & 2389.80 $\pm$ 106.08 & 11.74 $\pm$ 0.10 & 10.36 $\pm$ 0.03 & 8.77 $\pm$ 0.04 & 7.95 $\pm$ 0.05\
G025.3970+00.5614 & 24.32 $\pm$ 1.55 & 93.97 $\pm$ 1.42 & 381.40 $\pm$ 10.54 & 913.01 $\pm$ 35.13 & 12.94 $\pm$ 0.07 & 11.47 $\pm$ 0.02 & 9.95 $\pm$ 0.03 & 9.00 $\pm$ 0.04\
G025.3981$-$00.1411 & 389.84 $\pm$ 110.58 & 1001.96 $\pm$ 120.32 & 5658.69 $\pm$ 1120.13 & 16138.67 $\pm$ 3107.68 & 9.92 $\pm$ 0.31 & 8.90 $\pm$ 0.13 & 7.02 $\pm$ 0.21 & 5.88 $\pm$ 0.21\
G025.3983+00.5617 & 40.72 $\pm$ 1.85 & 105.87 $\pm$ 1.81 & 506.75 $\pm$ 11.99 & 1177.70 $\pm$ 34.80 & 12.38 $\pm$ 0.05 & 11.34 $\pm$ 0.02 & 9.64 $\pm$ 0.03 & 8.72 $\pm$ 0.03\
G025.3991$-$00.1366 & 49.22 $\pm$ 15.25 & 94.48 $\pm$ 22.34 & 811.57 $\pm$ 308.98 & 2525.76 $\pm$ 1045.67 & 12.17 $\pm$ 0.33 & 11.46 $\pm$ 0.26 & 9.13 $\pm$ 0.41 & 7.89 $\pm$ 0.45\
G025.7157+00.0487 & 119.78 $\pm$ 2.88 & 165.50 $\pm$ 3.60 & 818.64 $\pm$ 18.74 & 2358.09 $\pm$ 56.92 & 11.20 $\pm$ 0.03 & 10.85 $\pm$ 0.02 & 9.12 $\pm$ 0.02 & 7.97 $\pm$ 0.03\
G026.0916$-$00.0565 & 23.51 $\pm$ 3.67 & 32.99 $\pm$ 3.56 & 277.79 $\pm$ 20.92 & 722.48 $\pm$ 55.99 & 12.97 $\pm$ 0.17 & 12.60 $\pm$ 0.12 & 10.29 $\pm$ 0.08 & 9.25 $\pm$ 0.08\
G026.1094$-$00.0937 & 26.39 $\pm$ 2.19 & 47.37 $\pm$ 2.12 & 150.53 $\pm$ 12.19 & 342.55 $\pm$ 34.26 & 12.85 $\pm$ 0.09 & 12.21 $\pm$ 0.05 & 10.96 $\pm$ 0.09 & 10.06 $\pm$ 0.11\
G026.5976$-$00.0236 & 59.31 $\pm$ 3.18 & 232.09 $\pm$ 3.27 & 849.86 $\pm$ 26.07 & 1416.62 $\pm$ 35.68 & 11.97 $\pm$ 0.06 & 10.49 $\pm$ 0.02 & 9.08 $\pm$ 0.03 & 8.52 $\pm$ 0.03\
G026.6089$-$00.2121 & 130.91 $\pm$ 9.34 & 147.16 $\pm$ 7.69 & 611.47 $\pm$ 27.40 & 1534.05 $\pm$ 63.55 & 11.11 $\pm$ 0.08 & 10.98 $\pm$ 0.06 & 9.43 $\pm$ 0.05 & 8.44 $\pm$ 0.04\
G026.8304$-$00.2067 & 18.09 $\pm$ 2.36 & 22.84 $\pm$ 1.98 & 231.67 $\pm$ 8.11 & 635.05 $\pm$ 19.28 & 13.26 $\pm$ 0.14 & 13.00 $\pm$ 0.09 & 10.49 $\pm$ 0.04 & 9.39 $\pm$ 0.03\
G027.1859$-$00.0816 & 30.24 $\pm$ 6.31 & 66.50 $\pm$ 5.97 & 339.28 $\pm$ 50.65 & 921.01 $\pm$ 105.02 & 12.70 $\pm$ 0.23 & 11.84 $\pm$ 0.10 & 10.07 $\pm$ 0.16 & 8.99 $\pm$ 0.12\
G027.2800+00.1447 & 66.41 $\pm$ 15.56 & 118.31 $\pm$ 14.43 & 711.42 $\pm$ 123.84 & 2160.30 $\pm$ 356.56 & 11.84 $\pm$ 0.25 & 11.22 $\pm$ 0.13 & 9.27 $\pm$ 0.19 & 8.06 $\pm$ 0.18\
G027.3644$-$00.1657 & 31.39 $\pm$ 6.31 & 60.12 $\pm$ 5.52 & 409.05 $\pm$ 66.10 & 1311.50 $\pm$ 216.18 & 12.66 $\pm$ 0.22 & 11.95 $\pm$ 0.10 & 9.87 $\pm$ 0.17 & 8.61 $\pm$ 0.18\
G027.5637+00.0845 & 113.44 $\pm$ 17.22 & 121.82 $\pm$ 14.81 & 620.61 $\pm$ 73.59 & 1633.34 $\pm$ 212.29 & 11.26 $\pm$ 0.16 & 11.19 $\pm$ 0.13 & 9.42 $\pm$ 0.13 & 8.37 $\pm$ 0.14\
G028.4518+00.0027 & 7.35 $\pm$ 1.67 & 14.40 $\pm$ 1.59 & 157.58 $\pm$ 11.77 & 426.25 $\pm$ 39.14 & 14.23 $\pm$ 0.25 & 13.50 $\pm$ 0.12 & 10.91 $\pm$ 0.08 & 9.83 $\pm$ 0.10\
G028.5816+00.1447 & 22.57 $\pm$ 3.24 & 35.81 $\pm$ 3.48 & 272.12 $\pm$ 15.05 & 688.78 $\pm$ 26.33 & 13.02 $\pm$ 0.16 & 12.51 $\pm$ 0.10 & 10.31 $\pm$ 0.06 & 9.30 $\pm$ 0.04\
G028.6082+00.0185 & 108.13 $\pm$ 7.08 & 194.96 $\pm$ 7.79 & 1265.97 $\pm$ 78.57 & 3237.33 $\pm$ 218.20 & 11.32 $\pm$ 0.07 & 10.68 $\pm$ 0.04 & 8.64 $\pm$ 0.07 & 7.62 $\pm$ 0.07\
G028.6523+00.0273 & 51.05 $\pm$ 4.75 & 113.25 $\pm$ 5.15 & 649.34 $\pm$ 28.60 & 1720.71 $\pm$ 73.23 & 12.13 $\pm$ 0.10 & 11.26 $\pm$ 0.05 & 9.37 $\pm$ 0.05 & 8.31 $\pm$ 0.05\
G028.6869+00.1770 & 110.74 $\pm$ 6.41 & 147.02 $\pm$ 5.85 & 1280.73 $\pm$ 64.02 & 3364.81 $\pm$ 216.00 & 11.29 $\pm$ 0.06 & 10.98 $\pm$ 0.04 & 8.63 $\pm$ 0.05 & 7.58 $\pm$ 0.07\
G029.7704+00.2189 & 26.98 $\pm$ 4.21 & 37.95 $\pm$ 3.52 & 180.00 $\pm$ 16.38 & 487.54 $\pm$ 32.77 & 12.82 $\pm$ 0.17 & 12.45 $\pm$ 0.10 & 10.76 $\pm$ 0.10 & 9.68 $\pm$ 0.07\
G030.0096$-$00.2734 & & 8.83 $\pm$ 1.04 & 23.89 $\pm$ 4.75 & 40.24 $\pm$ 11.39 & & 14.03 $\pm$ 0.13 & 12.95 $\pm$ 0.21 & 12.39 $\pm$ 0.31\
G030.2527+00.0540 & 42.90 $\pm$ 6.66 & 74.83 $\pm$ 6.14 & 647.62 $\pm$ 26.15 & 1848.75 $\pm$ 65.95 & 12.32 $\pm$ 0.17 & 11.71 $\pm$ 0.09 & 9.37 $\pm$ 0.04 & 8.23 $\pm$ 0.04\
G030.5313+00.0205 & 221.93 $\pm$ 5.69 & 209.73 $\pm$ 6.48 & 1066.10 $\pm$ 82.44 & 3049.80 $\pm$ 220.26 & 10.53 $\pm$ 0.03 & 10.60 $\pm$ 0.03 & 8.83 $\pm$ 0.08 & 7.69 $\pm$ 0.08\
G030.5353+00.0204 & 280.41 $\pm$ 13.63 & 403.72 $\pm$ 13.23 & 2221.58 $\pm$ 119.77 & 5625.50 $\pm$ 322.46 & 10.28 $\pm$ 0.05 & 9.88 $\pm$ 0.04 & 8.03 $\pm$ 0.06 & 7.02 $\pm$ 0.06\
G030.5887$-$00.0428 & 101.76 $\pm$ 1.28 & 409.13 $\pm$ 1.52 & 1366.37 $\pm$ 11.89 & 1795.32 $\pm$ 30.59 & 11.38 $\pm$ 0.01 & 9.87 $\pm$ 0.00 & 8.56 $\pm$ 0.01 & 8.26 $\pm$ 0.02\
G030.6881$-$00.0718 & 193.14 $\pm$ 16.70 & 360.17 $\pm$ 19.81 & 1842.89 $\pm$ 156.50 & 5928.66 $\pm$ 525.02 & 10.69 $\pm$ 0.09 & 10.01 $\pm$ 0.06 & 8.24 $\pm$ 0.09 & 6.97 $\pm$ 0.10\
G030.7197$-$00.0829 & 103.00 $\pm$ 11.38 & 102.44 $\pm$ 9.37 & 422.69 $\pm$ 86.84 & 1044.04 $\pm$ 112.78 & 11.37 $\pm$ 0.12 & 11.37 $\pm$ 0.10 & 9.83 $\pm$ 0.22 & 8.85 $\pm$ 0.12\
G030.7532$-$00.0511 & & & 894.17 $\pm$ 375.99 & & & & 9.02 $\pm$ 0.45 &\
G030.7579+00.2042 & 21.78 $\pm$ 3.47 & 53.65 $\pm$ 3.73 & 494.34 $\pm$ 25.26 & 1557.61 $\pm$ 73.18 & 13.05 $\pm$ 0.17 & 12.08 $\pm$ 0.08 & 9.66 $\pm$ 0.06 & 8.42 $\pm$ 0.05\
G030.7661$-$00.0348 & 821.01 $\pm$ 25.07 & 1030.24 $\pm$ 57.64 & 2357.18 $\pm$ 415.24 & 5130.34 $\pm$ 1666.61 & 9.11 $\pm$ 0.03 & 8.87 $\pm$ 0.06 & 7.97 $\pm$ 0.19 & 7.12 $\pm$ 0.35\
G030.9581+00.0869 & 94.82 $\pm$ 2.69 & 338.15 $\pm$ 3.75 & 1289.53 $\pm$ 20.84 & 2303.79 $\pm$ 13.07 & 11.46 $\pm$ 0.03 & 10.08 $\pm$ 0.01 & 8.62 $\pm$ 0.02 & 7.99 $\pm$ 0.01\
G031.0495+00.4697 & 15.35 $\pm$ 1.05 & 22.91 $\pm$ 1.33 & 179.98 $\pm$ 7.20 & 452.47 $\pm$ 15.55 & 13.43 $\pm$ 0.07 & 13.00 $\pm$ 0.06 & 10.76 $\pm$ 0.04 & 9.76 $\pm$ 0.04\
G031.0595+00.0922 & 13.25 $\pm$ 2.00 & 21.25 $\pm$ 2.38 & 237.44 $\pm$ 8.74 & 652.68 $\pm$ 28.83 & 13.59 $\pm$ 0.16 & 13.08 $\pm$ 0.12 & 10.46 $\pm$ 0.04 & 9.36 $\pm$ 0.05\
G031.1590+00.0465 & 12.23 $\pm$ 1.64 & 26.60 $\pm$ 1.62 & 163.52 $\pm$ 9.23 & 473.97 $\pm$ 22.71 & 13.68 $\pm$ 0.14 & 12.84 $\pm$ 0.07 & 10.87 $\pm$ 0.06 & 9.71 $\pm$ 0.05\
G031.1596+00.0448 & 11.28 $\pm$ 1.54 & 24.03 $\pm$ 1.64 & 141.49 $\pm$ 9.89 & 386.67 $\pm$ 29.22 & 13.77 $\pm$ 0.15 & 12.95 $\pm$ 0.07 & 11.02 $\pm$ 0.08 & 9.93 $\pm$ 0.08\
G031.2420$-$00.1106 & 102.54 $\pm$ 9.58 & 149.96 $\pm$ 7.36 & 894.50 $\pm$ 47.67 & 2332.76 $\pm$ 116.65 & 11.37 $\pm$ 0.10 & 10.96 $\pm$ 0.05 & 9.02 $\pm$ 0.06 & 7.98 $\pm$ 0.05\
G031.2435$-$00.1103 & 297.10 $\pm$ 3.89 & 212.20 $\pm$ 3.94 & 513.08 $\pm$ 30.06 & 1265.34 $\pm$ 99.72 & 10.22 $\pm$ 0.01 & 10.58 $\pm$ 0.02 & 9.62 $\pm$ 0.06 & 8.64 $\pm$ 0.09\
G031.2448$-$00.1132 & 30.57 $\pm$ 3.65 & 44.75 $\pm$ 3.57 & 305.07 $\pm$ 25.61 & 847.95 $\pm$ 73.93 & 12.69 $\pm$ 0.13 & 12.27 $\pm$ 0.09 & 10.19 $\pm$ 0.09 & 9.08 $\pm$ 0.09\
G031.3959$-$00.2570 & 255.27 $\pm$ 33.12 & 399.77 $\pm$ 28.76 & 1789.22 $\pm$ 277.93 & 4898.04 $\pm$ 845.34 & 10.38 $\pm$ 0.14 & 9.90 $\pm$ 0.08 & 8.27 $\pm$ 0.17 & 7.17 $\pm$ 0.19\
G031.4130+00.3065 & 40.42 $\pm$ 11.84 & 121.78 $\pm$ 13.66 & 1058.07 $\pm$ 59.31 & 3157.47 $\pm$ 103.40 & 12.38 $\pm$ 0.32 & 11.19 $\pm$ 0.12 & 8.84 $\pm$ 0.06 & 7.65 $\pm$ 0.04\
G031.5815+00.0744 & 12.10 $\pm$ 2.51 & 38.12 $\pm$ 3.11 & 280.50 $\pm$ 12.78 & 781.33 $\pm$ 32.52 & 13.69 $\pm$ 0.22 & 12.45 $\pm$ 0.09 & 10.28 $\pm$ 0.05 & 9.17 $\pm$ 0.04\
G032.0297+00.0491 & 195.53 $\pm$ 6.02 & 196.68 $\pm$ 5.18 & 1479.13 $\pm$ 25.79 & 3958.75 $\pm$ 77.25 & 10.67 $\pm$ 0.03 & 10.67 $\pm$ 0.03 & 8.47 $\pm$ 0.02 & 7.41 $\pm$ 0.02\
G032.1502+00.1329 & 714.72 $\pm$ 85.40 & 718.35 $\pm$ 69.54 & 4660.34 $\pm$ 631.16 & 13715.66 $\pm$ 1894.70 & 9.26 $\pm$ 0.13 & 9.26 $\pm$ 0.10 & 7.23 $\pm$ 0.15 & 6.06 $\pm$ 0.15\
G032.2730$-$00.2258 & 234.33 $\pm$ 21.21 & 227.17 $\pm$ 17.66 & 1293.33 $\pm$ 78.37 & 3634.94 $\pm$ 260.50 & 10.48 $\pm$ 0.10 & 10.51 $\pm$ 0.08 & 8.62 $\pm$ 0.07 & 7.50 $\pm$ 0.08\
G032.4727+00.2036 & 205.35 $\pm$ 1.91 & 346.68 $\pm$ 1.96 & 784.93 $\pm$ 9.67 & 1495.90 $\pm$ 28.72 & 10.62 $\pm$ 0.01 & 10.05 $\pm$ 0.01 & 9.16 $\pm$ 0.01 & 8.46 $\pm$ 0.02\
G032.7398+00.1940 & 9.64 $\pm$ 0.98 & 17.09 $\pm$ 0.92 & 126.21 $\pm$ 5.59 & 335.04 $\pm$ 15.87 & 13.94 $\pm$ 0.11 & 13.32 $\pm$ 0.06 & 11.15 $\pm$ 0.05 & 10.09 $\pm$ 0.05\
G032.7492$-$00.0643$^\dagger$ & 288.71 $\pm$ 1.33 & 537.69 $\pm$ 1.63 & 1380.26 $\pm$ 7.30 & 1616.34 $\pm$ 8.32 & 10.25 $\pm$ 0.00 & 9.57 $\pm$ 0.00 & 8.55 $\pm$ 0.01 & 8.38 $\pm$ 0.01\
G032.9273+00.6060 & 146.00 $\pm$ 11.41 & 182.68 $\pm$ 8.58 & 936.85 $\pm$ 48.02 & 2626.01 $\pm$ 125.84 & 10.99 $\pm$ 0.08 & 10.75 $\pm$ 0.05 & 8.97 $\pm$ 0.06 & 7.85 $\pm$ 0.05\
G033.1328$-$00.0923 & 19.10 $\pm$ 4.59 & 79.33 $\pm$ 4.52 & 265.90 $\pm$ 37.60 & 586.27 $\pm$ 128.20 & 13.20 $\pm$ 0.26 & 11.65 $\pm$ 0.06 & 10.34 $\pm$ 0.15 & 9.48 $\pm$ 0.24\
G033.4163$-$00.0036 & 274.35 $\pm$ 9.22 & 259.63 $\pm$ 10.10 & 997.27 $\pm$ 103.59 & 2321.88 $\pm$ 261.59 & 10.30 $\pm$ 0.04 & 10.36 $\pm$ 0.04 & 8.90 $\pm$ 0.11 & 7.99 $\pm$ 0.12\
G033.8100$-$00.1864 & 318.18 $\pm$ 4.74 & 394.98 $\pm$ 7.29 & 3830.11 $\pm$ 54.64 & 6988.93 $\pm$ 76.97 & 10.14 $\pm$ 0.02 & 9.91 $\pm$ 0.02 & 7.44 $\pm$ 0.02 & 6.79 $\pm$ 0.01\
G033.9145+00.1105 & 711.40 $\pm$ 20.66 & 925.40 $\pm$ 18.30 & 5839.29 $\pm$ 124.65 & 17553.89 $\pm$ 276.96 & 9.27 $\pm$ 0.03 & 8.98 $\pm$ 0.02 & 6.98 $\pm$ 0.02 & 5.79 $\pm$ 0.02\
G034.0901+00.4365 & 37.11 $\pm$ 1.92 & 40.69 $\pm$ 2.60 & 354.66 $\pm$ 21.97 & 1060.27 $\pm$ 72.87 & 12.48 $\pm$ 0.06 & 12.38 $\pm$ 0.07 & 10.03 $\pm$ 0.07 & 8.84 $\pm$ 0.07\
G034.1978$-$00.5912 & 53.29 $\pm$ 2.72 & 76.23 $\pm$ 2.71 & 717.16 $\pm$ 22.77 & 2058.83 $\pm$ 69.59 & 12.08 $\pm$ 0.06 & 11.69 $\pm$ 0.04 & 9.26 $\pm$ 0.03 & 8.12 $\pm$ 0.04\
G034.4032+00.2277 & 49.67 $\pm$ 0.79 & 130.38 $\pm$ 1.44 & 412.32 $\pm$ 7.94 & 931.48 $\pm$ 19.27 & 12.16 $\pm$ 0.02 & 11.11 $\pm$ 0.01 & 9.86 $\pm$ 0.02 & 8.98 $\pm$ 0.02\
G034.5920+00.2434 & 14.42 $\pm$ 1.54 & 23.93 $\pm$ 1.52 & 182.79 $\pm$ 8.15 & 482.14 $\pm$ 25.29 & 13.50 $\pm$ 0.12 & 12.95 $\pm$ 0.07 & 10.75 $\pm$ 0.05 & 9.69 $\pm$ 0.06\
G035.0242+00.3502 & 17.56 $\pm$ 1.21 & 116.41 $\pm$ 1.77 & 171.48 $\pm$ 10.37 & & 13.29 $\pm$ 0.07 & 11.24 $\pm$ 0.02 & 10.81 $\pm$ 0.07 &\
G035.0524$-$00.5177 & 90.38 $\pm$ 6.22 & 127.87 $\pm$ 7.94 & 785.46 $\pm$ 60.15 & 2342.59 $\pm$ 202.82 & 11.51 $\pm$ 0.07 & 11.13 $\pm$ 0.07 & 9.16 $\pm$ 0.08 & 7.98 $\pm$ 0.09\
G035.4570$-$00.1791 & 1.65 $\pm$ 0.96 & 3.02 $\pm$ 0.90 & 20.94 $\pm$ 3.35 & 56.69 $\pm$ 6.23 & 15.86 $\pm$ 0.63 & 15.20 $\pm$ 0.32 & 13.10 $\pm$ 0.17 & 12.02 $\pm$ 0.12\
G035.4669+00.1394 & 484.27 $\pm$ 3.12 & 668.88 $\pm$ 5.42 & 4448.55 $\pm$ 41.55 & 13779.98 $\pm$ 108.29 & 9.69 $\pm$ 0.01 & 9.34 $\pm$ 0.01 & 7.28 $\pm$ 0.01 & 6.05 $\pm$ 0.01\
G035.5781$-$00.0305 & 63.64 $\pm$ 5.65 & 142.12 $\pm$ 5.79 & 506.18 $\pm$ 74.32 & 1380.06 $\pm$ 238.34 & 11.89 $\pm$ 0.10 & 11.02 $\pm$ 0.04 & 9.64 $\pm$ 0.16 & 8.55 $\pm$ 0.19\
G036.4057+00.0226 & 75.91 $\pm$ 1.86 & 124.39 $\pm$ 1.75 & 561.88 $\pm$ 12.78 & 1345.25 $\pm$ 6.63 & 11.70 $\pm$ 0.03 & 11.16 $\pm$ 0.02 & 9.53 $\pm$ 0.02 & 8.58 $\pm$ 0.01\
G036.4062+00.0221 & 55.10 $\pm$ 1.49 & 110.72 $\pm$ 1.64 & 501.24 $\pm$ 7.75 & 1244.41 $\pm$ 5.90 & 12.05 $\pm$ 0.03 & 11.29 $\pm$ 0.02 & 9.65 $\pm$ 0.02 & 8.66 $\pm$ 0.01\
G037.5457$-$00.1120 & 355.43 $\pm$ 28.74 & 517.81 $\pm$ 25.57 & 2548.83 $\pm$ 542.22 & 6332.31 $\pm$ 1325.82 & 10.02 $\pm$ 0.09 & 9.61 $\pm$ 0.05 & 7.88 $\pm$ 0.23 & 6.90 $\pm$ 0.23\
G037.7347$-$00.1128 & 6.36 $\pm$ 0.96 & 25.45 $\pm$ 1.11 & 79.50 $\pm$ 7.11 & 134.37 $\pm$ 15.19 & 14.39 $\pm$ 0.16 & 12.89 $\pm$ 0.05 & 11.65 $\pm$ 0.10 & 11.08 $\pm$ 0.12\
G037.7562+00.5605 & 43.19 $\pm$ 3.20 & 53.63 $\pm$ 3.39 & 511.81 $\pm$ 24.83 & 1453.70 $\pm$ 74.01 & 12.31 $\pm$ 0.08 & 12.08 $\pm$ 0.07 & 9.63 $\pm$ 0.05 & 8.49 $\pm$ 0.05\
G037.8197+00.4140 & 10.55 $\pm$ 1.60 & 18.52 $\pm$ 2.14 & 148.25 $\pm$ 7.57 & 470.25 $\pm$ 17.94 & 13.84 $\pm$ 0.16 & 13.23 $\pm$ 0.12 & 10.97 $\pm$ 0.06 & 9.72 $\pm$ 0.04\
G037.8209+00.4125 & 8.08 $\pm$ 1.26 & 15.24 $\pm$ 1.56 & 116.91 $\pm$ 5.58 & 346.93 $\pm$ 17.94 & 14.13 $\pm$ 0.17 & 13.44 $\pm$ 0.11 & 11.23 $\pm$ 0.05 & 10.05 $\pm$ 0.06\
G037.8683$-$00.6008 & 141.33 $\pm$ 2.80 & 212.42 $\pm$ 2.49 & 1104.76 $\pm$ 12.86 & 3111.53 $\pm$ 20.62 & 11.02 $\pm$ 0.02 & 10.58 $\pm$ 0.01 & 8.79 $\pm$ 0.01 & 7.67 $\pm$ 0.01\
G037.9723$-$00.0965 & 8.41 $\pm$ 0.96 & 16.18 $\pm$ 0.97 & 117.41 $\pm$ 4.89 & 329.87 $\pm$ 12.67 & 14.09 $\pm$ 0.12 & 13.38 $\pm$ 0.06 & 11.23 $\pm$ 0.04 & 10.10 $\pm$ 0.04\
G038.5493+00.1646 & 95.57 $\pm$ 5.16 & 104.48 $\pm$ 4.53 & 677.19 $\pm$ 27.61 & 2060.49 $\pm$ 37.39 & 11.45 $\pm$ 0.06 & 11.35 $\pm$ 0.05 & 9.32 $\pm$ 0.04 & 8.12 $\pm$ 0.02\
G038.6465$-$00.2260 & 24.02 $\pm$ 2.10 & 38.61 $\pm$ 2.42 & 341.05 $\pm$ 16.30 & 891.27 $\pm$ 44.66 & 12.95 $\pm$ 0.09 & 12.43 $\pm$ 0.07 & 10.07 $\pm$ 0.05 & 9.02 $\pm$ 0.05\
G038.6529+00.0875 & 2.44 $\pm$ 0.52 & 6.81 $\pm$ 0.67 & 56.82 $\pm$ 2.97 & 173.26 $\pm$ 5.79 & 15.43 $\pm$ 0.23 & 14.32 $\pm$ 0.11 & 12.01 $\pm$ 0.06 & 10.80 $\pm$ 0.04\
G038.6934$-$00.4524 & 16.84 $\pm$ 1.13 & 30.05 $\pm$ 1.30 & 234.87 $\pm$ 5.29 & 648.54 $\pm$ 8.02 & 13.33 $\pm$ 0.07 & 12.71 $\pm$ 0.05 & 10.47 $\pm$ 0.02 & 9.37 $\pm$ 0.01\
G038.8756+00.3080 & 48.16 $\pm$ 2.84 & 66.43 $\pm$ 3.90 & 396.82 $\pm$ 15.07 & 1100.65 $\pm$ 30.97 & 12.19 $\pm$ 0.06 & 11.84 $\pm$ 0.06 & 9.90 $\pm$ 0.04 & 8.80 $\pm$ 0.03\
G039.1956+00.2255 & 27.49 $\pm$ 0.96 & 48.95 $\pm$ 1.14 & 220.85 $\pm$ 4.14 & 630.23 $\pm$ 9.07 & 12.80 $\pm$ 0.04 & 12.18 $\pm$ 0.03 & 10.54 $\pm$ 0.02 & 9.40 $\pm$ 0.02\
G039.8824$-$00.3460 & 56.13 $\pm$ 2.88 & 115.25 $\pm$ 2.77 & 700.94 $\pm$ 21.96 & 1888.43 $\pm$ 63.74 & 12.03 $\pm$ 0.06 & 11.25 $\pm$ 0.03 & 9.29 $\pm$ 0.03 & 8.21 $\pm$ 0.04\
G040.4251+00.7002 & 18.42 $\pm$ 1.68 & 57.91 $\pm$ 1.61 & 231.50 $\pm$ 16.84 & 605.19 $\pm$ 55.68 & 13.24 $\pm$ 0.10 & 11.99 $\pm$ 0.03 & 10.49 $\pm$ 0.08 & 9.45 $\pm$ 0.10\
G041.7419+00.0973 & 135.80 $\pm$ 7.49 & 201.40 $\pm$ 7.00 & 1125.66 $\pm$ 61.24 & 3197.17 $\pm$ 144.94 & 11.07 $\pm$ 0.06 & 10.64 $\pm$ 0.04 & 8.77 $\pm$ 0.06 & 7.64 $\pm$ 0.05\
G042.1090$-$00.4469 & 102.82 $\pm$ 3.74 & 139.53 $\pm$ 5.59 & 693.42 $\pm$ 20.68 & 1759.70 $\pm$ 64.85 & 11.37 $\pm$ 0.04 & 11.04 $\pm$ 0.04 & 9.30 $\pm$ 0.03 & 8.29 $\pm$ 0.04\
G042.4345$-$00.2605 & 673.00 $\pm$ 3.59 & 884.65 $\pm$ 5.20 & 1998.88 $\pm$ 45.48 & 4439.75 $\pm$ 147.80 & 9.33 $\pm$ 0.01 & 9.03 $\pm$ 0.01 & 8.15 $\pm$ 0.02 & 7.28 $\pm$ 0.04\
G043.1489+00.0130 & 340.70 $\pm$ 10.94 & 925.01 $\pm$ 16.48 & 4357.36 $\pm$ 159.72 & 6808.83 $\pm$ 475.75 & 10.07 $\pm$ 0.03 & 8.98 $\pm$ 0.02 & 7.30 $\pm$ 0.04 & 6.82 $\pm$ 0.08\
G043.1520+00.0115 & 198.77 $\pm$ 10.16 & 406.81 $\pm$ 17.23 & 1666.28 $\pm$ 115.88 & 5187.89 $\pm$ 425.47 & 10.65 $\pm$ 0.06 & 9.88 $\pm$ 0.05 & 8.35 $\pm$ 0.08 & 7.11 $\pm$ 0.09\
G043.1657+00.0116 & & 140.18 $\pm$ 9.94 & 1391.33 $\pm$ 155.20 & 2510.54 $\pm$ 642.52 & & 11.03 $\pm$ 0.08 & 8.54 $\pm$ 0.12 & 7.90 $\pm$ 0.28\
G043.1665+00.0106 & & 182.69 $\pm$ 30.73 & 1663.19 $\pm$ 338.49 & 3472.96 $\pm$ 1164.28 & & 10.75 $\pm$ 0.18 & 8.35 $\pm$ 0.22 & 7.55 $\pm$ 0.36\
G043.1674+00.0128 & 43.36 $\pm$ 3.71 & 134.68 $\pm$ 4.65 & 861.46 $\pm$ 70.11 & 2752.10 $\pm$ 326.89 & 12.31 $\pm$ 0.09 & 11.08 $\pm$ 0.04 & 9.06 $\pm$ 0.09 & 7.80 $\pm$ 0.13\
G043.1677+00.0196 & 62.04 $\pm$ 7.91 & 141.54 $\pm$ 7.32 & 751.87 $\pm$ 126.99 & 2375.49 $\pm$ 389.76 & 11.92 $\pm$ 0.14 & 11.02 $\pm$ 0.06 & 9.21 $\pm$ 0.18 & 7.96 $\pm$ 0.18\
G043.1684+00.0124 & 182.89 $\pm$ 4.47 & 342.97 $\pm$ 5.12 & 900.96 $\pm$ 74.74 & 2483.86 $\pm$ 305.79 & 10.74 $\pm$ 0.03 & 10.06 $\pm$ 0.02 & 9.01 $\pm$ 0.09 & 7.91 $\pm$ 0.13\
G043.1699+00.0115 & 40.44 $\pm$ 8.92 & 105.66 $\pm$ 9.37 & 647.46 $\pm$ 102.37 & 1682.22 $\pm$ 327.76 & 12.38 $\pm$ 0.24 & 11.34 $\pm$ 0.10 & 9.37 $\pm$ 0.17 & 8.34 $\pm$ 0.21\
G043.1701+00.0078 & & & 1189.51 $\pm$ 770.12 & & & & 8.71 $\pm$ 0.70 &\
G043.1706$-$00.0003 & 23.93 $\pm$ 12.87 & 47.47 $\pm$ 24.79 & 447.07 $\pm$ 165.84 & & 12.95 $\pm$ 0.58 & 12.21 $\pm$ 0.57 & 9.77 $\pm$ 0.40 &\
G043.1716+00.0001 & 55.98 $\pm$ 18.44 & 103.09 $\pm$ 34.48 & 790.68 $\pm$ 219.15 & 2386.61 $\pm$ 1215.33 & 12.03 $\pm$ 0.36 & 11.37 $\pm$ 0.36 & 9.15 $\pm$ 0.30 & 7.96 $\pm$ 0.55\
G043.1763+00.0248 & 133.77 $\pm$ 11.47 & 263.68 $\pm$ 21.63 & 935.86 $\pm$ 219.71 & 3289.25 $\pm$ 719.00 & 11.08 $\pm$ 0.09 & 10.35 $\pm$ 0.09 & 8.97 $\pm$ 0.25 & 7.61 $\pm$ 0.24\
G043.1778$-$00.5181 & 160.36 $\pm$ 8.79 & 233.12 $\pm$ 7.11 & 1549.17 $\pm$ 91.66 & 4098.20 $\pm$ 282.81 & 10.89 $\pm$ 0.06 & 10.48 $\pm$ 0.03 & 8.42 $\pm$ 0.06 & 7.37 $\pm$ 0.07\
G043.2371$-$00.0453 & 25.99 $\pm$ 2.27 & 78.06 $\pm$ 2.75 & 479.57 $\pm$ 16.91 & 1437.11 $\pm$ 68.83 & 12.86 $\pm$ 0.09 & 11.67 $\pm$ 0.04 & 9.70 $\pm$ 0.04 & 8.51 $\pm$ 0.05\
G043.3064$-$00.2114 & 29.58 $\pm$ 0.51 & 68.73 $\pm$ 0.67 & 322.74 $\pm$ 2.92 & 807.95 $\pm$ 5.25 & 12.72 $\pm$ 0.02 & 11.81 $\pm$ 0.01 & 10.13 $\pm$ 0.01 & 9.13 $\pm$ 0.01\
G043.7954$-$00.1274 & 55.90 $\pm$ 1.00 & 293.62 $\pm$ 1.81 & 939.12 $\pm$ 14.85 & 1831.11 $\pm$ 34.62 & 12.03 $\pm$ 0.02 & 10.23 $\pm$ 0.01 & 8.97 $\pm$ 0.02 & 8.24 $\pm$ 0.02\
G043.7960$-$00.1286 & 59.29 $\pm$ 1.08 & 134.84 $\pm$ 1.47 & 798.18 $\pm$ 15.71 & 1934.54 $\pm$ 41.03 & 11.97 $\pm$ 0.02 & 11.08 $\pm$ 0.01 & 9.14 $\pm$ 0.02 & 8.18 $\pm$ 0.02\
G043.8894$-$00.7840 & 235.66 $\pm$ 6.39 & 375.67 $\pm$ 7.02 & 1818.43 $\pm$ 54.14 & 5420.25 $\pm$ 168.60 & 10.47 $\pm$ 0.03 & 9.96 $\pm$ 0.02 & 8.25 $\pm$ 0.03 & 7.06 $\pm$ 0.03\
G043.9675+00.9939 & 32.88 $\pm$ 0.84 & 46.75 $\pm$ 0.78 & 202.65 $\pm$ 4.89 & 613.68 $\pm$ 8.63 & 12.61 $\pm$ 0.03 & 12.23 $\pm$ 0.02 & 10.63 $\pm$ 0.03 & 9.43 $\pm$ 0.02\
G044.3103+00.0410 & 31.50 $\pm$ 0.58 & 41.65 $\pm$ 0.63 & 265.33 $\pm$ 3.55 & 653.53 $\pm$ 12.17 & 12.65 $\pm$ 0.02 & 12.35 $\pm$ 0.02 & 10.34 $\pm$ 0.01 & 9.36 $\pm$ 0.02\
G044.4228+00.5377 & 4.97 $\pm$ 1.25 & 6.76 $\pm$ 1.27 & 77.43 $\pm$ 9.20 & 272.30 $\pm$ 23.22 & 14.66 $\pm$ 0.27 & 14.32 $\pm$ 0.20 & 11.68 $\pm$ 0.13 & 10.31 $\pm$ 0.09\
G045.1242+00.1356 & 166.07 $\pm$ 32.99 & 260.89 $\pm$ 35.53 & 4049.91 $\pm$ 202.54 & 8943.36 $\pm$ 752.21 & 10.85 $\pm$ 0.21 & 10.36 $\pm$ 0.15 & 7.38 $\pm$ 0.05 & 6.52 $\pm$ 0.09\
G045.4545+00.0591 & 704.08 $\pm$ 70.16 & 1487.07 $\pm$ 69.14 & 5402.69 $\pm$ 541.82 & 16875.34 $\pm$ 1621.34 & 9.28 $\pm$ 0.11 & 8.47 $\pm$ 0.05 & 7.07 $\pm$ 0.11 & 5.83 $\pm$ 0.10\
G045.4559+00.0613 & 187.57 $\pm$ 32.14 & 393.95 $\pm$ 36.92 & 2661.24 $\pm$ 297.80 & 8733.53 $\pm$ 942.49 & 10.72 $\pm$ 0.18 & 9.91 $\pm$ 0.10 & 7.84 $\pm$ 0.12 & 6.55 $\pm$ 0.12\
G045.4656+00.0452 & 32.56 $\pm$ 1.46 & 103.05 $\pm$ 3.25 & 98.79 $\pm$ 29.93 & & 12.62 $\pm$ 0.05 & 11.37 $\pm$ 0.03 & 11.41 $\pm$ 0.33 &\
G045.5431$-$00.0073 & 52.16 $\pm$ 5.67 & 74.41 $\pm$ 6.75 & 726.58 $\pm$ 44.00 & 2139.10 $\pm$ 126.27 & 12.11 $\pm$ 0.12 & 11.72 $\pm$ 0.10 & 9.25 $\pm$ 0.07 & 8.07 $\pm$ 0.06\
G048.6099+00.0270 & 476.67 $\pm$ 12.12 & 427.73 $\pm$ 13.26 & 910.97 $\pm$ 153.20 & 2074.39 $\pm$ 506.61 & 9.70 $\pm$ 0.03 & 9.82 $\pm$ 0.03 & 9.00 $\pm$ 0.18 & 8.11 $\pm$ 0.26\
G048.9296$-$00.2793 & 72.36 $\pm$ 37.48 & 120.43 $\pm$ 28.84 & & & 11.75 $\pm$ 0.56 & 11.20 $\pm$ 0.26 & &\
G048.9901$-$00.2988 & 40.73 $\pm$ 2.64 & 98.80 $\pm$ 3.89 & 181.77 $\pm$ 18.48 & 667.17 $\pm$ 82.06 & 12.38 $\pm$ 0.07 & 11.41 $\pm$ 0.04 & 10.75 $\pm$ 0.11 & 9.34 $\pm$ 0.13\
G049.3666$-$00.3010 & 44.28 $\pm$ 27.27 & 116.96 $\pm$ 58.43 & & & 12.28 $\pm$ 0.67 & 11.23 $\pm$ 0.54 & &\
G049.4640$-$00.3511 & 46.41 $\pm$ 13.98 & 115.48 $\pm$ 19.44 & 734.54 $\pm$ 177.15 & 3201.85 $\pm$ 556.14 & 12.23 $\pm$ 0.33 & 11.24 $\pm$ 0.18 & 9.23 $\pm$ 0.26 & 7.64 $\pm$ 0.19\
G050.3152+00.6762 & 79.77 $\pm$ 3.10 & 106.53 $\pm$ 3.07 & 858.95 $\pm$ 32.26 & 2480.15 $\pm$ 122.16 & 11.65 $\pm$ 0.04 & 11.33 $\pm$ 0.03 & 9.07 $\pm$ 0.04 & 7.91 $\pm$ 0.05\
G050.3157+00.6747 & 191.53 $\pm$ 8.60 & 178.21 $\pm$ 8.04 & 1536.99 $\pm$ 67.06 & 4672.83 $\pm$ 195.93 & 10.69 $\pm$ 0.05 & 10.77 $\pm$ 0.05 & 8.43 $\pm$ 0.05 & 7.23 $\pm$ 0.05\
G052.7533+00.3340 & 246.92 $\pm$ 9.27 & 275.77 $\pm$ 10.00 & 1925.19 $\pm$ 62.50 & 5219.79 $\pm$ 141.53 & 10.42 $\pm$ 0.04 & 10.30 $\pm$ 0.04 & 8.19 $\pm$ 0.04 & 7.11 $\pm$ 0.03\
G053.9589+00.0320 & 72.29 $\pm$ 2.08 & 97.01 $\pm$ 1.67 & 674.40 $\pm$ 28.03 & 1980.35 $\pm$ 104.89 & 11.75 $\pm$ 0.03 & 11.43 $\pm$ 0.02 & 9.33 $\pm$ 0.04 & 8.16 $\pm$ 0.06\
G058.7739+00.6457 & 108.24 $\pm$ 12.26 & 160.66 $\pm$ 9.43 & 1232.48 $\pm$ 101.73 & 3168.97 $\pm$ 343.44 & 11.31 $\pm$ 0.12 & 10.89 $\pm$ 0.06 & 8.67 $\pm$ 0.09 & 7.65 $\pm$ 0.12\
G059.6027+00.9118 & 12.03 $\pm$ 2.75 & 39.88 $\pm$ 2.74 & 99.68 $\pm$ 8.52 & 244.34 $\pm$ 10.79 & 13.70 $\pm$ 0.25 & 12.40 $\pm$ 0.07 & 11.40 $\pm$ 0.09 & 10.43 $\pm$ 0.05\
G061.4763+00.0892 & & 752.61 $\pm$ 76.83 & 5192.64 $\pm$ 1661.68 & 15695.08 $\pm$ 3054.71 & & 9.21 $\pm$ 0.11 & 7.11 $\pm$ 0.35 & 5.91 $\pm$ 0.21\
G061.4770+00.0891 & & 198.75 $\pm$ 55.98 & & & & 10.65 $\pm$ 0.30 & &\
G061.7207+00.8630 & 82.01 $\pm$ 1.02 & 87.55 $\pm$ 0.96 & 520.03 $\pm$ 6.93 & 1465.15 $\pm$ 16.38 & 11.62 $\pm$ 0.01 & 11.54 $\pm$ 0.01 & 9.61 $\pm$ 0.01 & 8.49 $\pm$ 0.01\
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G010.3009$-$00.1477 & 1.47 & 3.54 & 4.64 $\pm$ 1.81 & 15.05 & 13.65 & 12.83 $\pm$ 0.42 & 1.40 & 0.81\
G010.6297$-$00.3380 & 0.89 & 2.18 & 1.38 $\pm$ 0.58 & 15.59 & 14.17 & 14.15 $\pm$ 0.45 & 1.41 & 0.02\
G010.9584+00.0221 & 0.66 & 1.86 & 1.63 $\pm$ 0.77 & 15.92 & 14.35 & 13.97 $\pm$ 0.51 & 1.57 & 0.38\
G011.1104$-$00.3985 & 2.68 & 4.51 & 26.25 $\pm$ 1.53 & 14.39 & 13.39 & 10.95 $\pm$ 0.06 & 1.01 & 2.43\
G011.9368$-$00.6158 & 1.94 & 3.90 & 6.65 $\pm$ 1.79 & 14.74 & 13.54 & 12.44 $\pm$ 0.29 & 1.20 & 1.10\
G011.9446$-$00.0369 & 20.54 & 33.35 $\pm$ 18.56 & 43.24 $\pm$ 15.87 & 12.18 & 11.21 $\pm$ 0.60 & 10.41 $\pm$ 0.40 & 0.97 & 0.8 $\pm$ 0.72\
G012.4317$-$01.1112 & & 4.48 & 5.24 $\pm$ 1.27 & & 13.39 & 12.70 $\pm$ 0.26 & & 0.69\
G012.8050$-$00.2007 & 5.61 & 7.15 $\pm$ 4.35 & 30.26 $\pm$ 5.86 & 13.59 & 12.88 $\pm$ 0.66 & 10.80 $\pm$ 0.21 & 0.70 & 2.09 $\pm$ 0.69\
G012.8131$-$00.1976 & 0.99 & 2.08 & 2.11 $\pm$ 0.86 & 15.47 & 14.22 & 13.69 $\pm$ 0.44 & 1.25 & 0.54\
G014.1046+00.0918 & 0.76 & 2.43 & 2.17 $\pm$ 0.96 & 15.77 & 14.06 & 13.66 $\pm$ 0.48 & 1.71 & 0.40\
G017.1141$-$00.1124 & 0.66 & 1.93 & 3.25 $\pm$ 0.85 & 15.91 & 14.31 & 13.22 $\pm$ 0.28 & 1.60 & 1.09\
G018.1460$-$00.2839 & 10.90 $\pm$ 0.31 & 69.19 $\pm$ 11.81 & 91.95 $\pm$ 12.47 & 12.87 $\pm$ 0.03 & 10.42 $\pm$ 0.18 & 9.59 $\pm$ 0.15 & 2.45 $\pm$ 0.19 & 0.83 $\pm$ 0.24\
G018.3024$-$00.3910 & 2.07 $\pm$ 0.10 & 17.23 $\pm$ 4.71 & 66.23 $\pm$ 4.80 & 14.67 $\pm$ 0.05 & 11.93 $\pm$ 0.29 & 9.95 $\pm$ 0.08 & 2.74 $\pm$ 0.30 & 1.98 $\pm$ 0.31\
G018.8250$-$00.4675 & 0.53 & 1.30 & 1.17 $\pm$ 0.60 & 16.16 & 14.74 & 14.33 $\pm$ 0.56 & 1.42 & 0.41\
G019.0754$-$00.2874 & 2.09 & 5.29 & 27.71 $\pm$ 2.33 & 14.66 & 13.21 & 10.89 $\pm$ 0.09 & 1.45 & 2.32\
G019.6087$-$00.2351 & 10.36 & 31.23 & 33.87 $\pm$ 8.43 & 12.92 & 11.28 & 10.68 $\pm$ 0.27 & 1.64 & 0.61\
G019.6090$-$00.2313 & 2.19 & 6.61 & 8.26 $\pm$ 2.05 & 14.61 & 12.97 & 12.21 $\pm$ 0.27 & 1.64 & 0.76\
G020.0720$-$00.1421 & 2.46 & 7.08 & 14.25 $\pm$ 2.46 & 14.49 & 12.90 & 11.62 $\pm$ 0.19 & 1.59 & 1.28\
G021.3571$-$00.1766 & 0.55 & 1.87 & 2.87 $\pm$ 0.70 & 16.12 & 14.34 & 13.35 $\pm$ 0.26 & 1.78 & 0.99\
G021.3855$-$00.2541 & 0.75 & 2.27 & 3.32 $\pm$ 0.81 & 15.78 & 14.13 & 13.20 $\pm$ 0.26 & 1.65 & 0.93\
G023.1974$-$00.0006 & 0.65 & 2.09 $\pm$ 0.51 & 3.95 $\pm$ 0.53 & 15.93 & 14.22 $\pm$ 0.26 & 13.01 $\pm$ 0.14 & 1.71 & 1.21 $\pm$ 0.30\
G023.2654+00.0765 & 1.13 & 2.45 & 2.66 $\pm$ 0.89 & 15.33 & 14.05 & 13.44 $\pm$ 0.36 & 1.29 & 0.61\
G023.7110+00.1705 & 0.87 & 6.19 $\pm$ 0.84 & 8.85 $\pm$ 0.92 & 15.62 & 13.04 $\pm$ 0.15 & 12.13 $\pm$ 0.11 & 2.57 & 0.91 $\pm$ 0.18\
G024.4698+00.4954 & 0.94 $\pm$ 0.22 & 2.41 $\pm$ 0.57 & 3.60 $\pm$ 0.77 & 15.53 $\pm$ 0.26 & 14.06 $\pm$ 0.25 & 13.11 $\pm$ 0.23 & 1.47 $\pm$ 0.36 & 0.95 $\pm$ 0.34\
G024.4721+00.4877 & 4.71 $\pm$ 0.31 & 12.92 $\pm$ 0.92 & 14.20 $\pm$ 1.34 & 13.78 $\pm$ 0.07 & 12.24 $\pm$ 0.08 & 11.62 $\pm$ 0.10 & 1.54 $\pm$ 0.11 & 0.62 $\pm$ 0.13\
G024.4736+00.4950 & 1.23 & 2.55 & 4.74 $\pm$ 1.12 & 15.24 & 14.00 & 12.81 $\pm$ 0.26 & 1.23 & 1.19\
G024.5065$-$00.2224 & 1.23 & 2.63 & 10.76 $\pm$ 1.36 & 15.23 & 13.97 & 11.92 $\pm$ 0.14 & 1.26 & 2.05\
G025.3809$-$00.1815 & 17.74 $\pm$ 1.51 & 76.01 $\pm$ 8.15 & 93.97 $\pm$ 5.22 & 12.34 $\pm$ 0.09 & 10.32 $\pm$ 0.12 & 9.57 $\pm$ 0.06 & 2.02 $\pm$ 0.15 & 0.75 $\pm$ 0.13\
G025.3824$-$00.1812 & 8.49 $\pm$ 0.41 & 51.78 $\pm$ 3.33 & 80.95 $\pm$ 2.67 & 13.14 $\pm$ 0.05 & 10.74 $\pm$ 0.07 & 9.73 $\pm$ 0.04 & 2.4 $\pm$ 0.09 & 1.01 $\pm$ 0.08\
G025.3981$-$00.1411 & 2.60 & 6.13 & 4.92 $\pm$ 2.38 & 14.43 & 13.05 & 12.77 $\pm$ 0.52 & 1.37 & 0.28\
G025.8011$-$00.1568 & 0.40 & 3.56 $\pm$ 0.31 & 8.45 $\pm$ 0.39 & 16.47 & 13.64 $\pm$ 0.09 & 12.18 $\pm$ 0.05 & 2.83 & 1.46 $\pm$ 0.11\
G027.2800+00.1447 & 3.45 & 9.12 & 8.15 $\pm$ 2.60 & 14.12 & 12.62 & 12.22 $\pm$ 0.34 & 1.50 & 0.40\
G028.2879$-$00.3641 & 9.16 $\pm$ 0.44 & 40.46 $\pm$ 1.35 & 57.53 $\pm$ 2.95 & 13.06 $\pm$ 0.05 & 11.00 $\pm$ 0.04 & 10.10 $\pm$ 0.06 & 2.05 $\pm$ 0.06 & 0.9 $\pm$ 0.07\
G028.6082+00.0185 & 0.34 & 0.79 $\pm$ 0.25 & 9.50 $\pm$ 0.66 & 16.63 & 15.28 $\pm$ 0.34 & 12.06 $\pm$ 0.07 & 1.35 & 3.22 $\pm$ 0.35\
G028.6869+00.1770 & 0.51 & 1.35 & 8.65 $\pm$ 0.99 & 16.19 & 14.70 & 12.16 $\pm$ 0.12 & 1.49 & 2.54\
G029.9559$-$00.0168 & 8.11 $\pm$ 1.34 & 50.68 $\pm$ 2.69 & 109.83 $\pm$ 3.11 & 13.19 $\pm$ 0.18 & 10.76 $\pm$ 0.06 & 9.40 $\pm$ 0.03 & 2.43 $\pm$ 0.19 & 1.36 $\pm$ 0.06\
G030.5353+00.0204 & 1.98 & 3.66 & 28.94 $\pm$ 1.32 & 14.72 & 13.61 & 10.85 $\pm$ 0.05 & 1.11 & 2.76\
G030.6881$-$00.0718 & 0.99 & 1.80 & 7.67 $\pm$ 1.25 & 15.48 & 14.39 & 12.29 $\pm$ 0.18 & 1.09 & 2.10\
G030.7532$-$00.0511 & 0.39 & 0.89 & 2.92 & 16.49 & 15.15 & 13.33 & 1.34 & 1.81\
G030.7661$-$00.0348 & 0.41 & 5.30 $\pm$ 0.46 & 36.20 $\pm$ 2.10 & 16.44 & 13.21 $\pm$ 0.09 & 10.60 $\pm$ 0.06 & 3.23 & 2.61 $\pm$ 0.11\
G030.8662+00.1143 & 0.48 & 0.88 & 1.95 & 16.26 & 15.16 & 13.77 & 1.10 & 1.39\
G031.0495+00.4697 & 0.17 & 0.32 & 0.97 & 17.37 & 16.25 & 14.53 & 1.12 & 1.72\
G031.2435$-$00.1103 & 0.59 $\pm$ 0.15 & 3.09 $\pm$ 0.46 & 6.20 $\pm$ 0.86 & 16.03 $\pm$ 0.28 & 13.80 $\pm$ 0.16 & 12.52 $\pm$ 0.15 & 2.23 $\pm$ 0.32 & 1.28 $\pm$ 0.22\
G031.3959$-$00.2570 & & & 12.52 $\pm$ 0.14 & & & 11.76 $\pm$ 0.01 & & $-$11.76\
G032.0297+00.0491 & 0.77 & 1.00 $\pm$ 0.43 & 5.16 $\pm$ 1.00 & 15.74 & 15.02 $\pm$ 0.46 & 12.72 $\pm$ 0.21 & 0.72 & 2.31 $\pm$ 0.51\
G032.1502+00.1329 & 6.45 $\pm$ 1.07 & 21.64 $\pm$ 2.28 & 45.80 $\pm$ 4.71 & 13.44 $\pm$ 0.18 & 11.68 $\pm$ 0.11 & 10.35 $\pm$ 0.11 & 1.75 $\pm$ 0.21 & 1.34 $\pm$ 0.16\
G032.4727+00.2036 & 0.30 & 4.54 $\pm$ 0.25 & 17.25 $\pm$ 0.65 & 16.76 & 13.38 $\pm$ 0.06 & 11.41 $\pm$ 0.04 & 3.38 & 1.97 $\pm$ 0.07\
G032.7492$-$00.0643 & 0.76 & 2.06 & 8.73 $\pm$ 0.71 & 15.76 & 14.23 & 12.15 $\pm$ 0.09 & 1.52 & 2.09\
G032.7966+00.1909 & 4.94 & 10.02 & 33.71 $\pm$ 4.00 & 13.73 & 12.52 & 10.68 $\pm$ 0.13 & 1.21 & 1.84\
G034.2544+00.1460 & 1.06 & 2.22 & 9.73 $\pm$ 1.31 & 15.40 & 14.15 & 12.03 $\pm$ 0.15 & 1.24 & 2.13\
G034.2571+00.1466 & 0.55 & 1.17 & 6.18 $\pm$ 0.78 & 16.12 & 14.85 & 12.52 $\pm$ 0.14 & 1.26 & 2.33\
G034.4032+00.2277 & 0.40 & 0.75 & 3.00 $\pm$ 0.30 & 16.47 & 15.33 & 13.31 $\pm$ 0.11 & 1.14 & 2.02\
G035.0524$-$00.5177 & 0.77 & 1.56 $\pm$ 0.53 & 4.34 $\pm$ 0.68 & 15.74 & 14.54 $\pm$ 0.37 & 12.91 $\pm$ 0.17 & 1.20 & 1.63 $\pm$ 0.40\
G035.5781$-$00.0305 & 0.63 & 1.71 & 3.23 $\pm$ 0.66 & 15.96 & 14.44 & 13.23 $\pm$ 0.22 & 1.52 & 1.21\
G036.4057+00.0226 & 0.36 & 0.79 & 3.06 $\pm$ 0.35 & 16.58 & 15.27 & 13.29 $\pm$ 0.12 & 1.31 & 1.99\
G037.5457$-$00.1120 & 2.57 & 8.69 $\pm$ 1.71 & 11.62 $\pm$ 2.00 & 14.44 & 12.67 $\pm$ 0.21 & 11.84 $\pm$ 0.19 & 1.76 & 0.84 $\pm$ 0.28\
G037.8683$-$00.6008 & 1.26 & 1.68 $\pm$ 1.09 & 5.26 $\pm$ 1.13 & 15.21 & 14.46 $\pm$ 0.70 & 12.70 $\pm$ 0.23 & 0.76 & 1.76 $\pm$ 0.74\
G039.8824$-$00.3460 & 1.08 & 2.09 & 1.46 $\pm$ 0.81 & 15.38 & 14.22 & 14.09 $\pm$ 0.60 & 1.16 & 0.13\
G042.1090$-$00.4469 & 0.39 & 1.15 $\pm$ 0.31 & 4.35 $\pm$ 0.45 & 16.50 & 14.87 $\pm$ 0.29 & 12.90 $\pm$ 0.11 & 1.63 & 1.97 $\pm$ 0.31\
G042.4345$-$00.2605 & 0.57 & 1.76 & 10.87 $\pm$ 0.74 & 16.06 & 14.41 & 11.91 $\pm$ 0.07 & 1.66 & 2.50\
G043.1651$-$00.0283 & & 2.62 $\pm$ 0.07 & 12.81 $\pm$ 0.25 & & 13.98 $\pm$ 0.03 & 11.73 $\pm$ 0.02 & & 2.25 $\pm$ 0.04\
G043.1677+00.0196 & 0.75 & 1.83 & 1.46 $\pm$ 0.72 & 15.78 & 14.37 & 14.09 $\pm$ 0.54 & 1.41 & 0.28\
G043.1763+00.0248 & 0.85 & 3.35 $\pm$ 0.67 & 7.96 $\pm$ 0.77 & 15.64 & 13.71 $\pm$ 0.22 & 12.25 $\pm$ 0.10 & 1.94 & 1.46 $\pm$ 0.24\
G043.7960$-$00.1286 & & 0.92 & 2.02 $\pm$ 0.30 & & 15.12 & 13.74 $\pm$ 0.16 & & 1.38\
G043.8894$-$00.7840 & 1.58 & 3.74 $\pm$ 1.01 & 11.72 $\pm$ 0.93 & 14.97 & 13.59 $\pm$ 0.29 & 11.83 $\pm$ 0.09 & 1.38 & 1.76 $\pm$ 0.30\
G043.9675+00.9939 & 0.25 & 0.55 & 2.30 $\pm$ 0.40 & 16.96 & 15.67 & 13.60 $\pm$ 0.19 & 1.29 & 2.07\
G044.3103+00.0410 & 0.31 & 0.71 & 1.60 $\pm$ 0.27 & 16.74 & 15.40 & 13.99 $\pm$ 0.18 & 1.34 & 1.40\
G045.0694+00.1323 & & & 1.55 $\pm$ 0.05 & & & 14.02 $\pm$ 0.04 & & $-$14.02\
G045.1223+00.1321 & 15.09 $\pm$ 1.19 & 32.43 $\pm$ 2.74 & 149.19 $\pm$ 2.81 & 12.52 $\pm$ 0.08 & 11.24 $\pm$ 0.09 & 9.07 $\pm$ 0.02 & 1.27 $\pm$ 0.12 & 2.18 $\pm$ 0.09\
G045.1242+00.1356 & 1.07 & 8.22 & 8.20 $\pm$ 1.06 & 15.39 & 12.73 & 12.22 $\pm$ 0.14 & 2.65 & 0.52\
G045.4545+00.0591 & 17.78 $\pm$ 0.94 & 32.43 $\pm$ 2.14 & 40.46 $\pm$ 3.23 & 12.34 $\pm$ 0.06 & 11.24 $\pm$ 0.07 & 10.48 $\pm$ 0.09 & 1.09 $\pm$ 0.09 & 0.76 $\pm$ 0.11\
G045.4559+00.0613 & 0.95 & 2.16 & 8.90 $\pm$ 1.20 & 15.52 & 14.18 & 12.13 $\pm$ 0.15 & 1.34 & 2.06\
G045.4656+00.0452 & 0.37 & 0.81 & 3.10 $\pm$ 0.39 & 16.53 & 15.24 & 13.27 $\pm$ 0.14 & 1.29 & 1.97\
G048.6099+00.0270 & 1.15 & 4.02 & 5.67 $\pm$ 3.93 & 15.31 & 13.51 & 12.62 $\pm$ 0.75 & 1.80 & 0.89\
G049.3666$-$00.3010 & 0.83 & 1.65 & 1.74 $\pm$ 0.66 & 15.67 & 14.48 & 13.90 $\pm$ 0.41 & 1.19 & 0.58\
G049.4905$-$00.3688 & 5.44 $\pm$ 0.41 & 47.46 $\pm$ 1.21 & 134.25 $\pm$ 1.77 & 13.62 $\pm$ 0.08 & 10.83 $\pm$ 0.03 & 9.18 $\pm$ 0.01 & 2.79 $\pm$ 0.09 & 1.65 $\pm$ 0.03\
G050.3152+00.6762 & 0.51 & 3.62 & 3.17 $\pm$ 0.56 & 16.18 & 13.63 & 13.25 $\pm$ 0.19 & 2.56 & 0.38\
G050.3157+00.6747 & 1.68 $\pm$ 0.68 & 7.27 $\pm$ 1.41 & 11.06 $\pm$ 1.47 & 14.90 $\pm$ 0.44 & 12.87 $\pm$ 0.21 & 11.89 $\pm$ 0.14 & 2.03 $\pm$ 0.49 & 0.98 $\pm$ 0.25\
G052.7533+00.3340 & 2.42 & 5.06 & 8.61 $\pm$ 1.77 & 14.50 & 13.26 & 12.16 $\pm$ 0.22 & 1.24 & 1.10\
G053.9589+00.0320 & 0.44 & 0.87 & 2.84 $\pm$ 0.42 & 16.35 & 15.17 & 13.37 $\pm$ 0.16 & 1.18 & 1.80\
G058.7739+00.6457 & 0.67 & 2.86 $\pm$ 0.62 & 5.15 $\pm$ 0.52 & 15.89 & 13.88 $\pm$ 0.23 & 12.72 $\pm$ 0.11 & 2.01 & 1.16 $\pm$ 0.26\
G060.8842$-$00.1286 & & 0.95 & 4.89 $\pm$ 0.64 & & 15.08 & 12.78 $\pm$ 0.14 & & 2.30\
G061.7207+00.8630 & 0.47 & 0.93 & 8.55 $\pm$ 0.35 & 16.27 & 15.10 & 12.17 $\pm$ 0.04 & 1.17 & 2.93\
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G010.3009$-$00.1477 & 4.16 & 4.03 $\pm$ 0.44 & 36.71 & 35.49 $\pm$ 3.85 & 19.60 & 9.37\
G010.6297$-$00.3380 & 1.24 & 1.9 $\pm$ 0.46 & 10.97 & 16.74 $\pm$ 4.08 & 19.74 & $-$1.89\
G010.9584+00.0221 & 3.59 & 3.89 $\pm$ 0.52 & 31.69 & 34.31 $\pm$ 4.6 & 21.31 & 3.17\
G011.1104$-$00.3985 & 3.11 & 1.36 $\pm$ 0.07 & 27.45 & 11.96 $\pm$ 0.65 & 15.63 & 32.41\
G011.9368$-$00.6158 & 4.72 & 4.29 $\pm$ 0.31 & 41.57 & 37.84 $\pm$ 2.77 & 17.57 & 13.43\
G011.9446$-$00.0369 & 2.17 $\pm$ 0.60 & 2.04 $\pm$ 0.40 & 19.1 $\pm$ 5.33 & 17.98 $\pm$ 3.54 & 15.24 $\pm$ 6.07 & 9.21 $\pm$ 10.24\
G012.4317$-$01.1112 & 1.51 & 1.49 $\pm$ 0.27 & 13.29 & 13.15 $\pm$ 2.38 & & 7.64\
G012.8050$-$00.2007 & 6.65 $\pm$ 0.68 & 5.24 $\pm$ 0.25 & 58.65 $\pm$ 6.00 & 46.22 $\pm$ 2.22 & 12.58 $\pm$ 6.65 & 27.48 $\pm$ 9.81\
G012.8131$-$00.1976 & 5.68 & 5.82 $\pm$ 0.47 & 50.08 & 51.32 $\pm$ 4.18 & 18.09 & 5.41\
G014.1046+00.0918 & 1.05 & 1.33 $\pm$ 0.48 & 9.27 & 11.72 $\pm$ 4.22 & 22.74 & 3.45\
G017.1141$-$00.1124 & 0.91 & 0.5 $\pm$ 0.28 & 8.06 & 4.44 $\pm$ 2.5 & 21.63 & 13.26\
G018.1460$-$00.2839 & 1.27 $\pm$ 0.19 & 1.12 $\pm$ 0.15 & 11.18 $\pm$ 1.66 & 9.83 $\pm$ 1.32 & 30.2 $\pm$ 1.89 & 9.59 $\pm$ 3.35\
G018.3024$-$00.3910 & 3.21 $\pm$ 0.31 & 1.91 $\pm$ 0.09 & 28.32 $\pm$ 2.71 & 16.81 $\pm$ 0.83 & 33.18 $\pm$ 3.03 & 26 $\pm$ 4.34\
G018.8250$-$00.4675 & 0.90 & 1.17 $\pm$ 0.56 & 7.90 & 10.28 $\pm$ 4.95 & 19.81 & 3.56\
G019.0754$-$00.2874 & 3.18 & 1.54 $\pm$ 0.10 & 28.03 & 13.55 $\pm$ 0.89 & 20.15 & 30.77\
G019.6087$-$00.2351 & 3.46 & 3.52 $\pm$ 0.29 & 30.48 & 31.07 $\pm$ 2.51 & 22.03 & 6.45\
G019.6090$-$00.2313 & 2.52 & 2.44 $\pm$ 0.28 & 22.26 & 21.48 $\pm$ 2.45 & 22.06 & 8.66\
G020.0720$-$00.1421 & 2.22 & 1.61 $\pm$ 0.19 & 19.56 & 14.23 $\pm$ 1.7 & 21.53 & 16.02\
G021.3571$-$00.1766 & 1.35 & 1.04 $\pm$ 0.26 & 11.91 & 9.16 $\pm$ 2.33 & 23.41 & 11.85\
G021.3855$-$00.2541 & 2.79 & 2.53 $\pm$ 0.28 & 24.57 & 22.32 $\pm$ 2.43 & 22.12 & 11.05\
G023.1974$-$00.0006 & 0.24 $\pm$ 0.26 & $-$0.3 $\pm$ 0.15 & 2.09 $\pm$ 2.30 & $-$2.63 $\pm$ 1.28 & 22.76 $\pm$ 2.64 & 15.03 $\pm$ 4.25\
G023.2654+00.0765 & 2.43 & 2.5 $\pm$ 0.37 & 21.43 & 22.02 $\pm$ 3.27 & 18.47 & 6.46\
G023.7110+00.1705 & 2.36 $\pm$ 0.16 & 2.12 $\pm$ 0.13 & 20.77 $\pm$ 1.43 & 18.72 $\pm$ 1.13 & 31.47 $\pm$ 1.49 & 10.72 $\pm$ 2.63\
G024.4698+00.4954 & 1.25 $\pm$ 0.26 & 0.98 $\pm$ 0.24 & 11.06 $\pm$ 2.29 & 8.62 $\pm$ 2.08 & 20.32 $\pm$ 3.64 & 11.35 $\pm$ 4.87\
G024.4721+00.4877 & 0.11 $\pm$ 0.08 & 0.17 $\pm$ 0.10 & 1.01 $\pm$ 0.68 & 1.47 $\pm$ 0.9 & 20.99 $\pm$ 1.06 & 6.66 $\pm$ 1.81\
G024.4736+00.4950 & 2.51 & 2 $\pm$ 0.27 & 22.16 & 17.6 $\pm$ 2.34 & 17.94 & 14.76\
G024.5065$-$00.2224 & 3.27 & 1.9 $\pm$ 0.15 & 28.82 & 16.71 $\pm$ 1.29 & 18.22 & 26.96\
G025.3809$-$00.1815 & 0.49 $\pm$ 0.12 & 0.42 $\pm$ 0.06 & 4.36 $\pm$ 1.03 & 3.69 $\pm$ 0.54 & 25.89 $\pm$ 1.49 & 8.48 $\pm$ 1.85\
G025.3824$-$00.1812 & 0.01 $\pm$ 0.07 & $-$0.32 $\pm$ 0.04 & 0.05 $\pm$ 0.61 & $-$2.86 $\pm$ 0.33 & 29.76 $\pm$ 0.88 & 12.1 $\pm$ 1.11\
G025.3981$-$00.1411 & 4.89 & 5.29 $\pm$ 0.54 & 43.12 & 46.6 $\pm$ 4.77 & 19.35 & 1.80\
G025.8011$-$00.1568 & 0.92 $\pm$ 0.10 & 0.14 $\pm$ 0.05 & 8.1 $\pm$ 0.85 & 1.21 $\pm$ 0.44 & 34.03 $\pm$ 0.94 & 18.54 $\pm$ 1.49\
G027.2800+00.1447 & 2.72 & 2.99 $\pm$ 0.36 & 23.95 & 26.39 $\pm$ 3.14 & 20.59 & 3.47\
G028.2879$-$00.3641 & 1.38 $\pm$ 0.05 & 1.15 $\pm$ 0.06 & 12.13 $\pm$ 0.47 & 10.13 $\pm$ 0.57 & 26.22 $\pm$ 0.64 & 10.64 $\pm$ 0.94\
G028.6082+00.0185 & 4.6 $\pm$ 0.36 & 2.05 $\pm$ 0.10 & 40.53 $\pm$ 3.21 & 18.11 $\pm$ 0.84 & 19.12 $\pm$ 3.42 & 43.6 $\pm$ 4.93\
G028.6869+00.1770 & 3.25 & 1.38 $\pm$ 0.13 & 28.62 & 12.18 $\pm$ 1.16 & 20.57 & 33.93\
G029.9559$-$00.0168 & 3.01 $\pm$ 0.10 & 2.33 $\pm$ 0.07 & 26.53 $\pm$ 0.91 & 20.5 $\pm$ 0.65 & 30.02 $\pm$ 1.90 & 17.15 $\pm$ 0.92\
G030.5353+00.0204 & 4.26 & 2.17 $\pm$ 0.08 & 37.52 & 19.11 $\pm$ 0.69 & 16.70 & 37.12\
G030.6881$-$00.0718 & 4.57 & 3.15 $\pm$ 0.20 & 40.33 & 27.81 $\pm$ 1.75 & 16.48 & 27.62\
G030.7532$-$00.0511 & 4.86 & 3.73 & 42.87 & 32.85 & 19.04 & 23.57\
G030.7661$-$00.0348 & 1.58 $\pm$ 0.12 & $-$0.35 $\pm$ 0.06 & 13.95 $\pm$ 1.09 & $-$3.07 $\pm$ 0.57 & 38.11 $\pm$ 0.96 & 34.88 $\pm$ 1.61\
G030.8662+00.1143 & 4.96 & 4.25 & 43.71 & 37.45 & 16.61 & 17.52\
G031.0495+00.4697 & 2.61 & 1.56 & 22.98 & 13.76 & 16.77 & 22.29\
G031.2435$-$00.1103 & 3.68 $\pm$ 0.19 & 3.08 $\pm$ 0.17 & 32.45 $\pm$ 1.68 & 27.17 $\pm$ 1.51 & 28.05 $\pm$ 3.27 & 15.94 $\pm$ 3.13\
G031.3959$-$00.2570 & $-$11.71 & 0.72 $\pm$ 0.03 & & 6.34 $\pm$ 0.26 & &\
G032.0297+00.0491 & 2.11 $\pm$ 0.47 & 0.48 $\pm$ 0.21 & 18.57 $\pm$ 4.14 & 4.21 $\pm$ 1.86 & 12.72 $\pm$ 4.66 & 30.58 $\pm$ 7.21\
G032.1502+00.1329 & 2.02 $\pm$ 0.13 & 1.36 $\pm$ 0.12 & 17.78 $\pm$ 1.17 & 11.98 $\pm$ 1.06 & 23.2 $\pm$ 2.15 & 16.78 $\pm$ 2.26\
G032.4727+00.2036 & 1.87 $\pm$ 0.08 & 0.57 $\pm$ 0.04 & 16.46 $\pm$ 0.72 & 5.04 $\pm$ 0.39 & 39.61 $\pm$ 0.61 & 25.83 $\pm$ 1.03\
G032.7492$-$00.0643 & 0.55 & $-$0.86 $\pm$ 0.09 & 4.81 & $-$7.62 $\pm$ 0.82 & 20.85 & 27.47\
G032.7966+00.1909 & 4.77 & 3.61 $\pm$ 0.16 & 42.07 & 31.82 $\pm$ 1.42 & 17.68 & 23.94\
G034.2544+00.1460 & 4.04 & 2.59 $\pm$ 0.17 & 35.60 & 22.83 $\pm$ 1.49 & 18.03 & 28.02\
G034.2571+00.1466 & 2.57 & 0.91 $\pm$ 0.15 & 22.65 & 8.06 $\pm$ 1.29 & 18.23 & 30.96\
G034.4032+00.2277 & 1.22 & $-$0.12 $\pm$ 0.11 & 10.76 & $-$1.08 $\pm$ 0.96 & 17.00 & 26.51\
G035.0524$-$00.5177 & 2.63 $\pm$ 0.38 & 1.68 $\pm$ 0.18 & 23.21 $\pm$ 3.33 & 14.77 $\pm$ 1.56 & 17.6 $\pm$ 3.71 & 21.03 $\pm$ 5.74\
G035.5781$-$00.0305 & 3.64 & 3.1 $\pm$ 0.24 & 32.09 & 27.36 $\pm$ 2.1 & 20.87 & 15.04\
G036.4057+00.0226 & 2.16 & 0.85 $\pm$ 0.13 & 19.05 & 7.5 $\pm$ 1.11 & 18.69 & 26.05\
G037.5457$-$00.1120 & 2.71 $\pm$ 0.23 & 2.55 $\pm$ 0.20 & 23.91 $\pm$ 2.02 & 22.5 $\pm$ 1.77 & 23.29 $\pm$ 2.15 & 9.69 $\pm$ 4.02\
G037.8683$-$00.6008 & 3.78 $\pm$ 0.71 & 2.7 $\pm$ 0.25 & 33.31 $\pm$ 6.25 & 23.78 $\pm$ 2.18 & 13.12 $\pm$ 7.07 & 22.8 $\pm$ 10.48\
G039.8824$-$00.3460 & 3.84 & 4.39 $\pm$ 0.61 & 33.87 & 38.69 $\pm$ 5.4 & 17.17 & $-$0.37\
G042.1090$-$00.4469 & 1.31 $\pm$ 0.29 & 0.02 $\pm$ 0.11 & 11.55 $\pm$ 2.59 & 0.19 $\pm$ 0.99 & 21.92 $\pm$ 2.94 & 25.75 $\pm$ 4.44\
G042.4345$-$00.2605 & 2.73 & 0.91 $\pm$ 0.08 & 24.07 & 8.01 $\pm$ 0.7 & 22.20 & 33.33\
G043.1651$-$00.0283 & 6.08 $\pm$ 0.18 & 4.51 $\pm$ 0.13 & 53.57 $\pm$ 1.58 & 39.74 $\pm$ 1.17 & & 29.73 $\pm$ 0.5\
G043.1677+00.0196 & 3.04 & 3.44 $\pm$ 0.56 & 26.78 & 30.31 $\pm$ 4.92 & 19.75 & 1.72\
G043.1763+00.0248 & 2.73 $\pm$ 0.24 & 1.95 $\pm$ 0.13 & 24.07 $\pm$ 2.15 & 17.16 $\pm$ 1.16 & 25.03 $\pm$ 2.19 & 18.57 $\pm$ 3.42\
G043.7960$-$00.1286 & 1.11 & 0.4 $\pm$ 0.16 & 9.78 & 3.57 $\pm$ 1.43 & & 17.43\
G043.8894$-$00.7840 & 3.91 $\pm$ 0.31 & 2.83 $\pm$ 0.12 & 34.49 $\pm$ 2.75 & 24.92 $\pm$ 1.02 & 19.41 $\pm$ 2.95 & 22.85 $\pm$ 4.33\
G043.9675+00.9939 & 3.23 & 1.83 $\pm$ 0.20 & 28.44 & 16.14 $\pm$ 1.73 & 18.49 & 27.25\
G044.3103+00.0410 & 0.76 & 0.03 $\pm$ 0.18 & 6.68 & 0.27 $\pm$ 1.59 & 19.04 & 17.76\
G045.0694+00.1323 & $-$12.32 & 2.38 $\pm$ 0.08 & & 20.94 $\pm$ 0.68 & &\
G045.1223+00.1321 & 3.45 $\pm$ 0.13 & 1.95 $\pm$ 0.06 & 30.39 $\pm$ 1.16 & 17.15 $\pm$ 0.51 & 18.32 $\pm$ 1.26 & 28.77 $\pm$ 1.33\
G045.1242+00.1356 & 0.74 & 0.9 $\pm$ 0.15 & 6.53 & 7.93 $\pm$ 1.31 & 32.30 & 5.15\
G045.4545+00.0591 & 2.29 $\pm$ 0.10 & 2.21 $\pm$ 0.11 & 20.2 $\pm$ 0.85 & 19.45 $\pm$ 0.94 & 16.52 $\pm$ 0.92 & 8.62 $\pm$ 1.59\
G045.4559+00.0613 & 1.98 & 0.6 $\pm$ 0.15 & 17.47 & 5.3 $\pm$ 1.3 & 18.99 & 27.05\
G045.4656+00.0452 & 3.25 & 1.95 $\pm$ 0.15 & 28.62 & 17.2 $\pm$ 1.31 & 18.49 & 25.83\
G048.6099+00.0270 & 2.32 & 2.1 $\pm$ 0.75 & 20.47 & 18.54 $\pm$ 6.64 & 23.69 & 10.51\
G049.3666$-$00.3010 & 3.64 & 3.73 $\pm$ 0.48 & 32.05 & 32.92 $\pm$ 4.21 & 17.54 & 6.01\
G049.4905$-$00.3688 & 3.3 $\pm$ 0.10 & 2.33 $\pm$ 0.07 & 29.11 $\pm$ 0.87 & 20.53 $\pm$ 0.6 & 33.7 $\pm$ 0.88 & 21.26 $\pm$ 0.44\
G050.3152+00.6762 & 1.92 & 2.22 $\pm$ 0.20 & 16.90 & 19.53 $\pm$ 1.78 & 31.33 & 3.17\
G050.3157+00.6747 & 1.05 $\pm$ 0.21 & 0.75 $\pm$ 0.15 & 9.22 $\pm$ 1.87 & 6.57 $\pm$ 1.29 & 25.98 $\pm$ 4.91 & 11.69 $\pm$ 3.62\
G052.7533+00.3340 & 3.24 & 2.82 $\pm$ 0.24 & 28.59 & 24.88 $\pm$ 2.08 & 18.04 & 13.40\
G053.9589+00.0320 & 2.84 & 1.72 $\pm$ 0.17 & 25.05 & 15.12 $\pm$ 1.48 & 17.40 & 23.43\
G058.7739+00.6457 & $-$0.91 $\pm$ 0.24 & $-$1.39 $\pm$ 0.15 & $-$8.03 $\pm$ 2.14 & $-$12.28 $\pm$ 1.33 & 25.82 $\pm$ 2.36 & 14.27 $\pm$ 3.67\
G060.8842$-$00.1286 & 1.77 & 0.15 $\pm$ 0.14 & 15.64 & 1.29 $\pm$ 1.24 & & 30.56\
G061.7207+00.8630 & 3.64 & 1.38 $\pm$ 0.06 & 32.04 & 12.14 $\pm$ 0.53 & 17.26 & 39.48\
Distances and physical properties {#appendixE}
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& & & & & & & & &\
\
G010.3009$-$00.1477 & 2.4 & 6.15 & 0.061 & 47.5 & 7.03 & 4.2 & 3.02 $\pm$ 0.04 & 0.11 &\
G010.3204$-$00.2328 & 12.6$^\dagger$ & 4.5 & 0.18 & 47.7 & 6.19 & 3.54 & 2.18 $\pm$ 0.07 & 0.015 & 4.51\
G010.3204$-$00.2586 & 3.5 & 5.1 & < 0.028 & 46.3 & 6.51 & 4.12 & 2.5 $\pm$ 0.06 & 0.032 & 4.18\
G010.4724+00.0275 & 11.0 & 3.06 & 0.091 & 47.6 & 6.58 & 3.84 & 2.57 $\pm$ 0.05 & 0.038 & 5.84\
G010.4736+00.0274 & 11.0 & 3.06 & 0.075 & 47.3 & 6.37 & 3.76 & 2.36 $\pm$ 0.05 & 0.023 &\
G010.6218$-$00.3848 & 2.4 & 6.16 & 0.02 & 46.3 & 6.55 & 4.15 & 2.54 $\pm$ 0.05 & 0.035 &\
G010.6223$-$00.3788 & 2.4 & 6.16 & 0.065 & 47.4 & 6.86 & 4.11 & 2.85 $\pm$ 0.04 & 0.074 &\
G010.6234$-$00.3837 & 2.4 & 6.16 & 0.051 & 48.0 & 7.66 & 4.55 & 3.65 $\pm$ 0.04 & 0.59 & 4.95\
G010.6240$-$00.3813 & 2.4 & 6.16 & 0.016 & 46.6 & 6.92 & 4.36 & 2.91 $\pm$ 0.04 & 0.084 &\
G010.6297$-$00.3380 & 4.95$^\ddagger$ & 3.75 & 0.074 & 46.8 & 6.05 & 3.65 & 2.03 $\pm$ 0.08 & 0.011 &\
G010.8519$-$00.4407 & & & & & & & 2.21 $\pm$ 0.061 & 0.016 &\
G010.9584+00.0221 & 13.7 & 5.59 & 0.11 & 48.5 & 7.31 & 4.16 & 3.3 $\pm$ 0.04 & 0.22 & 5.27\
G010.9656+00.0089 & 2.7 & 5.87 & 0.06 & 46.5 & 6.04 & 3.7 & 2.03 $\pm$ 0.05 & 0.011 & 3.39\
G011.0328+00.0274 & 2.6 & 5.97 & 0.015 & 45.5 & 5.9 & 3.85 & 1.89 $\pm$ 0.09 & 0.0078 & 2.87\
G011.1104$-$00.3985 & 16.8 & 8.62 & 0.67 & 48.9 & 6.34 & 3.34 & 2.33 $\pm$ 0.04 & 0.022 & 5.65\
G011.1712$-$00.0662 & 12.8$^\dagger$ & 4.76 & 0.66 & 48.2 & 5.65 & 3.0 & 1.64 $\pm$ 0.05 & 0.0043 & 4.43\
G011.9032$-$00.1407 & 4.1 & 4.57 & 0.056 & 46.6 & 6.1 & 3.73 & 2.09 $\pm$ 0.06 & 0.012 & 4.11\
G011.9039$-$00.1411 & 4.1 & 4.57 & 0.022 & 46.4 & 6.39 & 4.0 & 2.38 $\pm$ 0.05 & 0.024 &\
G011.9368$-$00.6158 & 4.0 & 4.66 & 0.11 & 48.2 & 7.22 & 4.17 & 3.21 $\pm$ 0.04 & 0.18 &\
G011.9446$-$00.0369 & 12.6 & 4.63 & 0.89 & 49.1 & 6.34 & 3.29 & 2.33 $\pm$ 0.04 & 0.022 & 5.01\
G011.9786$-$00.0973 & & & & & & & 1.85 $\pm$ 0.12 & 0.0071 &\
G012.1988$-$00.0345 & 11.9 & 4.02 & 0.13 & 47.9 & 6.64 & 3.82 & 2.63 $\pm$ 0.04 & 0.044 & 5.3\
G012.2081$-$00.1019 & 13.6 & 5.59 & 0.16 & 48.5 & 7.11 & 4.01 & 3.1 $\pm$ 0.04 & 0.13 & 5.87\
G012.4294$-$00.0479 & 13.9 & 5.89 & 0.15 & 47.9 & 6.49 & 3.7 & 2.48 $\pm$ 0.04 & 0.03 & 4.83\
G012.4317$-$01.1112 & 4.1 & 4.58 & 0.081 & 47.0 & 6.25 & 3.74 & 2.24 $\pm$ 0.06 & 0.018 &\
G012.8050$-$00.2007 & 2.1 & 6.47 & 0.16 & 48.7 & 7.38 & 4.17 & 3.37 $\pm$ 0.04 & 0.27 &\
G012.8131$-$00.1976 & 2.1 & 6.47 & 0.053 & 47.8 & 7.41 & 4.42 & 3.4 $\pm$ 0.04 & 0.29 &\
G012.9995$-$00.3583 & 1.9 & 6.66 & 0.025 & 45.8 & 6.03 & 3.88 & 2.02 $\pm$ 0.09 & 0.01 & 3.39\
G013.2099$-$00.1428 & 4.6 & 4.16 & 0.18 & 48.3 & 6.83 & 3.87 & 2.82 $\pm$ 0.04 & 0.069 & 4.14\
G013.3850+00.0684 & 1.9 & 6.67 & 0.18 & 47.3 & 5.92 & 3.42 & 1.91 $\pm$ 0.04 & 0.0081 & 3.55\
G013.8726+00.2818 & 4.4 & 4.36 & 0.33 & 48.4 & 6.48 & 3.57 & 2.47 $\pm$ 0.04 & 0.03 & 4.84\
G014.1046+00.0918 & & & & & & & 2.17 $\pm$ 0.055 & 0.015 &\
G014.1741+00.0245 & 14.7$^\dagger$ & 6.79 & 0.48 & 48.0 & 5.7 & 3.09 & 1.69 $\pm$ 0.07 & 0.0049 & 5.05\
G014.2460$-$00.0728 & 11.6 & 3.96 & 0.2 & 47.8 & 6.22 & 3.53 & 2.21 $\pm$ 0.05 & 0.016 &\
G014.5988+00.0198 & 2.8 & 5.83 & 0.015 & 45.5 & 5.81 & 3.79 & 1.8 $\pm$ 0.12 & 0.0063 &\
G014.7785$-$00.3328 & 13.1 & 5.34 & 0.11 & 47.5 & 6.24 & 3.62 & 2.23 $\pm$ 0.06 & 0.017 & 5.09\
G016.1448+00.0088 & 12.4 & 4.85 & < 0.096 & 47.3 & 6.46 & 3.83 & 2.45 $\pm$ 0.05 & 0.029 & 4.52\
G016.3913$-$00.1383 & 12.2$^\dagger$ & 4.7 & 0.69 & 48.2 & 5.66 & 3.0 & 1.64 $\pm$ 0.05 & 0.0044 & 4.25\
G016.9445$-$00.0738 & 17.0 & 9.21 & 0.26 & 49.1 & 7.34 & 4.03 & 3.33 $\pm$ 0.04 & 0.24 & 5.71\
G017.0299$-$00.0696 & 10.4 & 3.37 & 0.096 & 46.7 & 5.68 & 3.39 & 1.66 $\pm$ 0.10 & 0.0046 & 4.74\
G017.1141$-$00.1124 & 10.4 & 3.38 & 0.12 & 47.2 & 6.03 & 3.53 & 2.02 $\pm$ 0.06 & 0.011 & 4.27\
G017.5549+00.1654 & 14.1 & 6.52 & 0.11 & 47.1 & 5.87 & 3.43 & 1.85 $\pm$ 0.08 & 0.0072 & 4.77\
G017.9850+00.1266 & 13.7 & 6.2 & 0.21 & & 5.62 & 3.21 & 1.61 $\pm$ 0.1 & 0.01 &\
G018.1460$-$00.2839 & 4.3 & 4.61 & 0.49 & 48.2 & 5.89 & 3.19 & 1.88 $\pm$ 0.04 & 0.0077 & 4.1\
G018.3024$-$00.3910 & 3.0 & 5.73 & 0.21 & 48.0 & 6.48 & 3.66 & 2.47 $\pm$ 0.04 & 0.03 & 4.59\
G018.4433$-$00.0056 & 12.0 & 4.77 & 0.11 & 48.0 & 6.85 & 3.94 & 2.84 $\pm$ 0.04 & 0.071 & 4.89\
G018.4614$-$00.0038 & 12.1 & 4.85 & 0.13 & 48.7 & 7.35 & 4.16 & 3.34 $\pm$ 0.04 & 0.25 & 5.16\
G018.6654+00.0294 & 10.9 & 3.94 & < 0.092 & 46.8 & 5.97 & 3.59 & 1.96 $\pm$ 0.07 & 0.0092 &\
G018.7106+00.0002 & 2.6 & 6.09 & 0.018 & 46.8 & 7.11 & 4.44 & 3.1 $\pm$ 0.04 & 0.13 & 3.6\
G018.7612+00.2630 & 14.0 & 6.55 & < 0.12 & 48.0 & 6.91 & 4.0 & 2.89 $\pm$ 0.04 & 0.082 & 5.29\
G018.8250$-$00.4675 & 4.5 & 4.48 & 0.044 & 46.3 & 5.95 & 3.69 & 1.94 $\pm$ 0.09 & 0.0088 & 4.25\
G018.8338$-$00.3002 & 12.6 & 5.32 & 0.4 & 48.3 & 6.16 & 3.36 & 2.15 $\pm$ 0.04 & 0.014 & 5.42\
G019.0035+00.1280 & 11.3 & 4.28 & 0.13 & 46.9 & 5.63 & 3.31 & 1.62 $\pm$ 0.12 & 0.0042 & 4.72\
G019.0754$-$00.2874 & 4.6 & 4.42 & 0.19 & 47.9 & 6.42 & 3.66 & 2.41 $\pm$ 0.04 & 0.026 & 5.08\
G019.0767$-$00.2882 & 4.6 & 4.42 & 0.045 & 47.4 & 7.01 & 4.22 & 3 $\pm$ 0.04 & 0.1 &\
G019.4752+00.1728 & 14.1 & 6.71 & 0.23 & 47.8 & 6.12 & 3.45 & 2.11 $\pm$ 0.05 & 0.013 &\
G019.6087$-$00.2351 & 12.7 & 5.49 & 0.8 & 49.6 & 6.93 & 3.6 & 2.92 $\pm$ 0.04 & 0.086 & 5.01\
G019.6090$-$00.2313 & 12.7 & 5.49 & 0.22 & 48.6 & 6.95 & 3.88 & 2.94 $\pm$ 0.04 & 0.09 &\
G019.7281$-$00.1135 & 4.3 & 4.68 & 0.063 & 46.6 & 6.07 & 3.7 & 2.05 $\pm$ 0.05 & 0.011 & 3.84\
G019.7549$-$00.1282 & 9.3 & 3.15 & < 0.073 & 47.5 & 6.84 & 4.07 & 2.82 $\pm$ 0.04 & 0.069 & 5.47\
G020.0720$-$00.1421 & 12.6 & 5.46 & 0.32 & 48.5 & 6.56 & 3.61 & 2.55 $\pm$ 0.04 & 0.036 &\
G020.0797$-$00.1337 & 12.6 & 5.46 & 0.098 & 47.3 & 6.18 & 3.62 & 2.17 $\pm$ 0.06 & 0.015 &\
G020.0809$-$00.1362 & 12.6 & 5.46 & 0.16 & 48.9 & 7.45 & 4.18 & 3.44 $\pm$ 0.04 & 0.32 & 5.2\
G020.3633$-$00.0136 & 3.9 & 5.03 & 0.047 & 46.9 & 6.51 & 3.97 & 2.5 $\pm$ 0.05 & 0.032 & 3.47\
G020.7619$-$00.0646 & 11.8 & 4.89 & 0.11 & 47.1 & 5.92 & 3.47 & 1.91 $\pm$ 0.1 & 0.0081 & 4.65\
G020.9636$-$00.0744 & 13.2 & 6.08 & 0.13 & 47.2 & 5.96 & 3.47 & 1.95 $\pm$ 0.08 & 0.009 & 4.79\
G021.3571$-$00.1766 & 10.3 & 3.91 & 0.06 & 47.4 & 6.53 & 3.87 & 2.52 $\pm$ 0.04 & 0.034 & 5.45\
G021.3855$-$00.2541 & 10.3 & 3.91 & 0.08 & 48.0 & 7.06 & 4.1 & 3.05 $\pm$ 0.04 & 0.12 & 5.62\
G021.6034$-$00.1685 & 16.2 & 8.87 & 0.27 & 47.7 & 5.85 & 3.28 & 1.84 $\pm$ 0.07 & 0.007 & 4.41\
G021.8751+00.0075 & 13.8 & 6.71 & 0.76 & 49.0 & 6.33 & 3.31 & 2.32 $\pm$ 0.04 & 0.021 & 4.94\
G023.1974$-$00.0006 & 4.8 & 4.5 & 0.04 & 46.3 & 5.99 & 3.72 & 1.98 $\pm$ 0.07 & 0.0095 & 3.95\
G023.2654+00.0765 & 4.9 & 4.44 & 0.1 & 47.3 & 6.32 & 3.73 & 2.31 $\pm$ 0.05 & 0.021 & 3.99\
G023.4553$-$00.2010 & 5.8 & 3.93 & < 0.048 & 46.6 & 6.39 & 3.94 & 2.38 $\pm$ 0.05 & 0.024 & 4.24\
G023.4835+00.0964 & 5.2 & 4.27 & 0.038 & 46.3 & 5.95 & 3.7 & 1.94 $\pm$ 0.08 & 0.0088 & 4.04\
G023.7110+00.1705 & 6.5 & 3.65 & 0.14 & 47.9 & 6.68 & 3.84 & 2.67 $\pm$ 0.04 & 0.048 & 5.05\
G023.8618$-$00.1250 & 10.7 & 4.52 & 0.2 & 47.6 & 6.08 & 3.47 & 2.07 $\pm$ 0.07 & 0.012 & 4.81\
G023.8985+00.0647 & 12.6 & 5.93 & 0.23 & 47.8 & 6.11 & 3.45 & 2.1 $\pm$ 0.06 & 0.013 & 5.2\
G023.9564+00.1493 & 5.0 & 4.42 & 0.32 & 48.4 & 6.53 & 3.6 & 2.52 $\pm$ 0.04 & 0.033 & 5.28\
G024.1839+00.1199 & 7.8 & 3.48 & < 0.064 & 46.3 & 5.83 & 3.6 & 1.81 $\pm$ 0.10 & 0.0065 & 4.14\
G024.4698+00.4954 & 5.51$^\star$ & 4.17 & 0.064 & 46.9 & 6.26 & 3.78 & 2.25 $\pm$ 0.07 & 0.018 &\
G024.4721+00.4877 & 5.51$^\star$ & 4.17 & 0.096 & 47.2 & 6.27 & 3.72 & 2.26 $\pm$ 0.06 & 0.018 & 4.87\
G024.4736+00.4950 & 5.5$^\star$ & 4.17 & 0.14 & 47.4 & 6.2 & 3.6 & 2.19 $\pm$ 0.05 & 0.016 & 4.32\
G024.4921$-$00.0386 & 6.3 & 3.8 & 0.13 & 47.7 & 6.55 & 3.8 & 2.54 $\pm$ 0.04 & 0.035 & 4.44\
G024.5065$-$00.2224 & 5.8 & 4.02 & 0.17 & 47.8 & 6.43 & 3.68 & 2.42 $\pm$ 0.04 & 0.027 & 4.39\
G024.8497+00.0881 & 5.77$^\star$ & 4.07 & 0.092 & 46.8 & 5.87 & 3.52 & 1.86 $\pm$ 0.10 & 0.0073 & 5.14\
G025.3809$-$00.1815 & 2.7 & 6.17 & 0.11 & 47.5 & 6.51 & 3.82 & 2.5 $\pm$ 0.04 & 0.032 &\
G025.3824$-$00.1812 & 2.7 & 6.17 & 0.041 & 47.1 & 6.93 & 4.23 & 2.92 $\pm$ 0.04 & 0.087 & 4.82\
G025.3948+00.0332 & 17.1 & 10.1 & 0.36 & 48.9 & 6.84 & 3.72 & 2.83 $\pm$ 0.04 & 0.07 & 5.78\
G025.3970+00.5614 & 14.0 & 7.3 & 0.095 & 48.3 & 7.17 & 4.1 & 3.16 $\pm$ 0.04 & 0.15 & 5.29\
G025.3981$-$00.1411 & 5.6 & 4.2 & 0.23 & 48.8 & 7.16 & 3.99 & 3.15 $\pm$ 0.04 & 0.15 & 5.1\
G025.3983+00.5617 & 14.0 & 7.3 & 0.14 & 48.0 & 6.62 & 3.78 & 2.61 $\pm$ 0.04 & 0.041 &\
G025.3991$-$00.1366 & 5.6 & 4.2 & 0.081 & 46.9 & 6.12 & 3.67 & 2.11 $\pm$ 0.09 & 0.013 &\
G025.7157+00.0487 & 9.5 & 4.12 & 0.083 & 47.2 & 6.27 & 3.71 & 2.26 $\pm$ 0.06 & 0.018 & 5.41\
G025.8011$-$00.1568 & 5.6 & 4.23 & < 0.048 & 47.0 & 6.7 & 4.1 & 2.69 $\pm$ 0.04 & 0.05 & 4.13\
G026.0083+00.1369 & 13.8$^\star$ & 7.17 & 0.073 & 47.1 & 5.99 & 3.54 & 1.98 $\pm$ 0.07 & 0.0096 & 3.07\
G026.0916$-$00.0565 & 13.1 & 6.62 & 0.13 & 47.3 & 5.95 & 3.46 & 1.94 $\pm$ 0.09 & 0.0087 & 5.09\
G026.1094$-$00.0937 & 13.3 & 6.79 & < 0.1 & 46.9 & 5.98 & 3.57 & 1.97 $\pm$ 0.08 & 0.0093 & 4.93\
G026.5976$-$00.0236 & 13.4 & 6.94 & 0.14 & 48.1 & 6.7 & 3.82 & 2.69 $\pm$ 0.05 & 0.05 & 5.19\
G026.6089$-$00.2121 & 7.6 & 3.81 & 0.24 & 48.0 & 6.38 & 3.59 & 2.36 $\pm$ 0.05 & 0.023 & 4.02\
G026.8304$-$00.2067 & 11.9$^\star$ & 5.77 & 0.075 & 47.2 & 6.2 & 3.66 & 2.19 $\pm$ 0.06 & 0.015 & 4.19\
G027.1859$-$00.0816 & 13.2 & 6.85 & 0.09 & 47.5 & 6.38 & 3.72 & 2.37 $\pm$ 0.05 & 0.024 & 5.65\
G027.2800+00.1447 & 12.8 & 6.53 & 0.34 & 48.8 & 6.81 & 3.72 & 2.8 $\pm$ 0.04 & 0.066 & 5.1\
G027.3644$-$00.1657 & 8.0 & 3.93 & 0.066 & 47.5 & 6.77 & 4.0 & 2.76 $\pm$ 0.04 & 0.059 & 5.37\
G027.5637+00.0845 & 9.9 & 4.59 & 0.59 & 48.2 & 5.73 & 3.07 & 1.72 $\pm$ 0.05 & 0.0053 & 4.22\
G027.9352+00.2056 & 12.1 & 6.08 & < 0.094 & 46.8 & 5.94 & 3.57 & 1.93 $\pm$ 0.11 & 0.0085 & 5.16\
G028.1985$-$00.0503 & 5.8 & 4.36 & 0.076 & 47.6 & 6.87 & 4.06 & 2.86 $\pm$ 0.04 & 0.075 &\
G028.2003$-$00.0494 & 5.8 & 4.36 & 0.045 & 47.7 & 7.23 & 4.31 & 3.22 $\pm$ 0.04 & 0.18 & 5.73\
G028.2879$-$00.3641 & 11.6 & 5.76 & 0.25 & 48.8 & 7.12 & 3.94 & 3.11 $\pm$ 0.04 & 0.14 & 5.68\
G028.4518+00.0027 & 16.5 & 9.89 & 0.14 & 47.9 & 6.5 & 3.7 & 2.49 $\pm$ 0.04 & 0.031 & 5.03\
G028.5816+00.1447 & 16.5 & 9.91 & 0.41 & 48.0 & 5.85 & 3.2 & 1.84 $\pm$ 0.08 & 0.007 & 4.38\
G028.6082+00.0185 & 7.5 & 4.07 & 0.12 & 48.0 & 6.91 & 3.98 & 2.89 $\pm$ 0.04 & 0.082 & 5.06\
G028.6523+00.0273 & 7.5 & 4.08 & 0.2 & 48.1 & 6.55 & 3.71 & 2.54 $\pm$ 0.04 & 0.036 & 5.02\
G028.6869+00.1770 & 9.8 & 4.71 & 0.24 & 47.9 & 6.27 & 3.52 & 2.26 $\pm$ 0.05 & 0.018 & 4.45\
G029.7704+00.2189 & 9.84$^\star$ & 4.89 & 0.31 & 47.6 & 5.76 & 3.22 & 1.75 $\pm$ 0.08 & 0.0057 &\
G029.9559$-$00.0168 & 7.4 & 4.24 & 0.34 & 49.2 & 7.23 & 3.93 & 3.22 $\pm$ 0.04 & 0.18 & 5.54\
G030.0096$-$00.2734 & 7.4 & 4.25 & < 0.058 & 46.3 & 5.94 & 3.68 & 1.93 $\pm$ 0.10 & 0.0085 & 4.43\
G030.2527+00.0540 & 10.2 & 5.15 & 0.28 & 48.0 & 6.16 & 3.44 & 2.15 $\pm$ 0.05 & 0.014 & 4.39\
G030.5313+00.0205 & 11.4 & 5.94 & 0.29 & 48.0 & 6.15 & 3.42 & 2.13 $\pm$ 0.05 & 0.014 & 4.36\
G030.5353+00.0204 & 11.4 & 5.94 & 0.34 & 48.9 & 6.95 & 3.79 & 2.94 $\pm$ 0.04 & 0.092 & 5.58\
G030.5887$-$00.0428 & 11.8 & 6.23 & < 0.1 & 48.1 & 7.16 & 4.17 & 3.15 $\pm$ 0.04 & 0.15 & 5.4\
G030.6881$-$00.0718 & 4.9$^\ddagger$ & 4.96 & 0.25 & 48.0 & 6.3 & 3.54 & 2.29 $\pm$ 0.04 & 0.02 &\
G030.7197$-$00.0829 & 4.9 & 4.96 & 0.1 & 48.3 & 7.36 & 4.25 & 3.35 $\pm$ 0.04 & 0.25 &\
G030.7532$-$00.0511 & 4.9 & 4.97 & 0.069 & 47.8 & 7.15 & 4.22 & 3.13 $\pm$ 0.04 & 0.15 & 4.85\
G030.7579+00.2042 & 7.22$^\star$ & 4.35 & 0.14 & 47.1 & 5.88 & 3.45 & 1.87 $\pm$ 0.10 & 0.0075 & 4.28\
G030.7661$-$00.0348 & 7.22$^\star$ & 4.35 & 0.1 & 47.6 & 6.61 & 3.87 & 2.6 $\pm$ 0.08 & 0.041 &\
G030.8662+00.1143 & 11.9 & 6.34 & 0.16 & 48.6 & 7.23 & 4.08 & 3.22 $\pm$ 0.04 & 0.18 & 5.61\
G030.9581+00.0869 & 11.8 & 6.28 & 0.16 & 47.5 & 6.1 & 3.51 & 2.09 $\pm$ 0.08 & 0.012 & 4.78\
G031.0495+00.4697 & 12.2 & 6.59 & < 0.1 & 47.3 & 6.33 & 3.74 & 2.32 $\pm$ 0.05 & 0.021 & 4.23\
G031.0595+00.0922 & 13.4 & 7.53 & 0.084 & 47.3 & 6.16 & 3.61 & 2.15 $\pm$ 0.06 & 0.014 & 4.55\
G031.1590+00.0465 & 2.7 & 6.35 & 0.018 & 45.7 & 5.94 & 3.85 & 1.93 $\pm$ 0.08 & 0.0085 &\
G031.1596+00.0448 & 2.7 & 6.35 & < 0.022 & 46.2 & 6.62 & 4.23 & 2.61 $\pm$ 0.04 & 0.042 & 3.73\
G031.2420$-$00.1106 & 12.7$^\star$ & 7.03 & 0.48 & 48.6 & 6.39 & 3.44 & 2.38 $\pm$ 0.04 & 0.024 &\
G031.2435$-$00.1103 & 13.0 & 7.23 & 0.14 & 48.7 & 7.39 & 4.17 & 3.38 $\pm$ 0.04 & 0.27 & 5.59\
G031.2448$-$00.1132 & 13.0 & 7.23 & 0.18 & 47.8 & 6.28 & 3.58 & 2.27 $\pm$ 0.05 & 0.019 &\
G031.2801+00.0632 & 7.1 & 4.42 & 0.32 & 48.1 & 6.19 & 3.43 & 2.18 $\pm$ 0.04 & 0.015 & 4.75\
G031.3959$-$00.2570 & 9.1 & 4.8 & 0.44 & 47.8 & 5.61 & 3.07 & 1.6 $\pm$ 0.06 & 0.0039 & 5.67\
G031.4130+00.3065 & 5.6 & 4.73 & 0.25 & 48.4 & 6.74 & 3.75 & 2.73 $\pm$ 0.04 & 0.055 & 5.25\
G031.5815+00.0744 & 7.2 & 4.45 & 0.066 & 46.8 & 6.1 & 3.68 & 2.09 $\pm$ 0.06 & 0.012 & 5.22\
G032.0297+00.0491 & 4.9$^\ddagger$ & 5.06 & 0.095 & 46.8 & 5.87 & 3.52 & 1.86 $\pm$ 0.07 & 0.0072 & 4.03\
G032.1502+00.1329 & 7.2 & 4.52 & 0.43 & 48.4 & 6.24 & 3.39 & 2.23 $\pm$ 0.05 & 0.017 & 4.79\
G032.2730$-$00.2258 & 12.8 & 7.22 & 0.6 & 48.7 & 6.21 & 3.3 & 2.19 $\pm$ 0.05 & 0.016 & 4.59\
G032.4727+00.2036 & 11.1 & 6.02 & 0.091 & 48.0 & 6.98 & 4.04 & 2.97 $\pm$ 0.04 & 0.098 & 5.6\
G032.7398+00.1940 & 12.9 & 7.36 & < 0.099 & 46.7 & 5.83 & 3.51 & 1.82 $\pm$ 0.11 & 0.0066 & 4.08\
G032.7492$-$00.0643 & 11.7 & 6.47 & 0.085 & 47.2 & 6.16 & 3.63 & 2.15 $\pm$ 0.06 & 0.014 & 4.49\
G032.7966+00.1909 & 13.3 & 7.69 & 0.64 & 49.7 & 7.19 & 3.78 & 3.18 $\pm$ 0.04 & 0.17 & 5.05\
G032.7982+00.1937 & 13.3 & 7.69 & 0.084 & 47.4 & 6.34 & 3.71 & 2.33 $\pm$ 0.05 & 0.022 &\
G032.9273+00.6060 & 18.6 & 12.4 & 0.6 & 48.9 & 6.49 & 3.44 & 2.48 $\pm$ 0.05 & 0.031 & 5.54\
G033.1328$-$00.0923 & 9.4 & 5.18 & 0.17 & 48.5 & 7.07 & 3.99 & 3.06 $\pm$ 0.04 & 0.12 & 5.28\
G033.4163$-$00.0036 & 9.4 & 5.22 & 0.39 & 47.8 & 5.69 & 3.13 & 1.68 $\pm$ 0.05 & 0.0048 & 4.37\
G033.8100$-$00.1864 & 11.1 & 6.22 & 0.086 & 48.1 & 7.05 & 4.08 & 3.04 $\pm$ 0.04 & 0.12 & 5.71\
G033.9145+00.1105 & 6.7 & 4.76 & 0.32 & 48.5 & 6.61 & 3.64 & 2.6 $\pm$ 0.05 & 0.041 & 5.88\
G034.0901+00.4365 & 11.4 & 6.46 & 0.11 & 47.1 & 5.92 & 3.49 & 1.91 $\pm$ 0.09 & 0.0082 & 4.36\
G034.1978$-$00.5912 & 10.3 & 5.79 & 0.14 & 47.0 & 5.7 & 3.33 & 1.69 $\pm$ 0.12 & 0.0049 & 4.49\
G034.2544+00.1460 & 3.58$^\star$ & 5.9 & 0.099 & 47.6 & 6.71 & 3.94 & 2.7 $\pm$ 0.05 & 0.051 &\
G034.2571+00.1466 & 3.58$^\star$ & 5.9 & 0.033 & 46.7 & 6.61 & 4.08 & 2.6 $\pm$ 0.09 & 0.041 &\
G034.2572+00.1535 & 2.1 & 6.87 & 0.057 & 47.8 & 7.43 & 4.42 & 3.42 $\pm$ 0.04 & 0.3 &\
G034.2573+00.1523 & 2.1 & 6.87 & < 0.015 & 46.3 & 7.05 & 4.52 & 3.04 $\pm$ 0.05 & 0.12 &\
G034.2581+00.1533 & 2.1 & 6.87 & < 0.015 & 46.2 & 6.9 & 4.45 & 2.89 $\pm$ 0.05 & 0.081 &\
G034.4032+00.2277 & 2.1 & 6.87 & < 0.018 & 45.5 & 6.17 & 4.05 & 2.16 $\pm$ 0.06 & 0.014 & 3.82\
G034.5920+00.2434 & 16.4 & 10.6 & 0.18 & 47.7 & 6.14 & 3.49 & 2.13 $\pm$ 0.05 & 0.013 & 4.35\
G035.0242+00.3502 & 10.4 & 5.97 & < 0.078 & 47.0 & 6.38 & 3.83 & 2.37 $\pm$ 0.05 & 0.024 & 5.6\
G035.0524$-$00.5177 & 10.6$^\ddagger$ & 6.07 & 0.17 & 47.8 & 6.4 & 3.65 & 2.39 $\pm$ 0.05 & 0.025 & 4.97\
G035.4570$-$00.1791 & 9.72$^\star$ & 5.67 & < 0.079 & 46.8 & 6.13 & 3.7 & 2.12 $\pm$ 0.07 & 0.013 & 4.31\
G035.4669+00.1394 & 8.8 & 5.28 & 0.21 & 48.3 & 6.78 & 3.81 & 2.77 $\pm$ 0.04 & 0.061 & 5.7\
G035.5781$-$00.0305 & 10.4 & 6.05 & 0.1 & 48.3 & 7.17 & 4.12 & 3.16 $\pm$ 0.04 & 0.15 & 5.59\
G036.4057+00.0226 & 3.8 & 5.89 & 0.02 & 46.5 & 6.51 & 4.08 & 2.5 $\pm$ 0.04 & 0.032 & 4.28\
G036.4062+00.0221 & 3.8 & 5.89 & < 0.03 & 46.1 & 6.25 & 3.98 & 2.24 $\pm$ 0.06 & 0.017 &\
G037.5457$-$00.1120 & 10.0 & 6.12 & 0.39 & 48.6 & 6.48 & 3.53 & 2.47 $\pm$ 0.04 & 0.03 & 5.5\
G037.7347$-$00.1128 & 10.3 & 6.31 & < 0.088 & 47.2 & 6.42 & 3.83 & 2.41 $\pm$ 0.05 & 0.026 & 5.01\
G037.7562+00.5605 & 12.2$^\star$ & 7.55 & 0.41 & 47.7 & 5.56 & 3.06 & 1.55 $\pm$ 0.10 & 0.0036 & 4.5\
G037.8197+00.4140 & 12.2 & 7.57 & 0.24 & 47.5 & 5.86 & 3.32 & 1.85 $\pm$ 0.07 & 0.007 &\
G037.8209+00.4125 & 12.2 & 7.57 & 0.095 & 47.4 & 6.32 & 3.69 & 2.3 $\pm$ 0.06 & 0.02 & 5.17\
G037.8683$-$00.6008 & 10.0 & 6.17 & 0.24 & 48.3 & 6.59 & 3.69 & 2.58 $\pm$ 0.04 & 0.039 & 5.76\
G037.8731$-$00.3996 & 9.4 & 5.87 & 0.4 & 49.3 & 7.21 & 3.89 & 3.2 $\pm$ 0.04 & 0.17 & 5.73\
G037.9723$-$00.0965 & 9.74$^\star$ & 6.05 & 0.09 & 47.3 & 6.24 & 3.68 & 2.23 $\pm$ 0.06 & 0.017 & 3.84\
G038.5493+00.1646 & 2.07$^\ddagger$ & 7.0 & 0.081 & 46.5 & 5.81 & 3.53 & 1.8 $\pm$ 0.06 & 0.0063 &\
G038.6465$-$00.2260 & 4.7 & 5.65 & 0.046 & 46.4 & 5.96 & 3.69 & 1.95 $\pm$ 0.08 & 0.009 & 3.61\
G038.6529+00.0875 & 17.1 & 11.7 & < 0.13 & 47.3 & 6.19 & 3.62 & 2.18 $\pm$ 0.06 & 0.015 & 4.9\
G038.6934$-$00.4524 & 9.9 & 6.24 & 0.11 & 47.2 & 6.14 & 3.6 & 2.13 $\pm$ 0.06 & 0.014 & 4.22\
G038.8756+00.3080 & 15.9 & 10.7 & 0.25 & 48.9 & 7.09 & 3.91 & 3.08 $\pm$ 0.04 & 0.13 & 5.43\
G039.1956+00.2255 & 15.9 & 10.8 & 0.093 & 48.2 & 6.92 & 3.96 & 2.9 $\pm$ 0.04 & 0.084 & 5.49\
G039.8824$-$00.3460 & 9.1 & 6.03 & 0.15 & 48.3 & 6.98 & 3.97 & 2.97 $\pm$ 0.04 & 0.098 & 4.63\
G040.4251+00.7002 & 12.1 & 7.88 & 0.15 & 47.2 & 5.81 & 3.37 & 1.8 $\pm$ 0.09 & 0.0063 & 5.26\
G041.7419+00.0973 & 11.8 & 7.86 & 0.31 & 48.5 & 6.56 & 3.62 & 2.55 $\pm$ 0.05 & 0.036 & 4.79\
G042.1090$-$00.4469 & 8.8 & 6.22 & 0.047 & 47.0 & 6.34 & 3.81 & 2.33 $\pm$ 0.05 & 0.022 & 4.37\
G042.4345$-$00.2605 & 5.1 & 5.85 & 0.082 & 47.3 & 6.5 & 3.86 & 2.49 $\pm$ 0.05 & 0.032 & 5.21\
G043.1460+00.0139 & 11.1 & 7.6 & 0.32 & 48.9 & 6.95 & 3.8 & 2.94 $\pm$ 0.04 & 0.092 &\
G043.1489+00.0130 & 11.1 & 7.6 & < 0.089 & 47.7 & 6.91 & 4.07 & 2.9 $\pm$ 0.06 & 0.083 & 5.09\
G043.1520+00.0115 & 11.1 & 7.6 & 0.11 & 48.5 & 7.36 & 4.19 & 3.34 $\pm$ 0.05 & 0.25 &\
G043.1651$-$00.0283 & 11.1 & 7.6 & 0.51 & 49.5 & 7.17 & 3.82 & 3.16 $\pm$ 0.04 & 0.16 & 5.47\
G043.1652+00.0129 & 11.1 & 7.6 & 0.075 & 48.2 & 7.27 & 4.2 & 3.26 $\pm$ 0.05 & 0.2 &\
G043.1657+00.0116 & 11.1 & 7.6 & 0.059 & 48.0 & 7.16 & 4.17 & 3.15 $\pm$ 0.05 & 0.15 &\
G043.1665+00.0106 & 11.1 & 7.6 & 0.18 & 49.2 & 7.7 & 4.29 & 3.69 $\pm$ 0.04 & 0.68 & 5.68\
G043.1674+00.0128 & 11.1 & 7.6 & < 0.093 & 47.9 & 7.1 & 4.15 & 3.09 $\pm$ 0.05 & 0.13 &\
G043.1677+00.0196 & 11.1 & 7.6 & 0.21 & 48.1 & 6.53 & 3.68 & 2.51 $\pm$ 0.07 & 0.033 &\
G043.1684+00.0087 & 11.1 & 7.6 & 0.097 & 48.3 & 7.24 & 4.16 & 3.23 $\pm$ 0.05 & 0.19 &\
G043.1684+00.0124 & 11.1 & 7.6 & < 0.081 & 47.8 & 7.15 & 4.21 & 3.14 $\pm$ 0.05 & 0.15 &\
G043.1699+00.0115 & 11.1 & 7.6 & 0.081 & 47.7 & 6.68 & 3.9 & 2.67 $\pm$ 0.08 & 0.048 &\
G043.1701+00.0078 & 11.1 & 7.61 & 0.4 & 49.1 & 6.99 & 3.78 & 2.98 $\pm$ 0.04 & 0.1 &\
G043.1706$-$00.0003 & 11.1 & 7.61 & 0.054 & 48.3 & 7.41 & 4.3 & 3.4 $\pm$ 0.04 & 0.29 & 5.84\
G043.1716+00.0001 & 11.1 & 7.61 & 0.12 & 47.8 & 6.56 & 3.79 & 2.55 $\pm$ 0.11 & 0.036 &\
G043.1720+00.0080 & 11.1 & 7.61 & 0.11 & 48.0 & 6.86 & 3.94 & 2.85 $\pm$ 0.05 & 0.073 &\
G043.1763+00.0248 & 11.1 & 7.61 & 0.23 & 48.2 & 6.59 & 3.69 & 2.58 $\pm$ 0.06 & 0.039 & 5.24\
G043.1778$-$00.5181 & 8.2 & 6.15 & 0.28 & 48.0 & 6.25 & 3.49 & 2.24 $\pm$ 0.05 & 0.018 & 5.17\
G043.2371$-$00.0453 & 11.9 & 8.15 & 0.2 & 48.4 & 6.8 & 3.82 & 2.79 $\pm$ 0.05 & 0.063 & 5.49\
G043.3064$-$00.2114 & 4.4 & 6.1 & 0.043 & 46.5 & 6.2 & 3.82 & 2.19 $\pm$ 0.07 & 0.015 & 5.15\
G043.7954$-$00.1274 & 9.2 & 6.63 & 0.045 & 47.3 & 6.57 & 3.92 & 2.56 $\pm$ 0.05 & 0.037 & 5.51\
G043.7960$-$00.1286 & 9.2 & 6.63 & 0.08 & 46.9 & 5.94 & 3.54 & 1.93 $\pm$ 0.09 & 0.0085 &\
G043.8894$-$00.7840 & 4.4 & 6.14 & 0.11 & 48.0 & 6.96 & 4.04 & 2.95 $\pm$ 0.04 & 0.094 & 4.23\
G043.9675+00.9939 & 14.1 & 9.93 & < 0.12 & 47.9 & 6.85 & 3.98 & 2.84 $\pm$ 0.04 & 0.072 & 5.28\
G044.3103+00.0410 & 8.0 & 6.24 & < 0.065 & 46.5 & 5.99 & 3.68 & 1.98 $\pm$ 0.08 & 0.0096 & 5.42\
G044.4228+00.5377 & 17.9 & 13.2 & < 0.14 & 47.1 & 5.92 & 3.48 & 1.91 $\pm$ 0.09 & 0.0082 & 4.51\
G045.0694+00.1323 & 4.4 & 6.23 & 0.028 & 46.9 & 6.78 & 4.17 & 2.77 $\pm$ 0.04 & 0.061 &\
G045.0712+00.1321 & 4.4 & 6.23 & 0.026 & 47.4 & 7.31 & 4.44 & 3.3 $\pm$ 0.04 & 0.22 & 5.64\
G045.1223+00.1321 & 4.4 & 6.23 & 0.16 & 48.7 & 7.43 & 4.2 & 3.42 $\pm$ 0.04 & 0.3 & 5.31\
G045.1242+00.1356 & 4.4 & 6.23 & 0.098 & 47.0 & 6.13 & 3.65 & 2.12 $\pm$ 0.08 & 0.013 &\
G045.4545+00.0591 & 7.2 & 6.18 & 0.26 & 48.7 & 6.95 & 3.85 & 2.94 $\pm$ 0.04 & 0.091 & 5.63\
G045.4559+00.0613 & 7.2 & 6.18 & 0.1 & 47.4 & 6.36 & 3.73 & 2.35 $\pm$ 0.07 & 0.022 &\
G045.4656+00.0452 & 6.0 & 6.06 & < 0.049 & 47.3 & 7.03 & 4.26 & 3.02 $\pm$ 0.04 & 0.11 & 5.12\
G045.5431$-$00.0073 & 5.89$^\star$ & 6.07 & 0.17 & 47.2 & 5.84 & 3.39 & 1.83 $\pm$ 0.07 & 0.0068 & 3.96\
G048.6099+00.0270 & 10.8$^\ddagger$ & 8.18 & 0.36 & 48.1 & 6.12 & 3.37 & 2.11 $\pm$ 0.05 & 0.013 & 5.09\
G048.9296$-$00.2793 & 5.4 & 6.41 & 0.16 & 47.7 & 6.39 & 3.68 & 2.38 $\pm$ 0.04 & 0.024 & 4.36\
G048.9901$-$00.2988 & 5.4 & 6.42 & < 0.046 & 46.3 & 6.08 & 3.8 & 2.07 $\pm$ 0.10 & 0.012 & 4.93\
G049.2679$-$00.3374 & 5.4 & 6.44 & 0.16 & 47.4 & 6.12 & 3.55 & 2.11 $\pm$ 0.07 & 0.013 & 4.18\
G049.3666$-$00.3010 & 5.4 & 6.45 & 0.089 & 47.7 & 6.82 & 4.01 & 2.81 $\pm$ 0.11 & 0.067 &\
G049.4640$-$00.3511 & 5.46$^\star$ & 6.46 & 0.15 & 48.3 & 7.02 & 4.01 & 3.01 $\pm$ 0.06 & 0.11 &\
G049.4891$-$00.3763 & 5.46$^\star$ & 6.46 & 0.071 & 47.8 & 7.06 & 4.17 & 3.05 $\pm$ 0.07 & 0.12 & 5.67\
G049.4905$-$00.3688 & 5.4 & 6.46 & 0.15 & 49.0 & 7.76 & 4.38 & 3.75 $\pm$ 0.04 & 0.83 & 5.36\
G050.3152+00.6762 & 1.9 & 7.43 & 0.014 & 46.4 & 6.96 & 4.43 & 2.95 $\pm$ 0.04 & 0.094 & 3.93\
G050.3157+00.6747 & 8.58$^\star$ & 7.26 & 0.4 & 47.7 & 5.58 & 3.07 & 1.57 $\pm$ 0.05 & 0.0037 &\
G051.6785+00.7193 & 10.5 & 8.48 & < 0.091 & 47.3 & 6.55 & 3.88 & 2.54 $\pm$ 0.04 & 0.035 & 5.28\
G052.7533+00.3340 & 9.3 & 7.94 & 0.38 & 48.5 & 6.41 & 3.5 & 2.4 $\pm$ 0.04 & 0.026 & 4.54\
G053.9589+00.0320 & 5.0 & 6.87 & 0.044 & 47.0 & 6.63 & 4.03 & 2.62 $\pm$ 0.04 & 0.043 & 4.7\
G058.7739+00.6457 & 4.4 & 7.27 & 0.049 & 45.9 & 5.51 & 3.46 & 1.50 $\pm$ 0.12 & 0.0031 & 4.42\
G059.6027+00.9118 & 4.2 & 7.33 & < 0.034 & 47.0 & 7.08 & 4.36 & 3.07 $\pm$ 0.04 & 0.12 & 4.21\
G060.8842$-$00.1286 & 2.16$^\ddagger$ & 7.68 & < 0.016 & 45.9 & 6.62 & 4.3 & 2.61 $\pm$ 0.05 & 0.041 & 4.23\
G061.4763+00.0892 & 4.01$^\star$ & 7.47 & 0.12 & 48.0 & 6.93 & 4.0 & 2.92 $\pm$ 0.04 & 0.086 & 5.49\
G061.4770+00.0891 & 4.01$^\star$ & 7.47 & < 0.03 & 47.0 & 7.16 & 4.43 & 3.15 $\pm$ 0.04 & 0.15 &\
G061.7207+00.8630 & 14.0$^\ddagger$ & 12.5 & 0.13 & 48.3 & 6.94 & 3.95 & 2.92 $\pm$ 0.04 & 0.088 & 5.59\
[^1]: E-mail: [email protected]
[^2]: <http://cornish.leeds.ac.uk/public/index.php>
[^3]: The same method was tested on the CORNISH images, in good agreement with the catalogued values.
[^4]: <http://rms.leeds.ac.uk/cgi-bin/public/RMS_DATABASE.cgi>
[^5]: <http://www.bu.edu/galacticring/>
[^6]: All sources were detected at at least 9 times the median background value. Errors were computed following [@masci:2009] and include prior (noise-model) and derived uncertainties.
[^7]: G023.9564+00.1493 has a visible nebula in the near-IR, but photometric issues due to image quality lead to unreliable flux values, and the source is not included in the final results table.
[^8]: <https://vvvsurvey.org/>
[^9]: The x-ray – derived value is the same also for G025.3809$-$00.1815, as the two neighbouring sources are not resolved by $XMM-Newton$.
|
Introduction
============
Understanding dynamical principles and mechanisms behind the control of activity, signal and information processing that occur in neurobiological networks is hardly possible without numerical studies of collective dynamics of large networks of neurons. These simulations need to take into account the complex architecture of couplings among individual neurons that is suggested by data from biological experiments. One of the complicating factors in understanding the simulation results is the complexity of temporal behavior of individual biological neurons. This complexity is due to the large number of ionic currents involved in the nonlinear dynamics of neurons. As a result, realistic channel-based models proposed for a single neuron is usually a system of many nonlinear differential equations (see, for example [@HH52; @Chay85; @Chay90; @Golovasch92; @Gollomb93] and the review of the models in [@abarreview96]). The strong nonlinearity and high dimensionality of the phase space is a significant obstacle in understanding the collective behavior of such dynamical systems. The dynamical mechanisms responsible for restructuring collective behavior of a network of channel-based neuron models are difficult or impossible to analyze because these mechanisms are well hidden behind the complexity of the equations. If there is a chance to identify possible dynamical mechanisms behind a particular behavior of the network without the use of complex models, then this knowledge can be used to guide through numerical study of this behavior with the high-dimensional channel-based models. Based on these results one can specify the actual dynamical mechanism occurred in the network and understand the biological processes which contribute to it.
One way to reveal the dynamical mechanisms is to use a simplified or phenomenological model of the neuron. However, many experiments indicate the restructuring of collective behavior utilizes a variety of dynamical regimes generated by individual dynamics of a neuron. When neural dynamics include the regime chaotic spiking-bursting oscillations the system of differential equations describing each neuron should be at least a three-dimensional system (see for example [@HRModel]). Further simplification of phenomenological models for complex dynamics of the neuron can be obtain using dynamical systems in the form of a map. Despite the low-dimensional phase space of this nonlinear map, it is able to demonstrate a large variety of complex dynamical regimes.
The use of low-dimensional model maps can be useful for understanding the dynamical mechanisms if they mimic the dynamics of oscillations observed in real neurons, show correct restructuring of collective behavior, and are simple enough to study the reasons behind such restructuring. The development of a model map which is capable of describing various types of neural activity, including generation of tonic spikes, irregular spiking, and both regular and irregular bursts of spikes, is the goal of this paper.
It is known that constructing a low-dimensional system of differential equation which is capable of generating fast spikes bursts excited on top of the slow oscillations, one needs to consider a system which has both slow and fast dynamics (see for example [@HRModel; @Rinzel85; @Rinzel87; @Belykh00; @Izhikevich00]). Using the same approach one can construct a two-dimensional map, which can be written in the form
$$\begin{aligned}
x_{n+1}&=&f(x_n,y_n+\beta_n)~, \label{mapx} \\
y_{n+1}&=&y_n-\mu (x_n+1) + \mu \sigma_n~, \label{mapy}\end{aligned}$$
where $x_n$ is the fast and $y_n$ is the slow dynamical variable. Slow time evolution of $y_n$ is due to small values of the parameter $\mu=0.001$. Terms $\beta_n$ and $\sigma_n$ describe external influences applied to the map. These terms model the dynamics of the neuron under the action of the external DC bias current $I_{DC}$ and synaptic inputs $I^{syn}_n$. The term $\sigma_n$ can also be used as the control parameter to select the regime of individual behavior.
A model in the form of two-dimensional map, whose form is similar to ([1]{}) was used in the study of dynamical mechanisms behind the emergence and regularization of chaotic bursts in a group of synchronously bursting cells coupled through a mean field[@rul_prl2001; @deVries01]. The effect of anti-phase regularization was also modeled before with one-dimensional maps [@Cazelles01]. In both these models the oscillations during the burst were described by a chaotic trajectory. Map ([1]{}) improves these models by adding a feature that enables one to mimic the dynamics of individual spikes within the burst. This is achieved using a modification of the shape of the nonlinear function $f(x,y)$ which is now a discontinuous function of the form
$$\begin{aligned}
f(x,y)=\cases {
\alpha/(1-x)+y, & $x \leq 0$ \cr
\alpha+y, & $0<x<\alpha+y$ \cr
-1, & $x \geq \alpha+y$ \cr
}
\label{func}\end{aligned}$$
where $\alpha$ is a control parameter of the map. The dependence of $f(x,y)$ on $x$ computed for a fixed value of $y$ is shown in Fig.\[fig1\]. In this plot the values of $\alpha$ and $y$ are set to illustrate the possibility of coexistence of limit cycle, $P_k$, corresponding to spiking oscillation in (\[mapx\]), and fixed points $x_{ps}$ and $x_{pu}$. Note that when $y$ increases or decreases the graph of $f(x,y)$ moves up or down, respectively, except for the third interval $x\geq \alpha+y$, where the values of $f(x,y)$ always remain equal to -1.
The paper is organized as follows: Section \[Section1\] considers the features of the fast and slow dynamics which explain the formation of various types of behavior in the isolated map. It also discusses the bifurcations responsible for the qualitative change of activity. The dynamics of response generated by the map after it is excited or inhibited by an external pulse is discussed in Section \[Section2\]. The goal of this section is to illustrate dynamical mechanisms that control the response, and to show how the relation between the two inputs of the map influence the properties of the response. Section \[Section3\] presents the results of a synchronization study in two coupled maps.
Individual Dynamics of the Map {#Section1}
==============================
Consider the regimes of oscillations produced by the individual dynamics of the map. In this case the inputs $\beta_n=\beta$ and $\sigma_n=\sigma$ are constants. Note that if $\beta$ is a constant, then it can be omitted from the equations using the change of variable $y_n+\beta\rightarrow y_n^{new}$. Therefore, the individual dynamics of the map depends only on the control parameters $\alpha$ and $\sigma$, and the map can be rewritten in the form
$$\begin{aligned}
x_{n+1}&=&f(x_n,y_n),
\label{umapx} \\
y_{n+1}&=&y_n-\mu (x_n+1) + \mu \sigma~. \label{umapy}\end{aligned}$$
Typical regimes of temporal behavior of the map are shown in Figs \[fig2\] and \[fig3\]. When the value of $\alpha$ is less then 4.0 then, depending on the value of parameter $\sigma$, the map generates spikes or stays in a steady state (see Fig \[fig2\]). The frequency of the spikes increases as the value of parameter $\sigma$ is increased (see Fig \[fig2\]).
For $\alpha>4$ the map dynamics are capable of producing bursts of spikes. The spiking-bursting regimes are found in the intermediate region of parameter $\sigma$ between the regimes of continuous tonic spiking and steady state (silence). The spiking-bursting regimes include both periodic and chaotic bursting. A few typical bursting-spiking regimes computed for different values of $\sigma$ are presented in Fig \[fig3\].
Due to the two different time scales involved in the dynamics of the model, the mechanisms behind the restructuring of the dynamical behavior can be understood through the analysis of the fast and slow dynamics separately. In this case, time evolution of the fast variable, $x$, is studied with the one-dimensional map (\[umapx\]) where the slow variable, $y$, is treated as a control parameter whose value drifts slowly in accordance with equation (\[umapy\]). It follows from (\[umapy\]) that the value of $y$ remains unchanged only if $x=x_{s}$ given by
$$\begin{aligned}
x_{s}=-1 +\sigma.\label{xys}\end{aligned}$$
If $x<x_{s}$, then the value of $y$ slowly increases. If $x>x_{s}$, then $y$ decreases.
From (\[umapx\]) one can find the equation for the coordinate of fixed points $x_p$ of the fast map. This equation is of form $$\begin{aligned}
y=x_p-\frac{\alpha}{1-x_p},\label{yvsx}\end{aligned}$$ where $x_p \leq 0$. Equation (\[yvsx\]) defines the branches of slow motion in the two-dimensional phase space $(x_n,y_n)$ (see Fig \[fig4\]). The stable branch $S_{ps}(y)$ exists for $x_p<1-\sqrt{\alpha}$ and the unstable branch $S_{pu}(y)$ exists within $1-\sqrt{\alpha}\leq x_p \leq 0$
Considering the fast and slow dynamics together, one can see that, if $x_s$ is in the stable branch $S_{ps}(y)$, then the map ([3]{}) has a stable fixed point. The stable fixed point corresponds to the regime of silence in the neural dynamics. The oscillations in the map dynamics will appear when $x_s>1-\sqrt{\alpha}$. This is the threshold of excitation, which corresponds to the bifurcation values of $\sigma$ given by $$\begin{aligned}
\sigma_{th}=2-\sqrt{\alpha}.\label{sth}\end{aligned}$$
To understand the dynamics of excited neurons within this model one needs to consider branches that correspond to the spiking regime. To evaluate the location of the spiking branches, consider the mean value of $x_n$ computed for the periodic trajectory of the fast map (\[umapx\]) as a function of $y$. Here $y$ is treated as a parameter. Use of such an approximation is quite typical for an analysis involving fast and slow dynamics. It works well for small values of parameter $\mu$.
It follows from the shape of $f(x,y)$ that for any value of $y$, map (\[umapx\]) generates no more than one periodic trajectory $P_k$, where $k$ is the period (i.e. $x_n=x_{n+k}$). The cycles $P_k$ always contain the point $x_n=-1$ and the index $k$ increases stepwise ($k \rightarrow k+1$) as $y$ decreases. Since the trajectory $P_k$ always has a point in the flat interval of $f(x,y)$, all these cycles are super-stable except for bifurcation values of $y$ for which the trajectory contains the point $x_n=0$ (see Fig \[fig1\]). The location of the spiking branch, $S_{spikes}$, of “slow” motion in the phase plane ($x_n,y_n$) can be estimated as the mean value of $x$ computed for the period of cycle $P_k$, $$\begin{aligned}
x_{mean}= \frac{1}{k} \sum_{m=1}^{k} f^{(m)}(-1,y), \label{xmean}\end{aligned}$$ where $k$ is the period of $P_k$, and $f^{(m)}(x,y)$ is the $m$-th iterate of (\[umapx\]), started at point $x$ and computed for fixed $y$. The spiking branch of “slow” dynamics evaluated with (\[xmean\]) is shown in Fig\[fig4\]. One can see that this branch has many discontinuities caused by the bifurcations of the super-stable cycles $P_k$.
To complete the picture of fast and slow dynamics of the model for $\alpha>4$, one needs to consider the fast map bifurcation associated with the formation of homoclinic orbit $h_{pu}$ originating from the unstable fixed point, $x_{pu}$. This homoclinic orbit occurs when the coordinate of $x_{pu}$ become equal to -1. It can be easily shown that such a situation can take place only if $\alpha>4$. The homoclinic orbit forms at the value of $y$ where the unstable branch $S_{pu}(y)$ crosses the line $x=-1$, see Fig.\[fig4\]b. When the map is firing spikes and the value of $y$ gets to the bifurcation point, the cycle $P_k$ merges into the homoclinic orbit, disappears, and then the trajectory of the map jumps to the stable fixed point $x_{ps}$.
Typical phase portraits of the model, obtained under the assumptions made above, are presented in Fig.\[fig4\]. Fig.\[fig4\]a shows the typical behavior for $2<\alpha\leq4$. Here, only two regimes are generated. The first regime is the state of silence, when the operating point (OP), given by the intersection of $x_s=-1+\sigma$ and one of the branches, is on the branch $S_{ps}(y)$. The second regime is the regime of tonic spiking, when the operating point is selected on the spiking branch $S_{spikes}$.
When $\alpha>4$ the picture changes qualitatively (see Fig \[fig4\]b). Now, stable branches $S_{ps}(y)$ and $S_{spikes}$ are separated by the unstable branch $S_{pu}(y)$. If the operating point is selected on $S_{pu}(y)$, then the phase of silence, corresponding to slow motion along $S_{ps}(y)$, and spiking, when system moves along $S_{spikes}$, alternate forming the regime of spiking-bursting oscillations. The beginning of a burst of spikes corresponds to the bifurcation state of the fast map where fixed points $x_{ps}$ and $x_{pu}$ merge together and disappear. Before this bifurcation the system is in $x_{ps}$ and, therefore, $y$ increases. The termination of the burst is due to the bifurcation of the fast map associated with the formation of the homoclinic orbit $h_{pu}$. Here, the limit cycle of the spiking mode merges into the homoclinic orbit and disappears. After that the fast subsystem flips to the stable fixed point $x_{ps}$. Then the process repeats (see arrows in Fig \[fig4\]b).
It is clear from Fig.\[fig4\]a that when operating point is set on $S_{ps}(y)$ the model will be in the regime of silence. When the operating point is set on the branch $S_{spikes}$ the model produces tonic spiking, unless the point is set close to the formation of homoclinic orbit $h_{pu}$. One can see that, at the vicinity of this bifurcation, the branch $S_{spikes}$ becomes densely folded. As a result, the behavior of $y$, which is governed by the mean value of $x_n$, can become extremely sensitive to small perturbations and even lead to instability caused by high-gain feedback. This is one of the reasons for the irregular, chaotic spiking-bursting behavior which occurs in the map when the operating point is set close to the area of the branch $S_{spikes}$ where this branch is densely folded. The detailed and rigorous analysis of chaotic dynamics cannot be done within the approximations made above and require more precise computation of $S_{spikes}$ which is beyond the scope of this paper.
The results of the analysis presented above are summarized in the sketch of the bifurcation diagram plotted on the parameter plain ($\sigma,\alpha$) (see Fig \[fig5\]). The bifurcation curve $\sigma_{th}$ corresponds to the excitation threshold (\[sth\]) where the fixed point of the 2-d map becomes unstable and the map starts generating spikes. Curve $L_{ts}$ shows the approximate location of the border between spiking and spiking-bursting regimes, obtained in the numerical simulations. Note that separation of spiking and bursting regimes is not always obvious, especially in the regime of chaotic spiking. The regime of spiking-bursting oscillations takes place within the upper triangle formed by curves $\sigma_{th}$ and $L_{ts}$. The regimes of chaotic spiking or spiking-bursting behavior are found in the relatively narrow region of the parameters located around $L_{ts}$. This region contains complex structure of bifurcations, associated with multi-stable regimes, and is not shown in Fig \[fig5\].
The bifurcation diagram shows the role of control parameters in the selection of dynamical features of the considered neuron model. Both parameters can be used to mimic a particular type of neural behavior. Parameter $\sigma$ can be used to model the external DC current injection that depolarizes or hyperpolarizes the neuron. In this case $\sigma$ can be written as $$\begin{aligned}
\sigma=\sigma_u+I_{DC}, \label{sig_dc}\end{aligned}$$ where $\sigma_u$ is the parameter that selects the dynamics of isolated neuron, and $I_{DC}$ is the parameter that models the DC current injected into the cell. The changes in behavior caused by $I_{DC}$ are similar to the action of the parameter $I$ in the well known Hindmarsh-Rose model [@HRModel].
If the individual dynamics of the modelled neuron are capable of generating only regimes of silence and tonic spiking, and do not support the regime of bursts of spikes, then the value of $\alpha$ should be set below 4. In this case, no matter what the level of $I_{DC}$ injected is, the system will not show bursts of spikes (see Fig \[fig5\]).
Modeling of Response to the Injection of Current {#Section2}
================================================
To study the dynamical regimes of neuronal behavior in experiments, biologists change the type of neural activity using the injection of electrical current into the cell though an electrode. It was shown above that the injection of DC current can be modeled in the map ([1]{}) using parameter $\sigma_n=\sigma$ (see equation (\[sig\_dc\])). In this case, since the external influence does not vary in time, the role of parameter $\beta_n=\beta$ is not important, because the behavior of the map after the transient is independent of the value of $\beta$. However, when the injected current changes in time, it may be useful to consider the dynamics of parameter $\beta_n$ in order to provide more realistic modeled behavior during the transient. Taking this into account, the input of the model can be considered in the form: $$\begin{aligned}
\beta_n=\beta^e I_n~,~~ \sigma_n=\sigma^e I_n, \label{e_coupl}\end{aligned}$$ where $I_n$ is injected current, and coefficients $\beta^e$ and $\sigma^e$ are selected to achieve the desired properties of response behavior.
This Section briefly illustrates how the relation between these coefficients effects the dynamics of response to the pulse of $I_n$. To be specific, consider the map in the regime of tonic spiking with $\alpha=5.0$, $\sigma=0.33$, $\beta=0$. To study the response behavior, positive and negative pulses of amplitude 0.8 and duration of 100 iterations were applied to the continuously spiking map. In the simulations presented below the coefficient $\sigma^e$ was selected to be equal to one.
Figures \[fig6\] and \[fig7\] show the response to the positive and negative pulse, respectively, computed with $\beta^e=0$. In this case, during the action of a positive pulse, the value of $y_n$ increases monotonically because of the increased value of $\sigma_n$ (see (\[mapy\])). The increase of $y_n$ pushes the fast map up, see Fig \[fig1\]. This leads to an increase in the frequency of spiking. After the action of the pulse ends the value of $y_n$ monotonically decreases back to the original state (see Fig. \[fig6\]).
When a negative pulse is applied, it pushes the operating point down and, if the amplitude of the pulse is sufficiently large, shuts off the regime of spiking (see Fig.\[fig7\]). This happens because the trajectory of the fast map gets to the perturbed operating point which is now on the stable branch $S_{ps}(y)$, (see Fig \[fig4\]b). After the pulse is over the spiking does not re-appear immediately because the system spends some time drifting along the stable branch of slow motions $S_{ps}(y)$, during which the variable $y$ overshoots its original value for the spiking regime. As a result, after the system switches to the spiking branch $S_{spikes}$, $y_n$ monotonically drifts down to the unperturbed operating point. The dynamics of the slow evolution are clearly seen in the lower panel of Fig.\[fig7\].
Figures \[fig8\] and \[fig9\] illustrate how response to the pulse of $I_n$ changes when coefficient $\beta_n$ is not equal to zero. To be brief and specific consider the case of $\beta^e=1.0$ and $\sigma^e=1.0$. Comparing Fig.\[fig8\] with Fig.\[fig6\] one can see that the response dynamics to the positive pulse changes qualitatively. Indeed, when the pulse current is applied, then, acting through $\beta_n$, it immediately forces the fast map to shift up. As a result, the rate of spiking and mean value of $x_n$ increases sharply. Variable $y_n$ reacts to this change and decreases its value to compensate for the sudden change of the mean value of $x_n$. When the action of the pulse is over, the value of $\beta_n$ returns to its original value and, due to the updated levels of $y_n$, the fast map overshoots its original state. As a result, the trajectory of the fast map reaches the stable fixed point. To return to the original regime of spiking the system has to go all the way along the branches of slow motion $S_{ps}(y)$ and $S_{spikes}$ back to the original operating point (see Fig. \[fig4\]b). This type of response is observed in real neurobiological experiments, see for example [@Kleinfeld90].
Using the same analysis one can understand the new effects in the response to a negative pulse caused by the action of $\beta_n$. These new effects can be clearly seen comparing Fig.\[fig9\] with Fig. \[fig7\].
The presented results illustrate how 2-d map ([1]{}) along with equation (\[e\_coupl\]) can model a large variety of transient neural behavior induced by injected current. Due to the simplicity of the model one can clearly see the nonlinear mechanisms behind the response behavior and apply them to select the desired balance between $\sigma^e$ and $\beta^e$ to model a particular type of response.
Regimes of synchronization in two coupled maps {#Section3}
==============================================
This section presents the results of studies of synchronization regimes in coupled chaotically bursting 2-d maps. The goal of this study is to reproduce the main regimes of synchronous behavior found in a real neurobiological experiment [@elson98]. The experiment was carried out on two electrically coupled neurons (the pyloric dilators, PD) from the pyloric CPG of the lobster stomatogastric ganglion [@CPG]. The regimes found in the experiment were also reproduced in numerical simulations using Hindmarsh-Rose model [@Abarbanel96; @Pinto2000], one-dimensional map model [@Cazelles01], and in experiments with electronic neurons [@Pinto2000].
The equations used in the numerical simulations of the coupled maps are of form $$\begin{aligned}
x_{i,n+1}&=&f(x_{i,n},y_{i,n}+\beta_{i,n}),\nonumber
\\
y_{i,n+1}&=&y_{i,n}-\mu (x_{i,n}+1) + \mu \sigma_i+\mu
\sigma_{i,n},
\label{twocells}\end{aligned}$$ where index $i$ specifies the cell, and $\sigma_i$ is the parameter that defines the dynamics of the uncoupled cell. The coupling between the cells is provided by the current flowing from one cell to the other. This coupling is modeled by $$\begin{aligned}
\beta_{i,n}&=&g_{ji} \beta^e (x_{j,n}-x_{i,n}) \nonumber
\\
\sigma_{i,n}&=&g_{ji} \sigma^e (x_{j,n}-x_{i,n}) \label{twocoupl}\end{aligned}$$ where $i \neq j$, and $g_{ji}$ is the parameter characterizing the strength of the coupling. The coefficients $\beta^e$, $\sigma^e$ set the balance between the couplings for the fast and slow processes in the cells, respectively. In the numerical simulations the values of the coefficients are set to be equal: $\beta^e=1.0$ and $\sigma^e=1.0$. The other parameters of the coupled maps (\[twocells\]) that remain unchanged in the simulations have the following values: $\mu=0.001$, $\alpha_1=4.9$, $\alpha_2=5.0$. The coupling between the maps is symmetrical, $g_{ji}=g_{ij}=g$.
First, consider the main regimes of synchronization between the maps generating irregular, chaotic bursts of spikes. To set the individual dynamics of the maps to this regime, the parameters $\sigma_i$, that take into account the DC bias current injected into the neurons (see [@elson98] for details), are tuned to the following values: $\sigma_1=0.240$, $\sigma_2=0.245$. When these maps are uncoupled ($g=0$) they produce chaotic spiking-bursting oscillations shown in Fig \[fig10\]a. When the coupling becomes sufficiently large the slow components of the bursts synchronize while spikes within the bursts remain asynchronous see the waveforms in Fig. \[fig10\]b obtained with $g=0.043$. This regime of synchronization is typical for naturally coupled PD neurons (see Figure 2a in [@elson98]).
Introduction of negative coupling, $g<0$, also leads to synchronization of bursts, but in this regime of synchronization the systems burst in antiphase. Typical waveforms produced in this regime are presented in Fig. \[fig10\]c. This regime of synchronization of chaotic bursts is also observed in the experimental study of coupled PD neurons (see Figure 2c in [@elson98]). It is important to emphasize that, both in this simulation and in the experiment, the regime of antiphase synchronization characterized by the onset of regular bursts.
The simplicity of this model enables one to understand the possible cause for the onset of regular bursting in the antiphase synchronization. It is shown in Section \[Section1\] that chaotic dynamics of bursts occurs when the operating point of the system appears close to the leftmost area of the spiking branch $S_{spikes}$. In this region, $S_{spikes}$ is densely folded and, if system slows down in this area, the timing for the end of the burst becomes very sensitive to infinitesimal perturbations.
This fact is illustrated in Fig. \[fig11\]a. This figure shows the attractor computed from the waveforms $x_{1,n}$ and $y_{1,n}$ of the first cell when the cells are uncoupled (this regime is shown in Fig. \[fig10\]a). To plot the attractor, the waveforms of $x_{1,n}$ and $y_{1,n}$ were filtered by a fourth order low-pass filter with cutoff frequency 0.6. As a result, the attractor does not contain sharp spikes and the approximate location $S_{spikes}$, which is close to the center of filtered spiking oscillations, is easy to see. The level of coordinate $x_f$, corresponding to the operating point, is equal to $x_s=-1+\sigma_1=-0.76$. One can see from Fig. \[fig11\]a that when the center of oscillations gets close to that level the trajectory becomes rather complex, as shown by the dense set of trajectories in the left part of the attractor.
The numerical simulations show that this effect of regularization takes place even when $\sigma^e=0$. Therefore, the vertical shift of operating point, $x_s$, caused by the slow coupling current is not very important for this effect. The coupling term $\beta_{i,n}$ directly influences the fast dynamics by shifting the graph of the 1-d map (see Fig. \[fig1\]) up or down depending on the sign of $\beta_{i,n}$. In the regime of antiphase synchronization $\beta_{i,n}$ is positive, when the $i$-th cell is spiking, and negative, when it is silent. Therefore, from the viewpoint of fast dynamics, the coupling forces the cells to stay on the current branch of slow motion. One can understand these dynamics from the graph of $f(x,y)$ and (\[mapx\]). This effect is also seen as the formation of extended shape of attractor plotted for the regime of antiphase synchronization (see Fig. \[fig11\]b). Note that axes in Fig. \[fig11\]a and Fig. \[fig11\]b have different scales.
In the regime of chaotic bursts the level of $x_s$ is close to the stable branch of spiking $S_{spikes}$, and, therefore, the slow evolution along the branch of silence $S_{pu}$ is faster then along $S_{spikes}$. This means that duration of the phase of silence in the cell is shorter than duration of the burst of spikes. When the cell switches from silence to spiking it sharply changes the value and the sign of $\beta_{i,n}$. This change pushes $S_{spikes}$ of the spiking cell down to its original location, and, as a result, quickly drags the trajectory through the area of complex behavior toward the branch $S_{pu}$. Due to this fast change the duration of bursting is determined only by the dynamics of the silent phase, which is regular.
It was observed in the experiment that, in the regime of tonic spiking, spikes in PD neurons synchronize at low levels of coupling (i.e. without added artificial coupling, see [@elson98] for details). This property of synchronization in the regime of tonic spiking is also typical for the dynamics of the maps considered here (see Fig.\[fig12\]). In these numerical simulations, the uncoupled maps were tuned to generate spikes with slightly different rates. One can clearly see the beats between the waveforms plotted in Fig.\[fig12\]a. When small coupling, $g=0.008$, is introduced the beats disappear as soon as the spikes get synchronized (see Fig. \[fig12\]b).
Although the synchronization between continuously spiking maps is easily achieved, it is important to understand that the dynamics of such synchronization in the maps is a much more complex process than it is in the models with continuous time. Due to the discrete time, the periodic spikes can lock with different frequency ratios. This results in the existence of a complex multistable structure of synchronization regimes. This structure becomes more noticeable in the dynamics of synchronization as the number of iterations in the period of spikes decreases.
Discussion and outlook {#Section4}
======================
The simple phenomenological model of complex dynamics of spiking-bursting neural activity is proposed. The model is given by two-dimensional map ([1]{}). The first equation (\[mapx\]) of the map describes the fast dynamics. Its isolated dynamics is capable of generating stable limit cycles, which mimics the spiking activity of a neuron, and a stable fixed point, which corresponds to phase of silence. This 1-d map has a region of parameters were both these stable regimes coexist. Existence of such multistable regimes is the reason for generation of bursts, when the operating point, defined by the dynamics of the second equation (\[mapy\]), is properly set (see Fig \[fig4\]b).
The shape of the nonlinear function $f(x,y)$ used in (\[mapx\]) is selected in the form (\[func\]) based on the following considerations: (i) The fast map should generate limit cycles whose waveforms mimics those of the spikes. (ii) Each spike generated by the map always has a single iteration residing on the most right interval $x\geq\alpha+y$. Therefore, the moment of time corresponding to the appearance of a trajectory on this interval can be used to define the time of a spike. This feature is important for modeling the dynamics of chemical synapses in a group of coupled neurons. (iii) The analytic expression of the nonlinear function in interval $x<0$ should be simple enough to allow rigorous analysis of bifurcations of fixed points of the fast map. (iv) Use of the fixed level, $-1$, in the rightmost interval of the function simplifies the analysis of dynamics at the end of the bursts. The end of a burst is associated with the formation of a homoclinic orbit, which corresponds to the case when the trajectory from this interval maps into the unstable fixed point (see Section \[Section1\]).
It is clear that the shape of function (\[func\]) can be modified to take into account other dynamical features that need to be modeled. Due to the low-dimensionality of the model, the dynamical mechanisms behind its behavior are easy to understand using phase plane analysis. This allows one to see how introduced modifications influence the dynamics.
Some modification of the fast map is required to enhance the region of parameters where the model can still be used to mimic neural dynamics properties. For example, from (\[func\]) and Fig. \[fig1\] one can see that, due to the dynamics of coupling terms, the trajectory of the fast system can stay in the middle interval of $f(x,y)$ for several iterations. This can happen when the external influence monotonically pushes the function up while the trajectory is located in the middle interval. In this case the trajectory will map to the middle interval again and again, increasing the duration of a spike. This artifact can be removed using one of the following modifications. Introduction of a sufficient gap between the right end of this interval and the diagonal (see Fig. \[fig1\]) will help the map to terminate the spike despite the monotonic elevation of the function. Alternatively one can introduce an additional condition to the fast map (\[mapx\]) that forces the map always to iterate its trajectory from the middle interval to the rightmost one, despite the dynamics of $y$ (see (\[func\])). This can be achieved using the function $f(x_n,y)$ of the following form $$\begin{aligned}
f(x_n,y)=\cases {
\alpha/(1-x_n)+y, & $x_n \leq 0$ \cr
\alpha+y, & $0<x_n<\alpha+y$ \cr
-1, & $x_n \geq \alpha+y$ or $x_{n-1}>0$\cr
}
\nonumber
$$
An important feature of the model discussed here is that one can use two inputs, $\beta_n$ and $\sigma_n$, to achieve the desired response dynamics. Although these inputs are not directly related to dynamics of specific ionic currents, they can be used to capture the collective dynamics of these currents. Selecting a proper balance between these inputs, one can model a large variety of the responses that are seen in different neurons. Again, the simplicity of the model helps one to understand the dynamical properties of each input and set the proper balance between them.
Acknowledgment
==============
The author is grateful to M.I. Rabinovich, R. Elson, A. Selverston, H.D.I. Abarbanel, P. Abbott, V.S. Afraimovich and A.R. Volkovskii for helpful discussions. This work was supported in part by U.S. Department of Energy (grant DE-FG03-95ER14516), the U.S. Army Research Office (MURI grant DAAG55-98-1-0269), and by a grant from the University of California Institute for Mexico and the United States (UC MEXUS) and the Consejo Nacional de Ciencia y Tecnologia de México (CONACYT).
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---
abstract: 'A one-dimensional Hamiltonian system with exponential interactions perturbed by a conservative noise is considered. It is proved that energy superdiffuses and upper and lower bounds describing this anomalous diffusion are obtained.'
author:
- Cédric Bernardin
- Patrícia Gonçalves
title: '[Anomalous fluctuations for a perturbed Hamiltonian system with exponential interactions]{}'
---
Introduction
============
Over the last decade, transport properties of one-dimensional Hamiltonian systems consisting of coupled oscillators on a lattice have been the subject of many theoretical and numerical studies, see the review papers [@BLR; @D; @LLP]. Despite many efforts, our knowledge of the fundamental mechanisms necessary and/or sufficient to have a normal diffusion remains very limited. Nevertheless, it has been recognized that conservation of momentum plays a major role and numerical simulations provide a strong evidence of the fact that one dimensional chains of anharmonic oscillators conserving momentum are usually [[^1]]{} superdiffusive.
An interesting area of current research consists in studying this problem for hybrid models where a stochastic perturbation is superposed to the deterministic evolution. Even if the problem is considerably simplified, several open challenging questions can be addressed for these systems. In [@BBO2] it is proved that the thermal conductivity of an unpinned harmonic chain of oscillators perturbed by an energy-momentum conservative noise is infinite while if a pinning potential (destroying momentum conservation) is added it is finite. In the same paper, diverging upper bounds are provided when some nonlinearities are added. This does not, however, exclude the possibility of having a finite conductivity. Therefore much more interesting would be to obtain lower bounds showing that the conductivity is infinite and that energy superdiffuses, but this problem is left open in [@BBO2].
In [@BS], has been introduced and studied numerically, a class of Hamiltonian models for which anomalous diffusion is observed. There, the investigated systems present strong analogies with standard chains of oscillators. They can be described as follows. Let $V$ and $U$ be two non-negative potentials on ${{\mathbb R}}$ and consider the Hamiltonian system $( \, {\bf r} (t) , {\bf p} (t) \,)_{t \ge 0}$ whose equations of motion are given by $$\label{eq:generaldynamics}
\frac{dp_x}{dt} = V'(r_{x+1}) -V'(r_x),
\qquad \frac{dr_x}{dt} = U' (p_x) -U' (p_{x-1}),
\qquad x \in {{\mathbb Z}},$$ where $p_x$ is the momentum of the particle $x$, $q_x$ its position and $r_x=q_{x} -q_{x-1}$ is the “deformation” of the lattice at $x$. Standard chains of oscillators are recovered for a quadratic kinetic energy $U(p)=p^2 /2$. Now, take $V=U$, and call $\eta_{2x-1}=r_x$ and $\eta_{2x}=p_x$. The dynamics can be rewritten as: $$\label{eq:dyneq}
d\eta_{x} (t) =\Big(V' (\eta_{x+1}) - V' (\eta_{x-1})\Big) dt.$$ Notice that with these new variables the energy of the system is simply given by $\sum_{x\in {{\mathbb Z}}} V(\eta_x)$. In [@BS] an anomalous diffusion of energy is numerically observed for a generic potential $V$. Then, following the spirit of [@BBO2], the deterministic evolution is perturbed by adding a noise which consists to exchange $\eta_{x}$ with $\eta_{x+1}$ at random exponential times, independently for each bond $\{x,x+1\}$. The dynamics still conserves the energy $\sum_{x\in {{\mathbb Z}}} V(\eta_x)$ and the “volume” $\sum_{x\in {{\mathbb Z}}} \eta_x$ and destroys all other conserved quantities. As argued in [@BS], the volume conservation law is responsible for the anomalous energy diffusion observed for this class of energy-volume conserving dynamics. This can be shown for quadratic interactions ([@BS]) with a behavior similar to the one observed in [@BBO2]. For nonlinear interactions the problem is much more difficult.
The aim of this paper is to show that if the interacting potential is of exponential type then the energy superdiffuses. Therefore, for this class of related models, in a particular case, we answer to the open question stated in [@BBO2]. With some additional technical work we think that our methods could be carried out to the Toda lattice perturbed by an energy-momentum conserving noise (considered e.g. in [@ILOS]). The exponential form of the potential $V$ makes the deterministic dynamics given by (\[eq:dyneq\]) completely integrable. Nevertheless our proofs do not rely on this exceptional property of the dynamics and could be potentially generalized to other potentials $V$. The main ingredient used is the existence of explicit orthogonal polynomials for the equilibrium measures (see Section \[sec:dua\]).
The paper is organized as follows. In Section \[sec:model\] we define precisely the model. The results are stated in Section \[sec:results\]. To prove the theorems we first perform a microscopic change of variables (Section \[sec:cv\]) which permits to use a nice orthogonal decomposition of the generator (Section \[sec:dua\]). Roughly speaking the upper bound on the energy superdiffusion is proved in Section \[sec:triv\] and the lower bound in Section \[sec:diff\]. Section \[sec:pert\] contains a comment about the possible extensions and comparisons of our model to others. In the Appendix we prove the existence of the infinite dynamics.
[**Notations**: For any $a, b \in {{\mathbb R}}^2$, $a \cdot b $ stands for the standard scalar product between $a$ and $b$ and $|a|= \sqrt{a \cdot a}$ for the norm of $a$. The transpose of a matrix $M$ is denoted by $M^T$. If $u: {{\mathbf x}}=(x_1, \ldots,x_n)^T \in {{\mathbb R}}^n \to u({{\mathbf x}}) =(u_1({{\mathbf x}}), \ldots, u_d ({{\mathbf x}}))^T \in {{\mathbb R}}^d$ is a differentiable function then $\partial_{x_j} u_i ({{\mathbf x}})$ denotes the partial derivative of $u_j$ with respect to the $j$-th coordinate at ${{\mathbf x}}$ and $\nabla u( {{\mathbf x}})$ denotes the differential matrix (the gradient if $d=1$) of $u$ at ${{\mathbf x}}$, i.e. the $n \times d$ matrix whose $(i,j)$-th entry is $\partial_{x_j} u_i({{\mathbf x}})$; if $u:=(u_1, \ldots, u_d)^T:{{\mathbb Z}}\to {{\mathbb R}}^d$ then we adopt the same notation to denote the discrete gradient of $u$ defined by $\nabla u:= (\nabla u_1, \ldots,\nabla u_d)^T$ with $\nabla u_i (x)= u_i (x+1) -u_i (x)$. ]{}
The model {#sec:model}
=========
Let $b>0$ and $V_{b} (q)= e^{-bq} -1+bq $. We consider the system $\eta(t)=\{\eta_x(t):x\in{\mathbb{Z}}\}$ on ${{\mathbb R}}^{{{\mathbb Z}}}$ defined by its generator $L=A+\gamma S$, $\gamma>0$, where for local [[^2]]{} differentiable functions $f:{{\mathbb R}}^{{{\mathbb Z}}}\rightarrow{\mathbb{R}}$ we have that $$(Af)(\eta)=\sum_{x \in {{\mathbb Z}}} \Big(V_b^{\prime} (\eta_{x+1}) -V_b^{\prime} (\eta_{x-1}) \Big) (\partial_{\eta_x} f)(\eta)$$ and $$(Sf)(\eta)=\sum_{x \in {{\mathbb Z}}} \Big( f(\eta^{x,x+1}) -f(\eta) \Big),$$ where $\eta^{x,x+1}$ is obtained from $\eta$ by exchanging the variables $\eta_x$ and $\eta_{x+1}$, namely $$\label{etax,x+1}
\eta^{x,x+1}_y=\left\{\begin{array}{cl}
\eta_{x+1},& \mbox{if}\,\,\, y=x\,,\\
\eta_x,& \mbox{if} \,\,\,y=x+1\,,\\
\eta_y,& \mbox{otherwise}\,.
\end{array}
\right.$$ The deterministic system (\[eq:dyneq\]) with potential $V_b$ is well known in the integrable systems literature. It has been introduced in [@KVM] by Kac and van Moerbecke and was shown to be completely integrable. Consequently, the energy transport is ballistic ([@BS; @Z]). As we will see this is different when the noise is added: the energy transport is no more ballistic but superdiffusive.
The existence of the dynamics generated by $L$ is proved in the Appendix for a large set of initial conditions and in particular for a set of full measure w.r.t. any invariant state $\mu_{{\bar \beta},{\bar \lambda}}$ (see bellow for its definition).
The system conserves the energy $\sum_{x \in {{\mathbb Z}}} V_{b} (\eta_x)$ and the volume $\sum_{x \in {{\mathbb Z}}} \eta_x$. In fact, we have $${L} (V_{b} (\eta_x))=-\nabla {\bar j}_{x-1,x}(\eta), \quad L (\eta_x)=-\nabla {\bar j}^{\prime}_{x-1,x}(\eta),$$ where the microscopic currents are given by $${\bar j}_{x,x+1}(\eta)=-b^2 e^{-b(\eta_x + \eta_{x+1})}+b^2(e^{-b \eta_x} +e^{-b \eta_{x+1}})- \gamma \nabla V_{b} (\eta_{x})$$ and $${\bar j}^{\prime}_{x,x+1}(\eta)=b e^{-b \eta_x} +b e^{-b\eta_{x+1}} -\gamma \nabla \eta_x .\\$$
Every product probability measure $\mu_{\bar \beta, \bar \lambda}$ [[on ${{\mathbb R}}^{{{\mathbb Z}}}$]{}]{} in the form $$\mu_{{\bar \beta}, {\bar \lambda}} (d\eta) = \prod_{x\in {{\mathbb Z}}} {\bar Z}^{-1} ({\bar \beta},{\bar \lambda}) \exp \{ -{\bar \beta} e^{-b \eta_x} - {\bar\lambda} \eta_x\}d\eta_x, \quad {\bar \beta}>0\, , \, {\bar\lambda} >0$$ is invariant for the dynamics.
Let $\langle \cdot\rangle_{\mu_{\bar \beta, \bar \lambda}}$ denote the average with respect to $\mu_{\bar \beta, \bar \lambda}$. We define ${\bar e}:=\bar {e} ({\bar \beta}, {\bar \lambda}), {\bar v}:=\bar{v} ({\bar \beta}, {\bar \lambda})$ as the averages of the conserved quantities $V_b (\eta_x)$, $\eta_x$ with respect to $\mu_{\bar \beta, \bar \lambda}$, respectively, namely ${\bar e}=\langle V_b(\eta_x) \rangle_{\mu_{\bar \beta, \bar \lambda}}$ and ${\bar v}=\langle \eta_x \rangle_{\mu_{\bar \beta, \bar \lambda}}$.
A simple computation shows that $$\label{mean of micro currents}
\langle {\bar j}_{x,x+1} \rangle_{\mu_{\bar \beta, \bar \lambda}}=-b^2({\bar e} -b {\bar v})^2+b^2 \quad \textrm{ and} \quad \langle {\bar j}^{\prime}_{x,x+1} \rangle_{\mu_{\bar \beta, \bar \lambda}} =2b ({\bar e} -b {\bar v}+1).$$ Hence, in the hyperbolic scaling, the hydrodynamical equations are given by $$\label{eq:hl1euler}
\begin{cases}
\partial_t {{{\mathfrak e}}} -b^2 \, \partial_q ( ({{{\mathfrak e}}} - b{{{\mathfrak v}}})^2) =0\\
\partial_{t} {{{\mathfrak v}}} + 2b \, \partial_q ({{{\mathfrak e}} -b {{{\mathfrak v}}}}) =0
\end{cases}$$ and can be written in the compact form $\partial_t {\bar {{{\mathfrak X}}}} + \partial_q {\bar{ {{\mathfrak J}}}} ({\bar {{{\mathfrak X}}}}) =0$ with $$\label{eq:hl-ss00}
\bar {{{\mathfrak X}}}=
\left(
\begin{array}{c}
{{{\mathfrak e}}}\\
{{{\mathfrak v}}}
\end{array}
\right)
, \quad \textrm{and} \quad
\bar{ {{\mathfrak J}} }({\bar {{{\mathfrak X}}}})=
\left(
\begin{array}{c}
-b^2 ( {{{\mathfrak e}}} - b{{{\mathfrak v}}})^2\\
2b ( {{{\mathfrak e}}} - b{{{\mathfrak v}}})
\end{array}
\right).$$ This can be proved before the appearance of the shocks (see [@BS]). The differential matrix of $\bar{ {{\mathfrak J}}}$ is given by $$\nabla \bar{ {{\mathfrak J}}}(\bar {{{\mathfrak X}}})=
2b \left(
\begin{array}{cc}
-b ( {{{\mathfrak e}}} - b{{{\mathfrak v}}})& b^2 ( {{{\mathfrak e}}} - b{{{\mathfrak v}}}) \\
1 & -b
\end{array}
\right).$$ For given $({\bar e}, {\bar v})$ we denote by $({\bar T}^{+}_t )_{t \ge 0}$ (resp. $({\bar T}^{-}_t)_{t \ge 0}$) the semigroup on $S({{\mathbb R}}) \times S({{\mathbb R}})$ generated by $$\label{linearized system 1}
\partial_t \varepsilon + {\bar M}^T\, \partial_q \varepsilon =0, \quad ({\text{resp.}} \; \partial_t \varepsilon - {\bar M}^T \, \partial_q \varepsilon =0 ),$$ where $${\bar M} := {\bar M} ({\bar e}, {\bar v})= [\nabla \bar{ {{\mathfrak J}}}] (\bar \omega), \quad \bar \omega= \left(
\begin{array}{c}
{\bar e}\\
{\bar v}
\end{array}
\right).$$ We omit the dependence of these semigroups on $(\bar e , \bar v)$ for lightness of the notations. Above $S({{\mathbb R}})$ denotes the Schwartz space of smooth rapidly decreasing functions.
Statement of the results {#sec:results}
========================
For each integer $z \ge 0$, let $H_z (x) = (-1)^z e^{x^2} \cfrac{d^z}{dx^z} e^{-x^2}$ be the Hermite polynomial and $h_z (x) =(z! {\sqrt{2\pi}})^{-1} H_{z} (x) e^{-x^2}$ the Hermite function. The set $\{ h_z, z\ge 0\}$ is an orthonormal basis of ${{{\mathbb L}}}^2 ({{\mathbb R}})$. Consider in ${{{\mathbb L}}}^2({{\mathbb R}})$ the operator $K_0 = x^2-\Delta$, $\Delta$ being the Laplacian on ${{\mathbb R}}$. For an integer $k \ge 0$, denote by ${{{\mathbb H}}}_k$ the Hilbert space induced by $S ({{\mathbb R}})$ and the scalar product $\langle \cdot,\cdot\rangle_{k}$ defined by $\langle f, g \rangle_k= \langle f, K_0^k g \rangle_0$, where $\langle \cdot,\cdot\rangle_0$ denotes the inner product of ${{\mathbb L}}^2 ({{\mathbb R}})$ and denote by ${{{\mathbb H}}}_{-k}$ the dual of ${{{\mathbb H}}}_k$, relatively to this inner product. Let $\langle\cdot\rangle$ represent the average with respect to the Lebesgue measure.
We take the infinite system at equilibrium under the Gibbs measure $\mu_{\bar \beta,\bar \lambda}$ corresponding to a mean energy $\bar e$ and a mean volume $\bar v$. Our goal is to study the energy-volume fluctuation field in the time-scale $tn^{1+\alpha}$, $\alpha \ge0$: $$\label{eq:YY}
\mathcal{Y}^{n,\alpha}_t ({{\mathbf G}}) =\frac{1}{\sqrt{n}} \sum_{x\in {{\mathbb Z}}}
{{\mathbf G}}\left(x/n\right) \cdot \left({\bar \omega}_x (tn^{1+\alpha}) - {\bar \omega}\right),$$ where for $q\in{\mathbb{R}}$, $x \in {{\mathbb Z}}$, $${{\mathbf G}}(q) = \left(
\begin{array}{c}
G_{1} (q)\\
G_2 (q)
\end{array}
\right), \quad
{\bar \omega}_x= \left(
\begin{array}{c}
V_b (\eta_x) \\
\eta_x
\end{array}
\right)$$ and $G_1, G_2$ are test functions belonging to $S({{\mathbb R}})$.
If $E$ is a Polish space then $D({{\mathbb R}}^+, E)$ (resp. $C({{\mathbb R}}^+, E)$) denotes the space of $E$-valued functions, right continuous with left limits (resp. continuous), endowed with the Skorohod (resp. uniform) topology. Let $Q^{n,\alpha}$ be the probability measure on ${D}({{\mathbb R}}^+,{{{\mathbb H}}}_{-k} \times {{{\mathbb H}}}_{-k})$ induced by the fluctuation field ${{{\mathcal Y}}}^{n,\alpha}_t$ and $\mu_{\bar \beta,\bar \lambda}$. Let $\mathbb{P}_{\mu_{\bar \beta,\bar \lambda}}$ denote the probability measure on ${ D}({{\mathbb R}}^+, {{\mathbb R}}^{{{\mathbb Z}}})$ induced by $(\eta(t))_{t\geq{0}}$ and $\mu_{\bar \beta,\bar \lambda}$. Let $\mathbb{E}_{\mu_{\bar \beta,\bar \lambda}}$ denote the expectation with respect to $\mathbb{P}_{\mu_{\bar \beta,\bar \lambda}}$.
\[th:fluct-hs\] Fix an integer $k>2$. Denote by $Q$ the probability measure on $C({{\mathbb R}}^+, {{{\mathbb H}}}_{-k} \times {{{\mathbb H}}}_{-k})$ corresponding to a stationary Gaussian process with mean $0$ and covariance given by $${\mathbb E}_{Q} \left[ \mathcal{Y}_t ({{\mathbf H}}) \, \mathcal{ Y}_s ({{\mathbf G}}) \right] = \langle \,{\bar T}_t^{-} {{\mathbf H}}\; \cdot \; \bar \chi \; {\bar T}_s^{-}{{\mathbf G}} \, \rangle$$ for every $0 \le s \le t$ and ${{\mathbf H}}, {{\mathbf G}}$ in ${{{\mathbb H}}}_k \times {{{\mathbb H}}}_k$. Here ${\bar \chi}:={\bar \chi} ({\bar \beta}, {\bar \lambda})$ is the equilibrium covariance matrix [[^3]]{} of ${\bar \omega}_0$. Then, the sequence $(Q^{n,0})_{n \ge 1}$ converges weakly, as $n\to\infty$, to the probability measure $Q$.
A byproduct of Theorem \[th:fluct-hs\] is a Central Limit Theorem for the energy flux and for the volume flux through a fixed bond. Despite it is not directly related to the problem of anomalous diffusion it has a probabilistic interest. For that purpose, fix a site $x\in{{{\mathbb Z}}}$, let $\mathcal{E}_{x,x+1}^n(t)$ (resp. $\mathcal{V}_{x,x+1}^n(t)$) denote the energy (resp. volume) flux through the bond $\{x,x+1\}$ during the time interval $[0,tn]$. By conservation laws, for any $x\in{\mathbb{Z}}$ it holds that: $$\mathcal{E}_{x-1,x}^n(t)-\mathcal{E}_{x,x+1}^n(t):=V_b(\eta_x(tn))-V_b(\eta_x(0))$$ $$\Big( \text{resp. \; } \mathcal{V}^n_{x-1,x}(t)-\mathcal{V}_{x,x+1}^n(t):=\eta_x(tn)-\eta_x(0)\Big).$$ This, together with the previous result allow us to conclude that
\[CLT Energy flux\] Fix $x\in{{\mathbb Z}}$ and let $Z^{n,e}_t:=\frac{1}{\sqrt n}\{\mathcal{E}_{x,x+1}^n(t)- {{\mathbb E}}_{\mu_{\bar \beta,\bar \lambda}}[\mathcal{E}_{x,x+1}^n(t)]\}$. For every $t\geq{0}$, $(Z^{n,e}_{t})_{n\geq 1}$ converges in law in the sense of finite-dimensional distributions, as $n\to\infty$, to a Brownian motion $Z_{t}^e$ with mean zero and covariance given by $${\mathbb E}_{Q} [Z^e_tZ^e_s]=\frac{2}{\bar\beta^3}(\bar\lambda-b\bar\beta)^2s,$$ for all $s\leq{t}$.
\[CLT volume flux\] Fix $x\in{{\mathbb Z}}$ and let $Z^{n,v}_t:=\frac{1}{\sqrt n}\{\mathcal{V}_{x,x+1}^n(t)-{{ {{\mathbb E}}_{\mu_{\bar \beta,\bar \lambda}}}}[\mathcal{V}_{x,x+1}^n(t)]\}$. For every $t\geq 0$, $(Z^{n,v}_t)_{n\geq 1}$ converges in law in the sense of finite-dimensional distributions, as $n\to\infty$, to a Brownian motion $Z_{t}^v$ with mean zero and covariance given by $${\mathbb E}_{Q}[Z^v_tZ^v_s]=\frac{2}{\bar\beta}s,$$ for all $s\leq{t}$.
We notice that, according to Corollary \[CLT Energy flux\], the limiting [[energy flux]{}]{} $Z_t^e$ has a vanishing variance for $\bar\lambda=b\bar\beta$ which is equivalent to $\bar{e} = b\bar{v}$. Last equivalence is a consequence of and .
The theorem above means that in the hyperbolic scaling the fluctuations are trivial: the initial fluctuations are transported by the linearized system of (\[eq:hl1euler\]). To see a nontrivial behavior we have to study, in the transport frame, the fluctuations at a longer time scale $t n^{1+\alpha}$, with $\alpha>0$. Thus, we consider the fluctuation field ${\widehat {{{\mathcal Y}}}}_{\cdot}^{n,\alpha}$, $\alpha>0$, defined, for any ${{\mathbf G}}\in S({{\mathbb R}}) \times S({{\mathbb R}})$, by $$\label{longer density field}
{\widehat {{{\mathcal Y}}}}_t^{n,\alpha} ({{\mathbf G}})= {{{\mathcal Y}}}_t^{n, \alpha} \left( {\bar T}^+_{t n^{\alpha}} {{\mathbf G}}\right).$$ According to the fluctuating hydrodynamics theory ([@Sp], pp. 85-96), in the case of a normal (diffusive) behavior $\alpha=1$, the field $( {\widehat{{{\mathcal Y}}}}_t^{n,\alpha} )\,_{t \ge 0}$ should converge to the stationary field $({\widehat{{{\mathcal Y}}}}_t \,)\,_{t \ge 0}$ simply related to the solution $(\widehat{{{\mathcal Z}}}_t\,)\,_{t \ge 0}$ of the linear two dimensional vector valued (infinite-dimensional) stochastic partial differential equation $$\label{eq:OUeq}
\partial_t {\widehat {{{\mathcal Z}}_t}}= \nabla \cdot \left( \, {{{\mathcal D}}} \, \nabla {\widehat {{{\mathcal Z}}}_t} \, \, \right) + \sqrt{2 {{{\mathcal D}}} {\bar \chi}}\, \nabla \cdot W{{_t}}.$$ Here $W_t$ is a standard two-dimensional vector valued space-time white noise and the coefficient ${{{\mathcal D}}}:= {{{\mathcal D}}} ({\bar e},{\bar v})$ is expressed by a Green-Kubo formula ( see (\[eq:GK00\])). As above, let $\widehat Q^{n,\alpha}$ be the probability measure on ${D}({{\mathbb R}}^+,{{{\mathbb H}}}_{-k} \times {{{\mathbb H}}}_{-k})$ induced by the fluctuation field $\widehat{{{\mathcal Y}}}^{n,\alpha}_t$ and $\mu_{\bar \beta,\bar \lambda}$. Our second main theorem shows that the correct scaling exponent $\alpha$ is greater or equal than $1/3$:
\[th:fluct-ds\] Fix an integer $k>1$ and $\alpha<1/3$. Denote by $Q$ the probability measure on $C({{\mathbb R}}^+, {{{\mathbb H}}}_{-k} \times {{{\mathbb H}}}_{-k})$ corresponding to a stationary Gaussian process with mean $0$ and covariance given by $${\mathbb E}_{Q} \left[ \mathcal{Y}_t ({{\mathbf H}}) \, \mathcal{ Y}_s ({{\mathbf G}}) \right] = \langle \, {{\mathbf H}} \;\cdot \; {\bar \chi} \; {{\mathbf G}} \rangle$$ for every $0 \le s \le t$ and ${{\mathbf H}}, {{\mathbf G}}$ in ${{{\mathbb H}}}_k \times {{{\mathbb H}}}_k$. Then, the sequence $(\widehat Q^{n,\alpha})_{n \ge 1}$ converges weakly, as $n\to\infty$, to the probability measure $Q$.
As in the hyperbolic time scale from the previous result we obtain limiting results for the energy flux and volume flux. In this case, we need to define the energy and volume flux through the time dependent bond ${\{u_t^{x,\alpha}(n),u_t^{x,\alpha}(n)+1\}}$, where $u_t^{x,\alpha}(n):= \lfloor x-\frac{-2b\bar\lambda}{\bar \beta} tn^{1+\alpha}\rfloor$ and $\lfloor u\rfloor$ denotes the biggest integer number smaller or equal to $u$. The justification for taking this reference frame with precisely this velocity will be given ahead in Remark \[velocity\]. Now, fix a site $x\in{\mathbb{Z}}$ and let $\mathcal{E}^n_{u_t^{x,\alpha}(n)}$ (resp. $\mathcal{V}_{u_t^{x,\alpha}(n)}^n(t)$) denote the energy (resp. volume) flux through the bond $\{u_t^{x,\alpha}(n),u_t^{x,\alpha}(n)+1\}$ during the time interval $[0,tn^{1+\alpha}]$. Then, from the previous result we conclude that
\[vanishing of Energy flux\] Fix $t\geq{0}$, $x\in{{\mathbb Z}}$ and $\alpha<1/3$. Then $$\lim_{n \to \infty}{{ {{\mathbb E}}_{\mu_{\bar \beta,\bar \lambda}}}}\left[ \left( \frac{1}{\sqrt n}\Big\{\mathcal{E}_{u_t^{x,\alpha}(n)}^n(t)-{{ {{\mathbb E}}_{\mu_{\bar \beta,\bar \lambda}}}}[\mathcal{E}_{u_t^{x,\alpha}(n)}^n(t)]\Big\}\right)^2\right]=0.$$ and $$\lim_{n \to \infty} {{ {{\mathbb E}}_{\mu_{\bar \beta,\bar \lambda}}}} \left[ \left( \frac{1}{\sqrt n}\Big\{\mathcal{V}_{u_t^{x,\alpha}(n)}^n(t)-{{ {{\mathbb E}}_{\mu_{\bar \beta,\bar \lambda}}}}[\mathcal{V}_{u_t^{x,\alpha}(n)}^n(t)]\Big\}\right)^2\right]=0.$$
Similar results have been obtained in [@G] by one of the authors for the asymmetric simple exclusion. The proof of Corollaries \[CLT Energy flux\], \[CLT volume flux\] and \[vanishing of Energy flux\] follows the same arguments as in [@G] once the previous theorems are proved. For that reason we will only give a sketch of their proof. The proof of the theorems is more problematic since the multi-scale analysis performed in [@G] relies crucially on the existence of a spectral gap so that we cannot follow [@G]. Therefore we propose an alternative approach based on computations of some resolvent norms.
Theorem \[th:fluct-ds\] does not exclude the possibility of normal fluctuations, i.e. $\alpha=1$. In order to show that the system we consider is really superdiffusive we will show that the transport coefficient ${{{\mathcal D}}}$ which appears in (\[eq:OUeq\]) is infinite so that the correct scaling exponent $\alpha$ is strictly smaller than $1$. Our third result, stated bellow, shows it is in fact less than $3/4$.
With the notations introduced in the previous section, the [*normalized*]{} currents are defined by $$\label{norm. currents}
{\hat J}_{x,x+1}(\eta) =
\left(
\begin{array}{c}
{\bar j}_{x,x+1}(\eta)\\
{\bar j}^{\prime}_{x,x+1}(\eta)
\end{array}
\right)
- {\bar {{{\mathfrak J}}}} ({\bar \omega}) - (\nabla {\bar{{{\mathfrak J}}}}) ({\bar \omega})
\left(
\begin{array}{c}
V_b (\eta_x) -{\bar e}\\
\eta_x -{\bar v}
\end{array}
\right).$$
Up to a constant matrix coming from a martingale term (due to the noise) and thus irrelevant for us (see [@BBO2], [@BS]), the coefficient ${{{\mathcal D}}}$ is defined by the Green-Kubo formula $$\label{eq:GK00}
{{{\mathcal D}}} = \int_{0}^{\infty} C(t) \, dt,$$ where $$C(t):={\mathbb E}_{\mu_{\bar \beta, \bar \lambda}} \left[ \sum_{x \in {{\mathbb Z}}} {\hat J}_{x,x+1} (\eta(t)) \left[ {\hat J}_{0,1} (\eta(0)) \right]^T \right]$$ is the current-current correlation function. The signature of the superdiffusive behavior of the system is seen in the divergence of the integral defining ${{{\mathcal D}}}$, i.e. in a slow decay of the current-current correlation function. We introduce the Laplace transform function ${{{\mathcal F}}} (\gamma, \cdot)$ of the current-current correlation function. It is defined, for any $z>0$ by $${{{\mathcal F}}} (\gamma, z) = \int_{0}^{\infty} e^{-z t} \, C(t)\, dt.$$
Our third theorem is the following lower bound on ${{{\mathcal F}}} (\gamma, z)$. Observe that ${{{\mathcal F}}} (\gamma, z)$ is a square matrix of size $2$ whose $(i,j)$-th entry is denoted by ${{{\mathcal F}}}_{i,j}$.
\[th:diffusivity\] Fix $\gamma>0$. For any $(i,j)\ne (1,1)$ and any $z>0$ we have $${{{\mathcal F}}}_{i,j} (\gamma, z)=0.$$ There exists a positive constant $c:=c(\gamma)>0$ such that for any $z>0$, $${{{\mathcal F}}}_{1,1} (\gamma, z) \ge c z^{-1/4}.$$ Moreover, there exists a positive constant $C:=C(\gamma)$ such that for any $z>0$, $$\label{eq:F11c}
C^{-1} {{{\mathcal F}}}_{1,1} (1,z/\gamma) \le {{{\mathcal F}}}_{1,1} (\gamma, z) \le C {{{\mathcal F}}}_{1,1} (1,z/\gamma).$$
The lower bound ${{{\mathcal F}}}_{1,1} (\gamma, z) \ge c z^{1/4}$ means roughly that the current-current correlation function $C(t)$ is bounded by bellow by a constant times $t^{-3/4}$. The last part of the theorem is easy to prove but has an important consequence. In [@BS] numerical simulations are performed to detect the anomalous diffusion of energy. Since it is difficult to estimate numerically the time autocorrelation functions of the currents because of their expected long-time tails, a more tenable approach consists in studying a non equilibrium system in its steady state, i.e. considering a finite system in contact with two thermostats which fix the value of the energy at the boundaries. Then we estimate the dependence of the energy transport coefficient $\kappa (N)$ with the system size $N$. The latter is defined as $N$ times the average energy current. It turns out that $\kappa (N) \sim N^{\delta}$ with a parameter $\delta:=\delta(\gamma)>0$ increasing with the noise intensity $\gamma$ (except for the singular value $\delta=1$ when $\gamma=0$ which is a manifestation of the ballistic behavior of the Kac-van Moerbecke system). This result is very surprising since the more stochasticity in the model is introduced, the less the system is diffusive. The same has been observed for other anharmonic potentials in [@BS] and also for the Toda lattice perturbed by an energy-momentum conservative noise ([@ILOS]). It has been argued in [@ILOS] that this may be explained by the fact that some diffusive phenomena due to non-linearities, like localized breathers, are destroyed by the noise. In [@BDLLO] simulations have been performed directly with the Green-Kubo formula for other standard anharmonic chains with the same conclusion: current-current correlation function decreases slower when the noise intensity increases. If all these numerical simulations reproduce correctly the real behavior of the models investigated, they dismiss the theories which pretend that some universality holds, e.g. [@VB]. It is therefore very important to decide if the phenomena numerically observed are correct or not.
Assuming that the current-current correlation function $C(t)$ has the time decay $C(t) \sim_{t \to \infty} t^{- \delta' (\gamma)}$, the inequality (\[eq:F11c\]) shows that the exponent $\delta':=\delta' (\gamma)$ is independent of $\gamma$ (up to possible slowly varying functions corrections, i.e. in a Tauberian sense). It is usually argued but not proved (see e.g. the end of Section 5.3 in [@LLP]) that the exponent $\delta$ defined by the non-equilibrium stationary state is related to $\delta'$ by the relation $\delta= 1-\delta'$, and is consequently independent of $\gamma$ too. Therefore the numerical simulations do not seem to reflect the correct behavior of the system [[^4]]{}. A possible explanation of the inconsistency between the numerical observation and our result is simply that the relation $\delta=1-\delta'$ is not satisfied. Nevertheless, notice that the last part of our theorem is in fact valid for all the models cited above. It applies in particular to the models studied in [@BDLLO] and shows that the numerical observations of [that]{} paper, which are performed for the Green-Kubo formula, are not consistent with the real behavior of the system.
A change of variables {#sec:cv}
=====================
To study the energy-volume fluctuation field ${{{\mathcal Y}}}_{\cdot}^{n, \alpha}$, we introduce the following change of variables $\xi_x = e^{-b \eta_x}$, [for each $x\in {{\mathbb Z}}$]{}. Then, the previous Markovian system $(\eta (t))_{t \ge 0}$ defines a new Markovian system $(\xi (t))_{t \ge 0}$ with state space $(0,+\infty)^{{{\mathbb Z}}}$ whose generator ${{{\mathcal L}}}$ is equal to $b^2 {{{\mathcal A}}} +{\gamma} {{{\mathcal S}}}$, where for local differentiable functions $f:(0,+\infty)^{{{\mathbb Z}}}\rightarrow{\mathbb{R}}$ we have that $$({{{\mathcal A}}} f)(\xi)=\sum_{x \in {{\mathbb Z}}} \xi_x \Big( \xi_{x+1}- \xi_{x-1} \Big) (\partial_{\xi_x} f)(\xi)$$ and $$({{{\mathcal S}}} f)(\xi)=\sum_{x \in {{\mathbb Z}}} \Big( f(\xi^{x,x+1}) -f(\xi) \Big),$$ where $\xi^{x,x+1}$ is defined as in .
Observe that the energy and volume conservation laws correspond, for the process $(\xi (t))_{t \ge 0}$, to the conservation of the two following quantities $\sum_{x \in {{\mathbb Z}}} {\xi_x}$ and $\sum_{x \in {{\mathbb Z}}} \log (\xi_x).$ The corresponding microscopic currents are defined by the conservation law equations: $${{{\mathcal L}}} (\xi_x) = -\nabla j_{x-1,x}(\xi), \quad {{{\mathcal L}}} (\log \xi_x) = -\nabla j^{\prime}_{x-1,x}(\xi),$$ where $$j_{x,x+1}(\xi) = -b^2 \xi_{x} \xi_{x+1} - \gamma \nabla \xi_x,$$ and $$j^{\prime}_{x,x+1}(\xi)= -b^2 (\xi_{x}+ \xi_{x+1})- \gamma \nabla \log (\xi_x).$$ We will use the compact notation $$\label{flux for xi}
J_{x,x+1}(\xi) =
\left(
\begin{array}{c}
j_{x,x+1} (\xi)\\
j^{\prime}_{x,x+1}(\xi)
\end{array}
\right).$$
Since $V_b(\eta_x)=\xi_x-\log( \xi_x)+1$ and $\eta_x=-\frac{1}{b}\log(\xi_x)$, we have the following relations between the microscopic currents $$\label{eq:corr-jj}
{\bar j}_{x,x+1}(\eta)=j_{x,x+1}(\xi)-j^{\prime}_{x,x+1}(\xi), \quad\textrm{and}\quad {\bar j}_{x,x+1}^{\prime}(\eta)= -\cfrac{1}{b} j_{x,x+1}^{\prime}(\xi).$$
If $\eta$ is distributed according to $\mu_{\bar \beta, \bar \lambda}$ then $\xi$ defined by $\xi_x =e^{-b \eta_x}$ is distributed according to the probability measure $\nu_{\beta, \lambda}$ on $(0,+\infty)^{{{\mathbb Z}}}$ given by $$\nu_{\beta, \lambda} (d\xi) =\prod_{x\in {{\mathbb Z}}} {Z}^{-1} (\beta,\lambda) {\bf 1}_{\{\xi_{x} >0\}} \exp \{ -\beta \xi_x + \lambda \log (\xi_x)\}d\xi_x$$ with $Z(\beta,\lambda)$ the partition function, $$\label{eq:relblbl}
\beta={\bar \beta}, \quad\textrm{and}\quad \lambda= -1+{\bar \lambda}/{b}.$$
Remark that $\nu_{\beta,\lambda}$ is nothing but a product probability measure whose marginal follows a Gamma distribution $\gamma_{\lambda+1, \beta^{-1}}$ with parameter $(\lambda+1, \beta^{-1})$. In particular, we have $Z:=Z(\beta, \lambda)= \beta^{-(\lambda +1)} \, \Gamma (\lambda +1)$, where $\Gamma$ is the usual Gamma function.
Thus, the process $(\xi (t))_{t \ge 0}$ has a family of translation invariant measures $\nu_{\beta, \lambda}$ parameterized by the chemical potentials $(\beta, \lambda)\in (0,+\infty) \times (-1, +\infty)$.
Let $\mathbb{P}_{\nu_{\beta,\lambda}}$ be the probability measure on ${D}(\mathbb{R}^+,(0,+\infty)^{\mathbb{Z}})$ induced by $(\xi(t))_{t\geq{0}}$ and $\nu_{\beta,\lambda}$ and let $\mathbb{E}_{\nu_{ \beta,\lambda}}$ denote the expectation with respect to $\mathbb{P}_{\nu_{ \beta, \lambda}}$.
Let $\langle \cdot \rangle_{\nu_{\beta, \lambda}}$ denote the average with respect to ${\nu_{\beta, \lambda}}$. The averages $\rho:=\rho(\beta, \lambda)$ and $\theta:=\theta (\beta, \lambda)$ of the conserved quantities for $(\xi (t))_{t \ge 0}$ at equilibrium under $\nu_{\beta,\lambda}$ are defined by $\rho = \langle \xi_x \rangle_{\nu_{\beta, \lambda}}$ and $\theta = \langle \log(\xi_x) \rangle_{\nu_{\beta, \lambda}}.$ By a direct computation we get
$$\label{eq:chimpot}
\rho=1+{\bar e} -b {\bar v} =\cfrac{\lambda +1}{\beta}, \quad\textrm{and}\quad \theta=-b {\bar v}=\frac{\Gamma' (\lambda +1)}{\Gamma' (\lambda+1)}- \log(\beta).$$
It is understood, here and in the whole paper, that $(\beta, \lambda)$ are related to $(\bar \beta, \bar \lambda)$ through (\[eq:relblbl\]). We will use the following compact notation, for each $x\in {{\mathbb Z}}$, $${\omega}_x= \left(
\begin{array}{c}
\xi_x \\
\log (\xi_x)
\end{array}
\right),\quad \textrm{and} \quad
{\omega}= \left(
\begin{array}{c}
\rho \\
\theta
\end{array}
\right).$$
Observe that ${\bar \omega}_x = \Lambda \omega_x - \left( \begin{array}{c} 1\\ 0 \end{array} \right)$, where $$\label{lambda matrix}
\Lambda=
\left(
\begin{array}{cc}
1&-1 \\
0 & -1/b
\end{array}
\right).$$
The covariance matrix $\chi:=\chi (\beta, \lambda)$ of $\omega_0$ under $\nu_{\beta,\lambda}$ is given by $$\chi =
\left(
\begin{array}{cc}
\langle(\xi_0-\rho)^2 \rangle_{\nu_{\beta,\lambda}} & \langle(\xi_0-\rho)(\log(\xi_0)-\theta) \rangle_{\nu_{\beta,\lambda}}\\
\langle(\xi_0-\rho)(\log(\xi_0)-\theta) \rangle_{\nu_{\beta,\lambda}}& \langle(\log(\xi_0)-\theta)^2 \rangle_{\nu_{\beta,\lambda}}
\end{array}
\right).$$ A simple computation shows that $$\chi =
\left(
\begin{array}{cc}
\frac{\lambda+1}{ \beta^{2}} & \frac{1}{\beta}\\
\frac{1}{\beta} & (\log \Gamma)'' (\lambda+1)
\end{array}
\right)=
\left(
\begin{array}{cc}
\partial_\beta^2 \log (Z) & - \partial_{\beta, \lambda} \log (Z)\\
- \partial_{\beta, \lambda} \log (Z) & \partial^2_{\lambda} \log (Z)
\end{array}
\right).$$ Denote the covariance matrix of $\bar \omega_0$ under $\mu_{\bar\beta,\bar\lambda}$ by $\bar\chi:=\bar\chi(\bar\beta,\bar\lambda)$, which is defined by $$\bar\chi =
\left(
\begin{array}{cc}
\langle(V_b(\eta_0)-\bar e)^2 \rangle_{\mu_{\bar\beta,\bar\lambda}} & \langle(V_b(\eta_0)-\bar e)(\eta_0-\bar v) \rangle_{\nu_{\bar\beta,\bar\lambda}}\\
\langle(V_b(\eta_0)-\bar e)(\eta_0-\bar v) \rangle_{\mu_{\bar\beta,\bar\lambda}}& \langle(\eta_0-\bar v)^2 \rangle_{\mu_{\bar\beta,\bar\lambda}}
\end{array}
\right).$$
Thus, the covariance matrix $\chi$ of $\omega_0$ under $\nu_{\beta,\lambda}$ is related to the covariance matrix ${\bar \chi}$ of $\bar \omega_0$ under $\mu_{\bar \beta, \bar \lambda}$, by $$\label{eq:chibar2}
{\bar \chi} = \Lambda \chi \Lambda^T=
\left(
\begin{array}{cc}
\frac{\lambda+1}{ \beta^{2}} +\frac{2}{\beta}+(\log \Gamma)'' (\lambda+1) & \quad \frac{1}{b\beta}+\frac{(\log \Gamma)'' (\lambda+1)}{b}\\
\frac{1}{b\beta}+\frac{(\log \Gamma)'' (\lambda+1)}{b}& \frac{(\log \Gamma)'' (\lambda+1)}{b^2}
\end{array}
\right).$$
A simple computation shows that $\langle j_{x,x+1} \rangle_{\nu_{\beta, \lambda}}=-b^2\rho^2$ and $\langle j'_{x,x+1} \rangle_{\nu_{\beta, \lambda}}=-2b^2\rho$. The hydrodynamical equations for the process $(\xi (t))_{t \ge 0}$ are given by $$\label{eq:hl-ss001}
\begin{cases}
\partial_t \rho - b^2 \partial_q (\rho^2) =0\\
\partial_t \theta -2 b^2 \partial_q \rho =0
\end{cases}$$ and can be written in the compact form $\partial_t {{{\mathfrak X}}} + \partial_q {{{\mathfrak J}}} ({{{\mathfrak X}}}) =0$ with $${{{\mathfrak X}}}=
\left(
\begin{array}{c}
{\rho}\\
{\theta}
\end{array}
\right)
, \quad\textrm{and}\quad
{{{\mathfrak J}}} ({{{\mathfrak X}}})=
\left(
\begin{array}{c}
-b^2 \rho^2\\
-2b^2\rho
\end{array}
\right).$$ The differential matrix of ${{{\mathfrak J}}}$ is given by $$\nabla {{{\mathfrak J}}}({{{\mathfrak X}}})=
\left(
\begin{array}{cc}
-2b^2\rho& 0\\
-2b^2 & 0
\end{array}
\right).$$ As above, let $({T}^+_t )_{t \ge 0}$ (resp. $({T}^{-}_t)_{t \ge 0}$) denote the semigroup on $S({{\mathbb R}}) \times S({{\mathbb R}})$ generated by $$\label{eq:lin ss01}
\partial_t \varepsilon + M^T \, \partial_q \varepsilon =0, \quad ({\text{resp.}} \; \partial_t \varepsilon - M^T \, \partial_q \varepsilon =0 ).$$ where $$M:=M(\rho,\theta)= (\nabla {{{\mathfrak J}}})(\omega),$$ $\rho$ and $\theta$ are given by . We omit the dependence of these semigroups on $(\rho, \theta)$ for lightness of the notations. We remark that the transposed linearized system of (\[eq:hl-ss001\]) around the constant profiles $(\rho, \theta)$ is given by the first equation on the left hand side of . It is easy to show that ${\bar M}= \Lambda M \Lambda^{-1}$ and $\Lambda^T {\bar T}_t^- = T_t^- \Lambda^T$.
Orthogonal decomposition {#sec:dua}
========================
Observe that $\nu_{\beta, \lambda}$ is a product of Gamma distributions. Let us recall that the Gamma distribution $\gamma_{\alpha, k}$ with parameter $(\alpha, k)$ is the probability distribution on $(0,+\infty)$ absolutely continuous with respect to the Lebesgue measure with density $f_{\alpha, k}$ given by $$f_{\alpha,k} (q) = \Big(k^\alpha \Gamma (\alpha)\Big)^{-1} q^{\alpha-1} e^{-q/k}, \quad q>0.$$ Thus, we have $ \nu_{\beta, \lambda} (d\xi) = \prod_{x \in {{\mathbb Z}}} \Big( f_{\lambda+1, \beta^{-1}} (\xi_x) d\xi_x\Big) = \prod_{x \in {{\mathbb Z}}} \Big(\beta f_{\lambda+1, 1} (\beta \xi_x) d\xi_x \Big)$. The generalized Laguerre polynomials $(H_{n}^{(\lambda)})_{n \ge 0}$ form an orthogonal basis of the space ${\mathbb L}^{2} ( \gamma_{\lambda+1,1})$. They satisfy the following equations: $$\label{relations for Laguerre functions}
\begin{split}
&H_{0}^{(\lambda)}=1,\\
&q\cfrac{d}{dq} H_{n}^{(\lambda)} = nH_{n}^{(\lambda)} -(n+\lambda) H_{n-1}^{(\lambda)},\\
& \Big(q \cfrac{d^2}{dq^2} + (\lambda+1-q) \cfrac{d}{dq} +n \Big) H_{n}^{(\lambda)} =0, \\
&(n+1)H_{n+1}^{(\lambda)}(q) = (2n+1+ \lambda -q) H_{n}^{(\lambda)} (q) -(n +\lambda) {H}_{n-1}^{(\lambda)} (q)
\end{split}$$ and the normalization condition $$\int_{0}^{\infty} \Big(H_{n}^{(\lambda)} (q)\Big)^2 f_{\lambda+1,1} (q) \, dq = \cfrac{\Gamma (\lambda +n +1)}{\Gamma (\lambda +1)}\cfrac{1}{n!}.$$ In particular, we have $$\label{eq:fop}
\begin{split}
&H_1^{(\lambda)} (q) = -q + (\lambda +1),\\
&H_{2}^{(\lambda)} (q) = \cfrac{(2+\lambda)(1+ \lambda)}{2} - (\lambda +2) q +\cfrac{q^2}{2}.
\end{split}$$
Let $\Sigma$ be the set composed of configurations $\sigma=(\sigma_x)_{x\in {{\mathbb Z}}} \in {{\mathbb N}}^{{{\mathbb Z}}}$ such that $\sigma_{x} \ne 0$ only for a finite number of $x$. The number $\sum_{x \in {{\mathbb Z}}} \sigma_x$ is called the size of $\sigma$ and is denoted by $|\sigma|$. Let $\Sigma_n = \{ \sigma \in \Sigma \; ; \; |\sigma|=n\}$. On the set of $n$-tuples ${{\mathbf x}}:=(x_1, \ldots,x_n)$ of ${{\mathbb Z}}^n$, we introduce the equivalence relation ${{\mathbf x}}\sim {{\mathbf y}}$ if there exists a permutation $p$ on $\{1, \ldots,n\}$ such that $x_{p(i)} =y_i$ for all $i \in \{1, \ldots,n\}$. The class of ${{\mathbf x}}$ for the relation $\sim$ is denoted by $[{{\mathbf x}}]$ and its cardinal by $c({\bf x})$. Then the set of configurations of $\Sigma_n$ can be identified with the set of $n$-tuples classes for $\sim$ by the one-to-one application: $$[{\bf x}]=[(x_1,\ldots,x_n)] \in {{\mathbb Z}}^n/ \sim \; \rightarrow \sigma^{[{\bf x}]} \in \Sigma_n$$ where for any $y \in {{\mathbb Z}}$, $(\sigma^{[\bf x]})_y= \sum_{i=1}^n {\bf 1}_{y=x_i}$. We will identify $\sigma \in \Sigma_n$ with the occupation numbers of a configuration with $n$ particles, and $[\bf x]$ will correspond to the positions of those $n$ particles.
To any $\sigma \in \Sigma$, we associate the polynomial function $H^{\beta,\lambda}_{\sigma}$ given by $$H^{\beta, \lambda}_{\sigma} (\xi) = \prod_{x \in {{\mathbb Z}}} H_{\sigma_x}^{(\lambda)} (\beta \xi_x).$$ Then, the family $\left\{ H^{\beta,\lambda}_{\sigma} \; ; \; \sigma \in \Sigma \right\}$ forms an orthogonal basis of ${{{\mathbb L}}}^2 (\nu_{\beta, \lambda})$ such that $$\label{eq:prod Hsigma}
\int H^{\beta,\lambda}_\sigma \, H^{\beta,\lambda}_{\sigma'} \, d\nu_{\beta, \lambda} = \delta_{\sigma=\sigma'} {\prod}_{x \in {{\mathbb Z}}} \cfrac{\Gamma (\lambda +\sigma_x +1)}{\Gamma (\lambda+1)}\cfrac{1}{\sigma_x !} = \delta_{\sigma=\sigma'} {{{\mathcal W}}}^{ \lambda} (\sigma),$$ where $$\label{def W}
{{{\mathcal W}}}^{\lambda} (\sigma):={\prod}_{x \in {{\mathbb Z}}} \cfrac{\Gamma (\lambda +\sigma_x +1)}{\Gamma (\lambda+1)}\cfrac{1}{\sigma_x !}$$ and $\delta$ denotes the Kronecker function, so that $\delta_{\sigma=\sigma'}=1$ if $\sigma=\sigma'$, otherwise it is equal to zero.
A function $F:\Sigma \to {{\mathbb R}}$ such that $F(\sigma)=0$ if $\sigma \notin \Sigma_n$ is called a degree $n$ function. Thus, such a function is sometimes considered as a function defined only on $\Sigma_n$. A local function $f \in {\mathbb L}^2 (\nu_{\beta,\lambda})$ whose decomposition on the orthogonal basis $\{ H_{\sigma}^{\beta,\lambda} \, ; \, \sigma \in \Sigma \}$ is given by $f=\sum_{\sigma} F(\sigma) H_{\sigma}^{\beta,\lambda}$ is called of degree $n$ if and only if $F$ is of degree $n$. A function $F: \Sigma_n \to {{\mathbb R}}$ is nothing but a symmetric function $F:{{\mathbb Z}}^n \to {{\mathbb R}}$ through the identification of $\sigma$ with $[{{\mathbf x}}]$. We denote by $\langle \cdot, \cdot \rangle$ the scalar product on $\oplus {\mathbb L}^2 (\Sigma_n)$, each $\Sigma_n$ being equipped with the counting measure. Hence, if $F,G:\Sigma \to {{\mathbb R}}$, we have $$\langle F, G \rangle = \sum_{n\ge 0} \sum_{\sigma \in \Sigma_n} F_n (\sigma) G_n (\sigma) = \sum_{n \ge 0} \sum_{{{\mathbf x}}\in {{\mathbb Z}}^n} \frac{1}{c({\bf x})} \, F_n ({{\mathbf x}}) G_n ({{\mathbf x}}),$$ with $F_n, G_n$ the restrictions of $F,G$ to $\Sigma_n$. We recall that $c({\bf x})$ is the cardinal of $[{\bf x}]$. Since $(\beta,\lambda)$ are fixed through the paper we denote $H_{\sigma}^{\beta,\lambda}$ by $H_{\sigma}$ and ${{{\mathcal W}}}^\lambda (\sigma)$ by ${{{\mathcal W}}} (\sigma)$.
If a local function $f \in {{{\mathbb L}}}^{2} (\nu_{\beta, \lambda})$ is written in the form $f(\xi)=\sum_{\sigma \in \Sigma} F(\sigma) H_{\sigma}(\xi)$ then we have $$({{{\mathcal A}}} f)(\xi) = \sum_{\sigma\in\Sigma } ({{{\mathfrak A}}} F)(\sigma) H_{\sigma}(\xi), \quad ({{{\mathcal S}}} f)(\xi) = \sum_{\sigma\in\Sigma} ({{{\mathfrak S}}} F)(\sigma) H_{\sigma}(\xi)$$ with $$({{{\mathfrak S}}} F)(\sigma) = \sum_{x \in {{\mathbb Z}}} ( F(\sigma^{x,x+1}) - F(\sigma)),$$ where $\sigma^{x,x+1}$ is obtained from $\sigma$ by exchanging the occupation numbers $\sigma_x$ and $\sigma_{x+1}$.
Let us now compute the operator ${{{\mathfrak A}}}$. We have $$({{{\mathcal A}}} H_{\sigma})(\xi) = \sum_{x\in {{\mathbb Z}}} \xi_x (\xi_{x+1} -\xi_{x-1})\partial_{\xi_x} H_{\sigma} (\xi).$$ By the definition of $H_\sigma$ and by the second equality in , it follows that $$({{{\mathcal A}}} H_{\sigma})(\xi) = \beta \sum_{x\in{{\mathbb Z}}} (\xi_{x+1} -\xi_{x-1}) \Big(\sigma_{x} H_{\sigma} (\xi)- (\sigma_x +\lambda) H_{\sigma - \delta_x} (\xi)\Big),$$ where $\sigma-\delta_x$ is the configuration where a particle has been deleted at site $x$ (if there was no particle on site $x$, then $\sigma -\delta_x = \sigma$).
Now, noticing that the fourth equality in can be written as $$\beta q H_{n}^{(\lambda)} (\beta q) = (2n+1 + \lambda) H_{n}^{(\lambda)} (\beta q) - (n+\lambda) H_{n-1}^{(\lambda)} (\beta q) - (n+1) H_{n+1}^{(\lambda)} (\beta q)$$ and performing some change of variables, we have that $$\begin{split}
({{{\mathcal A}}} H_{\sigma})(\xi)& = \sum_{\substack{x,y \in {{\mathbb Z}}\\ |x-y|=1}} a(y-x) (\sigma_x + \lambda) (\sigma_y +1) H_{\sigma + \delta_{y} -\delta_x}(\xi)\\
& -\sum_{x\in{{\mathbb Z}}} (\sigma_x +\lambda) (\sigma_{x+1} -\sigma_{x-1}) \, H_{\sigma -\delta_{x}}(\xi) \\
& + \sum_{x\in{{\mathbb Z}}} (\sigma_x +1) (\sigma_{x+1} -\sigma_{x-1}) \, H_{\sigma + \delta_{x}}(\xi).
\end{split}$$
Here, $a(z)=-1$ if $z=-1$, $a(z)=1$ if $z=1$ and $0$ otherwise. It follows that $${{{\mathfrak A}}} ={{{\mathfrak A}}}_0 + {{{\mathfrak A}}}_- +{{{\mathfrak A}}}_+$$ with $$\begin{split}
&({{{\mathfrak A}}}_0 F) (\sigma) = -\sum_{\substack{x,y \in {{\mathbb Z}}\\ |x-y|=1}} a(y-x) \sigma_x (\sigma_y +1 + \lambda) F(\sigma+\delta_y -\delta_x),\\
&({{{\mathfrak A}}}_+ F)(\sigma)= -\sum_{x\in{{\mathbb Z}}} \sigma_x (\sigma_{x+1} -\sigma_{x-1}) F(\sigma-\delta_x),\\
&({{{\mathfrak A}}}_- F)(\sigma)= \sum_{x\in{{\mathbb Z}}} (\sigma_x-1+\lambda) (\sigma_{x+1} -\sigma_{x-1}) F(\sigma+\delta_x).
\end{split}$$ Observe that if $F$ vanishes outside of $\Sigma_n$ then ${{{\mathfrak A}}}_{\pm } F$ vanishes outside of $\Sigma_{n \mp 1}$ and ${{{\mathfrak A}}}_0$ vanishes outside of $\Sigma_n$. In other words, ${{{\mathfrak A}}}_0 $ keeps fixed the degree of a function, ${{{\mathfrak A}}}_+ $ raises the degree by one while ${{{\mathfrak A}}}_-$ lowers the degree by one.
The Dirichlet form ${{{\mathcal D}}} (f)$ of a local function $f \in {\mathbb L}^2 (\nu_{\beta,\lambda})$ is defined by $${{{\mathcal D}}} (f)=\langle f \,, (-{{{\mathcal S}}} f) \rangle_{\nu_{\beta,\lambda}}= \cfrac{1}{2} \sum_{x \in {{\mathbb Z}}} \int \left( f(\xi^{x,x+1}) - f(\xi) \right)^2\nu_{\beta,\lambda}(d \xi).$$ Recall that $\langle \cdot,\cdot\rangle_{\nu_{\beta,\lambda}}$ denotes the inner product of ${{\mathbb L}}^2(\nu_{\beta,\lambda})$.
Since $f$ has the decomposition $f= \sum_{\sigma \in \Sigma} F(\sigma) H_{\sigma}$ then $$\label{eq:df01}
{{{\mathcal D}}} (f) =\cfrac{1}{2} \sum_{x \in {{\mathbb Z}}} \sum_{\sigma \in \Sigma} {{{\mathcal W}}} (\sigma) \left( F(\sigma^{x,x+1}) -F(\sigma) \right)^2.$$ Let $\Delta_{+}= \left\{ (x,y) \in {{\mathbb Z}}^2 \, ; \, y \ge x+1\right\}$, $\Delta_- = \left\{ (x,y) \in {{\mathbb Z}}^2 \, ; \, y \le x-1\right\}$ and $\Delta_0 = \left\{ (x,x) \, ; \, x \in {{\mathbb Z}}\right\}$. We denote by ${{{\mathbb D}}}_1$ the Dirichlet form of a symmetric simple one dimensional random walk, i.e. $${{{\mathbb D}}}_1 (F) = \cfrac{1}{2} \sum_{x \in {{\mathbb Z}}} (F(x+1) - F(x))^2,$$ where $F: {{\mathbb Z}}\to {{\mathbb R}}$ is such that $\sum_{x\in {{\mathbb Z}}} F^2 (x) < \infty$.
We denote by ${{{\mathbb D}}}_2$ the Dirichlet form of a symmetric simple random walk on ${{\mathbb Z}}^2$ where jumps from $\Delta_{\pm}$ to $\Delta_0$ and from $\Delta_0$ to $\Delta_{\pm}$ have been suppressed and jumps from $(x,x) \in \Delta_0$ to $(x \pm 1, x \pm 1) \in \Delta_0$ have been added, i.e. $${{{\mathbb D}}}_2(F)=\cfrac{1}{2} \sum_{|{{\mathbf e}}|=1} \!\sum_{{{\mathbf x}}\in \Delta_{\pm}, {{\mathbf x}}+ {{\mathbf e}}\in \Delta_{\pm}}\!\! \!\!\!\!\left( F({{\mathbf x}}+ {{\mathbf e}}) -F({{\mathbf x}})\right)^2 + \cfrac{1}{2} \sum_{{{\mathbf x}}\in \Delta_0} \!\left( F({{\mathbf x}}\pm (1,1)) -F({{\mathbf x}})\right)^2,$$ where $F:{{\mathbb Z}}^2 \to {{\mathbb R}}$ is a symmetric function such that $\sum_{{{\mathbf x}}\in {{\mathbb Z}}^2} F^2 ({{\mathbf x}}) < \infty$.
\[lem:df12\] Let $f=\sum_{n=1}^2 \sum_{\sigma \in \Sigma_n} F_n (\sigma) H_{\sigma}$ be a local function such that $F_1$ (resp. $F_2$) is of degree $1$ (resp. degree $2$). There exists a positive constant $C:=C(\lambda)$, independent of $f$, such that $$C^{-1} \left[ {{{\mathbb D}}}_1 (F_1) +{{{\mathbb D}}}_2 (F_2) \right] \le {{{\mathcal D}}} (f) \le C \left[ {{{\mathbb D}}}_1 (F_1) +{{{\mathbb D}}}_2 (F_2) \right].$$
Observe that
- If $\sigma \in \Sigma_1$, then ${{{\mathcal W}}} (\sigma)=(\lambda+1)$.
- If $\sigma \in \Sigma_2$, $\sigma= \delta_x +\delta_y$, $x\ne y$, then ${{{\mathcal W}}} (\sigma)=(\lambda+1)^2$; if $\sigma \in \Sigma_2$, $\sigma=2 \delta_x$, then ${{{\mathcal W}}} (\sigma)= [(\lambda +2)(\lambda +1)]/2$.
This follows from the relation $\Gamma (z+1)=z\Gamma (z)$. Then, by using (\[eq:df01\]) and the identification of functions $F: \Sigma_n \to {{\mathbb R}}$ of degree $n$ with their representations as symmetric functions on ${{\mathbb Z}}^n$, the claim follows.
Triviality of the fluctuations {#sec:triv}
==============================
In this section we prove Theorems \[th:fluct-hs\] and \[th:fluct-ds\] and Corollaries \[CLT Energy flux\], \[CLT volume flux\] and \[vanishing of Energy flux\] above. The proof of Theorems \[th:fluct-hs\] and \[th:fluct-ds\] is standard and relies on a careful analysis of martingales associated to the respective density fields. For this reason we present only the sketch of their proofs. For the interested reader we refer to chapter 11 of [@KL]. We notice that the restrictions on $k$ appearing in the statement of those theorems come from tightness estimates, that we do not prove here since they follow from very similar computations to those presented in [@KL].
To approach the proof of theorems we notice that since $V_b (\eta_x) -1= \xi_x -\log (\xi_x)$, $\eta_x = -b^{-1} \log (\xi_x)$, the problem is reduced to study the fluctuation field of the conserved quantities for the process $(\xi (t))_{t \ge 0}$ at equilibrium under the probability measure $\nu_{\beta, \lambda}$. The fluctuation field for $(\xi (t))_{t \ge 0}$ is defined by $$\label{eq:ZZ}
\mathcal{Z}^{n,\alpha}_t ({{\mathbf G}}) =\frac{1}{\sqrt{n}} \sum_{x\in {{\mathbb Z}}}
{{\mathbf G}}\left(x/n\right) \cdot \left({\omega}_x (tn^{1+\alpha}) - {\omega}\right),$$ where ${{\mathbf G}}$ is a test function belonging to $S({{\mathbb R}}) \times S({{\mathbb R}})$. Recalling (\[lambda matrix\]) we have $$\label{rel Y and Z}
{{{\mathcal Y}}}_t^{n,\alpha} ({{\mathbf G}}) = \cfrac{1}{\sqrt n} \sum_{x \in {{\mathbb Z}}} (\Lambda^T{{\mathbf G}}) (x/n) \cdot (\omega_x (t n^{1+\alpha}) -\omega)={{{\mathcal Z}}}_t^{n, \alpha} (\Lambda^T {{\mathbf G}}).$$
By the relation ${\bar M}= \Lambda M \Lambda^{-1}$, we are able to translate any result about the convergence of ${{{\mathcal Z}}}_\cdot^{n,\alpha}$ into a corresponding result for ${{{\mathcal Y}}}_{\cdot}^{n,\alpha}$.
The hyperbolic scaling
----------------------
For any local function $g:= g(\xi)$ we define the projection ${{{\mathcal P}}}_{\rho, \theta} \, g$ of $g$ on the fields of the conserved quantities by $$({{{\mathcal P}}}_{\rho, \theta} g)(\xi)=(\nabla {\tilde g})(\rho, \theta) \cdot (\omega_0-\omega)$$ where ${\tilde g} (\rho,\theta) = \langle g \rangle_{\nu_{\beta, \lambda}}$ and $\nabla {\tilde g}$ is the gradient of the function ${\tilde g}$.
We have that
For every ${{\mathbf H}} \in S({{\mathbb R}})\times S({{\mathbb R}})$ and every $t>0$, $$\lim_{n \to \infty} {\mathbb E}_{\nu_{\beta,\lambda}} \left[ \left( \int_0^{t} \cfrac{1}{\sqrt{n}} \sum_{x \in {{\mathbb Z}}} {{\mathbf H}}\left(x/n\right)\cdot \left[\tau_x V_{J_{0,1}}(\xi(sn))\right] ds\right)^2\right] =0,$$ where for a local function $g$ we define $V_g(\xi):= g(\xi)-\tilde{g}(\rho,\theta)- {{{\mathcal P}}}_{\rho, \theta} g (\xi )$ and for $\xi\in{(0,+\infty)^{{\mathbb Z}}}$, $\tau_xg(\xi):=g(\tau_x\xi)$, $\tau_x\xi(y):=\xi(x+y)$ and $J_{0,1}$ is given in .
Since we prove a refined version of this proposition we omit its proof. As a consequence of last result, we get that the fluctuation field $({{{\mathcal Z}}}_{\cdot}^{n,0})_{n \ge 1}$ converges in law (in the sense of Theorem \[th:fluct-hs\]) to $\mathcal{Z}_\cdot^0$ solution of the equation at the right hand side of (\[eq:lin ss01\]). Theorem \[th:fluct-hs\] is a simple consequence of this fact.
In order to prove Corollary \[CLT Energy flux\] and Corollary \[CLT volume flux\] we follow the approach first presented in [@RV] and considered also in [@J.L.] (resp. [@G]) for the symmetric (resp. asymmetric) simple exclusion. For that reason we sketch the main steps of the proof. For more details we refer the reader to, for example, the proof of Theorem 4.2 of [@G]. The main goal is to related the energy and volume currents with the density field and to use the result of Theorem \[th:fluct-hs\]. For that purpose and whenever the total energy (resp. volume) at $\eta$ is finite we can write down the energy (resp. volume) flux through the bond $\{x,x+1\}$ during the time interval $[0,tn]$, as: $$\mathcal{E}_{x,x+1}^n(t):=\sum_{y\geq{x+1}}\Big\{V_b(\eta_y(tn))-V_b(\eta_y(0))\Big\}$$ $$\Big( \text{resp. \; } \mathcal{V}_{x,x+1}^n(t):=\sum_{y\geq{x+1}}\Big\{\eta_y(tn)-\eta_y(0)\Big\}\Big).$$
In such case, we can relate the energy (resp. volume) flux given above with the energy-volume fluctuation field as $$\mathcal{E}_{x,x+1}^n(t):=\mathcal{Y}_t^n(H_x^1)-\mathcal{Y}^n_0(H_x^1)$$ $$\Big( \textrm{resp. \; } \mathcal{V}_{x,x+1}^n(t):=\mathcal{Y}_t^n(H_x^2)-\mathcal{Y}^n_0(H_x^2),$$ where $$H_x^1(y) = \left(
\begin{array}{c}
\bf 1_{\{y\geq{x}\}} \\
0
\end{array}
\right), \quad
H_x^2(y) = \left(
\begin{array}{c}
0 \\
\bf 1_{\{y\geq{x}\}}
\end{array}
\right).$$ Since the function $\bf 1_{\{y\geq{x}\}}$ does not belong to our space of test functions for which we derived Theorem \[th:fluct-hs\] we first show that
\[prop nec\] For every $t\geq{0}$, $$\lim_{\ell\to\infty} \mathbb{E}_{\nu_{\beta,\lambda}}\Big[\Big(\mathcal{E}_{x,x+1}^n(t)-(\mathcal{Y}_t^n(G_{\ell,x}^1)-\mathcal{Y}^n_0(G_{\ell,x}^1))\Big)^2\Big]=0,$$ $$(\textrm{resp.} \lim_{\ell\to\infty} \mathbb{E}_{\nu_{\beta,\lambda}}\Big[\Big(\mathcal{V}_{x,x+1}^n(t)-(\mathcal{Y}_t^n(G_{\ell,x}^2)-\mathcal{Y}^n_0(G_{\ell,x}^2))\Big)^2\Big]=0,$$ where $$G_{\ell,x}^1(y) = \left(
\begin{array}{c}
G_{\ell,x}(y) \\
0
\end{array}
\right), \quad
G_{\ell,x}^2(y) = \left(
\begin{array}{c}
0 \\
G_{\ell,x}(y)
\end{array}
\right)$$ and $G_{\ell,x}(y):=(1-y/\ell)\bf 1_{\{x\leq{y}\leq{x+\ell}\}}$.
The proof of last result follows the same lines as in the proof of Proposition 4.1 of [@G] and for that reason we omitted it. We notice that, at this point we are still not able to apply Theorem \[th:fluct-hs\] since $G_{\ell,x}^1$ and $G_{\ell,x}^2$ are not functions in $\mathcal{S}({{\mathbb R}})$. Therefore, we approximate in $L^2({{\mathbb R}})$ each one of these functions by smooth functions for which Theorem \[th:fluct-hs\] holds. Then, the proof of Corollary \[CLT Energy flux\] and \[CLT volume flux\] follows combining the previous proposition with Theorem \[th:fluct-hs\]. For more details on this argument, we refer the reader to [@G].
Finally, in order to compute the limiting variance, for example for the energy flux, we do the following. Here we take $x=0$ to simplify the notation $$\begin{split}
{{{\mathbb E}}_{Q}}[Z^e_{t}Z^e_{s}]&={{{\mathbb E}}_{Q}}[\{\mathcal{Y}_{t}(H_0^1)-\mathcal{Y}_{0}(H_0^1)\}\{\mathcal{Y}_{s}(H_0^1)-\mathcal{Y}_{0}(H_0^1)\}]\\
&\hspace{0.1cm}\begin{split}=\lim_{\ell\rightarrow{\infty}}{{{\mathbb E}}_{Q}}\Big[\mathcal{Y}_{t}(G_{\ell,0}^1)\mathcal{Y}_{s}(G_{\ell,0}^1)-&\mathcal{Y}_{t}(G_{\ell,0}^1)\mathcal{Y}_{0}(G_{\ell,0}^1)\\
-\mathcal{Y}_{s}(G_{\ell,0}^1)&\mathcal{Y}_{0}(G_{\ell,0}^1)+\mathcal{Y}_{0}(G_{\ell,0}^1)\mathcal{Y}_{0}(G_{\ell,0}^1)\Big]\end{split}
\end{split}$$
Now, to compute last expectation we use the change of variables. Notice that for $H,G\in{\mathcal{S}({{\mathbb R}})}$ we have that ${{{\mathbb E}}_{Q}}[\mathcal{Z}^0_t(H)\mathcal{Z}^0_0(G)]:=\left<T^{-}_t H \cdot \chi G\right>$. Combining this with , it follows that ${{{\mathbb E}}_{Q}}[\mathcal{Y}_t(H)\mathcal{Y}_0(G)]:=\left<T^{-}_t(\Lambda^T H)\cdot \chi\Lambda^TG\right>$. By the definition of $(T_t^{-})_{t\geq{0}}$ we have for $G_1, G_2$ test functions in $S({{\mathbb R}})$: $$T^{-}_{t}\left(
\begin{array}{c}
G_1(x) \\
G_2(x)
\end{array}
\right)
= \left(
\begin{array}{c}
\frac{1}{\rho}\Big(G_2(x-2b^2\rho t)-G_2(x)\Big)+G_1(x-2b^2\rho t) \\
G_2(x)
\end{array}
\right). \quad$$ As a consequence we obtain that $$\begin{split}
{{{\mathbb E}}_{Q}}[&Z^e_{t}Z^e_{s}]=\Big(1-{\frac{1}{\rho}}\Big)^2\Big(\frac{\lambda+1}{\beta^2}\Big)\\
&\times\lim_{\ell\rightarrow{\infty}}\int_{\mathbb{R}}\Big(G_\ell^t(x)G_\ell^s(x)-G_\ell^t(x)G_\ell(x)-G_\ell^s(x)G_\ell(x)+G_\ell(x)G_\ell(x)\Big)dx,
\end{split}$$ where for $t\geq{0}$, $G_\ell^{\,t}(x):=G_{\ell,0}(x-2b^2\rho t)$. Now, using and the proof ends. Analogously, repeating the computations above, replacing $H_0^1$ by $H_0^2$ we get the covariance for the volume flux.
The longer time scale
---------------------
Since in the hyperbolic time scale the initial fluctuations for the field $\mathcal{Z}^{n,\alpha}_\cdot$ are transported by the transposed linearized system given on the right hand side of (\[eq:lin ss01\]), we redefine the fluctuation field ${\widehat {{{\mathcal Z}}}}_{\cdot}^{n,\alpha}$, $\alpha>0$, on ${{\mathbf G}}\in S({{\mathbb R}}) \times S({{\mathbb R}})$, by $${\widehat {{{\mathcal Z}}}}_t^{n,\alpha} ({{\mathbf G}})= {{{\mathcal Z}}}_t^{n, \alpha} \left( {T}^{+}_{tn^{\alpha}} {{\mathbf G}}\right).$$
By Dynkin’s formula, see for example Appendix 1, section 5 of [@KL] $${{{\mathcal M}}}_t^{n, \alpha} ({{\mathbf G}}) = {\widehat {{{\mathcal Z}}}}_t^{n,\alpha} ({{\mathbf G}}) - {\widehat {{{\mathcal Z}}}}_0^{n,\alpha} ({{\mathbf G}}) - \int_{0}^t \left\{ n^{1+\alpha} {{{\mathcal L}}} \left( {\widehat {{{\mathcal Z}}}}_s^{n,\alpha} ({{\mathbf G}})\right) + \partial_s {\widehat {{{\mathcal Z}}}}_s^{n,\alpha} ({{\mathbf G}})\right\} \, ds$$ is a martingale with quadratic variation given by $$\langle {{{\mathcal M}}}^{n, \alpha} \rangle_t=\int_{0}^t n^{1+\alpha} {{{\mathcal L}}} \left( {\widehat {{{\mathcal Z}}}}_s^{n,\alpha} ({{\mathbf G}})\right)^2-2n^{1+\alpha} \left( {\widehat {{{\mathcal Z}}}}_s^{n,\alpha} ({{\mathbf G}})\right){{{\mathcal L}}} \left( {\widehat {{{\mathcal Z}}}}_s^{n,\alpha} ({{\mathbf G}})\right) \, ds.$$ A simple computation shows that ${\mathbb E}_{\nu_{\beta,\lambda}}[\langle {{{\mathcal M}}}^{n, \alpha} \rangle_t]$ vanishes as $n$ goes to ${\infty}$ for $\alpha<1$. This is equivalent to saying that the martingale ${{{\mathcal M}}}^{n, \alpha}_t$ vanishes as $n$ goes to ${\infty}$ in ${\mathbb L}^{2} ({{\mathbb P}}_{\nu_{\beta,\lambda}})$, for $\alpha<1$. Observe that, by definition of $(T^{+}_t)_{t \ge 0}$, we have $$\begin{split}
\partial_s {\widehat {{{\mathcal Z}}}}_s^{n,\alpha} ({{\mathbf G}}) &= -\cfrac{n^\alpha}{\sqrt n} \sum_{x \in {{\mathbb Z}}} M^T \left[ \partial_q\left( T^+_{s n^{\alpha}} {{\mathbf G}}\right) (x/n) \right] \cdot (\omega_x (s n^{1+\alpha}) -\omega)\\
&=-\cfrac{n^\alpha}{\sqrt n} \sum_{x \in {{\mathbb Z}}} \left[ \partial_q\left( T^+_{s n^{\alpha}} {{\mathbf G}}\right) (x/n) \right] \cdot M(\omega_x (s n^{1+\alpha}) -\omega).
\end{split}$$ On the other hand, the first term in the integral part of the martingale ${{{\mathcal M}}}_t^{n, \alpha} ({{\mathbf G}})$ is equal to $$\cfrac{n^\alpha}{\sqrt n} \sum_{x \in {{\mathbb Z}}} n\Big( (T^+_{sn^{\alpha}} {{\mathbf G}}) \left( \cfrac{x+1}{n}\right) - (T^+_{sn^{\alpha}} {{\mathbf G}}) \left( \cfrac{x}{n} \right)\Big) \cdot \Big( J_{x,x+1} (\xi(sn^{1+\alpha})) \, - \, \langle J_{x,x+1} \rangle_{}\nu_{\beta,\lambda}\Big).$$ Performing a Taylor expansion, we can replace this term, up to a term vanishing as $n$ goes to $\infty$ in ${\mathbb L}^{2} ({{\mathbb P}}_{\nu_{\beta,\lambda}})$, by $$\cfrac{n^\alpha}{\sqrt n} \sum_{x \in {{\mathbb Z}}} \Big((\partial_q T^+_{s n^{\alpha}} {{\mathbf G}}) (x/n)\Big) \cdot \Big( J_{x,x+1} (\xi(sn^{1+\alpha})) \, - \, \langle J_{x,x+1} \rangle_{}\nu_{\beta,\lambda}\Big).$$
Thus, in order to show that $$\label{vanish Z field}
\lim_{n \to \infty}{\mathbb E}_{\nu_{\beta,\lambda}} \left[ \Big({\widehat {{{\mathcal Z}}}}_t^{n,\alpha} ({{\mathbf G}}) - {\widehat {{{\mathcal Z}}}}_0^{n,\alpha} ({{\mathbf G}})\Big)^2\right]=0,$$ it remains to show that $$\lim_{n \to \infty} {\mathbb E}_{\nu_{\beta,\lambda}} \left[ \left( \cfrac{n^\alpha}{\sqrt n} \int_0^t \, ds \, \sum_{x \in {{\mathbb Z}}} (\partial_q T^+_{s n^{\alpha}} {{\mathbf G}}) (x/n) \cdot \Theta_{x} (\xi(sn^{1+\alpha})) \right)^2\right]=0$$ where for $\xi\in{(0,+\infty)^{{\mathbb Z}}}$ $$\Theta_x(\xi)= J_{x,x+1}(\xi) \, - \, \langle J_{x,x+1} \rangle_{\nu_{\beta,\lambda}} -M\, (\omega_x -\omega).$$ Observe that in this formula, $M:=M(\rho, \theta)$ is the differential with respect to $(\rho,\theta)$ of the function $ \langle J_{x,x+1} \rangle_{\nu_{\beta,\lambda}}$ as computed below . A simple computation shows that for $\xi\in{(0,+\infty)^{{\mathbb Z}}}$ $$\Theta_x(\xi)=
\left(
\begin{array}{c}
-b^{2} (\xi_{x+1} -\rho)(\xi_x -\rho)- (\gamma + b^2 \rho) \nabla \xi_x \\
-\nabla (b^2 \xi_x + \gamma \log (\xi_x))
\end{array}
\right).$$
The discrete gradient terms appearing in the previous expression, permit to perform another discrete integration by parts and the resulting terms vanish in ${\mathbb L}^{2} ({{\mathbb P}}_{\nu_{\beta,\lambda}})$ as $n$ goes to $\infty$, for $\alpha<1$. Using the smoothness of the function ${{\mathbf G}}$, we see that it only remains to show the following theorem with $\varphi (s,q)$ equal to the first component of the column vector $\partial_q T^+_{s} {{\mathbf G}}$.
\[th:BGII\]
Fix $\alpha <1/3$ and let $\varphi:\mathbb{R}^+\times {{\mathbb R}}\rightarrow{{{\mathbb R}}}$ be such that for any $t \ge 0$, $\varphi(t,\cdot)\in S({{\mathbb R}})$. For every $t>0$ $$\lim_{n \to \infty} {\mathbb E}_{\nu_{\beta,\lambda}} \left[ \left(\int_0^t \cfrac{n^{\alpha}}{\sqrt{n}} \sum_{x \in {{\mathbb Z}}} \varphi (sn^\alpha,x/n) (\xi_x(sn^{1+\alpha}) -\rho) (\xi_{x+1}(sn^{1+\alpha}) -\rho) \, ds \right)^2\right] =0$$
In the following, $C, C_0, C_1, \ldots$ denote constants independent of $n$ whose values can change from line to line.
Let $f_s(\xi)$ be the function defined by $$f_s (\xi)= \sum_{x \in {{\mathbb Z}}} \varphi (s,x/n) H_{\delta_x +\delta_{x+1}}(\xi)=-\beta^2 \sum_{x \in {{\mathbb Z}}} \varphi (s,x/n) (\xi_x -\rho) (\xi_{x+1} -\rho).$$ The last equality follows from and .
We have the following upper bound $$\begin{split}
{\mathbb E}_{\nu_{\beta,\lambda}} \left[ \left( \int_0^t\!\! f_{sn^\alpha}(\xi (s n^{1+\alpha}))\,ds \right)^2\right] & \le C \int_0^t \langle\, f_{sn^\alpha}, (s^{-1} - n^{1+\alpha} {{{\mathcal L}}})^{-1} f_{sn^\alpha} \rangle_{\nu_{\beta,\lambda}}ds \\
&= \cfrac{C} {n^{1+\alpha}} \int_{0}^t \Big\langle f_{sn^\alpha}, \left( \cfrac{1}{s n^{1+\alpha}} - {{{\mathcal L}}} \right)^{-1}\!\!\! \!\!f_{sn^\alpha} \Big\rangle_{\nu_{\beta,\lambda}}\!\!ds\\
&\le \cfrac{C} {n^{1+\alpha}}\int_{0}^t \Big\langle f_{sn^\alpha}, \left( \cfrac{1}{s n^{1+\alpha}} - \gamma {{{\mathcal S}}} \right)^{-1}\!\!\!\!\! f_{sn^\alpha} \Big \rangle_{\nu_{\beta,\lambda}}\!\!ds.
\end{split}$$
In the first inequality above we used Lemma 3.9 of [@S.] applied to this setting. We notice that since our test functions depend on time, the lemma of [@S.] has to be modified as written here. To prove the last result one can simply adapt the proof of Lemma 4.3 of [@C.L.O.] to this case.
In order to simplify notations, let us define ${\varepsilon}= 1/sn^{1+\alpha}$.
We denote by $\Sigma_2^0$ the set of configurations $\sigma$ of $\Sigma_2$ such that $\sigma= 2 \delta_x$, $x \in {{\mathbb Z}}$, and $\Sigma_2^{\pm}$ the complementary set of $\Sigma_2^0$ in $\Sigma_2$, i.e. the set of configurations $\sigma \in \Sigma_2$ such that $\sigma=\delta_x + \delta_y$, $y \ne x \in {{\mathbb Z}}$. Observe that $f_{sn^\alpha}$ is a function of degree $2$ with a decomposition in the form $f_{sn^\alpha}=\sum_{\sigma \in \Sigma_2} \Phi _{sn^\alpha}(\sigma) H_{\sigma}$ which satisfies $\Phi_{sn^\alpha}(\sigma)=0$ if $\sigma \in \Sigma_2^0$. We have that (see e.g. [@S.]) $$\Big\langle f_{sn^\alpha}\, , \left( {\varepsilon}-\gamma {{{\mathcal S}}} \right)^{-1} f_{sn^\alpha} \Big \rangle_{\nu_{\beta,\lambda}} = \sup_{g} \Big\{ 2 \langle f_{sn^\alpha}, g\rangle_{\nu_{\beta,\lambda}} - {\varepsilon}\langle g\;,g\rangle_{\nu_{\beta,\lambda}} - \gamma {{{\mathcal D}}} (g) \Big\}$$ where the supremum is taken over local functions $g \in {\mathbb L}^2 (\nu_{\beta,\lambda})$. Decompose $g$ appearing in this variational formula as $g=\sum_{\sigma} G(\sigma) H_{\sigma}$. Recall that $\{H_{\sigma} \, ; \, \sigma \in \Sigma\}$ are orthogonal, that the function $f_{sn^\alpha}$ is a degree $2$ function such that $\Phi_{sn^\alpha}(\sigma)=0$ for any $\sigma \notin \Sigma_2^{\pm}$ and formula (\[eq:df01\]) for the Dirichlet form ${{{\mathcal D}}} (g)$. Thus, we can restrict this supremum over degree $2$ functions $g$ such that $G(\sigma)=0$ if $\sigma \in \Sigma_2^0$. Then, by Lemma \[lem:df12\], we have $$\begin{split}
\Big\langle f_{sn^\alpha}, \left( {\varepsilon}-\gamma {{{\mathcal S}}} \right)^{-1} f_{sn^\alpha}& \Big \rangle_{\nu_{\beta,\lambda}}\le \sup_{G} \left\{ \sum_{x \ne y} \Phi_{sn^\alpha}(x,y) G(x,y) - {\varepsilon}\sum_{\substack{(x,y) \in {{\mathbb Z}}^2\\ x\neq y}} G^2 (x,y) \right.\\
&\left.\quad \quad \quad-C \sum_{|{{\mathbf e}}| =1}\sum_{\substack{(x,y) \in \Delta^{\pm}\\ (x,y) + {{\mathbf e}}\in \Delta^{\pm}}} \Big( G((x,y)+{{\mathbf e}}) -G(x,y) \Big)^2\right\}
\end{split}$$ where $C:=C(\lambda, \gamma)$, $\Delta_{\pm} = \{ (x,y) \in {{\mathbb Z}}^2 \, ; \, x\ne y\}$ and as usual we identify the functions defined on $\Sigma_n$ with symmetric functions defined on ${{\mathbb Z}}^n$.
In order to get rid of the geometric constraints appearing in the last term of the variational formula, for any symmetric function $G$ defined on the set $\Delta_{\pm}$, we denote by ${\tilde G}$ its extension to ${{\mathbb Z}}^2$ defined by $${\tilde G} (x,y) =G (x,y) \; \text{if}\; x\ne y, \quad {\tilde G} (x,x) = \cfrac{1}{4} \sum_{|{{\mathbf e}}|=1} G((x,x) +{{\mathbf e}}).$$ It is trivial that $$\begin{split}
& \sum_{(x,y) \in {{\mathbb Z}}^2} {\tilde G}^2 (x,y)\le C\sum_{\substack{(x,y) \in {{\mathbb Z}}^2\\ x\neq y}} {G}^{2} (x,y) ,\\
\textrm{and}\\
& \sum_{|{{\mathbf e}}| =1}\sum_{(x,y) \in {{\mathbb Z}}^2} \Big({\tilde G} ((x,y)+{{\mathbf e}}) -{\tilde G} (x,y) \Big)^2 \\
&\quad \quad \quad \quad \quad \quad\quad \quad \quad\quad \quad \quad\le C \sum_{|{{\mathbf e}}| =1}\sum_{\substack{(x,y) \in \Delta^{\pm}\\ (x,y) + {{\mathbf e}}\in \Delta^{\pm}}} \Big(G((x,y)+{{\mathbf e}}) -G(x,y) \Big)^2 .
\end{split}$$
Thus, we have $$\begin{split}
\Big\langle f_{sn^\alpha}, \left( {\varepsilon}-\gamma {{{\mathcal S}}} \right)^{-1}& f_{sn^\alpha}\Big \rangle_{\nu_{\beta,\lambda}} \\
\le C_0 \sup_{G} &\left\{ \sum_{(x,y) \in {{\mathbb Z}}^2} \Phi _{sn^\alpha}(x,y) G(x,y) - C_1 {\varepsilon}\sum_{(x, y) \in {{\mathbb Z}}^2} G^2 (x,y) \right. \\
&\left. \quad \quad \quad -C_2 \sum_{|{{\mathbf e}}| =1}\sum_{(x,y) \in {{\mathbb Z}}^2 } \Big( G((x,y)+{{\mathbf e}}) -G(x,y) \Big)^2\right\}
\end{split}$$ where the supremum is now taken over all symmetric local functions $G:{{\mathbb Z}}^2 \to {{\mathbb R}}$. Notice that the last variational formula is equal to the resolvent norm, for a simple symmetric two dimensional random walk, of the function $\Phi_{sn^\alpha}$. By using Fourier transform one can easily show that this supremum is equal to $$\cfrac{C_0}{4} \int_{[0,1]^2} \cfrac{|{\hat \Phi}_{sn^\alpha}({{\mathbf k}})|^2}{C_1 {\varepsilon}+ 4C_2 \sum_{i=1}^2 \sin^{2} (\pi k_i) } d{{\mathbf k}}$$ where the Fourier transform ${\hat \Phi}_{sn^\alpha}$ of $\Phi_{sn^\alpha}$ is given by $${\hat \Phi}_{sn^\alpha}( {{\mathbf k}}) = \sum_{(x,y) \in {{\mathbb Z}}^2} \Phi_{sn^\alpha}(x,y) e^{2i\pi (k_1 x +k_2 y)}, \quad {{\mathbf k}}= (k_1,k_2) \in [0,1]^2.$$ By definition of $f_{sn^\alpha}$, we have $\Phi_{sn^\alpha}(x,y)=\cfrac{1}{2} \Big(\varphi (sn^\alpha,x/n) + \varphi(sn^\alpha,y/n)\Big)$ if $|x-y|=1$ and $0$ otherwise. Consequently, we have $$\begin{split}
\Big\langle f_{sn^\alpha}, &\left( {\varepsilon}-\gamma {{{\mathcal S}}} \right)^{-1} f_{sn^\alpha} \Big \rangle_{\nu_{\beta,\lambda}} \le \cfrac{C_0}{16} \int_{[0,1]^2} \cfrac{\left| \sum_{x\in {{\mathbb Z}}} \varphi (sn^\alpha,x/n) e^{2i \pi x (k_1 +k_2)} \right|^2}{C_1 {\varepsilon}+ 4C_2 \sum_{i=1}^2 \sin^{2} (\pi k_i) } d{{\mathbf k}}\\
&= \cfrac{C_0}{16} \int_{[0,2]}\!\!\Big(\int_{[0,1]} \cfrac{\textbf{1}_{[\sup(1-p,1),\inf(1,0)]}(p)\left| \sum_{x\in {{\mathbb Z}}} \varphi (sn^\alpha,x/n) e^{2i \pi x p} \right|^2}{C_1 {\varepsilon}+ 4C_2 \sin^{2} (\pi k_1) +4C_2 \sin^{2} (\pi (p-k_1)) } dk_1 \,\Big) dp\\
&= \cfrac{C_0}{16} \int_{[0,1]}\Big( \int_{[0,1]}\cfrac{\left| \sum_{x\in {{\mathbb Z}}} \varphi (sn^\alpha,x/n) e^{2i \pi x p} \right|^2}{C_1 {\varepsilon}+ 4C_2 \sin^{2} (\pi k_1) +4C_2 \sin^{2} (\pi (p-k_1)) } dk_1 \,\Big) dp\\
\end{split}$$ where we used the change of variables $p=k_2+k_1$ for the first equality and the periodicity of the functions involved for the second one. It follows that $$\begin{split}
\Big\langle f_{sn^\alpha}, \left( {\varepsilon}-\gamma {{{\mathcal S}}} \right) &^{-1} f_{sn^\alpha} \Big \rangle_{\nu_{\beta,\lambda}} \\
& \le \cfrac{C_0}{16} \int_{[0,1]} {\left| \sum_{x\in {{\mathbb Z}}} \varphi (sn^\alpha,x/n) e^{2i \pi x p} \right|^2} dp \int_{[0,1]} \cfrac{dk_1}{C_1 {\varepsilon}+ 4C_2 \sin^{2} (\pi k_1) } \\&\le \cfrac{C}{\sqrt{{\varepsilon}}} \int_{[0,1]} {\left| \sum_{x\in {{\mathbb Z}}} \varphi (sn^\alpha,x/n) e^{2i \pi x p} \right|^2} dp.
\end{split}$$ Observe now that $$\begin{split}
\int_{[0,1]} {\left| \sum_{x\in {{\mathbb Z}}} \varphi (sn^\alpha x/n) e^{2i \pi x p} \right|^2}dp &= \sum_{x\in {{\mathbb Z}}} \varphi^2 (sn^{\alpha}, x/n) \le C n.
\end{split}$$ Putting everything together, we get that $$\begin{split}
{\mathbb E}_{\nu_{\beta,\lambda}} \Big[ \Big(\int_0^t \cfrac{n^{\alpha}}{\sqrt{n}} \sum_{x \in {{\mathbb Z}}} \varphi (sn^\alpha,x/n) (\xi_x(sn^{1+\alpha}) -\rho) (\xi_{x+1}&(sn^{1+\alpha}) -\rho)\, ds\Big)^2 \Big] \\
&\le \frac{ Ct n^{2\alpha -1}}{n^{1+\alpha}} \int_{0}^{t} \cfrac{n}{\sqrt{{\varepsilon}}}\;ds.
\end{split}$$ Since ${\varepsilon}:=1/sn^{1+\alpha}$ last expression vanishes as $n$ goes to $\infty$, if $\alpha<1/3$.
Now, in order to prove Corollary \[vanishing of Energy flux\] we follow the same arguments as in the proof of Proposition 9.3 of [@G] and we proceed as follows. Whenever the total energy (resp. volume) at $\eta$ is finite we can write down: $$\label{longer energy}
\begin{split}
& \mathcal{E}_{u_t^{x,\alpha}(n)}^n(t):=\sum_{y\geq{u_t^{x,\alpha}(n)}}\Big\{V_b(\eta_y(tn^{1+\alpha}))-V_b(\eta_y(0))\Big\}, \\
\Big(&\text{ resp. \, } \mathcal{V}_{u_t^{x,\alpha}(n)}^n(t):=\sum_{y\geq{u_t^{x,\alpha}(n)}}\Big\{\eta_y(tn^{1+\alpha})-\eta_y(0)\Big\}\Big).
\end{split}$$ In order to justify the previous equalities one can repeat the same arguments as used in the hyperbolic scaling. Now, we use the change of variables to define the energy (resp. volume) flux through the time-dependent bond $\{u_t^{x,\alpha}(n),u_t^{x,\alpha}(n)+1\}$ during the time interval $[0,tn^{1+\alpha}]$. For that purpose, we define the flux fields in terms of $\xi_x$ such that $$\begin{split}
&\tilde{\mathcal E}_{x-1,x}^n(t)-\tilde{\mathcal E}_{x,x+1}^n(t):=\xi_x(tn^{1+\alpha})-\xi_x(0) \\
\Big( \text{resp. \; } &\tilde{\mathcal{V}}_{x-1,x}^n(t)-\tilde{\mathcal{V}}_{x,x+1}^n(t):=\log(\xi_x(tn^{1+\alpha}))-\log(\xi_x(0)).
\end{split}$$ As above, when it makes sense, we have that $$\begin{split}
&\tilde{\mathcal E}_{u_t^{x,\alpha}(n)}^n(t):=\sum_{y\geq{u_t^{x,\alpha}(n)}}\Big\{\xi_y(tn^{1+\alpha})-\xi_y(0)\Big\} \\
\Big( \text{resp. \; } &\tilde{\mathcal{V}}_{u_t^{x,\alpha}(n)}^n(t):=\sum_{y\geq{u_t^{x,\alpha}(n)}}\Big\{\log(\xi_y(tn^{1+\alpha}))-\log(\xi_y(0))\Big\}\Big)
\end{split}$$ and in this case we can write the previous fields in terms of $\widehat Z_t^{n,\alpha}$. A simple computation shows that Proposition \[prop nec\] can similarly stated for last fields. Combining this with we have that $$\mathcal{E}_{u_t^{x,\alpha}(n)}^n(t):=\tilde{\mathcal E}_{u_t^{x,\alpha}(n)}^n(t)-\tilde{\mathcal V}_{u_t^{x,\alpha}(n)}^n(t),
\quad \quad \mathcal{V}_{u_t^{x,\alpha}(n)}^n(t):=-\frac{1}{b}\tilde{\mathcal V}_{u_t^{x,\alpha}(n)}^n(t).$$ Then, applying to $G_{\ell,x}^1(y)=
\left(
\begin{array}{c}
G_{\ell,x}(y) \\
0
\end{array}
\right)$ we obtain that $$\lim_{n \to \infty} {\mathbb E}_{\nu_{\beta,\lambda}} \left[ \left( \frac{1}{\sqrt n}\Big\{\tilde{\mathcal E}_{u_t^{x,\alpha}(n)}^n(t)-{{\mathbb E}}_{{\nu_{\beta,\lambda}}}[\tilde{\mathcal E}_{u_t^{x,\alpha}(n)}^n(t)]\Big\}\right)^2\right]=0.$$ On the other hand, applying to $$\tilde G_{\ell,x}(y)=
\left(
\begin{array}{c}
\frac{1}{\rho}G_{\ell,x}(y-u_t^{x,\alpha}(n)) \\
-G_{\ell,x}(y-u_t^{x,\alpha}(n))
\end{array}
\right)$$ we obtain that $$\begin{split}
\lim_{n \to \infty} {\mathbb E}_{\nu_{\beta,\lambda}} &\left[ \left( \frac{1}{\rho}\frac{1}{\sqrt n}\Big\{\tilde{\mathcal E}_{u_t^{x,\alpha}(n)}^n(t)-{{\mathbb E}}_{{\nu_{\beta,\lambda}}}[\tilde{\mathcal E}_{u_t^{x,\alpha}(n)}^n(t)]\Big\}\right.\right.\\
&\left.\left.\quad \quad \quad\quad \quad \quad\quad \quad \quad-\frac{1}{\sqrt n}\Big\{\tilde{\mathcal V}_{u_t^{x,\alpha}(n)}^n(t)-{{\mathbb E}}_{{\nu_{\beta,\lambda}}}[\tilde{\mathcal V}_{u_t^{x,\alpha}(n)}^n(t)] \Big\}\right)^2\right]=0.
\end{split}$$ Now, Corollary \[vanishing of Energy flux\] follows easily from the previous results.
\[velocity\] From , the hydrodynamic equation of $\rho$ is independent of $\theta$ and it can be rewritten as $\partial_t \rho-2b^2\rho\partial_q\rho=0.$ Following the system along the characteristics for $\rho$, that is, removing the velocity $2b^2\rho$ from the system, we do not see a time evolution for $\rho$, and since $1/\rho\partial_t\rho-\partial_t\theta=0$, nor for $\theta$. Therefore, translating the velocity $2b^2\rho$ in terms of the original variables it corresponds to $2b\bar\lambda/\bar\beta$ and that is the reason why we took the time dependent bond as written in Corollary \[vanishing of Energy flux\].
Diffusivity {#sec:diff}
===========
In this section we prove Theorem \[th:diffusivity\]. Our proof is based on the resolvent methods introduced in [@B.; @12] and developed in few other contexts (e.g. [@Bermerde; @SS; @TTV]). Some differences with these previous works are the presence of two and not only one conserved quantity and the degeneracy of the symmetric part of the generator.
[The main steps of the proof are the following. First we use the microscopic change of variables and express the Laplace transform of the current-current correlation function as a resolvent norm in a suitable Hilbert space (see (\[eq:lapltransfW\])). Then, we rewrite this resolvent norm as the supremum over the set of local functions of a functional acting on these functions (see (\[eq:varformula11\])). To get a lower bound we restrict the supremum over degree two functions. The estimate of the value of the functional for a given degree two function remains in general very difficult. Thus we replace the functional restricted to the set of degree two functions by an equivalent functional simpler to estimate. This is accomplished through Lemma \[lem:compa\], Lemma \[lem:DF\] and Lemma \[lem:007\]. In the context of the asymmetric simple exclusion, this replacement step is called the “free particles approximation” ([@B.]) or the “hard core removal” ([@12]). It is then possible to estimate the value of this equivalent functional for a suitable degree two test function.]{}
We fix $\rho>0, \theta \in {{\mathbb R}}$ and denote by $\beta,\lambda$ the chemical potentials given by (\[eq:chimpot\]). Let also $({\bar \beta}, {\bar \lambda})$ be given in terms of $(\beta,\lambda)$ by (\[eq:relblbl\]).
Recall the definition of $\hat J_{x,x+1}$ given in . We introduce the normalized currents ${{{\mathbf j}}}_{x,x+1}$, ${{{\mathbf j}}}^{\prime}_{x,x+1}$ and ${{{\mathbf J}}}_{x,x+1}$ corresponding to the process $(\xi (t))_{t \ge 0}$, which are defined by $$\begin{split}
&{{{\mathbf j}}}_{x,x+1}(\xi) \\
&= j_{x,x+1}(\xi) - \langle j_{x,x+1} \rangle_{\nu_{\beta,\lambda}} - \partial_{\rho} \langle j_{x,x+1} \rangle_{\nu_{\beta,\lambda}} (\xi_x -\rho) - \partial_{\theta} \langle j_{x,x+1} \rangle_{\nu_{\beta,\lambda}} (\log (\xi_x) -\theta), \\
&{{{\mathbf j}}}^{\prime}_{x,x+1}(\xi) \\
&= j^{\prime}_{x,x+1}(\xi) - \langle j^{\prime}_{x,x+1} \rangle_{\nu_{\beta,\lambda}} - \partial_{\rho} \langle j^{\prime}_{x,x+1} \rangle_{\nu_{\beta,\lambda}} (\xi_x -\rho) - \partial_{\theta} \langle j^{\prime}_{x,x+1} \rangle_{\nu_{\beta,\lambda}} (\log (\xi_x) -\theta),\\
&{{{\mathbf J}}}_{x,x+1}(\xi) = {{{\mathbf j}}}_{x,x+1}(\xi) -{{{\mathbf j}}}^{\prime}_{x,x+1}(\xi).
\end{split}$$
Since $\langle j_{x,x+1} \rangle_{\nu_{\beta,\lambda}}=-b^2 \rho^2$ and $\langle j^{\prime}_{x,x+1} \rangle_{\nu_{\beta,\lambda}} = -2b^2 \rho$, we get $$\label{norm. currents 2}
\begin{split}
&{{{\mathbf j}}}_{x,x+1}(\xi) =-b^2 (\xi_{x} -\rho) (\xi_{x+1} -\rho) - (\gamma+b^2 \rho) \nabla \xi_x\\
&{{{\mathbf j}}}^{\prime}_{x,x+1}(\xi) = -\nabla ( b^2 \xi_x + \gamma \log (\xi_x) ).
\end{split}$$
For any local compactly supported functions $f,g:(0,+\infty)^{{{\mathbb Z}}}\rightarrow{{{\mathbb R}}}$ we define the semi-inner product $\ll f, g \gg:=\ll f,g\gg_{\beta,\lambda}$ of $f$ and $g$ by $$\begin{split}
&\ll f , g \gg \\
&= \sum_{x \in {{\mathbb Z}}} \left( \langle \tau_x f g \rangle_{\nu_{\beta,\lambda}} -\langle f\rangle_{\nu_{\beta,\lambda}} \langle g \rangle_{\nu_{\beta,\lambda}} \right) \\
&= \lim_{k \to \infty} \sum_{|x| \le k} \left( \langle \tau_x f g \rangle_{\nu_{\beta,\lambda}} -\langle f\rangle_{\nu_{\beta,\lambda}} \langle g \rangle_{\nu_{\beta,\lambda}} \right) \\
&= \lim_{k \to \infty} \frac{1}{2k+1} \sum_{|x| \le k} \left\{ \sum_{|y-x| \le k} \Big( \langle \tau_{x+y} f \,\tau_y g \rangle_{\nu_{\beta,\lambda}} -\langle f\rangle_{\nu_{\beta,\lambda}} \langle g \rangle_{\nu_{\beta,\lambda}} \Big) \right\} \\
&= \lim_{k \to \infty}\!\! \left\langle \left( \cfrac{1}{\sqrt{2k+1}}\sum_{|x| \le k} (\tau_x f -\langle f\rangle_{\nu_{\beta,\lambda}})\right) \!\!\!\left( \cfrac{1}{\sqrt{2k+1}}\sum_{|x| \le k} (\tau_x g -\langle g\rangle_{\nu_{\beta,\lambda}})\right) \right\rangle_{\nu_{\beta,\lambda}}
\end{split}$$ where the third equality follows from the invariance of $\nu_{\beta,\lambda}$ by the shift. Observe also that the first sum on ${{\mathbb Z}}$ is in fact a finite sum since $f$ and $g$ are assumed to be local functions. We denote by ${{{\mathcal H}}}_0$ the space generated by the local compactly supported functions and the semi-inner product $\ll\cdot, \cdot \gg$. Observe that any constant or gradient functions are equal to $0$ in ${{{\mathcal H}}}_0$.
By , the normalized current associated to the volume is a gradient and this shows that ${{{\mathcal F}}}_{i,j} (\gamma,z)=0$ if $(i,j) \ne (1,1)$. By the definition of $\hat J_{x,x+1}$ and by , we are only interested in the behavior, as $z \to 0$, of $${{{\mathfrak L}}} (z) = \ll {{{\mathbf J}}}_{0,1}, (z -{{{\mathcal L}}})^{-1} {{{\mathbf J}}}_{0,1} \gg =\int_{0}^{\infty} \, e^{-z t} \, \ll {{{\mathbf J}}}_{0,1} (t) \, , \, {{{\mathbf J}}}_{0,1} (0) \gg \, dt.$$ Since gradient functions are equal to $0$ in ${{{\mathcal H}}}_0$, this is equivalent to estimate $$\label{eq:lapltransfW}
{{{\mathfrak L}}} (z) = b^4 \ll W_{0,1}, (z -{{{\mathcal L}}})^{-1} W_{0,1} \gg$$ where $W_{x,y}$ is the local function $W_{x,y}=(\xi_{x} -\rho) (\xi_{y} -\rho)$.
In this section we prove that there exists a constant $C>0$ such that $$\label{eq:z14}
\ll W_{0,1}, (z -{{{\mathcal L}}})^{-1} W_{0,1} \gg\, \ge \, C z^{-1/4}.$$
But before proving (\[eq:z14\]) let us show (\[eq:F11c\]) which is a direct consequence of the following lemma.
For any $\gamma>0$, there exists a constant $C:=C(\gamma)$ such that $$\begin{split}
\ll W_{0,1}\, , \, (z/\gamma -b^2 {{{\mathcal A}}}-{{{\mathcal S}}})^{-1} W_{0,1} \gg \, \le \, C \ll W_{0,1}, (z -b^2 {{{\mathcal A}}}-\gamma {{{\mathcal S}}})^{-1} W_{0,1} \gg
\end{split}$$ and $$\ll W_{0,1}, (z -b^2 {{{\mathcal A}}}-\gamma {{{\mathcal S}}})^{-1} W_{0,1} \gg\, \le \, C \ll W_{0,1}\, ,\, (z/\gamma -b^2 {{{\mathcal A}}}- {{{\mathcal S}}})^{-1} W_{0,1} \gg.$$
Assume $\gamma>1$ the case $\gamma<1$ being similar. By Lemma 2.1 of [@B.] we have the variational formula for $\ll W_{0,1}, (z -{{\mathcal L}})^{-1} W_{0,1} \gg$, where $\mathcal{L}=b^2{{\mathcal A}}+\gamma \mathcal S$, given by $$\sup_{f} \left\{ 2 \ll W_{0,1}, f \gg - \ll f, (z -\gamma {{{\mathcal S}}}) f \gg - b^4 \ll {{{\mathcal A}}} f, (z -\gamma {{{\mathcal S}}})^{-1} {{{\mathcal A}}} f \gg \right\},$$ where the supremum is carried over functions $f$ belonging to the domain of the generator $\mathcal{L}$ or equivalently to a dense subspace included in this domain, say the space of smooth local compactly supported functions. We have that $$\begin{split}
&\sup_{f} \Big\{ 2 \ll W_{0,1}, f \gg - \ll f, (z -\gamma {{{\mathcal S}}}) f \gg - b^4 \ll {{{\mathcal A}}} f, (z -\gamma {{{\mathcal S}}})^{-1} {{{\mathcal A}}} f \gg \Big\}\\
=&\sup_{f} \Big\{ 2 \ll W_{0,1}, f \gg - \gamma \ll f, (z/\gamma - {{{\mathcal S}}}) f \gg - b^4 \gamma^{-1} \ll {{{\mathcal A}}} f, (z/\gamma - {{{\mathcal S}}})^{-1} {{{\mathcal A}}} f \gg \Big\}\\
\ge& \sup_{f} \Big\{ 2 \ll W_{0,1}, f \gg - \gamma \ll f, (z/\gamma - {{{\mathcal S}}}) f \gg - b^4 \gamma \ll {{{\mathcal A}}} f, (z/\gamma - {{{\mathcal S}}})^{-1} {{{\mathcal A}}} f \gg \Big\}\\
=& \sup_{f}\Big\{ 2 \gamma^{-1/2} \ll W_{0,1}, f \gg - \ll f, (z/\gamma - {{{\mathcal S}}}) f \gg - b^4 \ll {{{\mathcal A}}} f, (z/\gamma - {{{\mathcal S}}})^{-1} {{{\mathcal A}}} f \gg \Big\}
\end{split}$$ where the inequality comes from $\gamma>1$ and last equality is obtained by the change of $f$ into $\gamma^{-1/2} f$. The last term is equal to $$\gamma^{-1} \ll W_{0,1}\, ,\, (z/\gamma -b^2 {{{\mathcal A}}}- {{{\mathcal S}}})^{-1} W_{0,1} \gg$$ and this proves the first inequality of the lemma.
For the second one we proceed similarly: $$\begin{split}
&\sup_{f} \Big\{ 2 \ll W_{0,1}, f \gg - \ll f, (z -\gamma {{{\mathcal S}}}) f \gg - b^4 \ll {{{\mathcal A}}} f, (z -\gamma {{{\mathcal S}}})^{-1} {{{\mathcal A}}} f \gg \Big\}\\
=&\sup_{f} \Big\{ 2 \ll W_{0,1}, f \gg - \gamma \ll f, (z/\gamma - {{{\mathcal S}}}) f \gg - b^4 \gamma^{-1} \ll {{{\mathcal A}}} f, (z/\gamma - {{{\mathcal S}}})^{-1} {{{\mathcal A}}} f \gg \Big\}\\
\le& \sup_{f} \Big\{ 2 \ll W_{0,1}, f \gg - \gamma^{-1} \ll f, (z/\gamma - {{{\mathcal S}}}) f \gg - b^4 \gamma^{-1} \ll {{{\mathcal A}}} f, (z/\gamma - {{{\mathcal S}}})^{-1} {{{\mathcal A}}} f \gg \Big\}\\
=&\sup_{f} \Big\{ 2 \gamma^{1/2} \ll W_{0,1}, f \gg - \ll f, (z/\gamma - {{{\mathcal S}}}) f \gg - b^4 \ll {{{\mathcal A}}} f, (z/\gamma - {{{\mathcal S}}})^{-1} {{{\mathcal A}}} f \gg \Big\}\\
=&\gamma \ll W_{0,1}\, ,\, (z/\gamma -b^2 {{{\mathcal A}}}- {{{\mathcal S}}})^{-1} W_{0,1}\gg.
\end{split}$$
Recall the orthogonal decomposition described in Section \[sec:dua\]. Let $f=\sum_{\sigma} F(\sigma) H_{\sigma}$ and $g= \sum_{\sigma} G(\sigma) H_{\sigma}$ be two centered local functions. The configuration $\sigma$ shifted by $z \in {{\mathbb Z}}$ is denoted by $\tau_z \sigma$, that is $\tau_z\sigma(x)=\sigma(x-z)$. We identify $F_n, G_n$, the restrictions of $F,G$ to $\Sigma_n$, with symmetric functions on ${{\mathbb Z}}^n$. By we have that $$\ll f , g \gg = \sum_{z\in{{{\mathbb Z}}}}\sum_{\sigma \in \Sigma} F (\tau_z \sigma) G (\sigma) {{{\mathcal W}}} (\sigma),$$ where ${{{\mathcal W}}}$ was defined in .
With some abuse of notations, we denote by $\ll F, G \gg$ the scalar product defined by $$\ll F, G \gg= \sum_{z\in{{{\mathbb Z}}}}\sum_{\sigma \in \Sigma} F (\tau_z \sigma) G (\sigma) {{{\mathcal W}}} (\sigma).$$
We also introduce the inner product ${\ll{\cdot, \cdot}\gg_{\rm{free}}}$ defined by $$\begin{split}
&{\ll{F, G}\gg_{\rm{free}}} \;= \; \sum_{ y \in {{\mathbb Z}}}\sum_{\sigma \in \Sigma} F(\tau_y \sigma) G(\sigma).\\
\end{split}$$
Since the function ${{{\mathcal W}}}$ is invariant by the shift, we have a very simple relation between these two inner products: $$\label{eq:fspsp}
\ll F, G \gg \; =\; {\ll{{{{\mathcal W}}}^{1/2} F, {{{\mathcal W}}}^{1/2} G}\gg_{\rm{free}}}.$$
On the set $\Sigma_n$ we introduce the equivalence relation $\star$ defined by $\sigma \star \sigma'$ if and only if there exists $u \in {{\mathbb Z}}$ such that $\tau_u \sigma =\sigma'$. Let $\Sigma_n^\star = \Sigma_n/ \star$ be the set of classes for this relation and $\Sigma^\star = \cup_{n \ge 1} \Sigma_n^{\star}$. We can rewrite the scalar product ${\ll{\cdot, \cdot}\gg_{\rm{free}}}$ as $${\ll{F,G}\gg_{\rm{free}}} = \sum_{{\bar \sigma} \in \Sigma^\star} {\bar F} ({\bar \sigma}) {\bar G} ({\bar \sigma}).$$ Here $\bar F$ is defined by $\bar F ({\bar \sigma})=\left(\sum_{y \in {{\mathbb Z}}} \tau_y F\right) (\sigma)$ where $\sigma$ is any element of ${\bar \sigma}$. The function ${{{\mathcal W}}}$ being invariant by the shift, we define ${{{\mathcal W}}} (\bar \sigma)$ by ${{\mathcal W}} (\sigma)$, $\sigma \in {\bar \sigma}$, ${\bar \sigma} \in {\Sigma}^{\star}$. Then, we have $$\ll F , G \gg = \sum_{{\bar \sigma} \in \Sigma^\star} {{{\mathcal W}}} ({\bar \sigma}) {\bar F} ({\bar \sigma}) {\bar G} ({\bar \sigma}).$$
\[lem:compa\] There exists a constant $C:=C(n,\lambda)$ such that for any local function $F:\Sigma_n \to {{\mathbb R}}$ of degree $n$ it holds that
1. $$C^{-1} {\ll{F, F}\gg_{\rm{free}}} \; \le\; \ll F, F \gg \;\le \; C {\ll{F, F}\gg_{\rm{free}}}.$$
2. $$C^{-1} {\ll{F, -{{{\mathfrak S}}} F}\gg_{\rm{free}}} \; \le\; \ll F, - {{{\mathfrak S}}} F \gg \;\le \; C {\ll{F, -{{{\mathfrak S}}} F}\gg_{\rm{free}}}.$$
Moreover, for any positive real $z>0$ $$\begin{split}
\ll F, (z- \gamma {{{\mathfrak S}}})^{-1} F \gg ={\ll{{{{\mathcal W}}}^{1/2} F\, ,\, (z- \gamma {{{\mathfrak S}}})^{-1} \, {{{\mathcal W}}}^{1/2} F}\gg_{\rm{free}}}.
\end{split}$$
Recall the definition of ${{{\mathcal W}}}$ from . Thus, ${{{\mathcal W}}}$ is bounded from above (resp. from bellow) by a constant $C(n,\lambda)$ (resp. $C^{-1} (n,\lambda)$) independent of $\sigma \in \Sigma_n$. This is enough to conclude $\textit{(1)}$. In order to prove *(2)*, it is enough to use and the fact that for any local function $F: \Sigma \to {{\mathbb R}}$ we have that ${{{\mathfrak S}}} ({{{\mathcal W}}}^{1/2} F)= {{{\mathcal W}}}^{1/2} {{{\mathfrak S}}} F$. Finally, for a local function $F$ of degree $n$, we have by (\[eq:fspsp\]) and the fact that $$\ll F, (z- \gamma {{{\mathfrak S}}})^{-1} F \gg = \sup_{G {\text{ of degree $n$}} } \left\{ 2 \ll F, G \gg - \ll G, (z- \gamma {{{\mathfrak S}}}) G \gg \right\},$$ the following equality $$\begin{split}
\ll F, (z- \gamma {{{\mathfrak S}}})^{-1} F \gg ={\ll{{{{\mathcal W}}}^{1/2} F, (z- \gamma {{{\mathfrak S}}})^{-1} {{{\mathcal W}}}^{1/2} F}\gg_{\rm{free}}},
\end{split}$$ which proves the last assertion.
Our goal is to get a lower bound for $\ll W_{0,1}, (z -{{{\mathcal L}}})^{-1} W_{0,1} \gg$ which by Lemma 2.1 of [@B.] can be rewritten in the variational form $$\label{eq:varformula11}
\sup_{f} \Big\{ 2 \ll W_{0,1}, f \gg - \ll f, (z -\gamma {{{\mathcal S}}}) f \gg - b^4 \ll {{{\mathcal A}}} f, (z -\gamma {{{\mathcal S}}})^{-1} {{{\mathcal A}}} f \gg \Big\}.$$
Any element $\bar \sigma \in \Sigma_n^{\star}$ can be identified with an element of ${\mathbb N}^{n-1}$ through the application which associates to $(\alpha_1, \ldots, \alpha_{n-1}) \in {{{\mathbb N}}}^{n-1}$ the class of the configuration $\sigma =\delta_{0} + \delta_{\alpha_1} + \ldots +\delta_{\alpha_1 + \ldots+ \alpha_{n-1}}$.
Observe also that ${{{\mathfrak S}}}$ is a self-adjoint operator with respect to $\ll \cdot, \cdot \gg$ and with respect to ${\ll{\cdot,\cdot}\gg_{\rm{free}}}$. We restrict the previous supremum over degree $2$ functions $f=\sum_{(x,y) \in {{\mathbb Z}}^2} F([x,y]) H_{[x,y]}$. In order to keep notation simple, whenever we identify a configuration $\sigma\in\Sigma_n$ with $[\textbf{x}]\in \mathbb{Z}^n$ we will simply write $F(x)$, instead of $F([\textbf{x}])$.
Up to some irrelevant multiplicative constant, a lower bound is given by $$\sup_{F {\text{of degree $2$}} } \Big\{ 2 F(0,1) - \| F \|_{1,z}^2 -b^4 \| {{{\mathfrak A}}}_- F \|_{-1,z}^2 -b^4 \| {{{\mathfrak A}}}_+ F \|_{-1,z}^2 -b^4\| {{{\mathfrak A}}}_0 F \|_{-1,z}^2\Big\}$$ where $\| F \|_{\pm 1, z}^2 = \ll F, (z- \gamma {{{\mathfrak S}}})^{\pm 1} F \gg$. We also introduce the corresponding $H_{{\pm 1,z}}$-norms associated to ${\ll{\cdot,\cdot}\gg_{\rm{free}}}$: $\| F \|_{\pm 1, z,{\rm{free}}}^2 = {\ll{ F, (z- \gamma {{{\mathfrak S}}})^{\pm 1} F }\gg_{\rm{free}}}$, for $F:\Sigma\rightarrow{{{\mathbb R}}}$.
By Lemma \[lem:compa\], there exists a constant $C$ such that this lower bound is bounded from bellow by $$\begin{split}
&\sup_{F {\text{of degree $2$}} } \left\{ 2 F(0,1) - C \| F \|_{+1,z,{\rm{free}}}^2\right. \\
& \left. -b^4 \| {{{\mathcal W}}}^{1/2} {{{\mathfrak A}}}_- F \|_{-1,z, {\rm free}}^2 -b^4 \|{{{\mathcal W}}}^{1/2} {{{\mathfrak A}}}_+ F \|_{-1,z, {\rm free}}^2 -b^4\| {{{\mathcal W}}}^{1/2} {{{\mathfrak A}}}_0 F \|_{-1,z, {\rm free}}^2\right\}.
\end{split}$$
Let us first show that if $F$ is of degree $2$ then the contributions given by $ \| {{{\mathcal W}}}^{1/2} {{{\mathfrak A}}}_- F \|_{-1,z, {\rm free} }^2$ and $\| {{{\mathcal W}}}^{1/2} {{{\mathfrak A}}}_0 F \|_{-1,z, {\rm free} }^2$ are equal to zero.
The function ${{{\mathcal W}}}$ is constant and equal to $(\lambda+1)$ on $\Sigma_1$ so that ${{{\mathcal W}}}^{1/2} {{{\mathfrak A}}}_- F =\sqrt{\lambda+1} {{{\mathfrak A}}}_- F $. It is easy to check that the degree one function ${{{\mathfrak A}}}_- F$ satisfies $$({{{\mathfrak A}}}_- F)(u)=(\lambda-1)\Big( F(u-1,u) - F(u,u+1)\Big).$$
For any degree $1$ function $G$, we have $${\ll{ {{{\mathfrak A}}}_- F , G }\gg_{\rm{free}}}= \sum_{u,y \in {{\mathbb Z}}} G(u+y)(\lambda-1)\Big( F(u-1,u) - F(u,u+1)\Big) =0$$ by a telescopic sum argument. This shows that ${{{\mathfrak A}}}_- F$ is equal to zero in the Hilbert space generated by ${\ll{\cdot, \cdot}\gg_{\rm{free}}}$.
Recall that if $F$ is a degree $2$ function, i.e. a symmetric function on ${{\mathbb Z}}^2$, then ${F}$ is identified with a function $\bar{F}$ defined on ${{\mathbb N}}$ by $${\bar F} (\alpha) = \sum_{u \in {{\mathbb Z}}} F(u,u +\alpha)$$ and as a consequence, for $F$ and $G$ degree $2$ functions it holds that $$\label{eq: eqfree}
{\ll{F,G}\gg_{\rm{free}}}\; =\; \sum_{\alpha \in {{\mathbb N}}} {\bar F} (\alpha) {\bar G} (\alpha).$$
Observe that $({{{\mathfrak A}}}_0 F)(u,v)$ is equal to $$\begin{split}
\begin{cases}
2(1+\lambda) \Big( F(u-1,u) -F(u,u+1)\Big),\quad {\text{if}} \; u=v,\\
(1+\lambda) \Big( F(u-1,u+1) -F(u,u+2)\Big) +(2+\lambda)\Big(F(u,u) -F(u+1,u+1) \Big),\\
\quad \quad {\text{if}} \; (u,v)=(u,u+1),\\
(1+\lambda)\Big( F(u-1,v) - F(u+1,v) +F(u,v-1)-F(u,v+1) \Big), \quad {\text {if}} \; |u-v| \ge 2
\end{cases}
\end{split}$$ and $$\label{expression for W}
{{{\mathcal W}}} (u,u)= \cfrac{(\lambda+1)(\lambda+2)}{2}, \quad {{{\mathcal W}}} (u,v)= (\lambda+1)^2\quad \text{ for } u \ne v.$$
It is then easy to show that $$\overline{{{{\mathcal W}}}^{1/2} ({{{\mathfrak A}}}_0 F)} (\alpha)=0$$ for any $\alpha \in {{\mathbb N}}$. Putting together the previous result and it follows that:
$$\begin{split}
\|{{{\mathcal W}}}^{1/2} {{{\mathfrak A}}}_0 F\|_{-1,z, {\rm{free}}}^2&= {\ll{ {{{\mathcal W}}}^{1/2} {{{\mathfrak A}}}_0 F\; ,\; (z - \gamma {{{\mathfrak S}}})^{-1} ({{{\mathcal W}}}^{1/2} {{{\mathfrak A}}}_0 F) }\gg_{\rm{free}}}\\
&= \sum_{\alpha \in {{\mathbb N}}} \overline{{{{\mathcal W}}}^{1/2} ({{{\mathfrak A}}}_0 F)} (\alpha) \; \overline{(\lambda -\gamma {{{\mathfrak S}}})^{-1} [{{{\mathcal W}}^{1/2} ({{{\mathfrak A}}}_0 F)]}} (\alpha) =0.
\end{split}$$
\[lem:DF\] There exists a positive constant $C$ such that for every symmetric function $F$ of degree $2$, if ${\bar F} (\alpha) = \sum_{z\in {{\mathbb Z}}} F(z,z+\alpha)$, then $$C^{-1} \sum_{\substack{x,y \ne 0,\\ |x-y|=1}} \Big( {\bar F} (y) -{\bar F} (x) \Big)^2 \; \le \; {\ll{F, -{{{\mathfrak S}}} F }\gg_{\rm{free}}}\; \le \; C \sum_{\substack{x,y \ne 0,\\ |x-y|=1}} \Big({\bar F} (y) -{\bar F} (x) \Big)^2.$$
This follows easily from the following equalities together with : $$\begin{split}
\overline{{{{\mathfrak S}}} F} (0) &=\sum_{y\in{{{\mathbb Z}}}} ({{{\mathfrak S}}} F) (y,y)\\
&= \sum_{y\in{{{\mathbb Z}}}} \Big( F (y+1,y+1) -F(y,y) \Big) +\Big( F (y-1,y-1) -F(y,y) \Big)=0,
\end{split}$$ $$\begin{split}
\overline{{{{\mathfrak S}}} F} (1) &= \sum_{y\in{{{\mathbb Z}}}} ({{{\mathfrak S}}} F) (y,y+1)\\
&= \sum_{y\in{{{\mathbb Z}}}} \Big( F(y-1,y+1) -F(y,y+1)\Big) + \sum_{y\in{{{\mathbb Z}}}} \Big( F(y,y+2) -F(y,y+1)\Big)\\
&=2 \Big({\bar F} (2) - {\bar F} (1)\Big),
\end{split}$$ $$\begin{split}
\overline{{{{\mathfrak S}}} F} (\alpha)&= \sum_{y} ({{{\mathfrak S}}} F) (y,y+\alpha)\\
&= \sum_{y\in{{{\mathbb Z}}}} \Big( F(y-1,y+\alpha) -F(y,y+\alpha)\Big) + \sum_{y\in{{{\mathbb Z}}}}\Big( F(y+1,y+\alpha) -F(y,y+\alpha)\Big)\\
&+ \sum_{y\in{{{\mathbb Z}}}} \Big( F(y,y+\alpha+1) -F(y,y+\alpha)\Big) + \sum_{y\in{{{\mathbb Z}}}} \Big( F(y,y+\alpha-1) -F(y,y+\alpha)\Big)\\
&= 2 \Big( {\bar F} (\alpha + 1) -{\bar F} (\alpha)\Big)+2 \Big( {\bar F} (\alpha - 1) -{\bar F} (\alpha)\Big), \quad \alpha \ge 2.
\end{split}$$
To any degree $3$ function $G$, i.e. a symmetric function $G$ on ${{\mathbb Z}}^3$, the function ${\bar G}$ is identified with a function on ${{\mathbb N}}^2$: $${\bar G} (u,v) =\sum_{y \in {{\mathbb Z}}} G(y,u+y,u+v+y).$$
Since G is symmetric on ${{\mathbb Z}}^3$, then $\bar G$ is symmetric on ${{\mathbb Z}}^2$. As above, for $F$ and $G$ degree $3$ functions it holds that $$\label{eq: eqfree2}
{\ll{F,G}\gg_{\rm{free}}}\; =\; \sum_{(\alpha,\beta) \in {{\mathbb N}}^2} {\bar F} (\alpha,\beta) {\bar G} (\alpha,\beta).$$
Let ${{{\mathbb D}}}_3$, acting on the local functions on ${{\mathbb N}}^2$, be defined by $$\begin{split}
{{{\mathbb D}}}_3 (\bar G) &= \sum_{u \ge 1} \Big( {\bar G} (u+1,0) -{\bar G} (u,0) \Big)^2 + \sum_{v \ge 1} \Big( {\bar G} (0,v+1) -{\bar G} (0,v) \Big)^2 \\&
+ \Big({\bar G} (1,0) -{\bar G} (0,1)\Big)^2+ \sum_{u,v \ge 1} \Big( {\bar G} (u+1,v) -{\bar G} (u,v)\Big)^2 \\
&+ \Big( {\bar G} (u,v+1) -{\bar G} (u,v)\Big)^2.
\end{split}$$ This is the Dirichlet form of a symmetric nearest neighbors random walk on ${{\mathbb N}}^2$ where all the jumps between $\{0\} \times {{\mathbb N}}$ and ${{\mathbb N}}^*\times {{\mathbb N}}^*\footnote{Here and in the sequel ${{\mathbb N}}^*:={{\mathbb N}}\backslash{\{0\}}$}$, all the jumps from ${{\mathbb N}}\times \{0\}$ and ${{\mathbb N}}^*\times {{\mathbb N}}^*$ and all the jumps from $0$ have been suppressed, and a jump between $(0,1)$ and $(1,0)$ has been added.
\[lem:007\] There exists a constant $C>0$ such that for any symmetric function $G$ on ${{\mathbb Z}}^3$ $$C^{-1} {{{\mathbb D}}}_3 ({\bar G}) \; \le \; {\ll{ G, -{{{\mathfrak S}}} G }\gg_{\rm{free}}} \; \le \; C {\mathbb D}_3 ({\bar G}).$$
We have the following equalities $$\begin{split}
&\overline{{{{\mathfrak S}}} G } \, (0,0)=0,\\
&\overline{{{{\mathfrak S}}} G } \, (0,1)= 2 \Big( {\bar G} (0,2) -{\bar G} (0,1) \Big) + \Big({\bar G} (1,0) -{\bar G} (0,1)\Big),\\
&\overline{{{{\mathfrak S}}} G } \, (0,\beta)= 2 \Big( {\bar G} (0,\beta + 1) -{\bar G} (0,\beta) \Big)+2\Big( {\bar G} (0,\beta - 1) -{\bar G} (0,\beta)\Big), \quad \beta \ge 2,\\
&\overline{{{{\mathfrak S}}} G } \, (1,0)= 2 \Big( {\bar G} (2,0) -{\bar G} (1,0) \Big) + \Big({\bar G} (0,1) -{\bar G} (1,0)\Big)\\
&\overline{{{{\mathfrak S}}} G } \, (\alpha,0)= 2 \Big( {\bar G} (\alpha + 1) -{\bar G} (\alpha,0) \Big)+2 \Big({\bar G} (\alpha - 1) -{\bar G} (\alpha,0) \Big), \quad \alpha \ge 2,\\
&\overline{{{{\mathfrak S}}} G } \, (\alpha,\beta) = \Big( {\bar G} (\alpha+1, \beta) -{\bar G}(\alpha,\beta) \Big) + \Big( {\bar G} (\alpha, \beta+1) -{\bar G}(\alpha,\beta) \Big)\\
&+ {\bf 1}_{\{\alpha \ge 2\}}\Big( {\bar G} (\alpha-1, \beta +1) -{\bar G} (\alpha,\beta)\Big) + {\bf 1}_{\{\alpha \ge 2\}} \Big({\bar G} (\alpha-1, \beta) -{\bar G} (\alpha,\beta)\Big) \\
&+{\bf 1}_{\{\beta \ge 2\}}\Big({\bar G} (\alpha+1, \beta -1) -{\bar G} (\alpha,\beta)\Big) + {\bf 1}_{\{\beta \ge 2\}}\Big({\bar G} (\alpha, \beta-1) -{\bar G} (\alpha,\beta)\Big), \quad \alpha,\beta \ge 1.
\end{split}$$
We recognize in these expressions the generator of a symmetric nearest neighbors random walk on ${{\mathbb N}}^2$ where
- all the jumps between $\{0\} \times {{\mathbb N}}$ and ${{{\mathbb N}}}^*\times {{\mathbb N}}^*$, all the jumps between ${{\mathbb N}}\times \{0\}$ and ${{{\mathbb N}}}^*\times {{\mathbb N}}^*$, and all the jumps from $0$ have been suppressed;
- a jump between $(0,1)$ and $(1,0)$ with rate $1$ has been added;
- jumps between $(\alpha,\beta)$ and $(\alpha \pm 1, \beta \mp 1)$ for $(\alpha,\beta) \in {{\mathbb N}}^*\times{{\mathbb N}}^*$ with rate $1$ have been added.
- the non vanishing jumps on ${{\mathbb N}}\times \{0\}$ and on $\{0\} \times {{\mathbb N}}$ have been multiplied by $2$.
This together with , implies the lemma.
We choose a degree $2$ symmetric function $F$ such that $$\label{eq:FF}
\begin{split}
&{\overline F} (\alpha) = z^{-1/4} e^{-z^{3/4} (\alpha -1)}, \quad \alpha \ge 1,\\
&{\overline F} (0) ={\bar F} (1).
\end{split}$$
This function exists since given a function $G$ defined on ${{\mathbb N}}$ we can find a symmetric function $F$ defined in ${{\mathbb Z}}^2$ such that $\bar F=G$. For that purpose, take $F(x,y) ={G} (|y-x|) [ \phi (x) + \phi (y)]$ where the function $\phi$ is defined on ${{\mathbb Z}}$ and is such that $\sum_{x\in{{{\mathbb Z}}}} \phi (x) =1/2$. Then for any $\alpha \in {{\mathbb N}}$, ${\bar F} (\alpha) = \sum_{u \in {{\mathbb Z}}} F(u,u+\alpha)= G(\alpha) \sum_{u \in {{\mathbb Z}}} [\phi (u) + \phi (u+\alpha)]= G(\alpha)$.
Observe that with this choice, by Lemma \[lem:DF\], $$\label{eq:eraclio}
{\ll{ F, -{{{\mathfrak S}}} F }\gg_{\rm{free}}}\sim z^{1/4}, \quad {\bar F} (1) = z^{-1/4}, \quad z \sum_{\alpha \in {{\mathbb N}}} {\bar F}^2 (\alpha) \sim z^{-1/4}.$$
It remains to estimate the last contribution given by $\| {{{\mathcal W}}}^{1/2} G\|_{-1,z,{\rm{free}}}^2$ where $G={{{\mathfrak A}}}_{+} F $ is a degree $3$ function.
\[lem:hadrien\] Let $G={{{\mathfrak A}}}_{+} F $ where $F$ is defined by (\[eq:FF\]). There exists a constant $C>0$ such that $$\| {{{\mathcal W}}}^{1/2} G\|_{-1,z,{\rm{free}}}^2 \ge C z^{-1/4}.$$
For any $u,v,w \in {{\mathbb Z}}$, we have $$\begin{split}
& G(u,u+1,u+2)= F(u, u+1)- F(u+1,u+2) ,\\
& G(u,u+1,v) = F(u,v)-F(u+1,v), \quad v>u+1,\\
&G(v,u,u+1)= F(v,u)-F(v,u+1), \quad v<u,\\
& G(u,u,u+1)= 2\Big( F(u,u) - F(u,u+1)\Big),\\
& G(u,u,u-1) = 2 \Big(F(u-1,u) - F(u,u) \Big),\\
&G(u,v,w)=0 \quad {\text{otherwise.}}
\end{split}$$
Let us now compute ${\bar G}(u,v)$, $u,v \in {{\mathbb N}}$. We get $$\label{eq:GGG}
\begin{split}
&{\bar G} (0,1)=-{\bar G} (1,0)=2 {\bar F} (0) -2 {\bar F} (1),\\
&{\bar G} (1,v)= {\bar F} (v+1) -{\bar F} (v), \quad v \ge 2,\\
&{\bar G} (u,1)= {\bar F} (u) -{\bar F} (u+1) , \quad u \ge 2,\\
&{\bar G} (u,v)=0 \quad {\text{otherwise.}}
\end{split}$$ By (\[eq:FF\]) we have that ${\bar G} (0,1)= {\bar G} (1,0)=0$. Also notice that $\bar G(u,u)=0$ and by we have that ${{{\mathcal W}}}^{1/2} (u,v)=(1+\lambda)$ for $u\neq{v}$.
It follows, by Lemma [\[lem:007\]]{}, that $\|{{{\mathcal W}}}^{1/2} G \|_{-1,z,{\rm{free}}}^2$ is upper bounded by the variational formula:
$$\begin{split}
\| {{{\mathcal W}}}^{1/2} G \|_{-1,z, {\rm{free}}}^2 &= \sup_{R} \left\{ 2 \sum_{(u,v){\in{{{\mathbb N}}^2}}} R(u,v) {{\mathcal W}}^{1/2}(u,v){\bar G} (u,v) -C_{0} {{{\mathbb D}}}_3 (R) \right\}\\
&= \sup_{R} \left\{ 2 (1+\lambda)\sum_{(u,v)\in{{{\mathbb N}}^2}} R(u,v) {\bar G} (u,v) -C_{0} {{{\mathbb D}}}_3 (R) \right\}
\end{split}$$
where the supremum is taken over local functions on ${{\mathbb N}}^2$. By (\[eq:GGG\]), we have that $$\label{eq:trajan}
\begin{split}
&\sum_{(u,v)\in{{{\mathbb N}}^2}} R(u,v) {\bar G} (u,v)\\&
= \sum_{v \ge 2} R (1,v)\Big( {\bar F} (v+1) -{\bar F} (v) \Big)- \sum_{u \ge 2} R(u,1) \Big( {\bar F} (u+1) - {\bar F} (u) \Big)\\
&= \sum_{v \ge 3} {\bar F} (v) \Big( R(1,v-1) -R(1,v)\Big)- \sum_{u \ge 3} {\bar F} (u) \Big( R(u-1,1) -R(u,1)\Big) \\
&+ {\bar F} (2)\Big(R(2,1) -R(1,2) \Big)\\
&= \sum_{v \ge 2} {\bar F} (v) \Big( R(1,v-1) -R(1,v)\Big)-\sum_{u \ge 2} {\bar F} (u) \Big( R(u-1,1) -R(u,1)\Big).
\end{split}$$
We use now the following parametrization of $R$. For $k \ge 1$, $v \in {{\mathbb Z}}$, let us define $$R(k,v)=\phi (k-1,v-k), \quad v \ge k, \quad R(u,k)=\phi (k-1,-u+k), u \ge k,$$ where $\{\phi (k,\cdot) \; ;\; k \ge 0\}$ are functions from ${{\mathbb Z}}\to {{\mathbb R}}$. We have the following lower bound for ${{{\mathbb D}}}_3 (R)$: $${{{\mathbb D}}}_3 (R) \ge \sum_{u,v \ge 1} \Big( R (u+1,v) -R (u,v)\Big)^2 + \Big( R (u,v+1) -R (u,v)\Big)^2$$ which is nothing but the Dirichlet form of a random walk where only jumps connecting sites of ${{\mathbb N}}^* \times {{\mathbb N}}^*$ have been conserved. With the choice of the parametrization for $R$ and this lower bound, it is not difficult to show there exists a constant $C>0$ such that $${{{\mathbb D}}}_3 (R) \ge C \sum_{k \ge 0} \sum_{v \in {{\mathbb Z}}} \Big( \phi (k, v+1) -\phi (k,v) \Big)^2 +\Big( \phi (k+1, v) -\phi (k,v) \Big)^2.$$ The right hand side of the previous inequality is the Dirichlet form of a symmetric simple random walk on ${{\mathbb N}}\times {{\mathbb Z}}$.
By (\[eq:trajan\]), we get $$\sum_{(u,v)\in {{\mathbb N}}^2} R(u,v) {\bar G} (u,v)=
\sum_{u \in {{\mathbb Z}}} \phi (0,u) \Big( {\tilde F}(u-1) -{\tilde F} (u) \Big)$$ where ${\tilde F}: {{\mathbb Z}}\to {{\mathbb R}}$ is defined by ${\tilde F} (u)= -{\bar F} (u+2) {\bf 1}_{\{u \ge 0\}} -{\bar F} (1-u) {\bf 1}_{\{u \le -1\}}$. We extend the function $\phi$ defined on ${{\mathbb N}}\times {{\mathbb Z}}$ to ${{\mathbb Z}}^2$ by defining $\phi(-k, u)= \phi (k,u)$, $k \ge 1, u \in {{\mathbb Z}}$. Observe then that $$\begin{split}
{{{\mathbb D}}}_3 (R) &\ge C \sum_{k \ge 0} \sum_{v \in {{\mathbb Z}}}\Big( \phi (k, v+1) -\phi (k,v) \Big)^2 + \Big( \phi (k+1, v) -\phi (k,v)\Big)^2 \\
&= \cfrac{C}{2} \sum_{k \in {{\mathbb Z}}} \sum_{v \in {{\mathbb Z}}} \Big( \phi (k, v+1) -\phi (k,v) \Big)^2 + \Big( \phi (k+1, v) -\phi (k,v) \Big)^2. \\
\end{split}$$
Consequently we have, for suitable positive constants $C_1, C_2$: $$\label{eq:flavien}
\begin{split}
\| {{{\mathcal W}}}^{1/2} G \|_{-1,z, {\rm{free}}}^2 \le C_1 \sup_{\phi} \Big\{ 2 \sum_{u \in {{\mathbb Z}}} \phi(0,u)& \Big( {\tilde F} (u-1) -{\tilde F} (u) \Big)\\
&-C_{2} \sum_{\substack{({{\mathbf u}},{{\mathbf v}})\in {{\mathbb Z}}^2\\ |{{\mathbf u}}-{{\mathbf v}}|=1}} \Big( \phi ({{\mathbf u}}) -\phi ({{\mathbf v}}) \Big)^2\Big \}.
\end{split}$$
A standard Fourier computation shows this supremum is of order $z^{-1/4}$. Indeed, let ${\widehat u}$ be the Fourier transform of the function $u: {{\mathbb Z}}^n \to {{\mathbb R}}$, defined by $${\widehat u} ({{\mathbf k}}) = \sum_{{{\mathbf x}}\in {{\mathbb Z}}^n} e^{2i \pi {{\mathbf x}}\cdot {{\mathbf k}}} u ({{\mathbf x}}), \quad {{\mathbf k}}=(k_1, \ldots,k_n),$$ and denote by ${\widehat u}^* ({{\mathbf k}})$ the complex conjugate of ${\widehat u} ({{\mathbf k}})$. Using the expression of the sum of a convergent geometric series, we obtain the following expression for the Fourier transform $\Psi (k_1, k_2)$ of the function $(x,y) \in {{\mathbb Z}}^2 \to \delta_0 (y) {\tilde F} (x)$: $$\Psi (k_1, k_2) = - z^{-1/4} e^{-z^{3/4}} \left\{ \cfrac{1}{1- e^{2i\pi k_1} e^{-z^{3/4}}}-\cfrac{e^{-2i \pi k_1}} {1-e^{-2i\pi k_1} e^{-z^{3/4}} } \right\}$$ which satisfies $$\left| \Psi (k_1, k_2) \right| \le \cfrac{C_3 \sqrt{z}}{z^{3/2} + C_4 \sin^{2} (\pi k_1) }$$ for some positive constants $C_3, C_4$. The supremum appearing in (\[eq:flavien\]) is then given by $$C_2^{-1} \int_{[0,1]^2} \cfrac{| \Psi (k_1, k_2)|^2 }{z+ 4 \sin^2 (\pi k_1) + 4 \sin^{2} (\pi k_2) } dk_1 dk_2.$$ Then the result follows by a standard study of this integral.
To obtain (\[eq:z14\]), by (\[eq:eraclio\]) and Lemma \[lem:hadrien\], it suffices to take a test function in the form $aF$ with $F$ given by (\[eq:FF\]) and $a$ sufficiently small.
Stochastic perturbations of Hamiltonian systems {#sec:pert}
===============================================
In this section we discuss some other possible stochastic perturbations and make some connections with the recent models considered in [@BC]. Let us start with the Hamiltonian system (\[eq:dyneq\]) with potential $V$ and generator $A$ given by $$A =\sum_{x\in{{{\mathbb Z}}}} \Big(V' (\eta_{x+1}) -V' (\eta_{x-1}) \Big) \partial_{\eta_x}.$$ The energy $\sum_{x\in{{{\mathbb Z}}}} V(\eta_x)$ and the volume $\sum_{x\in{{{\mathbb Z}}}} \eta_x$ are conserved by these dynamics. Remark that in fact $\sum_{x\in{{{\mathbb Z}}}} \eta_{2x}$ and $\sum_{x\in{{{\mathbb Z}}}} \eta_{2x+1}$ are also conserved and that we cannot exclude the case that still many others exist. This is the case for example for the exponential interaction for which an infinite number of conserved quantities can be explicitly identified. Anyway, we are only interested in these two first quantities. The product probability measures $\mu_{\beta,\lambda}$ defined by $$\mu_{\beta,\lambda} (d\eta) = \prod_{x \in {{\mathbb Z}}} Z(\beta,\lambda)^{-1}
\exp\left\{ -\beta V( \eta_x) -\lambda \eta_x \right\} \, d\eta_x, \\$$ where $$Z(\beta, \lambda) = \int_{-\infty}^{+\infty} \exp\left( -\beta V(r) -\lambda r \right)\, dr.$$ are invariant for the infinite dynamics.
In [@BS] we proposed to perturb this deterministic dynamics by the Poissonian noise considered in this paper and conserving both the energy and the volume. One could also consider the “ Brownian” noise whose generator $S$ is given by $ S= \sum_{x\in{{{\mathbb Z}}}} Y_x^2$ where $$Y_x\!=\!(V'(\eta_{x+1})-V'(\eta_{x-1}))\! \partial_{\eta_{x}}\!+(V'(\eta_{x-1})-V'(\eta_{x}))\! \partial_{\eta_{x+1}}\! + (V'(\eta_{x})-V'(\eta_{x+1})) \! \partial_{\eta_{x-1}},$$ is the vector field tangent to the curve $$\Big\{ (\eta_{x-1}, \eta_x, \eta_{x+1}) \in {{\mathbb R}}^3 \, ; \sum_{y=x-1}^{x+1} \eta_y =0, \; \sum_{y=x-1}^{x+1} V (\eta_y) =1\Big\}.$$ It is easy to see that the process with generator $L=A+S$ conserves the energy and the volume and has $\mu_{\beta, \lambda}$ as invariant measures. A priori, it should be possible to extend our result to this system for $V$ of exponential type but the noise $S$ seems to have a quite complicated expression in the orthogonal basis we used in this paper. The advantage of the Poissonian noise is its very simple form. Notice also that the Poissonian noise is a weaker perturbation of the Hamiltonian dynamics than the Brownian noise in the sense it is less mixing. Indeed, consider the discrete torus ${{{\mathbb T}}}^N$ of length $N$ and the Brownian noise ${S}^N=\sum_{x\in {{\mathbb T}}^N} Y_x^2$ restricted to the manifold ${{{\mathcal M}}}_{\pi, E}^N$ defined by $${{{\mathcal M}}}^N_{\pi,E} = \left\{ \eta \in {{\mathbb R}}^{{{\mathbb T}}^N} \, ; \, \sum_{y \in {{\mathbb T}}^N} \eta_y =\pi, \; \; \sum_{y \in {{\mathbb T}}^N} V(\eta_y) =E \right\}, \quad E>0, \pi \in {{\mathbb R}}.$$ Then $S^N$ is ergodic on ${{{\mathcal M}}}_{\pi, E}^N$ but this is not true for the restriction of the Poissonian noise restricted to ${{{\mathcal M}}}_{\pi, E}^N$. We could also decide to conserve energy and not the volume by adding a suitable perturbation. The invariant states are then given by $\mu_{\beta,0}$, $\beta>0$. If $V$ is even, a simple Poissonian noise consists to change the sign of $\eta_x$ independently on each site $x$ at random exponential times. In this case one can prove, as in [@BO], that the energy diffuses in the sense that the Green-Kubo formula converges to a well defined finite value. For a generic $V$ a Brownian noise with generator $S$ given by $S=\sum_{x\in {{\mathbb Z}}} K_x^2$ with $K_x = V' (\eta_{x+1}) \partial_{\eta_x} - V' (\eta_x) \partial_{\eta_{x+1}}$ makes the job.
Consider now the case where we want to add a stochastic perturbation conserving only the volume. It does not seem to be easy to define a simple Poissonian noise with such a property. A Brownian noise is obtained by the following scheme. Fix $\beta>0$, consider the vector field $X_x = \partial_{\eta_{x+1}} -\partial_{\eta_x}$ which is tangent to the hyperplane $\{ (\eta_x, \eta_{x+1}) \in {{\mathbb R}}^2\; ; \; \eta_x + \eta_{x+1} =1\}$ and define the Langevin operator ${S}_{\beta}$ by $$\begin{split}
{S}_\beta &= \frac{1}{2} \sum_{x\in {{\mathbb Z}}} e^{-{{{\mathcal H}}}_{\beta, \lambda}} X_x (e^{{{{\mathcal H}}}_{\beta,\lambda}} X_x )\\
&= \frac{1}{2} \sum_{x\in {{\mathbb Z}}} X_x^2 + \frac{\beta}{2} \sum_{x\in{{{\mathbb Z}}}} \Big(V' (\eta_{x+1}) -V' (\eta_x)\Big) X_x
\end{split}$$ where ${{{\mathcal H}}}_{\beta, \lambda} = \beta \sum_{x\in {{\mathbb Z}}} V(\eta_x) +\lambda \sum_{x\in {{\mathbb Z}}} \eta_x$. Observe that ${S}_\beta$ depends on $\beta$ but is independent of $\lambda$. The operator ${S}_{\beta}$ is a nonpositive self-adjoint operator in ${{{\mathbb L}}}^{2} (\mu_{\beta, \lambda})$ for any $\lambda$ and ${S}_{\beta} (\sum_{x\in {{\mathbb Z}}} \eta_x) =0$. Then, the perturbed volume-conserving model has a generator $L^V_{\beta}$ given by $$L^V_{\beta} = A + \gamma S_\beta$$ where $\gamma>0$ is a parameter fixing the strength of the noise. By construction, the Markov process generated by $L^V_{\beta}$ has $\mu_{\beta, \lambda}$ as a set of invariant probability measures. In fact, using the same methods as in [@BS; @FFL] one can prove that the only space-time invariant probability measures with finite local entropy density are mixtures of the $(\mu_{\beta, \lambda})_{\lambda}$. We can also rewrite $L^V_{\beta}$ as $$\begin{split}
L^V_{\beta} & = \sum_{x\in {{\mathbb Z}}} \left\{ \left(1 -\frac{\gamma \beta}{2} \right) V' (\eta_{x+1}) + \gamma \beta V' (\eta_x) - \left(1+ \frac{\gamma \beta}{2} \right) V' (\eta_{x-1}) \right\} \partial_{\eta_x}\\
&+ \gamma \sum_{x\in {{\mathbb Z}}}( \partial_{\eta_x}^2 -\partial_{\eta_x, \eta_{x+1}}^2).
\end{split}$$
The microscopic flux $j_{x,x+1}$ associated to the volume conservation law is defined by $$L^V_{\beta} (\eta_x) = -\nabla j_{x-1,x}, \quad j_{x-1,x} = -\left( 1+\frac{\gamma \beta}{2}\right) V' (\eta_{x-1}) - \left( 1 - \frac{\gamma \beta}{2} \right) V' (\eta_x).$$ The semi-discrete directed polymer model considered in [@BC] is, up to an irrelevant scaling factor $2$, recovered by taking $V(\eta) =e^{-\eta}$, $\beta=1$ and $\gamma=2$ (see (3.7) in [@Sp3]). In [@BC] the authors show that for a particular non stationary initial condition (“wedge”), by developing a very nice theory of Macdonald processes, the system belongs to the Kardar-Parisi-Zhang universality class ([@Sp3]). Unfortunately one can not use their results or their methods to derive a more precise picture for the model with exponential interactions considered in this paper. For other potentials $V$ the theory developed by Borodin and Corwin in [@BC] can not be adapted but it would be very interesting to see if one can relate the models generated by $L^V_\beta$ to the semi-discrete directed polymer and deduce some qualitative information from the latter. The use of the variational formulas considered in this paper could be the way.
Existence of the infinite dynamics {#sec:exis}
==================================
In this section we prove existence of the infinite volume dynamics $(\xi(t))_{t \ge 0}$. We focus here on the process $\xi$ but the same proof can be carried for the process $\eta$ (or just define $\eta$ in terms of $\xi$ by $\eta_x (t) =- b^{-1} \log \xi_x (t) $, $x \in {{\mathbb Z}}$. To simplify notations we will assume $b=1$.
Since the interaction coming from the deterministic part is non-quadratic at infinity, proving the existence of the infinite dynamics is a non trivial task. Nevertheless nice sophisticated techniques have been introduced by Dobrushin and Fritz in [@DF]. Here, we follow closely the approach of [@F3] (see also [@F1; @F2]) adapted to our case. By itself, the strategy of the proof of existence of solutions is standard: we consider finite subsystems and prove compactness of this family by means of an a priori bound for a quantity ${\bar E}$ which plays the role of an energy density. The obtention of this a priori bound is however non trivial and is the main step to get the existence of the dynamics. The aim of this appendix is to show how to get such an a priori bound. The a priori bound we derive here for the infinite dynamics is also valid for finite subsystems corresponding to a finite set $\Lambda \subset {{\mathbb Z}}$ with a bound which is independent of the size of $\Lambda$. This proves then that the finite subsystems form a compact family from which one can extract a subsequence converging to the infinite dynamics.
We have first to specify the space of allowed configurations $\Omega \subset (0,+\infty)^{{{\mathbb Z}}}$. For $x\in{{\mathbb Z}}$, let $g(x) = 1 + \log (1+|x|)$ and denote by $E(\xi, \mu,\sigma)$, $\xi\in(0,+\infty)^{{{\mathbb Z}}}$, $\mu \in {{\mathbb Z}}$, $\sigma>0$, the quantities $$\begin{split}
&E(\xi,\mu,\sigma)= \sum_{|x - \mu| \le \sigma} (1 + 2 \xi_x - \log (\xi_x)),\\
&{\bar E} (\xi)= \sup_{\mu \in {{\mathbb Z}}} \, \sup_{ \sigma \ge g(\mu)} \, \sigma^{-1} E (\xi,\mu, \sigma).
\end{split}$$
The quantity ${\bar E}$ is called the logarithmic fluctuation of energy and the set $\Omega$ is defined as $$\Omega:=\{\xi\in{(0,+\infty)^{{\mathbb Z}}}: {\bar E} (\xi) <+ \infty\}.$$ The configuration space $\Omega$ is equipped with the product topology and with the associated Borel structure. It is easy to see that $ \nu_{\beta, \lambda} (\Omega) =1$ for any $\beta>0$ and $\lambda>-1$.
Let ${\bf N} (t) = \{N_{x,x+1} (t) \, ; \, x \in {{\mathbb Z}}\}$ be a collection of independent Poison processes of intensity $\gamma>0$. The equations of motion corresponding to the generator ${{{\mathcal L}}}$ read as $$\label{eq:app-stoch-eq}
{d \xi_x} = \xi_x (\xi_{x+1} -\xi_{x-1}) dt + \nabla \left( (\xi_{x} -\xi_{x-1}) dN_{x-1,x} (t)\right),\quad x \in {{\mathbb Z}}.$$
Let $D({{\mathbb R}}_+, {{\mathbb R}})$ denote the space of càdlàg functions of ${{\mathbb R}}_+$ into ${{\mathbb R}}$ with the Skorohod topology and let ${{{\mathbb D}}}= [D({{\mathbb R}}_+, {{\mathbb R}})]^{{{\mathbb Z}}}$ equipped with the product topology and the associated Borel field ${{{\mathcal B}}}$. The smallest $\sigma$-algebra on which all projectionsrestricted to the time interval $[0,t]$ are measurable will be denoted by ${{{\mathcal B}}}_t$. Finally, suppose that we are given a probability measure ${\bf P}$ on ${{{\mathcal B}}}$ such that our Poisson processes $N_{x,x+1}$ are realized as components of the random element of ${{{\mathbb D}}}$.
A ${{{\mathcal B}}}_t$-adapted mapping $\xi (t):= \xi(t, {\bf N})$ of ${{{\mathbb D}}}$ into itself is called a tempered solution of (\[eq:app-stoch-eq\]) with initial configuration $\xi^0 \in \Omega$ if $\xi (0) =\xi^0$, almost each trajectory $\xi (\cdot, {\bf N})$ satisfies the integral form of (\[eq:app-stoch-eq\]), and the logarithmic energy fluctuation ${\bar E} (\xi (t))$ is bounded on finite intervals of time with probability one.
\[th:existdyn2\] For any $\xi^0 \in \Omega$, there exists a unique tempered solution of (\[eq:app-stoch-eq\]) with initial configuration $\xi^0 \in \Omega$.
As explained above, the main step to prove this theorem is to obtain an a priori bound that we prove in Proposition \[prop:apb\]. For a complete proof, we refer to [@F3] ( or [@F1; @F2]).\
Now we notice that the Gibbs state $\nu_{\beta, \lambda}$, $(\beta, \lambda) \in (0, +\infty) \times (-1, +\infty)$ is formally invariant for the infinite dynamics generated by $(\xi (t))_{t \ge 0}$. This can be seen by observing that $ \int ({{{\mathcal L}}} f)(\xi) d\nu_{\beta, \lambda}(d\xi)=0$ for nice functions $f:\Omega\rightarrow{{{\mathbb R}}}$. Nevertheless, some care has to be taken to prove this. Indeed, we do not know that ${{{\mathcal L}}}$ is really the generator of the semigroup generated by $(\xi (t))_{ t\ge 0}$ on the space of bounded measurable functions on $\Omega$ in the usual Hille-Yosida theory. This can be a very difficult question that we prefer to avoid (see [@F2]). Instead we use the fact that the infinite dynamics can be approximated by finite subsystems.
For any $\beta>0, \lambda > -1$, the probability measure $\nu_{\beta, \lambda}$ is invariant for the process $(\xi (t))_{t \ge 0}$.
Let $n \ge 2$ and consider the local dynamics generated by the generator ${{{\mathcal L}}}_n = {{{\mathcal A}}}_n + \gamma {{{\mathcal S}}}_n $ where $$\begin{split}
({{{\mathcal A}}}_n f)(\xi) & = \sum_{x=-n}^n \xi_x (\xi_{x+1} -{\xi}_{x-1}) \partial_{\xi_x} f (\xi)\\
& \, -\, \xi_{n+1} \left( \xi_n +\frac{\lambda +1}{\beta}\right) \partial_{\xi_n} f(\xi) + \xi_{-n -1} \left( \xi_{-n} \, +\, \frac{\lambda +1}{\beta} \right) \partial_{\xi_{-n-1}} f (\xi),\\
({{{\mathcal S}}}_n f)(\xi) &= \sum_{x=-n}^{n} \Big( f(\xi^{x,x+1}) -f(\xi)\Big)
\end{split}$$ where $f: \Omega \to {{\mathbb R}}$ is a compactly supported continuously differentiable function. The dynamics is essentially finite-dimensional since the particles outside the box $\{-n-1, \ldots, n+1\}$ are frozen. Thus, the classical Hille-Yosida theory can be applied. The boundary conditions have been chosen to have $$\int ({{{\mathcal L}}}_n f) (\xi) d \nu_{\beta, \lambda}(\xi) =0$$ for any compactly supported continuously differentiable function $f$ which shows that $\nu_{\beta, \lambda}$ is invariant for the local dynamics. Since, as a consequence of the a priori bound, the infinite dynamics is obtained as a limit of finite local dynamics, this implies that $\nu_{\beta, \lambda}$ is invariant for the infinite dynamics.
Then this defines a strongly continuous semigroup of contractions $(P_t)_{t \ge 0}$ on the Hilbert space ${{{\mathbb L}}}^2 (\Omega, {{{\mathcal B}}}, \nu_{\beta, \lambda})$. Moreover, Itô’s formula shows that its generator is a closable extension of ${{{\mathcal L}}}$ given by ${{{\mathcal A}}} +\gamma {{{\mathcal S}}}$ since for any local compactly supported continuously differentiable function $f$, we have $$(P_t f) (\xi) = f (\xi) + \int_0^t ({P_s} {{{\mathcal L}}} f) (\xi) ds, \quad \xi \in \Omega, \quad t \ge 0.$$
Logarithmic energy fluctuation
------------------------------
We have first to consider a clever smooth modification of ${\bar E}$. Let $0<\lambda<1$ and consider a twice continuously differentiable nonincreasing function $\varphi: {{\mathbb R}}\to (0,1)$ such that $\varphi (u) =e^{\lambda (1-u)}$ if $u \ge 2$, $\varphi (u) =(1+\lambda + \lambda^2 /2) e^{-\lambda}$ if $u \le 1$, and $\varphi$ is concave for $u \le 3/2$, convex if $u\ge 3/2$. Finally, $0 \le - \varphi' (u) \le \lambda \varphi (u) \le e^{\lambda(1-u)}$, $\varphi (u) \ge e^{-\lambda (1+u)}$ and $|\varphi^{\prime \prime} (u)| \le \varphi (u)$ for all $u>0$.
For $x\in{{\mathbb Z}}$ and $\sigma \ge 1$ we define the function $f$ as $$f(x,\sigma)= \int_{{{\mathbb R}}} {\varphi} (|x-y| /\sigma) e^{-2\lambda|y|} dy.$$
In [@F2] are proved the following properties on $f$: $$\begin{aligned}
\begin{split}
\label{eq:equff1} &c_1 \exp (-\lambda |x| / \sigma) \le f(x,\sigma) \le c_2 \exp (-\lambda |x| /\sigma), \\
\label{eq:equff2} &f(x,\sigma) \le f(y,\sigma) e^{2 \lambda |x-y|}, \quad \partial_ \sigma f (x,\sigma) \le e^{2 \lambda | x-y|} \partial_{\sigma} f (y,\sigma). \\
\label{eq:equff3} &| \partial_x f (x,\sigma) | \le \min \{\partial_\sigma f (x,\sigma) , \sigma^{-1} f (x,\sigma)\},\\
\label{eq:equff4}&g(x) |\partial_x f (x-\mu, \sigma) | \le 4 g(|\mu| +\sigma)\, ( \partial_{\sigma} f)(x-\mu, \sigma).
\end{split}\end{aligned}$$
Here the constants depend only on $\lambda$.
For $\xi\in(0,+\infty)^{{{\mathbb Z}}}$, $\mu\in{{{\mathbb Z}}}$ and $\sigma>0$, consider the function $$W(\xi, \mu, \sigma) =\sum_{x \in {{\mathbb Z}}} f(x-\mu, \sigma) (1+ 2 \xi_x - \log \xi_x)$$ and let $${\bar W} (\xi) = \sup_{\mu \in {{\mathbb Z}}} \, \sup_{\sigma \ge g(\mu)} \,\Big\{ \sigma^{-1} W (\xi,\mu,\sigma)\Big\}.$$
Observe that by (\[eq:equff1\]), $$\label{eq:compWE}
W(\xi,\mu,\sigma) \ge c_1e^{-\lambda} E(\xi, \mu,\sigma),$$ for all $\xi\in(0,+\infty)^{{{\mathbb Z}}}$, $\mu\in{{{\mathbb Z}}}$ and $\sigma>0$.
For $\xi\in(0,+\infty)^{{{\mathbb Z}}}$, we also consider the function $${\widehat W} (\xi) = \sup_{\mu \in {{\mathbb Z}}} \Big\{\frac{W(\xi, \mu,g(\mu))}{g(\mu)}\Big\}.$$
The following lemma shows that these two modifications of the logarithmic energy fluctuation are equivalent to ${\bar E}$.
\[lem: comp6\]
There exists a constant $C$ such that for all $\xi\in(0,+\infty)^{{{\mathbb Z}}}$: $$C^{-1} {\widehat W} (\xi) \le {\bar W}(\xi)\le C {\widehat W}(\xi), \quad C^{-1} {\bar E} (\xi)\le {\bar W}(\xi) \le C {\bar E}(\xi).$$
The inequality ${\widehat W}(\xi) \le {\bar W}(\xi) $ for all $\xi\in(0,+\infty)^{{{\mathbb Z}}}$, is trivial. Let us prove the second one by taking $\sigma \ge g(\mu)$, $\mu \in {{\mathbb Z}}$ and denoting $1+ 2\xi_x -\log \xi_x$ by $H_x$. By (\[eq:equff1\]), we have $$\begin{split}
W(\xi, \mu,\sigma)& \le c_2 \sum_{x \in {{\mathbb Z}}} \exp \left( -\lambda |x-\mu|/\sigma \right) H_x =c_2 \sum_{n =0}^{\infty} e^{- \lambda n/\sigma} \sum_{|x-\mu|=n} H_x\\
&=c_2 (1- e^{- \lambda /\sigma}) \sum_{n=0}^{\infty} e^{- \lambda n /\sigma} \sum_{|x-\mu| \le n} H_x,
\end{split}$$ where the last equality follows from $\sum_{|x-\mu|=n} H_x = \sum_{|x-\mu| \le n} H_x - \sum_{|x-\mu| \le n-1} H_x $ and a discrete integration by parts. Let $r\ge 1$ be the integer such that $r-1 < g(\mu) \le r$ and decompose the set $\{x \in {{\mathbb Z}}\, ;\, |x-\mu| \le n\}$ as $\cup_{j=1}^{K+1} \Lambda_j $ where the $\Lambda_j$ are non intersecting intervals of length $r$ for $j=1, \ldots,K$ and $\Lambda_{K+1}$ is of length at most $r-1$. Observe that $K+1$ is of order $n/ g(\mu)$. By using (\[eq:compWE\]), we have easily that $$\sum_{x \in \Lambda_j} H_x \le C \, g(\mu)\, {\widehat W} (\xi)$$ where $C$ depends only on $\lambda$. Thus we get $$\begin{split}
W(\xi, \mu,\sigma)& \le C (1- e^{- \lambda /\sigma}) \sum_{n=0}^{\infty} e^{- \lambda n /\sigma} n {\widehat W} (\xi)\le C' \sigma {\widehat W} (\xi)
\end{split}$$ which concludes the proof of the second inequality.
The proof of $C^{-1} {\bar E}(\xi) \le {\bar W}(\xi) \le C {\bar E}(\xi) $ for all $\xi\in(0,+\infty)^{{{\mathbb Z}}}$, is the same. The first inequality follows from and the constant can be taken equal to $c_1e^{-\lambda}$. The second inequality follows from a similar argument to the one used above.
The [*a priori*]{} bound
------------------------
\[prop:apb\] For each $w \ge 1$ there exists a continuous function $q_{w} (t)$, $t\ge 0$, such that $${\bf P} \left\{ \sup_{0 \le s \le t} {\bar W} (\xi(s)) > \exp(q_{w} (t) g(u)) \right\} \le e^{-u}$$ for each $u \ge 1$, $t \ge 0$, whenever ${\bar W} (\xi^0) \le w$ and $(\xi (t))_{t \ge 0}$ is a tempered solution of (\[eq:app-stoch-eq\]) with initial condition $\xi^0$.
We consider a tempered solution $(\xi (t))_{t \ge 0}$ of (\[eq:app-stoch-eq\]) with initial configuration $\xi^0 \in \Omega$.
For each $k \ge 1$, $\mu \in {{\mathbb Z}}$ and $t\geq{0}$ we define the stochastic process $\rho_{k}$ by $$\rho_k (t) = k g (\mu) - C_0 \int_0^t g (|\mu| +|\rho_k (s)| ) Z' (s) ds$$ where $C_0:=C_0(\gamma, \lambda)$ is a positive constant that will be chosen later and $$Z(t)= \int_0^t {\bar W} (\xi (s)) ds.$$
Since the function $f(\cdot)$ is positive, $\bar W(\cdot)$ is also positive and this turns $Z(\cdot)$ positive. The trajectories of $\rho_k$ are differentiable, decreasing and satisfy $\rho_{k+1} (t) - \rho_k (t) \le g(\mu)$ a.s. for each $t\ge 0$. We consider also the sequence of stopping times $\tau_k =\inf \{ t \ge 0 \, ; \, \rho_k (t) \le g(\mu) \}$ which satisfy $\tau_k < \tau_{k+1} < +\infty$ and $\lim_{k \to \infty} \tau_k = \infty$ a.s. We evaluate now the stochastic differential of $t \to W (\xi (t), \mu, \rho_k (t))$ for $t \le \tau_k$ (so that $\rho_k (t) \ge 1$). This is given by $$\begin{split}
d \left[ W (\xi (t), \mu, \rho_k (t)) \right] &= I^{(k)}_0 (t) dt - C_0 (\partial_\sigma W) (\xi(t), \mu, \rho_k (t)) g (|\mu| +\rho_k (t)) {\bar W} (\xi (t)) dt\\
& + \; dI^{(k)}_1 (t)
\end{split}$$ where $$\label{eq:I0k}
\begin{split}
I_0^{(k)} (t) =&2 \sum_{x \in {{\mathbb Z}}} \Big( f(x- \mu, \rho_k (t) ) - f(x+1- \mu, \rho_k (t) )\Big) \xi_x (t) \xi_{x+1} (t) \\
+& \sum_{x \in {{\mathbb Z}}} \Big(f(x+1- \mu, \rho_k (t) ) - f (x-1-\mu, \rho_k (t) ) \Big) \xi_x (t)
\end{split}$$ and $$d I^{(k)}_1 =\sum_{x\in{{\mathbb Z}}} f(x- \mu, \rho_k ) \Big\{ 2 \nabla\Big((\xi_{x} -\xi_{x-1} ) dN_{x-1,x} \Big) - \nabla\Big( (\log \xi_{x} -\log \xi_{x-1}) dN_{x-1,x} \Big) \Big\}.$$
We first estimate the term $I_0^{(k)} (t)$ and we show that if $C_0$ is taken sufficiently large then, for $t \le \tau_k$ we have that $$\label{eq:I0kne}
I_{0}^{(k)} (t) \; - \; C_0 \, (\partial_\sigma W) (\xi(t), \mu, \rho_k (t)) \, g (|\mu| +\rho_k (t)) {\bar W} (\xi (t)) \le 0.$$
The second term on the right hand side of (\[eq:I0k\]) can be estimated, by using (\[eq:equff2\]) and (\[eq:equff3\]), to get to
$$\label{detailed estimate}
\begin{split}
\Big| f(x+1- \mu, \rho_k (t) ) - f (x-1-\mu, \rho_k (t) ) \Big| &= \Big| \int_{-1}^1 \, (\partial_x f) (x-\mu +\alpha, \rho_k (t)) d\alpha \Big|\\
& \le \int_{-1}^1 \, \Big| (\partial_x f) (x-\mu +\alpha, \rho_k (t))d\alpha \Big|\\
& \le \int_{-1}^1 \, (\partial_\sigma f) (x-\mu +\alpha, \rho_k (t))d\alpha \\
& \le 2 \sup_{[ x- \mu -1, x -\mu +1]}\Big\{ \partial_{\sigma} f( \cdot, \rho_{k} (t))\Big\} \\
&\le 2 e^{2 \lambda} \partial_{\sigma} f( x-\mu, \rho_{k} (t))
\end{split}$$
which gives us that $$\begin{split}
\sum_{x \in {{\mathbb Z}}} \Big(f(x+1- \mu, \rho_k (t) ) - f (x-1-\mu, \rho_k (t) ) \Big) \xi_x (t) &\le C \sum_{x \in {{\mathbb Z}}} \partial_\sigma f (x-\mu, \rho_k (t) ) \xi_x (t)\\
&\leq C (\partial_\sigma W) (\xi(t), \mu, \rho_k (t)).
\end{split}$$
Now, notice that for any $x \in {{\mathbb Z}}$ and for all $\xi\in{(0,\infty)^{{{\mathbb Z}}}}$ we have that $$\bar W(\xi)\geq{\hat W(\xi)}\geq{\frac{W(\xi,x,g(x))}{g(x)}}.$$ On the other hand, by and since for all $x>0$ it holds that $\log(x)\leq{1+x}$, then we have that $W(\xi,x,g(x))\geq{c_1e^{\lambda}E(\xi,x,g(x))}\geq{c_1e^{-\lambda}\xi_{x+1}}$. Then, we conclude that there exists a constant $C$ such that for all $x\in{{{\mathbb Z}}}$ and $\xi\in{(0,\infty)}^{{{\mathbb Z}}}$, $$\label{useful estimate}
\xi_{x+1} \le Cg(x) {\bar W} (\xi).$$
To estimate the first term on the right hand side of (\[eq:I0k\]) we use the previous estimate, (\[eq:equff4\]) and a similar argument as done in . It follows that $$\begin{split}
&\Big|\sum_{x \in {{\mathbb Z}}} \left( f(x- \mu, \rho_k (t) ) - f(x+1- \mu, \rho_k (t) )\right) \xi_x (t) \xi_{x+1} (t)\Big| \\
& \le C \, {\bar W} (\xi (t)) \, g(|\mu| +\rho_k (t) ) \sum_{x \in {{\mathbb Z}}} \partial_\sigma f (x-\mu, \rho_k (t) ) \xi_x (t)\\
&\leq C \, {\bar W} (\xi (t)) \, g(|\mu| +\rho_k (t) ) \, (\partial_\sigma W) (\xi(t), \mu, \rho_k (t)).
\end{split}$$
Then, (\[eq:I0kne\]) follows.
The term $dI^{(k)}_1$ can be written as $$\begin{split}
dI^{(k)}_1 \!\!&=\sum_{x \in {{\mathbb Z}}}\!f(x-\mu,\rho_k )\!\Big\{ 2 \nabla\left( (\xi_{x} -\xi_{x-1}) dN_{x-1,x} \right)\!-\!\nabla( (\log \xi_{x} -\log \xi_{x-1}) dN_{x-1,x})\Big\}\\
&= - \sum_{x \in {{\mathbb Z}}} \left( f(x+1-\mu,\rho_k ) -f(x- \mu, \rho_k ) \right) \left\{ 2 \nabla \xi_x -\nabla \log \xi_x \right\} dN_{x,x+1}\\
&= - \sum_{x \in {{\mathbb Z}}} \left( f(x+1-\mu,\rho_k ) -f(x- \mu, \rho_k) \right) \left\{ 2 \nabla \xi_x -\nabla \log \xi_x \right\} (dN_{x,x+1} -\gamma dt)\\
&- \gamma \sum_{x \in {{\mathbb Z}}} \left( f(x+1-\mu,\rho_k ) -f(x- \mu, \rho_k ) \right) \left\{ 2 \nabla \xi_x -\nabla \log \xi_x \right\} dt.
\end{split}$$
Since the compensated Poisson processes $N_{x,x+1} (t) -\gamma t$ are orthogonal martingales with quadratic variation $\gamma^2 t$, then $$dM^{(k)}_{\mu}\! =\! -\! \sum_{x \in {{\mathbb Z}}} \left( f(x+1-\mu,\rho_k) \!-\!f(x- \mu, \rho_k ) \right) \Big\{ 2 \nabla \xi_x\! -\!\nabla \log \xi_x \Big\} (dN_{x,x+1} -\gamma dt)$$ defines a martingale with a quadratic variation equal to $$d\langle M^{(k)}_{\mu} \rangle_t =\gamma^2 \sum_{x \in {{\mathbb Z}}} \left( f(x+1-\mu,\rho_k ) -f(x- \mu, \rho_k ) \right)^2 \Big\{ 2 \nabla \xi_x -\nabla \log \xi_x \Big\}^2 \, dt.$$
Using a similar argument to the one in , together with the fact that for all $x,y\in {{\mathbb Z}}$ such that $|x|, |y| \le C$ it holds that $|x-y|^2\leq 2C|x-y|$, the boundedness of the function $f$, (\[eq:equff2\]), (\[eq:equff3\]), (\[eq:equff4\]) and , one has that there exists a constant $C$ such that $$d \langle M^{(k)}_\mu \rangle_t \le C\; g (|\mu| + \rho_k (t) ) \; {\bar W} (\xi (t)) \; \partial_{\sigma} W(\xi (t), \mu, \rho_k (t)) \, dt.$$
Similarly we obtain that $$\begin{split}
&\left| \sum_{x \in {{\mathbb Z}}} \left[ f(x+1-\mu,\rho_k (t)) -f(x- \mu, \rho_k (t)) \right] \left\{ 2 \nabla \xi_x (t) -\nabla \log \xi_x (t) \right\} \right| \\
&\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \le C \, \partial_{\sigma} W (\xi (t), \mu, \rho_k (t)).
\end{split}$$
Thus, if the constant $C_0$ is chosen sufficiently large, we have $$\begin{split}
\sup_{t \ge 0} \Big\{W (\xi (t\wedge \tau_k ), \mu, \rho_k (t \wedge \tau_k )) \Big\}\le W (\xi (0), \mu, k g(\mu)) + \sup_{t \ge 0}\Big\{ N(\mu,k, t)\Big\}
\end{split}$$ where $N(\mu,k, t)= M^{(k)}_{\mu} (t \wedge \tau_k)- \frac{1}{2} \langle M^{(k)}_{\mu} \rangle_{t \wedge \tau_k}$. Observe that $\exp ( M^{(k)}_{\mu} (t \wedge \tau_k) -\frac12 \langle M^{(k)}_{\mu} \rangle_{t \wedge \tau_k})$ is a martingale with expectation equal to $1$. By the exponential supermartingale inequality, we have that $${\bf P } \Big(\sup_{t \ge 0}\Big\{ N (\mu, k, t) > u\Big\} \Big) \le e^{-u}.$$ Thus we proved that for each $k \ge 1$, $\mu \in {{\mathbb Z}}$ and $u>0$, $$\label{eq:diavolo}
\sup_{t \ge 0} \Big\{W (\xi (t \wedge \tau_k),\mu, \rho_k (t \wedge \tau_k))\Big\} \le W ( \xi (0), \mu, kg(\mu)) +u$$ with a probability greater than $1-e^{-u}$. Applying (\[eq:diavolo\]) for each $\mu \in {{\mathbb Z}}$ and $k \ge 1$ with $u$ replaced by $u +Ak g(\mu)$ where $A\ge 1$ is sufficiently large to have $\sum_{k \ge 1} \sum_{\mu \in {{\mathbb Z}}} e^{-Ak g(\mu)} \le 1$, we obtain $$\label{eq:diavolo2}
\begin{split}
\sup_{t \ge 0}\Big\{ W (\xi (t \wedge \tau_k),\mu, \rho_k (t \wedge \tau_k))\Big\} &\le W ( \xi (0), \mu, kg(\mu)) +Ak g(\mu) +u \\
& \le k g(\mu) {\bar W} (\xi (0)) + A k g(\mu) +u
\end{split}$$ with a probability greater than $1-e^{-u}$ uniformly in $k$ and $\mu$.
Define now $k:=k_t$, $t \ge 0$ as the smallest integer $k \ge 1$ for which $\rho_k (t) > g(\mu)$; then $\tau_k >t$ and $\rho_k (t) \le 2 g(\mu)$ as $\rho_{k-1} (t) \le g(\mu)$; thus choosing $k=k_t$ in (\[eq:diavolo2\]) and using that $W(\xi,\mu,\sigma)$ is increasing in $\sigma$ (since $\partial_{\sigma} f \ge 0$ by the conditions imposed on $\varphi$), we get $$\frac{W(\xi (t), \mu,g(\mu))}{g(\mu)} \!\le\! \frac{W(\xi (t), \mu, \rho_k (t) )}{g(\mu)} \!\le\! k {\bar W} (\xi (0))\! +Ak\! +\!\frac{u}{g(\mu)}\!\leq{k {\bar W} (\xi (0)) \!+Ak \!+{u}},$$ where in the last inequality we used the fact that $g(x)\geq{1}$ for all $x\in{{{\mathbb R}}}$. Taking the supremum over $\mu$ and using Lemma \[lem: comp6\], we obtain $${\bar W} (\xi (t)) \le C k_t {\bar W} (\xi (0)) +u$$ for each $t\ge 0$ with probability at least $1-e^{-u}$. On the other hand, $$2 g(\mu) \ge k_t g(\mu) - C_0 \int_0^t g (|\mu| + |\rho_k (s)|) Z' (s) ds$$ whence $$k_t \le 2 + C_0 \int_0^t \frac{g (|\mu| + |\rho_k (s)|)}{g(\mu)} Z' (s) ds.$$ Since $\rho_k (s) \le k_t g(\mu)$ for any $s \in [0,t]$ and $g$ is increasing, we have that $g (|\mu| + |\rho_k (s)|)\leq{g(\mu+k_tg(\mu))}$. On the other hand for $x \ge 2$, $g(x) \le x$ together with the fact that for $x,y\in {{\mathbb R}}$ $g(|x||y|) \le g(|x|) g(|y|)$ and since $g(1+x) \le 1 +g(x)$ for $x \ge 1$, we obtain that $g(\mu+k_tg(\mu))\leq{g(\mu)(1+g(k_t))}$. As a consequence we obtain that $$\label{final estimate}
k_t\leq{2+C_0Z(t) (1+ g(k_t))}.$$ Since for all $x\geq{1}$ we have that $g(x) \le 1+2 \sqrt{|x|}$, then
$$k_t \le 2 + C_0 Z(t) (2 +2 {\sqrt{k_t}}).$$
Finally, it follows that ${\sqrt{k_t}} \le 2 + 4C_0 Z(t)$. Then, since $g$ is increasing and by plugging the previous inequality in , we obtain that $$k_t \le 2 + C_0 Z (t) (1 +g((2+4C_0 Z(t))^2) ).$$ Recalling that $Z'(t)={\bar W} (\xi (t))$ we obtained that there exists a constant $M >0$ depending only on $\lambda$ such that for any $w \ge 1$ and any initial condition $\xi (0)$ satisfying ${\bar W} (\xi (0)) \le w$, $${\bf P} \left[ \sup_{t \ge 0} \Big\{M^{-1} Z' (t) - w (1 + Z(t) g(Z(t)))\Big\} \le u \right] \ge 1-e^{-u}.$$
The a priori bound follows from this last inequality (see [@F3], Proposition 1).
Acknowledgements {#acknowledgements .unnumbered}
================
The authors are very grateful to József Fritz for illuminating discussions on the existence of the infinite dynamics. We acknowledge the support of the French Ministry of Education through the grant ANR-10-BLAN 0108 (SHEPI). We are grateful to ' Egide and FCT for the research project FCT/1560/25/1/2012/S. We are grateful to FCT (Portugal) for support through the research project PTDC/MAT/109844/2009. PG thanks the Research Centre of Mathematics of the University of Minho, for the financial support provided by “FEDER” through the “Programa Operacional Factores de Competitividade – COMPETE” and by FCT through the research project PEst-C/MAT/UI0013/2011. PG thanks the warm hospitality of “Courant Institute of Mathematical Sciences", where part of this work was done.
[30]{}
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[^1]: See however the coupled-rotor model which displays normal behavior (see [@LLP], Section 6.4).
[^2]: A function $f$ defined on an infinite product space is said to be local if it depends only on its variable through a finite number of coordinates.
[^3]: See (\[eq:chibar2\]) for an explicit expression.
[^4]: It would be very interesting to understand why the numerical simulations are so sensitive to the noise.
|
---
abstract: 'Systems with bond disorder are defined through lattice Hamiltonians that are of pure nearest neighbour hopping type, i.e. do not contain on-site contributions. They stand representative for the general family of disordered systems with chiral symmetries. Application of the Dorokhov-Mello-Pereyra-Kumar transfer matrix technique \[P. W. Brouwer [*et al.*]{}, Phys. Rev. Lett [**81**]{}, 862 (1998); Phys. Rev. Lett. [**84**]{}, 1913 (2000)\] has shown that both spectral and transport properties of quasi one-dimensional systems belonging to this category are highly unusual. Most notably, regimes with absence of exponential Anderson localization are observed, the single particle density of states exhibits singular structure in the vicinity of the band centre, and the manifestation of these phenomena depends in an apparently topological manner on the even- or oddness of the channel number. In this paper we re-consider the problem from the complementary perspective of the non-linear $\sigma$-model. Relying on the standard analogy between one-dimensional statistical field theories and zero-dimensional quantum mechanics, we will relate the problem to the behaviour of a quantum point particle subject to an Aharonov-Bohm flux. We will build on this analogy to re-derive earlier DMPK results, identify a new class of even/odd staggering phenomena (now dependent on the total number of sites in the system) and trace back the anomalous behaviour of the bond disordered system to a simple physical mechanism, viz. the flux periodicity of the quantum Aharonov-Bohm system. We will also touch upon connections to the low energy physics of other lattice systems, notably disordered chiral systems in $0$ and $2$ dimensions and antiferromagnetic spin chains.'
address: |
${}^a$Theoretische Physik III, Ruhr-Universität-Bochum, 44780 Bochum, Germany\
${}^b$ Institut für theoretische Physik, Zülpicher Str. 77, 50937 Köln, Germany
author:
- 'Alexander Altland${}^a$ and Rainer Merkt${}^{b}$'
title: Spectral and Transport Properties of Quantum Wires with Bond Disorder
---
*Dedicated to E. Müller-Hartmann on the occasion of his 60th birthday.*
Introduction {#sec:intro}
============
In 1979, Wegner and Oppermann[@oppermann79] introduced a variant of the disordered lattice Anderson model they termed ’sublattice system’. The sublattice system differs from the generic Anderson model in that its Hamiltonian does not contain on-site matrix elements, i.e. is of [*pure*]{} hopping type. For a long time this species of disordered electronic systems went largely unnoticed. The status rapidly changed[@gade93; @miller96; @BMSA:98; @altland99:NPB_flux; @fukui99; @furusaki99; @biswas; @guruswamy00; @brouwer00:_nonun; @brouwer00_off; @brouwer:trans_dos_off; @favrizio] after two aspects became generally appreciated: first, models with sublattice structure occur in a number of physical applications. The random flux model, lattice QCD models[@verbaarschot00], random antiferromagnets[@fisher94:_random], models of gapless semiconductors[@ovchinikov77] and effective models of transport in manganese oxides[@mueller-hartmann96] are of sublattice type or at least acquire sublattice structure in limiting cases. Second, and contrary to naive expectations based on universality and insensitivity to details of the microscopic implementation of disorder, the low energy properties of the sublattice system differ drastically from those of the generic Anderson model:
- In contrast to the Anderson model, average spectral and transport properties of the sublattice system sensitively depend on the value of the Fermi energy, $E_F$. For Fermi energies far away from the centre of the tight binding band, $E_F = 0$, the sublattice system falls into the universality class of standard disordered electron systems (which simply follows from the fact that a Fermi energy $E_F \not=0$ can be interpreted as a constant non-vanishing on-site contribution to the Hamiltonian.) However, in the vicinity of zero energy drastic deviations from standard behaviour occur:
- In dimensions $d \le 2$, the average density of states (DoS), $\nu(E)$, exhibits singular behaviour upon $E$ approaching the band centre.
- Perturbative one-loop RG calculations[@gade93] as well as the analysis of Ref.[@guruswamy00] indicate that right in the middle of the band the $2d$ system is metallic, i.e. does not drive towards an exponentially localized phase.
- Several phenomena of apparent topological origin are observed. E.g., the DoS profile of sublattice quantum dots (systems in the ’zero-dimensional’ limit) sensitively depends on the total number of sites of the host lattice being even or odd. Similarly, the properties of quasi one-dimensional sublattice systems depend on the number of channels, $N$, being even or odd. For $N$ even the transport behaviour is conventional – conductance exponentially decreasing on the scale of a certain localization length $\xi$ – whereas the energy dependent DoS vanishes in a close to linear fashion. In contrast, for $N$ odd, the wire at $E=0$ is metallic, i.e. the length-dependent conductance decays only algebraically. At the same time, the DoS diverges upon approaching the band centre.
A schematic summary of the band centre phenomenology of sublattice models is displayed in table \[tab:1\].
All these phenomena root in the fact that the sublattice Hamiltonian possesses a discrete symmetry not present in the Anderson model: $[\sigma_3,\hat H]_+=0$, where $[\;,\;]_+$ is the anti-commutator and $\sigma_3$ a site-diagonal operator that takes values $+1/-1$ on alternating sites. The presence of this ’chiral’ symmetry implies that sublattice systems fall into symmetry classes different from the three standard Wigner-Dyson classes ’unitary’, ’orthogonal’ or ’symplectic’. To be specific, let us focus on the simplest case of a sublattice system with broken time reversal invariance but good spin rotation symmetry (the analogue of the Anderson model of unitary symmetry.) Chiral systems fulfilling these two extra symmetry criteria belong to a symmetry class denoted $A$III in the terminology of Ref.[@zirnbauer96]. An alternative denotation, coined in the paper[@verbaarschot94:RMT2], is ChGUE for ’Chiral GUE’.
[|c||c|c|c|c|c|]{}&&&$2d$staggering &$L$ even&$L$ odd& $N$ even &$N$ odd&Conductance ($\epsilon=0$) &vanishing& non-vanishing& localization & deloc.& deloc. Density of states &spectral gap & zero energy states &spectral gap & divergence & divergence
\
As with conventional disordered electronic systems of Wigner-Dyson symmetry, universal transport and thermodynamic properties of systems of class $A$III can be described in terms of effective low energy theories. E.g., the results for sublattice quantum wires summarized above have been obtained within a symmetry adapted variant of the Dorokhov-Mello-Kumar-Pereyra (DMPK) transfer matrix approach[@BMSA:98; @brouwer00_off; @brouwer00:_nonun]. This theory differs from the standard cases of unitary symmetry in that the transfer matrices describing the propagation through the system take values in a different target space. Similarly, the general field theory approach to disordered electronic systems, the nonlinear $\sigma$-model, has been extended to systems of class $A$III[@gade93; @altland99:NPB_flux; @favrizio; @fukui99], too. Like its conventional relatives, the $A$III variant of the $\sigma$-model is a matrix field theory whose internal structure depends on the specific implementation (boson replicas, fermion replicas or supersymmetry.) Previous studies of these models have focused on the two-dimensional case[@gade93; @guruswamy00], where information on long range behaviour can be obtained from renormalization group calculations, or on the zero-dimensional case[@verbaarschot96:susy] where the model can be evaluated rigorously by full integration over the zero-mode manifold.
It is the purpose of the present paper to analyse the intermediate, quasi one-dimensional variant of the field theory. As mentioned above key aspects of the phenomenology of quasi one-dimensional sublattice quantum wires have been discussed previously within the framework of the DMPK approach, and it is near at hand to ask what motivates revisiting the problem. Referring for a more substantial discussion to section \[sec:summary-results\] below, we here merely mention a lose collection of points. The field theory approach enables one to approach the problem from a comparatively broad perspective. Specifically, the one-dimensional variant of the model is but a representative of a larger family of ’chiral’ $\sigma$-models. This makes possible to relate the behaviour of the one-dimensional system to the extensively studied zero and two dimensional cases. Further, the Green function oriented $\sigma$-model formalism enables one to ’microscopically’ implement coupling operators connecting the wire to external leads. (Within previous DMPK formulations, the coupling has been treated somewhat implicitly, see however[@brouwer].) Unexpectedly, we will observe strong, albeit universal sensitivity to the modelling of the coupling and yet another class of staggering phenomena. The origin of these effects, and their connection to the channel number staggering mentioned before will be discussed below. Finally, the $\sigma$-model of the time reversal non-invariant sublattice is the by far most simple of all ten[@zirnbauer96] nonlinear $\sigma$-models of disordered systems: it has only four degrees of freedom, two Grassmann and two ordinary integration variables, the minimal set needed to construct a supersymmetric matrix model. This makes it an ideal tutorial system on which generic properties of the field theory approach to disordered quantum wires can be studied. We have tried to pedagogically expose several of these aspects, particularly the analogy one-dimensional field theories vs. point-particle quantum mechanics which plays an all important role in the present context. Nonetheless, the analysis below will be at times technical. To make its results and various qualitative connections generally accessible, the following section contains a synopsis of the paper.
Summary of Results and Qualitative Discussion {#sec:summary-results}
---------------------------------------------
Consider the system depicted in figure Fig. \[fig:figure1\]: a sequence of $L$ sites (alternatingly designated by crosses and circles) each of which supports $N$ electronic states, or orbitals (represented by the vertical stacks of ovals.) Nearest neighbour hopping is controlled by a regular tight binding contribution, diagonal in the orbital index, (the horizontal line segments) plus some bond randomness that connects different orbitals (the hatched areas). As in Refs. [@BMSA:98], we allow for some ’staggering’ in the tight binding amplitudes, i.e. the hopping amplitudes regularly alternate in strength (the alternating bond length.) At either end, a number of sites is coupled through some tunnelling barriers (horizontal hatched areas) to leads.
\
In this paper, three different regimes will be considered: (i) the ’quantum dot’ case, defined through the criterion that the time to diffusively propagate through the system is shorter than the Golden rule escape time into the leads, (ii) a diffusive regime, where the order of these time scales is inverted but localization effects do not yet play a role, and (iii) the regime of long systems with pronounced Anderson localization.
Beginning with the quantum dot case (i), we find that both transport and spectral properties sensitively depend on the number of sites $L$ being even or odd. Specifically, the DoS exhibits a singular spike in the band centre if the number of sites is odd (and the coupling to the leads switched off). Away from zero energy, $\nu(\epsilon)$ is strongly suppressed up to values $\nu \sim N\Delta$, where $\Delta$ is the mean level spacing. The conductance equals $\gamma N/2$, where $\gamma$ is a measure for the strength of the lead coupling, as for conventional quantum dots. In contrast, for an even number of sites, the DoS shows the spectral ’microgap’ of width $\Delta$ characteristic for finite random systems with chiral symmetries[@verbaarschot00]. Curiously, the conductance [*vanishes*]{} provided that only one site at either end is connected to the leads. The strong sensitivity to the total number of sites in the system disappears if more than one site in the terminal regions is coupled to leads (cf. Fig. \[fig:figure1\].) Summarizing, the tendency of sublattice systems to exhibit staggering phenomena pertains to the zero-dimensional limit. However, unlike in the localized regime, the relevant integer control parameter is the number of sites $L$ and not the number of channels $N$. A qualitative interpretation of these phenomena will be given below.
In this paper, only limited attention will be payed to the intermediate diffusive regime (ii). It is likely that the staggering phenomena observed in the zero-dimensional regime will have interesting, albeit non-universal extensions into the diffusive regime. We here avoid the confrontation with these effects by connecting several sites to the leads. This leads to equilibrated behaviour with Ohmic conductance, as for ordinary wires. As for the density of states, we have explored the influence of spatially fluctuating diffusion modes on the spectral microgaps discussed above. (Semiclassically, the spectral gaps observed in chiral or superconductor systems can be interpreted as due to an accumulation of diffusion modes (see Ref. [@altland99:NPB_flux] for a detailed discussion of this point). For large energies $\epsilon \gg \Delta$, these modes can be treated perturbatively by standard diagrammatic methods.) Surprisingly, no corrections are found up to and including three loop order. This is a speciality of the one-dimensional case. For a two dimensional system diffusive modes would lead to a modulation of the spectrum on the scale of the Thouless energy.
In the localized regime we reproduce the results found earlier within the DMPK approach: for an even number of channels, the conductance decays exponentially on the scale of a localization length $\xi \propto
N l$, where $l$ is the mean free path. In contrast, for an odd number of channels algebraic scaling, $g\sim L^{-1/2}$, supported by one delocalized mode is observed[@BMSA:98]. A comparably strong even/odd dependence is observed in the behaviour of the DoS. For $N$ even, the DoS vanishes at zero energy as $\nu(\epsilon) \sim -|s|
\ln(|s|)$, where $s=\pi \epsilon/\Delta_\xi$, and the characteristic scale of the gap, $\Delta_\xi$, is the level spacing of a system of length $L\sim \xi$. This behaviour has a simple interpretation: roughly speaking, a system with $L\gg \xi$ can be viewed as a sequence of $L/\xi$ decoupled ’localization volumes’. On small energy scales $\epsilon < \Delta_\xi$, dynamics within each of these is approximately ergodic. One would then expect the DoS to be gapped, as for the sublattice quantum dot, with $\Delta_\xi$ as the characteristic level spacing. Generalizing an earlier result[@theodorou76; @eggarter78] for the specific case $N=1$, it has been found that for an odd number of channels the DoS diverges as $\nu(\epsilon) \sim 1/ (|s|\ln^3|s|)$[@brouwer00_off]. This accumulation of spectral weight can be interpreted in two different directions. First, it is near at hand to view the algebraically decaying conductance observed in the odd case as a resonant tunnelling phenomenon: the principal tendency to localize is outweighed by the high density of states in the vicinity of zero energy. (This picture was first suggested by V. Kravtsov.) Second, it is tempting to interpret the zero energy peak as an generalization of the singular spike found in the $L$-odd zero-dimensional case. Unfortunately, we are not aware of a qualitative picture explaining this analogy. At least technically, however, both phenomena can be traced back to a common origin. Finally, a periodic modulation, or staggering, of the hopping amplitudes can be employed to continuously interpolate between the $N$-even and $N$-odd case, respectively.
We next briefly outline the field theory route to exploring the above phenomenology. Within the fieldtheoretical approach, long range properties of the system are described by a functional integral with action $$\label{eq:46}
S = \int_0^L dr \left[-{\xi\over 8} {{\,\rm str\,}}(\partial T\partial T^{-1})
- {N+f\over 2} {{\,\rm str\,}}(T\partial T^{-1}) -i {s\over 2} {{\,\rm str\,}}(T+T^{-1})\right] +
S_{\rm Gade} + S_{\rm T}.$$ Here we have introduced a continuum variable $r \in [0,L]$ replacing the formerly discrete site index, $f$ is a parameter related to the staggering strength, $s=\epsilon \pi \nu$, where $\nu$ is the bulk metallic density of states. Further, $T$ is a matrix field taking values in ${{\rm GL}(1|1)}$, i.e. the group of two-dimensional invertible supermatrices, and ’str’ is the supertrace. Finally, $S_{\rm
Gade}[T]\equiv C {{\,\rm str\,}}^2(T\partial T^{-1})$, where $C$ is some small constant and $S_{\rm T}$ is a contribution describing the coupling to the leads.
The action $S[T]$ defines a one-dimensional representative of the general family of ’chiral’ non-linear $\sigma$-models. In contrast to its well investigated zero- and two-dimensional relatives – much of the results summarized in table \[tab:1\] have been derived within these models – the $1d$ variant has not been explored so far.
Much of our analysis of this model will rely on the standard analogy between one-dimensional statistical field theory and zero-dimensional quantum mechanics: $S[T]$ can be interpreted as the quantummechanical action of a point particle propagating on the supersymmetric target manifold, in our case ${{\rm GL}(1|1)}$. The first term of the action represents a kinetic energy, the third term a potential, and the second term, linear in the ’velocity’, coupling to a constant vector potential of strength $(N+f)/2$. Quantum analogies of this type have been proven a powerful technical tool in previous analyses of the standard $\sigma$-models[@Ef:83]. However, the present case is special in that the target manifold is so simple that an [*intuitive*]{} interpretation of the quantum picture becomes straightforward. Indeed, the fields $T$ have the explicit matrix representation $T =
\left(\matrix{u&\rho\cr\sigma&v}\right)$, where $\rho,\sigma (u,v)$ are Grassmann (commuting) variables, to be compared with the four or eight dimensional matrices entering the standard $\sigma$-models. Later on we will see that convergence criteria constrain the component $u$ to be positive real, while $v=\exp(i\phi)$ is a pure phase. Thus, temporarily leaving the Grassmann variables aside, our model describes quantum propagation on a (half)line and on a circle, respectively. Notice that the latter component is topologically non-trivial.
At this stage, the role of the vector potential contribution to the action becomes evident. While inessential in the non-compact sector of the theory, in the compact, circular sector, it desribes the presence of an Aharonov-Bohm type magnetic flux. This analogy explains the presence of phenomenona periodic in the channel number. An even number of channels translates to an integer number of flux quanta through the ring, which has no effect. However for $N$ odd or, alternatively, a finite staggering parameter $f$, a fractional flux pierces the system and this influences both, spectrum and dynamics of the quantum system. To develop the picture somewhat further, notice that the conductance, essentially the transition probability through the system, maps onto the Green function of the quantum system evaluated at imaginary ’time’ $L$. Imagining the latter represented through a spectral decomposition, the large $L$ behaviour depends on the low lying portions of the quantum spectrum, in particular the discrete, and flux periodic level structure of the compact sector of the theory. Later on we will see that for half integer flux (i.e. $N$ odd) there is a zero-energy level ($\rightarrow$ absence of localization) while for all other magnetic configurations the spectrum is gapped ($\rightarrow$ exponential localization.) This mechanism, and its relevance for the localization behaviour of the system was first analysed by Martin Zirnbauer[@fn1]. Similar but slightly more elaborate arguments can be used to understand the profile of the DoS.
We finally mention some intriguing parallels to the physics of the antiferromagnetic spin chain. According to Haldanes conjecture, a chain of spins with half-integer (integer) $S$ is in a long range ordered (disordered) phase[@haldane83]. It has also been found (see Ref. [@affleck88] for review) that for a chain with staggered hopping amplitude $j$, the strict integer/half-integer pattern is violated, e.g. an integer chain can be fine-tuned to an ordered phase. Technically, the system is described by a $\sigma$-model with a topological term not dissimilar to the one above. The differences are that in the spin case (a) the base manifold is $1+1$-dimensional (a [*quantum*]{} chain), (b) the target manifold is the two-sphere $\simeq {\rm SU}(2)/{\rm U}(1)$, and (c) the topological term classifies field configurations according to the number of coverings of the sphere (instead of winding numbers around the circle, as in our case). In the spin case, the coupling constant of the topological term is given by $S+j$, i.e. spin replaces channel number and staggering plays a similar role as in our case. (In fact, the linear coupling of the topological term to the staggering amplitude follows, independently of the model, from parity-conservation arguments to be discussed below.) Beyond these apparent technical parallels, the connection between the quenched disordered sublattice and the spin chain, respectively, is not understood.
The rest of the paper is organized as follows. In section \[sec:defin-model\] we quantify the definition of the model. Its field theory representation is introduced in section \[sec:field-theory\]. In section \[sec:transf-matr-appr\] the $\sigma$-model transfer matrix approach, i.e. the representation of observables in terms of the quantum Green function is discussed. This formulation is then applied to the calculation of conductance (section \[sec:conductance\]) and density of states (section \[sec:density-states\]). We conclude in section \[sec:summary\].
Definition of the model {#sec:defin-model}
=======================
We begin this section by upgrading the above qualitative introduction of the sublattice system to a more quantitative formulation. Quantum transport through the bulk of the system is described by the Hamiltonian $$\label{eq:1}
\hat H = \sum_{\langle ll'\rangle} \, c_{l,\mu}^\dagger
\left(t_{ll'}\delta^{\mu\mu'}+ R^{\mu \mu'}_{ll'}\right) c_{l'\mu'},$$ where $c_{l\mu}^\dagger$ creates a spinless electron at site $l$ in state $\mu=1,\dots,N$, the sum extends over nearest neighbour sites, and $R_{ll'}$ are $N\times N$ Gaussian distributed random hopping matrices with moments $$\begin{aligned}
&&\langle R_{ll'}^{\mu\nu} \rangle = 0,\\
&&\langle R_{ll'}^{\mu\nu} R_{ll'}^{\nu'\mu'}\rangle = {\lambda^2\over
N}\delta_{\mu\nu}
\delta_{\mu'\nu'},\qquad \lambda \ll 1. \end{aligned}$$ Apart from the Hermiticity condition $R_{ll'} = R^\dagger_{l'l}$ matrices on different links are statistically independent. These random matrices compete with the regular contribution to the hopping matrix elements, $t_{ll'}$. To be specific, we set $t_{ll'}= 1
+(-) a$ if the smaller of the two neighbouring site indices $l$ and $l'$ is even (odd). The real parameter $a$ is a measure for the staggering strength. Notice that $t_{ll} = {\cal O}(1)\gg \lambda$ implies that we are dealing with a weakly disordered system.
At both ends, a number of sites is coupled to leads (see figure \[fig:figure1\]). Quantum propagation within these leads is assumed to be generic, i.e. governed by a Hamiltonian without sublattice symmetry. To describe the coupling, we add to our bulk Hamiltonian a tunnelling contribution $$\begin{aligned}
&& \hat H_T = \hat H_T^L + \hat H_T^R,\\
&& \hat H_T^L = \sum_{l=1,2,\dots} \left(c_{\alpha}^\dagger
W^L_{\alpha a} d^L_a +
{\rm h.c.}\right),\\
&& \hat H_T^R = \sum_{l=L,L-1,\dots} \left(c_{\alpha}^\dagger W^R_{\alpha
a} d^R_a +
{\rm h.c.}\right),\end{aligned}$$ where $d^{L(R)\dagger}_a$ creates an electron propagating in the left (right) lead in a certain state $a=1,\dots, M\gg 1$ and we have introduced a composite index $\alpha =(l,\mu)$ comprising site and orbital index of the bulk system. The coupling matrix elements $W^{L/R}_{\alpha}$ are subject to the orthogonality relation[@Hans:1] $$\begin{aligned}
\label{eq:6}
&&\sum_{\mu} W^L_{a,l\mu}W^L_{l' \mu,b}= f(l) \delta_{ab}\delta_{ll'},
\nonumber\\
&&\sum_{\mu} W^R_{a,(L-l)\mu}W^R_{(L-l') \mu,b}= f(l)
\delta_{ab}\delta_{ll'}, \end{aligned}$$ where $f(x)$ is some envelope function decaying on a scale of ${\cal
O}(1)$ and normalized through $\sum_l f(l) = \gamma\ll 1$. The function $f$ and parameter $\gamma$ describe profile and strength of the coupling, respectively. Why did we introduce the multi-site coupling operators (\[eq:6\]) instead of connecting just the two terminal sites $l=1,L$ to the lead continuum? Modelling the coupling in a more general way is motivated by the presence of the site number staggering phenomena mentioned in the introduction. The above implementation of the coupling operator is sufficiently flexible to selectively probe these effects (sections \[sec:cond-short-syst-1\] and \[sec:density-states-short\]) [*or*]{} to average over any boundary oscillatory structures (sections \[sec:reduction-problem\] and \[sec:heat-kernel-finite\].)
To conclude the definition of the problem, let us introduce the Green function, $$\label{eq:2}
G(z) = \left(z- \hat H + i\pi ({\rm sgn\, Im\,}z)\sum_{C=L,R}
\hat W^C\hat W^{CT}\right)^{-1},$$ where $\hat W^C\hat W^{CT} = \{ \sum_a W^C_{\mu l, a} W^C_{a,\mu' l}
\}$ is an operator describing the escape of electrons from the bulk system into the leads[@Hans:1; @Hans:2]. Expressed in terms of these objects, the Landauer conductance of the system assumes the form $$\label{eq:5}
g = (2\pi)^2 \sum_{ab=1}^M
W^L_{a \alpha} W^R_{\beta b} W^R_{b \beta'} W^L_{\alpha' a}
\;\;\left\langle G_{\alpha \beta}(0^+) G_{\beta'\alpha'}(0^-) \right \rangle.$$ Our second quantity of interest, the density of states per site, is given by the standard expression $$\label{eq:7}
\nu(\epsilon) = -{1\over \pi L} {\rm \, Im \,} \sum_\alpha
\langle G_{\alpha \alpha}(\epsilon^+)\rangle.$$
Finally, to prepare the field theory formulation, let us consider the symmetries of the problem. Expressed in the notation introduced above, the chiral symmetry assumes the form $[\hat H,\hat \sigma_3]_+=0$, where $\hat \sigma_3 = \{ (-)^l \delta_{ll'}\}$ and $\hat H$ denotes the [*bulk*]{} part of the Hamiltonian. (Coupling to a non-sublattice continuum breaks chirality.) The presence of this symmetry implies invariance under the two parameter family of transformations $$\begin{aligned}
\label{eq:42}
c^\dagger_l \to c^\dagger_l e^{-z_1},\qquad& c_l \to e^{-z_2} c_l,&\qquad l
\;\rm{even},\nonumber\\
c^\dagger_l \to c^\dagger_l e^{+z_2},\qquad& c_l \to e^{+z_1} c_l,&\qquad l
\;\rm{odd},\end{aligned}$$ where $z_{1,2}$ are complex numbers. Depending on the choice of these parameters, (\[eq:42\]) expresses the standard ${\rm
U}(1)$-invariance of a model with conserved charge ($z_1=-z_2$), or the axial symmetry characteristic for chiral systems ($z_1=z_2$). (For the time being we treat the transformation as purely formal i.e. ignore the fact that for a general choice $z_{1,2}$ the transformed operators are no longer mutually adjoint.) We will come back to discussing the role of these symmetries after the effective field theory of the system has been introduced.
Field Theory {#sec:field-theory}
============
The construction of the low energy effective field theory of the sublattice wire follows the standard route of deriving nonlinear $\sigma$-models of disordered fermion systems[@Efetbook], there are no conceptually new elements involved. Referring to Appendices \[sec:field-integr-form\] and \[sec:deriv-field-theory\] for technical details of this derivation, we here discuss structure and key features of the resulting model.
As shown in the Appendices, the field theory representation of conductance and DoS is given by $$\label{eq:12}
g= -\left({M\pi\gamma\over 2}\right)^2 \left\langle
(T-T^{-1})_{12}(0)(T-T^{-1})_{21}(L) \right\rangle,$$ and $$\label{eq:11}
\nu(\epsilon) = {\nu_0\over 2 L} {\rm \, Re \,}\int_0^L dr \left\langle
T_{11}(r)+T^{-1}_{11}(r)\right\rangle,$$ respectively. Here, the bulk DoS, $ \nu_0 = {N\over 2\pi}$, $T(r)$ is a field taking values in the supergroup ${\rm GL}(1|1)$ and the continuum variable $r\in [0,L]$ replaces the site index. The angular brackets stand for functional averaging $$\langle \dots \rangle \equiv \int {\cal D}T e^{-S[T]}\; (\dots)$$ over a functional integral with action $S\equiv \int_0^L dr {\cal L}$ and effective Lagrangian $$\label{eq:9}
{\cal L} \equiv {\cal L}_{\rm fl} + {\cal L}_{\rm top} + {\cal
L}_{\rm T} + {\cal L}_z +
{\cal L}_{\rm Gade}.$$ The individual contributions are given by $$\begin{aligned}
\label{eq:4}
&&{\cal L}_{\rm top} = -{N+f\over 2} {\rm \,str\,}(T\partial_r
T^{-1}),\nonumber\\
&&{\cal L}_{\rm fl} = -{\xi\over 16}
{{\,\rm str\,}}(\partial_r T\partial_r T^{-1}),\nonumber \\
&& {\cal L}_{z} = -i {z\pi\nu_0\over 2} {\,\rm
str\,}(T+T^{-1}),\nonumber\\
&& {\cal L}_{\rm T} = {\pi M \gamma \over 2}{\,\rm \,str\,}(
T(r) + T^{-1}(r)) \left[\delta(r)+\delta(L-r)\right],\nonumber \\
&& {\cal L}_{\rm Gade} \equiv C \left[{{\,\rm str\,}}(T\partial_r
T^{-1})\right]^2,\end{aligned}$$ where $C$ is a coupling constant that need not be specified other than that it is small, $C= {\cal O}(1)\ll (N,M)$ and the parameter $f\equiv{2Na\over \lambda^2}$ measures the degree of staggering. (Notice our $f$ is identical to the control parameter $f$ defined in Ref.[@BMSA:98].) Finally, we have introduced a parameter $$\xi \equiv {N\over 2\lambda^2}$$ which will later identify as the localization length of the system.
To prepare the further discussion of the functional expectation values, let us briefly discuss the internal structure of the field theory. We first note that save for the values of the coupling constants, the structure of the action (\[eq:9\]) can be anticipated from inspection of Eq. (\[eq:42\]) and its supersymmetric extension: the field theory approach starts out from a promotion of the fermionic operator representation (\[eq:1\]) to a supersymmetric formulation in terms of Bose and Fermi fields. Within the latter representation (cf. e.g. Eq. (\[eq:3\])), the space of permissible invariance transformations is enlarged. The formerly complex parameters $e^{z_i}$ are replaced by two-dimensional matrices $T_i$ acting on the bosonic and fermionic components of the theory. Any sensible subsequent manipulation done on the functional integral must respect this invariance property. On the level of the effective theory described by $S[T]$, the transformation acts as $T\to T_1 T
T_2$ and indeed one verifies that the contributions ${\cal L}_{\rm
top}, {\cal L}_{\rm fl}$ and ${\cal L}_{\phi}$ of (\[eq:9\]) are invariant under this operation. Further, the two building blocks ${{\,\rm str\,}}(T\partial T^{-1})$ and ${{\,\rm str\,}}(\partial T\partial T^{-1})$ are the [*only*]{} operators with $\le 2$ derivatives compatible with the global ${\rm GL}(1|1)$ symmetry. In other words, the gross structure of the bulk action readily follows from the invariance criterion. For finite energies or coupling to the leads chirality is broken and global ${\rm U}(1)$ remains the only symmetry of the Hamiltonian. Within the supersymmetry formulation, the set of allowed transformations is then reduced down to configurations with $T_1=T_2^{-1}$ (the super-generalization of ${\rm U}(1)$). The operator ${{\,\rm str\,}}(T+T^{-1})$ is the minimal positive (see below) choice compatible with the restricted symmetry. Summarizing, the three terms ${{\,\rm str\,}}(T\partial T^{-1})$, ${{\,\rm str\,}}(\partial T\partial T^{-1})$, and ${{\,\rm str\,}}(T+T^{-1})$ exhaust the set of operators compatible with the global transformation behaviour of the model.
As for the coupling constants of the theory – not specified by symmetry arguments but all derived in Appendix \[sec:deriv-field-theory\] – notice that contrary to the standard $\sigma$-models of systems with WD symmetry two, instead of just one second derivative operators appear in the action. [ *Mathematically*]{}, the reason for the appearance of two contributions is that the target manifold of the theory, ${{\rm GL}(1|1)}$ is a reducible symmetric space; it decomposes into two irreducible factors, a point discussed in detail in Ref. [@zirnbauer96]. Each of these factors can be endowed with its own metric which implies the existence of two independent second derivative operators in the theory. [ *Physically*]{}, the presence of the non-standard operator $\sim
{{\,\rm str\,}}^2(T\partial T^{-1})$ has profound consequences for the behaviour of the $2d$-version of the field theory[@gade93]: the RG analysis of the model shows that the coupling constant of this operator grows under renormalization while driving the coupling of the energy operator $\sim {{\,\rm str\,}}(T+T^{-1})$ to large values. At the same time the coupling constant of the standard gradient operator $\sim{{\,\rm str\,}}(\partial T\partial T^{-1})$, essentially the conductance, remains un-renormalized. In one dimension, the situation is different. Contrary to what one might expect, the contribution ${\cal L}_{\rm Gade}$ is not remotely as important as in two dimensions, and it is another operator that drives the localization behaviour of the model. In parentheses we remark that the target space of the transfer matrix approach to the problem, ${\rm GL}(M)/{\rm U}(M) \simeq {\rm
SL}(M)/{\rm SU}(M) \times {\cal R}^+$ factors into two components, too[@brouwer00:_nonun]. Accordingly, the Fokker-Planck equation governing the ’Brownian motion’ on that space is controlled by [ *two*]{} independent coupling constants, both determined by the microscopic definition of the model. The second of these contributions, essentially the analogue of our Lagrangian ${\cal
L}_{\phi}$, leads to non-universal corrections to the overall picture which perish in the limit of a large number of channels.
A second aspect discriminating the present model from its WD relatives is the appearance of a first order gradient operator in the action. In fact, the presence of this contribution might raise suspicion: although allowed by symmetry, ${{\,\rm str\,}}(T\partial_r T^{-1})$ is not invariant under space reflection $r\to -r$, in contrast to the microscopic parent model (for $a=0$.) The resolution of this puzzle lies in the fact that $S_{\rm top}\equiv \int {\cal L}$ is of topologcial origin and, although not manifestly so, [*does*]{} respect the space inversion property. We will discuss this point momentarily after the internal structure of the field manifold and the integration measure have been specified.
Functional integrals can only be defined on manifolds that are Riemannian, i.e. endowed with a positive metric. The supergroup ${\rm
GL}(1|1)$ (like all other supersymmetric spaces that appear in the context of field theories of disordered Fermi systems) does not fulfil this criterion, a point discussed in detail in Ref. [@zirnbauer96]. However, it does contain a maximally Riemannian [*sub*]{}manifold ${\cal M}$, defined as follows: the boson-boson block of ${\cal M}$, a one-dimensional manifold by itself, is isomorphic to the non-compact symmetric space ${\rm GL}(1)/{\rm
U}(1)\simeq {\cal R}^+$, i.e. the positive real numbers. (This space is trivially Riemannian.) The fermion-fermion block is isomorphic to the compact symmetric space ${\rm U}(1)$. No limitations in the Grassmann valued boson-fermion sectors of the theory are needed since the whole issue of convergence does not arise here. Summarizing, ${\cal M} = {\cal R}^{+} \times {\rm U}(1)$, where the notation is symbolic, specifying the boson-boson and fermion-fermion sector, respectively.
A convenient field representation respecting these convergence criteria is given by $$\begin{aligned}
\label{eq:8}
T = k a k^{-1},\qquad
k = \exp\left(\matrix{&\eta\cr\nu&}\right),\qquad
a = \exp\left(\matrix{x &\cr&iy}\right),
\qquad x,y\in {\cal R}.\end{aligned}$$ In this parameterization, the group integration $\int dT$ extends over the degrees of freedom $x,y,\eta,\nu$, without further constraints. The invariant group measure associated to the representation (\[eq:8\]) are defined in Eqs. (\[eq:22\]) and (\[eq:28\]).
We are now in a position to discuss the role of the contribution $S_{\rm top}$. First notice that for sufficiently strong lead coupling the boundary action ${\cal L}_{\rm T}$ projects the fields onto the group origin $T(x,y)=\openone$, i.e. enforces $x(0)=x(L)=0$ and $y(0)=2\pi k$, $y(L)=2\pi k'$, where $k,k'$ are integer. As a consequence, the first derivative operator can be written as $$\begin{aligned}
\label{eq:35}
&&S_{\rm top}[T] = -{N+f\over 2} \int dr {\rm \,str\,}(T\partial_r
T^{-1})=\nonumber\\
&&\qquad= {N+f\over 2} \int dr \partial_r {\rm \,str\,}\ln T=\nonumber\\
&&\qquad= {N+f\over 2} {\rm \,str\,}(\ln T(L)-\ln T(0))=\nonumber\\
&&\qquad= i \pi (N+f)(k-k').
\end{aligned}$$ This makes the topological nature of the term manifest: it counts winding numbers in the fermionic sector of the theory. The integer $k-k'$ is a topological invariant characterizing each field configuration $T(r)$. Further notice that for $f=0$, $$e^{-S_{\rm top}[T]} = e^{+i\pi N (k-k')} = e^{-i\pi N(k-k')}.$$ Since it is the exponentiated action and not the action itself that matters, the last identity tells us that the global [*sign*]{} of the first gradient operator is irrelevant (all for $f=0$). This settles the above raised issue of the behaviour of the model under space reflection: although the first order derivative is not invariant under $r \to -r$ the exponentiated action is.
Transfer Matrix Approach {#sec:transf-matr-appr}
========================
Principal aspects of the system properties we are going to analyse are non-perturbative, i.e. cannot be obtained as power series in the coupling constants of the action. Progress with such type of problems can be made by applying the $\sigma$-model transfer-matrix technique[@Ef:83], an approach conceptually similar to the DMPK formalism.
We begin by defining the two functions $$\begin{aligned}
&&Y_L(T_1,T_2,r) \equiv \int_{T(0)=T_1\atop T(r)=T_2} {\cal D}T e^{-\int_0^r dr {\cal
L}[T]},\\
&&Y_R(T_1,T_2,r) \equiv \int_{T(r)=T_1\atop T(L)=T_2} {\cal D}T
e^{-\int_r^L dr {\cal L}[T]}.\end{aligned}$$ Expressed in terms of these objects, the DoS assumes the form $$\begin{aligned}
\label{eq:13}
\nu(\epsilon) = {\nu_0 \over 2 L} {\,\rm Re\,} \int_0^L
dr
\int dT \; Y_L(\openone,T,r)\; (T_{11}+T^{-1}_{11})\; Y_R(T,\openone,L-r),\end{aligned}$$ where we have used that for sufficiently strong coupling to the leads, the boundary configurations $T$ are close to unity. (Throughout much of this paper we will consider the DoS of coupled systems. For sufficiently large systems, the choice of boundary conditions is inessential, a point to be verified below.) Similarly, the conductance obtains as $$\begin{aligned}
\label{eq:14}
&& g= -\left({\pi M\gamma\over 2}\right)^2 \int dT \;
dT'\;\times\nonumber\\
&&\hspace{2.0cm}\times e^{-{\pi M
\gamma \over 2} {{\,\rm str\,}}(T+T^{-1})}
(T-T^{-1})_{12}\; Y_L(T,T',L)\; (T'-T'^{-1})_{21} e^{-{\pi M
\gamma \over 2} {{\,\rm str\,}}(T'+T'^{-1})}.\end{aligned}$$ From Eqs. (\[eq:13\]) and (\[eq:14\]) it is clear that the functions $Y_{L,R}$ encode the essential system properties we are interested in.
As a first step towards the computation of these objects let us explore how the symmetries of the action translate to symmetries of $Y_{L,R}$. We first consider the case $z=0$, relevant to the analysis of the conductance, where the action is fully invariant under ${{\rm GL}(1|1)}$-transformations. The invariance ${\cal L}[T]={\cal L}[T_1 T
T_2]$, $T_{1,2}={\rm const.}$ then directly implies $Y_{L,R}(T,T',r)=Y_{L,R}((T_1T T_2),(T_1 T' T_2),r)$. From this identity one readily verifies that $$\begin{aligned}
&&Y_R[T,T',r]=Y_R[T T'^{-1},\openone,r]=
Y_R[T^{'-1} T,\openone,r],\\
&&Y_L[T,T',r]=Y_L[\openone,T' T^{-1},r]=
Y_L[\openone,T^{-1} T',r],\qquad(z=0). \end{aligned}$$ In other words, for $z=0$ the arguments of the heat kernels enter in an invariant product type form and it suffices to consider the reduced functions $$\begin{aligned}
\label{eq:16}
Y_{R}(T,r)\equiv Y_{R}(T,\openone,r),\qquad Y_{L}(T,r)\equiv
Y_{L}(\openone,T,r),\end{aligned}$$ depending on a single argument only. The same invariance property (now evaluated for $T_2=T_1^{-1}$) implies that $Y_{L,R}(T,r) = Y_{L,R}(T_1
T T_1^{-1},r)$. Imagining the argument $T$ to be represented in the polar decomposition (\[eq:8\]) and setting $T_1=k^{-1}$ the argument can be reduced to the diagonal matrix of eigenvalues $a$: $$\begin{aligned}
\label{eq:15}
Y_{L,R}(kak^{-1},r)=Y_{L,R}(a,r). \end{aligned}$$ For $z\not=0$, the invariance of the theory collapses down to transformations with $T_2=T_1^{-1}$. However, the representation of the DoS above implies that we are solely interested in functions of the type (\[eq:16\]), with second argument set to unity, anyway. Since transformations $T\to T_1 T T_1^{-1}$ are still permissible, these objects depend on the eigenvalues of the argument matrix only, as before for the case $z=0$. Summarizing, in the analysis of both conductance and DoS, it is sufficient to consider functions $Y_{L,R}$ depending on a single argument with invariance property (\[eq:15\]).
Following the basic philosophy of the transfer matrix approach, we will compute the functions $Y_{L,R}(T,r)$ iteratively, by asking how much they change under infinitesimal variation of the arguments $r\to
r+\epsilon$. Considering the function $Y_L$ for definiteness, we first notice that, by definition, $$Y_L(T,r+\epsilon)=\int {\cal D} T e^{-\int_r^{r+\epsilon}dr' {\cal
L}[T]} Y_L(T(r),r).$$ For sufficiently small $\epsilon$, the action can be expanded and we obtain $$\begin{aligned}
&& Y_L(T,r+\epsilon)-Y_L(T,r) = \int dT' e^{-{\xi\over 16 \epsilon}
{{\,\rm str\,}}(TT'^{-1} + T^{-1}T')}\times\\
&&\hspace{3.0cm}\times
e^{ {N+f\over 2} {{\,\rm str\,}}(T'T^{-1}) + i {z\pi\nu\epsilon \over 2} {{\,\rm str\,}}(T+T^{-1})}
\left(Y_L(T',r)-Y_L(T,r)\right),\end{aligned}$$ where we have used that, due to supersymmetry, $ \int {\cal D}T
\exp(-\int_r^{r+\epsilon} {\cal L}[T]) \times 1 = 1$. We have also temporarily set the coupling constant of the operator ${\cal L}_{\rm Gade}$ to zero. As mentioned above, this term does not have much relevance in the present context. We will briefly discuss its role in section \[sec:role-gade-term\].
From hereon, the derivation of an evolution equation for $Y_L$ is conceptually straightforward: the exponential weight factor in the first line of the equation enforces that $T'$ is close to $T$, symbolically, $T'T^{-1} = \openone + {\cal O}(\epsilon)$. We should thus expand $T'$ around $T$ and evaluate the integral $\int dT'$ perturbatively in $\epsilon$. This expansion is most conveniently done in the polar coordinates introduced above (because the heat kernel depends on the radial degrees of freedom, only.) As a result of a calculation similar but much more simple than the one for the standard $\sigma$-models with their larger matrix fields[@Efetbook] one obtains the Schrödinger type equation $$\begin{aligned}
\label{eq:17}
&& \left(\mp \partial_t - 4({{\bf D}}\pm {{\bf A}})^2 + V(x,y)
\right)Y_{L\atop R}(a,r)=0,\end{aligned}$$ where $ V(x,y)= -is (\cosh(x)-\cos(y)) $ and we have introduced the dimensionless parameters $$t \equiv {r\over \xi},\qquad s= \pi \nu\xi \epsilon.$$ Physically, $t$ is the length coordinate measured in units of the localization length $\xi$ and $s$ the energy measured in units of the single particle level spacing $\Delta_\xi \equiv \xi \nu$ of a system of length $\xi$. Further, the symbol ${{\bf D}}=(D_x,D_y)^T$ denotes a vector differential operator defined through $$\begin{aligned}
\label{eq:21}
&&D_{x} = \partial_{x} - {1\over 2}
\coth\left({x-iy\over 2}\right), \nonumber\\
&&D_{y} = \partial_{y} + {i\over 2}
\coth\left({x-iy\over 2}\right),\end{aligned}$$ where the constant vector $$\label{eq:24}
{{\bf A}}= {N+f\over 2}(1,-i)^T.$$ Finally, evaluation of (\[eq:17\]) for asymptotically small times $t
\searrow 0$ leads to the initial condition $$\label{eq:18}
\lim_{t\to 0} Y_{L,R}(x,y,t)=\delta(x,y) \equiv \lim_{t\to 0}e^{-{1
\over 8t} (x^2 + y^2)}.$$
It is very instructive to interpret the structure of the evolution equation (\[eq:17\]) in the light of the analogy between field theory and point particle quantum mechanics on ${{\rm GL}(1|1)}$ discussed in the introduction. Within this picture, the functions $Y_L[T,r]$ acquire the meaning of [*Green functions*]{}, i.e. transition amplitudes for the propagation between the origin of the manifold $T=\openone$ at time $0$ and a final configuration $T$ at time $t$. The evolution equations (\[eq:17\]) becomes a a time-dependent Schrödinger equation with quantum Hamiltonian, $H = -2 ({{\bf D}}-{{\bf A}})^2 + V(x,y)$. While the term $V(x,y) = -is (\cosh(x)-\cos(y)) = -{is\over 2}
{{\,\rm str\,}}(T+T^{-1})$ simply represents the potential inherited from the Lagrangian, the ’kinetic’ operator is more interesting. The covariant structure $({{\bf D}}-{{\bf A}})^2$ describes minimal coupling to the constant vector potential where the unfamiliarly looking structure of the derivative operator ${{\bf D}}$ is a consequence of the fact that our particle lives on a curved manifold. Indeed, it is straightforward to verify that for ${{\bf A}}=0$ $${{\bf D}}\cdot {{\bf D}}= \sum_{i=x,y} J^{-1}\partial_i J\partial_i,$$ where $J(x,y)$, specified in Eq. (\[eq:28\]), is the square root of the determinant of the metric tensor on ${{\rm GL}(1|1)}$. This structure identifies ${{\bf D}}\cdot {{\bf D}}$ as the radial part of the Laplace operator on ${{\rm GL}(1|1)}$. (’Radial’, because the two coordinates $x$ and $y$, spanning a maximally commutative sub-algebra of the Lie algebra of ${{\rm GL}(1|1)}$.)
Summarizing, we have identified $H$ as the Hamiltonian of a charged particle on the group manifold ${{\rm GL}(1|1)}$. Our next task will be to compute its Green functions $Y_{L/R}$. We begin by considering the case of a free particle, $V=0\leadsto \epsilon=0$. The solution of this problem will contain the information needed to compute the conductance.
Conductance {#sec:conductance}
===========
In this section the general formalism developed above is employed to compute the conductance $g$ of the system. Our main objective will be to understand the impact of topology on the localization behaviour of the system. However, before embarking on this analysis it is tempting to digress for a moment and to briefly consider the transport behaviour of [*short*]{} systems, specifically the aforementioned anomalous sensitivity to the coupling to the leads. Being not directly related to the mainstream of the paper, the technicalities of this discussion have been deferred to Appendix \[sec:cond-short-syst\] and we here restrict ourselves to a summary of results.
Digression: Conductance of Short Wires {#sec:cond-short-syst-1}
--------------------------------------
In the present context, the phrase ’short’ means that systems in the quantum dot regime $L<\xi/(M\gamma)$ are considered: the conductance is not so much determined by the bulk transport properties of the system but rather by the strength of the coupling to the leads. Moreover, and this is a speciality of the sublattice system, the [*parity*]{} of the coupling, i.e. the even- or oddness of the connecting sites, turns out to play a crucial role. More precisely we find that (a) for systems where at both ends a number of sites of alternating parity are connected – the setup considered in much of this paper – the short system conductance is given by $$g= {M\pi \gamma\over 2},$$ in accord with the behaviour of conventional quantum dots. The same result obtains (b) for systems where only one site at either end is coupled and these sites have opposite parity (even/odd or odd/even). However, (c) for single site coupling with even/even or odd/odd connectors, the conductance [*vanishes*]{}. To heuristically understand the phenomenon, it is instructive to consider the toy-model case of a strictly one-dimensional clean sublattice system. For zero energy, the wave length of current carrying excitations is commensurable with the lattice spacing. This means that the relevant quantum wave functions have nodes at alternating sites. E.g., for an even-even configuration a state entering from the right has zero quantum amplitude at the exit site on the right. This implies a total blockade of electric current. A less obvious fact is that this phenomenon survives generalization to many channels and disorder.
We repeat that all these results are obtained for short systems; field fluctuations, describing the propagation of spatially non-uniform diffusive excitations, are neglected. An interesting question, not considered in this paper, is how such modes would affect the conductance as the system size is increased.
Reduction of the Problem to (0-dimensional) quantum mechanics on ${{\rm GL}(1|1)}$ {#sec:reduction-problem}
----------------------------------------------------------------------------------
We next turn back to the discussion of large systems (equilibrated coupling of type (a) understood.) Inspection of the basic expression (\[eq:14\]) shows that the problem factorizes into doing the boundary integrals and analysing the bulk behaviour of the heat kernel, respectively.
We begin by discussing the boundary regions. Following a line of arguments developed in Ref.[@mirlin94], we first notice that due to the exponential weights $\sim \exp(-{\pi M \gamma\over
2}{{\,\rm str\,}}(T+T^{-1}))$, the integrands are confined to the immediate vicinity of the group origin. This suggests to write $T=\exp W$, and do the integrals over generators $W$ in a Gaussian approximation. Setting $W=\left(\matrix{u&\sigma \cr \rho & iv}\right)$ and expanding in coefficients we arrive at integrals of the structure, $$\begin{aligned}
g={\rm const.}\times \int du dv d\rho d\sigma e^{-{\pi M \gamma \over
4}(u^2 + v^2+ 2 \sigma
\rho)} \sigma F(T(u,v,\sigma,\rho)), \end{aligned}$$ where the symbol $F(T)$, represents the functional dependence of the heat kernel on the boundary field. Evaluation of the Gaussian superintegral leads to $$g = {\rm const.} \times {1\over \pi M \gamma} \partial_\rho
F(T)\big|_{T=\openone}.$$ Doing the same procedure for the second boundary integral and fixing factors we obtain $$g = \partial_\rho \partial_{\sigma'}Y_L(T,T',L)\big|_{T=T'=\openone}=
\partial_\rho \partial_{\sigma'}Y_L(TT'^{-1},L)\big|_{T=T'=\openone},$$ where the second expression contains the one-argument heat kernel introduced in Eq. (\[eq:16\]). According to this expression, the conductance is obtained by second order expansion of the heat kernel around the origin. We next note that due to the invariance property (\[eq:15\]) the expansion starts as $$\label{eq:23}
Y_L(\tilde T=e^{\tilde W},L) = 1 + c_1 {{\,\rm str\,}}(\tilde W) + c_2 {{\,\rm str\,}}(\tilde W^2)
+ c_3 [{{\,\rm str\,}}(\tilde W)]^2+\dots .$$ Now, in our case, $$\tilde W = \ln(TT'^{-1})= \ln\left(\exp\left(
\matrix{&0\cr\rho&}\right) \exp\left( \matrix{&-\sigma'\cr
0&}\right) \right)=\left(\matrix{{1\over 2}\sigma'\rho
&-\sigma'\cr \rho &{1\over 2}\sigma'\rho}\right).$$ Substitution of this expression into (\[eq:23\]) and differentiation leads to $$\label{eq:34}
g=2c_2,$$ i.e. the problem has been reduced to fixing the coefficients of the series expansion of $Y_L$. Notice that this series representation is totally determined by the features of the bulk system; all aspects related to the coupling to the leads have disappeared from the problem.
We now have to face up to the principal task, the calculation of the heat kernel. Its interpretation as the Green function of the problem suggests to begin by representing $Y_L$ through a formal spectral decomposition: consider Eq. (\[eq:17\]) at zero potential, $
\left(-\partial_t - 4 ({{\bf D}}-{{\bf A}})^2 \right)Y_{L}(x,y,t)=0$, and suppose we had managed to find a set of eigenfunctions, $$\label{eq:26}
-4({{\bf D}}-{{\bf A}})^2 \Psi^{(r)}_n = \epsilon_n \Psi^{(r)}_n.$$ Assuming completeness, we can then span the heat kernel as $$Y_L(x,y,t) = \sum_n c_n \Psi^{(r)}_n(x,y) e^{-\epsilon_n t},$$ where the expansion coefficients are determined through the initial condition (\[eq:18\]). Provided the spectrum is suitably structured (positive and gapped against some low lying levels), and keeping in mind that we are interested in asymptotically large values of $t$, the sum may be restricted to a limited set of $n$’s. The problem thus reduces to (a) exploring the low-lying spectral content of the operator ${{\bf D}}-{{\bf A}}$, and (b) fixing the expansion coefficients.
Spectrum and Eigenfunctions of the ’Kinetic Energy Operator’ on ${{\rm GL}(1|1)}$ {#sec:spectr-eigenf-kinet}
---------------------------------------------------------------------------------
In spite of the unfamiliarly looking coordinate representation of $({{\bf D}}-{{\bf A}})^2$ analytic progress with the problem is straightforward, the reason being that this operator is nothing but a plane wave operator in disguise. To make the hidden simplicity of the problem manifest, let us first remove the dependence of the differential operator on the (pure gauge) potential ${{\bf A}}$: transformation $\Psi_n(x,y) \to
e^{{N+f\over 2}(x-iy)} \Psi_n(x,y) \equiv \hat \Psi_n(x,y)$ brings the eigenvalue equation (\[eq:26\]) into the form $$-4 \Delta \hat \Psi_n = \epsilon_n \hat \Psi_n,$$ where $\Delta={{\bf D}}{{\bf D}}$ is the radial part of the Laplacian on ${{\rm GL}(1|1)}$. Notice that the structure of this equation does [*not*]{} imply that the vector potential has disappeared from the problem. It has merely been transferred from the differential operator to the boundary conditions attached to the differential equation. While irrelevant in the non-compact sector of the theory, changes of the boundary conditions in the compact sector generally cause qualitative effects. To appreciate this point, notice that the un-gauged Hilbert space of the problem has periodic boundary conditions, $ \Psi_n(x,y) =
\Psi_n(x,y+2\pi). $ Yet, after the gauge transformation, $ \hat
\Psi_n(x,y) = \hat \Psi_n(x,y+2\pi) (-)^{N+f} $, i.e. for $N$ odd or non-zero staggering a transmutation to twisted boundary conditions has taken place. Needless to say that this change bears consequences for the spectral structure of the problem and, therefore, for the transport behaviour of the system.
To make further progress, we subject the eigenvalue problem to the similarity transformation $$\begin{aligned}
&&\hat \Psi \to J^{1/2} \hat \Psi \equiv \tilde
\Psi,\nonumber\\
&&\Delta \to J^{1/2} \Delta J^{-1/2} \equiv \tilde \Delta =
\partial_x^2 + \partial_y^2.\end{aligned}$$ This change of representation entails an enormous simplification of the problem. The uninvitingly looking operator $\Delta$ has become a flat two-dimensional Laplace operator[@fn2]. However, notice that the transformation by $J^{1/2}$ effects yet another change in the boundary conditions: due to $J^{1/2}(x,0) = -J^{1/2}(x,2\pi)$, the transformed states are subject to the condition $$\tilde\Psi_n(x,y) = \tilde \Psi_n(x,y+2\pi) (-)^{N+f+1}.$$ At this point, the solution of the eigenvalue problem has become a triviality. The equation $$-4(\partial_x^2 + \partial_y^2)\tilde \Psi_n(x,y) =
\epsilon_n \tilde \Psi_n(x,y),$$ is solved by the exponentials $ \tilde \Psi_{k,l}(x,y) =
e^{ip_l x + ip_k y}$, where $(k,l)\equiv n$ are two ’quantum numbers’, $p_{k,l}$ the associated momenta and the eigenvalues $\epsilon_{kl} =
4(p_k^2 + p_l^2)$. From these states we obtain our un-gauged and un-transformed original wave functions as $$\label{eq:25}
\Psi_{kl}(x,y) = \sinh\left({x-iy\over
2}\right) e^{(ip_l -{N+f\over 2}) x + i(p_k + {N+f\over 2}) y}.$$ To give this set of solutions some meaning, we need to specify the range of permissible $k$’s and $l$’s. In the compact sector the situation is clear – the circular boundary condition specified above enforces $p_k= k- {N+f\over 2}$, with [*half*]{}integer $k$. The conditions to be imposed in the non-compact sector are tightly linked to the integrability properties of our wave functions:
The space of radial functions on ${{\rm GL}(1|1)}$ is endowed with a natural scalar product, viz. $$\label{eq:30}
\langle f,g\rangle \equiv \int_{-\infty}^\infty dx \int_0^{2\pi} dy
J(x,y) f(x,y) g(x,y).$$ We demand that the eigenfunctions contributing to the spectral decomposition of the heat kernel be square integrable w.r.t. $\langle
\;,\;\rangle$. Inspection of Eq. (\[eq:25\]) shows that this requirement enforces $p_l = l -i {N+f\over 2}$, where $l$ may be arbitrary and real.
Finally, we need to specify a set of functions sufficiently complete to generate an expansion of the heat kernel. The present problem does not come with a natural Hermitian or symmetric structure (i.e. for finite ${{\bf A}}$ the kinetic energy neither symmetric nor Hermitian.) However, defining a ’fake complex conjugation’ through $
\bar \Psi_{kl} = \Psi_{-k,-l} $ it is straightforward to show that $$\label{eq:31}
\langle \bar \Psi_{kl},\Psi_{k'l'} \rangle = (2\pi)^2 \delta(l-l')
\delta_{kk'}.$$ We may thus attempt to represent sufficiently well behaved (for the cautious formulation, see below) functions as $$\label{eq:32}
g(x,y) = \sum_k \int dl\; g_{kl} \Psi_{kl}(x,y), \qquad
g_{kl} = (2\pi)^{-2} \langle \bar \Psi_{kl},g \rangle.$$ Before applying this procedure to the heat kernel, let us summarize our main findings for clarity: the radial Laplacian on ${{\rm GL}(1|1)}$ is diagonalized by the set of functions $$\label{eq:33}
\Psi_{kl}(x,y) = \sinh\left({x-iy\over
2}\right) e^{i(lx + ky)}.$$ where $k\in {\cal Z}+1/2$, $l$ real and the eigenvalues are given by $$\label{eq:29}
\epsilon_{kl} = 4 \left[ \left(k - {N+f\over 2}\right)^2 + \left(l -
i {N+f\over 2}\right)^2\right].$$ (Notice that the appearance of an imaginary part proportional to the strength of the vector potential is due to the fact that for finite ${\bf A}$, the Hamiltonian of the theory is neither symmetric nor Hermitian.) The spectral decomposition of radial functions is defined through Eqs. (\[eq:30\]), (\[eq:31\]) and (\[eq:32\]).
Computation of the Conductance {#sec:comp-cond}
------------------------------
We now apply the machinery developed in the last section to the analysis of the heat kernel. In principle, the strategy seems to be prescribed by what was said above. We should determine the Fourier coefficients of the initial configuration (\[eq:18\]), $\delta_{kl}
= \langle \bar \Psi_{lk},\delta\rangle$, from where the heat kernel would follow as $Y_L(x,y,t) = (2\pi)^{-2} \sum_k \int dl\; \delta_{kl}
e^{-\epsilon_{kl} t} \Psi_{kl}(x,y)$. There is a problem with this procedure, viz. the expansion coefficients of the ’$\delta$-distribution’ $\delta(x,y)$ vanish. Indeed one verifies that in the limit $t\to 0$, the support of the Gaussian in (\[eq:18\]) shrinks to zero while its maximum remains limited by one. This readily implies $\langle \Psi,\delta\rangle=0$. The reason for this pathological behaviour is that our radial theory memorizes that it derived from a supersymmetric parent theory. In a sense, supersymmetry can be interpreted as a theory on a zero-dimensional background, i.e. there is no singular ’volume factor’ compensating for the vanishing support of the $\delta$-function, as would be the case in spaces with positive dimension.
The problem can be circumvented by a cute trick[@mirlin94]. Instead of Fourier expanding $Y_L$, we consider the function $Y_L -1$. Since unity by itself solves the heat equation, no harm has been done and all that has changed is the boundary condition: $\lim_{t\to 0}
(Y_{L}(x,y,t)-1)= \delta(x,y)-1$, a function that equals minus unity almost everywhere save for the origin where it vanishes.
We thus represent the heat kernel as $$\label{eq:39}
Y_L(x,y,r) = 1- (2\pi)^{-2} \sum_k \int dl \, 1_{kl} \Psi_{kl}(x,y)
e^{-\epsilon_n t},$$ where $$1_{kl} = \langle \bar \Psi_{kl},1 \rangle = \int dx \int dy
{e^{i(lx + ky)}\over \sinh\left({x-iy \over 2}\right)}=
{4\pi i \over l-ik}.$$ (To obtain the last equality, it is convenient to first do the $x$-integral. Closure of the integration contour in the upper/lower complex half plane for positive/negative $l$ yields a semi-infinite sum over residues of the $\sinh$-function, along with a $y$-integral that is of simple plane wave type. Doing sum and integral one obtains the result.)
Substitution of this result into the expansion of the heat kernel now yields $$\begin{aligned}
&& Y_L(x,y,t)-1 = -{i\over \pi} \sum_k \int dl \, {1\over l-ik} \sinh\left({x-iy\over
2}\right) e^{i(lx + ky)}
e^{-\epsilon_{kl} t}\stackrel{{\cal O}(x,y)^2}{\longrightarrow}\\
&&\qquad \stackrel{{\cal O}(x,y)^2}{\longrightarrow}
{1\over 4\pi}\sum_k \int dl \left(
{l+ik\over l-ik}(x-iy)^2 + (x^2+y^2)\right)e^{-\epsilon_{kl} t}=\\
&&\qquad = {1\over 4\pi }\sum_k \int dl \left(
{l+ik\over l-ik}({{\,\rm str\,}}(W))^2 + {{\,\rm str\,}}(W^2)\right)e^{-\epsilon_{kl} t},\end{aligned}$$ where in the last line we have switched back to a coordinate invariant representation. Comparison with Eqs. (\[eq:23\]) and (\[eq:34\]) finally leads to the identification $$\begin{aligned}
&&g = {1\over 2\pi}\sum_{k\in {\cal Z}+1/2} \int_{-\infty}^{\infty} dl
e^{- {4L\over \xi}\left[ \left(k - {N+f\over 2}\right)^2 + \left(l -
i {N+f\over 2}\right)^2\right]}=\\
&&\qquad = {1\over 2}\left({\xi \over 4\pi L}\right)^{1/2}
\sum_{k\in {\cal Z}+1/2}
e^{- {4L\over \xi} \left(k - {N+f\over 2}\right)^2}.\end{aligned}$$ where we have inserted the explicit form of the eigenvalues. Notice that all manipulations leading from the original $\sigma$-model representation to the above Gaussian integral representation have been exact.
We next evaluate this result in the two limiting cases of physical interest, $L\ll \xi$ (Ohmic regime) and $L\gg \xi$ (localized regime.) Beginning with the Ohmic case, we first notice that for $L \ll \xi$ many terms contribute to the $k$ summation implying that the sum can be approximated by an integral. Thus, $$\begin{aligned}
&& g \stackrel{L\ll \xi}{\approx}
{1\over 2}\left({\xi \over 4\pi L}\right)^{1/2}
\int dk
e^{- {4L\over \xi} \left(k - {N-f\over 2}\right)^2} = {\xi\over 16L}.\end{aligned}$$ As for any ordinary conductor, $g$ is inversely proportional to the system size; the parity of the channel number does not play a role.
In the opposite case, $L \gg \xi$, only those discrete indices that minimize the exponent contribute to the sum. Specifically, for an even channel number and no staggering, $$g \stackrel{L\gg \xi}{\approx}
\left({\xi \over 4\pi L}\right)^{1/2}
e^{- {L\over \xi}}, \qquad N\; \mbox{even}, f=0.$$ In contrast, for $N$ odd and $f$ still zero, the exponent vanishes for the half integer $k={N\over 2}$ and $$g \stackrel{L\gg \xi}{\approx}
{1\over 2}\left({\xi \over 4\pi L}\right)^{1/2}, \qquad N\;
\mbox{odd}, f=0$$ depends algebraically on the system size. Finally, it is clear that for non-vanishing staggering, $f\not=0$, intermediate types of behaviour are realized. E.g., an $N$ even chain with staggering $f=\pm 1$ behaves like a symmetric $N$ odd chain, etc.
The Role of the Gade Term {#sec:role-gade-term}
-------------------------
Before leaving this section let us briefly discuss the role of the, so far neglected, Gade operator $S_{\rm Gade}[T]$. The inclusion of this term in the derivation of the heat equation is straightforward. As a result, the planar Laplacian $\tilde \Delta = \partial_x^2 +
\partial_y^2$ gets replaced by $$\tilde \Delta =
\partial_x^2 + \partial_y^2 + \tilde \eta (\partial_x - i\partial_y)^2,$$ where $\tilde \eta \equiv {16 C \over \xi}\propto N^{-1}\ll 1$. This operator is still diagonalized by the plane waves discussed above. The eigenvalues change to $$\epsilon_{kl}=4 \left[(1-\tilde \eta)\left(k - {N+f\over 2}\right)^2 +
(1+\tilde \eta)\left(l - i {N+f\over 2}\right)^2\right].$$ Recapitulating the computation of the conductance, one finds that the small dilatation introduced by finite values of $\tilde \eta$ does not affect the long range transport behaviour of the system.
All this is compatible with the structure of the DMPK transfer matrix approach to the problem. As mentioned above, the DMPK evolution equation is controlled by [*two*]{} coupling constants. One of these, in Ref. [@brouwer00:_nonun] denoted by $\eta$, is small, $\eta\sim {\cal O}(M^{-1})$ and becomes inessential in the limit of a large number of channels. The analysis above suggests that the coupling constant of the Gade operator, $\propto \tilde \eta$, and the $\eta$ of the DMPK approach play the same role. This analogy is supported by the above discussed geometric structure of the target spaces of the two theories.
Density of States {#sec:density-states}
=================
We now turn our attention to the low energy density of states of the sublattice system. As with the conductance, we will first consider the behaviour of short wires, and then discuss the localized regime.
Density of states of short wires {#sec:density-states-short}
--------------------------------
As in section \[sec:cond-short-syst-1\] we consider a short sublattice wire of length $L<\xi /(M\gamma)$. The spectrum of such systems exhibits structure on the scale of the mean level spacing. In order not to blur these fine structures, the coupling to the external leads will be switched off throughout this section. Following the logics of section \[sec:cond-short-syst-1\], one would then conclude that the action reduces to $S[T]=\int dr {\cal L}_z[T] =
-i{s\over 2}{{\,\rm str\,}}(T+T^{-1})$, where the matrix $T$ parameterizes a zero mode configuration, $s={\pi \nu \over \Delta}$ and $\Delta = (\nu L)^{-1}$ is the level spacing of the isolated system. (Temporarily deviating from the convention of the rest of the paper, in this section $s$ measures the energy in units of the total level spacing and not the level spacing of a localization volume.) This presumption is almost but not quite correct. The point is that our so far discussion of the low energy action implicitly assumed that the number of sites of the system is even. In the opposite case, an extra contribution $S_{{\rm
top},2}[T] = N {{\,\rm str\,\ln}}(T(L))$, derived and discussed in section \[sec:goldst-mode-fluct\], appears. The structure of this term reflects the fact that due to the presence of one uncompensated site the global ${{\rm GL}(1|1)}$-invariance of the model is lost. ($S_{{\rm
top},2}[T]$ is not invariant under $T\to T_L T T_R$ even for constant $T_{L,R}$.) In the majority of cases this extra term is of little interest. However for the spectral properties of a short isolated system, the presence of $S_{{\rm top,2}}$ bears crucial effects. Indeed we will see that this term is responsible for the formation of staggering phenomena akin, and probably related to the effects discussed earlier in section \[sec:cond-short-syst-1\].
Adding the two contributions to the action and using Eq. (\[eq:11\]) we obtain $$\nu(\epsilon) = {1\over 2 \Delta} \int dT\; (T_{11}+T^{-1}_{11})
e^{i{s\over 2} (T+T^{-1}) - N {{\,\rm str\,\ln}}(T)},$$ for the $\sigma$-model representation of the zero-dimensional DoS. The task thus is to integrate over a single copy of the target manifold ${{\rm GL}(1|1)}$. For sufficiently small energies $\epsilon \sim
\Delta$, the action is not large enough to confine the integrand to the origin of the group manifold implying that the integral has to be done non-perturbatively. Referring to Ref. [@altland99:NPB_flux] and Appendix \[sec:cond-short-syst\] for technical details we here merely display the final result of this integration procedure, $$\begin{aligned}
\label{eq:36}
\nu(s) = {\pi s \over 2\Delta}
\left[ J_0^2(s) + J_1^2(s)\right],\qquad &&L\;\mbox{even}\nonumber\\
\nu(s) = N\delta(\epsilon) + {\pi s\over 2\Delta}
\left[ J_N^2(s) -
J_{N+1}(s)J_{N-1}(s)\right],\qquad&& L\;\mbox{odd}\end{aligned}$$ The structure of these DoS profiles is shown in Fig. \[fig:figure2\] for the example $N=3$.
Eqs. (\[eq:36\]) have been obtained earlier within pure random matrix theory[@verbaarschot93], and its supersymmetry implementation[@ast; @verbaarschot96:susy]. Studies of chiral random matrix ensembles were largely motivated by the relevance of the spectral structure of effectively zero dimensional chiral systems in [*finite size*]{} lattice QCD. More generally, ’microgaps’ of the type shown in Fig. \[fig:figure1\] are an omnipresent side effect seen in the spectrum of generic chiral systems with finite mean level spacing $\Delta$.
\
Qualitatively, origin and structure of these gaps can easily be understood. First, the chiral symmetry $[H,\sigma_3]_+=0$ entails that for any [*non-vanishing*]{} energy level $\epsilon_n$ its negative $-\epsilon_n$ is an eigenvalue, too. Disorder generated level repulsion prevents these states from coming close to each other (on the scale of the mean level spacing) which explains the presence of the spectral gap in the $L$-even case. For $L$ odd, this picture has to be modified. To understand what is happening, let us imagine the Hamiltonian as a block off-diagonal matrix in sublattice space: $$H = \left(\matrix{&Z\cr Z^\dagger&}\right)\qquad
\begin{array}{l}
\mbox{odd}\cr\mbox{even.}
\end{array}$$ Since $L$ is odd, the number of odd sites exceeds the number of even sides by one, i.e. the blocks $Z$ are rectangular with $N
(L-1)/2$ rows and $N(L+1)/2$ columns. Now, any block-off diagonal matrix with $k$ rows and $l$ columns has $|k-l|$ eigenvalues $0$. Applied to our system, this means that the $L$ odd system has $N$ zero modes for any realization of disorder (the $\delta$-function contribution in (\[eq:36\]).) Other levels repel from this concentrated accumulation of spectral weight which explains the bathtub type suppression of the DoS up to energies $\sim N\Delta$.
Summarizing we have seen that the $L$ even/odd staggering behaviour observed earlier in connection with the conductance pertains to spectral properties. Above we had argued that the vanishing of the conductance in the $L$ even case was due to the peculiar spatial profile of zero energy wave functions. In view of (\[eq:36\]) it is tempting to relate the same effect to the vanishing of spectral weight at zero energy, although we have not analysed this picture any further.
Keeping in mind the tendency towards buildup of zero energy spectral weight in systems with mis-matched sublattice structure we next turn back to the analysis of large systems.
Heat kernel for finite energies {#sec:heat-kernel-finite}
-------------------------------
The aim of this section is to compute the DoS of a large system with $L\gg \xi$. To facilitate comparison with the behaviour of the conductance we will drop the assumption of isolatedness and again couple the system to leads. Notice that this stands complementary to the analysis of Ref. [@brouwer00_off], where a closed system was discussed. That we will obtain identical scaling of the DoS is proof of the (in view of the existence of topological zero modes not entirely obvious) assertion [@brouwer00_off] that boundary conditions do not affect the bulk spectrum.
We will not be able to compute the DoS $\nu(\epsilon)$ for arbitrary $\epsilon$. Instead, an asymptotic expression valid for low energies will be derived. Remembering the behaviour found for small systems we anticipate nonanalytic behaviour, $\nu(\epsilon) \sim
\ln^n(|\epsilon|) |\epsilon|$, where the logarithmic factor is due to potentially existent localization corrections to the zero mode behaviour. Our objective is to identify the most singular contribution of this type.
Starting point of the analysis is the transfer matrix representation (\[eq:13\]) of $\nu(\epsilon)$. To evaluate this expression we need to compute the functions $Y^{L/R}$, now for finite potential $V$. We will do this following a procedure developed in Ref. [@AltF]. Starting out from a spectral decomposition of the type (\[eq:39\]), we first notice that only zero energy eigenfunctions contribute to long distance behaviour of $Y_{L/R}(r)$. This means that our ’time’ dependent ’Schrödinger equation’ (\[eq:17\]) can effectively be replaced by the stationary form $$\label{eq:40}
t\gg \xi:\qquad \left(-({{\bf D}}\pm{{\bf A}})^2 + \eta (\cosh(x)-\cos(y))
\right) Y_{L/R}(x,y)=0,$$ where we have substituted the explicit form of the potential and omitted the spatial argument of $Y_{L/R}$ for simplicity. For later convenience we have also analytically continued from real to imaginary energy arguments, $-is^+ \to 4 \eta >0$. Substitution of these functions into Eq. (\[eq:13\]) yields the reduced representation $$\begin{aligned}
\label{eq:41}
\nu(E) = \nu_0\left(1+ {\,\rm Re\,} {1\over 2\pi}
\int dxdy\;{\cosh(x)-\cos(y)\over \sinh^2 \left({x-iy\over
2}\right)}
Y_L(x,y)Y_R(x,y)\right).\end{aligned}$$ To obtain this equation we have expanded the pre-exponential sources in Grassmann variables and integrated over these. The constant contribution $\nu_0$ appears as a consequence of the Efetov-Wegner theorem. (The representation of the pre-exponential term in polar coordinates contains a purely non-Grassmann contribution. Integration over a term of this type obtains the integrand at the origin[@Efetbook] which, in our case, equals unity.)
We now turn to the actual computation of the functions $Y_{L/R}$. The first step is the derivation of a set of matching, or boundary conditions relating the heat kernel to its $\eta\to0$ asymptotics. To this end we evaluate the $\eta=0$ spectral decomposition (\[eq:39\]) in the limit $t\gg \xi$. Neglecting contributions that decay exponentially in $t/\xi$, it is straightforward to obtain the asymptotic expressions: $$\begin{aligned}
&& Y_{L \atop R}(x,y) \stackrel{\eta\to 0}{\longrightarrow} 1,
\qquad N\;\mbox{even},\nonumber\\
&& Y_{L \atop R}(x,y) \stackrel{\eta \to
0}{\longrightarrow}{1\over 2} (e^{\pm(x-iy)}+1),
\qquad N\;\mbox{odd},\end{aligned}$$ where we have temporarily neglected the staggering parameter $f$. The structure of these functions will be motivated shortly. Turning to the case $\eta\not=0$, we next gauge and transform Eq. (\[eq:40\]) as in section \[sec:spectr-eigenf-kinet\]. Transformation $ Y_{L\atop R}(x,y) \to
J^{1/2}(x,y)e^{\pm{1-{\cal P}_N\over 2}(x-iy)}Y_{L\atop R}(x,y)\equiv
\tilde Y_{L \atop R}(x,y)$, where ${\cal P}_N \equiv (1+(-)^N)/2$ brings the equation into the form $$\label{eq:37}
\left(\partial^2_x+\partial^2_y - \eta (\cosh(x)
-\cos(y))\right)
\tilde Y_{L/R}(x,y)=0,$$ while the transformed $\eta\to 0$ asymptotics read $$\begin{aligned}
\label{eq:38}
\tilde Y_{L/R}(x,y) \stackrel{\eta\to 0}{\longrightarrow}&
\sinh^{-1}\left({x-iy\over 2}\right) &
\stackrel{|x|\gg 1}{\longrightarrow} 2{{\,\rm sgn\,}}(x) e^{-{{\,\rm sgn\,}}x
\left({x-iy\over 2}\right)},
\qquad N\;\mbox{even},\nonumber\\
\tilde Y_{L/R}(x,y) \stackrel{\eta \to
0}{\longrightarrow}&\coth\left({x-iy\over 2}\right)&\stackrel{|x|\gg
1}{\longrightarrow}{{\,\rm sgn\,}}(x),
\qquad N\;\mbox{odd}.\end{aligned}$$ Here we have anticipated that the dominant contribution to the above double integral representation of the DoS will come from large values of the non-compact variable $x$. Eqs. (\[eq:37\]) and (\[eq:38\]) have the nice feature of full separability. Writing $\tilde Y(x,y) =
{\cal N} Y_1(x) Y_2(y)$, where ${\cal N}$ is a normalization factor and the subscript $L/R$ has been dropped for simplicity, Eq. (\[eq:37\]) can be traded for the set of decoupled ordinary differential equations $$\begin{aligned}
&& \left( \partial_x^2 - {\eta\over 2} e^{x}- {{\cal P}_N\over 4}\right)Y_1(x)
= 0+ {\cal O}(\eta),\\
&& \left(\partial_y^2 + {{\cal P}_N\over 4}\right) Y_2(y) = 0 + {\cal
O}(\eta).\end{aligned}$$ The second equation is trivially solved by $Y_2(y) \approx e^{\pm
i{\cal P}_N{y\over 2}}$. The first line is a Bessel equation. Its two solutions are given by $I_{{\cal P}_N}((2\eta)^{1/2}e^{|x|/2})$ and $K_{{\cal P}_N}((2\eta)^{1/2}e^{|x|/2})$. Discarding the exponentially divergent solutions $I_\nu$ and using that for small arguments, $K_0(z) \approx - \ln(z)$ and $K_1(z) \approx z^{-1} +
{z\over 2}\ln\left( {z\over 2}\right)$, the normalization constants ${\cal N}$ can now be fixed by matching to the zero energy asymptotics in the limit of small $\eta$. This obtains the approximate solution $$\begin{aligned}
& \tilde Y_{L/R}(x,y) \stackrel{|x|\gg 1}\approx (8\eta)^{1/2} K_1
\left((2\eta)^{1/2}e^{|x|/2}\right)e^{i{\,{{\,\rm sgn\,}}\,}x
{y\over 2}}, &\qquad N {\;\rm even},\\
&\tilde Y_{L/R}(x,y) \stackrel{|x|\gg 1}\approx -{2\over \ln \eta}
K_0\left((2\eta)^{1/2}e^{|x|/2}\right),&\qquad N {\;\rm odd}.\end{aligned}$$ It is now a straightforward matter to substitute this expression back into the above integral representation for $\nu$ and to integrate over coordinates. The $y$-integration, extending over a purely harmonic integrand, is trivially done. (Notice that the $\sinh^{-2}$-factor in (\[eq:41\]) cancels against the factor $(J^{1/2})^2$ from the similarity transformation). As for the $x$-integration, we note that due to the exponentially decaying asymptotics $K_\nu(z) \sim
\exp(-z/2)$ for $|z|\gg 1$, the integral can be cut off at $(2\eta)^{1/2}e^{|x|/2} \sim 1 \Rightarrow |x|\sim -\ln(2\eta)$. Within the domain of integration, the Bessel functions can be replaced by the small argument asymptotics specified above. Substituting these expressions it is then straightforward to obtain $\nu(\eta) \approx \nu_0
{\,\rm Re\,}\eta \ln^2\eta$ (even $N$) and $\nu(\eta)\approx \nu_0
{\,\rm Re\,} (\eta \ln^2 \eta)^{-1}$ (odd $N$) for the small $\eta$-asymptotics of the DoS. Analytic continuation back to real energies finally leads to the result $$\begin{aligned}
\label{eq:43}
\nu(s) \approx - \nu_0 |s| \ln|s|,\qquad N \,\mbox{even},&&\nonumber\\
\nu(s) \approx - {\nu_0\over |s|\ln^3|s|},\qquad N
\,\mbox{odd},&&\end{aligned}$$ for the low energy behaviour of the DoS. Eqs. (\[eq:43\]) agree with the results found earlier in Ref. [@brouwer00_off].
We finally discuss the extension of the above results to non-zero staggering. For non-vanishing $f$ and even $N$, the large argument asymptotics (\[eq:38\]) generalize to $$\label{eq:45}
\tilde Y_{L/R}(x,y) \stackrel{\eta\to 0,x\gg 1}{\longrightarrow}
2 e^{{1\over 2}
(\pm f +1)(x-iy)}.$$ One can now follow the same steps as in the non-staggered cases above to obtain the result $$\label{eq:44}
\nu=2\nu_0 {\Gamma(|f|)\over \Gamma(2+|f|)}
{2^{-2(1-|f|)}\over|f|}
\cos\left({\pi\over 2} (1-|f|)\right) |s|^{1-|f|}.$$ For non-zero $f$, the DoS vanishes in a more singular manner as in the non-staggered case. This behaviour provokes the question, how matching with the diverging profile in the case $N$ odd, $f=0$ might be obtained. The principal structure of the theory entails that ($N$ even/$f=1$) should be equivalent to ($N$ odd/$f=0$). On the other hand, the $f\nearrow 1$ version of Eq. (\[eq:44\]) certainly does not agree with the divergent result (\[eq:43\]).
To resolve this paradox, it is helpful to re-interpret the asymptotic expressions (\[eq:38\]) and (\[eq:45\]) within the quantum mechanical picture of the theory. Focusing on the compact sector and temporarily ignoring the factor $J^{1/2}$ from the transformation to a flat Laplacian, these functions acquire the meaning of ground state wave functions $\Psi_0$ of a one dimensional ring subject to a gauge flux ${N+f\over 2}$. For $N$ even and zero $f$, an integer number of ’flux quanta’ pierce the ring, and the ground state wave function carries zero ’momentum’, $\Psi_0(y) \propto
1$ which, after multiplication with $J^{1/2}$ leads to the first line of Eq. (\[eq:38\]). In contrast, for $N$ odd and $f$ still zero, a half integer flux pierces the ring. This is a special situation in the sense that the ground state wave function is two-fold degenerate, i.e. $\Psi_0(y)= c_+ e^{iy/2} + c_- e^{-iy/2}$. Our earlier analysis has fixed the a priori un-determined constants $c_\pm$ to a symmetric configuration. Inclusion of the non-compact variable and multiplication with $J^{1/2}$ then leads to the second line of (\[eq:38\]).
We can now understand what happens as $f$ is turned on for an $N$ even configuration: a flux $f/2$ is sent through the ring and the ground state wave function remains unique (cf. Eq. (\[eq:45\])) [*until*]{} $f$ comes close to the critical value $1$. In the immediate vicinity of the degeneracy point, the fact must no longer be neglected that our one-dimensional system is subject to a weak potential $\eta \cos(y)$. For values of $f$ such that the level splitting $\sim 1-f$ between the two nearly degenerate levels becomes comparable with the characteristic strength of the potential $\sim \eta$, the ’true’ ground state configuration is given by the symmetric superposition of the two levels, as in the $N$ odd case. For these values of $f$ the ground state configuration is given by the second line of Eq. (\[eq:38\]) and the DoS follows the $N$ odd asymptotics.
This qualitative argument predicts that for asymptotically small energies, the DoS scales as in Eq. (\[eq:44\]). However, for larger values of the energy, $\eta \sim |1-f|$, a crossover to the characteristics of the $N$ odd $f=0$ DoS profile takes place.
Summary {#sec:summary}
=======
In this paper, transport and spectral properties of weakly disordered quantum sublattice wires have been explored from a fieldtheoretical perspective. We re-derived results obtained previously within the DMPK transfer matrix formalism, observed a surprisingly strong sensitivity of system properties to the realization of the lead/device coupling, and found that conductance and DoS, at least of short systems, exhibit drastic dependence on the parity of the total site number in the sublattice chain. It is likely that both this phenomenon and the dependence of system properties on the parity of the channel number root in the same origin, i.e. the existence of zero energy states for effectively block off-diagonal Hamiltonians with rectangular (non-quadratic) block structure. Although we are not aware of an intuitive explanation for the channel number parity effects, this belief is supported by the observation that all staggering phenomena are controlled by the same topological term $S_{\rm top}$ in the action of the $\sigma$-model. From these findings one might expect that the delocalized band-centre behaviour exhibited by the [ *two*]{}-dimensional sublattice model, is driven by the $2d$-analogue of this operator. Curiously, this is not so. In the two-dimensional field theory, the so-called Gade term $S_{\rm Gade}$, a two-gradient operator contribution with small and non-universal coupling constant, drives the system towards de-localization. The operator $S_{\rm top}$ does have a generalization to two dimensions[@altland99:NPB_flux], but its role has not been investigated so far. Summarizing we find that comparable phenomenology (metallic behaviour and diverging DoS) in one and two dimensions is described by different operators in the $\sigma$-model action. This indicates that some ’deeper’ physical principle, not understood at present, lies beyond the visible structure of the field theory.
[*Acknowledgement:*]{} Discussions with P. Brouwer, J. Chalker, V. Kravtsov, A. Ludwig, C. Mudry, B. Simons, and J. Verbaarschot are greatfully acknowledged. Special thanks to M. Zirnbauer for discussions, numerous supportive hints, and for drawing our attention to the most important aspect of the field theory, the topological term. This work was partly supported by Sonderforschungbereich 237 of the Deutsche Forschungsgemeinschaft.
Field Integral Formulation {#sec:field-integr-form}
==========================
$\sigma$-model representations of zero- and two-dimensional systems with $A$III-symmetry have been constructed in different contexts before[@gade93; @verbaarschot96:susy; @altland99:NPB_flux; @guruswamy00]. That the following two sections discuss the construction of the field theory in some detail is motivated by non-generic features particular to the $1d$-system, most notably the coupling operators and the existence of topological structures. In the present section, we will derive a representation of the model and its correlations functions in terms of a supersymmetric field integral. The projection of this, a priori exact representation onto its low energy sector will be the subject of the subsequent Appendix \[sec:deriv-field-theory\].
Consider the two correlation functions $$\begin{aligned}
& C^{(1)}_{\alpha \alpha'}\equiv \langle G_{\alpha\alpha'}(E^+) \rangle
&\qquad \mbox{DoS}\\
& C^{(2)}_{\alpha \beta \alpha' \beta'}\equiv \langle
G_{\alpha \beta}(0^+) G_{\alpha' \beta'}(0^-)
\rangle&\qquad\mbox{conductance}, \end{aligned}$$ relevant for the computation of DoS and conductance, respectively. To compute these objects, we follow the now standard supersymmetry scheme for disordered electronic systems[@Efetbook] and represent the Green functions as $$G_{\alpha \alpha'}(z) = \left. {\delta \over \delta J^{\rm b}_{\alpha' \alpha}}\right|_{\hat
J=0}Z[\hat J] = - \left.
{\delta \over \delta J^{\rm f}_{\alpha '\alpha}}\right|_{\hat
J=0} Z[\hat J],$$ where $$\label{eq:3}
Z[\hat J] = \int {\cal D}(\bar\psi,\psi)
e^{i \bar \psi \left(z-\hat H + i\pi \sum_C
W^{C}W^{CT}- \hat J\right)\psi},$$ and we have assumed that ${\rm sgn\,Im\,} z>0$. (In the opposite case, the sign of both the total action and the coupling operator change.) Here $\psi = \{(S_\alpha,\chi_\alpha)^T\}$ and $\bar \psi = \{(\bar
S_\alpha,\bar\chi_\alpha)\}$ are two-component superfields where $\bar
S_\alpha$ is the complex conjugate of $S_\alpha$, while $\chi_\alpha $ and $\bar \chi_\alpha$ are independent Grassmann variables. The source field $$\hat J = \left(\matrix{\hat J^{\rm b}&0\cr 0&\hat J^{\rm f}}\right)$$ where $\hat J^{\rm b,f} = \{\hat J^{\rm b,f}_{\alpha \alpha'}\}$ are ordinary matrices in site and orbital space.
From this representation, the one-point correlation function obtains as $C^{(1)}_{\alpha \alpha'}= \left. \delta_{J^{\rm b}_{\alpha'
\alpha}}\right|_{\hat J=0}\langle Z[\hat J]\rangle$ for $z=E^+$. However, at first sight it looks like [*two*]{} Gaussian field integrals (\[eq:3\]) are needed to compute the two-particle correlator $C^{(2)}$, one for the Green function $G(0^+)$ the other for $G(0^-)$. Fortunately, this is not so, a direct consequence of the chiral symmetry of the Hamiltonian: the relation $[\sigma_3,\hat
H]_+=0$ implies that $$\hat G(z) = - \sigma_3 G(-z) \sigma_3$$ and thus $G_{\alpha \beta}(0^+) = (-)^{l+k+1} G_{\alpha
\beta}(0^-)$, where $l$ and $k$ are the site indices carried by the composite variables $\alpha$ and $\beta$, respectively. In other words, the retarded and the advanced Green function are not independent and we can obtain the correlation function $C^{(2)}$ as $$C^{(2)}_{\alpha \beta \alpha' \beta'} = (-)^{l'+k'}
\left. {\delta^2
\over \delta J^{\rm b}_{\beta \alpha} J^{\rm
f}_{\beta'\alpha'}}\right|_{\hat
J=0}\langle Z[\hat J]\rangle$$ from the comparatively simple generating functional of the one-point function (evaluated for $z=0^+)$.
Derivation of the field theory {#sec:deriv-field-theory}
==============================
In this Appendix we derive the effective Lagrangian (\[eq:9\]) from the basic representation (\[eq:3\]). To keep the discussion simple, we will suppress the source-field dependence of the partition function in much of what follows. The final results for the correlation functions $C^{(1,2)}$, obtained by straightforward expansion of the action to first and second order in $\hat J$ are displayed in the final Eq. (\[eq:10\]).
The derivation of the effective action essentially follows the standard [@Efetbook] construction route of field theories of disordered electronic systems. We begin by averaging the partition function over the Gaussian distribution of the random hopping matrices $R$: $$Z[0] = \int {\cal D}(\bar\psi,\psi)
e^{i \bar \psi \left(z-\hat H_0 + i\pi \sum_C
W^{C}W^{CT}\right)\psi - {\lambda^2 \over N} \sum_l {\rm \,str\,}(\psi_{l\mu}
\bar \psi_{l\mu} \psi_{(l+1)\nu}\bar \psi_{(l+1)\nu})},$$ where $\hat H_0 = \{t_{ll'} \delta_{\alpha \alpha'}\}$ is the clean part of the Hamiltonian. Next, the quartic contribution is decoupled by means of two auxiliary fields: $$\begin{aligned}
&&Z[0] = \int {\cal D}(Q,P)
e^{-N\sum_l {\,\rm str\,}(Q_{l,l+1}^2 + P_{l,l+1}^2)}
\int {\cal D}(\bar\psi,\psi)
e^{i \bar \psi \left(z-\hat H_0 + i\pi \sum_C
W^{C}W^{CT}\right)\psi}\times\\
&&\hspace{2.0cm} \times e^{
i\lambda \sum_{l,{\rm even}} \bar\psi_l
\left[Q^+_{l,l+1}+Q^+_{l,l-1}\right]\psi_l
-i\lambda \sum_{l,{\rm odd}} \bar\psi_l
\left[Q^-_{l,l+1}+Q^-_{l,l-1}\right]\psi_l
}, \end{aligned}$$ where $Q^\pm = Q \pm iP$, and $Q$ and $P$ are two-component supermatrix fields (reflecting the two-component matrix structure of the dyadic products $\psi \bar \psi$) living on the non-directed [ *links*]{} of our system. Both, internal structure and symmetry properties of these fields will be discussed momentarily. At this stage, we merely anticipate that the field configurations relevant to the long range behaviour of the system will be smooth. The structure of the action then suggests to define $Q^\pm_l \equiv {1\over
2}(Q^\pm_{l,l+1} + Q^\pm_{l,l-1})$ as a new field variable, which now again lives on the sites of our system.
To account for the staggered even/odd structure of the theory we next ’double the unit cell’ and define a two component field $$\Psi_j = \left(\matrix{\psi_{2j+1}\cr \psi_{2j}}\right)\equiv
\left(\matrix{\Psi_{1,j}\cr \Psi_{2,j}}\right)$$ Here we have introduced a new counting index $j=0,\dots,L/2$ enumerating the doubled unit cells of our system. To avoid confusion, we will systematically designate the index $0,\dots,L$ of the ’primitive’ sites by $l,l',\dots$ and the new index by $j,j',\dots$. Expressed in terms of $\Psi$ the functional integral assumes the form $$\begin{aligned}
&&Z[0] = \int {\cal D}(Q,P)
e^{-2 N\sum_j {\,\rm str\,}(Q_j^2 + P_j^2)}\int {\cal
D}(\bar\Psi,\Psi)\times\\
&&
\times
\exp\bigg( i \bar \Psi \bigg(z + 2
\lambda(Q+iP\sigma_3)+
i\pi \sum\limits_{C=L,R}
W^{C}W^{CT} +
\sigma_+ H^{12}_{0}
+\sigma_- H^{21}_{0}\bigg)\Psi\bigg).\end{aligned}$$ Here, $\sigma_\pm = \sigma_1 \pm \sigma_2$ where $\sigma_i, i=1,2,3$ are Pauli matrices acting in the space defined through the two-component structure of $\Psi$. In lack of better terminology, we will refer to this space as the ’chiral space’. The second line of the expression above, purely off diagonal in the chiral space, contains the clean Hamiltonian. The explicit lattice structure of the blocks $H_0^{21}=(H_0^{12})^\dagger$ is given by $$\begin{aligned}
&&H_{0,ll'}^{12}= \delta_{ll'}(1+a) + \delta_{ll'-1}(1-a),\\
&&H_{0,ll'}^{21}= \delta_{ll'}(1+a) + \delta_{ll'+1}(1-a),\end{aligned}$$ where $a$ is the staggering parameter introduced in (\[eq:1\]). Finally, notice that the chiral matrix structure of the coupling operator is given by $$W^C_j W^{CT}_j = \left(\matrix{W^C_{2j+1} W^{CT}_{2j+1}&\cr
&W^C_{2j} W^{CT}_{2j}}\right).$$ At this stage, the superfield $\Psi$ can be integrated out and we arrive at $$\begin{aligned}
&&Z[0] = \int {\cal D}(Q,P)
e^{-2 N\sum_j {\,\rm str\,}(Q_j^2 + P_j^2)}\times\\
&&\hspace{2.0cm}
\times
\exp \left(-N {\rm \, str\,\ln}\bigg(z+ i\pi \sum\limits_C
W^{C}W^{CT} + 2\lambda (Q+iP\sigma_3)+
\sigma_+ H^{12}_{0}
+\sigma_- H^{21}_{0}\bigg)\right).\end{aligned}$$ To make further progress, we subject the functional integral to a saddle point analysis. Following the standard scheme[@Efetbook], we seek for saddle point configurations $\bar Q$ and $\bar P$ that are matrix-diagonal and spatially uniform. Further, we temporarily set $a=W={\, \rm Re\,}z=0$ and neglect boundary effects due to the finite extent of the system. Making the ansatz $\bar Q = i q \cdot \openone$, $\bar P = p\cdot \openone$, where $q$ and $p$ are complex numbers and $\openone$ is the two-dimensional unit matrix in superspace, a variation of the action w.r.t. $\bar Q$ and $\bar P$ then generates the set of equations $$\begin{aligned}
q &=& {i\lambda\over 2} {\rm tr\,}\left(\hat G_0^{-1} + 2 i \lambda(q+p
\sigma_3)\right)_{jj}^{-1}\\
p &=& {i\lambda\over 2} {\rm tr\,}\left[\left(\hat G_0^{-1} +
2i \lambda(q+p\sigma_3 )\right)_{jj}^{-1}\sigma_3\right], \end{aligned}$$ where $G_0 \equiv \left. (i\delta + \sigma_+ H^{12}_{0} +\sigma_-
H^{21}_{0})^{-1}\right|_{a=0}$ and the trace extends over the two chiral components of the operators (but not over superspace). To evaluate the trace, we Fourier transform to momentum space. Defining the transform through $$\begin{aligned}
&&f(k) = \left({2\over L}\right)^{1/2} \sum_j e^{i2kj}f_j\\
&&f_j = \left({2\over L}\right)^{1/2} \sum_k e^{-i2kj}f(k) \end{aligned}$$ (where the factor of two in the exponent serves as a mnemonic indicating that we have doubled the unit cell of our system) one finds that the Green function $G_0$ is diagonal in momentum space with $
G_0(k) = \left(i\delta + (1 + e^{2ik}) \sigma_+ +
(1+e^{-2ik})\sigma_-\right)^{-1} $. It is now straightforward to verify that our equations are solved by $p=0$ and $q$ determined through the self consistency equation $$q= 2\lambda^2 {2\over L}\sum_k {1\over (2\lambda q)^2 + 2(1-\cos(2k))}.$$ Replacing the sum by an integral, $\sum_k \to {L\over 2\pi}
\int_0^\pi dk$, one finds that the equation is solved by $q=\lambda
/2$.
The presence of the chiral symmetry implies that the configuration $\bar Q = {i\lambda \over 2}\openone$ is but a particular representative of a whole manifold of solutions. To explore the morphology of that manifold, we notice that our restricted ($z=W=0$) action is invariant under the transformation $$\begin{aligned}
\bar \Psi_1 \to \bar \Psi_1 T_1^{-1},&\qquad&
\bar \Psi_2 \to \bar \Psi_2 T_2,\\
\Psi_1 \to \bar T_2^{-1} \Psi_1,&\qquad&
\Psi_2 \to T_1\Psi_2,\\
Q+iP \to T_1(Q+iP)T_2,&\qquad&Q-iP \to T_2^{-1}(Q-iP)T_1^{-1},\end{aligned}$$ where $T_1,T_2 \in {{\rm GL}(1|1)}$. This symmetry, the super-generalization of the fermion symmetry (\[eq:42\]), states that the model has ${{\rm GL}(1|1)}\times {{\rm GL}(1|1)}$ as a global invariance group. Applying the transformation to the diagonal stationary phase solution discussed above we find that $${i \lambda \over 2} \openone \to {i\lambda \over 2}
\left(\matrix{T&\cr&T^{-1}}\right),$$ where $T=T_1T_2$. Arguing in reverse, we conclude that any matrix $T$ defines a solution of the mean field equations, i.e. ${\rm
GL}(1|1)$ is the Goldstone mode manifold of the model. This is the celebrated mechanism of chiral symmetry breaking (see Ref. [@verbaarschot00] for review): field theory implementations of models with a discrete chiral symmetry on the microscopic level, possess continuous factor groups ${\rm G}\times {\rm G}$ as symmetry manifolds. (In our case ${\rm G}={\rm GL}(1|1)$.) This symmetry is spontaneously broken by the saddle point configurations of the model. What remains is a Goldstone mode isomorphic to a single factor ${\rm G}$.
Combining these results, we parameterize our field manifold as $$\left(\matrix{Q+iP&\cr&Q-iP}\right) = {i\lambda \over 2}
\left(\matrix{UT&\cr&T^{-1}U}\right),$$ where both $T,U \in {\rm GL}(1|1)$. Here, the matrices $T$ span the Goldstone model manifold whereas the $U$’s, incompatible with the global symmetry of the model, represent massive modes. The next logical step in the construction of the field theory is to substitute these configurations back into the action and to expand in (i) energy arguments $z$ and matrix elements $W$, (ii) long-ranged spatial fluctuations of the Goldstone modes, and (iii) massive modes.
Goldstone mode fluctuations {#sec:goldst-mode-fluct}
---------------------------
We begin with the second element of the program formulated above, the expansion of the action in long ranged spatial fluctuations of the Goldstone mode. Temporarily setting $z=W=0$, $U=\openone$, we re-organize the ’str ln’ of the action according to $$X\equiv N{\rm str\,\ln}\left(\matrix{i\lambda^2 T &
H_0^{12}\cr H_0^{21} & i\lambda^2 T^{-1}}\right) = N{\rm
str\,\ln}\left(\matrix{i\lambda^2 & H_0^{12}\cr T
H_0^{21}T^{-1} & i\lambda^2}\right),$$ where $T$ is a slowly fluctuating field of Goldstone modes. Writing $T H_0^{21}T^{-1} = H_0^{21} + T [H_0^{21},T^{-1}]$ and noticing that, due to the slow fluctuation of $T$, the commutator is small, we expand as $$X= N {\rm str\,}(\bar G^{12}T [H_0^{21},T^{-1}]) - {N\over 2} {\rm
str\,}(\bar G^{12}T [H_0^{21},T^{-1}]\bar G^{12}T
[H_0^{21},T^{-1}])+ \dots,$$ where we have defined $\bar G_0 = (G_0 + i\lambda)^{-1}$ and the ellipses stand for infrared irrelevant higher order commutator terms. One next Fourier transforms these expressions, substitutes the explicit momentum representation $G_0(k)$ and uses that the characteristic momentum $q$ carried by the transforms $T(q)$ is small. The subsequent integral over the ’fast momentum’ $k$ is most economically done by noticing that full integration over $k$ amounts to integrating the characteristic phases $\exp(i2k)$ once over the complex unit circle. This integral has a simple pole inside the integration contour, whose residues depend on the ’small’ momentum $q$. Expanding the residues to lowest non-vanishing order in $q$ and transforming back to coordinate space one obtains $X= S_{\rm top} +
S_{\rm fl}$, where the two contributions are displayed in Eq. (\[eq:4\]) and a continuum limit $\sum_j \to {1\over 2}\int_0^L dr$ has been taken.
Notice that the above discussion implicitly assumed that the number of sites of our system is even: the operator $X$ had a structure where each $T$ (living at an odd site) came with a partner $T^{-1}$ at the neighbouring even site. For a system with an [*odd*]{} number of sites $L$, however, there remains one uncompensated degree of freedom $T((L-1)/2)$ at the terminating site. Neglecting gradients, the action due to this extra contribution reads $S_{{\rm top},2} = N {{\,\rm str\,\ln}}T(L)$. In principle, this contribution is as relevant as the ’bulk’ contribution to the topological action: it is local in space but, on the other hand, does not contain derivatives. Indeed, the structure of $S_{{\rm top},2}$ is closely related to that of $S_{\rm top}$ as can be seen by representing the latter as a boundary action (cf. the third line of (\[eq:35\]).) Yet in the majority of cases, the extra contribution $S_{{\rm top},2}$ does not play much of a role, wherefore we have largely ignored it in main analysis. E.g., for a system coupled to the outside world, $S_{{\rm top},2}=N{{\,\rm str\,\ln}}(T(L))=N2\pi i$, evaluates to a phase of no physical effect (cf. the related discussion around Eq. (\[eq:35\]).) There is, however, one exception to that rule, viz. the physics of [*isolated*]{} short systems with an odd number of sites discussed in section \[sec:density-states-short\], where the presence of the extra contribution is of key relevance.
Finite energies and coupling to the leads {#sec:finite-energ-coupl}
-----------------------------------------
To obtain the action associated to finite $z$ and $WW^T$, we organize the action as, $$\begin{aligned}
&& {\,\rm str\,\ln}\left(
\matrix{z+i\pi WW^T + i \lambda^2 T & H_0^{12}\cr
H_0^{21}&z+i\pi WW^T + i\lambda^2 T^{-1}}\right) \approx \\
&&\hspace{2.0cm}\approx {\,\rm str\,\ln}\Bigg[\left(
\matrix{(z+i\pi WW^T) T^{-1}&\cr&(z+i\pi WW^T)T&}\right)
+ \underbrace{\left(\matrix{i \lambda^2 & H_0^{12}\cr
H_0^{21}& i\lambda^2}\right)}_{\displaystyle \bar G^{-1}}\Bigg]\approx\\.
&&\hspace{2.0cm}\approx \sum_j {\,\rm str\,}
\left[(z+i\pi (WW^T)_{2j+1}) T_{l}^{-1}\bar G^{11}_{jj} + (z+i\pi
(WW^T)_{2j})T_l\bar G^{22}_{jj}\right]=\\
&&\hspace{2.0cm}= -i z \pi\nu_0\sum_j {\,\rm
str\,}(T+T^{-1}) + {\pi \over 2} \sum_j {\,\rm \,str\,}
\left[(WW^T)_{2j+1} T_{l}^{-1}+
(WW^T)_{2j}T_l\right],\end{aligned}$$ where we have used that (from the saddle point equation) $\nu =
i{N\over \pi} \bar G^{11}_{jj}=i {N\over \pi} \bar G^{22}_{jj} =
{N \over \pi}$ and $\nu$ is the DoS per site evaluated for energies far away from the middle of the band. We next assume that in the coupling region to the leads, i.e. the region where the envelope function $f$ defined through (\[eq:6\]) is non-vanishing, fluctuations of the Goldstone modes are negligible. Application of the orthogonality relation (\[eq:6\]) then directly leads to $${\pi M \gamma \over 2} \sum_{j=0,L/2}{\,\rm \,str\,}
(T_j + T_{j}^{-1})$$ for the contribution of the coupling term. Combining terms and taking the continuum limit, we finally obtain the two expressions $S_z$ and $S_{\rm T}$ of Eq. (\[eq:4\]) for the contribution of energy and coupling operators to the effective action, respectively.
Integration over massive modes {#sec:integr-over-mass}
------------------------------
We finally turn to the discussion of the role played by the massive modes $U$. First, notice that due to the presence of a weight term $\sim \exp(-{N\over 2}\sum {\,\rm str\,}(Q^2 + P^2))$ in the action and $$N{\rm str\,}(Q^2 + P^2) = N{\rm str\,}((Q+iP)(Q-iP)) = -{N\lambda^2\over
8}{\rm\,str\,}(U^2)$$ fluctuations of these fields are strongly inhibited. Starting out from an ansatz $U = \exp i W$, where $W$ is some generator, we may thus perturbatively expand the action around $W=0$. The actual realization of this program is cumbersome (cf. [@altland99:NPB_flux] for a concrete example.) Since the result, an extra Goldstone mode operator weakly coupled to the action, will not play much of a role in the present analysis we restrict ourselves to a schematic outline of the calculation.
Perturbative expansion of the action in powers of $W$ obtains an expression like $$\begin{aligned}
&&Z[0] = \int {\cal D}T e^{-S[T]}
\left\langle e^{-S^{(1)}[T,W]+S^{(2)}[T,W]+\dots}\right\rangle_W\approx\\
&&\qquad \approx
\int {\cal D}T e^{-S[T]}
e^{-\langle S^{(2)}[T,W]\rangle_W + {1\over 2}\langle
(S^{(1)}[T,W])^2\rangle_W } \end{aligned}$$ where $\langle \dots \rangle_W \equiv \int {\cal D}W
e^{-{N\lambda^2\over 8}{{\,\rm str\,}}(W^2)}$ and $S^{(n)}[T,W]$ denotes the expansion of the action to $n$-th order in $W$. The ellipses stand for contributions of higher order in $W$ which can safely be neglected (due to the large overall factor $N\gg 1$). It is straightforward to verify that the explicit evaluation of the operators $S^{(n)}[W,T]$ obtains contributions of the structure $ c_1 N{{\,\rm str\,}}(W \Phi)$ and $c_2
N{{\,\rm str\,}}(W \Phi W \Phi)$, where $\Phi \equiv T \partial T^{-1}$ and the coupling constants $c_{1,2}$ are proportional to powers of $\lambda$. Substituting these expressions back into the action and performing the Gaussian integration over $W$, we arrive at $$\begin{aligned}
&& \langle {{\,\rm str\,}}(S^{(1)}[T,W])^2\rangle_W \propto N c_1^2
{{\,\rm str\,}}(\Phi^2) = N c_1^2 {{\,\rm str\,}}(\partial T \partial T^{-1}),\\
&& \langle {{\,\rm str\,}}( S^{(2)}[T,W])\rangle_W \propto c_2
\left[{{\,\rm str\,}}(\Phi)\right]^2 = c_2
\left[{{\,\rm str\,}}(T \partial T^{-1})\right]^2.\end{aligned}$$ What can be said about structure and relevance of these operators? First, the two expressions ${{\,\rm str\,}}(\Phi^2)$ and ${{\,\rm str\,}}^2(\Phi)$ are the only invariant Goldstone mode operators with two derivatives. The first of these is already contained in the action (cf. Eq. (\[eq:4\])) with a coupling constant parametrically larger than the constant $N c_1^2$ obtained above. Thus, the first of the two contributions coming from the massive mode integration is irrelevant. In contrast, the second contribution must be taken seriously and, after integration over spatial coordinates, gives the term $S_{\rm
Gade}$ of Eq. (\[eq:4\]).
This completes our derivation of the effective action of the model. Finally, to compute the correlation functions $C^{(1,2)}$ defined in the text, we have to add the source field $\hat J$ to the partition function, expand to first or second order and differentiate. The structure of the resulting expressions depends on the index configuration of the correlation functions. For the correlators relevant to the computation of conductance and DoS, respectively, we obtain $$\begin{aligned}
&& C^{(1)}_{\alpha\alpha'} = -{i\over 2}\delta_{\mu\mu'} \left\{
\begin{array}{ll}
\left\langle T_{{l\over2},11}\right \rangle &,\qquad l\;\mbox{even}\\
\left\langle T^{-1}_{{l-1 \over 2},11} \right\rangle
&,\qquad l\;\mbox{odd}
\end{array}\right.,\nonumber\\
&& C^{(2)}_{\alpha\alpha'\alpha'\alpha} = -{1\over 4}\delta_{\mu\mu'} \left\{
\begin{array}{ll}
\left\langle T_{{l\over2},12} T_{{l'\over2},21} \right \rangle &,\qquad
l\;\mbox{even},\;l'\;\mbox{even}\\
- \left\langle T_{{l\over2},12}T^{-1}_{{(l'-1)\over2},21}\right
\rangle &,\qquad
l\;\mbox{even},\;l'\;\mbox{odd}\\
- \left\langle T^{-1}_{{(l-1)\over 2},12}T_{{l'\over2},21} \right
\rangle &,\qquad
l\;\mbox{odd},\;l'\;\mbox{even}\\
\left\langle T^{-1}_{{(l-1)\over2},12}T_{{(l'-1)\over2},21}\right
\rangle &,\qquad
l\;\mbox{odd},\;l'\;\mbox{odd}
\end{array}\right.,
\label{eq:10}\end{aligned}$$ where the angular brackets stand for the functional average.
Geometry of ${{\rm GL}(1|1)}$ {#sec:geometry-gl}
=============================
The canonical metric on the supergroup ${{\rm GL}(1|1)}$ derives from the differential two-form $\omega \equiv - {{\,\rm str\,}}(dT dT^{-1})$. $\omega$ is not a [*positive*]{} two-form, but its restriction to the sub-manifold ${\cal M}\subset {{\rm GL}(1|1)}$ defined in the text is; ${\cal M}$ is the maximally [*Riemannian*]{} subset of ${{\rm GL}(1|1)}$. Parameterizing the group manifold in terms of some coordinates, $T=T(x_1,\dots,x_4)$ (half of which are anti-commuting), the metric two-form assumes the form $$\omega = \sum_{ij} g_{ij} dx_i dx_j,$$ where $\{g_{ij}\}$ defines the metric tensor (represented in the basis $\{ x_i\}$.) As with non-super Riemannian manifolds, the metric tensor determines the geometry of the manifold. Specifically, the invariant group integral is defined through $$\int dT = \int \prod_i dx_1 \;J(x_1,\dots,x_4),$$ where the Jacobian $J\propto {\rm sdet} g$. Similarly, the Laplacian has the standard structure $$\Delta = {1\over g^{1/2}}\partial_{x_i} g^{ij} g^{1/2} \partial_{x_j},$$ where $g^{ij}$ is the inverse of $g$, $g_{ij}g^{jk}=\delta_{i}^k$.
We next wish to represent the metric tensor, the invariant measure, and the Laplacian in the polar decomposition, $T=kak^{-1}$ defined in Eq. (\[eq:8\]). Due to the manifest invariance of $\omega$ under transformation $T\to k_0 T k_0^{-1}$, $k_0$ a fixed rotation matrix, it is sufficient to evaluate the metric tensor at $k=\openone$. Substitution of the decomposition $T=k a k^{-1}$ into the two form then yields the covariant structure $$\begin{aligned}
&&\omega = - {{\,\rm str\,}}\left.\left(d(k a k^{-1}) d(ka^{-1} k^{-1})\right)\right|_{k=\openone} =
- {{\,\rm str\,}}\left(([dk,a] + da)\;([dk,a^{-1}]+da^{-1})\right)=\\
&&\hspace{2.0cm}- {{\,\rm str\,}}\left([dk,a][dk,a^{-1}] + dada^{-1}\right).\end{aligned}$$ Using that $$dk = \left(\matrix{&d\eta\cr d\nu &}\right),\qquad
da = \left(\matrix{e^x dx&\cr & ie^{iy}dy}\right)$$ it is straightforward to verify by elementary matrix manipulation that the metric bilinear form assumes the form $${\bf g} = \left(\matrix{1&0&&\cr
0&1&&\cr
&&0& -4 \sinh^2({x-iy\over 2})\cr
&& +4 \sinh^2({x-iy\over 2})&0}\right),$$ where ${\bf g} = \{g_{ij}\}$, a vectorial structure $d{\bf
x}=(dx,dy,d\eta,d\nu)$ is understood and empty blocks are filled with zeros. The associated superdeterminant, $$g= {1\over \left(4\sinh^2\left({x-iy\over 2}\right)\right)^2}.$$ From $g$ derives the unit-normalized group integral $$\label{eq:22}
\int dT f(T) = \int_{-\infty}^{\infty} dx\int_0^{2\pi} dy \int d\eta
d\nu J(x,y) f(x,y,\eta,\nu),$$ with Jacobian $$\label{eq:28}
J(x,y) = \sinh^{-2}\left({x-iy\over 2}\right).$$ Further, the [*radial*]{} part of the Laplacian reads as $$\Delta = \sum_{i=x,y} g^{-1/2} \partial_i g^{1/2} \partial_i.$$
Conductance of Short Systems {#sec:cond-short-syst}
============================
Consider a short sublattice system of length $L<\xi/(M\gamma)$. This is the quantum dot regime, where the conductance is not Ohmic but rather determined by the coupling of the system to the leads. For a normal, non-sublattice system, $g\sim M \gamma$, reflecting that each of the $M$ channels contributes to transport with an efficiency set by the coupling. We here wish to explore to which extent this behaviour generalizes to the sublattice system. Specifically, three different cases will be discussed: (a) smoothened coupling where a set of sites at each end is connected – the type of coupling considered in the text, (b) coupling through a single site on either end where the two terminal sites are of the same parity, odd/odd, say. (c) One of the two terminating sites is even, the other odd,
That the system is short means that the functional integral is controlled by spatially uniform configurations $T={\rm const.}$; the stiffness introduced by the gradient term is too strong to allow for significant fluctuations. (More precisely, fluctuating field configurations lead to relative corrections of ${\cal
O}(LM\gamma/\xi)$ which we are not going to consider.)
Let us begin by considering case (a). Evaluation of the functional expectation value (\[eq:12\]) for a spatially uniform field configuration leads to the expression $$\begin{aligned}
g^{(a)}= -\left({M\pi\gamma\over 2}\right)^2 \int dT
(T-T^{-1})_{12}(T-T^{-1})_{21} e^{-\pi M \gamma {{\,\rm str\,}}(T+T^{-1})}. \end{aligned}$$ We next need to do the group integral. Inspection of the exponent shows that the integration is dominated by configurations close to the group origin, $T=\openone$. This suggests to represent our $T$’s as $T=\exp W$ and to integrate over the generators $W$ in a Gaussian approximation: $$\begin{aligned}
g^{(a)}\approx -(M \pi\gamma)^2 \int dW
W_{12}W_{21} e^{-\pi M \gamma{{\,\rm str\,}}W^2},\end{aligned}$$ where we have used that close to the group origin the integration measure is flat. Doing the integral over the components of $W$ one finds $$g^{(a)}= {M\pi \gamma \over 2}$$ in agreement with the behaviour of a non-sublattice system. The situation in case (b) is not much different. Noticing that fields $T$ sit at the odd sites of our system, while $T^{-1}$’s are attached to even sites, we find that the conductance is expressed as $$\begin{aligned}
g^{(b)}= (M\pi\gamma)^2 \int dT \;
T_{12}(T^{-1})_{21} e^{-\pi M \gamma {{\,\rm str\,}}(T+T^{-1})}. \end{aligned}$$ (The disappearance of the combination $T-T^{-1}$ in the pre-exponent reflects the fact that only [*single*]{} sites of different parity are coupled to the continuum. Again the integral is dominated by the group origin and similar reasoning as above leads to $$\begin{aligned}
g^{(b)}\approx -\left({M \pi\gamma\over 2}\right)^2 \int dW
W_{12}W_{21} e^{-{\pi M \gamma \over 4}{{\,\rm str\,}}W^2}= {M\pi \gamma\over 2}.\end{aligned}$$ In case (c), however, the situation is different. Owing to the identical parity of the terminating sites, the integral representation of the conductance now assumes the form $$\begin{aligned}
g^{(c)}= -(M\pi\gamma)^2 \int dT \;
T_{12}T_{21} e^{-\pi M \gamma {{\,\rm str\,}}(T^2)}. \end{aligned}$$ This is an unpleasant expression: the exponential weight no longer projects onto the group origin, in fact does not even have a stable saddle point. To compute the conductance we therefore have to integrate over the full group manifold, a task that is most efficiently done in polar coordinates. Using Eqs. (\[eq:8\]) and (\[eq:22\]) and integrating out Grassmann components it is straightforward to verify that the radial part of the integral assumes the form $$g^{(c)} =(M\pi\gamma)^2 \int_{-\infty}^{\infty} dx
\int_0^{2\pi} \; {(e^{x}-e^{iy})^2\over \sinh^2\left({x-iy\over
2}\right)}
e^{-(e^x-e^{iy})}=0.$$ To understand the vanishing of the integral, notice that $\int dy =
{1\over 2\pi i} \oint {dz\over z}$ can be transformed into the integral of the complex variable $z=e^{iy}$ over the unit circle. One verifies that, regardless of the value of $x$, the integrand is analytic and void of singularities inside the integration contour. Cauchies theorem then implies vanishing of the integral.
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|
---
abstract: 'Breaking of G-parity or new weak (second class) currents can be responsible for [$\tau^- \to \eta^{(\prime)} \pi^- \nu_{\tau}$ ]{}decays. Forthcoming measurements of $\tau$ lepton properties at the Belle II experiment will be able to measure this decay channel for the first time. Isolating new physics contributions from the measured rates will require a careful evaluation of G-parity breaking contributions. Here we evaluate the one-loop contribution to [$\tau^- \to \eta^{(\prime)} \pi^- \nu_{\tau}$ ]{}decays induced by the emission of two virtual photons and its later conversion into an $\eta^{(\prime)}$ meson. As expected, this contribution is very small and may be relevant only for new physics searches contributing at the $10^{-4}$ level to the decay rate of [$\tau^- \to \eta^{(\prime)} \pi^- \nu_{\tau}$ ]{}.'
author:
- 'G. Hernández-Tomé'
- 'G. López Castro'
- 'P. Roig'
title: 'G-parity breaking in [$\tau^- \to \eta^{(\prime)} \pi^- \nu_{\tau}$ ]{}decays induced by the $\eta^{(\prime)}\gamma\gamma$ form factor '
---
\[Intro\]
![$\tau^- \to \eta^{(\prime)} \pi^- \nu_{\tau}$ decay at one loop level induced by two-photon electromagnetic interaction. The contributions from diagrams (c), (e) and (f) are identically zero (see text). The effective weak $\tau^-\to\nu_\tau\pi^-$ vertex is depicted by a thick dot, while the effective electroweak $\tau^-\to\nu_\tau\pi^-\gamma$ vertex is represented by a thick square. \[dia\]](diagramas1.eps)
It is well known that electromagnetic and weak hadronic currents with isospin 0, 1 quantum numbers can be classified according to their G-parity [@Lee:1956sw] transformation properties into two classes [@Weinberg:1958ut]. The first class includes currents with quantum numbers $J^{PG}=0^{++}, 0^{--}, 1^{+-}, 1^{-+}$, whereas the second class currents (SCC) have opposite $G$-parity $J^{PG}=0^{+-}, 0^{-+}, 1^{++}, 1^{--}$. Since G-parity invariance is broken by isospin non-conservation, electromagnetic effects and the mass difference of $u-d$ quarks can induce the hadronization of the [standard model]{} (SM) currents into states that mimic the effects of SCC; therefore, $G$-parity violating processes are naturally suppressed. So far, no experimental evidence of SCC weak interactions has been reported. Similarly, isosinglet and isotriplet meson states have well defined G-parity quantum numbers; in this case isospin breaking can mix the neutral components of states with different G-parity giving rise to the well known $\pi^0-\eta-\eta'$ and $\omega-\rho^0$ mixing phenomena explaining the observed rates of $\omega\to \pi^+\pi^-$ or $\rho\to 3\pi$ decays [@Olive:2016xmw].
A clean test for the existence of SCC would be provided by the observation of the semileptonic transitions $\tau^-\to\eta^{(\prime)}\pi^-\nu_\tau$ [@Leroy:1977pq], since the G-parity of the hadronic system ($-1$) is opposite to the one of the charged weak current in the SM (G=$+1$). Currently, the most stringent bounds available are based on searches by the BaBar collaboration [@delAmoSanchez:2010pc] corresponding to $BR(\tau^-\to\eta\pi^-\nu_\tau )<9.9\times 10^{-5}$ and $BR(\tau^-\to\eta^{\prime}\pi^-\nu_\tau )<7.2\times 10^{-6}$ [@Aubert:2008nj], which lie close to the estimates based on isospin symmetry breaking [@iso] for the $BR$([$\tau^- \to \eta^{(\prime)} \pi^- \nu_{\tau}$ ]{}) decays mainly induced by the $u-d$ quark mass difference [@ChPT; @ChPT2]. Further, Belle II is expected to accumulate up to two orders of magnitude more $\tau$ lepton pairs than BaBar and Belle, which should make possible the discovery of SCC.
In addition to the $u-d$ quark mass difference, electromagnetic interactions also break isospin symmetry and will contribute to [$\tau^- \to \eta^{(\prime)} \pi^- \nu_{\tau}$ ]{}decays. This can occur at the one-loop level, [*via*]{} the emission of a pair of photons from $\tau^- \to \pi^-\nu_{\tau}$ decays and their later conversion into an $\eta^{(\prime)}$ meson through the anomalous vertex as shown in Figure \[dia\] [^1]. Despite this kind of processes are expected to give only a minor correction to the observables, it is very important to have a reliable estimate of these effects in order to eliminate a possible source of background for a genuine SCC (see Ref. [@Guevara:2016trs] for a dedicated study of the backgrounds given by radiative decays). As a reference to quantify the effect of the new contribution we are studying to the [$\tau^- \to \eta^{(\prime)} \pi^- \nu_{\tau}$ ]{}decays, we will use the results in Ref. [@Escribano:2016ntp], which employs a data-driven approach to the vector form factor contributions [@Fujikawa:2008ma] and the state-of-the-art analysis of meson-meson scattering within unitarized Chiral Perturbation Theory [@GuoOller] to obtain those of scalar form factors (see appendix A for definition of those tree-level form factors).
In order to analyze the different contributions for [$\tau^- \to \eta^{(\prime)} \pi^- \nu_{\tau}$ ]{}decays induced at one loop level [^2] by the electromagnetic interaction we consider, in the low-energy limit, a point-like interaction for the $\tau^-\to\nu_\tau\pi^-$ vertex, which can be described by the Lagrangian density
$$\begin{aligned}
\mathcal{L}&=&G_{F}V_{ud}f_{\pi}\bar{\nu}_{\ell}\gamma_{\mu}\left(1-\gamma_{5}\right)\ell\partial^{\mu}\pi^{+}+\textrm{h.c.}\,,\end{aligned}$$
where $f_{\pi}=F_{\pi}/\sqrt{2}=92.2 \ $ MeV. Form factors complying with the low- and high-energy limits of QCD are considered for the pion electromagnetic coupling and the two-photon coupling of the neutral meson, as it is discussed in appendix \[A2\], where we also explain the approximations adopted in the computation of the corresponding loop integrals.
As we mentioned before, we are interested in the study of the $\tau^-(p_{\tau}) \to \eta^{(\prime)}(p_{\eta})\pi^-(p_{\pi})\nu_{\tau}(p_{\nu})$ decays induced at one loop level (see Fig. \[dia\]) as a possible background for a genuine SCC. The decay amplitudes for diagrams Fig. \[dia\] (c) and (f) vanish owing to the conservation of P and CP by strong and electromagnetic interactions, whereas the contribution from diagram (e) vanishes when considering the loop integration because it is odd in the integration variable. After performing the loop integration for diagrams in Fig. \[dia\] (a), (b) and (d) and employing the Chisholm identity [^3] for the Levi-Civita tensor contracted with a gamma matrix, we get the following generic form for each of the non-vanishing amplitudes
$$\mathcal{M}^k =\frac{e^4G_FV_{ud}f_{\pi}}{16 \pi^2}g_{\gamma\gamma\eta^{(\prime)}}\bar{u}\left(p_{\nu}\right)\left[F^k_{0}P_{R}+F^k_{1}\slashed p_{\pi}P_{L}\right]u\left(p_{\tau}\right),\label{amplitud}$$
where the superindex $k=a,b,d$ labels the non-vanishing Feynman diagrams in Figure \[dia\].
The form factors $F^k_{0,1}$ are generated by the loop integration and will be discussed in more detail below eq. (\[M2\]). The factor $g_{\gamma\gamma\eta^{(\prime)}}$ corresponds to the value of the $\gamma^{(*)}\gamma^{(*)}\eta^{(\prime)}$ form factor for on-shell photons, which is a global dependence of the matrix element (\[amplitud\]). Its analogue for the $\pi^0$ case, $g_{\gamma\gamma\pi^0}$, is fixed by the ABJ anomaly [@ABJ]
$$g_{\gamma\gamma\pi^0}\,=\,\frac{N_C}{12\pi^2 f_\pi},$$
with $N_C=3$ in QCD. In terms of this $\pi^0\gamma\gamma$ coupling, $g_{\gamma\gamma\eta^{(\prime)}}$ are determined [@Roig:2014uja] considering Chiral Perturbation Theory in the large-$N_C$ limit [@LargeN; @ChPTLargeN], namely
$$\label{TFF}
g_{\gamma\gamma\eta}\,=\,\left(\frac{5}{3}C_q-\frac{\sqrt{2}}{3}C_s\right)g_{\gamma\gamma\pi^0}\,,\quad \ \ \ g_{\gamma\gamma\eta'}\,=\,g_{\gamma\gamma\eta}\left(C_q\to C_{q'},\,C_s\to-C_{s'}\right) \,,$$
where the input values for the mixing coefficients can be found in Ref. [@Roig:2014uja].
The square of the total decay amplitude (${\cal M}=\sum_k {\cal M}^k$) is given by $$\begin{aligned}
\label{M2}
|\mathcal{M}|^2 &=&\frac{\left(e^4|V_{ud}| G_Ff_{\pi}g_{\gamma\gamma\eta^{(\prime)}}\right)^2}{128 \pi^4} \left[p_\nu\cdot p_\tau \left( |F_0|^2-|F_1|^2 m_\pi^2 \right) \right. \frac{}{}
+2 \left. p_\nu\cdot p_\pi \left(|F_1|^2 p_\tau \cdot p_\pi + m_\tau \textrm{Re}[F_0 F_1^*]\right)\right]\,,\end{aligned}$$ where the $F_0=\sum_{k=a,b,d} F_0^k$ and $F_1=\sum_{k=a,b,d} F_1^k$ functions are given in terms of the invariant Passarino-Veltman (PaVe) scalar functions [@Passarino:1978jh]. The explicit expression for these form factors can be found in appendix B of [@arXiv_v1]. We will provide a Mathematica file with these results upon request; they are functions of two independent kinematical scalars which can be chosen as $s_{12}=(p_{\pi}+p_{\eta})^2=(p_{\tau}-p_{\nu})^2$, the square of the invariant-mass of the hadronic system, and $s_{13}=(p_{\pi}+p_{\nu})^2=(p_{\tau}-p_{\eta})^2$. The functions $F_{0,1}$ have been obtained using the Mathematica packages FeynCalc [@Mertig:1990an] and LoopTools [@vanOldenborgh:1989wn; @Hahn:1998yk].
The contribution of Eq. (\[M2\]) alone to the branching ratio of the considered decays can be calculated straightforwardly . Using the notation $BR_{P^0}^{\gamma\gamma} \equiv BR(\tau^-\to P^0\pi^-\nu_{\tau})$ when $P^0$ is produced from a $2\gamma$ intermediate state ($P=\pi,\eta,\eta'$), we obtain $$\label{BRgammagamma}
BR_{\pi^0}^{\gamma\gamma}\,=\,5.3\cdot10^{-13}\,,\quad BR_\eta^{\gamma\gamma}\,=\,5.2\cdot10^{-13}\,,\quad BR_{\eta'}^{\gamma\gamma}\,=\,0.8\cdot10^{-16}\,.$$ While for the $\eta^{(\prime)}$ modes the ratio between the numbers in eq. (\[BRgammagamma\]) and the corresponding branching fractions predicted by the tree level ($(\pi^0-)\eta-\eta'$ mixing) contributions in Ref. [@Escribano:2016ntp] ($BR^{{\rm tree}}_\eta\sim1.7\cdot10^{-5}$ and $10^{-7}\leq BR^{{\rm tree}}_{\eta'} \leq 10^{-8}$) is at the level of $10^{-8}$, it goes further down to $10^{-11}$ for the $\pi^0$ decay mode (with respect to its measured branching fraction $BR_{\pi^0}\sim25\%$), which validates neglecting the one-loop contribution in the $\tau^-\to\pi^-\pi^0\nu_\tau$ decays, as we anticipated.
Next we turn to the evaluation of the branching ratio including the sum of the tree level [@Escribano:2016ntp] and one-loop amplitudes. For this, we note the equivalence between the $F_{0,+}$ form factors appearing in Ref. [@Escribano:2016ntp] and our $F_{0,1}$ form factors in Eq. (\[amplitud\]). In order to avoid confusion between the $F_0$ form factors appearing in both, we will use an upper index $\gamma\gamma$ for the $F_{0,1}$ form factors defined in eq. (\[amplitud\]). This allows to include the electromagnetic contribution into the vector and scalar form factors by shifting $F_{+,0}^{\pi\eta^{(\prime)}}(s_{12}) \to F_{+,0}^{\pi\eta^{(\prime)}}(s_{12})+F_{+,0}^{\gamma\gamma}(s_{12},s_{13})$, where:
$$\label{relationFFs}
F_+^{\gamma\gamma}\,=\,-\frac{e^4f_\pi g_{\gamma\gamma\eta^{(\prime)}}F_1}{64\pi^2}\, , \ \ \ \ \ \quad
F_0^{\gamma\gamma}\,=\,\frac{\frac{\textstyle f_\pi}{\textstyle 16 \sqrt{2} \pi^2}e^4 g_{\gamma\gamma\eta^{(\prime)}}
\left[\frac{\textstyle F_0}{\textstyle m_\tau}+\frac{\textstyle F_1}{\textstyle 2}\left(1+\frac{\textstyle \Delta_{\pi\eta^{(\prime)}}}{\textstyle s_{12}}\right)\right]}{c^S_{\pi\eta}\frac{\textstyle \Delta_{K^0K^+}^{QCD}}{ \textstyle s_{12}}}\,,$$
with $c^S_{\pi\eta}=\sqrt{\frac{2}{3}}$ $\left(c^S_{\pi\eta^\prime}=\frac{\sqrt{2}}{3}\right)$ and $\Delta_{PQ}=M_P^2-M_Q^2$. The factor $\Delta_{K^0K^+}^{QCD}$ corresponds to the (squared) mass splitting of the $K^0K^+$ mesons which is due to strong interactions. Precisely $\Delta_{K^0K^+}^{QCD}\equiv m_{K^0}^2-m_{K^+}^2-(m_{\pi^0}^2-m_{\pi^+}^2)$ cancels the electromagnetic (squared) mass splitting between the kaon states and corresponds to the referred $QCD$ (squared) mass difference between neutral and charged kaons.
We have used the fortran version of LoopTools to compute $$\begin{aligned}
BR^{{\rm tree}+\gamma\gamma}_{\eta}-BR^{{\rm tree}}_{\eta} \, &\in &\,\left[-5\cdot10^{-9},2\cdot10^{-9}\right]\,, \\ BR^{{\rm tree}+\gamma\gamma}_{\eta'}-BR^{{\rm tree}}_{\eta^\prime}\, &\in & \,\left[-3\cdot10^{-12},3\cdot10^{-12}\right]\,,\end{aligned}$$ where the difference comes mainly from the interference or tree- and loop-level contributions and –as in Ref. [@Escribano:2016ntp]– the quoted errors only arise from the uncertainties in the overall normalization factor $F_+^{\pi\eta^{(\prime)}}(0)$.
As we stated previously, the two-photon mediated amplitude considered in this paper is negligibly small compared to the dominant contribution to the $\tau^- \to \pi^-\pi^0\nu_{\tau}$ branching fraction. However, it may affect the branching fraction of channels with $\eta^{(\prime)}$ meson at the $3\cdot10^{-4}$ level [^4]. The tree level contribution to the $\eta'$ decay channel has a big uncertainty [@Escribano:2016ntp], $\sim90\%$, which is dominated by the error on $F_+^{\pi\eta'}(0)$. The main modification given by the loop contributions comes from the interference between the former and the one-loop contribution studied here. Consequently, this interference is also affected by the big uncertainty on $F_+^{\pi\eta'}(0)$.
In this paper we have considered the one-loop contribution to the $\tau^-\to P^0\pi^-\nu_\tau$ decays induced by the $P^0\gamma\gamma$ form factors; to the best of our knowledge this is the first study of electromagnetic contributions to these decays. The transition form factors (and the $\pi$ electromagnetic form factor) have been modeled to fulfill the low- and high-energy limits of QCD (within $U(3)$ flavor symmetry for the lightest vector resonance multiplet). The proper form factors asymptotics has naturally rendered finite the computation of the loop integrals. We have verified that the contributions we are considering are negligible for the $\pi^0\pi^-$ channel. In the case of final state with an $\eta$ meson, the $2\gamma$ intermediate states contribute -at most- with corrections at the $10^{-4}$ level and in the case of a decay with an $\eta^\prime$ their maximum relative size can vary between $3\cdot10^{-4}$ and $3\cdot10^{-5}$ depending on the value of the tree level branching ratio.
It is clear that searches at forthcoming flavor factories will not be sensitive to effects of two-photon contributions in [$\tau^- \to \eta^{(\prime)} \pi^- \nu_{\tau}$ ]{}decays . On the one hand, SCC have not been discovered yet and even if [$\tau^- \to \eta^{(\prime)} \pi^- \nu_{\tau}$ ]{}decays are finally measured at Belle-II it will be very difficult to achieve a measurement with a few percent accuracy even with the complete Belle-II data sample. Moreover, current theoretical uncertainties are huge (see Ref. [@Escribano:2016ntp] and references therein), which prevents pinpointing New Physics effects below the $10^{-6}$ level in the branching fractions . The quoted analysis only includes the errors given by $F_+^{\pi\eta^{(\prime)}}(0)$, which imply a one-order of magnitude uncertainty for the $\eta'$ decays and some $5\%$ error on the $\eta$ decays. It is difficult to quantify the error on the branching ratio prediction for these modes induced by the uncertainty on the couplings entering the unitarized meson-meson scattering amplitudes (which affects the dominant scalar form factor contributions) but it will surely dominate over the previous one. In addition to measurements of the branching fraction, further information on the hadronic mass as well as on angular distributions will be helpful to disentangle New Physics effects . Conversely, we confirm that the contributions considered in these paper can be neglected in forthcoming SCC searches.
Acknowledgements {#acknowledgements .unnumbered}
================
This work has been supported by Conacyt Projects No. FOINS-296-2016 (’Fronteras de la Ciencia’) and 236394 and 250628 (’Ciencia Básica’). The authors have benefited from discussions with Gilberto Tavares and Sergi Gonzàlez Solís.
Form factors of the tree-level amplitude {#A1}
========================================
In this appendix we recall the main formulas obtained in Ref. [@Escribano:2016ntp] for the [$\tau^- \to \eta^{(\prime)} \pi^- \nu_{\tau}$ ]{}results at tree level, cf. eqs. (\[relationFFs\]).
The amplitude of the decay $\tau^{-}\to\pi^{-}\eta^{(\prime)}\nu_{\tau}$ reads $$\mathcal{M}=\frac{G_{F}}{\sqrt{2}}V_{ud}\bar{u}(p_{\nu_{\tau}})\gamma_{\mu}(1-\gamma_{5})u(p_{\tau})\langle\pi^{-}\eta^{(\prime)}|\bar{d}\gamma^{\mu}u|0\rangle\,.$$ where the hadron matrix element is $$\langle \pi^{-}\eta^{(\prime)}|\bar{d}\gamma^{\mu}u|0\rangle=\left[(p_{\eta^{(\prime)}}-p_{\pi})^{\mu}+\frac{\Delta_{\pi^{-}\eta^{(\prime)}}}{s}q^{\mu}\right]c^{V}_{\pi\eta^{(\prime)}}F_{+}^{\pi\eta^{(\prime)}}(s)+
\frac{\Delta^{QCD}_{K^{0}K^{+}}}{s}q^{\mu}c^{S}_{\pi^{-}\eta^{(\prime)}}F_{0}^{\pi^{-}\eta^{(\prime)}}(s)\,
\label{vectorcurrent}$$ and we have used $s=q^2=(p_\pi+p_{\eta^{(\prime)}})^2$, $c^{V}_{\pi\eta^{(\prime)}}=\sqrt{2}$, and definitions introduced after eqs. (\[relationFFs\]).
Thus, the differential partial decay width, as a function of the $\pi^{-}\eta^{(\prime)}$ invariant mass, is $$\begin{aligned}
\frac{d\Gamma\left(\tau^-\to\pi^-\eta^{(\prime)}\nu_\tau\right)}{d\sqrt{s}}\,&=&\,\frac{G_F^2M_\tau^3}{24\pi^3s}S_{EW} \left|V_{ud}F_+^{\pi^-\eta^{(\prime)}}(0)\right|^2
\left(1-\frac{s}{M_\tau^2}\right)^2\nonumber\\
& &\times \left\lbrace\left(1+\frac{2s}{M_\tau^2}\right)q_{\pi^-\eta^{(\prime)}}^3(s)|\widetilde{F}_+^{\pi^-\eta^{(\prime)}}(s)|^2+\frac{3\Delta_{\pi^-\eta^{(\prime)}}^2}{4s}q_{\pi^-\eta^{(\prime)}}(s)|\widetilde{F}_0^{\pi^-\eta^{(\prime)}}(s)|^2\right\rbrace,
\label{width}\end{aligned}$$ where $$\widetilde{F}_{+,0}^{\pi^-\eta^{(\prime)}}(s)=\frac{F_{+,0}^{\pi^-\eta^{(\prime)}}(s)}{F_{+,0}^{\pi^-\eta^{(\prime)}}(0)},$$ are the two form factors normalized to unity at the origin. In eq. (\[width\]) we have introduced the short-distance electroweak correction factor $S_{EW}=1.0201$ [@Erler:2002mv] and $$\label{q}
q_{PQ}(s)=\frac{\sqrt{s^2-2s\Sigma_{PQ}+\Delta_{PQ}^2}}{2\sqrt{s}}\,,\quad \Sigma_{PQ}=m_P^2+m_Q^2\,.$$
Meson form factors and approximations in the computation of the loop integrals {#A2}
==============================================================================
Expressions for the $\gamma\gamma\eta^{(\prime)}$ and charged pion electromagnetic form factors are required in the evaluation of the Feynman diagrams in Fig. \[dia\]. Noting that eq. (\[TFF\]) remains valid when including structure-dependent contributions if $U(3)$ flavor symmetry is assumed for the lightest resonance multiplet, the $\pi^0$ transition form factor encodes -under the discussed approximations- all dynamics needed to obtain the $\gamma\gamma\eta^{(\prime)}$ form factor [@Roig:2014uja; @Czyz:2012nq]. Therefore, we will only discuss the pion form factors in the following.
In these limits, the structure of these form factors is [@RChT; @Roig:2014uja]
$$\begin{aligned}
\label{FF}
\frac{F^\pi(s)}{F^\pi(0)}\,&=&\,\frac{M_V^2}{M_V^2-s}\,, \nonumber \\
\frac{F^{\pi\gamma\gamma}(p^2,q^2)}{F^{\pi\gamma\gamma}(0,0)}\,&=&\,\frac{1}{2}\left[2+\frac{p^2}{M_V^2-p^2}+
\frac{q^2}{M_V^2-q^2} + \frac{(p^2+q^2)M_V^2}{(M_V^2-p^2)(M_V^2-q^2)}\right]\,,\end{aligned}$$
where $M_V\sim M_\rho(770)$ is the $U(3)$ average mass of the lowest-lying vector resonance nonet, $F^\pi (0)=1$ with an excellent approximation and $F^{\pi\gamma\gamma}(0,0)=g_{\gamma\gamma\pi^0}$.
We point out that the purpose of including these form factors in the evaluation of the one-loop diagrams in Fig. \[dia\] is two-folded: on the one hand, they incorporate the finite size of the pions and their structure; on the other, they naturally regulate the ultraviolet divergences appearing in diagrams $(b)$ and $(d)$ through their Brodsky-Lepage behaviour [@BL].
In our computations, we have included $F^\pi(s)$ as given in eq. (\[FF\]). We have noted that ultraviolet divergences always arise when both $p^2$ and $q^2$ in $F^{\pi\gamma\gamma}(p^2,q^2)$ tend to $\infty$. This implies that neglecting the last term of $F^{\pi\gamma\gamma}(p^2,q^2)$ does not spoil the finiteness of the result. Since LoopTools could not handle the case when the term with the double propagator is included, we decided to neglect it. We justify this approximation in the following.
We have verified that the bulk of the contribution to the loop integrals comes from the region with low photon virtualities (we understand this because the kernel of the integration having photon propagators). This, by the way, makes negligible the corrections induced by the finite width of the $\rho$ meson and justifies neglecting the contributions from excited resonance multiplets. Taking all this into account, we considered the following simplified expression for the pion transition form factor (which warrants the cancellation of ultraviolet divergences in our loop integrations) $$\label{SimplifiedTFF}
\frac{F^{\pi\gamma\gamma}(p^2,q^2)}{F^{\pi\gamma\gamma}(0,0)}\,=\,\frac{1}{2}\left[2+\frac{a p^2 + b}{M_V^2-p^2}+\frac{a q^2 + b}{M_V^2-q^2}\right]\,,$$ where $b=0$ (in agreement with the ABJ prediction); $a=1$ would thus correspond to neglecting the contribution of the last term of this form factor in eq. (\[FF\]), which does not modify sizeably its value in the dominant integration regions. Given the above discussion, we judge eq. (\[SimplifiedTFF\]) a sufficient approximation for our computations and we will use $a=1$ and $b=0$ in the numerics. Nevertheless, our results in appendix B of Ref. [@arXiv_v1] are given in terms of $a$ and $b$. Varying $a$ will modify the coefficient of the $1/Q^2$ ($Q=p,q$) asymptotic damping of the form factor. A tiny value of $b\neq0$ would still be consistent with the very small error of the $\pi^0\to\gamma\gamma$ decay rate, which is in agreement with the ABJ prediction.
Finally, we remark that diagram $a)$ is finite even using $F^{\pi\gamma\gamma}(0,0)=g_{\gamma\gamma\pi^0}$, i. e. neglecting model-dependent contributions in the transition form factor. Being its topology more complicated than the one in diagrams $(b)$ and $(d)$ we have followed this procedure so as to be able to evaluate it with LoopTools. Adding structure-dependent terms to this point-like interaction will only reduce the strength of the coupling (and thus the contribution coming from this diagram) for larger photon virtualities. We also note that the contribution to the branching fraction of diagram $(a)$ is subdominant with respect to that of diagrams $b)$ and $d)$. This contribution of diagram $a)$ alone is two orders of magnitude smaller than the figures in eq. (\[BRgammagamma\]). Because of this, we disregard the error induced by considering $F^{\pi\gamma\gamma}(0,0)=g_{\gamma\gamma\pi^0}$ in the evaluation of diagram $a)$. Incidentally, diagrams $b)$ and $d)$ give very similar contributions to the branching ratio for the $\pi^0$ and $\eta$ channels. For the $\eta'$ channel, the contribution of diagram $d)$ approximates the total branching ratio within 10$\%$.
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[^1]: Obviously, this type of vertex would also contribute to the $\tau^-\to\pi^-\pi^0\nu_\tau$ decays. However, as we will check, it is negligible in this case because the tree level contribution is not suppressed.
[^2]: We recall the main formulas for the analyses of these decays at tree level in appendix \[A1\].
[^3]: Chisholm identity reads $\gamma_{\mu}\epsilon^{\mu\nu\rho\sigma}=i\left(\gamma^{\nu}\gamma^{\rho}\gamma^{\sigma}-g^{\nu\rho}\gamma^{\sigma}-g^{\rho\sigma}\gamma^{\nu}+g^{\nu\sigma}\gamma^{\rho}\right)\gamma^5$.
[^4]: This relative contribution may become larger if the dominant tree-level contribution to the [$\tau^- \to \eta^{(\prime)} \pi^- \nu_{\tau}$ ]{}decays, given by the scalar form factor, has a smoother energy distribution than the one shown in Ref. [@Escribano:2016ntp].
|
---
author:
- Freddy Bonnin
date: 8 avril 2004
title: Les groupements
---
|
---
abstract: 'In this paper a model incorporating chiral symmetry breaking and dynamical soft-wall AdS/QCD is established. The AdS/QCD background is introduced dynamically as suggested by Wayne de Paula etc al and chiral symmetry breaking is discussed by using a bulk scalar field including a cubic term. The mass spectrum of scalar, vector and axial vector mesons are obtained and a comparison with experimental data is presented.'
author:
- 'Qi Wang$^{1}$[^1], An Min Wang$^{2}$[^2]\'
title: 'Chiral Symmetry Breaking in the Dynamical Soft-Wall Model'
---
Introduction {#secIntro}
============
Quantum Chromodynamics (QCD) is an non-abelian gauge theory of the strong interactions. It has a coupling constant that highly depends on the energy scale, and the perturbation theory doesn’t work at low energies. The Anti-de Sitter/conformal field theory (AdS/CFT) correspondence [@Maldacena:1997re] provides a novel way to relate QCD with a 5-D gravitational theory and calculate many observables more efficiently. This leads to two types of AdS/QCD models: the top-down models that are constructed by branes in string theory and the bottom-up models by the phenomenological aspects. In the top-down model, mesons are identified as open strings with both ends on flavor branes [@Karch:2002sh]. The most successful top-down model in reproducing various facts of low energy QCD dynamics is D4-D8 system of Sakai and Sugimoto [@Sakai:2004cn] [@Sakai:2005yt].
The bottom-up approach assumes that QCD has suitable 5D dual and construct the dual model from a small number of operators that are influential. The first bottom-up model is the so-called hard-wall model [@EKSS] [@DP]; it simply place an infrared cutoff on the fifth dimension to break the chiral symmetry. But in hard-wall model the spectrum of mass square $m_n^2$ grow as $n^2$, which is contradict with the experimental data [@Shifman-Regge]. The analysis of experimental data indicates a Regge behavior of the highly resonance $m_n^2\sim n$. To compensate this difference between QCD and AdS/QCD, soft-wall model was proposed [@Karch:2006pv]. In this model the spacetime smoothly cap off instead of the hard-wall infrared cutoff by introducing a background dilaton field $\Phi$: $$S_{5}=-\int d^{5}x
\sqrt{-g}\,e^{-\Phi(z)}\mathscr{L},$$where the background field is parametrized to reproduce Regge-like mass spectrum. In spite of success in reproducing Regge-like mass spectrum, chiral symmetry breaking in this model is not QCD-like. Some further researches try to incorporate chiral symmetry breaking and confinement in this model by using high order terms in the potential for scalar field [@Gherghetta:2009ac] [@zhangpeng:2010prd].
But the dilaton field and scalar field in the soft-wall model are imposed by hand to and cannot be derived from any equation of motion. Some works on the dilaton-gravity coupled model show that a linear confining background is possible as a solution of the dilation-gravity coupled equations and Regge trajectories of meson spectrum is obtained [@dePaulaPRD09] [@dynamicalmetric].
In this paper, we try to incorporate soft-wall AdS/QCD and dynamical dilaton-gravity model. We introduce the meson sector action with a cubic term of the bulk field in the bulk scalar potential under the dynamical metric background. We derive a nonlinear differential equation related the vacuum expectation value(VEV) of bulk field, wrap factor and dilaton field. An analysis of the asymptotic behavior of the VEV is presented. We obtain that the VEV is a constant in the IR limit that suggests chiral symmetry restoration exists. The coupling constant can be determined by other parameters in this model.
This paper proceeds as follow: A brief review of dynamical gravity-dilaton background is presented in Sec. \[secMetric\]. In Sec. \[model\] we introduce the bulk field and meson sector action under dynamical metric background and obtain the differential equation of the bulk field VEV. An analysis of asymptotic behavior is presented and a parametrized solution is given. Using the parametrized solution we calculate the scalar, vector and axial vector mass spectrums in Sec. \[massspectrum\]. A conclusion is given in Sec. \[conclusion\].
Background Equations {#secMetric}
====================
We start from the Einstein-Hilbert action of five-dimensional gravity coupled to a dilaton $\Phi $ as proposed by de Paula, Frederico, Forkel and Beyer [@dePaulaPRD09]:
$$S=\frac{1}{2\kappa ^{2}}\int d^{5}x\sqrt{-g}\left( -%
\emph{R}+\frac{1}{2}g^{MN}\partial _{M}\Phi \partial _{N}\Phi
-V(\Phi )\right) \label{actiongd}$$
where $\kappa $ is the five-dimensional Newton constant and $V$ is a still general potential for the scalar field. Then we restrict the metric to the form $$g_{MN}=e^{-2A(z)}\eta_{MN} \label{metric}$$ where $\eta_{MN}$ is the Minkowski metric. We write the warp factor as $$A(z)=\ln z+C\left( z\right) \label{wrapfactor}$$
Variation of the action (\[actiongd\]) leads to the Einstein-dilaton equations for the background fields $A$ and $\Phi$: $$\begin{aligned}
6A^{\prime }{}^{2}-\frac{1}{2}\Phi ^{\prime 2}+e^{-2A}V(\Phi ) &=&0
\label{einsteinzz} \\
3A^{\prime \prime }-3A^{\prime }{}^{2}-\frac{1}{2}\Phi ^{\prime
2}-e^{-2A}V(\Phi ) &=&0 \label{einstein00} \\
\Phi ^{\prime \prime }-3A^{\prime }\Phi ^{\prime
}-e^{-2A}\frac{dV}{d\Phi } &=&0 \label{dilatonequation}\end{aligned}$$ Here the prime denotes the derivative with respect to $z$.
By adding the two Einstein equation ones can get the dilaton field directly: $$\Phi ^{\prime }=\sqrt{3}\sqrt{A^{\prime
}{}^{2}+A^{\prime \prime }}~ \label{constrain}$$
and by substituting (\[constrain\]) in (\[einsteinzz\]) or (\[einstein00\]): $$V(\Phi \left( z\right) )=\frac{3e^{2A\left( z\right) }}{2}\left[
A^{\prime \prime }\left( z\right) -3A^{\prime }{}^{2}\left( z\right)
\right] \label{vz}$$
If $A(z)=\log(z)+z^\lambda$ is set for simplicity, the asymptotic behavior of dilation field $\Phi$ in UV and IR limits can be obtained
$$\Phi(z)=\left\{
\begin{array}{ll}
\underrightarrow{z\rightarrow0} & 2\sqrt{3(1+\lambda^{-1})}z^{\frac{\lambda}{2}}\\
\underrightarrow{z\rightarrow\infty} & \sqrt{3}z^\lambda
\end{array}
\right.$$
The Model {#model}
=========
In this section, we will establish the soft-wall AdS/QCD model first introduced in [@Karch:2006pv] under the dynamical background mentioned in Sec. \[secMetric\]. The background geometry is chosen to be 5D AdS space with the metric $$ds^{2}=g_{MN}dx^{M}dx^{N}=e^{-2A}\left(
\eta_{\mu\nu}dx^{\mu}dx^{\nu}+dz^{2}\right)~ \label{background}$$ where $A$ is the warp factor, and $\eta_{\mu\nu}$ is Minkowski metric. Unlike [@Karch:2006pv] we introduce a background dilaton field $\Phi$ that is coupled with $A$ and $\Phi$ can be determined in terms of $A$ as shown in Sec. \[secMetric\].
In order to describe chiral symmetry breaking, a bifundamental scalar field $X$ is introduced and we add an cubic term potential instead of the quartic term as shown in [@Gherghetta:2009ac] $$S_{5}=-\int d^{5}x \sqrt{-g}\,e^{-\Phi(z)}{\rm
Tr}[|DX|^{2}+m_{X}^{2} |X|^{2} \nonumber$$ $$-\lambda|X|^{3}+\frac{1}{4g_{5}^{2}}(F_{L}^{2}+F_{R}^{2})]
\label{mesonaction}$$ where $m_{X}=-3$, $\lambda$ is a constant and $g_5^2=12\pi^2/N_c=4\pi^2$. The fields $F_{L,R}$ are defined by $$F_{L,R}^{MN} = \partial^{M} A_{L,R}^{N} - \partial^{N} A_{L,R}^{M} -
i [A_{L,R}^{M},A_{L,R}^{N}], \nonumber$$ here $A_{L,R}^{MN} = A_{L,R}^{MN} t^{a}$, $Tr[t^{a} t^{b}] =
\frac{1}{2} \delta^{ab}$, and the covariant derivative is $D^M
X=\partial^M X-i A_L^MX+iXA_R^M$.
Bulk scalar VEV solution and dilaton field
------------------------------------------
The scalar field $X$ is assumed to have a z-dependent VEV $$<X>=\frac{v(z)}{2}\left(\begin{array}{cc}1 & 0\\
0 & 1
\end{array}\right) \label{xvev}$$
We can obtain a nonlinear equation related $A$, $\Phi$ and $v(z)$ from the action (\[mesonaction\]) $$\partial_z( e^{-3A} e^{-\Phi} \partial_z v(z))+ e^{-5A} e^{-\Phi} (3 v(z)+\frac{3}{4}\lambda v^2(z))=0 \label{VEVequation}$$
If $A$ and $\Phi$ are given, this equation can be simplified to a second order differential equation of $v(z)$ $$v''(z)-(\Phi'+3A')v'(z)+e^{-2A}(3 v(z)+\frac{3}{4}\lambda v^2(z))=0
\label{VEVequation1}$$ where $ \Phi ^{\prime }=\sqrt{3}\sqrt{A^{\prime }{}^{2}+A^{\prime
\prime }}~ \label{constrain} $.
Firstly, we consider the limit $z\rightarrow \infty$. If wrap factor $C(z)$ has a asymptotic behavior of $z^2$ as $z\rightarrow \infty$, we can determine that $v(z)\rightarrow$ constant. This means that chiral symmetry restoration in the mass spectrum. But there are many controversies on whether such a restoration really exists [@Cohen:2005am] [@Wagenbrunn:2006cs].
Secondly, we analyze the asymptotic behavior of $v(z)$ as $z\rightarrow 0$. As shown in [@Witten:1998qj] [@Klebanov:1999tb], the VEV as $z\rightarrow 0$ is required to be $$v(z)=m_q\zeta z+\frac{\sigma}{\zeta}z^3$$ where $m_q$ is the quark mass, $\sigma$ is the chiral condensate and $\zeta=\frac{\sqrt{3}}{2\pi}$.
We assume that $C(z)$ takes the asymptotic form as $z\rightarrow 0$ $$C(z)=\alpha z^2+\beta z^3$$ Then we can get the asymptotic form of $\Phi'$ in the IR limit. Considering the coefficients of $\frac{1}{z}$ and constant terms at the left side of Equation (\[VEVequation1\]) should be eliminated as $z\rightarrow 0$, we can conclude that the cubic term of bulk field is necessary and a relation between $\alpha$, $\beta$ and $\lambda$ can be obtained $$\lambda m_q\zeta=4\sqrt{2\alpha}. \label{lambdarelation}$$
A parametrized solution and parameter setting
---------------------------------------------
The equation (\[VEVequation1\]) is a nonlinear differential equation and it is difficult to solve the VEV $v(z)$, so we choose to select a parametrized form of $v(z)$ that satisfies the asymptotic constraint instead of solving the equation directly.
We set the wrap factor as suggested in [@dePaulaPRD09] $$A(z)=\log z+\frac{(\xi\Lambda_{QCD} z)^2}{1+\exp(1-\xi\Lambda_{QCD}
z)} \label{wrapfactor}$$ where $\xi$ is a scale factor and $\Lambda_{QCD}=0.3\textrm{GeV}$ is the QCD scale.
Also we assume the VEV $v(z)$ asymptotically behaves as discussed in previous section $$v(z)=\left\{
\begin{array}{ll}
\underrightarrow{z\rightarrow\infty} & \gamma\\
\underrightarrow{z\rightarrow0} & m_q\zeta z+\frac{\sigma}{\zeta}z^3
\end{array}
\right. \label{VEVasymptotic}$$
A simple parametrized form for $v(z)$ that satisfies asymptotic constraint (\[VEVasymptotic\]) can be chosen as $$v(z)=\frac{z(A+Bz^2)}{\sqrt{1+(Cz)^6}}$$ The relations between $m_q$, $\sigma$, $\gamma$ and $A$, $B$, $C$ are: $$A=m_q\zeta, B=\frac{\sigma}{\zeta}, \frac{B}{C^3}=\gamma$$
Using the data of meson mass data, pion mass and pion decay constant we can get a good fitting of $A$, $B$, $C$, $\lambda$ as follows:
$$A=1.63\textrm{MeV}, B=(157\textrm{MeV})^3, C=10,
\lambda=\frac{4\sqrt{2}\xi \Lambda_{QCD}}{m_q\zeta\sqrt{1+e}}$$
In next section we will use this parametrized solution to calculate scalar, vector and axial-vector meson mass spectra and compare them with experimental data.
Meson Mass Spectrum {#massspectrum}
===================
Starting from the action (\[mesonaction\]), we can derive the Schr$\ddot{o}$dinger-like equations that describe the scalar, vector and axial vector mesons: $$-\partial_z^2S_n(z)+(\frac{1}{4}\omega'^2-\frac{1}{2}\omega''-e^{-2A}(3+\frac{3}{2}\lambda
v(z)))S_n(z)=m^2_{S_n}S_n(z) \nonumber$$
$$-\partial_z^2V_n(z)+(\frac{1}{4}\omega'^2-\frac{1}{2}\omega'')V_n(z)=m^2_{V_n}V_n
\nonumber$$
$$-\partial_z^2A_n(z)+(\frac{1}{4}\omega'^2-\frac{1}{2}\omega''+g_5^2e^{-2A}v^2(z))A_n(z)=m^2_{A_n}A_n(z)
\label{mesonquation}$$
where the prime denotes the derivative with respect to $z$ and $\omega=\Phi(z)+3A(z)$ for scalar mesons, $\omega=\Phi(z)+A(z)$ for vector and axial vector mesons.
The meson mass spectrum is obtained through calculating the eigenvalues of these equations. We will calculate the eigenvalues of these equations numerically using the parametrized solution introduced in Sec. \[model\].
Since the potential is complicated, we must solve the eigenvalue equation numerically. As suggest in [@dePaulaPRD09], we set $\xi=0.58$ for scalars and the numerical results are shown in Table I.
n $f_0 (Exp)$ $f_0 (Model)$
--- -- ------------- ---------------
0 550 897
1 980 1135
2 1350 1348
3 1505 1540
4 1724 1717
5 1992 1881
6 2103 2034
7 2134 2178
: Scalar mesons spectra in MeV.
For vectors, the value of $\xi$ is chosen to be 0.88 and the numerical results are listed in Table II.
n $\rho (Exp)$ $\rho (Model)$
--- -- -------------- ----------------
0 775.5 988.9
1 1465 1348
2 1720 1650
3 1909 1913
4 2149 2147
5 2265 2359
: Vector mesons spectra in MeV.
Sine $\Delta m^2=m^2_{A^2_n}-m^2_{V^2_n}=g^2_5e^{-2A}v^2(z)$ tends to zero as $z\rightarrow 0$, it means the mass of vector and axial-vector mesons are equal at high values.
Conclusion
==========
In this paper we incorporated chiral symmetry breaking into the soft-wall AdS/QCD model with a dynamical background. Then a discussion about the asymptotic behavior of the VEV of the bulk field was presented and a chiral symmetry restoration was suggested in the IR limit. We also introduced a parametrized solution of the VEV of the bulk field and fitted the parameters by using the meson mass spectra and pion mass and its decay constant. The numerical solution of meson mass spectra and comparison with experimental data was also considered. The agreement between the theoretical calculation and experimental data is good.
There are some issues worthy of further consideration. The differential equation of the VEV $v(z)$ needs further study, numerically and analytically, to obtain a more precise form of $v(z)$. The mass spectra of nucleons and some more complicated properties of mesons and nucleons in this model will be our following work.
Acknowledgement
===============
This work has been supported by the National Natural Science Foundation of China under Grant No. 10975125.
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[^1]: Email:[email protected]
[^2]: Email:[email protected]
|
---
author:
- 'I.E. Papadakis'
- 'P. Boumis'
- 'V. Samaritakis'
- 'J. Papamastorakis'
date: 'Received ?; accepted ?'
title: 'Multi-band optical micro-variability observations of BL Lacertae'
---
Introduction
============
BL Lac objects are one of the most peculiar classes of active galactic nuclei (AGN). They show high polarization (up to a few percent, as opposed to less than $\sim 1\%$ for most AGNs) and usually do not exhibit strong emission or absorption lines in their spectra (but see for example Corbett [et al. ]{}1996, and Vermeulen [et al. ]{}1995 for BL Lac itself, and Pian [et al. ]{}2002, for PKS 0537-441). They also show continuum variability at all wavelengths at which they have been observed, from X–rays to radio wavelengths. In the optical band they show large amplitude, short-time scale variations. The overall spectral energy distribution of BL Lacs shows two distinct components in the $\nu-\nu F_{\nu}$ representation. The first one peaks from mm to the X–rays, while the second component peaks at GeV–TeV energies (e.g. Fossati [et al. ]{}1998). The commonly accepted scenario assumes that the non-thermal emission from BL Lacs is synchrotron and inverse-Compton radiation produced by relativistic electrons in a jet oriented close to the line of sight (e.g. Ghisellini [et al. ]{}1998).
BL Lacertae, the object that was used to define this class of AGN, is a well-studied source that has been observed in the optical band for more than a century. Large amplitude variations have been observed on both long and short time scales (Villata [et al. ]{}2002, and references therein). [ ]{}was among the first BL Lac objects which showed large amplitude variations on time scales of $\sim$ hours (Racine, 1970; Miller [et al. ]{}, 1989). During the long, large amplitude 1997 optical outburst the intra-night variations of the object were studied in detail. Nesci [et al. ]{}(1998) and Speziali & Natali (1998) presented multi-band, micro-variability studies based on observations that were made in July 1997 and August 1997, respectively. In both cases, rapid variations were detected in all bands, with their amplitude increasing towards smaller wavelengths. Matsumoto [et al. ]{}(1999), presented $V$ band observations of [ ]{}during August and September, 1997, and reported the detection of a rapid flux increase of about 0.6 mag within 40 min. Ghosh [et al. ]{}(2000), presented $V$ band observations taken between August and October, 1997. They combined the optical fast, intra-night variations with simultaneous X–ray and $\gamma-$ray observations, and analyzed their results within the context of theoretical models. Clements & Carini (2001), presented $V$ and $R$ micro-variability observations that were obtained during the summer of 1997. They observed significant intra-night variations, which were associated with spectral variations as well. The spectrum of the source was becoming “bluer" as it brightened. Finally, Ravasio [et al. ]{}(2002) presented simultaneous X–ray (from [*Beppo-Sax*]{}) and optical observations of the source during June and December, 1999. The source has varied continuously in the optical band. No clear correlation between fast X–ray and optical variability could be found.
In this work, we present simultaneous, $B$, $R$ and $I$ band monitoring, intra-night observations of [ ]{}obtained during 1999 and 2001. The availability of observations in different bands and the almost evenly, dense sampling pattern of the light curves offer us the opportunity to study in detail the flux and spectral variations of the source on time scales of $\sim$ min to hours. The paper is organized as follows. In the next section we present our observations. In Sect. 3 we estimate the variability amplitude and compare the observed time scales between the different band light curves. In Sect. 4 we present the results from a power spectrum analysis, in Sect. 5 we discuss the spectral variability properties of the source, and in Sect. 6 we present the cross-correlation analysis results. A discussion follows in Sect. 7, while a summary of our work is presented in Sect. 8.
Observations and data reduction
===============================
BL Lac was observed for 2 nights in 1999 and 3 nights in 2001 from the 1.3 m, f/7.7 Ritchey-Cretien telescope at Skinakas Observatory in Crete, Greece. The observations were carried out through the standard Johnson $B$ and Cousins $R, I$ filters. The CCD used was a $1024 \times 1024$ SITe chip with a 24 $\mu$m$^{2}$ pixel size (corresponding to $0^{\prime\prime}.5$ on sky). The exposure time was 180, 60 and 30 sec for the $B,R $ and $I$ filters, respectively, during the 2001 observing run, and 120, 60 and 30 sec, respectively, during the 1999 run. In Table 1 we list the observation dates, and the number of the $B,R$ and $I$ frames that we obtained each night. During the observations, the seeing was between $\sim 1^{\prime\prime} - 2^{\prime\prime}$. Standard image processing (bias subtraction and flat fielding using twilight-sky exposures) was applied to all frames.
---------- ------- ------- -------
Date $B$ $R$ $I$
(nof) (nof) (nof)
28/07/99 39 39 39
29/07/99 37 37 37
05/07/01 40 37 38
06/07/01 38 38 38
08/07/01 42 44 42
---------- ------- ------- -------
: Date of observations, and number of frames (nof) obtained at each band.
We performed aperture photometry of [ ]{}and of the comparison stars B, C, H, and K by integrating counts within a circular aperture of radius $10^{\prime\prime}$ centered on the objects. The photometry of the comparison stars was taken from Smith [et al. ]{}(1985) and Fiorucci & Tosti (1996) for the $B$ and the $R,I$ bands, respectively. The calibrated magnitudes of [ ]{}were corrected for reddening and the contribution of the host galaxy as follows.
For the reddening correction of the nuclear fluxes, we used the relationship: $A_{V}=N_{H}/2.23\times10^{21}$ (Ryter 1996), in order to estimate the $V$-band extinction ($A_{V}$, in magnitudes). We assumed the column density value of $N_{H}=2.0\times 10^{21}$ cm$^{-2}$, derived from the X–ray measurements with ROSAT (Urry [et al. ]{}1996). This value should be representative of the nuclear flux absorption, and its use implies $A_{V}=0.907$. Using the $A_{\lambda}$ versus $\lambda$ relationship of Cardelli [et al. ]{}(1989) we found the extinction (in magnitudes) in the $B,
R,$ and $I$ filters: $A_{B}=1.21, A_{R}=0.73$, and $A_{I}=0.54$.
We converted the dereddened magnitudes into flux, and then we corrected for the contribution of the host galaxy to the measured flux in each band. According to Scarpa et al. (2000), the $R$ band magnitude of the [ ]{}host galaxy is $R_{host}=15.55\pm 0.02$. In order to correct $R_{host}$ for reddening, we used the value of $A_{R}=0.88$ (taken from NED [^1]). This value is derived from the Schlegel [et al. ]{}(1998) maps, and may be more representative of the Galactic absorption for the host galaxy, which is more extended than the active nucleus itself. We inferred the $B_{host}$ and $I_{host}$ magnitudes adopting the elliptical galaxy colors (at redshift zero) of $V-R=0.61, B-V=0.96,$ and $R-I=0.70$ (Fukugita [et al. ]{}1995). The resulting host galaxy fluxes in the $B, R,$ and $I$ bands are 1.40, 4.29, and 6.76 mJy, respectively. Using the results of Scarpa [et al. ]{}(2000) and a de Vaucouleurs $r^{1/4}$ profile, we estimated that the host galaxy contribution within the circular aperture of radius $10^{\prime\prime}$ centered on [ ]{}should be $70\%$ of the whole galaxy flux. Therefore, the final $B_{host}, R_{host},$ and $I_{host}$ fluxes that were subtracted from the observed [ ]{}fluxes were $0.98, 3.0,$ and $4.73$ mJy, respectively.
The observed light curves
=========================
---------- -------------- -------------- --------------
Date $f_{rms,B}$ $f_{rms,R}$ $f_{rms,I}$
(%) (%) (%)
28/07/99 7.2$\pm 0.7$ 6.2$\pm 0.4$ 5.7$\pm 0.3$
29/07/99 6.9$\pm 0.7$ 5.5$\pm 0.3$ 4.8$\pm 0.3$
05/07/01 4.2$\pm 0.4$ 3.9$\pm 0.3$ 3.5$\pm 0.3$
06/07/01 4.3$\pm 0.4$ 3.6$\pm 0.2$ 3.6$\pm 0.2$
08/07/01 9.1$\pm 0.8$ 8.8$\pm 0.7$ 8.5$\pm 0.6$
---------- -------------- -------------- --------------
: The fractional variability amplitude of the light curves at each band.
In Fig. 1 we show the final $B,R,$ and $I$ light curves of [ ]{}during our observations. In the same figure we also show the B band light curve of the comparison star B. While the light curve of this star (and of the other 3 comparison stars in all bands) does not show significant variations, significant intra-night variations can be observed in the [ ]{}light curves at all bands. During the July $28$ and $29$, 1999 observations we detected two flare-like events at the beginning and at the end of the observations, respectively. During the July 5 and 8, 2001 observations the flux decreased at the beginning of the night and then, after $2-4$ hours, it increased again. Finally, a well defined flare which lasted for $\sim 3$ hours was detected in July 6, 2001. Overall, the observed light curves show smooth variations which last for a few hours, with no significant variations on time scales of $\sim$ minutes.
In order to compare the amplitude of the variations that we observe in the different band light curves, we computed the “fractional variability amplitude" ($f_{rms}$) of all the light curves. The fractional variability amplitude is defined as: $f_{rms}=(\sigma^{2}-\sigma^{2}_{N})^{1/2}/\bar{x}$, where $\sigma^{2}$ is the sample variance of the light curve, $\sigma_{N}^{2}=\sum_{i=1}^{N}err_{i}^{2}/N$ is the variance introduced by the instrumental noise process ($err_{i}$ is the error associated with each point, $i$, in the light curve), and $\bar{x}$ is the light curve mean. The fractional variability amplitude represents the average amplitude of the observed variations as a percentage of the light curve mean. The $f_{rms}$ values are listed in Table 2. The errors on the $f_{rms}$ values were estimated using the bootstrap method of Peterson [et al. ]{}(1998). They represent the uncertainty associated with the flux uncertainties in the individual measurements and the uncertainty associated with the observational sampling of the light curves (i.e. the size of the intervals between observations and the length of the light curves). On average, the amplitude of the observed variations is similar for both the 1999 and 2001 observations. Furthermore, the variability amplitude increases from the $I$ to the $B$ band. The average $f_{rms}$ of the $B, R,$ and $I$ light curves is $6.3\pm0.3\%, 5.6\pm0.2\%$, and $5.2\pm0.2\%$, respectively.
In order to visualize how well the light curves in the three bands agree with each other, we normalized them to their mean, and plotted them together (Fig. 2, and 3 for the 1999 and 2001 light curves, respectively; for clarity reasons we have plotted only the $B$ and $I$ band light curves). These plots show that the light curves are well correlated. They also show that the “rising"/“decaying" parts of the $B$ band light curves are steeper than the respective parts in the $I$ band light curves.
In order to compare the variability time scales between the $B, R,$ and $I$ band light curves, we estimated their “Flux Variability Rate" ($FVR$) as follows. First, we identified the rising and decaying parts of the light curves. Then, we fitted them with a linear model of the form: $flux(t) = A+B\times t$ (where $t$ is time in hours). The linear model fits well the rising/decaying phases in all the light curves (as shown by the solid and dashed lines in Fig. 2 and 3), except for the rising part of the July 29, 1999 light curves, which was best fitted by an exponential function of the form: $flux(t)=A_{1}\times \exp (t/\tau)+A_{2}$. Based on the best model fitting results, we estimated the $FVR$ (in units of mJy/hrs) for the different phases of each light curve. In effect, the $FVR$ values show the amount of the flux variation per unit time. The results for the $B$ and $I$ band light curves are listed in Table 3 (similar model fitting results were also obtained for the $R$ band light curves as well).
The average $B$ and $I$ band $FVR$ values are: $\bar{FVR}_{B,{\rm
rise}}=7.8\pm0.6$, $\bar{FVR}_{B,{\rm decay}}=-7.4\pm0.5$, and $\bar{FVR}_{I,{\rm rise}}=6.1\pm0.3$, $\bar{FVR}_{I,{\rm
decay}}=-6.4\pm0.2$ (all values are in units of $\times 10^{-2}$ mJy/hr). These results show that the rising $FVR$ values are similar, in absolute magnitude, to the decaying $FVR$ values, in both the $B$ and $I$ bands. This result implies that the rising and decaying parts of the light curves are symmetric. However, when we compare the values between the two bands we find that the average $FVR_{B}$ values are larger (in absolute magnitude) than the $FVR_{I}$ band values: $\bar{FVR}_{B,{\rm
rise}}-\bar{FVR}_{I,{\rm rise}}=1.7\pm0.8$ $\times 10^{-2}$ mJy/hr, and $\bar{FVR}_{B,{\rm decay}}-\bar{FVR}_{I,{\rm decay}}=-1.0\pm0.5$ $\times
10^{-2}$ mJy/hr. Therefore, the rising/decaying time scales in the $B$ band are [*shorter*]{} than in the $I$ band. However this difference is significant only at the $2\sigma$ level.
Finally, apart from the rising/decaying parts, we can identify at least two flares with a broad peak during the July 6, 2001 and July 28, 1999 observations. This “flare plateau" state lasted for $\sim 1$ hr in both cases and in all light curves. It is possible that the last part of the July 29, 1999 observation may also represent the plateau state of yet another flare. However, we cannot be certain because of the lack of observations of the decaying phase of the flare.
---------- --------------- -------------- ------------- --------------- -- --
Date $FVR_{B}$ $FVR_{B}$ $FVR_{I}$ $FVR_{I}$
(rise) (decay) (rise) (decay)
28/07/99 $7.1\pm 1.7$ $-8.7\pm0.9$ $6.0\pm0.7$ $-7.1\pm0.5$
29/07/99 $4.1\pm 0.4$ $-$ $3.1\pm0.3$ $-$
05/07/01 $6.5\pm 0.5$ $-5.0\pm0.4$ $5.9\pm0.3$ $-3.6\pm0.2$
06/07/01 $13.4\pm 1.7$ $-8.5\pm1.5$ $9.5\pm0.1$ $ -9.4\pm0.6$
08/07/01 $-$ $-7.5\pm0.6$ $-$ $-5.7\pm0.5$
---------- --------------- -------------- ------------- --------------- -- --
: The “Flux Variability Rate" ($FVR$, in units of $\times
10^{-2}$ mJy/hr) during the rising and decaying parts of the $B$ and $I$ band light curves.
Power spectrum analysis
=======================
In order to quantify the variability seen in the optical light curves of [ ]{}we estimated their power spectrum as follows. For each individual light curve, we computed the periodogram as:
$$\hat{I}(\nu_{i})=(\Delta t/N) \{
\sum_{i=1}^{N}[x(t_{i})/\bar{x}-1]e^{-i2\pi\nu_{i}t_{i}} \}^{2},$$
where $\bar{x}, \Delta t,$ and $N$ are the mean value, bin size, and number of points of each light curve, respectively, and $\nu_{i}=i/(N\Delta t), i=1,2,\ldots, (N/2)-1$ (e.g. Papadakis & Lawrence, 1993). The periodogram calculated in this way (i.e. with the points normalized to the light curve mean) has the units of Hz$^{-1}$. This normalization is necessary in order to combine the periodograms of the different light curves (see below). As $\Delta t$ we accepted the mean interval between the points in each light curve. We note that, due to the continuous monitoring of the source, the light curves are almost evenly sampled. The intervals between successive points are almost equal, and any minor difference cannot affect seriously the estimation of the power spectrum. Finally, we combined the periodograms of all the light curves in each band, we sorted them in order of increasing frequency, we computed their logarithm and grouped them into bins of size 15 following the method of Papadakis & Lawrence (1993).
The resulting $B, R,$ and $I$ band power spectra are shown in the upper plot of Fig. 4 (filled squares, open squares and open circles, respectively). They all show a similar red noise component, i.e. they all increase logarithmically towards lower frequencies. The power spectrum normalization increases slightly from the $I$ to the $B$ band power spectrum, as expected from the fact that the $B$ band light curves show the largest $f_{rms}$ values.
Since the three power spectra show a similar shape, we combined the periodograms of all the light curves in order to estimate an “average optical band" power spectrum of the source. Since the Poisson noise power level is different in each light curve (due to the fact that the experimental errors are different), we subtracted the expected Poisson noise level from each periodogram, we sorted them in order of increasing frequency, and grouped the periodogram estimates into bins of size 20.
The average optical power spectrum of [ ]{}is plotted in the lower plot of Fig. 4. Using standard $\chi^{2}$ statistics, we fitted the power spectrum with a power law model of the form $P(\nu)=A\nu^{-a}$. The model provides a good fit to the data ($\chi^{2}=10.6$ for 12 dof). The best fitting model is also plotted in Fig. 4 (dashed line). The best fitting slope value is $a=1.87\pm0.16$ (all errors quoted in the paper correspond to the $68\%$ confidence region).
Spectral variability
====================
Since $f_{rms}$ is different in the $B,R,$ and $I$ band light curves, and there is an indication that the rise/decay time scales may be different as well, we expect the flux variations to be associated with spectral variations. In order to investigate this possibility we used the dereddened light curves, after correction for the host galaxy contribution, to calculate the two-point spectral indices ($B-R$, $\alpha_{BR}$, and $B-I$, $\alpha_{RI}$) using the equation, $\alpha_{12}=\log(F_{1}/F_{2})/\log(\nu_{1}/\nu_{2})$, where $F_{1}$ and $F_{2}$ are the flux densities at frequencies $\nu_{1}$ and $\nu_{2}$, respectively.
Figs. 5 and 6 show the $\alpha_{BR}$ and $\alpha_{BI}$ versus the $B$ band flux plots for the 1999 and 2001 observations, respectively. The $\chi^{2}$ values for a constant slope show that the variations are statistically significant. The $\alpha_{BR}$ and $\alpha_{BI}$ variations are broadly correlated with the source flux in a similar way. Overall, as the $B$ band flux increases, both $\alpha_{BR}$ and $\alpha_{BI}$ increase as well. This result implies that the spectrum becomes “harder" (i.e. “bluer") as the flux increases. However, the relation between flux and spectral index is not simple. For example, a linear model of the form $\alpha = A+B\times flux$, cannot fit well any of the the \[$\alpha_{BI},
B$\] plots in Fig. 5 and 6 (except for the \[$\alpha_{BI}, B$\] plot of the July 29, 1999 observations). Because of this reason, we considered the spectral slope variations during the different phases of the light curves (i.e. rising, decaying, plateau) and we present our results below. We focus our discussion on the $[\alpha_{BI}, B]$ plots mainly, since the errors on the $\alpha_{BI}$ points are smaller than the errors of the $\alpha_{BR}$ points.
During the July 28, 1999 observations, the flux increased at the beginning of the observations, it reached a plateau, then it decreased, and rose again (at a slower rate) towards the end of the observation (see Fig. 1 and 2). During the rising part of the flare, the spectral slope increased (solid circles in the upper right plot of Fig. 5), it reached its maximum value during the plateau state, and then decreased again (open circles in the same plot) as the source flux decreased. The flux related slope variations in the two parts of the flare follow the same linear trend (shown with a solid line in the same plot of Fig. 5). During the last part of the observation, the moderate flux increase is associated with a marked spectral slope increase (shown with solid squares). This is caused by the fact that the $B$ band flux increases faster than the $I$ band flux. During the rising part of the July 29, 1999 observation the spectral index increases linearly with the source flux (solid circles in the lower right plot in Fig. 5; the best fitting linear model is also shown with the solid line). It reaches its maximum value during the plateau state (shown with crosses in the same plot), which lasts until the end of the observation.
A linear relation between spectral slope and source flux is also observed during the July 5, 2001 observation (upper right plot, Fig. 6). As the flux decreased during the first part of the observation, the spectral slope decreased as well (open circles in this plot). Then, when the flux increased during the second part of the observation, the spectral slope also increased (solid circles). The linear trend is similar for both the rise/decay parts of the observation, although the normalization is different, with the rising part resulting in systematically larger spectral slope values. A similar behaviour is observed during the July 8, 2001 observation. During the first, long, decaying part of the observation, the spectral slope decreased (open circles, bottom plot in Fig. 6). A linear model describes well the overall trend (shown with solid line in the same plot), but it cannot fit well the data. This is mainly due to the fact that a few, small amplitude sub-flares can be seen on top of the long, decreasing phase of the light curves. The moderate flux increase at the end of the observation is associated with a steep spectral slope increase (shown with solid squares in the plot). This behaviour is similar to what was observed at the end of the July 28, 1999 observation. Finally, during the July 6, 2001 observations, the spectral slope remained constant ($\alpha_{BI}\sim -0.97$, crosses in the middle plot of Fig. 6) before and after the flare which started two hours after the beginning of the observations. The spectral slope increased/decreased together with the flux in the rise/decay flare phases (shown with solid and open circles, respectively). The spectral slope reached its maximum value during the plateau phase.
Cross-correlation analysis
==========================
Date $CCF_{max}$ $k_{max}$ (hrs)
---------- ------------- -------------------------
28/07/99 1.0 $-0.07\pm 0.20$
29/07/99 1.0 $+0.10\pm 0.28$
05/07/01 0.90 $+0.23^{+0.19}_{-0.15}$
06/07/01 0.96 $+0.09^{+0.15}_{-0.12}$
08/07/01 0.76 $+0.17^{+0.29}_{-0.26}$
: The cross-correlation analysis results.
In order to compare the cross-links between the observed variations in the different bands we estimated the cross-correlation function (CCF) using the Discrete Correlation Function (DCF) of Edelson & Krolik (1988). Fig. 7 shows the DCF between the $B$ and $I$ band light curves during the 1999 and 2001 observations (lower and upper plot, respectively). In these plots, a positive lag means that the $B$ band leads the $I$ band variations. In order to quantify the maximum cross-correlation ($CCF_{max}$) and the “delay" between the light curves (i.e. the time lag, say $k_{max}$, at which this maximum occurs) we fitted the DCF points around zero lag with a parabola, and accepted the best fitting values of the parabola peak as our best estimate of $CCF_{max}$ and $k_{max}$. Our results are listed in Table 4 (columns 2 and 3 for $CCF_{max}$ and $k_{max}$, respectively). The uncertainties in the observed lag values were estimated using the Monte Carlo simulation techniques of Peterson [et al. ]{}(1998).
The DCF plots in Fig. 7 and the results on $CCF_{max}$ show that the $B$ and $I$ band light curves are highly correlated, as expected from the visual good agreement between the different band light curves (e.g. Fig. 2 and 3). The estimated lag values ($k_{max}$) imply that the delay between the two band light curves is consistent with zero, within the errors. Using the cross-correlation results from all the light curves (except for the July 5, 2001; see below), we estimate an average lag of $k_{max}=0.07$ hrs, while the $90\%$ confidence limits are $|k_{max}|<
0.42$ hrs (which is $\sim 4$ times the mean sampling rate of the light curves).
The CCF of the July 5, 2001 light curves, shows an indication that the $I$ band light curve [*is*]{} delayed with respect to the $B$ band light curve. We find a delay of $k_{max}=0.23$ hrs ($\sim$ twice the mean sampling interval of the light curves) with a $68\%$ confidence region of \[$0.08 - 0.42$ hrs\]. In fact, the probability that the $I$ band is delayed with respect to the $B$ band light curve (i.e. the probability that $k_{max}>0$) is $95.4\%$.
Finally, all cross-correlation functions appear to be asymmetric (except for the July 28, 1999 CCF). The asymmetry is in the sense that, on time scales larger than $ 0.5$ hours, the correlation at positive lags is larger than the correlation at negative lags. In fact, at positive lags, the $B$ band light curves are better correlated with the $I$ band light curves than with themselves. This result implies that the variability components with periods larger than $\sim 0.5$ hours in the $B$ band light curves lead the respective components in the $I$ band light curves.
Discussion
==========
Although we cannot estimate systematically time scales from the data presented in this work, the fact that the duration of the rising/decaying parts of the light curves is different during our observations suggests the presence of different variability time scales in [ ]{}. For example, we observe a flare which lasted for $\sim 3$ hours during the July 6, 2001 observations, while longer flares are detected during the July 28, and July 29, 1999 observations. Furthermore, the smooth, “long"-term trends observed during the July 5, and 8, 2001 observations, are probably parts of even longer flare-like events. If the difference in time scales correspond to differences in the size of the emitting source, then perhaps physically different regions/parts of the jet contribute to the optical emission of [ ]{}. Even if this is the case, the similarity in the flux and spectral variability properties suggests that the same physical mechanism operates in all cases.
The rapid, optical variability properties of [ ]{}are similar to the [*X–ray*]{} variability properties of well studied objects like Mkn 421 and PKS 2155-30. For example, the fractional variability amplitude of the X–ray light curves in Mkn 421 and PKS 2155-304 varies between $\sim 2\%$ and $\sim 15\%$ (Edelson [et al. ]{}2001, Sembay [et al. ]{}2002). These values are comparable to the $f_{rms}$ values listed in Table 2 for [ ]{}. Furthermore, the X–ray power spectrum of PKS 2155-304 and Mkn 421 shows a red noise character with a slope of $\sim 2-3$ (Zhang [et al. ]{}2002, Kataoka [et al. ]{}2001), consistent with the slope of the optical power spectrum of [ ]{}. Asymmetric CCFs, with no measurable time lags, similar to the CCFs shown in Fig. 7, have also been observed in PKS 2155-304 and Mkn 421 (Edelson [et al. ]{}2001, Sembay [et al. ]{}2002). Finally, there are similarities between the X–ray spectral variations observed in Mkn 421 and PKS 2155-304, and the optical band spectral variability of [ ]{}as well. Fossati [et al. ]{}(2000) find that, in the case of MKN 421, the spectral slope at 5 keV decreases (i.e. the spectrum flattens) as the source flux increases (Figure 4a in their paper). Their result is based on [*Beppo-Sax*]{} observations in late April/early May 1997, during which no individual flares could be observed. Zhang [et al. ]{}(2002) also find a similar behaviour between the spectral slope at 0.5 keV and the source flux in PKS 2155-304, again during observations with no obvious, single flares (see for example the spectral slope vs flux plot of the “1997 \#3" data-set in Figure 11 of their paper). However, clear clockwise and anti-clockwise “loop-like" variations of the spectral slope with respect to the source flux have also been observed in these two X–ray bright BL Lacs. During the April 1998 [*Beppo-Sax*]{} observations of MKN 421, the hard X–ray spectral slope varied with respect to the source flux following an “anti-clockwise" loop (Fossati [et al. ]{}2000). Similar loops with a well defined, quasi-circular form have also observed in PKS 2155-304 (Zhang [et al. ]{}2002), mainly during observations dominated by well sampled, individual flares. In our case, single flares dominate the observed light curves only during the July 28, 1999 and (mainly) the July 06, 2001 observations. Although in the first case the spectral variations follow the normal “flux increase - spectrum flattening" pattern, it is possible that during the July 06, 2001 observations the variation of the $\alpha_{BI}$ as a function of the $B$ band flux follows an anti- clockwise path during the rising and decaying parts of the flare (see middle panel in Fig. 6). Due to the large errors, we cannot be certain about the reality of this loop-like variation. Longer and better sampled observations (with the collaboration of more than one telescope) are necessary in order to detect and investigate the spectral variations during individual flares and confirm the existence of loop-like variations in the optical band “spectral slope vs flux" plots of BL Lac.
The similarity in the properties of the fast X–ray variations of MKN 421 and PKS 2155-304 and optical variations of BL Lac, may be due to some physical reason. The multi-frequency spectral energy distribution of [ ]{}shows a peak at $\sim 10^{14.5}$ Hz while the peak of the synchrotron emission in the other two objects is located at higher frequencies (e.g. Sambruna [et al. ]{}, 1996). Therefore, the optical and X–ray bands correspond to frequencies that are located around/above the peak of the spectral energy distribution of [ ]{}on one hand, and Mkn 421 and PKS 2155-30, on the other. This fact, together with the similarity in their fast variability properties imply that the intra-night variations in [ ]{}are probably a direct result of the acceleration/cooling mechanism of relativistic electrons which represent the highest energy tail of the synchrotron component, as is generally accepted for the rapid X-ray variations in the other two sources. In this case, by studying the optical, rapid variations of [ ]{}we can obtain useful information on the acceleration ($t_{acc}$) and cooling ($t_{cool}$) time scales of the most energetic electrons.
Our results suggest that the acceleration process of the energetic particles does [*not*]{} dominate the observed variations. Since the acceleration time scale should be shorter for lower energy particles, we should expect the lower energy (i.e the $I$ band) light curves to show steeper rising phases than the $B$ band light curves. However, we observe the opposite effect. The $FVR_{B{\rm rise}}$ values are consistently larger than the $FVR_{R,I{\rm rise}}$ values. Therefore, the $B$ band light curves rise [*faster*]{} than $I$ band light curves. Furthermore, the asymmetry in the CCFs towards positive lags is opposite to what we would expect if the emission propagates from lower to higher energy. We conclude that our observations imply that the injection of radiating electrons in the optical emitting region of [ ]{}is almost instantaneous.
In this case emission should propagate from higher (i.e. the $B$ band) to lower energy (the $I$ band) and the higher energy should lead the lower energy photons. This is consistent with the results from the cross-correlation analysis. Although in most cases we do not detect a significant delay (except for the July 5, 2001 observations, see below), the CCF asymmetry towards positive lags implies complex delays between long period components in the two light curves, in the expected direction. If the $B$ band variations during the July 05, 2001 observations do lead the $I$ band variations with $k_{max}\sim 0.2$ hrs, then we can obtain an estimate of the magnetic field strength in the source. Assuming that the delay is due to synchrotron losses of the high energy electrons, i.e. $k_{max}=t_{cool}(I)-t_{cool}(B)$, we can use the following formula (e.g. Chiappetti [et al. ]{}1999),
$$B\delta^{1/3}\sim
300(\frac{1+z}{\nu_{I}})^{1/3}[\frac{1-(\nu_{I}/\nu_{B})^{1/2}}{k_{max}}]^
{2/3},$$
to estimate $B$ (in G, when frequencies are given in units of $10^{17}$ Hz). Using $k_{max}=1.7\times 10^{4}$ s$ (=0.2$ hrs), $z=0.0688$, and $\nu_{B}=0.0068, \nu_{I}=0.0034$ (in units of $10^{17}$ Hz), we find that $B\delta^{1/3}\sim 1.3$ G, or $B\sim 0.4-0.6$ G, assuming that $10
<\delta<30$ (e.g. Vermeulen & Cohen, 1994).
The fact that the rising/decaying time scales of the fully resolved flare during the July 6, 2001 observations are comparable and roughly similar to plateau state duration, bears interesting consequences. If $t_{cool}<<R/c$ (where $R$ is the source radius), the energetic particles would reach equilibrium very fast. The corresponding synchrotron emission should be switched on and off for a short time and, due to light travel effects, after a short period (controlled by the short injection time scale, $t_{inj}$) the observer should see a constant flux, produced by a single “switched on" slice running across the source (Chiaberge & Ghisellini, 1999). After the last region of the source is switched on and off, a fast decline, controlled by $t_{cool}$, should be observed. Therefore, when both $t_{inj}$ and $t_{cool}$ are much smaller than $R/c$, we expect to observe a plateau phase which will last much longer than the rise and decay phase of the flare. The fact that the July 6, 2001 flare is symmetric, with rising/decaying/plateau phases which last roughly the same period, implies that $t_{cool}$ is not much smaller than $R/c$. In this case, we expect the flux to increase as the observer receives photons from an increasing volume. At time $t\sim t_{cool}$, the parts of the source that were seen first will stop emitting, and the flux will remain $\sim$ constant as new parts of the source start to switch on while the parts closer to the observer are switched off. After $t\sim R/c$, the flux will start decreasing, as the whole source volume has been activated, and the front parts of the source keep switching off.
The hypothesis of a jet perturbation with $t_{inj}<<R/c$ and $t_{cool}
\sim R/c$, can also explain the observed spectral variations. As emission propagates from higher to lower frequencies, it takes time for the higher energy electrons to cool and start emitting at lower frequencies (i.e. in the $I$ band). If the increase in the $B$ band emitting volume is faster than $t_{cool}$, then the higher energy light curves will rise steeper than the lower energy light curves. As a result, as the $B$ band flux rises, the observed spectrum will flatten systematically. The flattest spectral index will correspond to the maximum $B$ band flux, when the $B$ band emitting region has reached its maximum size, and the volume of the $I$ band emitting region is still increasing. Furthermore, since $t_{cool}$ is faster for the $B$ band than the $I$ band emitting electrons, the decaying phases of the light curves are expected to be steeper in the $B$ band, as observed. At the same time, the spectrum will become “redder" as the $B$ band flux decreases.
Conclusions
===========
We have observed [ ]{}in three bands, namely $B$, $R$, and $I$, for 2 nights in July, 1999 and 3 nights in July, 2001. On average, each light curve lasts for $\sim 6$ hours. There are $\sim 40$ points in each of them, almost evenly spaced, with an average sampling interval of $\sim
0.1$ hrs. Because of the dense sampling and the availability of light curves in three bands, we were able to study in detail the intra-night flux and spectral variations of the source. Our results can be summarized as follows:
1\) The source is highly variable in all bands. The variations are smooth, showing rising/decaying phases which, in some cases, last longer than the length of each individual light curve. We have also detected 3 “flare-like" events. In particular, during the July 6, 2001 observation, we observed the whole cycle of a flare which appears symmetric, with a plateau, and lasted for $\sim$ 3 hours (in all bands). In general, the rising time scales are comparable to the decaying time scales within each band. However, these time scales are shorter in the $B$ than the other two band light curves.
2\) The variability amplitude decreases from $\sim 6.5\%$ in the $B$ band, to $\sim 5.5\%$ and $\sim 5\%$ in the $R$, and $I$ band light curves, respectively. The average, optical power spectrum of the source has a red noise character, with a slope of $\sim -2$ in the frequency range between \[5.5 (hrs$)^{-1} - 15$ (min)$^{-1}$\].
3\) The light curves in the three bands are well correlated. The variations occur almost simultaneously in all of them, in the sense that the delay between the $I$ and $B$ band variations is smaller than $\sim
\pm0.4$ hrs. However, we also find that during the July 5, 2001 observation, there is a $95\%$ probability that the $I$ band light curve variations are delayed with respect to the $B$ band variations by $\sim
0.2$ hrs. Furthermore, most of the CCFs are significantly asymmetric, implying complex delays of the $I$ band variations, in all cases.
4\) Finally, the source shows significant intra-night spectral slope variations. These variations are associated with the source flux, in the sense that the spectrum becomes “bluer" as the flux increases. The flattest spectral slope corresponds to the maximum $B$ band flux. The rate of the spectral slope changes is different for different rising/decaying parts of the light curves.
Assuming that the variations are caused by perturbations which activate the jet, the observation of variations with different duration implies that the perturbations affect different regions of the jet. The fact that the rising time scales are steeper in the $B$ band light curves and the CCF asymmetry towards positive lags imply that the injection time scales are very short (i.e. shorter than our average sampling rate which is $\sim 3-6$ minutes). These results, together with the observed spectral variability pattern, imply that the observed variations are governed by the cooling time scale of the relativistic particles and the light crossing time scale. The detection of symmetric flares with a plateau state implies that these time scales are comparable. Finally, the detection of a soft lag in one of the observations, allows us to obtain an estimate of the magnetic field strength, $B\sim 0.5$ G.
We believe that our results demonstrate that well sampled, multi-band optical, [*intra-night*]{} observations of [ ]{}objects, whose peak of the emitted power is at mm/IR wavelengths so that the optical emission corresponds to the emission from the most energetic, synchrotron emitting electrons in the jet, will offer us important clues on the acceleration and cooling mechanism of these particles. Since the injection and cooling times of the particles are very short, light curves with an average sampling of no more than a few minutes are necessary to this end.
0.4cm
We would like to thank E. Pian, the referee, for helpful comments. Skinakas Observatory is a collaborative project of the University of Crete, the Foundation for Research and Technology-Hellas, and the Max–Planck–Institut für extraterrestrische Physik.
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[^1]: The NASA/IPAC Extragalactic Database (NED) is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration.
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abstract: 'We study relations between the quadraticity of the Kuranishi family of a coherent sheaf on a complex projective scheme and the formality of the DG-Lie algebra of its derived endomorphisms. In particular, we prove that for a polystable coherent sheaf of a smooth complex projective surface the DG-Lie algebra of derived endomorphisms is formal if and only if the Kuranishi family is quadratic.'
address: 'Università degli studi di Roma La Sapienza,Dipartimento di Matematica Guido Castelnuovo,P.le Aldo Moro 5, I-00185 Roma, Italy.'
author:
- Ruggero Bandiera
- Marco Manetti
- Francesco Meazzini
date: 'February 17, 2019'
title: 'Deformations of polystable sheaves on surfaces: quadraticity implies formality'
---
\[section\] \[theorem\][Theorem]{} \[theorem\][Lemma]{} \[theorem\][Proposition]{} \[theorem\][Corollary]{}
\[theorem\][Definition]{} \[theorem\][Example]{}
\[theorem\][Remark]{}
Introduction
============
Recall that for a coherent sheaf ${\mathcal{F}}$ over a projective scheme, we say that the Kuranishi family is quadratic if the base space of the Kuranishi family (i.e., the semiuniversal deformation) of ${\mathcal{F}}$ is a quadratic singularity. One of the main goals of this paper is to prove the following result.
\[thm.maintheorem1\] Let ${\mathcal{F}}$ be a polystable coherent sheaf of a smooth complex projective surface. Then the differential graded (DG) Lie algebra ${R\mspace{-2mu}\operatorname{Hom}}({\mathcal{F}},{\mathcal{F}})$ of derived endomorphisms is formal if and only if the Kuranishi family of ${\mathcal{F}}$ is quadratic.
The initial motivation was to provide a proof of the Kaledin-Lehn formality conjecture for polystable sheaves on projective K3 surfaces as a consequence of the quadraticity theorem of Kuranishi space proved by Arbarello and Saccà [@AS] and by Yoshioka [@yoshioka].
In the meantime the above mentioned formality conjecture has been recently proved by Budur and Zhang [@budur]; we also refer to [@budur] for a detailed description of the formality conjecture and its implication in the theory of moduli spaces of sheaves on K3 surfaces. However, the proof of Budur and Zhang is done along a completely different approach, and our result applies to every smooth projective surface, not necessarily K3.
Since the automorphisms group of a polystable sheaf is a product of groups ${\operatorname{GL}}_n({\mathbb{C}})$ (see e.g. [@HL]), Theorem \[thm.maintheorem1\] is an immediate consequence of the following more general result.
\[thm.maintheorem2\] Let ${\mathcal{F}}$ be a coherent sheaf of finite projective dimension on a complex projective scheme $X$. Assume that:
1. the algebraic group ${\operatorname{Aut}}_X({\mathcal{F}})$ of automorphisms of ${\mathcal{F}}$ is linearly reductive;
2. the trace map ${\operatorname{Tr}}\colon {{\operatorname{Ext}}}^i_X({\mathcal{F}},{\mathcal{F}})\to H^i({\mathcal{O}}_X)$ is injective for every $i\ge 3$.
Then the DG-Lie algebra ${R\mspace{-2mu}\operatorname{Hom}}({\mathcal{F}},{\mathcal{F}})$ of derived endomorphisms is formal if and only if the Kuranishi family of ${\mathcal{F}}$ is quadratic.
The DG-Lie algebra of derived endomorphisms of ${\mathcal{F}}$ controls the deformation theory of ${\mathcal{F}}$ (see e.g. [@AS; @budur; @FIM; @FM]) in the usual way, via Maurer-Cartan modulus gauge action, see e.g. [@GoMil1; @Man], and its cohomology compute the derived functors of ${\operatorname{Hom}}_X$, i.e., $$H^i({R\mspace{-2mu}\operatorname{Hom}}({\mathcal{F}},{\mathcal{F}}))={{\operatorname{Ext}}}^i_X({\mathcal{F}},{\mathcal{F}})\,.$$
The “only if” implication in Theorem \[thm.maintheorem2\] is a completely standard consequence of the homotopy invariance of the Kuranishi family: a detailed proof of this fact is given in [@GoMil1; @GoMil2], where it is used in order to prove that certain moduli spaces of representations of the fundamental group of a compact Kähler manifold have quadratic singularities, cf. also [@MartPadova] and references therein.
The idea of proof of Theorem \[thm.maintheorem2\] is the following. Assume for simplicity that ${{\operatorname{Ext}}}^i_X({\mathcal{F}},{\mathcal{F}})=0$ for every $i\ge 3$ and suppose first heuristically that ${\operatorname{Aut}}_X({\mathcal{F}})$ is a discrete group (clearly this is true only for ${\mathcal{F}}=0$), then $H^i({R\mspace{-2mu}\operatorname{Hom}}({\mathcal{F}},{\mathcal{F}}))=0$ for every $i\not=1,2$; under these assumptions quadraticity implies formality by the well known fact that the Kuranishi moduli space uniquely determines the minimal $L_{\infty}[1]$-model of ${R\mspace{-2mu}\operatorname{Hom}}({\mathcal{F}},{\mathcal{F}})$ up to (non-canonical) isomorphism; for reference purposes we give a proof of this fact in Subsection \[subsec.formalityquadra\].
In the general case, the first step is to show that there exists a DG-Lie representative $L$ of ${R\mspace{-2mu}\operatorname{Hom}}({\mathcal{F}},{\mathcal{F}})$ equipped with a rational algebraic action of the group ${\operatorname{Aut}}_X({\mathcal{F}})$, compatible with the natural induced action on deformations of ${\mathcal{F}}$ and with the adjoint action on the graded vector space ${{\operatorname{Ext}}}^*_X({\mathcal{F}},{\mathcal{F}})$. The second step is to show that the above DG-Lie algebra $L$ can be chosen as the semidirect product of the Lie algebra ${{\operatorname{Ext}}}^0_X({\mathcal{F}},{\mathcal{F}})$ and a DG-Lie algebra $K$ concentrated in strictly positive degrees. By general facts $L$ and $K$ have the same Kuranishi family, hence $L$ is quadratic iff $K$ is quadratic iff $K$ is formal. The last step is to use the reductivity of ${\operatorname{Aut}}_X({\mathcal{F}})$ and some argument of [@Man4] for proving that $K$ is formal if and only if $L$ is formal.
The construction of an equivariant representative of derived endomorphisms requires some care, since in general the rationality of (infinite dimensional) representations is not preserved by ${\operatorname{Hom}}$ functors: for instance, given a rational representation $V$ of the algebraic group ${\operatorname{GL}}_1({\mathbb{C}})={\mathbb{C}}^*$, the dual representation ${\operatorname{Hom}}_{{\mathbb{C}}}(V,{\mathbb{C}})$ is rational if and only if $V$ has a finite number of isotypic components. This suggests to restrict our attention to rational finitely supported representations: a detailed introduction to finitely supported representations is given in Section \[sec.finitelysupp\]; here we only anticipate that a semisimple representation is finitely supported if and only if it has a finite number of isotypic components.
This restriction forces to avoid injective resolutions of ${\mathcal{F}}$ and to work, following [@FM], with local projective resolutions with respect to a finite affine cover of $X$. Then at the end of Section \[sec.three\] we shall prove the following result.
\[thm.maintheorem3\] Let ${\mathcal{F}}$ be a coherent sheaf of finite projective dimension on a complex projective scheme $X$. Then there exists a DG-Lie algebra $L$, representing ${R\mspace{-2mu}\operatorname{Hom}}({\mathcal{F}},{\mathcal{F}})$, together with a DG-Lie action of ${\operatorname{Aut}}_X({\mathcal{F}})$ on $L$ (i.e., a morphism of groups ${\operatorname{Aut}}_X({\mathcal{F}})\to {\operatorname{Aut}}_{DG-Lie}(L)$) such that:
1. the action of ${\operatorname{Aut}}_X({\mathcal{F}})$ on every homogeneous component $L^i$, $i\in {\mathbb{Z}}$, is rational algebraic and finitely supported;
2. there exists an invariant Lie subalgebra $\mathfrak{g}\subset Z^0(L)$, canonically isomorphic to ${{\operatorname{Ext}}}^0_X({\mathcal{F}},{\mathcal{F}})$ via the projection $Z^0(L)\to H^0(L)$;
3. via the above canonical isomorphism $\mathfrak{g}\cong{{\operatorname{Ext}}}^0_X({\mathcal{F}},{\mathcal{F}})$ the adjoint action of $\mathfrak{g}$ on $L$ is the same as the infinitesimal action of ${\operatorname{Aut}}_X({\mathcal{F}})$.
It is interesting to point out that, as a consequence of Theorem \[thm.maintheorem3\], we have that if ${\operatorname{Aut}}_X({\mathcal{F}})$ is linearly reductive, then $L$ admits an equivariant Hodge decomposition and the usual Kuranishi argument, see e.g. [@NijRich64 Sec. 4] and [@GoMil2 Sec. 2], gives an equivariant Kuranishi family.
Finitely supported and rational representations {#sec.finitelysupp}
===============================================
Let $G$ be a group and ${\mathbb{K}\,}$ a fixed field. Recall (see e.g., [@procesi]) that a ${\mathbb{K}\,}$-linear representation $G\to GL(V,{\mathbb{K}\,})$, equivalently a left ${\mathbb{K}\,}[G]$-module, where $V$ is a (possibly infinite dimensional) vector space, is called:
- *trivial* if $gv=v$ for every $g\in G$, $v\in V$;
- *irreducible* if the only $G$-invariant subspaces of $V$ are $0,V$;
- *semisimple* if $V$ is a direct sum of irreducible representations.
Given two representations $V,W$ we shall denote by ${\operatorname{Hom}}_{{\mathbb{K}\,}}(V,W)^G\subseteq {\operatorname{Hom}}_{{\mathbb{K}\,}}(V,W)$ the vector subspace of $G$-equivariant linear maps. By definition, ${\operatorname{Hom}}_{{\mathbb{K}\,}}(V,W)^G$ is the set of morphisms from $V$ to $W$ in the category of ${\mathbb{K}\,}$-linear representations of $G$.
\[def.isotypic\] Denote by ${\mathbb{K}\,}[G]^\vee$ the set of isomorphism classes of irreducible representations of $G$. Given $\alpha\in {\mathbb{K}\,}[G]^\vee$, the *isotypic component* $V_\alpha$ of a representation $V$ is the sum of all irreducible $G$-invariant subspaces $H\subset V$ in the class $\alpha$.
Notice that if $H$ is an irreducible representation in the class $\alpha$, then $V_{\alpha}\not=0$ if and only if ${\operatorname{Hom}}_{{\mathbb{K}\,}}(H,V)^G\not=0$.
\[lem.sottomoduloisotipica\] In the above situation, if $\alpha\in {\mathbb{K}\,}[G]^\vee$, $K\subset V$ is an irreducible submodule and $K\cap V_\alpha\not=0$, then $K$ is in the class $\alpha$.
If $K\cap V_\alpha\not=0$, then there exists a finite number of irreducible submodules $H_1,\ldots,H_n$ in the class $\alpha$ such that $K\cap \sum H_i\not=0$. Thus there exists an index $1\le r\le n$ such that $$K\cap \sum_{i<r}H_i=0,\qquad K\cap \sum_{i\le r}H_i\not=0,$$ and then $K\subset \sum_{i\le r}H_i$. A fortiori $\sum_{i<r}H_i\not=\sum_{i\le r}H_i$ and the restrictions to $K$ and $H_r$ of the projection to the quotient give two non-trivial maps $$\xymatrix{&H_r\ar[d]^-{p}\\
K\ar[r]^-{i}&\dfrac{\;\sum_{i\le r}H_i\;}{\;\;\sum_{i<r}H_i\;\;}}$$ with $p$ surjective. Since both $K$ and $H_r$ are irreducible it follows that both $i$ and $p$ are isomorphisms, and hence $K$ belongs to the class $\alpha$.
\[lem.isotipicapertriviale\] In the above situation, if $W$ is a trivial representation and $M= V\otimes W$, then $M_{\alpha}= V_{\alpha}\otimes W$ for every class $\alpha\in {\mathbb{K}\,}[G]^\vee$.
If $H\subset V$ is an irreducible submodule in the class $\alpha$, then $H\otimes W$ is a direct sum of copies of $H$, therefore $H\otimes W\subset M_{\alpha}$ and this implies $V_{\alpha}\otimes W\subset M_{\alpha}$. Conversely, if $H\subset V\otimes W$ is a irreducible submodule in the class $\alpha$, we need to prove that $h\in V_{\alpha}\otimes W$ for every $h\in H$. Writing $h=\sum_{i=1}^n v_i\otimes w_i$ with $v_i\in V\setminus\{0\}$, $w_i\in W$ and $w_1,\ldots,w_n$ linearly independent, for every $i$ we can find a linear map $f_i\colon W\to {\mathbb{K}\,}$ such that $f(w_i)=1$ and $f(w_j)=0$ for $j\not=i$; the images of $H$ under the $G$-equivariant maps $$V\otimes W\to V,\qquad v\otimes w\mapsto f_i(w)v,$$ are non trivial and hence isomorphic to $H$. Therefore $v_i\in V_{\alpha}$ for every $i$ and $h\in V_{\alpha}\otimes W$.
The *support* $S_G(V)\subset {\mathbb{K}\,}[G]^\vee$ of a representation $V$ of $G$ is the set of isomorphism classes of irreducible representations $H$ such that ${\operatorname{Hom}}_{{\mathbb{K}\,}}(H,V/W)^G\not=0$ for some $G$-invariant subspace $W\subset V$. A representation $V$ is called *finitely supported* if its support is finite.
If $U\subset V$ is a $G$-invariant subspace then it is obvious that $S_G(V/U)\subseteq S_G(V)$ and it is easy to see that $S_G(U)\subseteq S_G(V)$: in fact if $W\subset U$ is an invariant subspace and $H$ an irreducible representation such that ${\operatorname{Hom}}_{{\mathbb{K}\,}}(H,U/W)^G\not=0$ then $U/W\subset V/W$ and therefore ${\operatorname{Hom}}_{{\mathbb{K}\,}}(H,V/W)^G\not=0$. Moreover $S_G(V\oplus W)=S_G(V)\cup S_G(W)$: it is clear that $S_G(V)\cup S_G(W)\subset S_G(V\oplus W)$; conversely if $V\oplus W\xrightarrow{p}U$ is a surjective morphism of representations and $K\subset U$ is irreducible, then either $K\subset p(V)$, and then $K\in S_G(V)$, or $K\subset U/p(V)$, and then $K\in S_G(W)$.
\[lem.calcolosupporto\] Let $V$ be a representation of $G$ and $\alpha\in {\mathbb{K}\,}[G]^\vee$ such that $V_{\alpha}\not=0$. Then $S_G(V)=\{\alpha\}\cup S_G(V/V_\alpha)$.
The only nontrivial inclusion to prove is $S_G(V)\subset \{\alpha\}\cup S_G(V/V_\alpha)$. Let $p\colon V\to W$ be a surjective morphism of representations and let $K\subset W$ be an irreducible submodule; we need to prove that if $K$ is not in the class $\alpha$ then its isomorphism class belongs to $S_G(V/V_\alpha)$. Since $W/p(V_{\alpha})$ is a quotient of $V/V_\alpha$, it is sufficient to prove that $K\not\subset p(V_{\alpha})$ and therefore that ${\operatorname{Hom}}_{{\mathbb{K}\,}}(K,W/p(V_{\alpha}))^G\not=0$.
By definition $V_{\alpha}=\sum_{i}H_i$, where every $H_i\subset V$ is irreducible in the class $\alpha$, thus $p(V_\alpha)=\sum_{i}p(H_i)$ is contained in the isotypic component $W_{\alpha}$. If $K\subset p(V_{\alpha})\subset W_\alpha$ then by Lemma \[lem.sottomoduloisotipica\] the isomorphism class of $K$ is $\alpha$.
Let $V$ be a finite dimensional representation and $W$ a trivial representation. Then $S_G(V\otimes W)=S_G(V)$ is finite.
We prove the result by induction on the dimension of $V$. If $V=0$ there is nothing to prove; if $V\not=0$, since $\dim V<\infty$ there exists an irreducible submodule $H\subset V$, and then also a nontrivial isotypic component $V_{\alpha}\subset V$. Denoting $M=V\otimes W$, by Lemma \[lem.isotipicapertriviale\] we have $M_{\alpha}=V_\alpha\otimes W$. By Lemma \[lem.calcolosupporto\] and the inductive assumption we have that $S_G(V/V_{\alpha}\otimes W)=S_G(V/V_{\alpha})$ is finite and then $$S_G(M)=\{\alpha\}\cup S_G(M/M_{\alpha})=\{\alpha\}\cup S_G(V/V_{\alpha}\otimes W)=
\{\alpha\}\cup S_G(V/V_{\alpha})=S_G(V)\,$$ is also finite.
The following result is clear.
\[lemma.finsup\] A representation $V$ is semisimple and finitely supported if and only if there exists a finite number of irreducible representations $H_1,\ldots,H_n$ and trivial representations $W_1,\ldots,W_n$ such that $$V=(H_1\otimes W_1)\oplus\cdots\oplus (H_n\otimes W_n)\,.$$
\[rmk.subobjectquotient\] Assume now that $G$ is an algebraic group over ${\mathbb{K}\,}$. Recall that a representation $\phi\colon G\to GL(V,{\mathbb{K}\,})$ is called *rational* if every vector $v\in V$ belongs to a finite dimensional $G$-invariant subspace $U\subset V$ and the group homomorphism $G\to GL(U,{\mathbb{K}\,})$ is algebraic, cf. [@procesi]. Notice that:
1. every irreducible rational representation is finite dimensional;
2. let $U\subset V$ be an invariant subspace of a representation, by general representation theory if $V$ is semisimple (resp.: rational), then also $U$ and $V/U$ are semisimple (resp.: rational);
3. if $V$ is a finite dimensional rational representation and $W$ is a trivial representation, then $V\otimes W$ is a rational representation that is finitely supported for every subgroup of $G$.
A rational and finitely supported model for ${R\mspace{-2mu}\operatorname{Hom}}({\mathcal{F}},{\mathcal{F}})$ {#sec.three}
=============================================================================================================
Preliminaries and notation
--------------------------
Our aim is to study derived endomorphisms of a quasi-coherent sheaf ${\mathcal{F}}$ on a projective scheme $X$ over a field ${\mathbb{K}\,}$ of characteristic $0$. Even if most of the following results hold under mild assumptions, for the sake of simplifying the exposition we decided to fix the above hypothesis on $X$ throughout all this section. We begin by introducing the geometric framework we shall work with, and by briefly recalling the results needed.
The main tool we will make use of is the category ${\mathbf{Mod}}(A_{{\boldsymbol{\cdot}}})$ of modules over (the diagram $A_{{\boldsymbol{\cdot}}}$ representing) the scheme $X$. Fix an open affine covering ${\mathcal{U}}=\{U_h\}$ for $X$, and consider its nerve $${\mathcal{N}}=\{ \alpha=\{h_0,\dots,h_k\}\,\vert\,U_{\alpha}=U_{h_0}\cap\dots \cap U_{h_k}\neq\emptyset \}$$ which carries a *degree function* $\deg\colon{\mathcal{N}}\to{\mathbb{N}}$ defined by $\deg(\{h_0,\dots,h_k\}) = k$; moreover ${\mathcal{N}}$ can be seen as a poset by setting $\alpha\leq\beta$ if and only if $\alpha\subseteq\beta$. Then to $X$ it is associated the diagram $A_{{\boldsymbol{\cdot}}}$ of commutative unitary ${\mathbb{K}\,}$-algebras defined as $$A_{{\boldsymbol{\cdot}}}\colon {\mathcal{N}}\to {\mathbf{Alg}}_{{\mathbb{K}\,}} \; ,\qquad \qquad \alpha\mapsto A_{\alpha}=\Gamma(U_{\alpha},{\mathcal{O}}_X)$$ where the map $A_{\alpha}\to A_{\beta}$ is induced by the inclusion $U_{\beta}={\operatorname{Spec}}(A_{\beta})\hookrightarrow {\operatorname{Spec}}(A_{\beta})=U_{\alpha}$ for any $\alpha\leq\beta$ in ${\mathcal{N}}$.
For every unitary commutative ring $A$ we shall denote by ${\operatorname{DGMod}}(A)$ the category of cochain complexes of $A$-modules, equipped with the projective model structure [@Hov99 Sec. 2.3].
For the moment we need neither the covering nor the nerve to be finite, so that we could keep working on an arbitrary ${\mathbb{K}\,}$-scheme $X$ which is only assumed to be separated, which is sufficient for $A_{{\boldsymbol{\cdot}}}$ to be well-defined (i.e. intersections of affines are affines).
\[def.pseudo-module\] An $A_{{\boldsymbol{\cdot}}}$**-module** $\mathcal{F}$ over $X$, with respect to the fixed covering ${\mathcal{U}}$, consists of the following data:
1. an object ${\mathcal{F}}_{\alpha}\in{\operatorname{DGMod}}(A_{\alpha})$, for every $\alpha\in{\mathcal{N}}$,
2. a morphism $f_{\alpha\beta}\colon {\mathcal{F}}_{\alpha}\otimes_{A_{\alpha}}A_{\beta}\to {\mathcal{F}}_{\beta}$ in ${\operatorname{DGMod}}(A_{\beta})$, for every $\alpha\leq\beta$ in ${\mathcal{N}}$,
satisfying the *cocycle condition* $f_{\beta\gamma}\circ\left(f_{\alpha\beta}\otimes{\operatorname{Id}}_{A_{\gamma}}\right) = f_{\alpha\gamma}$, for every $\alpha\leq\beta\leq\gamma$ in ${\mathcal{N}}$.
Similar notions were considered in [@EE; @FK; @GS; @Sto]. Taking advantage of the standard projective model structure on DG-modules, the category ${\mathbf{Mod}}(A_{{\boldsymbol{\cdot}}})$ has been endowed with a cofibrantly generated model structure, see [@FM Theorem 3.9], where weak equivalences and fibrations are detected pointwise. In order to work with quasi-coherent sheaves, we need a (homotopical) version of quasi-coherence for $A_{{\boldsymbol{\cdot}}}$-modules: ${\mathcal{F}}\in{\mathbf{Mod}}(A_{{\boldsymbol{\cdot}}})$ is called *quasi-coherent* if all the maps $f_{\alpha\beta}$ introduced above are quasi-isomorphisms, see [@FM Definition 3.12].
Now, denote by ${\operatorname{Ho}}({\mathbf{QCoh}}(A_{{\boldsymbol{\cdot}}}))$ the category of quasi-coherent $A_{{\boldsymbol{\cdot}}}$-modules localised with respect to the weak equivalences. Then there exists an equivalence of triangulated categories $$\Psi\colon {\operatorname{D}}({\mathbf{QCoh}}(X)) \to {\operatorname{Ho}}({\mathbf{QCoh}}(A_{{\boldsymbol{\cdot}}}))$$ where ${\operatorname{D}}({\mathbf{QCoh}}(X))$ denotes the unbounded derived category of quasi-coherent sheaves on $X$, see [@FM Theorem 5.7]. A partial result in this direction was previously proven in [@BF Proposition 2.28].
The functor $\Psi$ introduced above may be easily defined as follows. Given a complex ${\mathcal{G}}^{\ast}$ of quasi-coherent sheaves on $X$, with a little ambiguity of notation we shall denote by $\Psi{\mathcal{G}}^{\ast}$ the $A_{{\boldsymbol{\cdot}}}$-module defined by $(\Psi{\mathcal{G}}^{\ast})_{\alpha}={\mathcal{G}}^{\ast}(U_{\alpha})$, where for any $\alpha\leq\beta$ in ${\mathcal{N}}$ the map $$(\Psi{\mathcal{G}}^{\ast})_{\alpha}\otimes_{A_{\alpha}}A_{\beta} \to (\Psi{\mathcal{G}}^{\ast})_{\beta}$$ is the natural isomorphism given degreewise by the restriction maps of the sheaves ${\mathcal{G}}^k$, $k\in\mathbb{Z}$. Then define $$\Psi[{\mathcal{G}}^{\ast}] = [\Psi{\mathcal{G}}^{\ast}] \in {\operatorname{Ho}}({\mathbf{QCoh}}(A_{{\boldsymbol{\cdot}}}))$$ for any $[{\mathcal{G}}^{\ast}] \in {\operatorname{D}}({\mathbf{QCoh}}(X))$.
The aim of the next section is to describe ${R\mspace{-2mu}\operatorname{Hom}}({\mathcal{F}},{\mathcal{F}})$ for a given quasi-coherent sheaf, in terms of a cofibrant replacement of $\Psi{\mathcal{F}}$, where ${\mathcal{F}}$ has to be thought of as a complex concentrated in degree $0$.
Derived endomorphisms of complexes of locally free sheaves {#section.derivedEnd}
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Throughout this subsection we shall consider a fixed *bounded above* complex of locally free sheaves ${\mathcal{E}}^{\ast}$. The aim is to give an explicit description of derived endomorphisms ${R\mspace{-2mu}\operatorname{Hom}}({\mathcal{E}}^{\ast},{\mathcal{E}}^{\ast})$. Following [@FM], the idea is to replace the $A_{{\boldsymbol{\cdot}}}$-module $\Psi{\mathcal{E}}^{\ast}$ by a cofibrant replacement, whose endomorphisms in the homotopy category ${\operatorname{Ho}}({\mathbf{QCoh}}(A_{{\boldsymbol{\cdot}}}))\simeq {\operatorname{D}}({\mathbf{QCoh}}(X))$ will represent the desired derived endomorphisms of the given complex ${\mathcal{E}}^{\ast}$.
Take an open affine cover $\{U_h\}_{h\in H}$ and let ${\mathcal{N}}$ be its nerve. Fix a total order on $H$, and for each $\alpha\in{\mathcal{N}}$ denote by $\Delta_{\alpha}$ the oriented abstract simplicial complex whose faces are defined as the non-empty subsets of $\alpha$. The associated chain complex will be denoted by $C_{\ast}(\Delta_{\alpha})$; recall that for any $r\in{\mathbb{Z}}$ the rank of the free ${\mathbb{Z}}$-module $C_{r}(\Delta_{\alpha})$ counts the number of $r$-faces of $\Delta_{\alpha}$, i.e. ${\operatorname{rk}}\left(C_r(\Delta_{\alpha})\right)={\binom{\deg(\alpha)+1}{r+1}}$. More precisely: $$\begin{aligned}
C_r(\Delta_{\alpha})=\bigoplus_{\substack{\beta\leq\alpha \\ \vert\beta\vert =r+1}}{\mathbb{Z}}\cdot\beta &\longrightarrow \bigoplus_{\substack{\gamma\leq\alpha \\ \vert\gamma\vert =r}}{\mathbb{Z}}\cdot\gamma= C_{r-1}(\Delta_{\alpha}) \\
\beta=(i_0,\dots,i_r) &\mapsto \sum_{j=0}^r(-1)^j(i_0,\dots,\widehat{i_j},\dots,i_r)
\end{aligned}$$ Concerning its homology we have $H_0(C_{\ast}(\Delta_{\alpha};{\mathbb{Z}}))={\mathbb{Z}}$ and $H_r(C_{\ast}(\Delta_{\alpha};{\mathbb{Z}}))=0$ for every $r\neq 0$.
We now introduce the $A_{{\boldsymbol{\cdot}}}$-module $C^{\ast}_{{\boldsymbol{\cdot}}}({\mathcal{E}}^{\ast})$, which we will prove to be the above mentioned cofibrant replacement of $\Psi{\mathcal{E}}^{\ast}$. In order to avoid possible confusion we shall always denote by ${\boldsymbol{\cdot}}$ the dependence on ${\mathcal{N}}$ and respectively by $\ast$ the degrees of the complexes; moreover, following the standard notation we shall use labels on the top to denote degrees of *cochain* complexes and on the bottom for *chain* complexes. Therefore, for any $\alpha\in{\mathcal{N}}$ we can define the cochain complex $\left(\check{C}^{\ast}(\Delta_{\alpha}), \check{\partial}\right)$ as $$\check{C}^{r}(\Delta_{\alpha}) = C_{-r}(\Delta_{\alpha}) \; ; \qquad \qquad \check{\partial}^r = \partial_{-r}\colon \check{C}^{r}(\Delta_{\alpha}) \to \check{C}^{r+1}(\Delta_{\alpha})$$ and eventually $C^{\ast}_{\alpha}({\mathcal{E}}^{\ast})= \check{C}^{\ast}(\Delta_{\alpha})\otimes_{{\mathbb{Z}}}{\mathcal{E}}^{\ast}(U_{\alpha})$. By definition, the cohomology of $C^{\ast}_{\alpha}({\mathcal{E}}^{\ast})$ is non trivial only in degree zero: $H^0\left( C^{\ast}_{\alpha}({\mathcal{E}}^{\ast}) \right) \cong {\mathcal{E}}(U_{\alpha})$.
Our next goal is to prove that the $A_{{\boldsymbol{\cdot}}}$-module $C^{\ast}_{{\boldsymbol{\cdot}}}({\mathcal{E}}^{\ast})$ is cofibrant in ${\mathbf{Mod}}(A_{{\boldsymbol{\cdot}}})$. By [@FM Theorem 3.9], this is equivalent to prove that for every choice of $\alpha\in{\mathcal{N}}$ the natural *latching* map $$l_{\alpha}\colon \operatorname*{colim}_{\gamma<\alpha}\left(C^{\ast}_{\gamma}({\mathcal{E}}^{\ast})\otimes_{A_{\gamma}}A_{\alpha}\right)\to C^{\ast}_{\alpha}({\mathcal{E}}^{\ast})$$ is a cofibration of DG-modules over $A_{\alpha}$. To this aim, consider the short exact sequence $$0\to \operatorname*{colim}_{\gamma<\alpha} \check{C}^{\ast}(\Delta_{\gamma}) \xrightarrow{\iota_{\alpha}} \check{C}^{\ast}(\Delta_{\alpha}) \to K_{\alpha}\to 0$$ and notice that $K_{\alpha}={\operatorname{coker}}(\iota_{\alpha})$ is a cochain complex concentrated in degree $-\deg(\alpha)$: $K_{\alpha}~=~{\mathbb{Z}}[\deg(\alpha)]\cdot\alpha$. Now, the latching map $$\l_{\alpha} \colon \operatorname*{colim}_{\gamma<\alpha}\left(C^{\ast}_{\gamma}({\mathcal{E}}^{\ast})\otimes_{A_{\gamma}}A_{\alpha}\right) \cong \operatorname*{colim}_{\gamma<\alpha}\left( \check{C}^{\ast}(\Delta_{\gamma}) \right)\otimes_{{\mathbb{Z}}}{\mathcal{E}}^{\ast}(U_{\alpha}) \longrightarrow \check{C}^{\ast}(\Delta_{\alpha})\otimes_{{\mathbb{Z}}}{\mathcal{E}}^{\ast}(U_{\alpha}) = C^{\ast}_{\alpha}({\mathcal{E}}^{\ast})$$ equals $\iota_{\alpha}\otimes {\operatorname{Id}}_{{\mathcal{E}}^{\ast}(U_{\alpha})}$ and hence it has cofibrant cokernel (because we assumed ${\mathcal{E}}^{\ast}$ to be bounded above) and it is degreewise split injective, hence by [@Hov99 Lemma 2.3.6] it is a cofibration of differential graded $A_{\alpha}$-modules as required.
In order to show that $C^{\ast}_{{\boldsymbol{\cdot}}}({\mathcal{E}}^{\ast})$ is a cofibrant replacement of $\Psi{\mathcal{E}}^{\ast}$ in ${\mathbf{Mod}}(A_{{\boldsymbol{\cdot}}})$, we are only left with the proof of the existence of a trivial fibration $C^{\ast}_{{\boldsymbol{\cdot}}}({\mathcal{E}}^{\ast})\to \Psi{\mathcal{E}}^{\ast}$ of $A_{{\boldsymbol{\cdot}}}$-modules. To this aim, it is sufficient to consider the natural projections $C^{\ast}_{\alpha}({\mathcal{E}}^{\ast})\to H^0\left( C^{\ast}_{\alpha}({\mathcal{E}}^{\ast}) \right) = {\mathcal{E}}(U_{\alpha})$ for every $\alpha~\in~{\mathcal{N}}$, which by functoriality give the desired map of $A_{{\boldsymbol{\cdot}}}$-modules. Moreover, again by [@FM Theorem 3.9] the morphism $C^{\ast}_{{\boldsymbol{\cdot}}}({\mathcal{E}}^{\ast})\to \Psi{\mathcal{E}}^{\ast}$ is a trivial fibration of $A_{{\boldsymbol{\cdot}}}$-modules being pointwise a surjective quasi-isomorphism. We conclude that the derived endomorphisms ${R\mspace{-2mu}\operatorname{Hom}}_X({\mathcal{E}}^{\ast},{\mathcal{E}}^{\ast})$ are represented by the DG-Lie algebra ${\operatorname{Hom}}^{\ast}_{A_{{\boldsymbol{\cdot}}}}\left(C^{\ast}_{{\boldsymbol{\cdot}}}({\mathcal{E}}^{\ast}),C^{\ast}_{{\boldsymbol{\cdot}}}({\mathcal{E}}^{\ast})\right)$, cf. [@FM Theorem 6.4].
A locally free ${\operatorname{Aut}}_X({\mathcal{F}})$-equivariant resolution for ${\mathcal{F}}$ {#subsec.locfreeres}
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Throughout this subsection ${\mathcal{F}}$ will be a fixed coherent sheaf of finite projective dimension on $X$. Moreover, we shall denote by ${\operatorname{Aut}}_{X}({\mathcal{F}})$ the group of automorphisms of ${\mathcal{F}}$.
Recall that for a given global section $s\in H^0(X,{\mathcal{F}})$ there exists a unique map of ${\mathcal{O}}_X$-modules $\varphi\colon{\mathcal{O}}_X\to{\mathcal{F}}$ defined by $f\mapsto fs$. Notice that $s=\varphi(1)$. Under our hypothesis on $X$ and ${\mathcal{F}}$ Serre’s theorem applies, see e.g. [@Har Theorem 5.17], so that for $n\in{\mathbb{N}}$ sufficiently large the sheaf ${\mathcal{F}}(n)$ is generated by global sections, i.e. the map $H^0(X,{\mathcal{F}}(n)) \otimes {\mathcal{O}}_X \to {\mathcal{F}}(n)$ is surjective, where $H^0(X,{\mathcal{F}}(n))$ has finite dimension. Hence, since tensoring by ${\mathcal{O}}_X(-n)$ is an exact functor, we obtain a surjective morphism $$\varphi\colon H^0(X,{\mathcal{F}}(n)) \otimes {\mathcal{O}}_X(-n) \to {\mathcal{F}}$$ of coherent ${\mathcal{O}}_X$-modules, which stays surjective when computed on any affine $U\subseteq X$: $$\varphi_U\colon H^0(X,{\mathcal{F}}(n)) \otimes {\mathcal{O}}_X(-n)(U) \to {\mathcal{F}}(U) \to 0 \; ; \qquad \qquad \sum_i s_i\otimes f_i \mapsto \sum_i s_i\vert_U \cdot f_i \; .$$
In particular, we have an injective homomorphism of groups ${\operatorname{Aut}}_{X}({\mathcal{F}})\to {\operatorname{GL}}(H^0(X,{\mathcal{F}}(n)))$ together with a short exact sequence of coherent sheaves $$0\to {\mathcal{G}}\to H^0(X,{\mathcal{F}}(n))\otimes {\mathcal{O}}_X(-n)\xrightarrow{\varphi} {\mathcal{F}}\to 0$$ and then ${\operatorname{Aut}}_{X}({\mathcal{F}})$ is the stabilizer of ${\mathcal{G}}$ under the action of ${\operatorname{GL}}(H^0(X,{\mathcal{F}}(n)))$ on the sheaf $H^0(X,{\mathcal{F}}(n))\otimes {\mathcal{O}}(-n)$. Notice that the action is trivial on ${\mathcal{O}}_X(-n)(U)$ since each element of ${\operatorname{Aut}}_{X}({\mathcal{F}})$ is ${\mathcal{O}}_X$-linear and $\varphi_U$ is ${\operatorname{Aut}}_{X}({\mathcal{F}})$-equivariant.
Notice that the above construction is clearly functorial in the sense that if $\alpha\leq\beta$ in the nerve ${\mathcal{N}}$, then we have a commutative square: $$\xymatrix{ H^0(X,{\mathcal{F}}(n))\otimes {\mathcal{O}}(-n)(U_{\alpha})\ar@{->}[d] \ar@{->}[r] & {\mathcal{F}}(U_{\alpha})\ar@{->}[d] \\
H^0(X,{\mathcal{F}}(n))\otimes {\mathcal{O}}(-n)(U_{\beta}) \ar@{->}[r] & {\mathcal{F}}(U_{\beta}) }$$ where the horizontal arrows are surjective and the vertical arrows are the restriction maps.
\[lemma.coherentRFS\] Let ${\mathcal{F}}$ be a coherent sheaf on a projective ${\mathbb{K}\,}$-scheme $X$. Then for every open subset $U\subset X$, the space of sections $\Gamma(U,{\mathcal{F}})$ is a rational representation of ${\operatorname{Aut}}_X({\mathcal{F}})$ that is finitely supported for every subgroup $G\subseteq{\operatorname{Aut}}_X({\mathcal{F}})$.
If $U=\bigcup U_i$ is a finite open affine cover we have $\Gamma(U,{\mathcal{F}})\subset \oplus_i \Gamma(U_i,{\mathcal{F}})$ and then by Remark \[rmk.subobjectquotient\] it is not restrictive to prove the statement assuming $U$ to be an open affine subset. We have already proved that there exists $n\in{\mathbb{N}}$ sufficiently large such that for every affine open subset $U\subset X$ there exists a surjective equivariant map $$H^0(X,{\mathcal{F}}(n))\otimes \Gamma(U,{\mathcal{O}}(-n))\to \Gamma(U,{\mathcal{F}})\,.$$ The conclusion follows by Remark \[rmk.subobjectquotient\].
\[prop.resolution\] Let $X$ be a projective ${\mathbb{K}\,}$-scheme, and let ${\mathcal{F}}$ be a coherent sheaf of finite projective dimension on $X$. Then there exists a finite locally free ${\operatorname{Aut}}_{X}({\mathcal{F}})$-equivariant resolution ${\mathcal{E}}^{\ast}\to {\mathcal{F}}$ such that for any open subset $U\subseteq X$ the complex $\Gamma(U,{\mathcal{E}}^{\ast})$ is a degreewise rational representation that is finitely supported for every subgroup $G\subseteq {\operatorname{Aut}}_{X}({\mathcal{F}})$.
Choose $n$ sufficiently large giving a short exact sequence $$0\to {\mathcal{F}}_1 \to H^0(X,{\mathcal{F}}(n)) \otimes {\mathcal{O}}_X(-n) \xrightarrow{\varphi} {\mathcal{F}}\to 0$$ where each map is ${\operatorname{Aut}}_{X}({\mathcal{F}})$-equivariant. Since ${\mathcal{F}}_1$ is a coherent sheaf, we can reproduce the same argument and by the hypothesis on the projective dimension of ${\mathcal{F}}$ we obtain a resolution of ${\mathcal{F}}$ of the form $$\begin{gathered}
\quad 0\to {\mathcal{F}}_{k} \to H^0(X,{\mathcal{F}}_{k-1}(n_{k-1})) \otimes {\mathcal{O}}_X(-n_{k-1}) \to \cdots\\
\cdots \to H^0(X,{\mathcal{F}}(n_0)) \otimes {\mathcal{O}}_X(-n_0) \to {\mathcal{F}}\to 0\quad\end{gathered}$$ where ${\mathcal{F}}_k$ is locally free and each map is ${\operatorname{Aut}}_{X}({\mathcal{F}})$-equivariant. To conclude, applying the functor $\Gamma(U,-)$ to the above resolution we obtain a complex of finitely supported representations because of Lemma \[lemma.coherentRFS\].
Notice that the associated infinitesimal action of ${\operatorname{Aut}}_{X}({\mathcal{F}})$ on ${\mathcal{E}}^*$ gives a morphism of Lie algebras ${{\operatorname{Ext}}}^0_X({\mathcal{F}},{\mathcal{F}})\to Z^0({\operatorname{Hom}}_X^*({\mathcal{E}}^*,{\mathcal{E}}^*))$; in other words, every endomorphism of ${\mathcal{F}}$ lifts functorially to an endomorphism of the complex of sheaves ${\mathcal{E}}^*$.
\[lemma.HomClosure\] Let ${\mathcal{F}}$ be a coherent sheaf and ${\mathcal{E}}$ a locally free sheaf on a projective scheme $X$ over the field ${\mathbb{K}\,}$. Then for every open affine subset $U\subset X$, the ${\mathcal{O}}_X(U)$-module $${\operatorname{Hom}}_U({\mathcal{F}},{\mathcal{E}})={\operatorname{Hom}}_{{\mathcal{O}}_X(U)}({\mathcal{F}}(U),{\mathcal{E}}(U))$$ is a rational representation of ${\operatorname{Aut}}_{X}({\mathcal{F}})\times {\operatorname{Aut}}_{X}({\mathcal{E}})$ that is finitely supported with respect any subgroup.
We have already seen that there exists a sufficiently large $n\in{\mathbb{N}}$ such that the sheaves ${\mathcal{F}}(n)$ and ${\mathcal{E}}^{\vee}(n)={\operatorname{\mathcal H}\!\!om}_X({\mathcal{E}},{\mathcal{O}}_X)(n)$ are generated by global sections, i.e. there exist surjective morphisms of sheaves $$H^0(X,{\mathcal{F}}(n))\otimes {\mathcal{O}}(-n)\to {\mathcal{F}},\qquad
H^0(X,{\mathcal{E}}^{\vee}(n))\otimes {\mathcal{O}}(-n)\to {\mathcal{E}}^\vee\to 0\,.$$ Dualizing the second sequence we get an injective morphism of ${\mathcal{O}}_X$-modules $$0\to {\mathcal{E}}\to H^0(X,{\mathcal{E}}^{\vee}(n))^\vee\otimes {\mathcal{O}}(n)$$ and then for every open affine subset $U\subseteq X$ we get two exact sequences $$\Gamma(X,{\mathcal{F}}(n))\otimes \Gamma(U,{\mathcal{O}}(-n))\to {\mathcal{F}}(U)\to 0,\qquad
0\to{\mathcal{E}}(U)\subset H^0(X,{\mathcal{E}}^{\vee}(n))^\vee\otimes \Gamma(U,{\mathcal{O}}(n))\,.$$ To conclude it is sufficient to observe that ${\operatorname{Hom}}_{{\mathcal{O}}_X(U)}({\mathcal{F}}(U),{\mathcal{E}}(U))$ is a subrepresentation of ${\operatorname{Hom}}_{{\mathbb{K}\,}}({\mathcal{F}}(U),{\mathcal{E}}(U))$ which in turn is a subrepresentation of the rational representation $$\begin{split}
&{\operatorname{Hom}}_{{\mathbb{K}\,}}\left(\Gamma(X,{\mathcal{F}}(n)))\otimes \Gamma(U,{\mathcal{O}}(-n)), \vphantom{I^1_1}
{\operatorname{Hom}}_X({\mathcal{E}},{\mathcal{O}}(n))^\vee\otimes \Gamma(U,{\mathcal{O}}(n))\right)\\
&\qquad={\operatorname{Hom}}_{{\mathbb{K}\,}}(\Gamma(X,{\mathcal{F}}(n)),H^0(X,{\mathcal{E}}^{\vee}(n))^\vee)\otimes {\operatorname{Hom}}_{{\mathbb{K}\,}}(\Gamma(U,{\mathcal{O}}(-n)),\Gamma(U,{\mathcal{O}}(n)))\,.
\end{split}$$
\[theorem:goodmodel\] Let $X$ be a projective ${\mathbb{K}\,}$-scheme, ${\mathcal{F}}$ a coherent sheaf of finite projective dimension on $X$. Consider a resolution ${\mathcal{E}}^{\ast}\to {\mathcal{F}}$ as in Proposition \[prop.resolution\]. Then the derived endomorphisms of ${\mathcal{F}}$ are represented by the DG-Lie algebra ${\operatorname{Hom}}^{\ast}_{A_{{\boldsymbol{\cdot}}}}(C_{{\boldsymbol{\cdot}}}^{\ast}({\mathcal{E}}^{\ast}),C_{{\boldsymbol{\cdot}}}^{\ast}({\mathcal{E}}^{\ast}))$. Moreover, each degree of such complex is a rational representation with respect to the inherited ${\operatorname{Aut}}_{X}({\mathcal{F}})$-action, and it is finitely supported for every subgroup.
We are only left with the proof of the last part of the statement, since it was proven in Section \[section.derivedEnd\] that ${\operatorname{Hom}}^{\ast}_{A_{{\boldsymbol{\cdot}}}}(C_{{\boldsymbol{\cdot}}}^{\ast}({\mathcal{E}}^{\ast}),C_{{\boldsymbol{\cdot}}}^{\ast}({\mathcal{E}}^{\ast}))$ represents ${R\mspace{-2mu}\operatorname{Hom}}({\mathcal{F}},{\mathcal{F}})$. First notice that for every $k\in{\mathbb{Z}}$ we have $${\operatorname{Hom}}^{k}_{A_{{\boldsymbol{\cdot}}}}(C_{{\boldsymbol{\cdot}}}^{\ast}({\mathcal{E}}^{\ast}),C_{{\boldsymbol{\cdot}}}^{\ast}({\mathcal{E}}^{\ast})) \subseteq \prod_{i\in{\mathbb{Z}}} {\operatorname{Hom}}_{{\mathbb{K}\,}}(C_{{\boldsymbol{\cdot}}}^{i}({\mathcal{E}}^{\ast}),C_{{\boldsymbol{\cdot}}}^{i+k}({\mathcal{E}}^{\ast}))$$ and that by Remark \[rmk.subobjectquotient\] and Lemma \[lemma.HomClosure\] rational and finitely supported representations are closed under taking subobjects, and homomorphisms. Now recall that ${\mathcal{E}}^{\ast}$ is bounded by assumption, and since the scheme is assumed to be projective then the covering can be chosen to be finite, so that $C_{{\boldsymbol{\cdot}}}^{\ast}({\mathcal{E}}^{\ast})$ is bounded too. Hence the product above is finite and the statement follows.
Proof of Theorem \[thm.maintheorem3\]. {#proof-of-theoremthm.maintheorem3. .unnumbered}
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Finally, combining the above results we can easily prove Theorem \[thm.maintheorem3\]. To this aim, by Theorem \[theorem:goodmodel\] it is sufficient to show that the DG-Lie algebra $$L={\operatorname{Hom}}^{\ast}_{A_{{\boldsymbol{\cdot}}}}(C_{{\boldsymbol{\cdot}}}^{\ast}({\mathcal{E}}^{\ast}),C_{{\boldsymbol{\cdot}}}^{\ast}({\mathcal{E}}^{\ast}))$$ satisfies (2) and (3) of Theorem \[thm.maintheorem3\]. We already noticed that endomorphisms of ${\mathcal{F}}$ lift functorially to endomorphisms of the complex ${\mathcal{E}}^*$ and hence of the $A_{{\boldsymbol{\cdot}}}$-module $\Psi{\mathcal{E}}^*$. Then it is clear from the explicit construction of the cofibrant replacement $C_{{\boldsymbol{\cdot}}}^{\ast}({\mathcal{E}}^{\ast})\to \Psi{\mathcal{E}}^*$ that every endomorphism of $\Psi{\mathcal{E}}^*$ lifts canonically to an endomorphism of $C_{{\boldsymbol{\cdot}}}^{\ast}({\mathcal{E}}^{\ast})$. In conclusion we have an injective morphism of Lie algebras $${\operatorname{Hom}}_X({\mathcal{F}},{\mathcal{F}})={{\operatorname{Ext}}}^0_X({\mathcal{F}},{\mathcal{F}})\to Z^0({\operatorname{Hom}}^{\ast}_{A_{{\boldsymbol{\cdot}}}}(C_{{\boldsymbol{\cdot}}}^{\ast}({\mathcal{E}}^{\ast}),C_{{\boldsymbol{\cdot}}}^{\ast}({\mathcal{E}}^{\ast}))$$ and we can choose its image as the required $\mathfrak{g}$.
Review of $L_\infty[1]$ algebras and formality
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$L_\infty[1]$ algebras
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In this subsection we review some basic facts and notations concerning $L_\infty[1]$ algebras, following the paper [@Man4], to which we refer for more details.
Given a graded ${\mathbb{K}\,}$-vector space $V=\oplus_{i\in{\mathbb{Z}}} V^i$, we denote by $S^c(V)=\oplus_{n\geq0}V^{\odot n}$ the symmetric coalgebra over $V$ (see [@Man4 §4]). Denoting by $p:S^c(V)\to V$ the natural projection, corestriction induces an isomorphism of graded spaces $${\operatorname{Coder}}(S^c(V))\to{\operatorname{Hom}}(S^c(V),V)=\prod_{n\geq0}{\operatorname{Hom}}(V^{\odot n},V)\colon Q\to p\circ Q= (q_0,q_1,\ldots,q_n,\ldots)$$ (see [@Man4 Proposition 4.2] for an explicit description of the inverse), where ${\operatorname{Coder}}(S^c(V))$ is the graded Lie algebra of coderivations of $S^c(V)$. We call the components $q_n:V^{\odot n}\to V$ of $Q\in{\operatorname{Coder}}(S^c(V))$ under the corestriction isomorphisms the *Taylor coefficients* of $Q$, and $q_0$, $q_1$ respectively the *constant* and the *linear part* of $Q$. Via the above isomorphism, the natural commutator bracket on ${\operatorname{Coder}}(S^c(V))$ induces a Lie bracket on ${\operatorname{Hom}}(S^c(V),V)$, which is called the *Nijenhuis-Richardson bracket* and denoted by $[-,-]_{NR}$.
An $L_\infty[1]$ algebra structure on $V$ is a degree $+1$ coderivation $$Q\in{\operatorname{Coder}}^1(S^c(V)),\qquad q_0=0,\qquad Q\circ Q=0,$$ with vanishing constant part and squaring to zero. In particular, the linear part $q_1:V\to V$ squares to zero: the complex $(V,q_1)$ is called the *tangent complex* of the $L_\infty[1]$ algebra $(V,Q)$. The $L_\infty[1]$ algebra $(V,Q)$ is said to be *minimal* if $q_1=0$.
Given a second $L_\infty[1]$-algebra $(W,R)$, an $L_\infty[1]$ morphism $F\colon V\dashrightarrow W$ from $(V,Q)$ to $(W,R)$ is a morphism of (counital, coaugmented) DG coalgebras $F\colon (S^c(V),Q)\to (S^c(W),R)$. As a morphism of graded (coaugmented, i.e., $F(1)=1$) coalgebras, $F$ is completely determined by its corestriction $$p\circ F = (0,f_1,\ldots,f_n,\ldots)\in{\operatorname{Hom}}^0(S^c(V),W)=\prod_{n\geq0}{\operatorname{Hom}}^0(V^{\odot n},W).$$ As for coderivations, we call the $f_n:V^{\odot n}\to W$ the *Taylor coefficients* of $F$, and $f_1:V\to W$ the *linear part* of $F$. The compatibility with the codifferentials translates into a bunch of identities involving the Taylor coefficients $f_i,q_j,r_k$: we won’t need to write these down explicitly. In particular, $f_1:(V,q_1)\to (W,r_1)$ is a morphism between the tangent complexes: we say that $F$ is a *weak equivalence*, if $f_1$ is a quasi-isomorphism. Finally, an $L_\infty[1]$ morphism $F\colon V\dashrightarrow W$ is *strict* if the only non-vanishing Taylor coefficient is the linear one $f_1$.
The category of $L_\infty[1]$ algebras is equivalent, up to a shift, to the category of $L_\infty$ algebras, or strong homotopy Lie algebras, see [@bibid], and in particular contains the usual category of DG Lie algebras as a subcategory. More precisely there is a bijective correspondence between $L_\infty[1]$ algebra structures on a graded space $V$ and $L_\infty$ algebra structures on its suspension $V[-1]$: under this correspondence, DG Lie algebra structures correspond to those $L_\infty[1]$ algebra structures $Q$ such that $q_n=0$ for $n\ge3$.
\[rem: r2\] Given an $L_\infty[1]$ algebra $(V,Q)$, the relation $Q\circ Q$ translates into a bunch of relations involving the Taylor coefficients $q_n$, which in particular imply that the quadratic bracket $q_2$ descends to a quadratic bracket $r_2\colon H(V,q_1)^{\odot 2}\to H(V,q_1)$ on the tangent cohomology, and the latter satisfies $[r_2,r_2]_{NR}=0$ (in other words, it corresponds to a graded Lie algebra structure on $H(V,q_1)[-1]$).
A very important and useful fact about $L_\infty[1]$ algebra structures is that they can be transferred along contractions. For a proof of the following result we refer to [@HueSta; @fuka; @BerglundHPT].
\[th: transfer\] Let $(V,Q)$ be an $L_\infty[1]$ algebra, and $(f,g,h)$ be a contraction of the tangent complex $(V,q_1)$ onto some complex $(W,r_1)$, i.e., $f:W\to V$ and $g:V\to W$ are DG maps and $h\colon V\to V[-1]$ a contracting homotopy such that the following conditions are satisfied:$$gf = {\operatorname{id}}_W,\qquad fg = {\operatorname{id}}_V + q_1h + hq_1,\qquad gh = h^2 = fh = 0.$$ Then there is an induced $L_\infty[1]$ structure $R$ on $W$ with linear part $r_1$, together with $L_\infty[1]$ morphisms $F\colon W\dashrightarrow V$ and $G\colon V\dashrightarrow W$ with linear parts $f_1=f$ and $g_1=g$ respectively. The higher Taylor coefficients $r_n,f_n,g_n$, $n\ge2$, are recursively determined by the Taylor coefficients $q_n$, $n\ge2$, and the contraction data $(f,g,h)$ (see e.g. [@descent Theorem 1.9] for explicit recursive formulas).
An immediate and fundamental consequence of the above theorem is the existence (and, with a little more work, uniqueness) of minimal models.
Given an $L_\infty[1]$ algebra $(V,Q)$, a *minimal model* of $(V,Q)$ is the datum of a minimal $L_\infty[1]$ algebra $(W,R)$ together with a weak equivalence $F:W\dashrightarrow V$ of $L_\infty[1]$ algebras. General structure theory of $L_\infty[1]$ algebras says that minimal models always exist, and are well defined up to $L_\infty[1]$ isomorphisms: furthermore, two $L_\infty[1]$ algebras are weakly equivalent if and only if they have isomorphic minimal models.
\[rem:hodge\] In order to obtain an explicit minimal model of $(V,Q)$ it is sufficient to choose an abstract *Hodge decomposition* for the complex $(V,q_1)$. By this we mean the choice of a splitting of the sequence of inclusion $B^i(V)\subset Z^i(V)\subset V^i$, or in other words, of vector space decompositions $Z^i(V)=B^i(V)\oplus H^i$, $V^i=Z^i(V)\oplus W^i=B^i(V)\oplus H^i\oplus W^i$, for all $i\in{\mathbb{Z}}$. We notice that the differential $q_1$ vanishes on $H:=\oplus_{i\in{\mathbb{Z}}}H^i$, and it restricts to an isomorphism from $W[-1]=\oplus_{i\in{\mathbb{Z}}}W^{i-1}$ to $B(V)=\oplus_{i\in{\mathbb{Z}}}B^i(V)$. For any such a choice, there is a canonically induced contraction $(f,g,h)$ of $(V,q_1)$ onto $(H,0)$: the inclusion $f:H\to V$ and the projection $g\colon V\to H$ are induced by the decomposition $V=B(V)\oplus H\oplus W$, and the contracting homotopy is the composition $h\colon V\twoheadrightarrow B(V)\xrightarrow{(q_1)^{-1}} W[-1]\hookrightarrow V[-1]$. Via homotopy transfer along this contraction, there is an induced minimal $L_\infty[1]$ algebra structure $(H,R)=(H,0,r_2,\ldots,r_n,\ldots)$ on $H\cong H(V,q_1)$, together with quasi-inverses weak equivalences $F\colon H\dashrightarrow V$, $G\colon V\dashrightarrow H$ such that $GF$ is the identity on $H$. We notice that the quadratic bracket $r_2$ identifies with the one from Remark \[rem: r2\].
An $L_{\infty}[1]$ algebra is called homotopy abelian if it is weakly equivalent to a graded vector space, considered as an $L_{\infty}[1]$ algebra with trivial bracket: it is plain that an $L_{\infty}[1]$ algebra is homotopy abelian if and only if its minimal model carries the trivial $L_{\infty}[1]$ structure.
\[lem.4sette\] Let $F\colon (V,q_1,\ldots)\dashrightarrow (W,r_1,\ldots)$ be a morphism of $L_{\infty}[1]$ algebra with $W$ homotopy abelian. Then there exists a minimal model $(H^*(V,q_1),0,s_2,\ldots)$ for $V$ such that the image of every map $s_r$ is contained in the kernel of $f_1\colon H^*(V,q_1)\to H^*(W,r_1)$.
It is not restrictive to assume both $V$ and $W$ minimal, i.e., $q_1=r_1=0$, and then $r_n=0$ for every $n$ since $W$ is assumed homotopy abelian. Denote by $U\subset W$ the image of $f_1$ and choose a projection $\pi\colon W\to U$. Then the composition $\pi F\colon V\dashrightarrow U$ is an $L_{\infty}[1]$ morphism with linear component $\pi f_1$ surjective. By general theory of $L_{\infty}[1]$ algebra (for a proof see e.g. [@yukawate Lemma 7.2]) there exists an $L_{\infty}[1]$ isomorphism $G\colon (H,0,s_2,\ldots) \dashrightarrow V$ such that the composition $\pi FG\colon (H,0,s_2,\ldots) \dashrightarrow (U,0,0,\ldots)$ is strict and this implies that the image of every $s_r$ is contained in the kernel of the linear part of $\pi FG$.
We finally come to the main object of interest in this paper, namely, formal $L_\infty[1]$ algebras.
Given an $L_\infty[1]$ algebra $(V,Q)$, we denote by $r_2:H(V,q_1)^{\odot 2} \to H(V,q_1)$ the induced bracket on tangent cohomology, as in Remark \[rem: r2\]. $V$ is said to be *formal* if there exists a weak equivalence $$F\colon (V,Q)=(V,q_1,q_2,q_3,\ldots,q_n,\ldots)\to(H(V,q_1),r_2)=(H(V,q_1),0,r_2,0,\ldots,0,\ldots)$$ of $L_\infty[1]$ algebras, or in other words, if every minimal model of $(V,Q)$ is $L_\infty[1]$ isomorphic to $(H(V,q_1),r_2)$.
Formality versus quadraticity {#subsec.formalityquadra}
-----------------------------
We denote by $\mathbf{Art}_{{\mathbb{K}\,}}$ the category of local Artin ${\mathbb{K}\,}$-algebras with residue field equal to ${\mathbb{K}\,}$. To any $L_\infty[1]$ algebra $V$ is associated a deformation functor ${\operatorname{Def}}_V\colon{\mathbf{Art}}_{{\mathbb{K}\,}}\to{\mathbf{Set}}$, sending $A\in{\mathbf{Art}}_{{\mathbb{K}\,}}$ with maximal ideal ${\mathfrak{m}_{A}}$ to the set of Maurer-Cartan elements in $V\otimes{\mathfrak{m}_{A}}$ modulo homotopy equivalence, see [@ManRendiconti] for details. Moreover, weakly equivalent $L_\infty[1]$ algebras yield isomorphic deformation functors. According to a well known general philosophy[^1], over a field of characteristic zero every deformation problem is controlled in the above manner by some weak equivalence type of $L_\infty[1]$ algebras, or equivalently, by some isomorphism type of minimal $L_\infty[1]$ algebras. Knowing a controlling $L_\infty[1]$ algebra, it is easy to recover several important features of the deformation problem at hand: for instance, the tangent space $T^1{\operatorname{Def}}_V$ is isomorphic to $H^0(V,q_1)$, and there is a complete obstruction theory with values in $H^1(V,q_1)$.
When $\dim\,H^0(V,q_1)<+\infty$, the deformation functor ${\operatorname{Def}}_V$ satisfies conditions (H1), (H2) and (H3) from Schlessinger’s paper [@Sch Theorem 2.11], and in particular, it admits a *hull*. In order to read this off directly from $(V,Q)$ we set $H=H(V,q_1)$ and fix a minimal model $(H,R)$ of $(V,Q)$. We denote by $(H^0)^\vee,(H^1)^\vee$ the dual vector spaces. As $H^0$ is finite dimensional, the Taylor coefficients $r_n:(H^0)^{\odot n}\to H^1$ induce, via transposition, maps $r^\vee_n:(H^1)^{\vee}\to((H^0)^\vee)^{\odot n}$, which together assemble to a map $r^\vee=r_2^\vee+\cdots+r_n^\vee+\cdots:(H^1)^\vee\to\widehat{S}((H^0)^\vee)=\prod_{n\geq0}((H^0)^\vee)^{\odot n}$, where we denote by $\widehat{S}((H^0)^\vee)$ the completed symmetric algebra over $(H^0)^\vee$. If $\dim\,H^0=n$, we can identify $\widehat{S}((H^0)^\vee)$ with the usual algebra of formal power series ${\mathbb{K}\,}[[x_1,\ldots,x_n]]$: moreover, we denote by $\mathfrak{m}\subset{\mathbb{K}\,}[[x_1,\ldots,x_n]]$ the maximal ideal, and by $I\subset\mathfrak{m}^2$ the ideal generated by the image of $r^\vee$.
In the above setup, we call the local noetherian complete ${\mathbb{K}\,}$-algebra $$A =\frac{{\mathbb{K}\,}[[x_1,\ldots,x_n]]}{I}$$ the *Kuranishi algebra* of $(H,R)$. A bit improperly, we shall also refer to $A$ as the Kuranishi algebra of $(V,Q)$, but notice that it is determined by $(V,Q)$ only up to a non-canonical isomorphism.
The following result is essentially shown in [@fuka].
Given an $L_\infty[1]$ algebra $(V,Q)$ with $\dim\,H^0(V,q_1)<+\infty$, the associated Kuranishi algebra is a hull for ${\operatorname{Def}}_V$.
We denote by $\widehat{{\mathbf{Art}}}_{{\mathbb{K}\,}}$ the category of local noetherian complete ${\mathbb{K}\,}$-algebras with residue field equal to ${\mathbb{K}\,}$. Every such an algebra, up to isomorphism, can be presented as a quotient $$A=\frac{{\mathbb{K}\,}[[x_1,\ldots,x_n]]}{(f_1,\ldots,f_m)}$$ where every $f_i$ has multiplicity $\mu(f_i)\ge 2$. Here $n$ is the embedding dimension of $A$ (the dimension of the Zariski tangent space).
An algebra $A\in\widehat{{\mathbf{Art}}}_{{\mathbb{K}\,}}$ is called *quadratic* if it is isomorphic to an algebra of type ${\mathbb{K}\,}[[x_1,\ldots,x_n]]/I$, where the ideal $I$ is generated by homogeneous polynomials of degree two. Given an $L_\infty[1]$ algebra $(V,Q)$ such that $\dim\,H^0(V,q_1)<+\infty$, we say that it satisfies the *quadraticity property* if the associated Kuranishi algebra is quadratic.
Obviously, by homotopy invariance of the Kuranishi algebra, if an $L_\infty[1]$ algebra $(V,Q)$ is formal it satisfies the quadraticity property. The converse is in general not true: formality is a stronger property, putting strong constraints on the full derived deformation functor associated to $V$, while the Kuranishi algebra only remembers the classical part ${\operatorname{Def}}_V$. On the other hand, in several situations the quadraticity property might be easier to verify: for instance, we have the following result, which follows from [@MartPadova Theorem 2.11, Proposition 2.14 and Theorem 2.16].
Let $F:(V,Q)\dashrightarrow(W,R)$ be and $L_\infty[1]$ morphism, and assume that $H^0(f_1):H^0(V,q_1)\to H^0(W,r_1)$ is surjective, $H^1(f_1):H^1(V,q_1)\to H^1(W,r_1)$ is injective and $\dim\, H^0(V,q_1)<+\infty$. Then the $L_\infty[1]$ algebra $(V,Q)$ satisfies the quadraticity property if and only if so does $(W,R)$.
The above proposition fails if in its statement we replace the quadraticity property by the property of being formal: to the best of the authors’ knowledge, the only result one can find in the literature going in a somewhat similar direction is the formality transfer theorem from [@Man4 Theorem 6.8], whose hypotheses are much harder to verify. With respect to the discussion in the introduction of [@budur], this is essentially the reason why the quadraticity conjecture [@budur Conjecture 1.1] by Kaledin and Lehn [@KaLe] has been easier to handle than the full formality conjecture [@budur Conjecture 1.2].
Our aim in the remainder of this subsection is to show that in the special case when $H(V,q_1)=:H=H^0\oplus H^1$ is concentrated in degrees zero and one and $\dim\,H=\dim\,H^0+\dim\,H^1<+\infty$, then $V$ is formal if and only if the associated Kuranishi algebra is quadratic (the assumption $\dim\,H^1<+\infty$ is actually unnecessary, we keep it for simplicity and since it is usually satisfied in concrete examples).
First, we shall look more closely at quadratic algebras in $\widehat{{\mathbf{Art}}}_{\mathbb{K}\,}$.
**Notation.** For every $f\in {\mathbb{K}\,}[[x_1,\ldots,x_n]]$ we denote by $\mu(f)$ its multiplicity, and by $f^{(n)}$ its homogeneous component of degree $n$, hence $f=f^{(\mu(f))}+f^{(\mu(f)+1)}+\cdots$.
Let $$A=\frac{{\mathbb{K}\,}[[x_1,\ldots,x_n]]}{(f_1,\ldots,f_m)},\qquad \mu(f_i)\ge 2,$$ be a quadratic algebra. Then there exists an isomorphism $$\phi\colon {\mathbb{K}\,}[[x_1,\ldots,x_n]]\to {\mathbb{K}\,}[[x_1,\ldots,x_n]]$$ with differential (i.e. the linear part) equal to the identity such that the ideal $$(\phi(f_1),\ldots,\phi(f_m))$$ is generated by the quadrics $\phi(f_i)^{(2)}=f_i^{(2)}$, $i=1,\ldots,m$.
By assumption there exists an isomorphism $$\psi\colon {\mathbb{K}\,}[[x_1,\ldots,x_n]]\to {\mathbb{K}\,}[[x_1,\ldots,x_n]]$$ such that the ideal $\psi(f_1,\ldots,f_m)$ is generated by quadrics. Define $\phi=\psi_1^{-1}\psi$, where $\psi_1$ is the automorphism induced by the linear part of $\psi$. Then $(\phi(f_1),\ldots,\phi(f_m))$ is generated by quadrics: $$(\phi(f_1),\ldots,\phi(f_m))=(q_1,\ldots,q_r),\qquad q_i=q_i^{(2)}\,.$$ Since $\mu(f_i)\ge 2$ for every $i$, every $q_i$ is a linear combination of $f_1^{(2)},\ldots,f_m^{(2)}$ and conversely. Thus $f_1^{(2)},\ldots,f_m^{(2)}$ and $q_1,\ldots,q_r$ generate the same vector space and therefore also the same ideal.
For simplicity of notation, we denote by $P={\mathbb{K}\,}[[x_1,\ldots,x_n]]$ and by $\mathfrak{m}\subset P$ its maximal ideal.
With the above notations, let $f_1,\ldots,f_m\in \mathfrak{m}^2$ such that the ideal $(f_1,\ldots,f_n)$ is generated by quadrics. Then there exists a matrix $A=(a_{ij})\in M_{m,m}(P)$ such that $(a_{ij})\equiv {\operatorname{Id}}$ $\pmod{\mathfrak{m}}$, and $$\sum_{j=1}^m a_{ij}\,f_j^{(2)}=f_i,\qquad i=1,\ldots,m\,.$$
There exists an invertible matrix $C=(c_{ij})\in M_{m,m}({\mathbb{K}\,})$ such that the power series $g_i=\sum_{j=1}^m c_{ij}\,f_j$ have the property that $g_1^{(2)},\ldots,g_r^{(2)}$ are linearly independent over ${\mathbb{K}\,}$ and $g_{r+1}^{(2)}=\cdots=g_m^{(2)}=0$ for some $0\le r\le m$.
Thus we have $(g_1,\ldots,g_m)=(g_1^{(2)},\ldots,g_r^{(2)})$ and for every $i=1,\ldots,m$ there exists power series $b_{i1},\ldots,b_{ir}\in P$ such that $$\sum_{j=1}^r b_{ij}g_j^{(2)}=g_i.\,$$ In particular $$\sum_{j=1}^r b_{ij}(0)g_j^{(2)}=g_i^{(2)}$$ and then $b_{ij}(0)=1$ if $i=j\le r$ and $b_{ij}(0)=0$ otherwise. For every $j>r$ define $b_{ij}=1$ if $i=j$ and $b_{ij}=0$ if $i\not=j$. Then the matrix $B=(b_{ij})\in M_{m,m}(P)$ is such that $(b_{ij}(0))={\operatorname{Id}}$ and $$\sum_{j=1}^m b_{ij}\,g_j^{(2)}=g_i,\qquad i=1,\ldots,m\,.$$ Finally take $A=C^{-1}BC$.
\[prop:quad\] Notation as above. For a sequence $f_1,\ldots,f_m\in \mathfrak{m}^2$ the following conditions are equivalent:
1. the algebra $P/(f_1,\ldots,f_m)$ is quadratic;
2. there exists an isomorphism of algebras $\phi\colon P\to P$ with linear part equal the identity and a matrix $(a_{ij})\in M_{m,m}(P)$ with $(a_{ij}(0))={\operatorname{Id}}$ such that $$\sum_{j=1}^m a_{ij}\,f_j^{(2)}=\phi(f_i)\,.$$
Immediate from lemmas.
Next, we shall look more closely at the category of minimal $L_\infty[1]$ algebras $(H=H^0\oplus H^1,R)$ concentrated in degrees zero and one and of finite total dimension $\dim\,H=\dim\,H^0+\dim\,H^1<+\infty$. Since $\dim\,H<+\infty$, denoting by $H^\vee$ the graded dual ($(H^\vee)^i=(H^{-i})^{\vee}$), transposition induces an anti-isomorphism of graded Lie algebras $${\operatorname{Coder}}(S^c(H))\cong\prod_{n\ge0} {\operatorname{Hom}}(H^{\odot n},H) \xrightarrow{\cong} \prod_{n\geq0} {\operatorname{Hom}}(H^\vee,(H^\vee)^{\odot n})\cong{\operatorname{Der}}(\widehat{S}(H^\vee))$$ from ${\operatorname{Coder}}(S^c(H))$ to the graded Lie algebra of derivations of the completed symmetric algebra $\widehat{S}(H^\vee)=\prod_{n\geq0}(H^\vee)^{\odot n}$ over $H^\vee$. In particular, the $L_\infty[1]$ algebra structure $R$ on $H$ corresponds, under transposition, to a DG algebra structure $R^\vee$ on $\widehat{S}(H^\vee)$: we call the DG algebra $(\widehat{S}(H^\vee),R^\vee)$ the *Chevalley-Eilenberg* algebra of $(H,R)$, and denote it by ${\operatorname{CE}}(H,R)^\vee$. In this context, the Kuranishi algebra of $(H,R)$ is just $H^0(\operatorname{CE}(H,R)^\vee,R^\vee)$.
We fix bases $e_1,\ldots,e_n$ and $\epsilon_1,\ldots,\epsilon_m$ of $H^0$ and $H^1$ respectively, together with the dual bases $x_1,\ldots,x_n$ and $\eta_1,\ldots,\eta_m$ of $(H^0)^\vee$ and $(H^1)^\vee$. We shall denote the graded algebra $\widehat{S}(H^\vee)$ by $\mathbb{K}[[x_1,\ldots,x_n,\eta_1,\ldots,\eta_m]]$, that is, the free complete graded commutative algebra generated by $x_1,\ldots,x_n$ in degree zero and $\eta_1,\ldots,\eta_m$ in degree minus one: in particular, its degree zero component is the usual algebra $\mathbb{K}[[x_1,\ldots,x_n]]$ of formal power series. As before, we denote by $\mathfrak{m}\subset\mathbb{K}[[x_1,\ldots,x_n]]$ the maximal ideal. The DG algebra structure $R^\vee$ on $\mathbb{K}[[x_1,\ldots,x_n,\eta_1,\ldots,\eta_m]]$ is completely determined by the $m$-tuple of formal power series $R^\vee(\eta_1)=:f_1,\ldots,R^\vee(\eta_m)=:f_m$, and since $R$ is a minimal $L_\infty[1]$ algebra structure $f_1,\ldots,f_m\in\mathfrak{m}^2$. More explicitly, if we denote by $I=(i_1,\ldots,i_n)\in\mathbb{N}^n$ a multi-index, by $x^I:=x_1^{i_1}\cdots x_n^{i_n}$, $e_I:=e_1^{\odot i_1}\odot\cdots\odot e_n^{\odot i_n}$ and $I!=i_1!\cdots i_n!$, if the corestriction $r\in{\operatorname{Hom}}(S^c(H^0),H^1)$ of $R$ is given by $r(e_I)=\sum_{j=1}^m r_I^j\epsilon_j$, then the corresponding formal power series are given by $f_j=\sum_{I\in\mathbb{N}^n}\frac{r_I^j}{I!}x^I$. Let $(K=K^0\oplus K^1,Q)$ be another finite dimensional minimal $L_\infty[1]$ algebra in degrees zero and one. We fix bases $e'_1,\ldots,e'_s$ and $\epsilon'_1,\ldots,\epsilon'_r$ of $K^0$ and $K^1$ respectively, together with the dual bases $y_1,\ldots,y_s$ and $\theta_1,\ldots,\theta_r$ of $(K^0)^\vee$ and $(K^1)^\vee$. Let the corresponding DG algebra structure on $\mathbb{K}[[y_1,\ldots,y_s,\theta_1,\ldots,\theta_r]]$ be given by $Q^\vee(\theta_1)=g_1,\ldots, Q^\vee(\theta_r)=g_r\in\mathbb{K}[[y_1,\ldots,y_s]]$. A morphism of graded coalgebras $F:S^c(H)\to S^c(K)$ corresponds dually to a morphism of graded algebras $$F^\vee: \mathbb{K}[[y_1,\ldots,y_s,\theta_1,\ldots,\theta_r]] \to \mathbb{K}[[x_1,\ldots,x_n,\eta_1,\ldots,\eta_m]]$$ The latter is completely determined by its restriction to the degree zero components, which we denote by $\phi:\mathbb{K}[[y_1,\ldots,y_s]]\to\mathbb{K}[[x_1,\ldots,x_n]]$, and the formal power series $a_{ij}\in\mathbb{K}[[x_1,\ldots,x_n]]$ defined by $F^\vee(\theta_i)=\sum_{j=1}^m a_{ij}\eta_j$, $i=1,\ldots,r$. The requirement that $F$ in an $L_\infty[1]$ morphism, that is, $F^\vee$ is a morphism of DG algebras, becomes $$\label{eq:morp} \phi(g_i) = F^\vee Q^\vee(\theta_i) = R^\vee F^\vee(\theta_i) = R^\vee\left(\sum_{j=1}^m a_{ij}\eta_j\right) = \sum_{j=1}^m a_{ij}f_j,\qquad\forall\, i=1,\ldots,r.$$
Putting together the previous considerations, the desired result follows easily.
\[thm.4sedici\] Let $(V,Q)$ be an $L_\infty[1]$ algebra with finite dimensional tangent cohomology $H(V,q_1)=:H=H^0\oplus H^1$ concentrated in degrees zero and one. Then $V$ is formal if and only if it satisfies the quadraticity property.
The only if implication is clear. Conversely, let $(H,R)\dashrightarrow (V,Q)$ be a minimal model of $(V,Q)$. As before, we fix bases of $H^0, H^1$, and the dual bases $x_1\ldots,x_n$, $\eta_1,\ldots,\eta_m$ of $(H^0)^\vee$, $(H^1)^\vee$ respectively: then the $L_\infty[1]$ algebra structure $R$ is determined by the formal power series $f_i=R^\vee(\eta_i)\in\mathfrak{m}^2$, $i=1,\ldots,m$, and the Kuranishi algebra of $(V,Q)$ is $$\frac{{\mathbb{K}\,}[[x_1,\ldots,x_n]]}{(f_1,\ldots,f_m)}.$$ If the latter is quadratic, we can construct $\phi:{\mathbb{K}\,}[[x_1,\ldots,x_n]]\to{\mathbb{K}\,}[[x_1,\ldots,x_n]]$ and $A=(a_{ij})$ as in Proposition \[prop:quad\]. According to the discussion preceding Equation , the datum of $\phi$ and $A$ is equivalent to the datum of an $L_\infty[1]$ isomorphism $$(H,R)=(H,0,r_2,r_3.\ldots,r_k,\ldots)\dashrightarrow(H,r_2)=(H,0,r_2,0,\ldots,0,\ldots)$$ with linear part the identity, hence $(V,Q)$ is formal.
\[cor.4sedici\] Let $(V,Q)$ be an $L_\infty[1]$ algebra with tangent cohomology in nonnegative degree. Assume that:
1. $H^0(V,q_1)$ and $H^1(V,q_1)$ are finite dimensional vector spaces;
2. there exists an $L_{\infty}$-morphism $f\colon (V,Q)\dashrightarrow (W,R)$ into a homotopy abelian $L_\infty[1]$ algebra $(W,R)$ such that $f\colon H^i(V,q_1)\to H^i(W,r_1)$ is injective for every $i\ge 2$.
Then $V$ is formal if and only if it satisfies the quadraticity property.
By Lemma \[lem.4sette\] there exists a minimal model $(H,0,s_2,\ldots)$ such that $H^i=0$ for every $i<0$ and the image of $s_r$ is contained in $H^0\oplus H^1$ for every $r$.
Hence $H$ is the direct product of two $L_{\infty}[1]$ algebras, namely $$(H^0\oplus H^1,0,s_2,\ldots)\times (\oplus_{i\ge 2}H^i,0,0,\ldots)\,.$$ Thus if $H^0\oplus H^1$ is formal then also $H$ and $V$ are formal; on the other side $V$, $H$ and $H^0\oplus H^1$ have isomorphic Kuranishi algebras and the conclusion follows by Theorem \[thm.4sedici\].
Equivariant formality
---------------------
Let $(g,v)\to gv$ be a representation of a group $G$ on a graded vector space $V$.
In the above setup, an $L_\infty[1]$ algebra structure $Q$ on $V$ is *$G$-equivariant* if so are its Taylor coefficients, i.e., $$gq_n(v_1,\ldots,v_n)=q_n(gv_1,\ldots,gv_n),\qquad\forall\,g\in G,\, n\ge1,\,v_1,\ldots,v_n\in V.$$ Similarly, given a second $G$-equivariant $L_\infty[1]$ algebra $(W,R)$, an $L_\infty[1]$ morphism $F:(V,Q)\dashrightarrow(W,R)$ is $G$-equivariant if $$gf_n(v_1,\ldots,v_n)=f_n(gv_1,\ldots,gv_n),\qquad\forall\,g\in G,\, n\ge1,\,v_1,\ldots,v_n\in V.$$
Given a $G$-equivariant $L_\infty[1]$ algebra $(V,Q)$, the $G$-action descends on the tangent cohomology $H(V,q_1)$, and the induced bracket $r_2\colon H(V,q_1)^{\odot2}\to H(V,q_1)$, as in Remark \[rem: r2\], is $G$-equivariant. We say that $(V,Q)$ is *$G$-equivariantly formal* if there exists a $G$-equivariant weak equivalence $(H(V,q_1),r_2)\dashrightarrow(V,Q)$. The aim of this subsection is to prove that under some mild assumptions a $G$-equivariant $L_\infty[1]$ algebra is $G$-equivariantly formal if and only if it is formal in the usual sense.
\[th:Gformality\] Let $(V,Q)$ be a $G$-equivariant $L_\infty[1]$ algebra. Assume that the following $G$-modules are semisimple:
- $V^i$, for all $i\in{\mathbb{Z}}$; and
- ${\operatorname{Hom}}^j(H(V,q_1)^{\odot n}, H(V,q_1) )$ for $j=0,1$ and all $n\ge2$.
Then, if $V$ is formal it is also $G$-equivariantly formal.
\[rem:Gformality\] For instance, the hypotheses of the theorem are automatically satisfied in the following situations:
- when $G$ is finite;
- when $G$ is a linearly reductive algebraic group and $V^i$ is a finitely supported rational $G$-module for all $i\in{\mathbb{Z}}$;
- when $G$ is a linearly reductive algebraic group, $V^i$ is a rational $G$-module for all $i\in{\mathbb{Z}}$ and moreover $\dim H(V,q_1)=\sum_{i\in{\mathbb{Z}}}\dim\,H^i(V,q_1) <+\infty$.
The first item follows from the well known fact that any representation of a finite group is semisimple. The second and third items from the fact that, since $G$ is assumed linearly reductive, any rational representation of $G$ is semisimple, and the class of rational representations (as well as the subclass of finitely supported rational representations) is closed under subobjects and quotients, see Remark \[rmk.subobjectquotient\]. Moreover, the class of finite dimensional rational representations is further closed under tensor products and Hom spaces, and Lemma \[lemma.finsup\] shows that the same is true for the class of finitely supported rational representations.
(of Theorem \[th:Gformality\].) The proof is broken into two steps: the first step is to show the existence of a $G$-equivariant minimal model of $(V,Q)$. The assumption that $V^i$ is semisimple, $\forall\,i\in{\mathbb{Z}}$, assures the existence of a $G$-invariant abstract Hodge decomposition of the complex $(V,q_1)$ (cf. with Remark \[rem:hodge\]), that is, direct sum decompositions $Z^i(V)=B^i(V)\oplus H^i$, $V^i=Z^i(V)\oplus W^i=B^i(V)\oplus H^i\oplus W^i$ such that $H^i,W^i\subset V^i$ are $G$-invariant subspaces. It follows that $H=\oplus_{i\in{\mathbb{Z}}} H^i$ is a $G$-module, and the induced contraction $(f,g,h)$ of $(V,q_1)$ onto $(H,0)$, as in Remark \[rem:hodge\], is $G$-equivariant. It is now a straightforward consequence of the explicit recursive formulas for homotopy transfer (for which, once again, we may refer to [@descent Theorem 1.9]) that the induced minimal $L_\infty[1]$ algebra structure $R$ on $H$ and the induced $L_\infty[1]$ weak equivalences $F:(H,,R)\dashrightarrow(V,Q)$ and $G:(V,Q)\dashrightarrow(H,R)$ are all $G$-equivariant.
The second step is to prove that if $(V,Q)$ is formal, then $(H,R)$ constructed as in the previous paragraph is $G$-equivariantly $L_\infty[1]$ isomorphic to $(H,r_2)$: recall that formality of $(V,Q)$ is equivalent to the existence of an $L_\infty[1]$ isomorphism $(H,R)\dashrightarrow(H,r_2)$ with linear part the identity, but a priori the latter might not be $G$-equivariant. We reason inductively, following closely the proof of [@Man4 Theorem 6.3]: the inductive step will depend on the following key lemma, which is a $G$-equivariant version of [@Man4 Lemma 6.2].
\[lem:man4\] Given $k\geq3$ and a minimal $G$-equivariant $L_\infty[1]$ algebra of the form $(H,R')=(H,0,r'_2,0,\ldots,0,r'_k,r'_{k+1},\ldots)$, if $(H,R')$ is formal in the usual sense and the $G$-module ${\operatorname{Hom}}^j(H^{\odot n}, H)$ is semisimple for $j=0,1$ and $n=k-1,k$, then there exists a degree zero $G$-equivariant map $\alpha_{k-1}:H^{\odot k-1}\to H$ such that $r'_k=[r'_2,\alpha_{k-1}]_{NR}$.
According to [@Man4 Lemma 6.2 (3)] there exists a not necessarily $G$-equivariant map $\widetilde{\alpha}_{k-1}:H^{\odot k-1}\to H$ having the desired property $[r'_2,\widetilde{\alpha}_{k-1}]_{NR}=r'_k$. Furthermore, the hypothesis that ${\operatorname{Hom}}^j(H^{\odot n},H)$ is semisimple ($j=0,1$, $n=k-1,k$) implies the existence of *Reynolds operators* (cf. [@procesi §6.2.4]), that is, projectors $R\colon {\operatorname{Hom}}^j(H^{\odot n},H)\twoheadrightarrow{\operatorname{Hom}}^j(H^{\odot n}, H)^G$ onto the invariants, which are natural with respect to maps of $G$-modules (see [@procesi §6.2.3, Proposition 2]). Finally, we notice that the Nijenhuis-Richardson bracket $[-,-]_{NR}$ is equivariant with respect to the natural action of $G$ on ${\operatorname{Hom}}^j(H^{\odot n}, H)$ given by $(gq_n)(h_1,\ldots,h_n)= gq_n(g^{-1}h_1,\ldots,g^{-1}h_n)$. Since $r'_2$ is supposed to be $G$-equivariant, $[r'_2,-]_{NR}:{\operatorname{Hom}}^0(H^{\odot k-1},H)\to {\operatorname{Hom}}^1(H^{\odot k},H)$ is a morphism of $G$-modules, and then the diagram $$\xymatrix{ {\operatorname{Hom}}^0(H^{\odot k-1},H)\ar[d]_{[r'_2,-]_{NR}}\ar[r]^-R & {\operatorname{Hom}}^0(H^{\odot k-1},H)^G\ar[d]_{[r'_2,-]_{NR}} \\ {\operatorname{Hom}}^1(H^{\odot k},H)\ar[r]^-R & {\operatorname{Hom}}^1(H^{\odot k},H)^G}$$ is commutative. In particular, since $r'_k$ is also supposed to be $G$-equivariant, we have $$r'_k = R(r'_k) = R([r'_2,\widetilde{\alpha}_{k-1}]_{NR}) = [r'_2, R(\widetilde{\alpha}_{k-1})]_{NR},$$ and the proof is concluded by setting $\alpha_{k-1}=R(\widetilde{\alpha}_{k-1})$.
Going back to the proof of the theorem, given $k\geq3$ we assume inductively to have constructed a minimal $L_\infty[1]$ algebra $(H,R')=(H,0,r'_2,0,\ldots,0,r'_{k},r'_{k+1},\ldots)$ as in the hypotheses of the previous lemma and a $G$-equivariant $L_\infty[1]$-isomorphism $F':(H,R)\to(H,R')$ with linear part the identity. In particular, this implies $r'_2=r_2$. The basis of the induction is $k=3$, $R'=R$ and $(f'_1,f'_2,\ldots,f'_n,\ldots)=({\operatorname{id}}_H,0,\ldots,0,\ldots)$. Since $(H,R)$ is supposed to be formal, so is $(H,R')$, and we can find $\alpha_{k-1}$ as in Lemma \[lem:man4\]. As in the proof of [@Man4 Theorem 6.3], we denote by $\widehat{\alpha}_{k-1}$ the $G$-equivariant coderivation $\widehat{\alpha}_{k-1}=(0,0,\ldots,0,\alpha_{k-1},0,\ldots)$: its exponential $e^{\widehat{\alpha}_{k-1}}$ is a well defined $G$-equivariant automorphism of the symmetric coalgebra $S^c(H)$, acting as the identity on $\oplus_{i<k-1}H^{\odot i}\subset S^c(H)$. Finally, we put $R'' = e^{\widehat{\alpha}_{k-1}}\circ R'\circ e^{-\widehat{\alpha}_{k-1}}$ and $F''=e^{\widehat{\alpha}_{k-1}}\circ F'$. It is clear that $F'':(H,R)\to(H,R'')$ is an $L_\infty[1]$ isomorphism with linear part the identity, and the same computations as in the proof of [@Man4 Theorem 6.3] show that $R''$ has the form $R''=(0,r_2,0,\ldots,0,r''_{k+1},r''_{k+2},\ldots)$: moreover, since $R'$, $F'$ and $e^{\widehat{\alpha}_{k-1}}$ are $G$-equivariant, so are $R''$ and $F''$, and we can proceed with the induction.
In order to conclude, it is sufficient to observe that the infinite composition $$F:=\cdots\circ e^{\widehat{\alpha}_{k-1}}\circ\cdots\circ e^{\widehat{\alpha}_{3}}\circ e^{\widehat{\alpha}_{2}}\colon S^c(H)\to S^c(H)$$ is well defined, since $e^{\widehat{\alpha}_{k-1}}$ acts as the identity on $\oplus_{i<k-1}H^{\odot i}\subset S^c(H)$, and by construction $F$ is an $L_\infty[1]$ isomorphism $F\colon(H,0,r_2,r_3,\ldots,r_n,\ldots)\dashrightarrow(H,0,r_2,0,\ldots,0,\ldots)$ with linear part the identity.
\[cor.Gformality\] Let $(L,d,[-,-])$ be a DG Lie algebra. Assume the following hypotheses:
- $H^i(L,d)=0$ for $i<0$ and $H^0(L,d)=:\mathfrak{g}$ is the Lie algebra of a reductive algebraic group $G$;
- there is a Lie algebra embedding $\imath\colon\mathfrak{g}\hookrightarrow Z^0(L)$ that is a section of the natural projection map $Z^0(L)\to H^0(L,d)$;
- the adjoint action of $\mathfrak{g}$ on $L$ is induced by an action of $G$ by DG Lie algebra automorphisms which is degreewise rational and finitely supported.
Define a DG Lie subalgebra $K\subset L$ by setting $K^i=0$ for $i\leq0$, $K^1\subset L^1$ a $\mathfrak{g}$-invariant complement of $B^1(L)$ in $L^1$ (this exists because the hypotheses imply that $L^i$ is a semisimple $\mathfrak{g}$-module for all $i\in{\mathbb{Z}}$) and finally $K^i=L^i$ for $i\geq2$. Then, the DG Lie algebra $L$ is formal if and only if so is the one $K$.
We introduce a second DG Lie subalgebra $M:=\imath(\mathfrak{g})\oplus K\subset L$: clearly the inclusion $M\hookrightarrow L$ is a quasi-isomorphism, thus $L$ is formal if and only if so is $M$, and we need to prove that this is the case if and only if so is $K$.
We denote the $L_\infty[1]$ algebras corresponding to $M$ and $K$ respectively by $$(M[1],Q)=(L[1],q_1,q_2,0,\ldots,0,\ldots)\qquad\mbox{and}\qquad(K[1],Q)=(K[1],q_1,q_2,0,\ldots,0,\ldots),$$ and by $(H(M,d)[1],r_2)$ and $(H(K,d)[1],r_2)$ the $L_\infty[1]$ algebras corresponding to their cohomology graded Lie algebras (explicitly, if $M\to M[1]: x\to s^{-1}x$ is the desuspension isomorphism and $|x|$ is the degree of $x$ in $M$, then $q_1(s^{-1}x)=-s^{-1}dx$, $q_2(s^{-1}x\odot s^{-1}y)=(-1)^{|x|}s^{-1}[x,y]$: similarly for $r_2$). We notice that by construction $H(K,d)=H^{>0}(M,d)=\oplus_{i>0} H^i(M,d)$.
If $M$ is formal and $F\colon (H(M,d)[1],r_2)\dashrightarrow (M[1],Q)$ is an $L_\infty[1]$ weak equivalence, for trivial degree reasons when $x_1,\ldots,x_n\in H(K,d)[1]=H^{>0}(M,d)[1]\subset H(M,d)[1]$ then $f_n(x_1,\ldots,x_n)\in M^{>0}[1]=K[1]$, that is, $F$ restricts to an $L_\infty[1]$ weak equivalence $F\colon(H(K,d)[1],r_2)\dashrightarrow(K[1],Q)$ and $K$ is formal.
Conversely, we assume that $K$ is formal. By the hypotheses, $(K[1],Q)$ is a $G$-equivariant $L_\infty[1]$ algebra satisfying the assumptions of Theorem \[th:Gformality\] (see also Remark \[rem:Gformality\]), hence it is also $G$-equivariantly formal. Moreover, we fix a $G$-equivariant $L_\infty[1]$ weak equivalence $\widetilde{F}:(H(K,d)[1],r_2)\dashrightarrow(K[1],Q)$, and then we can define an $L_\infty[1]$ weak equivalence $F:(H(M,d)[1],r_2)\dashrightarrow(M[1],Q)$ as follows. Using the natural identification $H(M,d)\cong\mathfrak{g}\oplus H(K,d)$, we put $f_1(x)=\widetilde{f}_1(x)$ if $x\in H(K,d)[1]\subset H(M,d)[1]$ and $f_1(x)=\imath(x)$ if $x\in\mathfrak{g}[1]\subset H(M,d)[1]$: for $n\geq2$, we put $f_n(x_1,\ldots,x_n)=\widetilde{f}_n(x_1,\ldots,x_n)$ if $x_1,\ldots,x_n\in H(K,d)[1]\subset H(M,d)[1]$, and $f_n(x_1,\ldots,x_n)=0$ otherwise. It is clear that $f_1$ is a quasi-isomorphism. The remaining relations that have to be satisfied in order for $F$ to define an $L_\infty[1]$ morphism read $$\begin{gathered}
\label{eq:Loomorp} q_1f_n(x_1,\ldots,x_n) +\frac{1}{2}\sum_{i=1}^{n-1}\sum_{\sigma\in S(i,n-i)}\pm q_2(f_i(x_{\sigma(1)},\ldots,x_{\sigma(i)}),f_{n-i}(x_{\sigma(i+1)},\ldots,x_{\sigma(n)})) = \\ \sum_{\sigma\in S(2,n-2)}\pm f_{n-1}(r_2(x_{\sigma(1)},x_{\sigma(2)}),x_{\sigma(3)},\ldots,x_{\sigma(n)})
\end{gathered}$$ for all $n\geq2$ and $x_1,\ldots,x_n\in H(M,d)[1]$, where $\pm$ is the appropriate Koszul sign and $S(i,n-i)$ is the set of $(i,n-i)$-unshuffles (i.e., permutations $\sigma\in S_n$ such that $\sigma(j)<\sigma(j+1)$ for $j\neq i$). When $x_1,\ldots,x_n\in H(K,d)[1]\subset H(M,d)[1]$ these relations follow from the corresponding ones for the $\widetilde{f}_n$, whereas both sides of are obviously zero whenever $x_i,x_j\in\mathfrak{g}[1]\subset H(M,d)[1]$ for some $i\neq j$ and $n\geq3$. For $n=2$ and $x_1,x_2\in\mathfrak{g}[1]$ follows from the fact that $\imath\colon \mathfrak{g}\to Z^0(L)$ is a morphism of Lie algebras. In the remaining case, it is not restrictive to assume $x_1\in\mathfrak{g}[1]$ and $x_2,\ldots,x_n\in H(K,d)[1]$: in this situation reduces to $$q_2(\imath(x_1),\widetilde{f}_{n-1}(x_2,\ldots,x_n)) = \sum_{j=2}^{n}\pm\widetilde{f}_{n-1}(r_2(x_1,x_j),x_2,\ldots,\widehat{x_j},\ldots,x_n),$$ which says that the Taylor coefficients $\widetilde{f}_{n-1}\colon H(K,d)[1]^{\odot n-1}\to K[1]$ are $\mathfrak{g}$-equivariant maps. This is clear, since by assumption the $\mathfrak{g}$-module structures are induced by $G$-module structures and the $\widetilde{f}_{n-1}$ are $G$-equivariant.
Proof of Theorem \[thm.maintheorem2\]. {#proof-of-theoremthm.maintheorem2. .unnumbered}
--------------------------------------
We only need to prove that if the Kuranishi family of ${\mathcal{F}}$ is quadratic then ${R\mspace{-2mu}\operatorname{Hom}}({\mathcal{F}},{\mathcal{F}})$ is formal, since the converse is true by general theory.
In the situation of Corollary \[cor.Gformality\], the DG-Lie algebras $L$ and $K$ have the same Kuranishi family [@GoMil2; @ManRendiconti; @NijRich64]. If $H^i(L)=0$ for every $i\not=0,1,2$, then $H^i(K)=0$ for every $i\not=1,2$. Therefore by Corollary \[cor.Gformality\] $L$ is formal if and only if so is $K$, while by Corollary \[cor.4sedici\] $K$ is formal if its Kuranishi algebra is quadratic and there exists an $L_{\infty}$ morphism from $K$ to a homotopy abelian $L_{\infty}$ algebra that is injective on the tangent cohomology groups $H^i$ for every $i\ge 3$.
According to [@DMcoppie] the trace maps ${\operatorname{Tr}}\colon {{\operatorname{Ext}}}^i_X({\mathcal{F}},{\mathcal{F}})\to {{\operatorname{Ext}}}^i_X({\mathcal{O}}_X,{\mathcal{O}}_X)=H^i({\mathcal{O}}_X)$ are induced by an $L_{\infty}$ morphism ${\operatorname{Tr}}\colon {R\mspace{-2mu}\operatorname{Hom}}({\mathcal{F}},{\mathcal{F}})\to
{R\mspace{-2mu}\operatorname{Hom}}({\mathcal{O}}_X,{\mathcal{O}}_X)$ and ${R\mspace{-2mu}\operatorname{Hom}}({\mathcal{O}}_X,{\mathcal{O}}_X)$ may be represented by the abelian DG-Lie algebra given by the Thom-Whitney totalisation of the sheaf of abelian Lie algebras ${\operatorname{\mathcal H}\!\!om}_{{\mathcal{O}}_X}({\mathcal{O}}_X,{\mathcal{O}}_X)$ with respect to a given affine open cover, see also [@FMM pag. 2255]. We also refer to [@FM] for an explicit description of a quasi-isomomorphism between the Thom-Whitney totalisations and the representative of derived endomorphisms described here in Subsection \[section.derivedEnd\].
Now Theorem \[thm.maintheorem2\] follows immediately from the previous considerations applied to the DG-Lie algebra of Theorem \[thm.maintheorem3\]
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[^1]: Introduced by Nijenhuis and Richardson [@NijRich64] in the ’60s and sponsored by Deligne, Drinfeld and others during the ’80s in the form of private communications [@DtM; @DtS], further developed in the works of Goldman-Millson [@GoMil1], Kontsevich [@Kont94], Hinich [@hinichdescent; @hinichDGC] and Manetti [@EDF], among others, during the ’90s, and finally made into a rigorous theorem – after passing to the setting of derived deformation functors – in the recent works of Lurie [@lurie2] and Pridham [@pridham].
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abstract: 'We analyze and present an effective solution to the minimal Gorenstein cover problem: given a local Artin $\res$-algebra $A=\res[\![x_1,\dots x_n]\!]/I$, compute an Artin Gorenstein $\res$-algebra $G=\res[\![x_1,\dots x_n]\!]/J$ such that $\ell(G)-\ell(A)$ is minimal. We approach the problem by using Macaulay’s inverse systems and a modification of the integration method for inverse systems to compute Gorenstein covers. We propose new characterizations of the minimal Gorenstein cover and present a new algorithm for the effective computation of the variety of all minimal Gorenstein covers of $A$ for low Gorenstein colength. Experimentation illustrates the practical behavior of the method.'
address:
- 'Juan Elias Departament de Matemàtiques i Informàtica Facultat de Matemàtiques Universitat de Barcelona Gran Via 585, 08007 Barcelona, Spain'
- 'Roser Homs Departament de Matemàtiques i Informàtica Facultat de Matemàtiques Universitat de Barcelona Gran Via 585, 08007 Barcelona, Spain'
- 'Bernard Mourrain AROMATH INRIA - Sophia Antipolis Méditeranée 2004 route des Lucioles, 06902 Sophia Antipolis, France '
author:
- 'J. Elias ${}^{*}$'
- 'R. Homs ${}^{**}$'
- 'B. Mourrain ${}^{***}$'
title: '[**Computing minimal Gorenstein covers**]{}'
---
[^1]
[^2]
[^3]
Introduction
============
Given a local Artin $\res$-algebra $A=R/I$, with $R=\res[\![x_1,\dots x_n]\!]$, an interesting problem is to find how far is it from being Gorenstein. In [@Ana08], Ananthnarayan introduces for the first time the notion of Gorenstein colength, denoted by $\gcl(A)$, as the minimum of $\ell(G)-\ell(A)$ among all Gorenstein Artin $\res$-algebras $G=R/J$ mapping onto $A$. Two natural questions arise immediately:
[Question A]{}: How we can explicitly compute the Gorenstein colength of a given local Artin $\res$-algebra $A$?
[Question B]{}: Which are its minimal Gorenstein covers, that is, all Gorenstein rings $G$ reaching the minimum $\gcl(A)=\ell(G)-\ell(A)$?
Ananthnarayan generalizes some results by Teter [@Tet74] and Huneke-Vraciu [@HV06] and provides a characterization of rings of $\gcl(A)\leq 2$ in terms of the existence of certain self-dual ideals $\mathfrak{q}\in A$ with respect to the canonical module $\omega_A$ of $A$ satisfying $\ell(A/\mathfrak{q})\leq 2$. For more information on this, see [@Ana08] or [@EH18 Section 4], for a reinterpretation in terms of inverse systems. Later on, Elias and Silva ([@ES17]) address the problem of the colength from the perspective of Macaulay’s inverse systems. In this setting, the goal is to find polynomials $F\in S$ such that $I^\perp\subset \langle F\rangle$ and $\ell(\langle F\rangle)-\ell(I^\perp)$ is minimal. Then the Gorenstein $\res$-algebra $G=R/\ann F$ is a minimal Gorenstein cover of $A$. A precise characterization of such polynomials $F\in S$ is provided for $\gcl(A)=1$ in [@ES17] and for $\gcl(A)=2$ in [@EH18].
However, the explicit computation of the Gorenstein colength of a given ring $A$ is not an easy task even for low colength - meaning $\gcl(A)$ equal or less than 2 - in the general case. For examples of computation of colength of certain families of rings, see [@Ana09] and [@EH18].
On the other hand, if $\gcl(A)=1$, the Teter variety introduced in [@ES17 Proposition 4.2] is precisely the variety of all minimal Gorenstein covers of $A$ and [@ES17 Proposition 4.5] already suggests that a method to compute such covers is possible.
In this paper we address questions A and B by extending the previous definition of Teter variety of a ring of Gorenstein colength 1 to the variety of minimal Gorenstein covers $MGC(A)$ where $A$ has arbitrary Gorenstein colength $t$. We use a constructive approach based on the integration method to compute inverse systems proposed by Mourrain in [@Mou96].
In section 2 we recall the basic definitions of inverse systems and introduce the notion of integral of an $R$-module $M$ of $S$ with respect to an ideal $K$ of $R$, denoted by $\int_K M$. Section 3 links generators $F\in S$ of inverse systems $J^\perp$ of Gorenstein covers $G=R/J$ of $A=R/I$ with elements in the integral $\int_{{\mathfrak m}^t} I^\perp$, where ${\mathfrak m}$ is the maximal ideal of $R$ and $t=\gcl(A)$. This relation is described in and sets the theoretical background to compute a $\res$-basis of the integral of a module extending Mourrain’s integration method.
In section 4, proves the existence of a quasi-projective sub-variety $MGC^n(A)$ whose set of closed points are associated to polynomials $F\in S$ such that $G=R/\ann F$ is a minimal Gorenstein cover of $A$. Section 5 is devoted to algorithms: explicit methods to compute a $\res$-basis of $\int_{{\mathfrak m}^t}I^\perp$ and $MGC(A)$ for colengths 1 and 2. Finally, in section 6 we provide several examples of the minimal Gorenstein covers variety and list the comptutation times of $MGC(A)$ for all analytic types of $\res$-algebras with $\gcl(A)\leq 2$ appearing in Poonen’s classification in [@Poo08a].
All algorithms appearing in this paper have been implemented in *Singular*, [@DGPS], and the library [@E-InvSyst14] for inverse system has also been used.
The second author wants to thank the third author for the opportunity to stay at INRIA Sophia Antipolis - Méditerranée (France) and his hospitality during her visit on the fall of 2017, where part of this project was carried out. This stay was financed by the Spanish Ministry of Economy and Competitiveness through the Estancias Breves programme (EEBB-I-17-12700).
Integrals and inverse systems
=============================
Set $R=\res[\![x_1,\dots x_n]\!]$ and $S=\res[y_1,\dots y_n]$. Recall that $S$ can be given an $R$-module structure by contraction: $$\begin{array}{cccc}
R\times S & \longrightarrow & S & \\
(x^\alpha, y^\beta) & \mapsto & x^\alpha \circ y^\beta = &
\left\{
\begin{array}{ll}
y^{\beta-\alpha}, & \beta \ge \alpha; \\
0, & \mbox{otherwise.}
\end{array}
\right.
\end{array}$$
The Macaulay inverse system of $A=R/I$ is the sub-$R$-module $I^\perp=\lbrace g\in S\mid I\circ g=0\rbrace$ of $S$. This provides the order-reversing bijection between ${\mathfrak m}$-primary ideals $I$ of $R$ and finitely generated sub-$R$-modules $M$ of $S$ described in Macaulay’s duality. As for the reverse correspondence, given a sub-$R$-module $M$ of $S$, the module $M^\perp$ is the ideal $\ann M=\lbrace f\in R\mid f\circ g=0\,\forall g\in M\rbrace$ of $R$. Moreover, it characterizes zero-dimensional Gorenstein rings $G=R/J$ as those with cyclic inverse system $J^\perp=\langle F\rangle$, where $\langle F\rangle$ is the $\res$-vector space $\langle x^\alpha\circ F:\vert\alpha\vert\leq \deg F\rangle_\res$. For more details on this construction, see [@ES17] and [@EH18].
Consider an Artin local ring $A=R/I$ of socle degree $s$ and inverse system $I^\perp$. We are interested in finding Artin local rings $R/\ann F$ that cover $R/I$, that is $I^\perp\subset \langle F\rangle$, but we also want to control how apart are those two inverse systems. In other words, given an ideal $K$, we want to find a Gorenstein cover $\langle F\rangle$ such that $K\circ \langle F\rangle\subset I^\perp$. Therefore it makes sense to think of an inverse operation to contraction.
Consider an $R$-submodule $M$ of $S$. We define the integral of $M$ with respect to the ideal $K$, denoted by $\int_K M$, as $$\int_K M=\lbrace G\in S\mid K\circ G\subset M\rbrace.$$
Note that the set $N=\lbrace G\in S\mid K\circ G\subset M\rbrace$ is, in fact, an $R$-submodule $N$ of $S$ endowed with the contraction structure. Indeed, given $G_1,G_2\in N$ then $K\circ (G_1+G_2)=K\circ G_1+K\circ G_2\subset M$, hence $G_1+G_2\in N$. For all $a\in R$ and $G\in N$ we have $K\circ (a\circ G)=aK\circ G=a\circ (K\circ G)\subset M$, hence $a\circ G\in N$.
\[integral\] With the above notations it holds $$\int_K M=\left(KM^\perp\right)^\perp.$$
Let $G\in\left(KM^\perp\right)^\perp$. Then $\left(KM^\perp\right)\circ G=0$, so $M^\perp\circ\left(K\circ G\right)=0$. Hence $K\circ G\subset M$, i.e. $G\in\int_K M$. We have proved that $\left(KM^\perp\right)^\perp\subseteq\int_K M$. Now let $G\in\int_K M$. By definition, $K\circ G\subset M$, so $M^\perp\circ\left(K\circ G\right)=0$ and hence $\left(M^\perp K\right)\circ G=0$. Therefore, $G\in\left(M^\perp K\right)^\perp$.
One of the key results of this paper is the effective computation of $\int_K M$ (see ). Last result gives a method for the computation of this module by computing two Macaulay duals. However, since computing Macaulay duals is expensive, avoids the computation of such duals.
\[incl\] The following properties hold: $(1)$ Given $K\subset L$ ideals of $R$ and $M$ $R$-module, if $K\subset L$, then $\int_L M\subset\int_K M.$ $(2)$ Given $K$ ideal of $R$ and $M\subset N$ $R$-modules, if $M\subset N$, then $\int_K M\subset\int_K N.$ $(3)$ Given any $R$-module $M$, $\int_ R M=M$.
The inclusion $K\circ \int_K M\subset M$ follows directly from the definition of integral. However, the equality does not hold in the general case:
\[Ex1\] Let us consider $R=\res[\![x_1,x_2,x_3]\!]$, $K=(x_1,x_2,x_3)$, $S=\res[y_1,y_2,y_3]$, and $M=\langle y_1y_2,y_3^3\rangle$. We have $\int_K M=\left(K M^\perp\right)^\perp=\langle y_1^2,y_1y_2,y_1y_3,y_2^2,y_2y_3,y_3^4 \rangle,$ and $K\circ \int_K M={\mathfrak m}\circ\langle y_1^2,y_1y_2,y_1y_3,y_2^2,y_2y_3,y_3^4\rangle=\langle y_1,y_2,y_3^3\rangle\subsetneq M.$
We also have the inclusion $M\subset\int_K K\circ M$. Indeed, for any $F\in M$, $K\circ F\subset K\circ M$ and hence $F\in\int_K K\circ M=\lbrace G\in S\mid K\circ G\subset K\circ M\rbrace$. Again, the equality does not hold.
Using the same example as in , we get $K\circ M={\mathfrak m}\circ \langle y_1y_2,y_3^3\rangle=\langle y_1,y_2,y_3^2\rangle,$ and $\int_K( K\circ M)=\left(K (K\circ M)^\perp\right)^\perp=\langle y_1^2,y_1y_2,y_1y_3,y_2^2,y_2y_3,y_3^2\rangle\nsubseteq M.$
In the particular case where we integrate with respect to a principal monomial ideal $K=(\underline{x}^{\underline{\alpha}})$, with $\underline{x}=(x_1,\dots,x_n)$ and $\underline{\alpha}=(\alpha_1,\dots,\alpha_n)$, the expected equality for integrals $$\underline{x}^{\underline{\alpha}}\circ\int_{\underline{x}^{\underline{\alpha}}}M=M$$ holds. Indeed, for any $m\in M$, take $G=\underline{y}^{\underline{\alpha}}m$. Since $\underline{x}^{\underline{\alpha}}\circ \underline{y}^{\underline{\alpha}}=1$, then $\underline{x}^{\underline{\alpha}}\circ \underline{y}^{\underline{\alpha}}m=m$ and the equality is reached.
In general we cannot extend the above identity to linear forms. We consider $\res= \mathbb C$, $L=x_1+ix_2$ and $P=y_1+iy_2$. Then $L\circ \int_L \langle P\rangle \subsetneq \langle P\rangle$.
Let us now consider an even more particular case: the integral of a cyclic module $M=\langle F\rangle$ with respect to the variable $x_i$. Since the equality $x_i\circ \int_{x_i}M=M$ holds, there exists $G\in S$ such that $x_i\circ G=F$. This polynomial $G$ is not unique because it can have any constant term with respect to $x_i$, that is $G=y_iF+p(y_1,\dots,\hat{y}_i,\dots,y_n)$. However, if we restrict to the non-constant polynomial we can define the following:
The $i$-primitive of a polynomial $f\in S$ is the polynomial $g\in S$, denoted by $\int_i f$, such that
1. $x_i\circ g=f$,
2. $g\vert_{y_i=0}=0$.
This notion of $i$-primitive of a polynomial $f\in S$ with respect to the variable $x_i\in R$ was provided in [@EM07] using the derivation structure:
The $i$-primitive of a polynomial $f\in S$ is the polynomial $g\in S$, denoted by $\int_i f$, such that
1. $\partial_{y_i} g=f$,
2. $g\vert_{y_i=0}=0$.
Therefore, we can think of the integral of a module with respect to an ideal as a generalization of the $i$-primitive proposed by Elkadi and Mourrain.
From now on, when we use the notation $\int_{x_i} f$ it refers to the contraction case. Since we are considering the $R$-module structure given by contraction instead of derivation, the $i$-primitive is precisely $$\int_i f=y_if.$$
Indeed, $x_i\circ(y_if)=f$ and $(y_if)\mid_{y_i=0}=0$, hence (i) and (ii) hold. Uniqueness can be easily proved. Consider $g_1,g_2$ to be $i$-primitives of $f$. Then $x_i\circ (g_1-g_2)=0$ and hence $g_1-g_2=p(y_1,\dots,\hat{y}_i,\dots,y_n)$. Clearly $(g_1-g_2)\vert_{y_i=0}=p(y_1,\dots,\hat{y}_i,\dots,y_n)$. On the other hand, $(g_1-g_2)\vert_{y_i=0}=g_1\vert_{y_i=0}-g_2\vert_{y_i=0}=0$. Hence $p=0$ and $g_1=g_2$.
\[r1\] Note that, by definition, $x_k\circ \int_k f=f$. Any $f$ can be decomposed in $f=f_1+f_2$, where the first term is a multiple of $y_k$ and the second has no appearances of this variable. Then $$\int_k x_k\circ f=\int_k x_k\circ f_1+\int_k x_k\circ f_2=f_1+\int_k 0$$ Therefore, in general, $$f_1=\int_k x_k\circ f\neq x_k\circ \int_k f=f.$$ However, for all $l\neq k$, $$\int_l x_k\circ f=\frac{y_lf_1}{y_k}=x_k\circ \int_l f.$$
Let us now recall Theorem 7.36 of Elkadi-Mourrain in [@EM07], which describes the elements of the inverse system $I^\perp$ up to a certain degree $d$. We define $\mathcal{D}_d=I^\perp\cap S_{\leq d}$, for any $1\leq d\leq s$, where $s=\socdeg(A)$. Since $\mathcal{D}_s=I^\perp$, this result leads to an algorithm proposed by the same author to obtain a $\res$-basis of an inverse system. We rewrite the theorem using the contraction setting instead of derivation.
\[EM\] Given an ideal $I=(f_1,\dots,f_m)$ and $d>1$. Let $\lbrace b_1,\dots,b_{t_{d-1}}\rbrace$ be a $\res$-basis of $\mathcal{D}_{d-1}$. The polynomials of $\mathcal{D}_d$ with no constant term are of the form $$\label{thm}
\Lambda=\sum_{j=1}^{t_{d-1}}\lambda_j^1\int_1 b_j\vert_{y_2=\cdots=y_n=0}+\sum_{j=1}^{t_{d-1}}\lambda_j^2\int_2 b_j\vert_{y_3=\cdots=y_n=0}+\dots+\sum_{j=1}^{t_{d-1}}\lambda_j^n\int_n b_j,\quad\lambda_j^k\in\res,$$ such that $$\label{cond1}
\sum_{j=1}^s\lambda_j^k (x_l\circ b_j)-\sum_{j=1}^s\lambda_j^l(x_k\circ b_j)=0, 1\leq k<l\leq n,$$ and $$\label{cond2}
\left(f_i\circ\Lambda\right)(0)=0, \mbox{ for } 1\leq i\leq m.$$
See [@Mou96] or [@EM07] for a proof.
Using integrals to obtain Gorenstein covers of Artin rings
==========================================================
Let us start by recalling the definitions of Gorenstein cover and Gorenstein colength of a local equicharacteristic Artin ring $A=R/I$ from [@EH18]:
We say that $G=R/J$, with $J=\ann F$, is a Gorenstein cover of $A$ if and only if $I^\perp\subset\langle F\rangle$. The Gorenstein colength of $A$ is $$\gcl(A)=\min\{\ell(G)-\ell(A)\mid G \text{ is a Gorenstein cover of } A\}.$$ $A$ Gorenstein cover $G$ of an Artin ring $A$ is minimal if $\ell(G)=\ell(A)+\gcl(A)$.
For all $F\in S$ defining a Gorenstein cover of $A$ we consider the ideal $K_F$ of $R$ defined by $$K_F=(I^{\perp} : \langle F\rangle).$$
In general, we do not know what is a right choice for an ideal $K_F$ that provides a minimal cover. However, for a given colength, we do know a lot about the form of the ideals $K_F$ associated to a polynomial $F$ that reaches this minimum. In the following proposition, we summarize the basic results regarding ideals $K_F$ from [@EH18]:
\[KF\] Let $A=R/I$ be a local Artin algebra and $G=R/J$, with $J=\ann F$, a minimal Gorenstein cover of $A$. Then,
1. $I^{\perp}= K_F \circ F$,
2. $\gcl(A)=\ell(R/K_F)$.
Moreover, $$K_F=
\left\{
\begin{array}{ll}
R, & \hbox{if \quad } \gcl(A)=0; \\
{\mathfrak m}, & \hbox{if \quad } \gcl(A)=1;\\
(L_1,\dots,L_{n-1},L_n^2), & \hbox{if \quad } \gcl(A)=2,
\end{array}
\right.$$ where $L_1,\dots,L_n$ are suitable independent linear forms in $R$.
Note that whereas in the case of colength 1 the ideal $K_F$ does not depend on the particular choice of $F$, this is no longer true for higher colengths. For colength higher that 2, things get more complicated since the $K_F$ can even have different analytic type. The simplest example is colength 3, where we have 2 possible non-isomorphic $K_F$’s: $(L_1,\dots,L_{n-1},L_n^3)$ and $(L_1,\dots,L_{n-2},L_{n-1}^2,L_{n-1}L_n,L_n^2)$. Therefore, although it is certainly true that $F\in\int_{K_F} I^\perp$, it will not be useful as a condition to check if $A$ has a certain Gorenstein colength.
The dependency of the integral on $F$ can be removed by imposing only the condition $F\in\int_{{\mathfrak m}^t} I^\perp$, for a suitable integer $t$. Later on we will see how to use this condition to find a minimal cover, but we first need to dig deeper into the structure of the integral of a module with respect to a power of the maximal ideal. The following result permits an iterative approach:
\[str\] Let $M$ be a finitely generated sub-$R$-module of $S$ and $d\geq 1$, then $$\int_{{\mathfrak m}}\left(\int_{{\mathfrak m}^{d-1}}M\right)=\int_{{\mathfrak m}^d} M.$$
Let us prove first the inclusion $\int_{{\mathfrak m}}\left(\int_{{\mathfrak m}^{d-1}}M\right)\subseteq \int_{{\mathfrak m}^d} M$. Take $\Lambda\in\int_{{\mathfrak m}}\left(\int_{{\mathfrak m}^{d-1}}M\right)$, then ${\mathfrak m}\circ\Lambda\subseteq \int_{{\mathfrak m}^{d-1}}M$ and hence ${\mathfrak m}^d\circ\Lambda={\mathfrak m}^{d-1}\circ\left({\mathfrak m}\circ\Lambda\right)\subseteq M$. Therefore, $\Lambda\in\int_{{\mathfrak m}^d} M$. To prove the reverse inclusion, consider $\Lambda\in\int_{{\mathfrak m}^d} M$, that is, ${\mathfrak m}^{d-1}\circ\left({\mathfrak m}\circ\Lambda\right)={\mathfrak m}^d\circ\Lambda\subseteq M$. In other words, ${\mathfrak m}\circ\Lambda\subseteq\int_{{\mathfrak m}^{d-1}}M$ and $\Lambda\in\int_{{\mathfrak m}}\left(\int_{{\mathfrak m}^{d-1}}M\right)$.
Since $\int_{{\mathfrak m}^t}M$ is a finitely dimensional $\res$-vector space that can be obtained by integrating $t$ times $M$ with respect to ${\mathfrak m}$, we can also consider a basis of $\int_{{\mathfrak m}^t}M$ which is built by extending the previous basis at each step.
\[adapted\] Let $M$ be a finitely generated sub-$R$-module of $S$. Given an integer $t$, we denote by $h_i$ the dimension of the $\res$-vector space $\int_{{\mathfrak m}^i}M/\int_{{\mathfrak m}^{i-1}}M$, $i=1,\cdots,t$. An adapted $\res$-basis of $\int_{{\mathfrak m}^t}M/M$ is a $\res$-basis $\overline{F}_j^i$, $i=1,\cdots, t$, $j=1,\cdots,h_i$, of $\int_{{\mathfrak m}^t}M/M$ such that $F_1^i,\cdots, F_{h_i}^i\in \int_{{\mathfrak m}^i}M$ and their cosets in $\int_{{\mathfrak m}^i}M/\int_{{\mathfrak m}^{i-1}}M$ form a $\res$-basis, $i=1\cdots,t$.
Let $A=R/I$ be an Artin ring, we denote by $\mathcal{L}_{A,t}$ the $R$-module $\int_{{\mathfrak m}^t}M/M$.
The following proposition is meant to overcome the obstacle of non-uniqueness of the ideals $K_F$:
\[F\] Given a ring $A=R/I$ of Gorenstein colength $t$ and a minimal Gorenstein cover $G=R/\ann F$ of $A$,
1. $F\in\int_{{\mathfrak m}^t}I^\perp$;
2. for any $H\in\int_{{\mathfrak m}^t}I^\perp$, the condition $I^\perp\subset\langle H\rangle$ does not depend on the representative of the class $\overline{H}$ in $\mathcal{L}_{A,t}$.
In particular, any $F'\in \int_{{\mathfrak m}^t}I^\perp$ such that $\overline{F'}=\overline{F}$ in $\mathcal{L}_{A,t}$ defines the same minimal Gorenstein cover $G=R/\ann F$.
\(i) By [@EH18 Proposition 3.8], we have $\gcl(A)=\ell(R/K_F)$, where $K_F\circ F=I^\perp$ for any polynomial $F$ that generates a minimal Gorenstein cover $G=R/\ann F$ of $A$. From the definition of integral we have $F\in\int_{K_F} I^\perp$. Since $\ell(R/K_F)=t$, then $\socdeg(R/K_F)\leq t-1$. Indeed, the extremal case corresponds to the most expanded Hilbert function $\lbrace 1,1,\dots,1\rbrace$, that is, a stretched algebra (see [@Sal79c],[@EV08]). Then $\HF_{R/K_F}(i)=0$, for any $i\geq t$, regardless of the particular form of $K_F$, and hence ${\mathfrak m}^t\subset K_F$. Therefore, $$F\in\int_{K_F} I^\perp\subset\int_{{\mathfrak m}^t} I^\perp.$$ (ii) Consider a polynomial $H\in\int_{{\mathfrak m}^t}I^\perp$ such that $I^\perp\subset\langle H\rangle$. By [@EH18 Proposition 3.8], $K_H\circ H=I^\perp$. Consider $H'\in\int_{{\mathfrak m}^t}I^\perp$ such that $\overline{H}=\overline{H'}$ in $\mathcal{L}_{A,t}$, so $H=H'+G$ for some $G\in I^\perp$. We want to prove that $$\label{nak}
K_H\circ H'+{\mathfrak m}\circ I^\perp=K_H\circ H+{\mathfrak m}\circ I^\perp=I^\perp.$$ The second equality is direct from $K_H\circ H=I^\perp$. Let us check the first. Take $h\circ H'+{\mathfrak m}\circ I^\perp\in K_H\circ H'+{\mathfrak m}\circ I^\perp$, with $h\in K_H \subset {\mathfrak m}$, $$h\circ H'+{\mathfrak m}\circ I^\perp=h\circ H-h\circ G+{\mathfrak m}\circ I^\perp=h\circ H+{\mathfrak m}\circ I^\perp\subset K_H\circ H+{\mathfrak m}\circ I^\perp.$$ The same argument holds for the reverse inclusion. Therefore, holds and we can apply Nakayama’s lemma to get $K_H\circ H'=I^\perp$. Hence $I^\perp\subset\langle H'\rangle$. In particular, $\langle H'\rangle=\langle H\rangle$. Indeed, since $H'=H-G$ and $\langle G\rangle\subset \langle I^\perp\rangle\subset \langle H\rangle$, then $H'\in \langle H\rangle + \langle G\rangle=\langle H\rangle$ and a similar argument gives $H\in\langle H'\rangle$.
Observe that the proposition says that, altought not all $F\in\int_{{\mathfrak m}^t} I^\perp$ correspond to covers $G=R/\ann F$ of $A=R/I$, if $F$ is actually a cover, then any $F'\in\int_{{\mathfrak m}^t} I^\perp$ such that $\overline{F'}=\overline{F}\in \mathcal{L}_{A,t}$ provides the exact same cover. That is, $\langle F'\rangle=\langle F\rangle$.
\[cor\] Let $A=R/I$ be an Artin ring of Gorenstein colength $t$ and let $\lbrace\overline{F}_j^i\rbrace_{1\leq i\leq t,1\leq j\leq h_i}$ be an adapted $\res$-basis of $\mathcal{L}_{A,t}$. Given a minimal Gorenstein cover $G=R/J$ there is a generator $F$ of $J^\perp$ such that $F$ can be written as $$F=a_1^1 F_1^1+\dots+a_{h_1}^1F_{h_1}^1+\dots+a_1^t F_1^t+\dots+a_{h_t}^t F_{h_t}^t\in\int_{{\mathfrak m}^t}I^\perp,\,a_i^j\in\res.$$
In $\mathcal{L}_{A,t}$ we have $\overline{F}=\sum_{i=1}^t\sum_{j=1}^{h_j}a_j^i\overline{F_j^i}$ and hence $F=\sum_{i=1}^t\sum_{j=1}^{h_i}a_j^iF_j^i+G$ with $G\in I^\perp.$ By proposition , any representative of the class $\overline{F}$ provides the same Gorenstein cover. In particular, we can take $G=0$ and we are done.
Our goal now is to compute the integrals of the inverse system with respect to powers of the maximal ideal. Rephrasing it in a more general manner: we want an effective computation of $\int_{{\mathfrak m}^k} M$, where $M\subset S$ is a sub-$R$-module of $S$ and $k\geq 1$.
Recall that, via Macaulay’s duality, we have $I^\perp=M$, where $I=\ann M$ is an ideal in $R$. Therefore, the most natural approach is to integrate $M$ in a similar way as $I$ is integrated in by Elkadi-Mourrain but removing the condition of orthogonality with respect to the generators of the ideal $I$ ( of ). Without this restriction we will be allowed to go beyond the inverse system $I^\perp=M$ and up to the integral of $M$ with respect to ${\mathfrak m}$. The proof we present is very similar to the proof of Theorem 7.36 in [@Mou96] but we reproduce it below for the sake of completeness and to show the use of the contraction structure.
\[propint\] Consider a sub-$R$-module $M$ of $S$ and let $\lbrace b_1,\dots,b_s\rbrace$ be a $\res$-basis of $M$. Let $\Lambda\in S$ be a polynomial with no constant terms. Then $\Lambda\in\int_{{\mathfrak m}} M$ if and only if $$\label{prop}
\Lambda=\sum_{j=1}^s\lambda_j^1\int_1 b_j\vert_{y_2=\cdots=y_n=0}+\sum_{j=1}^s\lambda_j^2\int_2 b_j\vert_{y_3=\cdots=y_n=0}+\dots+\sum_{j=1}^s\lambda_j^n\int_n b_j,\quad\lambda_j^k\in\res,$$ such that $$\label{cond1}
\sum_{j=1}^s\lambda_j^k (x_l\circ b_j)-\sum_{j=1}^s\lambda_j^l(x_k\circ b_j)=0, 1\leq k<l\leq n.$$
First we will prove that any element $\Lambda$ in $\int_{{\mathfrak m}} M$ is as described in and satisfies . Note that ${\mathfrak m}\circ\Lambda\subset M=\langle b_1,\dots,b_s\rangle_\res$. Therefore, $x_1\circ\Lambda=\sum_{j=1}^s\lambda_j^1 b_j$ and we can obtain a decomposition $\Lambda=\Lambda_1+\dots+\Lambda_n$ such that $$\label{lambda1}
\Lambda_l=\sum_{j=1}^{t_{d-1}}\lambda_j^l\int_lb_j-\left(\sigma_{l-1}-\sigma_{l-1}\mid_{y_l=0}\right)\in\res[y_l,\dots,y_n]\backslash\res[y_{l+1},\dots,y_n],\quad 1\leq l\leq n,$$ where $$\label{sigma1}
\sigma_k=\sum_{i=1}^k\Lambda_i=\sum_{j=1}^{t_{d-1}}\lambda_j^1\int_1b_j\mid_{y_2=\dots=y_k=0}+\sum_{j=1}^{t_{d-1}}\lambda_j^2\int_2b_j\mid_{y_3=\dots=y_k=0}+\dots+\sum_{j=1}^{t_{d-1}}\lambda_j^k\int_kb_j.$$ Clearly, $\Lambda=\sigma_n$, which gives the desired expression in . We want to prove now that holds. Since $\Lambda_l\in\res[y_l,\dots,y_n]$, then $x_k\circ\Lambda_l=0$ for $1\leq k<l\leq n$. Hence contracting first by $x_k$ and then by $x_l$ we get $$\label{sigma3}
\sum_{j=1}^{t_{d-1}}\lambda_j^l(x_k\circ b_j)=x_l\circ\left(x_k\circ\sigma_{l-1}\right).$$ On one hand, $x_k\circ\sigma_{l-1}=x_k\circ(\sum_{i=1}^k\Lambda_i)+x_k\circ(\sum_{i=k+1}^{l-1}\Lambda_i)=x_k\circ(\sum_{i=1}^k\Lambda_i)=x_k\circ\sigma_k$, for $k<l$. On the other, when contracting by $x_k$, in all terms to which we do elimination the $k$-th variable vanishes and this provides $$x_k\circ\sigma_k=\sum_{j=1}^{t_{d-1}}\lambda_j^1\int_1x_k\circ b_j\mid_{y_2=\dots=y_k=0}+\dots+\sum_{j=1}^{t_{d-1}}\lambda_j^k\left(x_k\circ\int_k b_j\right)=\sum_{j=1}^{t_{d-1}}\lambda_j^k b_j.$$ Therefore, we can rewrite as $\sum_{j=1}^{t_{d-1}}\lambda_j^l(x_k\circ b_j)=\sum_{j=1}^{t_{d-1}}\lambda_j^k(x_l\circ b_j),$ hence is satisfied.
Conversely, we want to know if every element of the form of satisfying is in $\int_{{\mathfrak m}}M$. By definition, $\Lambda\in\int_{{\mathfrak m}} M$ if and only if ${\mathfrak m}\circ\Lambda\subset M$. Therefore, it is enough to prove that $x_k\circ\Lambda\in M$ for any $1\leq k\leq n$. Let us then contract the expression by $x_k$, $1\leq k\leq n$: $$x_k\circ\Lambda=\sum_{j=1}^s\lambda_j^kb_j\mid_{y_{k+1}=\dots=y_n=0}+\sum_{j=1}^s\lambda_j^{k+1}\int_{k+1} x_k\circ b_j\mid_{y_{k+2}=\dots=y_n=0}+\dots+\sum_{j=1}^s\lambda_j^n\int_n x_k\circ b_j.$$ Consider condition , then its $l$-primitive, for any $k<l\leq n$, gives $$\sum_{j=1}^s\lambda_j^k \int_l x_l\circ b_j=\sum_{j=1}^s\lambda_j^l\int_l x_k\circ b_j.$$ It can be checked that $\int_l x_k\circ H\mid_{y_{l+1}=\dots=y_n=0}=\left(\int_l x_k\circ H\right)\mid_{y_{l+1}=\dots=y_n=0}$. Therefore, $$x_k\circ\Lambda=\sum_{j=1}^s\lambda_j^k\left(b_j\mid_{y_{k+1}=\dots=y_n=0}+\int_{k+1} x_{k+1}\circ b_j\mid_{y_{k+2}=\dots=y_n=0}+\dots+\int_n x_n\circ b_j\right).$$ We can also prove that $$b_j\mid_{y_{k+1}=\dots=y_n=0}+\int_{k+1} x_{k+1}\circ b_j\mid_{y_{k+2}=\dots=y_n=0}+\dots+\int_n x_n\circ b_j=b_j$$ is true for any $1\leq j\leq n$. Then $x_k\circ\Lambda=\sum_{j=1}^s\lambda_j^kb_j\in M$ and we are done.
From the previous theorem and the next corollary follows directly.
\[thmint\] Consider a sub-$R$-module $M$ of $S$ and $d\geq 1$. Let $\lbrace b_1,\dots,b_{t_{d-1}}\rbrace$ be a $\res$-basis of $\int_{{\mathfrak m}^{d-1}} M$ and let $\Lambda$ be a polynomial with no constant terms. Then $\Lambda\in\int_{{\mathfrak m}^d} M$ if and only if it is of the form $$\label{thm2}
\Lambda=\sum_{j=1}^{t_{d-1}}\lambda_j^1\int_1 b_j\vert_{y_2=\cdots=y_n=0}+\sum_{j=1}^{t_{d-1}}\lambda_j^2\int_2 b_j\vert_{y_3=\cdots=y_n=0}+\dots+\sum_{j=1}^{t_{d-1}}\lambda_j^n\int_n b_j,\quad\lambda_j^k\in\res,$$ such that $$\label{cond}
\sum_{j=1}^{t_{d-1}}\lambda_j^k (x_l\circ b_j)-\sum_{j=1}^{t_{d-1}}\lambda_j^l(x_k\circ b_j)=0,\quad 1\leq k<l\leq n.$$
Note that, using the notations of , it can be proved that $$\mathcal{D}_d=I^\perp\cap\int_{{\mathfrak m}}\mathcal{D}_{d-1},$$
for any $1<d\leq s$. Indeed, says that any element $\Lambda\in\mathcal{D}_d$ is of the form of , and because of , we know that it satisfies . Hence, by , $\Lambda\in\int_{{\mathfrak m}}\mathcal{D}_{d-1}$. Since $\Lambda\in\mathcal{D}_d=I^\perp\cap S_{\leq d}$, then $\Lambda\in I^\perp\cap\int_{{\mathfrak m}}\mathcal{D}_{d-1}$. Conversely, any element $\Lambda$ in $\left(\int_{{\mathfrak m}}\mathcal{D}_{d-1}\right)\cap I^\perp$ satisfies, in particular, ${\mathfrak m}\circ\Lambda\subseteq \mathcal{D}_{d-1}=I^\perp\cap S_{\leq d-1}$. Therefore $\deg\left({\mathfrak m}\circ\Lambda\right)\leq d-1$ and hence $\deg\Lambda\leq d$. Since $\Lambda\in I^\perp$, then $\Lambda\in I^\perp\cap S_{\leq d}=\mathcal{D}_d$.
We end this section by considering the low Gorenstein colength cases.
Teter rings
-----------
Let us remind that Teter rings are those $A=R/I$ such that $A\cong G/\soc(G)$ for some Gorenstein ring $G$. In [@ES17], the authors prove that $\gcl(A)=1$ whenever $\embd(A)\geq 2$. They are a special case to deal with because the $K_F$ associated to any generator $F\in S$ of a minimal cover is always the maximal ideal. We provide some additional criteria to characterize such rings:
\[propTeter\] Let $A=R/I$ be a non-Gorenstein local Artin ring of socle degree $s\geq 1$ and let $\lbrace\overline{F}_j\rbrace_{1\leq j\leq h}$ be an adapted $\res$-basis of $\mathcal{L}_{A,1}$. Then $\gcl(A)=1$ if and only if there exist a polynomial $F=\sum_{j=1}^h a_jF_j\in\int_{{\mathfrak m}}I^\perp$, $a_j\in\res$, such that $\dim_\res({\mathfrak m}\circ F)=\dim_\res I^\perp$.
The first implication is straightforward from and Teter rings characterization in [@ES17]. Reciprocally, if $F\in\int_{{\mathfrak m}}I^\perp$, then ${\mathfrak m}\circ F\subset I^\perp$ by definition, and from the equality of dimensions, it follows that ${\mathfrak m}\circ F=I^\perp$. Therefore, $0<\gcl(A)\leq\ell(R/{\mathfrak m})=1$ and we are done.
\[Ex3\] Recall with $I^\perp=\langle y_1y_2,y_3^3\rangle$ and $\int_{\mathfrak m}I^\perp=\langle y_1^2,y_1y_2,y_1y_3,y_2^2,y_2y_3,y_3^4\rangle$. Then $\overline{y_1^2},\overline{y_1y_3},\overline{y_2^2},\overline{y_2y_3},\overline{y_3^4}$ is a $\res$-basis of $\mathcal{L}_{A,1}$. As a consequence of , $A$ is Teter if and only if there exists a polynomial $$F=a_1y_1^2+a_2y_1y_3+a_3y_2^2+a_4y_2y_3+a_5y_3^4$$ such that ${\mathfrak m}\circ F=I^\perp$. But ${\mathfrak m}\circ F=\langle a_1y_1+a_2y_3,a_3y_2+a_4y_3,a_2y_1+a_4y_2+a_5y_3^3\rangle$ and clearly $y_1y_2$ does not belong here. Therefore, $\gcl(A)>1$.
Gorenstein colength 2
---------------------
By [@EH18], we know that $A$ is of Gorenstein colength 2 if and only if there exists a polynomial $F$ of degree $s+1$ or $s+2$ such that $K_F\circ F=I^\perp$, with $K_F=(L_1,\dots,L_{n-1},L_n^2)$, where $L_i$ are suitable independent linear forms.
Observe that a completely analogous characterization to the one we did for Teter rings is not possible. If $A=R/I$ has Gorenstein colength 2, by , there exists $F=\sum_{i=1}^2\sum_{j=1}^{h_i}a_j^iF_j^i\in\int_{{\mathfrak m}^2}I^\perp$, where $\lbrace\overline{F^i_j}\rbrace_{1\leq i\leq 2,1\leq j\leq h_i}$ is a $\res$-basis of $\mathcal{L}_{A,2}$, that generates a minimal Gorenstein cover of $A$ and then trivially $I^\perp\subset\langle F\rangle$. However, the reverse implication is not true.
Consider $A=R/{\mathfrak m}^3$, where $R$ is the ring of power series in 2 variables, and consider $F=y_1^2y_2^2$. It is easy to see that $F\in\int_{{\mathfrak m}^2}I^\perp=S_{\leq 4}$ and $I^\perp\subset\langle F\rangle$. However, it can be proved that $\gcl(A)=3$ using [@Ana09 Corollary 3.3]. Note that $K_F={\mathfrak m}^2$ and hence $\ell(R/K_F)=3$.
Therefore, given $F\in\int_{{\mathfrak m}^2}I^\perp$, the condition $I\subset\langle F\rangle$ is not sufficient to ensure that $\gcl(A)=2$. We must require that $\ell(R/K_F)=2$ as well.
\[gcl2\] Given a non-Gorenstein non-Teter local Artin ring $A=R/I$, $\gcl(A)=2$ if and only if there exist a polynomial $F=\sum_{i=1}^2\sum_{j=1}^{h_i} a_j^iF_j^i\in\int_{{\mathfrak m}^2}I^\perp$ such that $\lbrace\overline{F_j^i}\rbrace_{1\leq i\leq 2,1\leq j\leq h_i}$ is an adapted $\res$-basis of $\mathcal{L}_{A,2}$ and $(L_1,\dots,L_{n-1},L_n^2)\circ F=I^\perp$ for suitable independent linear forms $L_1,\dots,L_n$.
We will only prove that if $F$ satisfies the required conditions, then $\gcl(A)=2$. By definition of $K_F$, if $(L_1,\dots,L_{n-1},L_n^2)\circ F=I^\perp$, then $(L_1,\dots,L_{n-1},L_n^2)\subseteq K_F$. Again by [@EH18], $\gcl(A)\leq \ell(R/K_F)$ and hence $\gcl(A)\leq \ell\left(R/(L_1,\dots,L_{n-1},L_n^2)\right)=2$. Since $\gcl(A)\geq 2$ by hypothesis, then $\gcl(A)=2$.
Recall the ring $A=R/I$ in . Since $$\int_{{\mathfrak m}^2}I^\perp=\langle y_1^3,y_1^2y_2,y_1y_2^2,y_2^3,y_1^2y_3,y_1y_2y_3,y_2^2y_3,y_1y_3^2,y_2y_3^3,y_3^5\rangle$$ and $\gcl(A)>1$, its Gorenstein colength is 2 if and only if there exist some $$F\in\langle y_1^2,y_1y_2,y_1y_3,y_2^2,y_2y_3,y_3^4,y_1^3,y_1^2y_2,y_1y_2^2,y_2^3,y_1^2y_3,y_1y_2y_3,y_2^2y_3,y_1y_3^2,y_2y_3^3,y_3^5\rangle_\res$$ such that $(L_1,\dots,L_{n-1},L_n^2)\circ F=I^\perp$. Consider $F=y_3^4+y_1^2y_2$, then $$(x_1,x_2^2,x_3)\circ F=\langle x_1\circ F,x_2^2\circ F,x_3\circ F\rangle=\langle y_1y_2,y_3^3\rangle$$ and hence $\gcl(A)=2$.
Minimal Gorenstein covers varieties
===================================
We are now interested in providing a geometric interpretation of the set of all minimal Gorenstein covers $G=R/J$ of a given local Artin $\res$-algebra $A=R/I$. From now on, we will assume that $\res$ is an algebraically closed field. The following result is well known and it is an easy linear algebra exercise.
\[semic\] Let $\varphi_i:\res^a \longrightarrow \res^b$, $i=1\cdots,r$, be a family of Zariski continuous maps. Then the function $\varphi^*:\res^a\longrightarrow \mathbb N$ defined by $\varphi^*(z)=\dim_{\res} \langle \varphi_1(z),\cdots, \varphi_r(z)\rangle_{\res}$ is lower semicontinous, i.e. for all $z_0 \in \res^a$ there is a Zariski open set $z_0\in U \subset \res^a$ such that for all $z\in U$ it holds $\varphi^*(z)\geq \varphi^*(z_0)$.
\[ThMGC\] Let $A=R/I$ be an Artin ring of Gorenstein colength $t$. There exists a quasi-projective sub-variety $MGC^n(A)$, $n=\dim(R)$, of $\mathbb P_{\res}\left(\mathcal{L}_{A,t}\right)$ whose set of closed points are the points $[\overline{F}]$, $\overline{F}\in \mathcal{L}_{A,t}$, such that $G=R/\ann F$ is a minimal Gorenstein cover of $A$.
Let $E$ be a sub-$\res$-vector space of $\int_{{\mathfrak m}^t}I^\perp$ such that $$\int_{{\mathfrak m}^t}I^\perp \cong E\oplus I^{\perp},$$ we identify $\mathcal{L}_{A,t}$ with $E$. From , for all minimal Gorenstein cover $G=R/\ann F$ we may assume that $F\in E$. Given $F\in E$, the quotient $G=R/\ann F$ is a minimal cover of $A$ if and only if
1. $\dim_{\res}(\langle F\rangle)= \dim_{\res}(A)+t$, and
2. $I^{\perp}\subset \langle F \rangle$.
The second condition is equivalent to the numerical condition\
$(2')\quad\dim_{\res}(I^{\perp}+ \langle F \rangle) =\dim_{\res}\langle F \rangle$.\
Indeed, if $I^\perp\subset\langle F\rangle$, then $I^\perp\cap\langle F\rangle=I^\perp$ and hence $\dim_\res(I^\perp+\langle F\rangle)=\dim_\res\langle F\rangle$. Reciprocally, since $\dim_{\res}\langle F \rangle=\dim_\res(I^\perp+\langle F\rangle)=\dim_\res I^\perp+\dim_\res\langle F\rangle-\dim_\res(I^\perp\cap\langle F\rangle),$ then $\dim_\res I^\perp=\dim_\res(I^\perp\cap\langle F\rangle)$. But $I^\perp\cap\langle F\rangle\subset I^\perp$ and hence the equality $I^\perp\cap\langle F\rangle=I^\perp$ holds. Therefore, $I^\perp\subset \langle F\rangle$.
Define the family of Zariski continuous maps $\lbrace\varphi_{\underline{\alpha}}\rbrace_{\vert\underline{\alpha}\vert\leq\deg F}$, where $$\begin{array}{rrcl}
\varphi_{\underline{\alpha}}: & E & \longrightarrow & E\\
& F & \longmapsto & \underline{x}^{\underline{\alpha}}\circ F\\
\end{array}$$ In particular, $\varphi_0=Id_R$. We write $$\begin{array}{rrcl}
\varphi^\ast: & E & \longrightarrow & \mathbb{N}\\
& F & \longmapsto & \dim_\res\langle \underline{x}^{\underline{\alpha}}\circ F,\vert\underline{\alpha}\vert\leq\deg F\rangle_\res
\end{array}$$
Note that $\varphi^\ast(F)=\dim_\res \langle F\rangle$ and, by , $\varphi^\ast$ is a lower semicontinuous map. Hence $U_1=\lbrace F\in E\mid \dim_\res\langle F\rangle\geq\dim_\res A+t\rbrace$ is an open Zariski set in $E$. Using the same argument, $U_2=\lbrace F\in E\mid \dim_\res\langle F\rangle\geq\dim_\res A+t+1\rbrace$ is also an open Zariski set in $E$ and hence $Z_1=E\backslash U_2$ is a Zariski closed set such that $\dim_\res\langle F\rangle\leq\dim_\res A+t$ for any $F\in Z_1$. Then $Z_1\cap U_1=\lbrace F\in E\mid \dim_\res\langle F\rangle=\dim_\res A+t\rbrace$ is a locally closed set.
Let $G_1,\cdots,G_e$ be a $\res$-basis of $I^{\perp}$ and consider the constant map $$\begin{array}{rrcl}
\psi_{i}: & E & \longrightarrow & E\\
& F & \longmapsto & G_i\\
\end{array}$$ for any $i=1,\cdots,e$. By ,
$$\begin{array}{rrcl}
\psi^\ast: & E & \longrightarrow & \mathbb{N}\\
& F & \longmapsto & \dim_\res\left(\langle F\rangle+I^\perp\right)=\dim_\res \langle\lbrace\underline{x}^{\underline{\alpha}}\circ F\rbrace_{\vert\underline{\alpha}\vert\leq\deg F}, G_1,\dots, G_e\rangle_\res
\end{array}$$
is a lower semicontinuous map. Using an analogous argument, we can prove that $T=\lbrace F\in E\mid \dim_\res(I^\perp+\langle F\rangle)=\dim_\res A+t\rbrace$ is a locally closed set. Therefore, $$W=(Z_1\cap U_1)\cap T=\lbrace F\in E\mid \dim_\res A+t=\dim_\res(I^\perp+\langle F\rangle)=\dim_\res\langle F\rangle\rbrace$$ is a locally closed subset of $E$ whose set of closed points are all the $F$ in $E$ satisfying $(1)$ and $(2')$, i.e. defining a minimal Gorenstein cover $G=R/\ann F$ of $A$.
Moreover, since $\langle F\rangle=\langle \lambda F\rangle$ for any $\lambda\in\res^\ast$, conditions $(1)$ and $(2')$ are invariant under the multiplicative action of $\res^*$ on $F$ and hence $MGC^n(A)=\mathbb P_{\res}(W)\subset \mathbb P_{\res}(E)=\mathbb P_{\res}\left(\mathcal{L}_{A,t}\right)$.
Let $G=R/J$ be a minimal Gorenstein cover of $A=R/I$. Then $$\embd(G)\le \tau(A)+\gcl(A)+1.$$
Set $A=R/I$ and $G=R'/J$ such that $\embd(A)=\dim R$ and $\embd(G)=\dim R'$. We denote by ${\mathfrak m}$ and ${\mathfrak m}'$ the maximal ideals of $R$ and $R'$, respectively. From $.(i)$, it is easy to deduce that $K_F/({\mathfrak m}K_F+J)\simeq I^\perp/({\mathfrak m}\circ I^\perp)$. Hence $\tau(A)=\dim_\res K_F/({\mathfrak m}K_F+J)$ by [@ES17 Proposition 2.6]. Then $$\embd(G)-1=\dim_\res R'/({\mathfrak m}')^2\leq \dim_\res R'/({\mathfrak m}K_F+J)=\gcl(A)+\tau(A),$$ where the last equality follows from $.(ii)$.
Given an Artin ring $A=R/I$, the variety $MGC(A)=MGC^n(A)$, with $n=\tau(A)+\gcl(A)+1$, is called the minimal Gorenstein cover variety associated to $A$.
Let us recall that in [@EH18] we proved that for low Gorenstein colength of $A$, i.e. $\gcl(A)\le 2$, then $\embd(G)=\embd(A)$ for any minimal Gorenstein cover $G$ of $A$. In this situation we can define $MGC(A)$ as the variety $MGC^n(A)$ with $n=\embd(A)$.
Observe that this notion of minimal Gorenstein cover variety generalizes the definition of Teter variety introduced in [@ES17], which applies only to rings of Gorenstein colength 1, to any arbitrary colength.
Computing $MGC(A)$ for low Gorenstein colength {#s5}
==============================================
In this section we provide algorithms and examples to compute the variety of minimal Gorenstein covers of a given ring $A$ whenever its Gorenstein colength is 1 or 2. These algorithms can also be used to decide whether a ring has colength greater than 2, since it will correspond to empty varieties.
To start with, we provide the auxiliar algorithm to compute the integral of $I^\perp$ with respect to the $t$-th power of the maximal ideal of $R$. If there exist polynomials defining minimal Gorenstein covers of colength $t$, they must belong to this integral.
Computing integrals of modules
------------------------------
Consider a $\res$-basis $\mathbf{b}=(b_1,\dots,b_t)$ of a finitely generated sub-$R$-module $M$ of $S$ and consider $x_k\circ b_i=\sum_{j=1}^t a_j^i b_j$, for any $1\leq i\leq t$ and $1\leq k\leq n$. Let us define matrices $U_k=(a_j^i)_{1\leq j,i\leq t}$ for any $1\leq k\leq n$. Note that
$$\left(x_k\circ b_1 \cdots x_k\circ b_t\right)=
\left(b_1 \cdots b_t\right)
\left(\begin{array}{ccc}
a_1^1 & \dots & a_1^t\\
\vdots & & \vdots\\
a_t^1 & \dots & a_t^t
\end{array}\right)
.$$
Now consider any element $h\in M$. Then $$x_k\circ h=x_k\circ\sum_{i=1}^t h_ib_i=\sum_{i=1}^t (x_k\circ h_ib_i)=\sum_{i=1}^t (x_k\circ b_i)h_i=$$ $$=\left(x_k\circ b_1 \cdots x_k\circ b_t\right)
\left(\begin{array}{c}
h_1\\
\vdots\\
h_t
\end{array}\right)=\left(b_1 \cdots b_t\right)U_k
\left(\begin{array}{c}
h_1\\
\vdots\\
h_t
\end{array}\right),$$ where $h_1,\dots,h_t\in\res$.
Let $U_k$, $1\leq k\leq n$, be the square matrix of order $t$ such that $$x_k\circ h=\mathbf{b}\,U_k\,\mathbf{h}^t,$$ where $\mathbf{h}=(h_1,\dots,h_t)$ for any $h\in M$, with $h=\sum_{i=1}^t h_ib_i$. We call $U_k$ the contraction matrix of $M$ with respect to $x_k$ associated to a $\res$-basis $\mathbf{b}$ of $M$.
Since $x_kx_l\circ h=x_lx_k\circ h$ for any $h\in M$, we have $U_kU_l=U_lU_k$, with $1\leq k<l\leq n$.
In [@Mou96], Mourrain provides an effective algorithm based on that computes, along with a $\res$-basis of the inverse system $I^\perp$ of an ideal $I$ of $R$, the contraction matrices $U_1,\dots,U_n$ of $I^\perp$ associated to that basis.
Consider $A=R/I$, with $R=\res[\![x_1,x_2]\!]$ and $I={\mathfrak m}^2$. Then $\lbrace 1,y_1,y_2\rbrace$ is a $\res$-basis of $I^\perp$ and $U_1,U_2$ are its contraction matrices with respect to $x_1,x_2$, respectively: $$U_1=\left(\begin{array}{ccc}
0 & 1& 0\\
0 & 0& 0\\
0 & 0& 0
\end{array}\right),\quad
U_2=\left(\begin{array}{ccc}
0 & 0& 1\\
0 & 0& 0\\
0 & 0& 0
\end{array}\right).$$
Now we provide a modified algorithm based on that computes the integral of a finitely generated sub-$R$-module $M$ with respect to the maximal ideal. The algorithm can use the output of Mourrain’s integration method as initial data: a $\res$-basis of $I^\perp$ and the contraction matrices associated to this basis.
$b_1,\dots,b_t$ $\res$-basis of $M$;\
$U_1,\dots,U_n$ contraction matrices of $M$ associated to the $\res$-basis $b_1,\dots,b_t$. $D=b_1,\dots,b_t,b_{t+1},\dots,b_{t+h}$ $\res$-basis of $\int_{{\mathfrak m}} M$;\
$U'_1,\dots,U'_n$ contraction matrices of $\int_{{\mathfrak m}} M$ associated to the $\res$-basis $b_1,\dots,b_{t+h}$.
1. Solve the system of equations $U_k\,\mathbf{x_l}-U_l\,\mathbf{x_k}= 0$ for $1\leq k<l\leq n$.
2. Compute a basis $c_1,\dots,c_h$ of the solutions $\mathbf{X}=\left[\mathbf{x_1},\dots,\mathbf{x_n}\right]$ that provide new polynomials of the form $$\Lambda=\displaystyle\sum_{k=1}^n\left(\sum_{j=1}^t\lambda_j^k\int_k b_j\vert_{y_{k+1}=\cdots=y_n=0}\right)$$ which are linearly independent with respect to $b_1,\dots,b_t$.
3. Set $D:=b_1,\dots,b_t,b_{t+1},\dots,b_{t+h}$, where $b_{t+l}=c_l$ for any $1\leq l\leq h$.
4. Define square matrices $U'_k$ of order $t+h$ and set $U'_k[i]=U_k[i]$ for $1\leq i\leq t$.
5. Compute $x_k \circ b_i=\sum_{j=1}^t\mu_j^ib_j$ for $t+1\leq i\leq t+h$ and set $$U'_k[i]=\left(\begin{array}{cccccc}
\mu_1^i & \cdots & \mu_t^i & 0 & \cdots & 0
\end{array}\right)^t.$$
Observe that $\overline{b}_{t+1},\dots,\overline{b}_{t+h}$ is an adapted $\res$-basis of $\int_{{\mathfrak m}}M/M$. Also note that, since the algorithm returns the new contraction matrices, we can iterate the procedure in order to obtain the $\res$-basis of $\int_{{\mathfrak m}^t} M$, and hence an adapted basis of $\int_{{\mathfrak m}^t} M/M$.
Consider $A=R/I$, with $R=\res[\![x_1,x_2]\!]$ and $I={\mathfrak m}^2$. Then $\lbrace 1,y_1,y_2,y_2^2,y_1y_2,y_1^2\rbrace$ is a $\res$-basis of $\int_{{\mathfrak m}}I^\perp=S_{\leq 2}$ with the following contraction matrices: $$U'_1=\left(\begin{array}{cccccc}
0 & 1& 0 & 0 & 0 & 0\\
0 & 0& 0 & 0 & 0 & 1\\
0 & 0& 0 & 0 & 1 & 0\\
0 & 0& 0 & 0 & 0 & 0\\
0 & 0& 0 & 0 & 0 & 0\\
0 & 0& 0 & 0 & 0 & 0\\
\end{array}\right),\quad
U'_2=\left(\begin{array}{cccccc}
0 & 0& 1 & 0 & 0 & 0\\
0 & 0& 0 & 0 & 1 & 0\\
0 & 0& 0 & 1 & 0 & 0\\
0 & 0& 0 & 0 & 0 & 0\\
0 & 0& 0 & 0 & 0 & 0\\
0 & 0& 0 & 0 & 0 & 0\\
\end{array}\right).$$
Computing $MGC(A)$ for Teter rings
----------------------------------
The following algorithm provides a method to decide whether a non-Gorenstein ring $A=R/I$ has colength 1 and, if this is the case, it explicitly computes its $MGC(A)$.
$s$ socle degree of $A=R/I$;\
$b_1,\dots,b_t$ $\res$-basis of the inverse system $I^\perp$;\
$F_1,\dots,F_h$ such that $\overline{F_1},\dots,\overline{F_h}$ is an adapted $\res$-basis of $\int_{{\mathfrak m}}I^\perp/I^\perp$;\
$U_1,\dots,U_n$ contraction matrices of $\int_{\mathfrak m}I^\perp$. $F=a_1F_1+\dots+a_hF_h$ polynomial defining a minimal Gorenstein cover;\
$\mathfrak{a}$ ideal defining the elements not in $MGC(A)$.
1. Set $F=a_1F_1+\dots+a_hF_h$ and $\mathbf{F}=(a_1,\dots,a_h)^t$, where $a_1,\dots,a_h$ are variables in $\res$.
2. Build matrix $A=\left(\mu^\alpha_j\right)_{1\leq\vert\alpha\vert\leq s+1,1\leq j\leq t}$, where $$U^\alpha \textbf{F}=\sum_{j=1}^t\mu^\alpha_jb_j,\quad U^\alpha=U_1^{\alpha_1}\cdots U_n^{\alpha_n}.$$
3. Compute the ideal $\mathfrak{a}$ generated by all minors of order $t$ of the matrix $A$.
Let us consider a non-Gorenstein local Artin ring $A=R/I$ of socle degree $s$. Fix a $\res$-basis $b_1,\dots,b_t$ of $I^\perp$ and consider a polynomial $F=\sum_{j=1}^h a_jF_j\in \int_{{\mathfrak m}}I^\perp$, where $\overline{F}_1,\dots,\overline{F}_h$ is an adapted $\res$-basis of $\mathcal{L}_{A,1}$. According to , $F$ corresponds to a minimal Gorenstein cover if and only if $\dim_\res({\mathfrak m}\circ F)=t$. Therefore, we want to know for which values of $a_1,\dots,a_h$ this equality holds.
Note that $\deg F\leq s+1$ and $x_kx_l \circ F=x_lx_k\circ F$. Then ${\mathfrak m}\circ F=\langle x^{\alpha}\circ F: 1\leq\vert \alpha\vert\leq s+1\rangle_\res$, where $x^\alpha=x_1^{\alpha_1}\cdots x_n^{\alpha_n}$ and $\vert\alpha\vert=\alpha_1+\dots+\alpha_n$. Moreover, by definition of $F$, each $x^{\alpha}\circ F\in I^\perp$, hence $x^{\alpha}\circ F=\sum_{j=1}^t\mu_{\alpha}^jb_j$ for some $\mu_{\alpha}^j\in\res$.
Consider the matrix $A=(\mu_{\alpha}^j)_{1\leq\vert \alpha\vert\leq s+1,\,1\leq j\leq t}$, whose rows are the contractions $x^{\alpha}\circ F$ expressed in terms of the $\res$-basis $b_1,\dots,b_t$ of $I^\perp$. The rows of $A$ are a system of generators of ${\mathfrak m}\circ F$ as $\res$-vector space, hence $\dim_\res({\mathfrak m}\circ F)<t$ if and only if all order $t$ minors of $A$ vanish. Let $\mathfrak{a}$ be the ideal generated by all order $t$ minors $p_1,\dots,p_r$ of $A$. Note that the entries of matrix $A$ are homogeneous polynomials of degree 1 in $\res[a_1,\dots,a_h]$. Hence $\mathfrak{a}$ is generated by homogeneous polynomials of degree $t$ in $\res[a_1,\dots,a_h]$. Therefore, we can view the projective algebraic set $$\mathbb{V}_+(\mathfrak{a})=\lbrace [a_1:\dots:a_h]\in\mathbb{P}_\res^{h-1}\mid p_i(a_1,\dots,a_h)=0,\, 1\leq i\leq r\rbrace,$$ as the set of all points that do not correspond to Teter covers. We just proved the following result:
Let $A=R/I$ be an Artin ring with $\gcl(A)=1$, $h=\dim_\res\mathcal{L}_{A,1}$ and $\mathfrak{a}$ be the ideal of minors previously defined. Then $$MGC(A)=\mathbb{P}_\res^{h-1}\backslash\mathbb{V}(\mathfrak{a}).$$ Moreover, for any non-Gorenstein Artin ring $A$, $\gcl(A)=1$ if and only if $\mathfrak{a}\neq 0$.
The first part is already proved. On the other hand, if $\mathfrak{a}=0$, then $\mathbb{V}(\mathfrak{a})=\mathbb{P}_\res^{h-1}$ and $MGC(A)=\emptyset$. In other words, there exist no Teter covers, hence $\gcl(A)>1$.
With the following example we show how to interpret the output of the algorithm:
Consider $A=R/I$, with $R=\res[\![x_1,x_2]\!]$ and $I={\mathfrak m}^2$ [@ES17 Example 4.3].\
[Output]{}: $F=a_1y^4_2+a_2y_1y_2+a_3y_1^2$, $\rad(\mathfrak{a})=a_2^2-a_1a_3$.\
We consider points $(a_1:a_2:a_3)\in \mathbb{P}^2$. Then $MGC(A)=\mathbb{P}^2\backslash\lbrace a_2^2-a_1a_3=0\rbrace$ and any minimal Gorenstein cover $G=R/\ann F$ of $A$ is given by a polynomial $F=a_1y^4_2+a_2y_1y_2+a_3y_1^2$ such that $a_2^2-a_1a_3\neq 0$.
Computing $MGC(A)$ in colength 2
--------------------------------
Consider a $\res$-basis $b_1,\dots,b_t$ of $I^\perp$ and an adapted $\res$-basis $\overline{F}_1,\dots,\overline{F}_{h_1},\overline{G}_1,\dots,\overline{G}_{h_2}$ of $\mathcal{L}_{A,2}$ (see ) such that
- $b_1,\dots,b_t,F_1,\dots,F_{h_1}$ is a $\res$-basis of $\int_{\mathfrak m}I^\perp$,
- $b_1,\dots,b_t,F_1,\dots,F_{h_1},G_1,\dots,G_{h_2}$ is a $\res$-basis of $\int_{{\mathfrak m}^2} I^\perp$.
Consider a local Artin ring $A=R/I$. If a minimal Gorenstein cover $G=R/\ann H$ of colength 2 exists, then, by , $H$ is a polynomial of the form $$H=\sum_{i=1}^{h_1}\alpha_iF_i+\sum_{i=1}^{h_2}\beta_iG_i,\quad \alpha_i,\beta_i\in\res.$$ We want to obtain conditions on the $\alpha$’s and $\beta$’s under which $H$ actually generates a minimal Gorenstein cover of colength 2. By definition, $H\in\int_{{\mathfrak m}^2}I^\perp$, hence $x_k\circ H\in{\mathfrak m}\circ\int_{\mathfrak m}\left(\int_{\mathfrak m}I^\perp\right)\subseteq\int_{{\mathfrak m}}I^\perp$ and $$x_k\circ H=\sum_{j=1}^t\mu^j_kb_j+\sum_{j=1}^{h_1}\rho^j_kF_j,\quad \mu_k^j,\rho_k^j\in\res.$$ Set matrices $A_H=(\mu_k^j)$ and $B_H=(\rho_k^j)$. Let us describe matrix $B_H$ explicitly. We have $$x_k\circ H=\sum_{i=1}^{h_1}\alpha_i(x_k\circ F_i)+\sum_{i=1}^{h_2}\beta_i(x_k\circ G_i).$$ Note that each $x_k\circ G_i$, for any $1\leq i\leq h_2$, is in $\int_{{\mathfrak m}}I^\perp$ and hence it can be decomposed as $$x_k\circ G_i=\sum_{j=1}^t\lambda^{k,i}_jb_j+\sum_{j=1}^{h_1}a^{k,i}_jF_j,\quad \lambda^{k,i}_j,a_j^{k,i}\in\res.$$ Then $$x_k\circ H=\sum_{i=1}^{h_1}\alpha_i(x_k\circ F_i)+\sum_{i=1}^{h_2}\beta_i\left(\sum_{j=1}^t\lambda^{k,i}_jb_j+\sum_{j=1}^{h_1}a^{k,i}_jF_j\right)=b+\sum_{j=1}^{h_1}\left(\sum_{i=1}^{h_2}\beta_ia^{k,i}_j\right)F_j,$$ where $b:=\sum_{i=1}^{h_1}\alpha_i(x_k\circ F_i)+\sum_{i=1}^{h_2}\beta_i\left(\sum_{j=1}^t\lambda_j^{k,i}b_j\right)\in I^\perp$. Observe that $$\label{rho}
\rho_k^j=\sum_{i=1}^{h_2}a^{k,i}_j\beta_i,$$ hence the entries of matrix $B_H$ can be regarded as polynomials in variables $\beta_1,\dots,\beta_{h_2}$ with coefficients in $\res$.
\[rk(B)\] Consider the matrix $B_H=(\rho_k^j)$ as previously defined and let $B'_H=(\varrho^j_k)$ be the matrix of the coefficients of $\overline{L_k\circ H}=\sum_{j=1}^{h_1}\varrho_k^j\overline{F_j}\in\mathcal{L}_{A,1}$ where $L_1,\dots,L_n$ are independent linear forms. Then,
1. $\rk B_H=\dim_\res\left(\displaystyle\frac{{\mathfrak m}\circ H+I^\perp}{I^\perp}\right)$,
2. $\rk B'_H=\rk B_H$.
Note that $\overline{x_k\circ H}=\sum_{j=1}^{h_1}\rho_k^j\overline{F_j}\in\mathcal{L}_{A,1}$ and $\langle\overline{x_1\circ H},\dots,\overline{x_n\circ H}\rangle_\res=({\mathfrak m}\circ H+I^\perp)/I^\perp$. Since $\overline{F_1},\dots,\overline{F_{h_1}}$ is a $\res$-basis of $\mathcal{L}_{A,1}$ and $({\mathfrak m}\circ H+I^\perp)/I^\perp\subseteq \mathcal{L}_{A,1}$, then (i) holds.
For (ii) it will be enough to prove that $\langle \overline{x_1\circ H},\dots,\overline{x_n\circ H}\rangle_\res=\langle \overline{L_1\circ H},\dots,\overline{L_n\circ H}\rangle_\res$. Indeed, since $L_i=\sum_{j=1}^n\lambda^i_jx_j$ for any $1\leq i\leq n$, then $\overline{L_i\circ H}=\sum_{j=1}^n\lambda_j^i(\overline{x_j\circ H})\in\langle\overline{x_1\circ H},\dots,\overline{x_n\circ H}\rangle_\res$. The reverse inclusion comes from the fact that $(L_1,\dots,L_n)={\mathfrak m}$, hence $x_i$ can be expressed as a linear combination of $L_1,\dots,L_n$.
With the previous notation, consider a polynomial $H\in\int_{{\mathfrak m}^2}I^\perp$ with coefficients $\beta_1,\dots,\beta_{h_2}$ of $G_1,\dots,G_{h_2}$, respectively, and its corresponding matrix $B_H$. Then the following are equivalent:
1. $B_H\neq 0$,
2. ${\mathfrak m}\circ H\nsubseteq I^\perp$,
3. $(\beta_1,\dots,\beta_{h_2})\neq (0,\dots,0)$.
$(1)$ implies $(2)$. If $B_H\neq 0$, by , $({\mathfrak m}\circ H+I^\perp)/I^\perp\neq 0$ and hence ${\mathfrak m}\circ H\nsubseteq I^\perp$.
$(2)$ implies $(3)$. If ${\mathfrak m}\circ H\nsubseteq I^\perp$, by definition $H\notin \int_{\mathfrak m}I^\perp$ and hence $H\in\int_{{\mathfrak m}^2} I^\perp\backslash\int_{\mathfrak m}I^\perp$. Therefore, some $\beta_i$ must be non-zero.
$(3)$ implies $(1)$. Since $G_i\in\int_{{\mathfrak m}^2}I^\perp\backslash\int_{\mathfrak m}I^\perp$ for any $1\leq i\leq h_2$ and, by hypothesis, there is some non-zero $\beta_i$, we have that $H\in\int_{{\mathfrak m}^2}I^\perp\backslash \int_{\mathfrak m}I^\perp$. We claim that $x_k\circ H\in\int_{\mathfrak m}I^\perp\backslash I^\perp$ for some $k\in\lbrace 1,\dots,n\rbrace$. Suppose the claim is not true. Then $x_k\circ H\in I^\perp$ for any $1\leq k\leq n$, or equivalently, ${\mathfrak m}\circ H\subseteq I^\perp$ but, by definition, this means that $H\in\int_{\mathfrak m}I^\perp$, which is a contradiction. Since $$x_k\circ H=b+\sum_{j=1}^{h_1}\left(\sum_{i=1}^{h_2}\beta_ia^{k,i}_j\right)F_j\in\int_{\mathfrak m}I^\perp\backslash I^\perp,\quad b\in I^\perp,$$ for some $k\in\lbrace 1,\dots,n\rbrace$, then $\rho_k^j\neq 0$, for some $j\in\lbrace 1,\dots,h_1\rbrace$. Therefore, $B_H\neq 0$.
\[lemaB\] Consider the previous setting. If $B_H=0$, then either $\gcl(A)=0$ or $\gcl(A)=1$ or $R/\ann H$ is not a cover of $A$.
If $B_H=0$, then ${\mathfrak m}\circ H\subseteq I^\perp$ and hence $\ell(H)-1\leq \ell(I^\perp)$. If $I^\perp\subseteq \langle H\rangle$, then $G=R/\ann H$ is a Gorenstein cover of $A$ such that $\ell(G)-\ell(A)\leq 1$. Therefore, either $\gcl(A)\leq 1$ or $G$ is not a cover.
We already have techniques to check whether $A$ has colength 0 or 1. Therefore, we can assume $\gcl(A)\geq 2$. The previous two lemmas allow us to take into consideration only those polynomials $H$ such that $(\beta_1,\dots,\beta_{h_2})\neq (0,\dots,0)$ or, equivalently, $B_H\neq 0$. According to , $\gcl(A)=2$ if and only if $(L_1,\dots,L_{n-1},L_n^2)\circ H=I^\perp$ for some $H$ of the previously stated form and some independent linear forms $L_1,\dots,L_n$.
\[rk B\] Assume that $B_H\neq 0$. Then $\rk B_H=1$ if and only if $(L_1,\dots,L_{n-1},L_n^2)\circ H\subseteq I^\perp$ for some independent linear forms $L_1,\dots,L_n$.
Recall that, since we are under the assumption that $B_H\neq 0$, there exists $k$ such that $x_k\circ H\notin I^\perp$. Without loss of generality, we can assume that $x_n\circ H\notin I^\perp$. If $\rk B_H=1$, then any other row of $B_H$ must be a multiple of row $n$. Therefore, for any $1\leq i\leq n-1$, there exists $\lambda_i\in\res$ such that $(x_i-\lambda_ix_n)\circ H\in I^\perp$. Take $L_n:=x_k$ and $L_i:=x_i-\lambda_ix_n$. It is clear that $L_1,\dots,L_n$ are linearly independent and that $L_i\circ H\in I^\perp$ for any $1\leq i\leq n-1$. Moreover, $L_n^2\circ H=x_k^2\circ H\in {\mathfrak m}^2\circ\int_{m^2}I^\perp\subseteq I^\perp$.
Reciprocally, let $B'_H=(\varrho^j_k)$ be the matrix of the coefficients of $\overline{L_k\circ H}=\sum_{j=1}^{h_1}\varrho_k^j\overline{F_j}\in\mathcal{L}_{A,1}$. By , since $B_H\neq 0$, then $B'_H\neq 0$. We are assuming that $\overline{L_1\circ H}=\dots=\overline{L_{n-1}\circ H}=0$ but, since $B'_H\neq 0$, then $\overline{L_n\circ H}\neq 0$. It is clear that $\rk B'_H=1$ and hence, again by , $\rk B_H=1$.
Given $A=R/I$ such that $\gcl(A)\geq 2$, we can identify any polynomial $H=\sum_{i=1}^{h_1}\alpha_i F_i+\sum_{i=1}^{h_2}\beta_i G_i$ with a point $[H]$ in $\mathbb{P}_\res(\mathcal{L}_{A,2})$ by taking coordinates $(\alpha_1:\dots:\alpha_{h_1}:\beta_1:\dots:\beta_{h_2})$. Note that having $\gcl(A)\geq 2$ ensures that some $\beta_i$ is non-zero and hence it makes sure that $[H]$ is a well-defined point of the projective space.
On the other hand, by , any minor of $B_H=(\rho_k^j)$ is a homogeneous polynomial in variables $\beta_1,\dots,\beta_{h_2}$. Therefore, we can consider the ideal $\mathfrak{b}$ generated by all order-2-minors of $B_H$ in $\res[\alpha_1,\dots,\alpha_{h_1},\beta_1,\dots,\beta_{h_2}]$. Note that $\mathbb{V}_+(\mathfrak{b})$ can be considered as a projective variety in $\mathbb{P}_\res\left(\mathcal{L}_{A,2}\right)$. Observe that $\mathbb{P}_\res\left(\mathcal{L}_{A,2}\right)$ is a projective space over $\res$ of dimension $h_1+h_2-1$, hence we will denote it by $\mathbb{P}_\res^{h_1+h_2-1}$.
\[MGC1\] Let $A=R/I$ be an Artin ring such that $\gcl(A)=2$. Then $$MGC(A)\subseteq\mathbb{V}_+(\mathfrak{b})\subseteq\mathbb{P}_\res^{h_1+h_2-1}.$$
Note that $[H]\in\mathbb{V}_+(\mathfrak{b})$ if and only if all order-2-minors of $B_H$ vanish. In other words, if and only if $\rk B_H\leq 1$. On the other hand, if $[H]\in MGC(A)$, then $G=R/\ann H$ is a cover of colength 2 of $A$. Hence $I^\perp=(L_1,\dots,L_{n-1},L_n^2)\circ H$ for some $L_1,\dots,L_n$ and, by the previous proposition, $\rk B_H=1$. Therefore, $[H]\in\mathbb{V}_+(\mathfrak{b})$.
Note that the condition of $\rk B_H=1$ does not depend on $L_1,\dots,L_n$ and does not provide any information about what choices satisfy the inclusion. In fact, we are only interested in knowing which are the particular $L_n$ that meet the requirements. To that end, we fix $L_n=v_1x_1+\dots+v_nx_n$ and choose linear forms $L_i=\lambda_1^ix_1+\dots+\lambda_n^ix_n$, for $1\leq i\leq n-1$, such that $L_1,\dots,L_n$ is a $\res$-basis of $\langle x_1,\dots,x_n\rangle_\res$ and $\lambda_i\cdot v=0$, where $v=(v_1,\dots,v_n)$ and $\lambda_i=(\lambda_1^i,\dots,\lambda_n^i)$. We recall that $\langle L_1,\dots,L_{n-1}\rangle_\res=\langle\lbrace v_jx_i-v_ix_j\rbrace_{1\leq i<j\leq n}\rangle_\res$, which will be a useful fact in future propositions.
Define the following matrix: $$C_{H,v}=\left(\begin{array}{cccc}
\rho_1^1 & \dots & \rho_1^{h_1} & v_1\\
\vdots & & \vdots & \vdots\\
\rho_n^1 &\dots & \rho_n^{h_1} & v_n\\
\end{array}\right)$$
\[rk C\] Assume $B_H\neq 0$. $(L_1,\dots,L_{n-1},L_n^2)\circ H\subseteq I^\perp$ if and only if $\rk C_{H,v}=1$.
By , $(L_1,\dots,L_{n-1},L_n^2)\circ H\subseteq I^\perp$ if and only if $\rk B_H=1$. Therefore, we need to prove that $\rk C_{H,v}=1$ if and only $\rk B_H=1$.
On one hand, from $\rk C_{H,v}=1$ and $B_H\neq 0$ we directly deduce that $\rk B_H=1$. On the other hand, if $\rk B_H=1$, then we have $\rk C_{H,v}=1$ if and only if $\left\vert\begin{array}{cc}
\rho_k^j & v_k\\
\rho_l^j & v_l
\end{array}\right\vert=0$ for any $1\leq k<l\leq n$ and $1\leq j\leq h_1$. Let us check that these minors indeed vanish. Since $\langle L_1,\dots,L_{n-1}\rangle_\res=\langle\lbrace v_lx_k-v_kx_l\rbrace_{1\leq k<l\leq n}\rangle_\res$, then $v_l(x_k\circ H)-v_k(x_l\circ H)\in I^\perp$ for any $\quad 1\leq k<l\leq n.$ But $$v_l(x_k\circ H)-v_k(x_l\circ H)=b+\sum_{j=1}^{h_1}\left(v_l\rho_k^j-v_k\rho_l^j\right)F_j\in I^\perp,\quad \mbox{for some }b\in I^\perp,$$ hence $v_l\rho_k^j-v_k\rho_l^j=0$.
\[adm\] We say that $v=(v_1,\dots,v_n)$ is an admissible vector of $H$ if $v\neq 0$ and $v_l\rho_k^j-v_k\rho_l^j=0$ for any $1\leq k<l\leq n$ and $1\leq j\leq h_1$.
\[UniqueV\] Given a polynomial $H$ of the previous form:
1. there always exists an admissible vector $v\in\res^n$ of $H$;
2. if $w\in\res^n$ such that $w=\lambda v$, with $\lambda\in\res^\ast$, then $w$ is an admissible vector of $H$;
3. the admissible vector of $H$ is unique up to multiplication by elements of $\res^\ast$.
$(i)$ The existence of $v$ is deduced directly from the construction of $L_n=v_1x_1+\dots+v_nx_n$ in .\
$(ii)$ Clearly $w=\lambda v\neq 0$ and $w_l\rho_k^j-w_k\rho_l^j=\lambda(v_l\rho_k^j-v_k\rho_l^j)=0$.\
$(iii)$ Since we can assume that $B_H\neq 0$, there exists $\rho_k^j\neq 0$ for some $1\leq j\leq h_1$ and $1\leq k\leq n$. We will first prove that $v_k\neq 0$. Suppose that $v_k=0$. By , there exists $v_i\neq 0$, $i\neq k$, and $v_i\rho_k^j-v_k\rho_i^j=0$. Then $v_i\rho_k^j=0$ and we reach a contradiction. Consider now $w=(w_1,\dots,w_n)$ admissible with respect to $H$. From $\rho_k^jv_l-\rho_l^jv_k=0$ and $\rho_k^jw_l-\rho_l^jw_k=0$, we get $v_l=\left(\rho_l^j/\rho_k^j\right)v_k$ and $w_l=\left(\rho_l^j/\rho_k^j\right)w_k$. Set $\lambda_l:=\rho_l^j/\rho_k^j$. For any $1\leq l\leq n$, with $l\neq k$, from $v_l=\lambda_lv_k$ and $w_l=\lambda_lw_k$, we deduce that $w_l=\left(w_k/v_k\right)v_l$. Hence $w=\lambda v$, where $\lambda=w_k/v_k$, and any two admissible vectors of $H$ are linearly dependent.
We now want to provide a geometric interpretation of pairs of polynomials and admissible vectors and describe the variety where they lay. On one hand, as a consequence of , we can identify an admissible vector $v$ with a point $[v]$ of the projective space $\mathbb{P}^{n-1}_\res$ with homogeneous coordinates $(v_1:\dots:v_n)$. On the other hand, recall that the entries $\rho_j^k$ of matrix $B_H$ can be regarded as polynomials in variables $\beta_1,\dots,\beta_{h_2}$ and coefficients in $\res$, by . Then consider the ideal generated in $\res[\alpha_1,\dots,\alpha_{h_1},\beta_1,\dots,\beta_{h_2},v_1,\dots,v_n]$ by polynomials of the form $$\rho_k^j\rho_m^l-\rho_k^l\rho_m^j,\quad 1\leq k<m\leq n,1\leq j<l\leq h_1$$ and $$v_l\rho_k^j-v_k\rho_l^j,\quad 1\leq k<l\leq n,1\leq j\leq h_1.$$ It can be checked that all these polynomials are bihomogeneous polynomials in the sets of variables $\lbrace \alpha_1,\dots,\alpha_{h_1},\beta_1,\dots,\beta_{h_2}\rbrace$ and $\lbrace v_1,\dots,v_n\rbrace$. Therefore this ideal defines a variety in $\mathbb{P}_\res^{h_1+h_2-1}\times \mathbb{P}_\res^{n-1}$ the points of which satisfy the following equations: $$\label{Gens1}
\rho_k^j\rho_m^l-\rho_k^l\rho_m^j=0,\quad 1\leq k<m\leq n,1\leq j<l\leq h_1;$$ $$\label{Gens2}
v_l\rho_k^j-v_k\rho_l^j=0,\quad 1\leq k<l\leq n,1\leq j\leq h_1.$$
We denote by $\mathfrak{c}$ the ideal in $\res[\alpha_1,\dots,\alpha_{h_1},\beta_1,\dots,\beta_{h_2},v_1,\dots,v_n]$ generated by all order 2 minors of $C_{H,v}$. We denote by $\mathbb{V}_+(\mathfrak{c})$ the variety defined by $\mathfrak{c}$ in $\mathbb{P}_\res^{h_1+h_2-1}\times \mathbb{P}_\res^{n-1}$.
\[vc\] With the previous definitions, the set of points of $\mathbb{V}_+(\mathfrak{c})$ is $$\left\lbrace ([H],[v])\in \mathbb{P}_\res^{h_1+h_2-1}\times \mathbb{P}_\res^{n-1}\mid [H]\in\mathbb{V}_+(\mathfrak{b})\mbox{ and $v$ admissible with respect to $H$}\right\rbrace.$$
Consider a point $[p]$ in $\mathbb{V}_+(\mathfrak{c})\subseteq \mathbb{P}_\res^{h_1+h_2-1}\times \mathbb{P}_\res^{n-1}$. Then $[p]=([p_\beta],[p_v])$, with homogeneous coordinates $[p_\beta]=(\alpha_1:\dots:\alpha_{h_1}:\beta_1:\dots:\beta_{h_2})\in\mathbb{P}_\res^{h_1+h_2-1}$ and $[p_v]=(v_1:\dots:v_n)\in\mathbb{P}_\res^{n-1}$.\
On one hand, the coordinates of $[p_\beta]$ satisfy , which are precisely the conditions required to have $[p_\beta]\in\mathbb{V}_+(\mathfrak{b})$. On the other hand, coordinates of $[p_\beta]$ and $[p_v]$ satisfy , hence $v=(v_1,\dots,v_n)$ is admissible with respect to the polynomial $H=\sum_{i=1}^{h_1}\alpha_iF_i+\sum_{i=1}^{h_2}\beta_iG_i$. Therefore, $([p_\beta],[p_v])=([H],[v])$ for some $[H]\in\mathbb{V}_+(\mathfrak{b})$ and some $v$ admissible.\
Conversely, any pair of the form $([H],[v])$ with $[H]\in\mathbb{V}_+(\mathfrak{b})$ and $v$ admissible satisfy and by definition.
\[proj\] Let $\pi_1$ be the projection map from $\mathbb{P}_\res^{h_1+h_2-1}\times \mathbb{P}_\res^{n-1}\longrightarrow \mathbb{P}_\res^{h_1+h_2-1}$. Then,
1. $\pi_1(\mathbb{V}_+(\mathfrak{c}))=\mathbb{V}_+(\mathfrak{b})$;
2. there is a one-to-one correspondence between points $[H]\in\mathbb{V}_+(\mathfrak{b})$ and points $([H],[v])\in \mathbb{V}_+(\mathfrak{c})$.
$(i)$ Any element of $\mathbb{V}_+(\mathfrak{c})$ is of the form $([H],[v])$ described in . Then $\pi_1([H],[v])=[H]\in\mathbb{V}_+(\mathfrak{b})$. Conversely, given an element $[H]\in\mathbb{V}_+(\mathfrak{b})$, by there exist independent linear forms $L_1,\dots,L_n$ such that $(L_1,\dots,L_{n-1},L_n^2)\circ H\subseteq I^\perp$. Hence, taking $v$ as the vector of coefficients of $L_n$ and applying , we get an admissible $v$ with respect to $H$. Therefore, $[H]=\pi_1([H],[v])\in\pi_1(\mathbb{V}_+(\mathfrak{c}))$.\
$(ii)$ For any $[H]\in\mathbb{V}_+(\mathfrak{b})$, applies to $H$ and there always exists an admissible vector $v\in\res^n$ of $H$. By definition, $([H],[v])\in \mathbb{V}_+(\mathfrak{c})$. By the same lemma, any admissible vector of $H$ is of the form $w=\lambda v$, with $\lambda\in\res^\ast$, and hence $[w]=[v]\in \mathbb{P}_\res^{n-1}$. Then the assignation $i_1([H])=([H],[v])$ is a well-defined map from $\mathbb{V}_+(\mathfrak{b})$ to $\mathbb{V}_+(\mathfrak{c})$. Note that $\pi_1$ is the inverse map and hence we have a bijection.
From , we know that all covers $G=R/\ann H$ of colength 2 correspond to points $[H]\in\mathbb{V}_+(\mathfrak{b})$ but, in general, not all points of $\mathbb{V}_+(\mathfrak{b})$ correspond to covers of $A=R/I$. Therefore, we need to identify and remove those $[H]$ such that $(L_1,\dots,L_{n-1},L_n^2)\circ H\subsetneq I^\perp$.\
As $\res$-vector space, $(L_1,\dots,L_{n-1},L_n^2)\circ H$ is generated by
- $(v_l x_k-v_k x_l)\circ H$, $1\leq k<l\leq n$;
- $\underline{x}^{\underline{\theta}}\circ H$, $2\leq\vert\underline{\theta}\vert\leq s+2$.
Since $(L_1,\dots,L_{n-1},L_n^2)\circ H\subseteq I^\perp$, we can provide an explicit description of these generators with respect to the $\res$-basis $b_1,\dots,b_t$ of $I^\perp$ as follows: $$(x_kv_l-x_lv_k)\circ H=\sum_{j=1}^t\left(v_l\sum_{i=1}^{h_1}\alpha_i\mu_j^{k,i}-v_k\sum_{i=1}^{h_1}\alpha_i\mu_j^{l,i}+v_l\sum_{i=1}^{h_2}\beta_i\lambda_j^{k,i}-v_k\sum_{i=1}^{h_2}\beta_i\lambda_j^{l,i}\right)b_j,$$ for $1\leq l<k\leq n$, with $x_k\circ F_i=\sum_{j=1}^t\mu_j^{k,i}b_j$ and $x_k\circ G_i=\sum_{j=1}^t\lambda_j^{k,i}b_j+\sum_{j=1}^{h_1}a_j^{k,i}F_j$, $\mu_j^{k,i},\lambda_j^{k,i},a_j^{k,i}\in\res$; $$\underline{x}^{\underline{\theta}}\circ H=\sum_{j=1}^t\left(\sum_{i=1}^{h_1}\mu_j^{\underline{\theta},i}\alpha_i+\sum_{i=1}^{h_2}\lambda_j^{\underline{\theta},i}\beta_i\right)b_j,$$ where $2\leq\vert\underline{\theta}\vert\leq s+2$, $\underline{x}^{\underline{\theta}}\circ F_i=\sum_{j=1}^t\mu_j^{\underline{\theta},i}b_j$ and $\underline{x}^{\underline{\theta}}\circ G_i=\sum_{j=1}^t\lambda_j^{\underline{\theta},i}b_j$, $\mu_j^{\underline{\theta},i},\lambda_j^{\underline{\theta},i}\in\res$.
We now define matrix $U_{H,v}$ such that its rows are the coefficients of each generator of $(L_1,\dots,L_{n-1},L_n^2)\circ H$ with respect to the $\res$-basis $b_1,\dots,b_t$ of $I^\perp$: $$\begin{array}{r|ccc}
• & b_1 & \dots & b_t\\
\hline
(x_2v_1-x_1v_2)\circ H & \varrho^1_{1,2} & \cdots & \varrho^t_{1,2}\\
\vdots & \vdots & & \vdots \\
(x_nv_{n-1}-x_{n-1}v_n)\circ H & \varrho^1_{n-1,n} & \cdots & \varrho^t_{n-1,n} \\
x_1^2\circ H & \varsigma^1_{(2,0,\dots,0)} & \cdots & \varsigma^t_{(2,0,\dots,0)} \\
x_1x_2\circ H & \varsigma^1_{(1,1,0,\dots,0)} & \cdots & \varsigma^t_{(1,1,0,\dots,0)} \\
\vdots & \vdots & & \vdots \\
x_n^2\circ H & \varsigma^1_{(0,\dots,0,2)} & \cdots & \varsigma^t_{(0,\dots,0,2)} \\
\vdots & \vdots & & \vdots \\
x_n^{s+2}\circ H & \varsigma^1_{(0,\dots,0,s+2)} & \cdots & \varsigma^t_{(0,\dots,0,s+2)}\\
\end{array}$$ where $$\varrho^j_{l,k}:=v_l\sum_{i=1}^{h_1}\alpha_i\mu_j^{k,i}-v_k\sum_{i=1}^{h_1}\alpha_i\mu_j^{l,i}+v_l\sum_{i=1}^{h_2}\beta_i\lambda_j^{k,i}-v_k\sum_{i=1}^{h_2}\beta_i\lambda_j^{l,i}$$ and $$\varsigma^j_{\underline{\theta}}:=\sum_{i=1}^{h_1}\mu_j^{\underline{\theta},i}\alpha_i+\sum_{i=1}^{h_2}\lambda_j^{\underline{\theta},i}\beta_i.$$ It can be easily checked that the entries of this matrix are either bihomogeneous polynomials $\varrho^j_{l,k}$ in variables $((\underline{\alpha},\underline{\beta}),\underline{v})$ of bidegree $(1,1)$ or homogeneous polynomials $\varsigma^j_{\underline{\theta}}$ in variables $(\underline{\alpha},\underline{\beta})$ of degree 1. Let $\mathfrak{a}$ be the ideal in $\res[\alpha_1,\dots,\alpha_{h_1},\beta_1,\dots,\beta_{h_2},v_1,\dots,v_n]$ generated by all minors of $U_{H,v}$ of order $t=\dim_\res I^\perp$. It can be checked that it is a bihomogeneous ideal in variables $((\underline{\alpha},\underline{\beta}),\underline{v})$, hence we can think of $\mathbb{V}_+(\mathfrak{a})$ as a variety in $\mathbb{P}^{h_1+h_2-1}\times\mathbb{P}^{n-1}$. Note that the variety can be described as follows: $$\mathbb{V}_+(\mathfrak{a})=\lbrace ([H],[v])\in\mathbb{P}^{h_1+h_2-1}\times\mathbb{P}^{n-1}\mid \rk U_{H,v}<t\rbrace.$$
\[MGC2\] Given $[H]\in\mathbb{V}_+(\mathfrak{b})$ and $([H],[v])\in\mathbb{V}_+(\mathfrak{c})$ its image by the bijection defined in , then $$[H]\in MGC(A)\Longleftrightarrow ([H],[v])\notin \mathbb{V}_+(\mathfrak{a}),$$
By , $([H],[v])\in\mathbb{V}_+(\mathfrak{c})$ if and only if $(L_1,\dots,L_{n-1},L_n^2)\circ H\subseteq I^\perp$, where $L_n=v_1x_1+\dots+v_nx_n$. Under such conditions, $G=R/\ann H$ is a cover if and only if $\dim_\res (L_1,\dots,L_{n-1},L_n^2)\circ H=\dim_\res I^\perp=t$. In other words, $[H]\in MGC(A)$ if and only if $\rk U_{H,v}=t$.
With the previous definitions, $$MGC(A)=\mathbb{V}_+(\mathfrak{b})\backslash \pi_1\left(\mathbb{V}_+(\mathfrak{c})\cap\mathbb{V}_+(\mathfrak{a})\right).$$
Consider $[H]\in MGC(A)$. Since $MGC(A)\subseteq \mathbb{V}_+(\mathfrak{b})$ by , to prove the right inclusion it will be enough to check that $[H]\notin \pi_1\left(\mathbb{V}_+(\mathfrak{c})\cap\mathbb{V}_+(\mathfrak{a})\right)$. Suppose it is not true, that is, $[H]\in \pi_1\left(\mathbb{V}_+(\mathfrak{c})\cap\mathbb{V}_+(\mathfrak{a})\right)$. Note that the map $i_1$ of is still the inverse of $\pi_1$ when restricted to $\mathbb{V}_+(\mathfrak{c})\cap\mathbb{V}_+(\mathfrak{a})\subseteq \mathbb{V}_+(\mathfrak{c})$. Therefore, $$i_1([H])=([H],[v])\in i_1\left(\pi_1\left(\mathbb{V}_+(\mathfrak{c})\cap\mathbb{V}_+(\mathfrak{a})\right)\right)=\mathbb{V}_+(\mathfrak{c})\cap\mathbb{V}_+(\mathfrak{a}).$$ But then $([H],[v])\in\mathbb{V}_+(\mathfrak{a})$ and, by , $G=R/\ann H$ is not a cover, hence we reach a contradiction.\
Conversely, if $[H]$ is in $\mathbb{V}_+(\mathfrak{b})$ but not in $\pi_1\left(\mathbb{V}_+(\mathfrak{c})\cap\mathbb{V}_+(\mathfrak{a})\right)$, then $([H],[v])\notin \mathbb{V}_+(\mathfrak{c})\cap\mathbb{V}_+(\mathfrak{a})$. But $([H],[v])$ is in $\mathbb{V}_+(\mathfrak{c})$ by definition, hence the only possibility is that $([H],[v])\notin \mathbb{V}_+(\mathfrak{a})$. Therefore, $[H]\in MGC(A)$ by .
Finally, let us recall the following result for bihomogeneous ideals:
[@CLO97 Exercise 16, Section 8.5] Let ideals $\mathfrak{a},\mathfrak{c}$ be as previously defined, $\mathfrak{d}=\mathfrak{a}+\mathfrak{c}$ the sum ideal and $\pi_1:\mathbb{P}_\res^{h_1+h_2-1}\times \mathbb{P}_\res^{n-1}\longrightarrow \mathbb{P}_\res^{h_1+h_2-1}$ be the projection map. Let $\widehat{\mathfrak{d}}$ be the projective elimination of the ideal $\mathfrak{d}$ with respect to variables $v_1,\dots,v_n$. Then, $$\pi_1(\mathbb{V}_+(\mathbb{\mathfrak{a}})\cap\mathbb{V}_+(\mathbb{\mathfrak{c}}))=\mathbb{V}_+(\widehat{\mathfrak{d}}).$$
$s$ socle degree of $A=R/I$; $b_1,\dots,b_t$ $\res$-basis of the inverse system $I^\perp$; $F_1,\dots,F_{h_1},G_1,\dots,G_{h_2}$ an adapted $\res$-basis of $\mathcal{L}_{A,2}$; $U_1,\dots,U_n$ contraction matrices of $\int_{{\mathfrak m}^2} I^\perp$. $H=\alpha_1F_1+\dots+\alpha_{h_1}F_{h_1}+\beta_1G_1+\dots+\beta_{h_2}G_{h_2}$ polynomial defining a minimal Gorenstein cover, $\mathfrak{b}$ ideal such that $MGC(A)\subset\mathbb{V}_+(\mathfrak{b})$, $\widehat{\mathfrak{d}}$ ideal defining the elements of $\mathbb{V}_+(\mathfrak{b})$ not in $MGC(A)$.
1. Set $H=\alpha_1F_1+\dots+\alpha_{h_1}F_{h_1}+\beta_1G_1+\dots+\beta_{h_2}G_{h_2}$, where $\underline{\alpha},\underline{\beta}$ are variables in $\res$. Set column vectors $\textbf{H}=(0,\dots,0,\underline{\alpha},\underline{\beta})^t$ and $v=(v_1,\dots,v_n)^t$ in $R=\res[\underline{\alpha},\underline{\beta},\underline{v}]$, where the first $t$ components of $\textbf{H}$ are zero.
2. Build matrix $B_H=(\rho_i^j)_{1\leq i\leq n,\,1\leq j\leq h_1}$, where $U_i\textbf{H}$ is the column vector $(\mu_i^1,\dots,\mu_i^t,\rho_i^1,\dots,\rho_i^{h_1},0,\dots,0)^t$.
3. Build matrix $C_{H,v}=\left(\begin{array}{c|c}
B_H & v\\
\end{array}\right)$ as an horizontal concatenation of $B_H$ and the column vector $v$.
4. Compute the ideal $\mathfrak{c}\subseteq R$ generated by all minors of order 2 of $B_H$.
5. Build matrix $U_{H,v}$ as a vertical concatenation of matrices $(\varrho^j_{l,k})_{1\leq j\leq h_1,\,1\leq l<k\leq n}$ and $(\varsigma_\theta^j)_{2\leq\vert\theta\vert\leq s+2,\,1\leq j\leq h_1}$, such that $(v_l U_k-v_k U_l)\textbf{H}=(\varrho_{l,k}^1,\cdots,\varrho_{l,k}^{h_1},0,\cdots,0)^t$ and $U^\theta \textbf{H}=(\varsigma_\theta^1,\cdots,\varsigma_\theta^{h_1},0,\cdots,0)^t$, with $1\leq k<l\leq n$ and $2\leq\vert\theta\vert\leq s+2$.
6. Compute the ideal $\mathfrak{a}\subseteq R$ generated by all minors of order $t$ of $U_{H,v}$ and the ideal $\mathfrak{d}=\mathfrak{a}+\mathfrak{c}\subseteq R$ .
7. Compute $\widehat{\mathfrak{d}}\subseteq R'=\res[\underline{\alpha},\underline{\beta}]$, where $\widehat{\cdot}$ denotes the projective elimination of the ideal in $R$ with respect to variables $v_1,\dots,v_n$.
8. Compute the ideal $\mathfrak{b}:=\widehat{\mathfrak{c}}\subseteq R'$.
The output of can be interpreted as $MGC(A)=\mathbb{V}_+(\mathfrak{b})\backslash \mathbb{V}_+(\mathfrak{\widehat{d}})$. Moreover, any point $[\alpha_1:\dots:\alpha_{h_1}:\beta_1:\dots:\beta_{h_2}]\in MGC(A)$ corresponds to a minimal Gorenstein cover $G=R/\ann H$ of colength 2 of $A$, where $H=\alpha_1F_1+\dots+\alpha_{h_1}F_{h_1}+\beta_1G_1+\dots+\beta_{h_2}G_{h_2}$.
Consider $A=R/I$, with $R=\res[\![x_1,x_2]\!]$ and $I=(x_1^2,x_1x_2^2,x_2^4)$.\
[Output]{}: $H=a_1y_2^4+a_2y_1y_2^2+a_3y_1^2+b_1y_1^2y_2+b_2y_1y_2^3+b_3y_2^5+b_4y_1^3$, $\mathfrak{b}=(b_3b_4,b_2b_4)$, $\mathfrak{\widehat{d}}=(b_3b_4,b_2b_4,b_2^2-b_1b_3)$.\
Then $MGC(A)=\mathbb{V}_+(b_3b_4,b_2b_4)\backslash\mathbb{V}_+ (b_3b_4,b_2b_4,b_2^2-b_1b_3)=\mathbb{V}_+(b_3b_4,b_2b_4)\backslash\mathbb{V}_+ (b_2^2-b_1b_3).$ Note that if $b_3b_4=b_2b_4=0$ and $b_4\neq 0$, then both $b_2$ and $b_3$ are zero and the condition $b_2^2-b_1b_3=0$ always holds. Therefore, $$MGC(A)=\mathbb{V}_+(b_4)\backslash\mathbb{V}_+ (b_2^2-b_1b_3)\simeq \mathbb{P}^5\backslash\mathbb{V}_+ (b_2^2-b_1b_3),$$ where $(a_1:a_2:a_3:b_1:b_2:b_3)$ are the coordinates of the points in $\mathbb{P}^5$. Moreover, any minimal Gorenstein cover is of the form $G=R/\ann H$, where $$H=a_1y_2^4+a_2y_1y_2^2+a_3y_1^2+b_1y_1^2y_2+b_2y_1y_2^3+b_3y_2^5$$ satisfies $b_2^2-b_1b_3\neq 0$. All such covers admit $(x_1,x_2^2)$ as the corresponding $K_H$.
Computations
============
The first aim of this section is to provide a wide range of examples of the computation of the minimal Gorenstein cover variety of a local ring $A$. In [@Poo08a], Poonen provides a complete classification of local algebras over an algebraically closed field of length equal or less than 6. Note that, for higher lengths, the number of isomorphism classes is no longer finite. We will go through all algebras of Poonen’s list and restrict, for the sake of simplicity, to fields of characteristic zero.
On the other hand, we also intend to test the efficiency of the algorithms by collecting the computation times. We have implemented algorithms 1, 2 and 3 of in the commutative algebra software *Singular* [@DGPS]. The computer we use runs into the operating system Microsoft Windows 10 Pro and its technical specifications are the following: Surface Pro 3; Processor: 1.90 GHz Intel Core i5-4300U 3 MB SmartCache; Memory: 4GB 1600MHz DDR3.
Teter varieties
---------------
In this first part of the section we are interested in the computation of Teter varieties, that is, the $MGC(A)$ variety for local $\res$-algebras $A$ of Gorenstein colength 1. All the results are obtained by running in *Singular*.
Consider $A=R/I$, with $R=\res[\![x_1,x_2,x_3]\!]$ and $I=(x_1^2,x_1x_2,x_1x_3,x_2x_3,x_2^3,x_3^3)$. Note that $\HF_A=\lbrace 1,3,2\rbrace$ and $\tau(A)=3$. The output provided by our implementation of the algorithm in *Singular* [@DGPS] is the following: We consider points with coordinates $(a_1:a_2:a_3:a_4:a_5:a_6)\in\mathbb{P}^5$. Therefore, $MGC(A)=\mathbb{P}^5\backslash\mathbb{V}_+(a_1a_4a_6)$ and any minimal Gorenstein cover is of the form $G=R/\ann H$, where $H=a_1y_3^3+a_2y_2y_3+a_3y_1y_3+a_4y_2^3+a_5y_1y_2+a_6y_1^2$ with $a_1a_4a_6\neq 0$.
In below we show the computation time (in seconds) of all isomorphism classes of local $\res$-algebras $A$ of $\gcl(A)=1$ appearing in Poonen’s classification [@Poo08a]. In this table, we list the Hilbert function of $A=R/I$, the expression of the ideal $I$ up to linear isomorphism, the dimension $h-1$ of the projective space $\mathbb{P}^{h-1}$ where the variety $MGC(A)$ lies and the computation time. Note that our implementation of includes also the computation of the $\res$-basis of $\int_{\mathfrak m}I^\perp$, hence the computation time corresponds to the total.
Note that also allows us to prove that all the other non-Gorenstein local rings appearing in Poonen’s list have Gorenstein colength at least 2.
Minimal Gorenstein covers variety in colength 2
-----------------------------------------------
Now we want to compute $MGC(A)$ for $\gcl(A)=2$. All the examples are obtained by running in *Singular*.
Consider $A=R/I$, with $R=\res[\![x_1,x_2,x_3]\!]$ and $I=(x_1^2,x_2^2,x_3^2,x_1x_2,x_1x_3)$. Note that $\HF_A=\lbrace 1,3,1\rbrace$ and $\tau(A)=2$. The output provided by our implementation of the algorithm in *Singular* [@DGPS] is the following: We can simplify the output by using the primary decomposition of the ideal $\mathfrak{b}=\bigcap_{i=1}^k \mathfrak{b}_i$. Then, $$MGC(A)=\left(\bigcup_{i=1}^k \mathbb{V}_+(\mathfrak{b}_i)\right)\backslash\mathbb{V}_+(\mathfrak{\widehat{d}})=\bigcup_{i=1}^k \left(\mathbb{V}_+(\mathfrak{b}_i)\backslash\mathbb{V}_+(\mathfrak{\widehat{d}})\right).$$ *Singular* [@DGPS] provides a primary decomposition $\mathfrak{b}=\mathfrak{b}_1\cap \mathfrak{b}_2$ that satisfies $\mathbb{V}_+(\mathfrak{b}_2)\backslash\mathbb{V}_+(\mathfrak{\widehat{d}})=\emptyset$. Therefore, we get $$MGC(A)=\mathbb{V}_+(b_1,b_2,b_4,b_7,b_8,b_{10},b_3b_6-b_5b_9)\backslash\left(\mathbb{V}_+(a_5)\cup\mathbb{V}_+ (-b_6^3+b_5^2b_9,b_3b_5-b_6^2,b_3^2-b_6b_9)\right).$$ in $\mathbb{P}^{14}$. We can eliminate some of the variables and consider $MGC(A)$ to be the following variety: $$MGC(A)=\mathbb{V}_+(b_3b_6-b_5b_9)\backslash\left(\mathbb{V}_+(a_5)\cup\mathbb{V}_+ (b_5^2b_9-b_6^3,b_3b_5-b_6^2,b_3^2-b_6b_9)\right)\subset \mathbb{P}^8.$$ Therefore, any minimal Gorenstein cover is of the form $G=R/\ann H$, where $$H=a_1y_3^2+a_2y_1y_3+a_3y_2^2+a_4y_1y_2+a_5y_1^2+b_3y_2^2y_3+b_5y_3^3+b_6y_2y_3^2+b_9y_2^3$$ satisfies $b_3b_6-b_5b_9=0$, $a_5\neq 0$ and at least one of the following conditions: $b_5^2b_9-b_6^3\neq 0, b_3b_5-b_6^2\neq 0, b_3^2-b_6b_9\neq 0$.
Moreover, note that $\mathbb{V}_+(\mathfrak{c})\backslash\mathbb{V}_+(\mathfrak{a})=\mathbb{V}_+(\mathfrak{c_1})\backslash\mathbb{V}_+(\mathfrak{a})$, where $\mathfrak{c}=\mathfrak{c}_1\cap \mathfrak{c}_2$ is the primary decomposition of $\mathfrak{c}$ and $\mathfrak{c}_1=\mathfrak{b}_1+(v_1,v_2b_5-v_3b_6,v_2b_3-v_3b_9)$. Hence, any $K_H$ such that $K_H\circ H=I^\perp$ will be of the form $K_H=(L_1,L_2,L_3^2)$, where $L_1,L_2,L_3$ are independent linear forms in $R$ such that $L_3=v_2x_2+v_3x_3$, with $v_2b_5-v_3b_6=v_2b_3-v_3b_9=0$.
Consider $A=R/I$, with $R=\res[\![x_1,x_2,x_3]\!]$ and $I=(x_1x_2,x_1x_3,x_2x_3,x_2^2,x_3^2-x_1^3)$. Note that $\HF_A=\lbrace 1,3,1,1\rbrace$ and $\tau(A)=2$. The output provided by our implementation of the algorithm in *Singular* [@DGPS] is the following: *Singular* [@DGPS] provides a primary decomposition $\mathfrak{b}=\mathfrak{b}_1\cap \mathfrak{b}_2\cap\mathfrak{b}_3$ such that $\mathbb{V}_+(\mathfrak{b})\backslash\mathbb{V}_+(\mathfrak{\widehat{d}})=\mathbb{V}_+(\mathfrak{b}_2)\backslash\mathbb{V}_+(\mathfrak{\widehat{d}})$. Therefore, we get $$MGC(A)=\mathbb{V}_+(b_1,b_2,b_4,b_7,b_8,b_9,b_3^2-b_5b_6+b_3b_{10})\backslash\left(\mathbb{V}_+(a_4)\cup\mathbb{V}_+(b_{10})\cup\mathbb{V}_+ (b_3,b_5)\right).$$ in $\mathbb{P}^{14}$. We can eliminate some of the variables and consider $MGC(A)$ to be the following variety: $$MGC(A)=\mathbb{V}_+(b_3^2-b_5b_6+b_3b_{10})\backslash\left(\mathbb{V}_+(a_4)\cup\mathbb{V}_+(b_{10})\cup\mathbb{V}_+ (b_3,b_5)\right)\subset \mathbb{P}^8.$$ Therefore, any minimal Gorenstein cover is of the form $G=R/\ann H$, where $$H=a_1y_3^2+a_2y_2y_3+a_3y_1y_3+a_4y_2^2+a_5y_1y_2+b_3y_1y_3^2+b_5y_3^3+b_6y_1^2y_3-b_{10}y_1^4$$ satisfies $b_3^2-b_5b_6+b_3b_{10}=0$, $a_4\neq 0$, $b_{10}\neq 0$ and either $b_3\neq 0$ or $b_5\neq 0$ (or both).
Moreover, note that $\mathbb{V}_+(\mathfrak{c})\backslash\mathbb{V}_+(\mathfrak{a})=\mathbb{V}_+(\mathfrak{c_2})\backslash\mathbb{V}_+(\mathfrak{a})$, where $\mathfrak{c}=\mathfrak{c}_1\cap \mathfrak{c}_2\cap \mathfrak{c}_3$ is the primary decomposition of $\mathfrak{c}$ and $\mathfrak{c}_2=\mathfrak{b}_2+(v_2,v_1b_5-v_3b_3-v_3b_{10},v_1b_3-v_3b_6)$. Hence, any $K_H$ such that $K_H\circ H=I^\perp$ will be of the form $K_H=(L_1,L_2,L_3^2)$, where $L_1,L_2,L_3$ are independent linear forms in $R$ such that $L_3=v_1x_1+v_3x_3$, with $v_1b_5-v_3b_3-v_3b_{10}=v_1b_3-v_3b_6=0$.
Consider $A=R/I$, with $R=\res[\![x_1,x_2,x_3]\!]$ and $I=(x_1^2,x_2^2,x_3^2,x_1x_2)$. Note that $\HF_A=\lbrace 1,3,2\rbrace$ and $\tau(A)=2$. Doing analogous computations to the previous examples, *Singular* provides the following variety: $$MGC(A)=\mathbb{P}^7\backslash\mathbb{V}_+(b_2^2-b_1b_3)$$ The coordinates of points in $MGC(A)$ are of the form $(a_1:\dots:a_4:b_1:b_2:b_3:b_4)\in \mathbb{P}^7$ and they correspond to a polynomial $$H=b_1y_1^2y_3+b_2y_1y_2y_3+b_3y_2^2y_3+b_4y_3^3+a_1y_3^2+a_2y_2^2+a_3y_1y_2+a_4y_1^2$$ such that $b_2^2-b_1b_3\neq 0$. Any $G=R/\ann H$ is a minimal Gorenstein cover of colength 2 of $A$ and all such covers admit $(x_1,x_2,x_3^2)$ as the corresponding $K_H$.
Consider $A=R/I$, with $R=\res[\![x_1,x_2,x_3,x_4]\!]$ and $I=(x_1^2,x_2^2,x_3^2,x_4^2,x_1x_2,x_1x_3,x_1x_4,x_2x_3,x_2x_4)$. Note that $\HF_A=\lbrace 1,4,1\rbrace$ and $\tau(A)=3$. Doing analogous computations to the previous examples, *Singular* provides the following variety: $$MGC(A)=\mathbb{V}_+(b_6b_{10}-b_9b_{16})\backslash\left(\mathbb{V}_+(d_1)\cup\mathbb{V}_+(d_2)\right)\subset \mathbb{P}^{12},$$ where $d_1=(a_7a_9-a_8^2)$ and $d_2=(b_9^2b_{16}-b_{10}^3,b_6b_9-b_{10}^2,b_6^2-b_{10}b_{16})$. The coordinates of points in $MGC(A)$ are of the form $(a_1:\dots:a_9:b_6:b_9:b_{10}:b_{16})\in \mathbb{P}^{12}$ and they correspond to a polynomial $$H=b_{16}y_3^3+b_6y_3^2y_4+b_{10}y_3y_4^2+b_9y_4^3+a_9y_1^2+a_8y_1y_2+a_7y_2^2+$$ $$+a_6y_1y_3+a_5y_2y_3+a_4y_3^2+a_3y_1y_4+a_2y_2y_4+a_1y_4^2$$ such that $G=R/\ann H$ is a minimal Gorenstein cover of colength 2 of $A$. Moreover, any $K_H$ such that $K_H\circ H=I^\perp$ will be of the form $K_H=(L_1,L_2,L_3,L_4^2)$, where $L_1,L_2,L_3,L_4$ are independent linear forms in $R$ such that $L_4=v_3x_3+v_4x_4$, with $v_3b_9-v_4b_{10}=v_3b_6-v_4b_{16}=0$.
As in the case of colength 1, we now provide a table for the computation times of $MGC(A)$ of all isomorphism classes of local $\res$-algebras $A$ of length equal or less than 6 such that $\gcl(A)=2$.
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[^1]: ${}^{*}$ Partially supported by MTM2016-78881-P
[^2]: ${}^{**}$ Partially supported by MTM2016-78881-P, BES-2014-069364 and EEBB-I-17-12700.
[^3]: ${}^{***}$ Partially supported by the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 813211 of POEMA”.\
2010 MSC: Primary 13H10; Secondary 13H15; 13P99
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*Albion Lawrence[^1][Supported in part by an NSF Graduate Fellowship.]{} and Emil Martinec[^2][Supported in part by Dept. of Energy grant DEFG02-90ER-40560.]{}*
Enrico Fermi Inst. and Dept. of Physics
University of Chicago, Chicago, IL 60637
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We develop the quantization of a macroscopic string which extends radially from a Schwarzschild black hole. The Hawking process excites a thermal bath of string modes that causes the black hole to lose mass. The resulting typical string configuration is a random walk in the angular coordinates. We show that the energy flux in string excitations is approximately that of spacetime field modes. 1.5cm
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The collision of quantum mechanics and black hole physics has captured the interest of the theoretical physics community. The discovery of two-dimensional model systems \[\] sharing many of the features of the Hawking problem has spurred the hope that one might learn something about information storage and retrieval in black holes, and perhaps about some essential features of quantum gravity. Before this hope evaporates (with or without a remnant) we would like to point out another context in which such two-dimensional systems crop up: the quantization of the collective modes of a macroscopic string trapped by a black hole.
Often one ties two-dimensional model systems to four-dimensional physics when, at the core of a spherically symmetric object, all but S-wave modes of the four-dimensional system are effectively massive \[\] \[\] and may be integrated out. For a macroscopic string, a somewhat different mechanism occurs. The internal excitations of the string core are irrelevant to the physics of the Nambu-Goldstone transverse oscillation modes (for a macroscopic fundamental string, there are no such core excitations). The dominant dynamical effects come from the projection of the spacetime geometry onto the world sheet of the propagating string. When the string is trapped by the event horizon of a black hole, the event horizon is projected onto the world sheet. Hence one expects Hawking radiation of transverse string oscillations to occur upon quantization. Since the essence of the Hawking paradox is whether information follows energy, it would seem that one ought to understand it in this somewhat simpler two-dimensional context. Any remnant scenario would entail a huge expansion of the string density of states, since the remnant would be threaded by the string (although such remnants might be hard to produce, as may be the case for recently considered species of remnant \[\]). If black holes violate quantum mechanical unitarity, we will have to face the problem already in the quantization of a single string. Hawking radiation along the string leads to a number of interesting questions, of which we shall address two, which involve effects of the radiation on the string and its back-reaction on spacetime. We will show that the thermal bath of emitted string modes causes the mean square transverse extent of the string to grow linearly with its length (or some other infrared cutoff) $L$ $$\vev{:(\xi_\perp)^2:}\sim \frac{\ell_s^2 L}{r_h}\ ,$$ and that the power radiated in the string Hawking process is where $r_{h}$ is the horizon radius. This is to be compared with, , the total radiated power a spacetime massless scalar field in the presence of a black hole, which is the same within numerical factors.
We begin in section 2 with the analysis of the small fluctuations of a single straight, infinitely long, bosonic string in a Schwarzschild black hole background in the critical dimension of string theory by expanding the $\sigma$-model action to quadratic order in Riemannian normal coordinates. In section 3 we discuss the linearized classical equations of motion and show that the physical solutions are stable to this order. In section 4 we quantize the transverse fluctuations in a physical gauge, demonstrating both through this gauge fixing and through path integral techniques in conformal gauge that there are ($d_{cr}$–2) independent physical coordinates in the critical dimension. We choose the Unruh vacuum as the string state and construct the Bogolubov transformations used to describe Hawking radiation. Those who are familiar with this material may wish to skip directly to section 5, where we compute some observable manifestations of the Hawking process: the mean square deviation from the equilibrium position and the radiation of spacetime energy onto the string, both characteristic of a string thermally excited at the Hawking temperature.
The metric of the Schwarzschild black hole in D dimensions is derivable in the same fashion as the 4-dimensional case, and differs only in the power of the radial coordinate dependence of the metric for the $r-t$ plane. (see for example \[\]): Here $d\Omega^{2}$ is the metric for the D-2 sphere; $C$ is a constant of integration arising from solving Einstein’s equations and is related to the mass via where $A_{D-2}$ is the area of a unit D-2 sphere and G is the D-dimensional generalization of Newton’s constant, defined via the Einstein action with dimensions $[{\rm length}]^{D-2}$ ($\hbar~=~c~=~1$). The radius of the horizon is of course $r_{h}=C^{1/(D-3)}$.
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Penrose diagram for the x-y plane of the extended Kruskal manifold.
In this paper we will be switching between Schwarzschild null coordinates and Kruskal null coordinates. The former are found by solving for $ds^{2} = 0$ in the r-t plane, and are where $\rst$ is the D-dimensional generalization of the Regge-Wheeler tortoise coordinate \[\] The indefinite integral may be computed exactly by expanding the denominator in partial fractions, with the result: The metric may then be written as: Kruskal null coordinates are used here in dimensionless form, the $r-t$ metric is, in these coordinates, Figure 1 shows the standard Penrose diagram for the “r-t” or “x-y” plane of the black hole, labelled with our notational conventions.
We wish to study an infinitely long string fluctuating in this background; we will assume that the string and its fluctuations have a negligible effect on the spacetime. We will also assume that the black hole has a large mass or equivalently a large horizon radius , which will allow us to expand the action out in inverse powers of ; in addition, since the Hawking temperature goes as the inverse of the horizon radius, we may treat adiabatically the change in the background metric due to the shrinking of the black hole. The action may be written as the -model action The ghost action will arise from fixing to conformal gauge with some fiducial metric ${\hat h}_{\alpha\beta}$ multiplied by an arbitrary conformal factor $\rho$. The ghost action will be the same as that of a string in Minkowski spacetime; it arises purely from the worldsheet geometry, and will depend explicitly on the fiducial metic and implicitly (through ultraviolet regulation) on $\rho$. Note that since the space-time metric is Ricci-flat, this theory is free of the conformal anomaly in 26 dimensions \[\]. The generalization to the supersymmetric -model in this background is straightforward.
We would like to examine small fluctuations around an equilibrium position of the string, about some solution to the classical equations of motion; thus, we want some sort of Taylor expansion of $X^{\mu}$ around the equlibrium configuration $X^{\mu}_{\rm cl}$. We find that, in Kruskal coordinates, is the coordinate embedding for an infinitely long string in conformal gauge. This embedding defines a one-to-one mapping between the $x-y$ plane of spacetime and the classical worldsheet of the string. We may also pull back the metric for the spacetime $x-y$ plane coordinates onto worldsheet, fixing the fiducial metric as:
Riemannian normal coordinates are a standard and natural way to carry out a Taylor expansion of the -model action in a manifestly coordinate invariant fashion (see \[\] for an explanation of this expansion in the bosonic and supersymmetric -models). In particular, it is an expansion in derivatives of the metric; one finds that a term in the action of order $n$ in small fluctuations is accompanied by geometric coefficients (derivatives of and combinations of derivatives of the Riemann tensor) containing $n$ derivatives of the metric. Since the only dimensionful quantity in the metric is $r_{h}$, we may regard this expansion as an expansion in the fluctuation size divided by ; then the expansion is good if the fluctuations are small on the scale of the black hole or if one is in the asymptotic region. Classically, this means simply that we wish to perturb the string (shake it) with small amplitude. Quantum mechanically, the strength of worldsheet fluctuations is given by $1/\ell_{s}^{2}$ which appears in front of the nonlinear -model action; thus, perturbations to expectation values coming from nonlinear terms will be of higher order in $\ell_{s}/r_{h}$.
Following , let us expand the coordinate fields and the spacetime metric in normal coordinates, which are contravariant vectors $\xi^{\mu}$ in spacetime. We may rewrite these as spacetime tangent frame quantities, where $e^{m}_{\mu}(X_{\rm cl})$ is the [vielbein]{} field of the spacetime manifold at the spacetime coordinate $X_{\rm cl}$. By working with these [vielbein ]{} fields we will have a more standard kinetic term and thus a more standard propagator. The final form of the action for small coordinate fluctuations to quadratic order is: Here $\eta$ is a flat Minkowski metric for the tangent frames, and the derivatives $D$ are defined by where $\omega_{\mu}^{mn}$ is the spacetime spin connection.
The [vielbein]{} components are: ($r$ is defined implicitly via - ). We may use these to calculate $\omega_{\mu}^{mn}$ and $R_{\mu\nu\rho\sigma}$. The explicit form of in Kruskal coordinates is found to be: (In fact, if we let , , then is the action in Schwartzchild null coordinates as well.) Note that this action has the standard kinetic term for the transverse coordinates; in general, the curvature terms will be some finite r-dependent piece (vanishing as a power of $r_h/r$ asymptotically) times a factor of $1/r_{h}^{2}$. Hence the kinetic term dominates the action far from the horizon. On the other hand, near the horizon the modes relevant to Hawking radiation are highly blue-shifted and also dominate the curvature terms. The world sheet stress tensor comes from varying with respect to $h_{\alpha\beta}$: Using to expand this to quadratic order, we find that: where $k$ is summed over the angular variables. Here we have written the longitudinal modes as coordinate vector fields rather than as tangent frame vectors, as we will find this form to be easier to use in explicitly solving the constraints. For the Schwarzschild metric, the curvature term couples to the longitudinal modes only, and the explicit form of $T_{\pm\pm}$ is
The equations of motion for the angular coordinates are One may write the second (curvature) term more explicitly: again with $r=r(x=\sigma^{-},y=\sigma^{+})$. It is easy to see that vanishes for large $r$. For $r$ close to $r_h$, we may substitute in and , and use , to find that the zero in $\sigma^{+}\sigma^{-}$ cancels the zero in $[1 - (r_{h}/r)^{D-3}]$, and that the result is a finite number. As we shall see, all the modes will be blueshifted infinitely at the horizon, so that the kinetic term will then dominate close to the horizon. Thus we will, after ensuring stability, use the zeroth-order approximation that the transverse modes satisfy the 2-d massless Klein-Gordon equation for the purpose of computing Bogolubov transformations between mode bases. The inaccuracy of this procedure lies in the fact that there is mixing between left- and right-moving modes due to scattering off the background curvature. Hence the Bogolubov coefficients will mix left- and right-moving creation and annihilation operators.
We were unable to solve exactly, but one can find solutions that work asymptotically as () and as . To do this, transform the variables as follows: From one can see that Note that these coordinates are only defined in one or the other regions of Figure I which are outside the horizon; we will define an appropriate analytic continuation below. In these coordinates the equations of motion separate: One finds that at null infinity and in towards the horizon, the last term vanishes. Thus, in this asymptotic sense, the wave equation for the transverse coordinates is where $\omega$ should be real for stable solutions. Solutions to this approximate equation are for any $\omega$. Translated back into Kruskal coordinates, the solutions become
These mode solutions are standard in Hawking radiation calculations \[\] \[\], and following Unruh and Birrell and Davies, we put the logarithmic branch cut for in the upper half plane by making the choice With this choice, the solutions in are analytic in the lower half plane of $x$, and thus their Fourier transforms vanish for $\omega < 0$, so that the modes shown are positive frequency in $\sigma^{\pm}$ for all $\omega$. Note also that for $r \rightarrow \infty$, these look like plane wave solutions in Schwarzchild coordinates.
The equations of motion for the longitudinal modes are most easily solved in their coordinate vector form. The equations are: with the solutions where $\phi^{x,y}$ and $\psi^{x,y}$ are arbitrary functions. The actual classical solutions to this equation should include the satisfaction of the Virasoro constraints. If the gradients of the coordinate fluctuation fields are moderate (as they are asymptotically), the constraints will be self-consistently solved by equating the linear term in longitudinal modes to the quadratic term in transverse modes; the terms quadratic in longitudinal modes will be of fourth order in the fluctuation size and should be dealt with in the next order of the calculation. Thus we find that in the asymptotic region where $\xi^{k}~\sim~(\sigma^{+})^{i\omega}$, $$\eqalign{\psi^{x}\sim &(\sigma^{+})^{2i\omega-2}\cr
\xi^{x}\sim&\sigma^{-}(\sigma^{+})^{2i\omega}\quad,
\qquad r\rightarrow\infty\quad.}$$ As $r$ approaches , however, $f\d\xi^{x}$ will become large and the term in which is second order in derivatives of longitudinal fluctuations will become even larger and will become the dominant contribution to the longitudinal stress tensor. Thus, near the horizon we should equate this quadratic term with $\d_{\pm}\xi^{k}\d_{\pm}\xi^{k}$. Specifically, the piece of the stress tensor quadratic in derivatives of longitudinal fluctuations is Now As , we can find by writing out $\sigma^{\pm}$ in terms of $r$ and $t$ that $\d_{\pm}f/f$ vanishes as $\sqrt{r~-~r_{h}}$ as we approach the horizon; thus in this limit, and similarly for , . Matching these forms to $\d_{\pm}\xi^{k}\d_{\pm}\xi^{k}~\sim~(\sigma^{\pm})^{2i\omega-2}$, we can solve the constraints to lowest order by setting Note that with this solution the term quadratic in the derivatives of longitudinal fluctuations indeed diverges faster than both the term linear in the derivatives and the non-derivative quadratic term.
Before we quantize the quadratic theory , we need to know whether it is classically consistent – that is, if small fluctuations stay small over time. We should note that simply looking at the size of $\xi^{\mu}$ over space and time is incorrect since the metric is invariably singular in some region of spacetime for whatever coordinate system we might choose – in $H^{\pm}$ for Schwarzschild coordinates and in $\cal{I}^{\pm}$ for Kruskal coordinates. The natural quantity to look at is the scalar product of the fluctuations,
For the transverse fluctuations, in the absence of exact solutions to , we shall try to cast the equations of motion into a Sturm-Liouville form where stability is obvious. If we make an additional change of variables then the equations of motion become where with $r$ used as an implicit function of $R$. Roughly, is the Klein-Gordon equation with a spatially varying mass term. Since the “mass squared” $g(R)$ is always positive, the normalizable transverse solutions are stable.
Knowing this, we may use $(\sigma^{\pm})^{i\omega}$ for real $\omega$ as a complete set of asymptotic solutions for the transverse equations of motion, and we may then use the results of the previous subsection to examine the form of the longitudinal solutions in the asymptotic regions. For the $r~\rightarrow~\infty$ region, so that the scalar product of the longitudinal fluctuations is $$g_{xy}\xi^{x}\xi^{y} \sim (\sigma^{+}\sigma^{-})^{2i\omega}\ .$$ For the $r \rightarrow r_{h}$ region, f is a regular, finite quantity, the zero in $1/(\sigma^{+}\sigma^{-})$ cancelling the zero in $[1 - (r/r_{h})^{D-3}]$. The form of $\xi^{x,y}$ near the horizon will be and similarly for $x\leftrightarrow y$, $\sigma^{+}\leftrightarrow\sigma^{-}$. Naive power counting tells us that the second term should be regular near the horizon, so that $g_{xy}\xi^{x}\xi^{y}$ is also regular near the horizon.
In this section we would like to address two separate issues: the imposition of the Virasoro constraints on the theory defined by , and the problems of selecting physically motivated “in” and “out” states. We will first discuss in general the conformal invariance of the system, and argue that the angular fluctuations are a good representation of the physical Hilbert space of the string by showing that the partition function depends only on the determinants of the transverse fluctuation operators. We will then discuss the possible definitions of the positive frequency modes of the transverse fluctuations and construct the desired vacuum states and their Bogolubov transformations. Finally, we shall argue that the Hawking radiation does not include unphysical modes if the BRST cohomology theorem holds for this system.
Now the full nonlinear -model action satisfies the $\beta$-function equations of in the standard critical dimensions to first order in , since the metric coupling satisfies Einstein’s equations. Thus we should be able to use the conformal invariance to decouple the negative-norm states in the Hilbert space of coordinates plus ghosts, and we expect that in principle one should be able to embark upon a standard covariant quantization program by imposing the Virasoro conditions or the corresponding BRST condition. We have not been able to successfully use this formalism to prove a no-ghost theorem due to the subtleties of constructing the vacuum, which involves correlating the fluctuations of the string near the horizon and in the asymptotic region.
If the spectrum is free of negative norm states and conformal invariance may be imposed on the quantum level, then the only physical degrees of freedom are the transverse oscillations of the string, as in flat space. We will support this supposition by showing that the partition function depends only on the determinants of the transverse fluctuations; upon integrating out the conformal ghosts and (at quadratic order) the longitudinal bosonic coordinates, we will find that determinants from these integrals will formally cancel each other leaving only the partition function for the transverse coordinates.
Consider the longitudinal $(\xi^{0}, \xi^{1})$ part of and integrate it by parts to put the lagrangian in the form ${\cal L} = \xi^{m}D_{mn}\xi^{n}$. The result is where $m$ and $n$ are restricted to 0 and 1 (which the [vielbeins]{} map back to the $x-y$ plane). Integrating over $\xi^{0}$ and $\xi^{1}$ leads to the determinant of the above quadratic operator. On the other hand, Polyakov\[\] showed that in writing an arbitrary metric as where $\nabla$ is the worldsheet covariant derivative, the functional integration measure is with If we change variables from the coordinate vectors $\varepsilon_{\alpha}$ to the tangent space vectors $\varepsilon^{a}~=~e_{\alpha}^{a}\varepsilon^{\alpha}$, where $e_{\alpha}^{a}$ is the [zweibein]{} for the worldsheet with metric $\hat{h}_{\alpha\beta}$, then we get with the indices on the right hand side changed to tangent space indices and the tangent space operator $L_{ab}$ identical to the operator in , so that the Fadeev-Popov determinant formally cancels the determinant arising from integrating out the longitudinal modes.
In performing the normal coordinate expansion, the fields became contravariant vectors in spacetime, and so the kinetic terms were endowed with a spin connection induced by the spacetime geometry. This allowed us to use the determinants which arose from integrating out the longitudinal fluctuations to cancel the Fadeev-Popov determinant written with worldsheet covariant derivatives acting on worldsheet vectors and spinors; the worldsheet metric is pulled back from the spacetime metric, so it is reasonable to state that the worldsheet tangent space structure, and thus the spin connection, is induced by the spacetime tangent space structure. Thus this cancellation is a consequence of our choice of equilibrium classical solution and of the particluar choice of fiducial metric shown in equation .
It is important to keep straight in the above argument the distinction between the intrinsic metric $h_{\alpha\beta}$ and the induced metric $\gamma_{\alpha\beta}=\d_\alpha X^\mu\d_\beta X^\nu G_{\mu\nu}(X)$. The determinants that arise at quadratic order in the functional integral have the structure where $\Delta_0$ is the quadratic fluctuation operator of the transverse modes. The arguments of the differential operators indicate their explicit dependences; of course each determinant implicitly depends on $h$ through the regulator. Since we are in the critical dimension, the [*derivative*]{} of this product of determinants with respect to $h$ vanishes. On the other hand, if we are interested in the [*value*]{} of the determinants at the point $h=\gamma$, then the above treatment shows that the first two determinants cancel and the partition function will be the determinant of the transverse fluctuation operator regulated with the induced metric.
In constructing the Hilbert space of the transverse fluctuations we are faced with the well-known problem of finding a reasonable physical vacuum state (see for example for a general discussion) given curved worldsheet and spacetime manifolds. In the absence of an exact quantization we are restricted to constructing this vacuum perturbatively in the normal coordinate expansion. Based on the analysis of the previous section, this expansion is consistent with the solution of the Virasoro constraints in conformal gauge. Moreover we expect to be able to dress any state in the Hilbert space of the transverse modes into a physical state whose dependence on the longitudinal coordinates is a small deviation from the classical identification $\sigma^{-} = X^{x},
\sigma^{+} = X^{y}$ which may be corrected for systematically (order by order). In particular, we may use the classical worldsheet $\leftrightarrow$ spacetime correspondence to ask what worldsheet fluctuations might be positive frequency with respect to a spacetime observer. Here we will assume that the aforementioned solution to the Virasoro constraints may be arrived at quantum mechanically in fixing to a physical, light-cone-like gauge along the lines of section 3.1. The problem then reduces, in lowest order, to that of considering the transverse fluctuations in as a field theory in a 2d spacetime corresponding to the $r$ - $t$ or $x$ - $y$ plane of a Schwarzschild black hole; scalar field theories in this 2-D spacetime have been well studied (see for example and references therein).
We wish to assume that the “in” vacuum consists of no excitations in the far past in Schwarzschild time. On the Kruskal manifold, this corresponds to asking that there are no field excitations leaving $H^{-}$ and $\scrim$, in region I of figure 1, and no excitations leaving $H^{+}$ and $\scrip$ in region II of figure 1. An observer at $\scrim_{I}$, living in an asymptotically flat region, will be a Schwarzschild observer and thus will define positive frequency modes with respect to the Schwarzschild time $t$. An observer leaving or falling into the horizon, or at least one for whom the horizon is a finite time away, will be a Kruskal observer and thus will define positive frequency modes with respect to a characteristic Kruskal time; in particular, on $H^{\pm}$ and $\cal{I}^{\pm}$, the timelike Killing vector becomes lightlike so that the characteristic Kruskal time will be $\sigma^{+}$ or $\sigma^{-}$ depending on the surface. Thus, the “in” vacuum is in region I annihilated by left-moving modes which are positive-frequency with respect to $t$ and right-moving modes which are positive frequency with respect to Kruskal time. We are only asking questions about observers in region I of our Kruskal diagram, so we have chosen the modes in region II for mathematical convenience – phenomena in region I will not depend on how we treat region II. This vacuum is the “Unruh vacuum,” used to mock up gravitational collapse \[\]. We might use other vacuums, for example by asking that there is no radiation entering or leaving the horizon $H^{-}_{I}\cup H^{+}_{I}$ – the “Hartle-Hawking vacuum” – but the Unruh vacuum is physically compelling and at any rate the extension of our calculations to the Kruskal vacuum is not difficult.
The “out” vacuum we choose corresponds to an observer in the far future ($\scrip_{I}$) who, living in an asymptotically flat region, will define positive frequency modes in terms of Schwarzschild time; thus the “out” vacuum is that annihilated by all modes which are positive frequency with respect to $t$. This is known as the “Schwarzschild vacuum.”
For the right-moving modes used to define the Unruh vacuum we use the solutions $(\sigma^{-})^{i\omega}$ which are, as we have stated below , are positive frequency for all (real) $\omega$. Negative frequency modes are defined by taking the complex conjugate, which is most easily computed by writing $x^{i \omega}$ properly in terms of Schwarzschild modes in region I and II, where $u_{I}$ and $u_{II}$ are the Schwarzschild null coordinates in region I and II, respectively, of figure I, and $\theta$ is the Heaviside function. Note that due to the real exponential factor, the complex conjugate of $x^{i\omega}$ is different from $x^{-i\omega}$. More formally, complex conjugation will move the logarithmic branch cut in $x^{i\omega}$ from the upper half plane as defined in to the lower half plane, so that the complex conjugate is upper half plane analytic and thus is composed of negative frequency modes.
The transverse fluctuation operators may now be written in terms of the two different mode decompositions. The frequency-dependent coefficients in the sums over modes arise from requiring that the single-particle wavefunction in front of each operator is normalized to a delta function under the Klein-Gordon norm , . These decompositions are: in the modes by which we define the Unruh vacuum, and in the modes by which we define the Schwarzschild vacuum. The operators $a_{I,II}, b, c_{I,II}$ each satisfy canonical commutation relations The modes describing left-moving excitations, created by $a_{I}^{k\dagger}$ and $a_{II}^{k\dagger}$ are the same for both decompositions so that the only difference is in our definition of left-moving excitations. The Bogolubov transformations between the $b$-operators defining the Unruh vacuum and the $c_{I,II}$-operators defining the Schwarzschild vacuum (we shall call these “Kruskal” and “Schwarzschild” operators, respectively) are easily found by expanding $x^{i\omega a}$ as in , and are :
We shall use these transformations in the next section to compute the Hawking radiation of the black hole into spatial fluctuations of the string.
One of the original motivations of this work was to figure out whether the Virasoro constraints could be consistently imposed in the covariant formalism in the presence of Hawking radiation: one might think that the Hawking radiation would populate all modes – ghosts, longitudinal modes, – thermally and so there is no way to guarantee that an asymptotic observer would not see unphysical modes radiated by the black hole. However, in order to ask physical questions, one must either work in a physical gauge or work with physical operators in a covariant gauge, and define the vacuum in a physical fashion.
The first question that should be asked is whether one may legitimately fix to the physical gauge we have chosen, or in covariant language, whether one may consistently impose the BRST condition $$\qbrs \ket{{\rm physical}} = 0\$$ and find an appropriate representation for the BRST cohomology classes. We have not been able to rigorously prove either, but given that classically one may fix a physical gauge; that the theory is anomaly free to one loop (since the one-loop -model beta functional vanishes); and that the longitudinal vacuum fluctuations explicitly cancel off the ghost vacuum fluctuations in the partition function (again, to one loop), it seems that we have sufficient circumstantial evidence that the constraints may be solved explicitly and consistently.
Given a consistent physical gauge, we may simply work with the transverse oscillators and solve for the longitudinal oscillators order by order. This is fine for computing quantities such the size of transverse fluctuations $\langle \xi^{2} \rangle$ or the expectation value of the transverse stress tensor, at least to lowest order in the normal coordinate expansion, as we shall below. Alternatively, if BRST quantization is consistent, we expect that we can represent the BRST cohomology classes by building states with DDF-like operators\[\]\[\]\[\]\[\]. This also amounts to dressing the transverse oscillators with longitudinal modes to make a physical state. In building such states we are faced with questions of what a physical (spacetime) observer sees as positive frequency; the worldsheet coordinates have no meaning in covariant gauge, so in particular the question of what is a positive frequency mode on the worldsheet has no inherent spacetime meaning. However, since the frequency of a DDF operator is a spacetime quantity, we use it to define the vacuum directly with these manifestly physical operators. (This is exactly what Schoutens \[\] have done for their model of 2-D black hole evaporation.) We see why our original fear was naive; a proper formulation of the theory avoids any question of unphysical modes before Hawking radiation ever becomes an issue.
This argument can be recast in terms of more physical or heuristic descriptions of Hawking radiation. If one considers Hawking radiation as due to pair production near the horizon then the argument means essentially that pair production involves only physical modes, so that only physical states are radiated to $\scrip$. Similarly, if one describes Hawking radiation via a density matrix due to a summation over states at or behind the horizon, hidden from our asymptotic observer at ${\cal I}^{+}$, then one will sum only over states in the physical Hilbert space – the full Hilbert space including negative norm states being in some sense a fiction arising from the gauge freedom remaining after passing to a covariant gauge.
Having discussed quantization and the difference between the “in” and “out” vacuums, one may begin to ask some physical questions about the string. The Hawking radiation in the 2d worldsheet field theory will manifest itself in some thermal population of the transverse modes in the “out” basis, which means that an asymptotic (Schwarzschild) observer will see the string fluctuating as if it was thermally excited with the worldsheet temperature equal to the Hawking temperature The three most apparent (to us) detectable manifestations of this radiation are the thermal wandering of the string, the energy that is radiated in string fluctuations out of the black hole towards ${\cal I}^{+}$, and the production of physical spacetime modes (microscopic strings) due to transitions of the excited string. We discuss the first two of these for the bosonic string in the following subsections.
As an estimation of the wandering of the string, we may calculate the mean square deviation from the classical background solution, We need to fix both the state $\ket{\Psi}$ and the normal ordering prescription. $\ket{\Psi}$ we take to be the Unruh vacuum, but we wish to normal order this expression with respect to the Schwarzschild modes, on the assumption that normal ordering is a function of the observer. This means that we begin by expanding the operators $\xi^{k}$ according to , pushing the Schwarzschild creation operators to the left of the destruction operators, and then using to expand out $c^{k}$ in terms of $b^{k}$, so as to be able to calculate the expectation value in the “in” state which is annihilated by $a^{k}$ and $b^{k}$. The normal ordered correlator is where we have put in a factor of $\ell_s^{2} = \hbar^2 c^2 2\pi\alpha' = \hbar c/T$, where $T$ is the string tension \[\], to give $\langle \xi^{2}\rangle$ the correct dimensions. We note immediately that the normal ordering has removed the ultraviolet divergences but a nasty infrared divergence remains. Eq. has the form $\int d\omega D(\omega) \langle n(\omega, T_{H})\rangle$ (with $D$ being the density of states) of a 2d Bose gas of $D-2$ massless particles, where $k_{B} T_{H} = 1/\beta = 1/(2\pi a)$. The IR divergences arising from the density of states and from the Planck distribution function are characteristic of this general statistical mechanical system in the infinite volume limit, and are entirely physical. Also note that the transverse stress tensor $\d \xi \d \xi$ is infrared finite at this order. To make sense of , we need a physically motivated finite-size or finite-time cutoff – an example of the latter is the finite lifetime of the black hole.
Thus if one regulates with an infrared momentum cutoff at momentum $\epsilon$, one finds that $\langle \xi^{2} \rangle$ diverges linearly with the cutoff, The deviation then goes as the square root of the finite-size or finite-time cutoff $1/\epsilon$, so that the average string configuration is that of a random walk. This is a result we might have expected from studies of average string configurations at finite temperature; in particular, Mitchell and Turok \[\] found that for closed bosonic strings at finite temperature, the mean square radius of the string scaled as the length $L$ of the string, $$\langle \Delta r^{2} \rangle \sim L\ .$$ Nevertheless the angular variation $\delta\theta\sim \delta\xi/r$ is well-behaved, so although the string fluctuates far from its center of mass, it does not cover a large solid angle of the black hole.
The presence of this infrared catastrophe indicates that although the equations of motion derived from are stable, we expect that quantum fluctuations will require us to include some nonlinear effects in the worldsheet effective action. One might worry that, since the thermal propagator for the transverse string fluctuations is quadratically infrared divergent, that higher order corrections to the stress tensor will be much larger than the effects we have calculated (see below). We believe that what is breaking down is not the order of magnitude of the radiated Hawking flux, but rather the Taylor expansion of certain quantities in the action. For example, part of the small fluctuation expansion involves the Taylor expansion of terms like $G(X_{\rm cl}+\xi)\partial\xi\partial\xi$, which has a radius of convergence of order $r_{\rm cl}$. So instead of treating perturbatively the quantum fluctuations of all the modes, we should introduce an infrared cutoff and treat the low frequency oscillations adiabatically and classically (since they have large occupation numbers) while quantizing all the higher frequency modes. This scheme should work quite well if we stay far from the horizon.
We would like some idea of the energy radiated onto the string in the Hawking process. The world-sheet stress tensor measures the world-sheet energy radiated out along the string, so a definite relation between world-sheet and spacetime time coordinates will relate world-sheet and spacetime energy densities and fluxes. Such a relation is provided by the classical solution – hence we expect that the spacetime and world-sheet energy fluxes should coincide.
We shall work entirely in the asymptotically flat regime of spacetime, in Schwartzchild null coordinates. In this region the Lagrangian is manifestly (approximately) Lorentz invariant and so there is an asymptotically well defined Noether current where $T$ is the string tension. The total momentum radiated to future infinity will then be Expanding in normal coordinates, we find that Now $-P^{\mu}P_{\mu}$ should be the total mass-squared radiated away to $\scrip$, so that where the subscript $c$ denotes “classical.” We use the classical mass shell condition and our explicit values for $X^{\mu}_{c}$ to write The integrals in used to define the total momentum $P^{\mu}$ will be infinite and will furthermore select out the zero modes of $\d_{-} X^{\mu}$. We thus find an expression for mass per unit interval along $\scrip$: where the subscript $0$ indicates the space-independent part. We will drop the $\delta M^{2}$ term as being higher order in our expansion. Substituting in the lowest order light-cone condition in Schwarzchild null coordinates, we find that $$2 f \d_{-} X^{u}_{c} \d_{-} \xi^{v} = T^{\perp}_{--}$$ (where here we have set $\sm = u$, $\sp = v$), and thus The second term on the left hand side should vanish since we are looking at oscillating transverse solutions with zero total transverse momentum. Now ($M/{{\rm length}}$) is just the string tension, so that, converting energy density to energy flux, In order to compute the Hawking radiation, we wish to take the vacuum expectation value of $:T^{{\perp}}:$ in the Unruh vacuum, with normal ordering performed as before with respect to the Schwarzchild vacuum. With this vacuum and normal ordering prescription, only $T^{\perp}_{--}$ will be non-zero, and we find that to lowest order, The mass per unit time radiated out onto the string is thus found to be: where we have put all of the physical constants back in.
We may ask when the energy radiated onto the string modes is larger than the radiation into spacetime modes. The energy flux into, for example, a scalar field such as the string dilaton is essentially the same as calculated above since the radiation is mostly into S-wave states: Thus the energy flux of radiation onto string modes is proportional to the energy flux of spacetime scalars.
In fact both the world-sheet and spacetime field energy fluxes differ from the estimates and in several ways. Both the spacetime field modes and the angular string fluctuations will suffer backscattering from the gravitational field near the horizon. This backscattering will be frequency-dependent, and for energies typical of the Hawking radiation leads to a non-negligible fraction of the radiation to be reabsorbed by the black hole. Curiously, although the shape of this barrier is in general different for spacetime S-waves and string angular fluctuations, the two barriers are identical in $D=4$. The spacetime field modes of nonzero angular momentum also contribute some flux (decreasing exponentially in the angular momentum) which does not significantly alter the total flux. Finally, it should be emphasized that the string is generally propagating in a spacetime $\MM_{\rm schw}^{D}\times\KK^{d_{cr}-D}$, the product of the $D$-dimensional Schwarzschild geometry we have been discussing and an unspecified internal manifold $\KK$ that together with $\MM_{\rm schw}$ makes up the total effective central charge $d_{cr}$ of the sigma model in which the string propagates. The string coordinates of this internal space will also Hawking radiate, and in this case there is no barrier from the Schwarzschild curvature. The corrections to the flux in internal coordinates come from the difference between the density of states of the sigma model on $\KK$ and that of free fields. If the sigma model on $\KK$ is weakly coupled, in the normal coordinate expansion the corrections to the stress tensor will be down by powers of $\ell_s/r_\KK$ where $r_\KK$ is the typical radius of curvature of $\KK$. Thus the total flux in spacetime modes will be roughly proportional to the number of massless spacetime fields (remembering to count polarization states and correct for the angular momentum barrier in case of intrinsic spin), while the flux in string modes will be roughly proportional to $d_{cr}-2$.
Finally, one should remember that when the black hole entraps the macroscopic string, two semi-infinite pieces protrude from the horizon. This doubles the flux into string modes – or does it? For chirally asymmetric strings, one of the two pieces propagates ‘left-moving’ string modes in the radially outward direction; the other piece propagates ‘right-moving’ string modes radially outward. Thus the energy-momentum flux in general differs for the two halves. This effect is most pronounced for the heterotic string in the critical dimension, where the left-moving (bosonic string) flux is that of 24 bosons, while that of the right-moving (fermionic string) flux is effectively that of 12 bosons ( by bosonizing the eight transverse fermions). Curiously, the back-reaction of the Hawking radiation will not only evaporate the black hole, but due to the asymmetric radiation pressure will accelerate the black hole down the string!
One issue we have not really settled is how to restrict the string functional integral to the region of field space outside the horizon. In principle the string fluctuates away from the classical solution in such a way that part of the string dips into the black hole. If we take seriously the ultraviolet fluctuations of the world-sheet, these ‘dips’ are ubiquitous (and indeed extend to the singularity), although incoherent. A bit of averaging over short distances replaces such fluctuations by a renormalized wandering of order the renormalization scale. We leave a detailed investigation of the structure of physical states near the horizon to future work.
Another issue is the relation between the Bekenstein-Hawking entropy as measured by the world sheet, in comparison to that measured by spacetime physics. The former will be either $\log M$ or constant as in the two-dimensional model system of , whereas the latter will be proportional to the $(D-2)$-dimensional area of the event horizon; what states of the black hole does the string have access to? The naive calculation would suggest only the ‘states near the horizon’ where the string is attached to the black hole. It has been claimed recently that the divergent quantum corrections to the entropy are cut off in string theory \[\] because of string theory’s soft spacetime high energy behavior. However string theory still has divergences on the world sheet, and those presumably still contribute a logarithmic divergence to the macroscopic string’s entropy, unless there is a relation between the world sheet cutoff and the spacetime cutoff provided by the soft high-energy behavior of strings. But how does such a relation come about? Again it is not clear.
A logical extension of the present work (currently being contemplated) involves the construction of the physical vacuum. According to our discussion of section (4.3), this vacuum is not an incoherent superposition of all modes, but rather of a thermal distribution of transverse modes which are [*coherently*]{} dressed by ghosts and longitudinal modes. We intend to build such a state out of DDF-style operators for the dressed transverse oscillation modes, with the ultimate goal of calculating the amplitudes for emission and scattering of microscopic strings.
One may also contemplate a number of other model calculations which go beyond the scope of the present work. One would like to study not just the evaporation problem but also the formation of black holes in the macroscopic string model, as in the two-dimensional system of Callan . This might be achieved by an appropriate calculation of the gravitational field back-reaction of a macroscopic string. Naively the left- and right-moving modes of a fundamental string in flat space are free fields. However, we expect two counterpropagating pulses (‘shock waves’) to interact gravitationally, and if sufficiently energetic, to form a black hole upon collision. Generally one must also take into account the other long-range fields carried by the string ( the axion and dilaton charge per unit length carried by macroscopic fundamental strings \[\]). The appropriate starting point for such a calculation might be the family of exact string solutions of Garfinkle \[\]\[\]2[J. Gauntlett, J. Harvey, and D. Waldram, to appear.]{}.
The proper relation between the string and spacetime fields were only partially addressed here. It would be interesting to exhibit a self-consistent world-sheet calculation that would incorporate the back-reaction of the macroscopic string dynamics on the spacetime fields, for instance to show in a world-sheet beta function calculation how the energy radiated along the string contributes to a decrease in the black hole mass. We imagine such effects should appear in a term of the Polyakov type $(\partial \log[f(X)])^2$ in the world-sheet effective action, which then couples to the spacetime effective action in the manner described in . This mechanism is somewhat confusing in that the macroscopic string is a single quantum from the point of view of the string field theory, and we do not ordinarily incorporate the classical field of a single quantum in the classical background ( we do not treat single electrons as small black holes or background Coulomb charges). Presumably one needs to resum the emission of soft gravitons from the world sheet into an effective classical field, and such nearly on-shell gravitons will give logarithmic almost-divergences in the world-sheet theory which resum to a modified classical field potential. From the worlsheet point of view these microscopic strings that dress the spacetime geometry are a resummation of ‘baby universe’ processes. It would be nice to understand the details.
We wish to thank J. Harvey, V. Iyer, D. Kutasov, and L. Susskind for a number of discussions, explanations, and useful comments.
[^1]: $\dagger$
[^2]: \*
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abstract: 'We investigate from first principles the optoelectronic properties of nanometer-sized armchair graphene nanoribbons (GNRs). We show that many-body effects are essential to correctly describe both energy gaps and optical response. As a signature of the confined geometry, we observe strongly bound excitons dominating the optical spectra, with a clear family dependent binding energy. Our results demonstrate that GNRs constitute 1D nanostructures whose absorption and luminescence performance can be controlled by changing both family and edge termination.'
author:
- Deborah
- Daniele
- Alice
- Andrea
- Elisa
title: 'Optical properties of graphene nanoribbons: the role of many-body effects'
---
Graphite-related nanoscale materials, such as fullerenes and nanotubes, have long been the subject of an intense research for their remarkable properties [@dres+96book]. The recent discovery of stable, single-layer graphene [@novo+05nat; @zhan+05nat; @berger] has prompted the attention on a different graphitic quasi-1D nanostructure, i.e. graphene nanoribbons (GNRs). These systems have been theoretically studied in the past decade [@fuji+96jpsj; @naka+96prb; @waka+99prb; @kawa+00prb; @kusa-maru03prb] as simplified models of defective nanotubes and graphite nano-fragments. However, only very recently isolated nanometer-sized GNRs have been actually synthetized by etching larger graphene samples, or by CVD growth on suitably patterned surfaces [@chen+07condmat; @han+07condmat; @tana+02ssc]. The production techniques advanced in these pioneering works are expected to become highly controllable, opening up new avenues for both fundamental nanoscience and nanotechnology applications.
One of the most striking features of GNRs is the high sensitivity of their properties to the details of the atomic structure [@fuji+96jpsj; @naka+96prb; @waka+99prb; @ezaw06prb; @baro+06nl; @son+06prl; @pisa+07prb]. In particular, the edge shape dictates their classification in armchair (A), zigzag (Z) or chiral (C) ones, thus determining their energy band gaps. In addition to an overall decrease of energy gaps with increasing ribbon width, also observed experimentally [@han+07condmat], theoretical studies predict a superimposed oscillation feature [@baro+06nl; @ezaw06prb; @son+06prl], which is maximized for A-GNRs. According to this behaviour, A-GNRs are further classified in three distinct families, i. e. $N=3p-1$, $N=3p$, $N=3p+1$, with $p$ integer, where $N$ indicates the number of dimer lines across the ribbon width. This fine sensitivity to the atomic configuration raise the opportunity to tailor the optoelectronic properties of A-GNRs by appropriately selecting both ribbon family and width.
In spite of this interest, previous theoretical studies of the electronic (see e.g. Refs. ) and optical properties [@baro+06nl] of GNRs were only based on the independent-particle approximation or on semi-empirical calculations. However, many body effects are expected to play a key role in low dimensional systems [@ruin+02prl; @rohl-loui99prl; @chan+04prl; @spat+04prl; @brun+07prl] due to enhanced electron-electron correlation. Motivated by this theoretical issue and by recent experimental progress [@chen+07condmat; @han+07condmat; @tana+02ssc] pursuing the potential of GNRs for nanotechnolgy applications, we have carried out [*ab initio*]{} calculations to study the effects of many-body interactions on the optical spectra of 1-nm-wide A-GNRs belonging to different families.
In this Letter, we show that a sound and accurate description of the optoelectronic properties of A-GNRs must include many-body effects. We will demonstrate that there are many signatures of the non-local correlations occurring in these confined systems. First of all, quasiparticle corrections are found to be strongly state-dependent. Moreover, the optical response of A-GNRs is dominated by prominent excitonic peaks, with a complex bright-dark structure which would not have been even expected from an independent-particle framework. Both quasi-particle corrections and exciton binding energies are found to exhibit an oscillating behaviour, according to the family classification. Finally, the electronic and optical properties of hydrogen passivated A-GNRs are compared with those of clean-edge ribbons: including many-body effects allows us to single out the impact of this edge modification on absorption and luminescence.
The first-principles calculation of the optical excitations is carried out using a many-body perturbation theory approach, based on a three-step procedure [@note-review]. As a preliminary step, we obtain the ground state electronic properties of the relaxed system, by performing a density-functional theory (DFT) supercell calculation, within the local density approximation (LDA) [@Pwscf; @note-dft]. Second, the quasiparticle corrections to the LDA eigenvalues are evaluated within the $G_0W_0$ approximation for the self-energy operator, where the LDA wavefunctions are used as good approximations for the quasiparticle ones, and the screening is treated within the plasmon-pole approximation [@godb-need89prl]. Third, the electron-hole interaction is included by solving the Bethe-Salpeter (BS) equation in the basis set of quasielectron and quasihole states, where the static screening in the direct term is calculated within the random-phase approximation (RPA). Only the resonant part of the BS hamiltonian is taken into account throughout the calculations (Tamm-Dancoff approximation), since we have verified that the inclusion of the coupling part does not affect significantly the absorption spectra [@note-coupl]. Moreover, only the case of light polarized along the ribbon axis is examined, as a significant quenching of optical absorption is known to occur in 1D systems for polarization perpendicular to the principal axis [@mari+03prl]. All the $GW$-BS calculations are performed with the code SELF [@Self; @note-bse].
To treat an isolated system in the supercell approach, we consider a separation of 40 a.u. between images in the directions perpendicular to the ribbon axis. Moreover, in both $GW$ and BS calculations, we truncate the long-range screened Coulomb interaction between periodic images, in order to avoid non-physical interactions [@rozz+06prb]. Due to the rectangular geometry of the system, we use a box-shaped truncation [@vars-mari07unp].
![ (a) Optical absorption spectra of 1 nm wide hydrogen-passivated GNRs: $N=8$ (1.05 nm wide), $N=9$ (1.17 nm) and $N=10$ (1.29 nm). In each panel, the solid line represents the spectrum with electron-hole interaction, while the spectrum in the single-particle picture is in grey. All the spectra are computed introducing a lorentzian broadening. (b) Quasiparticle bandstructures. []{data-label="spectra-band"}](fig1.eps){width=".40\textwidth"}
![ In-plane spatial distribution of the electron for a fixed hole position (black dot), corresponding to the lowest excitonic peak in the $N=9$ case. The spatial density is averaged over the direction orthogonal to the ribbon plane. Dimension of the panel: $1.2 \times 6.4$ nm. []{data-label="exc-wfc"}](fig2.eps){width=".45\textwidth"}
We start by considering 1 nm wide hydrogen-passivated A-GNRs belonging to different families, namely $N=8$, 9, 10. Figure \[spectra-band\] (a) depicts their calculated optical absorption spectra, while the quasiparticle bandstructures are shown in Fig. \[spectra-band\] (b). All the results are summarized in Table \[tab\]. The quasiparticle $GW$ corrections open the LDA energy gaps at $\Gamma$ by 0.72, 1.32 and 1.66 eV for $N=8$, 9 and 10, respectively. These energy corrections are larger than those of bulk semiconductor with similar LDA gaps, due to the enhanced Coulomb interaction in low dimensional systems. In addition, a family modulation of the corrections can be noticed, with larger corrections for the GNRs with larger LDA gaps. The gap opening is accompanied by an overall stretching of the banstructure of $17-22\%$, similar to the value found for graphene (about $20\%$) [@angel].
In the absence of e-h interaction, such a bandstructure would result in the optical absorption spectra depicted in grey \[Fig. \[spectra-band\] (a)\], characterized by prominent 1-D van Hove singularities. The inclusion of the excitonic effects (solid black line) dramatically modifies both the peak position and absorption line-shape, giving rise to individual excitonic states below the onset of the continuum, with binding energy of the order of the eV.
The lowest-energy absorption peaks for $N=8$ and 9, labelled $A_8$ and $A_9$, have the same character: in both cases, the principal contribution comes from optical transitions between the last valence and first conduction bands, localized in k-space near the $\Gamma$ point \[Fig. \[spectra-band\] (b)\]. The binding energies for these lowest optically active excitons are 0.58 and 1.11 eV for $N=8$ and 9, respectively. As compared to the first two systems, the $N=10$ GNR shows a richer low-energy spectrum. Each noninteracting peak gives rise to a bright excitonic state \[arrows $A_{10}$ and $B_{10}$ in Fig. \[spectra-band\] (b)\], with binding energies of 1.31 and 0.95 eV. In addition, the mixing of dipole forbidden transitions between the same bands \[arrows $D$ in Fig \[spectra-band\] (b)\] is responsible for an optically inactive exciton degenerate in energy with $A_{10}$. The $D$ state thus provides a competing path for non radiative decay of optical excitations, which could affect the luminescence yield of the system. This feature results from transitions between pairs of bands very close in energy to each other, and is therefore expected to be a common outcome for all $N=3p+1$ GNRs.
A further insight in the effects of electron-hole interaction is provided by the evaluation of the resulting spatial correlations. In Fig. \[exc-wfc\], we plot the in-plane probability distribution of the electron for a fixed hole position (black dot), corresponding to the lowest excitonic state in the $N=9$ case. While the electron distribution extends over the whole ribbon width, the modulation of the exciton wavefunction $|\psi({\bf r_e}; {\bf r_h})|^2$ along the ribbon axis is entirely determined by the Coulomb interaction. Similar wavefunctions (not reported here) for the lowest excitons have been obtained for GNRs of different families, with spatial extentions [@note-ext] of about 34, 23 and 18 Å for $N=8$, 9 and 10, respectively.
$N$ LDA $GW$ BS $E_b$
------ ------ ------ ------------ ------------
8-H 0.28 1.00 0.42 0.58
8 0.50 1.59 0.71 0.88
9-H 0.78 2.10 0.99 1.11
9 0.56 1.50 0.64 0.86
10-H 1.16 2.82 1.51, 1.87 1.31, 0.95
10 1.09 2.64 1.46, 1.68 1.18, 0.96
: Energy gap (2nd and 3rd columns) and peak position (4th column) for $N=8$, 9 and 10 A-GNRs, with (-H) and without hydrogen passivation of the edge sites. The relative binding energies are reported in the last column. All the values are in eV. []{data-label="tab"}
![ (a) Quasiparticle bandstructure of the $N=9$ hydrogen-free GNR. Arrows indicate the edge related single particle bands. (b) Plot of the $GW$ quasiparticle energies vs the LDA energies. (c) Optical absorption spectrum, with (solid black) and without (grey) excitonic effects. The black arrow indicates the energy position of the optically forbidden edge-related exciton. Its excitonic wavefunction is depicted in panel (d), whose dimension is 1.0$\times$2.2 nm. []{data-label="clean"}](fig3.eps){width=".42\textwidth"}
We now consider the case of clean-edge nanoribbons, since this simple variation of the structure has been often suggested for ribbons obtained by high-temperature treatments or by dehydrogenation of hydrocarbons [@kawa+00prb; @baro+06nl; @rade-bock05jac]. This analysis allows us to further explore the role played by edge effects in the optoelectronic properties. Our results are summarized in Fig. \[clean\] and Table \[tab\]. As expected, the hydrogen removal leads to a major edge reconstruction, with the appearence of carbyne-like structures. In fact, the bond length for the edge dimers reduces from 1.36 for the passivated ribbons to 1.23 Å for the clean ones, pointing to the formation of C-C triple bonds at the edges. This edge modification leads to a variation of the energy gaps, such that the distinction between $N=3p-1$ and $N=3p$ families vanishes, in agreement with previous results [@baro+06nl].
In Fig. \[clean\] (a), we report the quasiparticle bandstructure for the $N=9$ bare ribbon. The main difference with respect to its passivated counterpart is the presence of edge-related bands (see arrows) in the low-energy optical window. Hence, we focus our attention on the properties of these edge states and their influence on the optical response. These states show the same energy dispersion and real-space localization, irrespective of both family and size, already in the LDA framework [@note-check]: due to this independence on bulk properties, their presence is reasonably expected for all non-passivated ribbons. The self-energy corrections to the LDA eigenvalues are similar to those of the passivated systems for the $\pi$ and $\pi^*$ bulk states. The edge states show quite a different correction, being deeper in energy and with a smoothed stretching with respect to the other bands \[Fig.\[clean\] (b)\]. This behaviour is to be ascribed to the different degree of real-space localization between bulk and edge states, and it can be singled out by virtue of the non-local character of the self-energy operator in the $GW$ framework, which is not correctly described within LDA.
The aformentioned modification of the bandstructure results in a correspondent blueshift ($N=8$) or redshift ($N=9$) of the lowest excitonic peak, with $A^{'}_8$ and $A^{'}_9$ becoming almost degenerate, with binding energies of about 0.9 eV. For the case of $N=10$, we find an inversion of the first and second conduction bands, which results in the $B^{'}_{10}$ peak lying below $A^{'}_{10}$ and $D^{'}$ almost degenerate in energy with $B^{'}_{10}$. In addition, the edge states introduce an optically inactive exciton, which arises from transitions among several bulk valence bands and the conduction edge states over the whole Brillouin zone. This [*edge exciton*]{} is present in all the studied nanoribbons and is located at about 1.4-1.7 eV (black arrow in Fig \[clean\] (d)), with very little dependence on family and size [@note-check]. This results in the edge exciton being above the first excitonic peak for $N=8$ and 9, and between the first and the second peaks for $N=10$. We remark that the accurate evaluation of quasi-particle corrections within $GW$, i.e. beyond the usual approximation based on a uniform band stretching on top of a rigid energy shift, is crucial to determine the exact energy position of the dark edge excitons relative to the bright ones.
To better understand the character of the edge-related dark state, we plot its excitonic wavefunction for the case $N=9$ in Fig. \[clean\] (d). The mixing of transitions over the whole Brillouin zone induces a strong localization of the edge exciton along the ribbon axis, with an extent of only $\sim 5$ Å, that is 4-7 times smaller than the Wannier-like [*bulk excitons*]{} (see Fig. \[exc-wfc\]).
In summary, we have found that the analysis of the electronic and optical features of GNRs requires a state-of-the-art approach within the many-body perturbation theory, and beyond the DFT framework. Many-body effects reveal that nanosized A-GNRs retain a quasi-1D character, which induces the suppression of the van Hove singularity, typical of non-interacting 1D systems, and the appearence of strong excitonic peaks in the optical absorption spectrum. The lowest excited states in GNRs are Wannier-like excitons and their binding energy as well as their luminescence properties are strongly dependent on the ribbon family. We investigate the role of many-body effects on the edge-states arising in non-passivated GNRs: our analysis could provide a practical tool for revealing the nature of the edges in realistic samples. We demonstrate that GNRs are intriguing systems with tunable optoelectronic features, that we quantitatively evaluate through our calculations. The present study calls for experiments addressing the optical response of GNRs: A combined theoretical and experimental understanding of ribbon size, family and edge-termination as control parameters for their performance can be considered as the first step towards the design of graphene-based applications in nanoscale optoelectronics.
We are grateful to A. Rubio, A. C. Ferrari, S. Piscanec, B. Montanari, T. Weller, M. Rontani and C. Cavazzoni for stimulating discussions. We acknoweledge CINECA CPU time granted through INFM-CNR. D. V. and A. M. thank the European Nanoquanta NoE (NMP4-CT-2004-500198) and the European Theoretical Spectroscopy Facility (ETSF).
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abstract: 'We use models of the rates of Type Ia supernovae (SNe Ia) and core-collapse supernovae, built in such a way that both are consistent with recent observational constraints at $z \la 1.6$ and can reproduce the measured cosmic star formation rate, to recover the history of the metals accumulation in the intra-cluster medium. We show that these SN rates, in unit of SN number per comoving volume and rest-frame year, provide on average a total amount of Iron that is marginally consistent with the value measured in galaxy clusters in the redshift range $0-1$, and a relative evolution with redshift that is in agreement with the observational constraints up to $z \approx 1.2$. Moreover, we verify that the predicted metals to Iron ratios reproduce the measurements obtained in nearby clusters through X-ray analysis, implying that (1) about half of the Iron mass and $\ga$ 75 per cent of the Nickel mass observed locally are produced by SN Ia ejecta, (2) the SN Ia contribution to the metal budget decreases steeply with redshift and by $z \approx 1$ is already less than half of the local amount and (3) a transition in the abundance ratios relative to the Iron is present between redshifts $\sim 0.5$ and $1.4$, with core-collapse SN products becoming dominant at higher redshifts.'
date: 'MNRAS, in press'
title: Brief history of the metal accumulation in the intracluster medium
---
galaxies: cluster: general – X-ray: galaxies – intergalactic medium – cosmology: observations.
INTRODUCTION
============
The primordial cosmic gas is composed by Hydrogen (about 75 per cent by mass), Helium ($\sim 24$ per cent), and traces of other light elements, like Deuterium, Helium-3, Lithium and Berillium. When this gas collapses into the dark matter halos typical of galaxy clusters ($\ga 10^{14} M_{\odot}$), it undergoes shocks and adiabatic compression, reaching densities of about $10^{-3}$ particles cm$^{-3}$ and temperatures of the order of $10^8$ K. For these physical conditions, the plasma is optically thin and its X-ray emission is dominated by thermal bremsstrahlung processes. Its enrichment by metals (i.e. elements with atomic number larger than 5) proceeds by the releases to the medium of the products of the star formation activities that take place in the member galaxies.
An effective way to estimate the metal abundance by number relative to Hydrogen is to measure the equivalent width of the emission lines above the X-ray thermal continuum. Nowadays, the Iron abundance is routinely determined in nearby systems thanks, in particular, to the prominence of the K-shell (i.e. the lower level of the electron transition is at the principal quantum number $n=1$) Iron line emission at rest-frame energies of 6.6-7.0 keV (Fe XXV and Fe XXVI). Furthermore, the few X-ray galaxy clusters known at $z>1$ have shown well detected Fe line, given sufficiently long ($\ga 200$ ksec) [*Chandra*]{} and [*XMM-Newton*]{} exposures (Rosati et al. 2004, Hashimoto et al. 2004, Tozzi et al. 2003). It is more difficult is to assess the abundance of the other prominent metals that should appear in an X-ray spectrum at energies (observer rest frame) between $\sim 0.5$ and $10$ keV, such as Oxygen (O VIII) at (cluster rest frame) 0.65 keV, Silicon (Si XIV) at 2.0 keV, Sulfur (S XVI) at 2.6 keV, and Nickel (Ni) at 7.8 keV.
A direct application of the detection of emission lines from these highly ionized elements is the possibility to describe the way the intracluster medium (ICM) is enriched of metals, whether by products from explosions of supernovae (SN) type Ia, mainly rich in Fe and Ni, or by outputs of collapsed massive stars (Core Collapsed SN), with relative higher contributions of $\alpha-$elements like O, Si, S. However, the ability to resolve and measure elemental abundances other than Iron, which was observed in the early days of the X-ray analysis of galaxy clusters (e.g. Mitchell et al. 1976, Serlemitsos et al. 1977) has required a significant improvement in the spectral resolution and sufficiently large telescope effective areas that have been reached initially with the X-ray satellite [*ASCA*]{} (e.g. Mushotzky et al. 1996, Fukazawa et al. 1998, Finoguenov, David & Ponman 2000) and then with the observatories of the new generation, such as [*Chandra*]{} (e.g. Ettori et al. 2002, Sanders et al. 2004) and [*XMM-Newton*]{} (e.g. Böhringer et al. 2001, Gastaldello & Molendi 2002, Finoguenov et al. 2002).
In the present work, we reverse the problem and infer from observed and modeled SN rates the total and relative amount of metals that should be present in the ICM both locally and at high redshifts. Through our phenomenological approach, we adopt the models of SN rates as a function of redshift that reproduce well both the very recent observational determinations of SN rates at $z \ga 0.3$ (Dahlen et al. 2004, Cappellaro et al. 2005) and the measurements of the star formation rate derived from UV-luminosity densities and IR data sets. We then compare the products of the enrichment process to the constraints obtained in the X-ray band for galaxy clusters observed up to $z \la 1.2$. We note that previous work on the production of Iron in galaxy clusters by, e.g., Matteucci & Vettolani (1988), Arnaud et al. (1992), and Renzini et al. (1993) made use of SN rates that were known, with large uncertainties, only at $z \approx 0$. However, they concluded that either the SN Ia rates were higher by a factor of 10 in the past or the production of Iron in clusters by core-collapsed supernovae (SNe CC) would require a very flat Initial Mass Function (IMF). An alternative scenario is that SN CC produce 1/4 of the Iron and the remnants 3/4 is released from SN Ia, which would also be in accordance with the standard chemical model for the galactic halo and disk, allowing an IMF in elliptical galaxies similar to the one in the solar neighborhood.
The main purpose of this work is to infer the mass of the most relevant and X-ray detectable elements present in the ICM and their relative abundance as function of redshift by using reliable (i.e. in agreement with several different observational constraints) models of SN explosion rates that include a dependence on the cosmological time. The fact that these rates are in unit of number per comoving volume and rest-frame time allows us to use them also over cosmological distances typical of high-redshift systems. The method presented here is a first attempt to provide well-motivated and analytic predictions of the expected metal abundances at $z \ga 0.5$ and of the relative role played by SNe Ia and SNe CC in polluting the ICM over time.
[c c c c c c c]{}\
[*metal*]{} & $W$ & $A$ & $m_{Ia}$ & $Y_{Ia}$ & $m_{CC}$ & $Y_{CC}$\
& & & $M_{\odot}$ & & $M_{\odot}$\
Fe & 55.845 & 4.68e-5 & 0.743 & $-$ & 0.091 & $-$\
O & 15.999 & 8.51e-4 & 0.143 & 0.037 & 1.805 & 3.818\
Si & 28.086 & 3.55e-5 & 0.153 & 0.538 & 0.122 & 3.526\
S & 32.065 & 1.62e-5 & 0.086 & 0.585 & 0.041 & 2.284\
Ni & 58.693 & 1.78e-6 & 0.141 & 4.758 & 0.006 & 1.647\
\
Observational constraints on the metal content of galaxy clusters
=================================================================
The observed Iron mass is obtained as $$M_{\rm Fe, obs} = 4 \pi A_{\rm Fe} W_{\rm Fe} \int_0^R Z_{\rm Fe}(r) \rho_{\rm H}(r) r^2 dr,$$ where $Z_{\rm Fe}(r)$ is the Iron abundance relative to the solar value $A_{\rm Fe}$ measured in the ICM as function of radius, $\rho_{\rm H}(r) = \rho_{\rm gas}(r) / (2.2 \mu)$ is the Hydrogen density, $\mu$ is the mean molecular weight and has a value of 0.60 for a typical fully ionized cluster plasma with a number density $n_{\rm gas} = n_{\rm H} +n_{\rm e} = 2.2 n_{\rm H}$. Our estimates refer to a cosmology with parameters $(H_0, \Omega_{\rm m}, \Omega_{\Lambda})$ equal to $(70, 0.3, 0.7)$.
De Grandi et al. (2004) have correlated the total amount of $M_{\rm Fe}$ observed with the X-ray satellite [*BeppoSAX*]{} in 22 nearby hot ($kT > 2$ keV) galaxy clusters with the gas temperature. For the assumed cosmology, we obtain a best-fitting result of $\log(M_{\rm Fe}/10^{10} M_{\odot})
= 0.33 (\pm 0.07) +1.62 (\pm 0.53) \times \log(kT/5 {\rm keV})$ (scatter of 0.28). This fit considers quantities estimated at the radius $R_{500}$, where the cluster overdensity is 500 times the critical density for an Einstein-de Sitter universe (i.e. an overdensity of 285 at $z=0$ for the cosmology assumed here) and corresponds to about $0.6 \times R_{\rm vir}$. The overall results presented in this paper do not depend upon the cluster gas temperature, although a value of $5$ keV is hereafter adopted as reference. Therefore, an Iron mass of $(2.1 \pm 0.3) \times 10^{10} M_{\odot}$ is associated with a $5$-keV cluster within $R_{500}$ at a median redshift of $0.05$, with a scatter in the range of $(1.1, 3.9) \times 10^{10} M_{\odot}$. At redshift between $0.3$ and $1.3$, for a sample of 19 objects observed with [*Chandra*]{} and presented in Tozzi et al. (2003) and Ettori et al. (2004a), that have a detection significant at the level of $2 \sigma$ of the Iron line emission and for which a single emission-weighted temperature and the radial profile of the gas density were measured, we obtain a best-fitting $\log(M_{\rm Fe}/10^{10} M_{\odot}) = 0.66 (\pm 0.06) +1.47 (\pm 0.48)
\times \log(kT/5 {\rm keV})$ (scatter of 0.23) and infer a total Iron mass for a $5$ keV cluster at the median $z=0.63$ of $M_{\rm Fe} (<R_{500}) = (2.7 \pm 0.4) \times 10^{10} M_{\odot}$, with a scatter limited within the values of $(1.6, 4.6) \times 10^{10} M_{\odot}$. These values are plotted in Fig. \[fig:mfe\_z\]. We assume that most (if not all) of the mass of the metals is located within $R_{500}$, even though this radius encloses just about 13 per cent of the virial volume ($R_{500} \approx 0.5 R_{\rm vir}$). A further contribution from the outer cluster regions is however marginal: from numerical simulation (e.g. Ettori et al. 2004b) and extrapolated data (e.g. Arnaud, Pointecouteau & Pratt 2005), one can estimate that $M_{500} / M_{\rm vir} \approx 0.6$ and, by assuming that in the outskirts (1) the metallicity is $<0.1$ times the solar value, $A_{\odot}$, whereas its mean value in the central regions is about $0.3 A_{\odot}$, and (2) the gas follows the dark matter distribution, one can evaluate that $M_{\rm Fe} (R_{500} < r < R_{\rm vir}) / M_{\rm Fe} (r < R_{500})
\la 0.2$. Apart from Fe, the most prominent lines detectable in X-ray spectra, like O, Si, S and Ni, were investigated initially with data from [*ASCA*]{} (an analysis of the average abundance of all the galaxy clusters in the [*ASCA*]{} archive is presented in Baumgartner et al. 2005) and more recently with the larger effective area and improved spectral and spatial capabilities of [*XMM-Newton*]{} (e.g. Tamura et al. 2004 present statistics of abundance ratios, also resolved spatially, for a sample of 19 nearby galaxy clusters). In Fig. \[fig:yfe\_z0\], we summarize these results by plotting the relative number abundance ratios. The Iron abundance is about $0.3 A_{\odot}$, although the [*ASCA*]{} measurements have a very narrow distribution around the mean of $0.27$ ($\pm 0.01$ at 90 per cent level of confidence) that departs significantly from the spatially resolved [*XMM-Newton*]{} estimates (only in the outer radius, between 200 and 500 $h^{-1}$ kpc, where the Fe/H decreases by about a factor of 2 with respect to the central estimate, it becomes consistent with the [*ASCA*]{} mean with a value of $0.32 \pm 0.08$). Moreover, the O/Fe ratio is about solar, whereas the Si/Fe and S/Fe ratios are roughly super-solar and sub-solar, respectively. The average Ni/Fe ratio in the [*ASCA*]{} sample is about $3.4 A_{\odot}$, well in agreement with the original [*ASCA*]{} determination in the data of the Perseus cluster by Dupke & Arnaud (2001) and consistent with the [*XMM-Newton*]{} constraints in Gastaldello & Molendi (2004) and with the [*BeppoSAX*]{} results for a sample of 22 objects (Fig. 6 in De Grandi & Molendi 2002). On the other hand, it is worth mentioning that the measurements of the Nickel abundance becomes reliable only in hot ($kT \ga 4$ keV) systems where the excitation of (mainly) K-shell lines makes them detectable in the spectrum around 8 keV, in a region less contaminated from blends with Iron but highly affected from a proper background subtraction. These difficulties put the actual limitations on solid Ni measurements for a large data set.
Given the differences between the two data sets, and the fact that the [*XMM-Newton*]{} estimates are spatially resolved and thus sensitive to the presence of any gradient that instead is washed out in the global measurement shown here from [*ASCA*]{}, they are considered independently in the following analysis.
Accumulating the metals
=======================
The total predicted Iron mass is obtained as $$\begin{aligned}
M_{\rm Fe, SN} & = & M_{\rm Fe, Ia} +M_{\rm Fe, CC} \nonumber \\
& = & \sum_{dt, dV} m_{\rm Fe, Ia} \times r_{\rm Ia}(dt) \times dt \times dV \nonumber \\
& & +\sum_{dt, dV} m_{\rm Fe, CC} \times r_{\rm CC}(dt) \times dt \times dV \end{aligned}$$ where we assume $m_{\rm Fe, Ia}$ and $m_{\rm Fe, CC}$ as quoted in Table 1 (from Nomoto et al. 1997) and an interval time at given redshift equals to $dt(z0,z1) = H_0^{-1} \int_{z0}^{z1} (1+z)^{-1}
\left[\Omega_{\rm m} (1+z)^3 +(1-\Omega_{\rm m}-\Omega_{\Lambda}) (1+z)^2
+\Omega_{\Lambda} \right]^{-0.5} dz$. As cluster volume, we define the volume corresponding to the spherical region that encompasses the cosmic background density, $\rho_{\rm b} = 3 H_0^2 / (8 \pi G) \times \Omega_{\rm m}$ ($\approx 4 \times 10^{10} M_{\odot}$ Mpc$^{-3}$ for the assumed cosmology), with a cumulative mass of $M_{\rm vir} \approx 6.8
(kT / 5{\rm keV})^{3/2} \times 10^{14} M_{\odot}$, as inferred from the best-fit results on the observed properties of X-ray clusters with $kT > 3.5$ keV in Arnaud et al. (2005) and adopting $M_{500}/ M_{\rm vir} \approx 0.6$: $dV = M_{\rm vir} / \rho_{\rm b} = 24,514 \ \frac{M_{\rm vir}}{10^{15} M_{\odot}}
\left( \frac{kT}{5{\rm keV}} \right)^{3/2} \ h_{70}^{-3}$ Mpc$^3$. It is worth noticing that the $M_{\rm vir}-T$ relation for simulated galaxy clusters (e.g. Ettori et al. 2004b), with an emission-weighted temperature greater than $3.5$ keV, provides a typical virial mass (and corresponding volume) that is larger than the above value by about 20 per cent. We discuss in Section 4 the effect of this discrepancy on the overall results. The volume measured at $z=0$ is considered as the dimension of the region involved in the formation of a typical galaxy cluster and is maintained constant in redshift.
The supernovae rates, i.e. [*the number of SN per rest-frame time and comoving volume*]{}, come from models that fit well the observed values and reproduce properly the Star Formation Rate (SFR) as widely discussed in Strolger et al. (2004) and Dahlen et al. (2004). In summary, once a functional form is defined to describe the evolution with time of the SFR (see equation 5 in Strolger et al. 2004 with parameters for the extinction-corrected model, called M1; the uncorrected M2 model just reduces the total amount of metals released by a factor of $\sim 4$), the corresponding SN rates can be defined as $$\begin{aligned}
r_{\rm Ia}(t) & = & \nu \int_{t_F}^t SFR(t') \times \phi(t-t') \ dt' \nonumber \\
r_{\rm CC}(t) & = & k \times h^2 \times SFR(t)\end{aligned}$$ where the normalization $\nu$ indicates how many SNe Ia explode per unit formed stellar mass and is function of the delay time distribution function $\phi(t_d)$ that represents the relative number exploded at a time $t_d$ since a single burst of star formation; $t_F$ is the time of formation of the first stars and corresponds to $z_F = 10$; the normalization $k$ is the number of CC progenitors per unit of formed stellar mass; $h$ is the Hubble constant in unit of 100 km s$^{-1}$ Mpc$^{-1}$. In the following analysis, we adopt a “narrow" Gaussian form for $\phi(t_d)$, with a delay time $\tau = 4$ Gyr and $\sigma_{t_d} = 0.2 \tau$ that better fit both the local ($z \la 0.1$) rates and the GOODS data (Strongler et al. 2004, Dahlen et al. 2004) and fix $\nu = 0.0010$. We assume $k = 0.0069 M_{\odot}^{-1}$ as appropriated for CC progenitors masses in the range $8 M_{\odot} \la M \la 50 M_{\odot}$ with a Salpeter (1955) Initial Mass Function (IMF). We show in Fig. \[fig:r\_cc\_ia\] the relative rates of SNe CC and Ia as function of redshift: this ratio ranges between 2 and 3 up to $z \approx 1$ and increases steeply at higher redshifts, with the “narrow" Gaussian function that provides a SN Ia rate at $z \approx 2$ lower by an order of magnitude than the other delay time distribution functions.
Regarding the adopted delay time, we mention here that Gal-Yam & Maoz (2004) and Maoz & Gal-Yam (2004) support the argument that some current data on the cosmic (field) star formation history require for the observed rate of SNe Ia a delay time larger than 3 Gyr. Combining this with the low observed rate of cluster SN Ia at $z<1$, they conclude that the measured amount of Iron in clusters cannot be produced mainly by SNe Ia explosions but, more probably, must be produced via outputs of SNe CC originating from a top-heavy IMF. The reader is refereed to the above-mentioned work for a detailed discussion on the effects of the assumed IMF on the SN rates. Our approach here is to use the best phenomenological models of cosmic star formation history to infer the corresponding cluster metal accumulation history and to compare some predictions with the observational constraints available.
To produce the cluster metal accumulation history, we have to take into account the fraction of the total produced Iron mass that remains locked up in the stars. The fraction of the total Iron that is then released to the ICM is estimated by considering that $M_{\rm Fe, tot} \approx Z_{\rm ICM} M_{\rm ICM} +
Z_{\rm star} M_{\rm star}$. We adopt $Z_{\rm ICM} \approx 0.3 A_{\odot}$ (see Fig. \[fig:yfe\_z0\]), $M_{\rm star} \approx 0.13 M_{\rm ICM}$ (from equations 1 and 10 in Lin, Mohr & Stanford 2003 for a 5 keV cluster) and $Z_{\rm star} = 1.38 A_{\odot}$ (from interpolation of the results in Lin, Mohr & Stanford 2003; see also Renzini 2003, Portinari et al. 2004). Therefore, $M_{\rm Fe, ICM} = M_{\rm Fe, tot} / (1 + f_Z)$, where $f_Z =
M_{\rm star} / M_{\rm ICM} \times Z_{\rm star} / Z_{\rm ICM} \approx 0.59$. Considering its weak dependence upon time (see, e.g., Table 2 and Fig. 3 in Portinari et al. 2004), we neglect the evolution of the locked-up fraction.
In Fig. \[fig:mfe\_z\], we plot the results of the expected Iron mass compared to the observational results discussed in Section 2. The predicted amount is well in agreement with the observed Iron mass in local clusters. At a median redshift of 0.05, the De Grandi et al. sample has a mean $M_{\rm Fe}$ associated to a 5 keV object that is between 30 (when the $M_{\rm vir}-T$ relation from simulations is adopted) and 60 (with the observed $M_{\rm vir}-T$ relation) per cent higher than the Iron mass accumulated in the ICM through SN activities from $z=10$. The latter value is, however, well within the scatter in the observed distribution. At higher redshift, the observed central value in the Tozzi et al. sample of Fe abundance measurements is a factor of about 3 higher than the predicted one. It is worth noticing that this result does not depend significantly upon either the delay time distribution function adopted to calculate $r_{\rm Ia}$ or the SN CC and Ia compilations assumed. The changes in the total Iron mass accumulated at $z=0.05$ are in the order of 10 per cent when different $\phi(t_d)$ are considered and of about 40 per cent (i.e. the ratio of the observed and the predicted values ranges between 1.0–1.2 and 1.4–1.8; see Table \[tab:test\]) when we use the calculations of SN CC explosions in two extreme cases presented in Woosley & Weaver (1995) and described at the end of this Section.
Note that, if the SN Ia rate in number per comoving volume and rest-frame year is converted to supernova unit (1 SNu = 1 SN per 100 year per $10^{10} L_{\odot, B}$) with an assumed local B-band luminosity density of $2 \times 10^8 h L_{\odot}$ Mpc$^{-3}$ which evolves as $(1+z)^{1.9}$ (see Dahlen et al. 2004 for the caveats in using such conversion for rates estimated on cosmological distances), we obtain a local ($z \approx 0.05$) rate of $0.25$ and $0.79$ SNu for SN Ia and CC, respectively. By integrating the SNu values over the cosmic time, we measure a local $M_{\rm Fe} / L_B$ of $\sim 0.0020$ and $0.0010 h_{70}^2$ as due to SN Ia and CC, respectively, with a dependence upon the redshift very similar to what is shown in Fig. \[fig:mfe\_z\]. The sum of these values is still a factor between 3 and 5 lower than the present estimate in galaxy clusters: this estimate, however, is affected from the extension of the cluster region considered to measure B-band luminosities and Iron mass distribution (see discussion on the uncertainties of these measurements in De Grandi et al. 2004, Sect.3.2). Therefore, even though we solve most of the discrepancy between predicted and measured Iron mass in typical galaxy clusters through models of SN rates that make use of the number of SN per comoving volume and rest-frame time, some difficulties remain in recovering the observed $M_{\rm Fe} / L_B$ ratio through a direct, but not straightforward (given the evolution with redshift of the B-band luminosity density), conversion to the number of SN per unit of B-band luminosity (e.g. Arnaud et al. 1992, Renzini et al. 1993). Regarding the latter issue, note that the suggested solution of an increase of the SN Ia rate in E/S0 galaxies by a factor of 5–10 at higher$-z$ is partially taken into account in the models presented here (see, e.g., Fig. \[fig:r\_cc\_ia\]), where the SN Ia rate is enhanced by a factor of $\sim 5$ from $z=0$ to $z \approx 0.8$, but then decreases rapidly beyond it. We can now extend these considerations to the enrichment history of the metals other than Fe. To infer these, we adopt, as done in the X-ray analyses presented above, the solar abundances tabulated in Anders & Grevesse (1989) and the most recent and widely used theoretical metal yields of Nomoto et al. (1997) for (i) SNe CC integrated between $10$ and $50 M_{\odot}$ for a Salpeter IMF (see Nakamura et al. 1999 on the uncertainties related to the adopted mass cut that can change the Nickel yield by a factor of 2) and (ii) SNe Ia exploding with fast deflagration according to the W7 model. All the numerical values considered here are presented in Table 1. We note that the values presented in Nomoto et al. (1997) are plotted relative to the solar photospheric abundance in Anders & Grevesse (1989). The latter values have been revised to match the meteoritic determinations, as summarized in Grevesse & Sauval (1998), and require conversion factors of $(0.676, 0.794, 1, 1.321, 1)$ to correct the original photospheric abundance for Fe, O, Si, S and Ni, respectively. We have, however, adopted the Anders & Grevesse values for consistency between models and observational constraints.
Given a metal $i$, its total mass is estimated as $$\begin{aligned}
M_i & = & M_{i, Ia} + M_{i, CC} \nonumber \\
& = & \sum_{dt, dV} m_{\rm i, Ia} \times r_{\rm Ia}(dt) \times dt \times dV \nonumber\\
& & +\sum_{dt, dV} m_{\rm i, CC} \times r_{\rm CC}(dt) \times dt \times dV \nonumber\\
& = & \left(M_{{\rm Fe}, Ia} \ Y_{Ia} + M_{{\rm Fe}, CC} \ Y_{CC}
\right) \frac{W_i \ A_i}{W_{\rm Fe} \ A_{\rm Fe}}, \end{aligned}$$ where the synthesized masses for each element of interest, $m_i$, or the corresponding abundance yields, $Y$, are presented in Table 1.
One direct way to check the consistency of these models is to compare the ratio between some of the most prominent elements detectable in X-ray spectra and the Iron. To this purpose, we overplot in Fig. \[fig:yfe\_z0\] the abundance ratios inferred from the models to the observational constraints on the O/Fe, Si/Fe, S/Fe and Ni/Fe discussed in Section 2. The agreement is remarkably good for what concerns the [*XMM-Newton*]{} means and the [*ASCA*]{} Ni/Fe measurement. A larger departure appears between the model prediction and the result for the S/Fe estimate from [*ASCA*]{}. We can evaluate the agreement by calculating the $\chi^2$ given the observational constraints (with the relative errors) and the predicted values. We obtain a $\chi^2 = 3.1$ (3 degrees of freedom) for the [*XMM-Newton*]{} data, which is statistically acceptable \[$P(\chi^2 > \chi^2_{\rm obs}) = 0.63$\] and is, however, the largest value measured for any delay time distribution functions: the “wide" Gaussian and the $e-$folding form (see caption of Fig. \[fig:r\_cc\_ia\]) give a $\chi^2$ of $2.2$ and $2.3$, respectively (see Table \[tab:test\]). It is worth noting that the Si/Fe ratio alone contributes about $2.2$ to the total $\chi^2$, owing to the over-predicted amount of Silicon by $\sim 40$ per cent. On the other hand, the [*ASCA*]{} measurements provide a $\chi^2$ of about 10.0 \[3 degrees of freedom, $P(\chi^2 > \chi^2_{\rm obs}) = 0.98$\], mainly due to the over-prediction of the S/Fe ratio by 80 per cent. We have also considered alternative SN CC and Ia compilations to the Nomoto et al. one, adopted here as reference. Following the discussion in Gibson, Loewenstein & Mushotzky (1997) on the uncertainties related to the physics of exploding massive stars, we consider two extreme conditions for the explosion of SN CC as indicated in Woosley and Weaver (1995), namely case A with metallicity equals to $10^{-4}$ solar and case B with solar metallicity. Both the models provide best-fit results worse than the Nomoto et al. one, with a $\chi^2$ of 9.3 and 3.1 (case A and B, respectively) when compared to the Tamura et al. abundance ratios and 42.5 and 21.0 when compared to the Baumgartner et al. values (see Table \[tab:test\]). Moreover, we consider SN Ia models with a delayed detonation induced from deflagration at low density layers (model WDD2 in Nomoto et al. 1997) that seem to reproduce well the observed yields in the outer regions of M87 (Finoguenov et al. 2002). This model doubles the amount of Si ($m_{\rm Si} =
0.27 M_{\odot}, Y = 1.01$) and S ($m_{\rm S} = 0.17 M_{\odot}, Y = 1.20$) and reduces the Ni ejecta by 70 per cent ($m_{\rm Ni} = 0.04 M_{\odot}, Y = 1.40$) with respect to W7, amplifying any mismatch with the observational mean values so that the $\chi^2$ increases to $\sim$ 8 and 38 when model predictions are compared with data from Tamura et al. and Baumgartner et al., respectively. We thus exclude the possibility that this model of SN Ia explosions can explain the present results on the overall enrichment history of galaxy clusters better than W7.
[l c c c c c c c]{}\
models & & $\frac{\hat{M}_{\rm Fe}}{M_{\rm Fe}}$ & $\frac{\hat{M}_{\rm Fe}}{M_{\rm Fe}}$ & & $\chi^2$ (dof) & $\chi^2$ (dof) & $\chi^2$ (dof)\
(Ia, CC, $\phi(t_d)$) & & $z=0.05$ & $z=0.63$ & & vs $Fe(z)$ & vs [*XMM*]{} & vs [*ASCA*]{}\
&\
\
(W7, N97, narrow) & & 1.29–1.58 & 2.58–3.16 & & 0.64 (5) & 3.1 (3) & 10.0 (3)\
& &\
(WDD2, ..., ...) & & 1.33–1.62 & 2.64–3.22 & & 0.64 (5) & 8.1 (3) & 37.5 (3)\
(..., WW95A, ...) & & 1.43–1.75 & 2.94–3.59 & & 0.66 (5) & 9.3 (3) & 42.5 (3)\
(..., WW95B, ...) & & 1.02–1.24 & 1.92–2.35 & & 0.66 (5) & 3.1 (3) & 21.0 (3)\
(..., ..., wide) & & 1.20–1.47 & 2.48–3.03 & & 0.68 (5) & 2.2 (3) & 8.9 (3)\
(..., ..., $e-$fol) & & 1.22–1.49 & 2.33–2.85 & & 0.74 (5) & 2.3 (3) & 9.0 (3)\
\
\[tab:test\]
We can now trace back the abundance ratios as function of redshift (see Fig. \[fig:yfe\_z\] and \[fig:fe\_z\]) and, given this history of the metals accumulation in galaxy clusters, plot in Fig. \[fig:mia\_z\] the relative contribution in mass of the SN Ia products to the total material of each elements investigated (i.e. O, Si, S, Fe and Ni) available in the cluster baryon budget as function of redshift.
Locally, 51 per cent of Iron, 1 per cent of Oxygen, 14 per cent of Silicon, 21 per cent of Sulfur and 75 per cent of Nichel are produced from SN Ia explosions. These values change only by few ($\la 3$) per cent by adopting different delay time distribution functions and by $\la 10$ per cent when the two other extreme cases of SN CC explosion are considered. We conclude that these predictions for nearby clusters are particularly stable and robust. As expected from the relative increase of SNe CC, the SN Ia contribution to the metal budget decreases significantly at higher redshifts, becoming responsible for less than half of the metal masses observed locally at $z \approx 1$. More interestingly, the predicted evolution in the Iron abundance (Fig. \[fig:fe\_z\]), although significantly steepens up to $z \approx 1$ where the expected value is half of the local one, is still consistent with observational constraints obtained from a sample of [*Chandra*]{} exposures of 49 clusters at redshift $>0.3$ (Fig. \[fig:fe\_z\]).
Summary and Discussion
======================
We summarize here our main findings on the history of the metals accumulation in the ICM. By using the rates of SNe Ia and CC as observed (at $z<1.6$ and $z<0.7$, respectively) and modelled from the cosmic star formation rate derived from UV-luminosity densities and IR data sets (Dahlen et al. 2004, Strolger et al. 2004) and adopting theoretical yields (as described in Table 1), we infer how the metals masses in the ICM are expected to accumulate as a function of the redshift.
We find that these models predict that the total Iron mass accumulated in massive galaxy clusters through SN activities is between 24 and 75 per cent (235–359 per cent) lower than the Iron mass estimated in local (high$-z$) systems as determined through the equivalent width of the emission lines detected in X-ray spectra (e.g. De Grandi et al. 2004, Tozzi et al. 2003). This discrepancy is reduced by about 20 per cent when the cluster volume used to accumulate the SN products as a function of time is defined by adopting an average virial mass from numerical simulations instead of the value extrapolated from the observed X-ray scaling relationships (e.g. Ettori et al. 2004b, Arnaud et al. 2005). Because of this kind of uncertainty in relating the cluster gas temperature to the associated virial and Iron masses, we conclude that the agreement between the expected and measured Iron mass, even though marginal, is acceptable within the observed scatter. Moreover, we can reproduce the relative number abundance of the most prominent metals detectable in an X-ray spectrum, such as Oxygen, Silicon, Sulfur and Nickel with respect to the Iron as estimated for nearby bright objects observed both with [*ASCA*]{} (Baumgartner et al. 2005) and [*XMM-Newton*]{} (Tamura et al. 2004). We also show that the predicted evolution in the Iron abundance, which should decrease by a factor of 2 at $z \approx 1$ with respect to the local value, is in good agreement with the current observational constraints obtained from a sample of 49 clusters at $z>0.3$ (see Fig. \[fig:fe\_z\]).
By using these models to describe the ICM enrichment, we can infer the relative SN contribution to the amount of elements present in galaxy clusters as function of the cosmic time. At increasing redshift, the products from SNe CC become dominant, owing to the steep rise of their relative rate with respect to SNe Ia. The transition occurs between $z=0.5$ and $1.4$, with an enhancement of the $\alpha-$elements with respect to Fe (e.g. O/Fe and Si/Fe ratios increase by a factor of more than 2) and a drastic decrease of the Ni/Fe ratio (see Fig. \[fig:yfe\_z\]). Then, the fractions of metals mass due to SN Ia outputs, which are locally about 51, 75, 1, 14 and 21 per cent of the total for Iron, Nickel, Oxygen, Silicon and Sulfur, respectively, halve at $z \approx 1$ (Fig. \[fig:mia\_z\]), almost independently from the adopted delay time distribution functions and models for SN CC explosions. When the assumed SN rates (number per comoving volume per rest-frame year) are converted to units of the B-band luminosity (but see caveats in Dahlen et al. 2004), which is well mapped only in nearby galaxy clusters, we show that the expected total $M_{\rm Fe} / L_B$ is still below the local measurement by a factor between 3 and 5 and that SN Ia metal production contributes by $\sim$70 per cent. We conclude that this well-known (e.g. Arnaud et al. 1992, Renzini et al. 1993) underestimate of the local $M_{\rm Fe} / L_B$ value cannot be explained with an increase in the SN rates at higher redshift, according to the present models.
The changes by a factor of 2 in the abundance ratios at $z>0.5$ arising from the predominance of the enrichment through SNe CC might be investigated in the near future with X-ray spectroscopically resolved metal abundance estimates in high-redshift galaxy clusters. We are attempting this with our [*Chandra*]{} and [*XMM-Newton*]{} data set of high$-z$ objects (Tozzi et al. 2003, Ettori et al. 2004a), with expected relative uncertainties on the abundance ratios of larger than $\sim20$ per cent at the $1 \sigma$ level (see, for example, Fig. \[fig:fe\_z\]). These X-ray satellites offer the best compromise available at present between field-of-view, effective area, and the spatial and spectral resolution required to pursue such a study. Only with [*XEUS*]{}[^1] and its sensitivity greater by two orders of magnitude than that of [*XMM-Newton*]{} and a spectral resolution of the order of 10 eV or less, will it be possible to investigate with significantly higher accuracy the metal budget of the ICM in high$-z$ systems.
ACKNOWLEDGEMENTS {#acknowledgements .unnumbered}
================
I thank Stefano Borgani for insightful comments and suggestions, Alexis Finoguenov and Paolo Tozzi for useful discussions. Italo Balestra is thanked for the spectral analysis of the data presented in Fig. \[fig:fe\_z\]. I thank an anonymous referee for helpful remarks relevant to improving the presentation of this work.
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[^1]: http://www.rssd.esa.int/index.php?project=XEUS
|
---
abstract: 'RoboCup Junior is a project-oriented educational initiative that sponsors regional, national and international robotic events for young students in primary and secondary school. It leads children to the fundamentals of teamwork and complex problem solving through step-by-step logical thinking using computers and robots. The Faculty of Engineering and Built Environment at the University of Newcastle in Australia has hosted and organized the Hunter regional tournament since 2012. This paper presents an analysis of data collected from RoboCup Junior in the Hunter Region, New South Wales, Australia, for a period of six years 2012-2017 inclusive. Our study evaluates the effectiveness of the competition in terms of geographical spread, participation numbers, and gender balance. We also present a case study about current university students who have previously participated in RoboCup Junior.'
author:
- 'Aaron S.W. Wong'
- Ryan Jeffery
- Peter Turner
- Scott Sleap
- 'Stephan K. Chalup'
bibliography:
- 'RJCHR.bib'
title: |
RoboCup Junior in the Hunter Region:\
Driving the Future of Robotic STEM Education
---
INTRODUCTION
============
The National Innovation and Science Agenda [@Commonwealth2016] is the Australian government’s initiative to improve and promote technologically related fields of commercial, industrial, and technical skill development for Australian citizens with a component focusing on the preparation of young school-aged students for studies related to the fields of Science, Technology, Engineering, and Mathematics (STEM). This component has been recognised as the next step in the evolution of the Australian education system. It is understood that the next generation of graduating students will have to be “STEM-ready” to cope with the challenges of a future technologically advanced and internationally competitive workforce.
Educating a STEM-ready workforce is not a trivial task as there are several behavioural factors inherent to modern Australian culture which impede this goal. These include fear of failure [@Michou2014] (of both students and their teachers), and mathematics anxiety [@wigfield1988]. Students in general should be encouraged at an early stage to take intellectual risk in order to gain life-skills that could be of value for a STEM-related career path [@AIG2015]. For students, pathways into a STEM related field can be increased by diversifying opportunities and options, so that there is a higher probability of attracting their attention. They could start, for example, with programming websites or building electronic components. In order to help students to access STEM, it is important to explain practical applications and give STEM a purpose. It also should be made clear that mathematics consists of many different disciplines and that they may require very different ways of thinking. There are different options within STEM and similarly within the general area of mathematics, e.g., not everyone who is talented in geometry and visualisations may also be good at memory and number tasks. With these critical thoughts in mind, a STEM-ready workforce has the potential to advance many different aspects of technology.
One aspect of an advanced technology can be found in the field of robotics. Many STEM-related fields and skills are required to develop an autonomous robot. These include sound knowledge of the fundamental concepts of science and mathematics as taught at school. These are also the basis to a successful development of the practical skills required by professional engineers. Similarly as the Personal Computer (PC) and the Internet had revolutionary impact on our culture, now mobile devices and robots are predicted to be basis of the next technological explosion. Hence, it is important to encourage young students to consider gaining skills in professions related to robotics or to pursue a other career paths that also can lead to a technologically skilled future, i.e., a “STEM-ready” future. For the Hunter region in New South Wales (NSW), Australia, RoboCup Junior [@RobocupJuniorWeb] is the only annual robotics event that targets young students to actively compete and perform as a team. RoboCup Junior is a project-oriented educational initiative that sponsors regional, state, national and international robotic events for young students, with the goal to encourage the next generation to pursue and take an interest in scientific and technological fields. In this paper we will investigate the following question: How can the current society be prepared for a sustainable STEM-minded future? How can a community engagement project such as RoboCup Junior Hunter contribute and how can its success be measured? The subsequent sections show how RoboCup Junior served as a platform for community engagement and for promoting STEM in the Hunter region. By detailing demographic information about the Hunter region it highlights the importance and impact of RoboCup Junior. The results section provides quantitative measures derived from data collected from RoboCup Junior events over the past 6 years. Then a case study of current University of Newcastle students who have previously participated in the competition is presented. The penultimate section discusses the importance and roles of several stakeholders that collaboratively supported RoboCup Junior in the Hunter region and how this led to one of the most successful regional initiatives of its kind in Australia.
ROBOCUP JUNIOR IN THE HUNTER REGION
===================================
RoboCup (est. 1997) is an international competition that fosters research in robotics [@Kitano1997], and the advancement of artificial intelligence within a competitive environment. RoboCup has seen a globally increasing trend in the past decade, see Fig. \[fig:1\]. The goal of RoboCup, in the near future, is to have designed and programmed a team of bipedal humanoid autonomous soccer playing robots, to win against the human world champion soccer team [@Veloso2012]. This goal is yet to mature, and may require some generations of research to achieve, and hence we have RoboCup Junior; the establishment of the next generation of “technologists”, with a focus on robotics.
![The number of participants who have attended the RoboCup International tournament since its inception [@Veloso2012]: There is a strong linearly increasing trend ($R^2 > 0.90$) over time with the number of people globally participating in this tournament.[]{data-label="fig:1"}](Figure1.PNG){width="\textwidth"}
RoboCup Junior is designed to introduce primary and secondary school-aged children to the fundamentals of teamwork and complex problem solving by employing step-by-step logical, rational processes using computers (robots) as a tool to complete a set task. The main objective of RoboCup Junior is to encourage the next generation to pursue and take an interest in scientific and technological fields; to cultivate their interests through a hands-on approach in robot design and creation using platforms including, but not limited to, Lego Mindstorm educational kits. Students are invited to compete in three distinct disciplines; soccer, dance, and rescue.
With the RoboCup Junior initiative, it is possible to create an environment of light-mindedness, experimentation, fun and teamwork that inspires and educates students to expand their horizon through STEM experiences. In this context, there are countless opportunities to establish links to other associated STEM disciplines. It is important that students feel respected as individuals and that they have, at an early stage, access to demonstrations and practical hands-on aspects of STEM careers in a broad manner where they can explore their own career goals. This is one of the key and defining ideals of RoboCup Junior in the Hunter region, NSW.
The Hunter region, NSW, resides approximately two hours north of Sydney and has a substantial rural demography as well as large urban population centres in Newcastle and Lake Macquarie. Although, the Hunter region has been noted to be an innovation hub, e.g. Newcastle as “Smart City” [@hunterbusinessreview_2016], where new smart technologies are being developed as applied solutions to the problems of the world today, the general population includes negatively skewed low social economic status (SES) indicators, when compared to the state’s capital, Sydney. For example, Higher School Certificate (HSC) completion rates in the Cessnock Area, within the Hunter region, are with 44% substantially lower than the NSW average of 75% [@CesnockCC2015]. These low-SES indicators have led the organisers of RoboCup Junior in the Hunter to follow an approach that maintains the core rules of the competition while incorporating additional coaching to promote participation. This allows children to have an attitude of “having-a-go”, to have fun, and to enjoy themselves while avoiding anxieties associated with STEM subjects and while subconsciously having a positive experience with STEM and gaining important skills required for a potential career path in STEM.
A career path in STEM does not require direct entry into a university degree, as there are many different pathways into a STEM-related career. However, traditional entry into a university STEM-related degree has generally been perceived to be the fastest arrangement to refine skills, achieve, advance and progress in a competitive STEM workforce. For this reason, the following results section presents information obtained in a case study of currently enrolled students (with their permission) who were past participants of RoboCup Junior, Hunter. This case study together with quantitative results recorded over the past six years corroborates the view that the Hunter RoboCup Junior initiative had substantial positive impact on driving STEM education in the Hunter Region.
RESULTS
=======
QUANTITATIVE ANALYSIS
---------------------
{width="\textwidth"}
Over the past six years, 2012 to 2017 (see Fig. \[fig:2\]) there has been a general growth in the number of students, teams, as well as schools, with a total of 1443 student participants in the Hunter Region RoboCup Junior competition. In 2015 the ME program (Section \[sec:ME\]) was not able to support the competition as usual. While in 2017, the date of the competition was significantly earlier then previous years. These factors caused a temporary decline in participation. Students were as young as 8 years of age in their 3rd school year, and the oldest participating students were up to 18 years of age. The majority of the participating students was aged 14 and 15 (in school years 8 and 9). About 22% of the 1443 participating students were female (2012=43, 2013=18, 2014=46, 2015=32, 2016=106 and 2017=71). For an extra-curricular school activity this fraction can be considered to be relatively high, given that only 13% of all engineers are female [@engineersaustralia]. As the geographical representation of participating schools in Fig. \[fig:2\] shows the geographical distribution of schools participating in the competition increased as the initiative matured over time. Participating schools were from the Central Coast in the South up to Camden Haven in the North. In addition some schools travelled up from the Sydney Region to attend the Hunter tournament in 2014.
With respect to the disciplines offered at RoboCup Junior, the local Hunter region competition comprises all available disciplines that are currently accessible at both state and national tournaments. As a result, the discipline participation distribution follows the identical ranking in proportion with the difficulty of discipline. The ranking from easiest to hardest is as follows; rescue, dance, and soccer. The data from our largest local competition, in 2016, is shown in Fig. \[fig:3\].
![Distributions of registered participants for 2016, RoboCup Junior, Hunter Region: Top Left, a histogram representing the number of teams registered per discipline. Top Middle, a histogram representing the number of participants by age. Top Right, gender distribution for the competition. Bottom, a histogram representing teams registered by school.[]{data-label="fig:3"}](Figure3_Update_20180522_PSv2.png){width="\textwidth"}
RoboCup Junior rescue consists of three sub-disciplines; primary, secondary, and open. Simplest sub-category is primary rescue, predominantly for students who are currently in primary school, while secondary rescue is for participants who are attending high school, or are between 13 and 18 years old. Advanced participants who have participated in at least two tournaments of rescue in previous years compete in open rescue. This hierarchy of sub-categories allows not only for an increase in difficulty, but for different challenges for the different categories of ages. The rescue discipline is very structured, and the task is set by the rules of the competition. This structure was the motivating factor that drove its popularity, with 80 (14 primary, 68 secondary) teams registered to participate in this discipline in 2016.
Like rescue, RoboCup Junior dance also has three sub-categories; primary, secondary and open, with similar age restrictions. This category leaves many open-ended opportunities and flexibility that allow students to explore and experiment with different implementations for their performance. This openness makes it somewhat further challenging when compared to the rescue discipline. Hence, the number of participants in RoboCup Junior dance is smaller than in the rescue category. The number of teams participating in each sub-category is fairly distributed, with approximately 20 teams for primary dance, and 14 teams for secondary dance, with one team for open dance.
Lastly, in RoboCup Junior soccer, the students are required to build and program a small robot team to autonomously play soccer. It is essential for the robots to autonomously adapt to the environment while playing soccer by the rules. This discipline is the utmost challenging of the disciplines offered at the local Hunter regional tournament. However, it is also the most rewarding of all disciplines, as it allows the students to program an autonomous agent, with a requirement for multivariate control algorithm be used. The other disciplines can be, but are not necessarily, much simpler. Due to its perceived difficulty, participation in the soccer discipline has dwindled throughout the history of our local tournament. RoboCup Junior soccer, like RoboCup Junior rescue, consists of three sub-disciplines; GENII, lightweight, and open. The sub-category of GENII only allows Lego Mindstorm NXT or EV3 Robots to compete, whereas lightweight allows modified self-built robots to be used. This includes Arduino based hardware under a certain weight and size limit. The open sub-discipline allows any hardware to be used in the build of robots, which could be of any weight, but within a size limitation. For 2016, we saw the largest cohort, with 18 teams in soccer discipline total (9 GENII, 7 Lightweight, and 2 Open).
A CASE STUDY
------------
A report released in 2015 by the Australian Industry Group, explains a set of key recommendations that can be implemented in order to further encourage school students to study and explore future careers in STEM [@AIG2015]. A particular key recommendation highlights the need for teachers and schools to be further supported in harnessing students’ interest, which is the primary aim of RoboCup Junior and the number of community partnerships involved. Burgher et al. [@burgher2015implementation] suggest that a hands-on and practical approach to education, results in students becoming more aware of the conceptual theory rather than students being traditionally educated in the format of lectures and traditional classroom exercises. Exploring this suggestion this study has taken the shape of a questionnaire to current university students (n=3), who have competed in the tournament in the past. The purpose of investigating this qualitatively was to understand what key factors associated with their participation in RoboCup Junior led the students to develop a deeper appreciation of STEM and finally resulted in their decision to pursue a career in STEM. To begin, all participants had indicated in the questionnaire that they had been encouraged to participate in RoboCup Junior by their schools and teachers. This exemplifies how critical and significant partnerships formed between the organising committee and schools are. Three prevalent themes in the responses consisted of the following areas; Problem solving, Conceptual thinking and Rewarding.
#### Problem Solving:
Responses indicated that students had found the nature of the problems presented in RoboCup Junior to be of a “broad nature”, which provided further incentive to aid the design of a solution they were presented with during the competition. A critical skill that has been cited is that students had to assess the abstract problem independently which leads into the second theme.
#### Conceptual thinking:
In alignment with Burgher et al. [@burgher2015implementation] RoboCup Junior being a practical activity, allowed students to gain an additional direct approach to conceptual learning. Results from the questionnaire indicated that the critical skills gained were of a separate nature to a school curriculum. Students also indicated that open-ended problems allowed them to focus on concepts rather than on textbook knowledge. Students also suggested that as a result of it being separate from a school curriculum it allowed focus to bridge gaps between abstract and practical nature.
#### Rewards:
Within RoboCup Junior, students are encouraged to use technology to solve a given problem. The nature of responses to the questionnaires indicates that participants find the solutions to be the most rewarding and therefore one of the most encouraging aspects. As STEM is centrally focused around problem solving, students who experienced this during the competition found it further encouraging to seek employment within a career that offers that same sense of reward for solving a broad problem.
DISCUSSION: PARTNERSHIPS
========================
The future of STEM in the Hunter region is deemed important by many STEM-related stakeholders of the local region. Several factors that have contributed to the success of the tournament are associated with the partnerships that have been developed with key stakeholders in the period 2012-2017. Some of these key relationships and their impact will be discussed in the following sub-sections.
THE UNIVERSITY OF NEWCASTLE, FACULTY OF ENGINEERING AND BUILT ENVIRONMENT
-------------------------------------------------------------------------
The Faculty of Engineering and Built Environment of the University of Newcastle has continuously been the main stakeholder of RoboCup Junior in the Hunter Region since the project’s re-inception in 2012. With the university acting as the host, the competition is held on campus at the university’s gymnasium, The Forum. The university is also the key supplier of human resources to organise and manage the competition. The faculty has supported the competition with expertise in management, in the form of faculty administrative staff. The faculty manages aspects such as registrations, budget, and covered the majority of costs to run the competition.
Technical aspects of the tournament were administered by members affiliated with the NUbots, the Newcastle University RoboCup team. The NUbots are a senior RoboCup team and comprise several university students and academics [@NUbots2018]. They competed in the Kidsize Humanoid League, and now in 2018, the TeenSize Humanoid League at RoboCup. They are part of the Newcastle Robotics Laboratory, situated in the Faculty of Engineering and Built Environment. The NUbots have participated in RoboCup since 2002. They became world champions in the Standard Platform League using the Aldebaran NAO Robots, in 2008, and were world champions in the 4-Four-Legged League in 2006 using the Sony AIBO robots. This internationally well-recognised team brings over a decade of robotics experience to the local RoboCup Junior Hunter region competition. NUbot members are members of the committee, deliver workshops, and play a crucial role on the Hunter Region competition day in roles such as technical refereeing and judging. Over the past decade, the faculty has had a strong interest in community engagement. With a particular interest in low social-economical-status areas, the faculty has deployed intensive training programs and funding for robotic kits at schools in areas such as, Raymond Terrace in Port Stephens, NSW.
REGIONAL DEVELOPMENT AUSTRALIA (RDA) HUNTER – MANUFACTURING ENGINEERING (ME) PROGRAM {#sec:ME}
------------------------------------------------------------------------------------
RDA – Hunter’s ME Program has been a highly successful STEM outreach program that has delivered tangible outcomes in terms of student uptake of STEM-based subjects in upper secondary schools [@Sleap2014]. The ME Program in the Hunter region has supported running of the RoboCup competition whenever possible during 2012- 2014 and 2016. In addition to supporting RoboCup Junior, the ME Program actively supports all aspects of STEM in the Hunter and has produced an innovative school curriculum which integrates the silos of STEM into a Year 9 and 10 elective subject (iSTEM), which was endorsed by BOSTES NSW in 2012. In 2017, there were over 100 schools across NSW teaching iSTEM, which includes robotics programs in a standard curriculum, which also includes RoboCup Junior preparation. As a result of the broader ME Program funding for local schools, it has delivered a substantial quantity of STEM equipment and training (e.g. professional learning for teachers and through the support of Robogals for schools). The hardware provided includes 3D printers, and of course, robotic kits that could be used as part of the RoboCup Junior competition. The 2016 ME Program has included a caveat for any school receiving Lego EV3 robots that they must compete in the RoboCup Junior, Hunter region competition. During 2016, there was a significant increase in the number of schools that received robotic kits as part of the ME program. During 2015-2016, over one hundred EV3 robots were provided to 22 local schools. As a consequence, there was a significant increase in registered participants for the local tournament in 2016.
ROBOGALS NEWCASTLE INITIATIVE
-----------------------------
Robogals is an international initiative aimed at promoting gender equality in the fields of STEM through the use of robots and robotic education. Volunteers of the initiative consistently visit different schools and perform their robots at local public events. In addition, they offer free short beginner classes in robotics in using the Lego NXT and EV3 Robots in many school classes. The local chapter of Robogals in the Hunter region is no exception. It consists of many enthusiastic individuals, who are always ready to assist and share when required. The local chapter of Robogals initiative has worked closely with the RoboCup Junior Hunter Region Competition, since the inception of the local chapter in 2013. Robogals have recently signed a Memorandum of Understanding with the ME Program and BAE Systems Williamtown and have been working with ME Program high school and their feeder primary schools. The ME Program also provided 10 EV3 Robots to complement their fleet of NXT units. Volunteers of Robogals have sat on the organising committee for RoboCup Junior, Hunter region, and have also assisted at the events with judging, and holding workshops on the competitions behalf while the competition was running. In addition, training material used to teach classes was shared between RoboCup Junior and Robogals, so that Robogals could concentrate on their goal to achieve gender equality in the fields of STEM. The Robogals initiative in the Hunter region is growing successfully. They have repeatedly reported that there are more schools on their waiting list then they can handle. The result of this partnership, and its growing success, can be seen in the increased female to male ratios (approximately 24% in Figure \[fig:3\]). It shows the number of females is relatively higher at RoboCup Junior when compared to the number of females enrolled in an engineering course at a later stage, e.g., at university level.
TRIBOTIX
--------
Tribotix is a local robotics company in the Hunter region that sells and builds various robots, mostly for educational purposes. It has a strong interest in the success of RoboCup Junior. The director of Tribotix has personally been involved with the RoboCup Junior since its inception and has also mentored teams in local schools, using a different style of robots than the standard Lego platform. Tribotix also partners with the national RoboCup Junior Australia committee in the development of state-of-the-art robotic educational kits. This includes, e.g., the DARwIn-MINI, for future use in a possible new Rescue league and a small humanoid league for RoboCup Junior. As more students start earlier with the competition, it would not be too long before these advancing students seek knowledge, information, and new hardware to fulfil their requirements. Tribotix has been a competent partner and helpful supplier throughout all years of the competition.
COMMUNITY SPONSORSHIP AND MEMBERSHIP
------------------------------------
Community support was vital for running the event. Support was supplied in terms of funding obtained from community grants, such as Orica (2014), AGL (2015), Newcastle Coal Infrastructure Group (NCIG) (2015), Newcastle City Council (2016), and the Kirby Foundation (2017). Without this funding, the competition itself could not have happened, and therefore no success could have been achieved. Members of the general community represented by teachers and parents of the participants were involved in all aspects of organising the competition. The success of the students comes directly from interacting with their mentors, some of which advise and attend monthly organising meetings which allow us to hear feedback and to incorporate and implement suggestions to make the competition run smoothly. Members of the community are an important part of the Hunter Region RoboCup Junior organising committee and have steered the competition to its current successful state. We acknowledge Mr. Jason Flood, Chair of Local Committee (all years, excluding 2014), for his commitment and extraordinary effort that added to the project’s success.
CONCLUSION
==========
With decreasing levels of participation in mathematics and science within Australian schools, winning students’ interest in the fields of STEM has become an uphill battle. Nonetheless, for the local region of the Hunter, the RoboCup Junior competition gained outstanding success. The partnership of RoboCup Junior Hunter Region with the University of Newcastle, and other key stakeholders such as the RDA Hunter’s ME program, stands as a project that will transform the landscape for STEM education in the future. Success of the project to this point is reflected by the increasing number of student participants, a growing geographical distribution, and an improvement of gender balance. In addition, qualitative evidence of the positive influence on students participating in RoboCup Junior explains what impact the competition can play on students’ path to a STEM career.
|
---
abstract: 'We show that the problem of directed percolation on an arbitrary lattice is equivalent to the problem of $m$ directed random walkers with rather general attractive interactions, when suitably continued to $m=0$. In 1+1 dimensions, this is dual to a model of interacting steps on a vicinal surface. A similar correspondence with interacting self-avoiding walks is constructed for isotropic percolation.'
address: |
$^1$University of Oxford, Department of Physics – Theoretical Physics, 1 Keble Road, Oxford OX1 3NP, U.K.\
$^2$All Souls College, Oxford.
author:
- 'John Cardy$^{1,2}$ and Francesca Colaiori$^{1}$'
title: Directed Percolation and Generalized Friendly Walkers
---
= 10000
[2]{} The problem of directed percolation (DP), first introduced by Broadbent and Hammersley [@Broadbent], continues to attract interest even though it has so far defied all attempts at an exact solution, even in two dimensions. Although the problem was originally formulated statically on a lattice with a preferred direction, when the latter is interpreted as time the universal behavior close to the percolation threshold is also believed to describe the transition from a noiseless absorbing state to a noisy, active one, which occurs in a wide class of stochastic processes [@Dickman]. It also maps onto reggeon field theory, which describes high-energy diffraction scattering in particle physics [@CS].
Some time ago, Arrowsmith, Mason and Essam [@AME] argued that the pair connectedness probability $G(r,r')$ for directed bond percolation on a two-dimensional diagonal square lattice can be related to the partition function for the weighted paths of $m$ ‘friendly’ walkers which all begin at $r$ and end at $r'$, when suitably continued to $m=0$. These are directed random walks which may share bonds of the lattice but do not cross each other (see Fig. \[fig1\]). In fact, Arrowsmith et al [@AME] represented these configurations in other ways: either as *vicious *walkers, which never intersect, by moving the friendly walkers each one lattice spacing apart horizontally; or as *integer flows *on the directed lattice, to be defined explicitly below. Arrowsmith and Essam [@AE] showed that $G(r,r')$ is also related to a partition function for a $\lambda$-state chiral Potts model on the dual lattice, on setting $\lambda=1$, thus generalizing the well-known result of Fortuin and Kasteleyn[@KF] for ordinary percolation.****
In a more recent paper, Tsuchiya and Katori [@TK] have considered instead the order parameter of the DP problem, and have shown that in d=2 it is related to a certain partition function of the same $\lambda=1$ chiral Potts model, and also that, for arbitrary $\lambda$, the latter is equivalent to a partition function for $m=(\lambda-1)/2$ friendly walkers.
It is the purpose of this Letter to describe a broad generalization of these results. We demonstrate, in particular, a direct connection between a general connectedness function of DP and a corresponding partition function for $m$ friendly walkers when continued to $m=0$. We show that this holds on an arbitrary directed lattice in any number of dimensions, and for all variant models of DP, whether bond, site or correlated. Moreover the weights for a given number of walkers passing along a given bond or through a given site may be chosen in a remarkably arbitrary fashion, still yielding the same result at $m=0$.
We now describe the correspondence between these two problems in detail. A *directed lattice *is composed of a set of points in ${\bf R}^d$ with a privileged coordinate $t$, which we may think of as time. Pairs of these sites $(r_i,r_j)$ are connected by fixed bonds, oriented in the direction of increasing $t$, to form a directed lattice. In the directed *bond *problem, each bond is open with a probability $p$ and closed with a probability $1-p$, and in the *site *problem it is the sites which have this property. In principle the probabilities $p$ could be inhomogeneous, and we could also consider site-bond percolation and situations in which different bonds and sites are correlated. Our general result applies to all these cases, but for clarity we shall restrict the argument to independent homogeneous directed bond percolation. The pair connectedness $G(r,r')$ is the probability that the points $r$ and $r'$ (with $t<t'$) are connected by a continuous path of bonds, always following the direction of increasing $t$.******
On the same lattice, let us define the corresponding *integer flow *problem. Assign a non-negative integer-valued current $n(r_i,r_j)$ to each bond, in such a way that it always flows in the direction of increasing $t$, and is conserved at the vertices. At the point $r$ there is a source of strength $m\geq1$, and at $r'$ a sink of the same strength. There is no flow at times earlier than that of $r$ or later than that of $r'$. Such a configuration may be thought of as representing the worldlines of $m$ particles, or walkers, where more than one walker may share the same bond. The configurations are labeled by distinct allowed values of the $n(r_i,r_j)$, so that they are counted in the same way as are those of identical bosons. Alternatively, in $1+1$ dimensions, we may regard the walkers as distinct but with worldlines which are not allowed to cross. In the partition sum, each bond is counted with a weight $p(n(r_i,r_j))$. In the simplest case we take $p(0)=1$ and $p(n)=p$ for $n\geq1$ (although we shall show later that this may be generalized). Since $p>p^n$ for $n>1$, there is an effective attraction between the walkers, leading to the description ‘friendly’. The partition function is then $$Z(r;r';m)\equiv\sum_{\rm allowed\ configs}\,\prod_{(r_i,r_j)}p(n(r_i,r_j))$$ This expression is a polynomial in $m$ and so may be evaluated at $m=0$. The statement of the correspondence between DP and the integer flow problem for the case of the pair connectedness is then $$G(r;r')=Z(r;r';0).$$ Note that since the weights $p(n)$ behave non-uniformly as $n\to0$, the continuation of $Z(r;r';m)$ to $m=0$ is not simply the result of taking zero walkers (which would be $Z=1$): rather it is the non-trivial answer $G$. Similar results hold for more generalized connectivities. For example, if we have points $(r'_1,r'_2,\ldots,r'_l)$ all at the same time $t'>t$, we may consider the probability $G(r;r'_1,r'_2,\ldots,r'_l)$ that all these points, irrespective of any others, are connected to $r$. The corresponding integer flow problem has a source of strength $m\geq l$ at $r$, and sinks of arbitrary (but non-zero) strength at each point $r'_j$. In this case $$G(r;r'_1,r'_2,\ldots,r'_l)=(-1)^{l-1}Z(r;r'_1,r'_2,\ldots,r'_l;m=0)$$ where the partition function is defined with the same weights as before. Since the order parameter for DP may be defined as the limit as $t'-t\to\infty$ of $P(t'-t)$, the probability that *any *site at time $t'$ is connected to $r$, and this may be written using an inclusion-exclusion argument as $$P(t'-t)=\sum_{r'}G(r;r')-\sum_{r'_1,r'_2}G(r;r'_1,r'_2)+\cdots$$ (where the sums over the $r'_j$ are all restricted to the fixed time $t'$), we see that it is in fact given by the $m=0$ evaluation of the partition function for *all *configurations of $m$ walkers which begin at $r$ and end at time $t'$. This generalizes the result of Tsuchiya and Katori [@TK] to an arbitrary lattice. Although this continuation to $m=0$ is reminiscent of the replica trick, it is in fact quite different. Moreover it is mathematically well-defined, since, as we argue below, $Z$ is a finite sum of terms, each of which, with the simple weights given above, is a polynomial in $m$.******
We now give a summary of the proof, which is elementary. The connectedness function $G(r;r'_1,r'_2,\ldots,r'_l)$ is given [@Essam] by the weighted sum of all graphs $\cal G$ which have the property that each vertex may be connected backwards to $r$ and forwards to at least one of the $r'_j$. (Alternatively, $\cal G$ is a union of directed paths from $r$ to one of the $r'_j$.) Each such graph is weighted by a factor $p$ for each bond and $(-1)$ for each closed loop. A simple example is shown in Fig. \[fig2\]. A given graph corresponds to summing over all configurations in which the bonds in $\cal G$ are open, irrespective of all other bonds in the lattice. The factors of $(-1)$ are needed to eliminate double-counting. It is useful to decompose vertices in $\cal G$ with coordination number $>3$ by inserting permanently open bonds into them in such a way that the only vertices are those in which two directed bonds merge to form one ($2\to1$), and vice versa. This does not affect the connectedness properties. We may then associate the factors of $(-1)$ with each $1\to2$ vertex in $\cal
G$, as long as we incorporate an overall factor $(-1)^{l-1}$ in $G$. With each graph $\cal G$ we associate a restricted set of integer flows, called proper flows, such that $n\geq1$ for each bond in $\cal G$, and $n=0$ on each bond not in $\cal G$. Those corresponding to the graphs in Fig. \[fig2\] are shown in Fig. \[fig3\]. Note that the last graph corresponds to $m-1$ configurations of integer flows, which gives precisely the required factor of $(-1)$ when we set $m=0$. In general, summing over all allowed integer flows will generate the sum over all allowed $\cal G$, with correct weights $p$: the non-trivial part is to show that we recover the correct factors of $(-1)$ when we set $m=0$.
This follows from the following simple lemma: if $A(n)$ is a polynomial in $n$, and we define the polynomial $B(m)\equiv\sum_{n=1}^{m-1}A(n)$, then $B(0)=-A(0)$. We give a proof which shows that the result may be generalized to other functions: write $A(n)$ as a Laplace transform $A(n)=\int_C(ds/2\pi i)e^{ns}\tilde A(s)$. Then $B(m)=\int_C(ds/2\pi i)\big((e^s-e^{ms})/(1-e^s)\big)\tilde A(s)$, so that $B(0)=-\int_C(ds/2\pi i)\tilde A(s)=-A(0)$. An immediate corollary is that if $A(n_1,n_2,\ldots)$ is a polynomial in several variables, and $B(m)\equiv\sum_{n=1}^{m-1}A(n,m-n,\ldots)$, then $B(0)=-A(0,0,\ldots)$. We use this to proceed by induction on the number of $1\to2$ vertices in $\cal G$. Beginning with the vertex which occurs at the earliest time, the contribution to $Z$ from the proper flow on $\cal G$, when evaluated at $m=0$, is, apart from a factor $(-1)$, equal to that for another graph $\cal G'$ which will have one fewer $1\to2$ vertex. However, $\cal G'$ differs from the previously allowed set of graphs $\cal G$ in that it may have more than one vertex at which current may flow into the graph. For this reason we extend the definition of the allowed set of graphs to include those in which every vertex is connected to at least one ‘input’ point $(r_1,r_2,\ldots)$ and at least one ‘output’ point $(r'_1,r'_2,\ldots)$. In the corresponding integer flow problem, currents $(m_1,m_2,\ldots)$ flow in at the inputs, whereas the only restriction on the outputs is that non-zero current should flow out. The partition function is then the weighted sum over all such allowed integer flows. Induction on the number of $1\to2$ vertices then shows that this partition function, evaluated at $m_j=0$, gives the corresponding DP graph correctly weighted. (The induction starts from graphs with no $1\to2$ vertices which involve no summations and for which the result is trivial.)
Since our main result relies only on the lemma it follows also for rather general weights $p(n)$. The only requirement is that $p(n)$ grow no faster than an exponential at large $n$, and that, when continued to $n=0$, it give the value $p\not=1$. In this case, $Z$ will no longer be a polynomial in $m$, but, since by the inductive argument above it is given by a sum of convolutions of $p(n)$, its continuation to $m=0$ will be well-defined through its Laplace transform representation. For example, we could take $p(n)=p^{1-n}$ for $n\geq1$. This raises the possibility of choosing some suitable set of weights for which the integer flow problem, at least in $1+1$ dimensions, is integrable, for example by Bethe ansatz methods. Unfortunately our results in this direction are, so far, negative. In the case of bond percolation on a diagonal square lattice let $Z(x_1,x_2, \dots ,x_m;t)$ be the partition function under the constraint that the walkers arrive at $\{x_1,x_2,\dots ,x_m\}$ at time $t$, the physical region being $\{ x_1 \leq x_2 \leq \dots \leq x_m \}$. Turning the master equation for $Z$ in an eigenvalue problem [@note] and writing the eigenfunction $\psi_m(x_1,x_2,\dots x_m)$ in the usual Bethe ansatz form, one gets for $\psi_2(x_1,x_2)\!=\!A_{12}e^{i(x_1k_1+x_2k_2)}+
A_{21} e^{i(x_1k_2+x_2k_1)}$ the following condition on the amplitudes: $$\frac{A_{21}}{A_{12}}=
-\frac{e^{\,\,i(k_1-k_2)}-\epsilon
\left(e^{i(k_1+k_2)}+e^{-i(k_1+k_2)}\right)}
{e^{\!-i(k_1-k_2)}-\epsilon
\left(e^{i(k_1+k_2)}+e^{-i(k_1+k_2)}\right)}$$ (the same as that which appears in the XXZ spin chain [@Yang]) where $\epsilon=p(2)/p(1)^2-1$. Requiring that the $m$-particle scattering should factorise into a product of these two-body $S$-matrices places constraints on the weights $p(n)$. In general these equations appear too difficult to solve, except in the weak interaction limit ($\epsilon\simeq 1/2$), where we find $$2^n=2q(n)+\sum_{s=1}^{n-1} q(n-s) q(s) (1-\lambda s(n-s))+O(\lambda^2)$$ where $q(s)=p(s)/(p(1))^s$, $\lambda=2\epsilon-1$. This may be solved for successive $q(n)$, but it is easy to see by applying the above lemma that, when continued to $n=0$, it will always yield the value $1$, rather than $p$ as required. We conclude that the $m=0$ continuation of this integrable case does not correspond to DP. It is nevertheless interesting that integrable models of such interacting walkers can be formulated.
In 1+1 dimensions, our generalized friendly walker model maps naturally onto a model of a step of total height $m$ on a vicinal surface, by assigning integer height variables $h(R)$ to the sites $R$ of the dual lattice, such that $h=0$ for $x\to-\infty$, $h=m$ for $x\to+\infty$, and $h$ increases by unity every time the path of a walker is crossed. The weights for neighboring dual sites $R$ and $R'$ are $p(h(R')-h(R))$. This is slightly different from, and simpler than, the chiral Potts model studied in [@AE; @TK].
A similar correspondence between percolation and interacting random walks is valid also for the isotropic case. The pair connectedness $G(r,r')$ may be represented by a sum of graphs $\cal G$, just as in DP [@Essam]. Each graph consists of a union of oriented paths from $r$ to $r'$, As before, each bond is counted with weight $p$ and each loop carries a factor $(-1)$. Note that graphs which contain a closed loop of oriented bonds are excluded. Such contributions cannot occur in DP because of the time-ordering. The correspondence with integer flows or friendly walkers follows as before. The latter picture is particularly simple. $m$ walkers begin at $r$ and end at $r'$. When two or more walkers occupy the same bond, they must flow parallel to each other. Since they cannot form closed loops, they are *self-avoiding*. Moreover, walkers other than those which begin and end at $r$ and $r'$, which could also form closed loops, are not allowed. Each occupied bond has weight $p(n)$ as before, and the separate configurations are counted using Bose statistics. $G(r,r')$ is then given by the continuation to $m=0$ of the partition function. We conclude that ordinary percolation is equivalent to the continuation to $m=0$ of a problem of $m$ oriented self-avoiding walks, with infinite repulsive interactions between anti-parallel segments on the same bond, but attractive parallel interactions. In two dimensions, this is again dual to an interesting height model, in which neighboring heights satisfy $|h(R')-h(R)|\leq m$, but local maxima or minima of $h(R)$ are excluded. For example, the order parameter of percolation is given by the continuation to $m=0$ of the partition function for a screw dislocation of strength $m$ in this model.**
To summarize, we have shown that the DP problem is simply related to the integer flow problem, or equivalently that of $m$ bosonic ‘friendly’ walkers, when suitably continued to $m=0$. This holds on an arbitrary directed lattice in any number of dimensions, and with rather general weights. It is to be hoped that this correspondence might provide a new avenue of attack on the unsolved problem of directed percolation.
The authors acknowledge useful discussions with F. Essler and A. J. Guttmann, and thank T. Tsuchiya and M. Katori for sending a copy of their paper prior to publication. This research was supported in part by the Engineering and Physical Sciences Research Council under Grant GR/J78327.
S. R. Broadbent and J. M. Hammersley, Camb. Philos. [**53**]{}, 629 (1957). See, for example, R. Dickman, Int. J. Mod. Phys. C [**4**]{}, 271 (1993). J. L. Cardy and R. L. Sugar, J. Phys. A [**13**]{}, L423 (1980). D. K. Arrowsmith, P. Mason and J. W. Essam, Physica A [**177**]{}, 267 (1991). The proof given in this paper is incomplete, since it relies on the (false) assertion that the minimal integer flow is unique for every graph $\cal
G$. D. K. Arrowsmith and J. W. Essam, Phys. Rev. Lett. [**25**]{}, 3068 (1990); Jnl. Comb. Thy. B [**62**]{}, 349 (1994). C. M. Fortuin and P. W. Kasteleyn, Physics [**57**]{}, 536 (1972). T. Tsuchiya and M. Katori, J. Phys. Soc. Japan [**67**]{}, 1655 (1998). J. W. Essam, Rep. Prog. Phys. [**43**]{}, 833 (1980). We note that is necessary to make further assumptions in concluding that the *eigenvalues *of the DP transfer matrix correspond to the continuation to $m=0$ of those of the $m$-walkers problem, since the eigenvalues do not necessarily have the same analytic properties as the partition functions evaluated at fixed $t$. C. N. Yang and C. P. Yang, Phys. Rev. [**150**]{}, 321 (1966).**
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abstract: 'Using a three neutrino framework we investigate bounds for the effective Majorana neutrino mass matrix. The mass measured in neutrinoless double beta decay is its (11) element. Lepton–number and –flavor violating processes sensitive to each element are considered and limits on branching ratios or cross sections are given. Those processes include $\mu^- e^+$ conversion, $K^+ \to \pi^- \mu^+ \mu^+$ or recently proposed high–energy scattering processes at HERA. Including all possible mass schemes, the three solar solutions and other allowed possibilities, there is a total of 80 mass matrices. The obtained indirect limits are up to 14 orders of magnitude more stringent than direct ones. It is investigated how neutrinoless double beta decay may judge between different mass and mixing schemes as well as solar solutions. Prospects of detecting processes depending on elements of the mass matrix are also discussed.'
author:
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Werner Rodejohann[^1]\
[*Lehrstuhl für Theoretische Physik III,*]{}\
[*Universität Dortmund, Otto–Hahn Str. 4,*]{}\
[*44221 Dortmund, Germany*]{}
title: |
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\
1.5cm **Neutrino oscillation experiments and limits on lepton–number and lepton–flavor violating processes**
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[Keywords: Oscillation, Lepton number violation, Double beta decay]{}
[^1]: Email address: [email protected]
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abstract: |
We report on the AAT-AAOmega LRG Pilot observing run to establish the feasibility of a large spectroscopic survey using the new AAOmega instrument. We have selected Luminous Red Galaxies (LRGs) using single epoch SDSS $riz$-photometry to $i<20.5$ and $z<20.2$. We have observed in 3 fields including the COSMOS field and the COMBO-17 S11 field, obtaining a sample of $\sim$600 redshift $z\gtrsim0.5$ LRGs. Exposure times varied from 1 - 4 hours to determine the minimum exposure for AAOmega to make an essentially complete LRG redshift survey in average conditions. We show that LRG redshifts to $i<20.5$ can measured in $\approx$1.5hr exposures and present comparisons with 2SLAQ and COMBO-17 (photo-)redshifts. Crucially, the $riz$ selection coupled with the 3-4$\times$ improved AAOmega throughput is shown to extend the LRG mean redshift from $z$=0.55 for 2SLAQ to $z=0.681\pm
0.005$ for $riz$-selected LRGs. This extended range is vital for maximising the S/N for the detection of the baryon acoustic oscillations (BAOs). Furthermore, we show that the amplitude of LRG clustering is $s_0 = 9.9\pm0.7\hmpc$, as high as that seen in the 2SLAQ LRG Survey. Consistent results for this clustering amplitude are found from the projected and semi-projected correlation functions. This high amplitude is consistent with a long-lived population whose bias evolves as predicted by a simple “high-peaks” model. We conclude that a redshift survey of LRGs over 3000 deg$^{2}$, with an effective volume some $4\times$ bigger than previously used to detect BAO with LRGs, is possible with AAOmega in 170 nights.
author:
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Nicholas P. Ross[^1]$^{1,2}$, T. Shanks$^1$, Russell D. Cannon$^3$, D.A. Wake$^1$, R. G. Sharp$^{3}$, S. M. Croom$^{4}$ and John A. Peacock$^{5}$\
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$^1$Physics Department, Durham University, South Road, Durham, DH1 3LE, UK\
$^2$Department of Astronomy and Astrophysics, The Pennsylvania State University, 525 Davey Laboratory, University Park, PA 16802, U.S.A.\
$^3$Anglo-Australian Observatory, PO Box 296, Epping, NSW 1710, Australia\
$^4$School of Physics, University of Sydney, Sydney, NSW 2006, Australia\
$^5$SUPA, Institute for Astronomy, University of Edinburgh, Royal Observatory, Blackford Hill, Edinburgh EH9 3HJ
title: 'Luminous Red Galaxy Clustering at $z\simeq0.7$ - First Results using AAOmega'
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galaxies - luminous red, surveys: clustering - large-scale structure: evolution - clustering.
Introduction
============
Large-scale structure (LSS) studies are one road into investigating “Dark Energy” (DE) and its potential evolution [e.g. @BlakeGlazebrook03; @Seo03; @Seo05; @Seo07; @Angulo08]. This has been powerfully demonstrated by recent results from the Luminous Red Galaxy (LRG) Sloan Digital Sky Survey (SDSS), [e.g. @Eisenstein05; @Tegmark06; @Percival07a; @Percival07b] and indeed the 2dFGRS [@Cole05]. Luminous Red Galaxies (LRGs) are predominantly massive early-type galaxies and are intrinsically luminous ($\gtrsim 3L^{*}$) [@Eisenstein03; @Loh06; @Wake06]. They are strongly biased objects, having values of $b\sim2$, [@Padmanabhan07] where $b$ is the linear bias and relates, in the linear regime, the underlying mass density distribution to that of the luminous tracers via $\delta_{g} = b \, \delta_{m}$. As such and coupled to their very clean and efficient selection, LRGs are excellent tracers of large-scale structure and can be used as cosmological probes. @Eisenstein05, @Tegmark06, @Hutsi06, @Percival07a and @Percival07b use positions and spectroscopic redshifts from the SDSS LRG Survey in order to accurately measure the correlation function and the Power Spectrum. Specifically, a detection of the baryon acoustic oscillations (BAOs) in the galaxy distribution is made. BAOs in the galaxy distribution are caused by sound waves propagating through the baryon-photon plasma in the early ($z > 1100$) Universe. At recombination, these sound waves are “frozen” into the distribution of matter at a preferred scale [see e.g. @Eisenstein98; @Meiksin99; @Yamamoto06; @Eisenstein07 for further BAO details]. With measurements of the BAOs now starting to appear feasible, there is a push to carry out large galaxy surveys at higher redshift, with the primary goal of tracking the evolution of dark energy and the related equation of state parameter, $w_{\rm DE}(z)$, over cosmic time. As such, several new galaxy redshift surveys have been proposed.
Field Name R.A. (J2000) Dec (J2000) No. of exposures
------------------------------ --------------- ------------------ ------------------ ----- ----- ----- ----- ----- -- ------ ------ ------ ------ ------
COSMOS 10h 00m 28.6s 02d 12m 21.0s 0+7+0+6+0 – 2.0 – 3.0 – – 1.39 – 1.27 –
COMBO-17 S11 11h 42m 58.0s $-$01d 42m 50.0s 2+6+4+0+9 2.0 1.8 1.7 – 1.9 1.15 1.19 1.21 – 1.19
2SLAQ d05 13h 21m 36.0s $-$00d 12m 35.0s 8+0+0+5+0 1.9 – – 1.6 – 1.22 – – 1.19 –
\[tab:The\_AAOmega\_fields\]
One possibility is to use the AAOmega spectrograph at the AAT to make a spectroscopic redshift survey of high redshift LRGs based on both SDSS Equatorial imaging, as well as new imaging from the 2.6m VLT Survey Telescope (VST). AAOmega retains the fibre-fed multi-object capability across a wide field-of-view from the old 2dF instrument but the top-end spectrographs have been replaced with a new single bench mounted spectrograph, with a red and a blue arm. @Sharp06 gives complete instrument details. In this paper we present the results from an AAOmega LRG redshift survey. Although the primary driver for this survey is as a “Pilot” study to investigate the nature of dark energy at high redshift via the BAOs, there are also several other areas of interest. By comparing clustering results at $1 < r < 10 \hmpc$ scales from low ($z<0.4$), intermediate ($z=0.55$), and high ($z\sim0.7$), redshift LRG studies [@Zehavi05a; @Ross07 and this study respectively] we can begin to learn about the formation and evolution of the most massive galaxies, and hence, potentially the most massive dark matter haloes, from high redshift.
The layout of the paper is as follows. In Section 2 we describe the selection criteria used to select our high redshift LRGs. In Section 3 we give a brief overview of the instrument set-up used and report on the redshift statistics for our survey, including example spectra. In Section 4 we present our clustering results and in Section 5 we discuss our results in the context of other recent results using a simple Halo Occupation Distribution (HOD) model. We conclude in Section 6. We assume a flat $\Lambda$CDM cosmology, with ($\Omm,
\Omlam$)=(0.3,0.7) throughout, unless otherwise explicitly stated. We quote distances in terms of $\hmpc$, where $h$ is the dimensionless Hubble constant such that $\ho=100h\kmsmpc$.
SDSS LRG Selection
==================
At its heart the AAOmega LRG Pilot relies on single-epoch photometric data from the SDSS [@York00; @Gunn06] to provide targets for the recently commissioned AAOmega instrument on the 3.9m Anglo-Australian Telescope (AAT).
The target selection was designed to select high-redshift LRGs out to $z\simeq1$ with a mean redshift of $z\simeq0.7$. Using the SDSS Data Release 4 [DR4; @Adelman-McCarthy06], we extracted photometric data for objects classified as galaxies. Three different selections were then applied to the downloaded data, with the selections being designed to recover a target sky density of $\sim90$ objects per square degree.
First, we repeat the $gri$-band based selection that was used in the 2SLAQ LRG Survey. We will not repeat the full selection criteria here (the reader is referred to @Cannon06 for further details) but note that LRGs are selected in the $(g-r)$-$(r-i)$ colour-colour plane with $17.5 < i_{\rm deV} < 19.8$, where $i_{\rm deV}$ is the $i$-band de Vaucouleurs magnitude.
Now with the aim of measuring significantly higher redshifts than the 2SLAQ LRG Survey ($\bar{z}_{\rm 2SLAQ}=0.55$), two further selections were carried out, this time in the $(r-i)$-$(i-z)$ colour-colour plane. The first $riz$-selection had objects in the magnitude range $19.8 < i_{\rm deV} < 20.5$, while the second $riz$-selection had objects in the magnitude range $19.5 < z < 20.2$, where $z$ is the SDSS “Model” magnitude [@Fukugita96; @Stoughton02]. These magnitude ranges were based on experience gained from the 2SLAQ LRG Survey as well as the expected performance of the new AAOmega instrument, such that LRGs with a significantly higher redshift than the previous survey could be selected and observed in a relatively short exposure ($\sim1.5$ hours). Within these two $riz$-band selections, objects were assigned different observational priorities. The line “$e_{\parallel}$”was defined (continuing on from, but not directly related to $c_{\parallel}$ in @Eisenstein01 and $d_{\parallel}$ in [@Cannon06]), as $$e_{\parallel} = (i-z) + \frac{9}{7}(r-i) \ge 2.0.
\label{eqn:epara}$$ and is used to define a boundary in the $riz$-plane. (All colours reported here, such as those given in Equation \[eqn:epara\], are again based on “Model” magnitudes). A higher priority $riz$-plane cut was imposed with $$0.5 \le (r - i) \le 1.8,$$ $$0.6 \le (i - z) \le 1.5,$$ $$e_{\parallel} \ge 2.0.$$ The lower priority cut has $$0.2 \le (i - z) \le 0.6,$$ $$x \le (r - i) \le 1.8,$$ where $x$ was the smaller of $e_{\parallel}$ and 1.2 at the given $(i-z)$. These cuts can be seen in Figure \[fig:riz\_plane\] where the two priorities are shown by the regions marked A and B. The two evolutionary tracks in Figure \[fig:riz\_plane\] the stellar population synthesis code based on @BC03. The solid line being a “single burst” model, where star formation occurs in a single instantaneous burst at high redshift and then has the stellar population evolving passively. The dashed line on the other hand is based on a model with continuous star formation, with the timescale of star formation given as $\tau$ = 1 Gyr, where $\tau$ is a decay constant in that the star formation rate (SFR) is $\propto
\exp^{-t/\tau}$. Both models assume a Salpeter IMF [@Salpeter55] with solar metallicity and a galaxy formation redshift of $z_{\rm form}
= 10$. The evolutionary tracks start near $(r-i) = (i-z) = 0.4$ for zero redshift, turn upwards near $(r-i) = 1.3$ corresponding to redshift $z=0.7$ and then turn down again near $(i-z) \sim 1.1$ corresponding to redshift $z =1.0$. These turning points correspond to the CaII H+K 4000Å break moving into the $i$- and $z$-bands respectively. The solid circles show the colour evolution at redshift $z=$0.0, 0.5, 1.0 and 1.5.
![The selection of $z \sim 0.7$ LRGs using the SDSS $riz$-bands. The (red) dots are objects with confirmed spectroscopic redshifts for both the $19.8 < i_{\rm deV} < 20.5$ and $19.5 < z < 20.2$ magnitude selections. The tracks are Bruzual & Charlot models, details given in the text with the solid (cyan) line being a “single burst” model and the dashed (magenta) line having being a $\tau$=1 Gyr model. The diagonal lines are $e_{\parallel}=2.0$. The area labelled “A” in the top right redshift $z<0.5$ panel gives the colour-colour space for the higher priority sample, while area “B” is for the lower priority sample. []{data-label="fig:riz_plane"}](ps/riz_AAOmega_6redshift_panels_igt19pnt8_PlanA_tracks_v3pnt0.ps){height="12.0cm" width="8.0cm"}
---------------------------- ------- ---------- ----- -- ------- ---------------- ---------- ----- -- ------- ---------- ----- ---------------
Field [**Survey**]{}
Selection $gri$ $i<20.5$ all $gri$ $i<20.5$ $z<20.2$ all $gri$ $i<20.5$ all [**total**]{}
Spectra Obtained 98 223 321 70 262 271 603 68 278 346 [**1270**]{}
$Q{\rm op} \geq 3$ 71 129 200 61 163 143 367 57 180 237 [**804**]{}
LRGs 67 89 156 55 119 80 254 50 127 177 [**587**]{}
\[tab:Target\_Statistics\]
---------------------------- ------- ---------- ----- -- ------- ---------------- ---------- ----- -- ------- ---------- ----- ---------------
LRG Sample/ Field (Seeing) d05 ($1.''6$) S11 ($1.''8$) COSMOS ($2.''1$)
---------------------------- --------------- --------------- ------------------
$gri$ $i<19.8$ (2SLAQ) $88\pm19$ $70\pm22$ $64\pm24$
$riz$ $19.8<i<20.5$ $84\pm13$ $60\pm11$ $50\pm9$
AAOmega Spectroscopy
====================
Observational Details
---------------------
Observations were made on the nights of 03 March 2006 to 07 March 2006 inclusive; the first three nights were Dark nights, the last two were Grey nights. Of these nights, a total of $\simeq 2$ were lost to cloud and seeing was frequently poor on the others (see Table \[tab:The\_AAOmega\_fields\]). We observed in 3 fields, with a total area of $\simeq10$ deg$^{2}$, including the COSMOS field [@Scoville07], the COMBO-17 S11 field [@Wolf03] and a previously observed 2SLAQ Survey field, d05 [@Cannon06], the coordinates of which are also given in Table \[tab:The\_AAOmega\_fields\]. For reference, the COSMOS Survey has an area of 2 deg$^{2}$, the COMBO-17 S11 field is 0.26 deg$^{2}$ in coverage, while the 2SLAQ LRG Survey has an effective area of 135 deg$^{2}$ [Sec. 7.2, @Cannon06].
All data were taken with the same spectrograph set-up. The 5700Å dichroic was used. For the red arm spectrograph the 385R grating was centred at 7625Å; for the blue arm spectrograph the 580V grating was centred at 4800Å. However, no blue arm data was used in our analysis as the S/N was low, as expected for red galaxies.
Data reduction was performed using the 2dF data reduction pipeline software, [2dfdr]{} [@Bailey05] and the redshifts were derived using [Zcode]{} developed by Will Sutherland and others for the 2dFGRS Survey [@Colless01 and references therein]. The modifications to [Zcode]{} originally made for the higher redshift $z\sim0.5$ galaxies in the 2SLAQ LRG Survey were retained. The final catalogue from the AAOmega LRG Pilot contains 1270 unique galaxy spectra with 804 objects having reliable “$Q{\rm op}
\geq 3$”[^2] redshifts, see Table \[tab:Target\_Statistics\]. Of these, 217 objects had M-type stellar spectra leaving 587 high-redshift LRGs. The COSMOS field contributed 156 LRGs out of 321 obtained spectra, the 2SLAQ d05 field 177/345 and the S11 field 254/604. The greater number of spectra obtained in S11 was due to the fact that objects in the field were targeted not only with the $19.8 < i < 20.5$ selection but also with the $19.5 < z <20.2$ $z$-band selection.
We present the catalogue for the first 40 objects in ascending RA in Appendix A, with the entire catalogue to be published online with the publication of this paper. In the next Section we report in more detail on the properties of the high-redshift LRGs.
Redshift Completeness
---------------------
![Examples of typical AAOmega spectra in 1.67hr exposures, from the $riz$ selected, $19.8 < i < 20.5$ LRG sample. The top six panels show spetra of confirmed, $Q{\rm op} \geq 3$ LRGs, with ranging magnitudes and redshifts. The second bottom panel shows an unconfirmed, $Q{\rm op} < 3$, spectrum, while the bottom spectrum is for a confirmed stellar source. []{data-label="fig:example_spectra_8"}](ps/d05_060306_red_all_spec_paper2.ps){height="12.0cm" width="8.5cm"}
The LRG redshift completeness statistics for each field can be calculated from Table \[tab:Target\_Statistics\] for the full, $\approx$ 4 hour, exposures and are given in Table \[tab:AAOmega\_completeness\] for a subset of data using 1.67 hour exposures. Our overall completeness was relatively low, compared to the 2SLAQ LRG Survey [@Cannon06], but one of the main reasons for this was due to the several technical issues associated with the new AAOmega instrument, which have since been corrected. When checks were made on the d05 field, we found that the redshift completeness rates for our $riz$, $19.8 < i_{\rm deV} < 20.5$ targets as estimated from $\approx 80$ “unfringed” fibres were $90\pm9\%$ in $\approx$4 hour exposures, $84\pm13\%$ in 1.67 hour exposures in 1.$''$6 seeing. Thus, using the full number of sub-exposures we found no significant increase in redshift completeness compared to a 1.67 hour exposure, although this may still be due to conditions varying within the 3 hour exposure time. But our general conclusion is that with reasonable seeing and transparency, we achieve 85-90% redshift completeness in a 1.67 hour exposure. We show a selection of spectra from the subset of data taken in the d05 field in Figure \[fig:example\_spectra\_8\]. The top six panels show spetra of confirmed, $Q{\rm op} \geq 3$ LRGs, with ranging magnitudes and redshifts, including a high redshift confirmed LRG at $z\approx0.9$. The second bottom panel shows an unconfirmed, $Q{\rm
op} < 3$, spectrum, while the bottom spectrum is for a confirmed M-star. The improved AAOmega throughput and sky subtraction enables us to work further into the near-infrared, allowing us to probe higher redshifts. Note the prominent CaII H+K 4000Å break appears in all the confirmed spectra, as expected for an old stellar population.
We also confirmed that the exposure time needed to obtain reliable redshifts of LRGs selected in the same manner as the 2SLAQ survey (using a $gri$-band, $i<19.8$ selection) was cut by a factor of $\sim
4$ from the old 2dF instrument. We note from Table 3 that at least in the more reasonable observing conditions for the d05 field that the completeness of the 1.67hr LRG sample is consistent with the high, 90%, completeness achieved for 2SLAQ LRGs.
Redshift Distribution
---------------------
![The $N(z)$ of $Q{\rm op} \geq 3$ LRGs from the AAOmega LRG Pilot Run, showing that $0.5\leq z \leq0.9$ can be readily selected using SDSS $riz$ photometry. The dotted (blue) histogram shows the distribution for the $i_{\rm deV} < 19.8$ $gri$-selection, while the solid (red) and the dashed (cyan) histograms show the $riz$ selections with $19.8 < i_{\rm deV} < 20.5$ and $19.5 < z < 20.2$ respectively. We also plot the polynomial fit (red line) that is used to model the $N(z)$ distribution for the $riz$, $19.8 < i_{\rm deV} < 20.5$ selection in Section 4.2.[]{data-label="fig:AAOmega_Nofz"}](ps/Nofz_AAOmega_Letter_v5.ps){height="6.5cm" width="6.5cm"}
![Star-Galaxy Separation using SDSS $z$-band magnitudes. All objects with $Q{\rm op} \geq 3$ and $19.8 < i_{\rm deV} <20.5$ are shown, with objects having stellar spectra plotted as (red) stars and objects having high-redshift LRG spectra plotted as (black) open squares. The ordinate gives the difference between the “PSF” and “Model” $z$-band magnitudes as given from the SDSS DR4 imaging.[]{data-label="fig:star_gal"}](ps/star_galaxy_sep_qop3_19pnt8i20pnt5.ps){height="6.5cm" width="6.5cm"}
The [*raison d’$\hat{e}$tre*]{} of the AAOmega LRG Pilot run was to test if we could readily select $z\sim 0.7$ LRGs using single-epoch SDSS $riz$-photometry. As can be seen in Figure \[fig:AAOmega\_Nofz\], where we plot the redshift distributions for confirmed $Q{\rm op} \geq 3$ LRGs, this proved feasible. The mean redshift of our $19.8 < i_{\rm deV} < 20.5$ magnitude sample was $z=0.681\pm 0.005$, with a strong tail out to redshift $z=0.8$ and indeed some objects at $z=0.9$. We found that there was no major difference between the samples with different priorities (areas “A” and “B” in Figure \[fig:riz\_plane\]). Also shown in Figure \[fig:riz\_plane\] are the $riz$-band colours for the objects with spectroscopically confirmed redshifts. When the magnitude limits applied were changed from $19.8 < i_{\rm deV} < 20.5$ to $19.5 < z < 20.2$, the mean redshift increased to $z = 0.698 \pm
0.015$. The mean redshift for our $gri$-band, $17.7 < i_{\rm deV} <
19.8$ selection was very comparable to the 2SLAQ LRG Survey at $z=0.578\pm0.006$.
However, since we found that even though we were able to obtain LRG spectra for $z<20.2$ objects from SDSS single-epoch imaging (and get the increase in redshift one might expect based on galaxy colours from evolutionary models), we find that the completeness of this sample dropped significantly and longer, $\geq2$ hours, exposures would be required in order to obtain $Q{\rm op} \geq 3$ redshifts. This is not surprising considering that with a $z < 20.2$ magnitude limit, we are selecting objects with $i_{\rm deV}\sim$20.8 given a $(i-z)$ colour of $\sim$0.6 (as seen in Fig. 1). Thus for the remainder of this analysis, and the eventual strategy for a large LRG-BAO Survey, we only consider objects with $19.8 < i_{\rm deV} <
20.5$.
As can be seen from Table \[tab:Target\_Statistics\], a significant fraction ($27\%$) of our $Q{\rm op} \geq 3$ objects were M-type stars. However, as shown in Figure \[fig:star\_gal\], [*a posteriori*]{} checking shows that we can reject 40% of these stars using a star-galaxy separation in the $z$-band, rather than the standard SDSS separation performed in the $r$-band. The stellar contamination drops to $16\%$, with very few high-redshift galaxies being lost. Employing near-IR imaging data, specifically a $J-K >1.3$ cut, would dramatically reduce the stellar contamination further, to the levels of a few percent.
2SLAQ, COMBO-17 and AAOmega Comparison
--------------------------------------
![COMBO-17 photometric redshifts vs. AAOmega spectroscopic redshifts. The solid line is the 1:1 relation. The insert shows the histogram of $\Delta z = z_{\rm spec} - z_{\rm phot}$ for AAOmega and COMBO-17 redshifts respectively.[]{data-label="fig:COMBO17_comparison"}](ps/COMBO17_comparison_v3pnt0.ps){height="6.5cm" width="6.5cm"}
In Figure \[fig:COMBO17\_comparison\] we show a comparison between the spectroscopic redshifts we recorded from our AAOmega observations and those measured photometrically by the Classifying Objects by Medium-Band Observations (COMBO-17) survey [e.g. @Wolf03; @Bell04a; @Phleps06]. As can be seen, the 43 common photometric and spectroscopic redshifts match extremely well for the objects for which we have secure redshifts ($Q{\rm op} \geq 3$). There seems to be a slight trend for the photometric redshifts to underestimate the spectroscopic redshift. Why this is the case is not well understood. Excluding 5 “catastrophic failures”, where $| \Delta z | \geq 0.2$, the average offset between the COMBO-17 photometric and AAOmega spectroscopic redshifts is $\overline{\Delta z}=0.026 \pm 0.005$, in the sense that COMBO-17 redshifts are too small. There are 3 spectroscopically confirmed stars that COMBO-17 classified as redshift $z\sim 0.7$ galaxies.
We also compare the spectroscopic redshifts measured by AAOmega with those obtained in the 2SLAQ LRG Survey. We find, for the $Q{\rm op} \geq 3$ LRGs common in both, the mean $\Delta z = 8.4\times10^{-4}$ with the spread on the difference in redshifts being $1.24\times10^{-3}$ i.e. $
370 \kms$. If the error is split evenly between the two surveys, then the error on AAOmega LRG redshifts is $\pm \, 370/\sqrt{2} = \pm 260 \kms$.
LRG Clustering Results
======================
AAOmega LRG Angular Correlation Function, $w(\theta)$ {#sec:AAOmega_wtheta}
-----------------------------------------------------
![The AAOmega LRG Pilot angular correlation function, $w(\theta)$, is given by the solid (blue) triangles. objects were used with magnitudes in the range $19.8 < i_{\rm deV} < 20.5$. The solid (black) line is a estimation of $w(\theta)$ given our redshift distribution and projecting using Limber’s Formula, with the associated $r_{0}$ and $\gamma$ jackknifed values given in Table \[tab:w\_theta\_values\].[]{data-label="fig:AAOmega_w_theta"}](ps/w_theta_AAOmega_v4.ps){height="6.5cm" width="6.5cm"}
Using the procedure described by @Ross07, the projected angular correlation function, $w(\theta)$, for the AAOmega LRG Pilot Survey is presented in Figure \[fig:AAOmega\_w\_theta\]. The solid (blue) triangles are for the measurements made utilising the “Input Catalogue” from which objects were selected as potential high-redshift LRG candidates. Approximately objects were used in this measurement from 6 fields that were observed by the 2SLAQ Survey, each $\pi$ deg$^{2}$ in area. All these objects were potential targets having passed the $riz$-cuts discussed above. Field centres of the 6 fields are given in Table \[tab:w\_theta\_fields\]. It should also be noted that the star-galaxy separation discussed above was applied to this input sample.
------------------------------------------------------------------
Field Name R.A. (J2000) DEC (J2000)
---------------------------------- -------------- ------------- --
2SLAQ c05 12h 38m 18s -00 12 35
“ c07 & 12h 47m 54s & -00 12 35\ 13h 31m 12s -00 12 35
” d07
“ e01 & 14h 34m 00s & -00 12 35\ 14h 42m 48s -00 12 35
” e03
" c07 & 12h 47m 54s & -00 12 35\
------------------------------------------------------------------
: Details of the 2dF fields that were used for the $w(\theta)$ measurements. Note, d05 was also used and details of this field are given in Table 1. All 6 fields were observed by the 2SLAQ Survey. []{data-label="tab:w_theta_fields"}
The error bars associated with the AAOmega LRG $w(\theta)$ measurement are [*field-to-field*]{} errors [see @Ross07] and do not take into account the fact that the clustering measurements are correlated and therefore, the errors on these points should only be regarded as indicative. When we come to calculate the errors on the fitted power-law parameters, defined in equation \[eqn:pl\_fit\], we perform a jackknife analysis on our measurements in the attempt to take into account these covariances. This involves removing one field at a time from our sample and recomputing and refitting the angular correlation function, weighting by the number of $DR$ pairs. As such, we present these jackknife errors for our measurements in Table \[tab:w\_theta\_values\].
2SLAQ LRG AAOmega LRG
------------------------ --------------- --------------- -- --
$r_{0, \rm ss} /\hmpc$ 5.47$\pm$0.40 5.0$\pm$0.34
$\gamma_{\rm ss}$ 2.16$\pm$0.07 2.28$\pm$0.04
$r_{0, \rm ls} /\hmpc$ 8.0$\pm$0.8 10.2$\pm$0.7
$\gamma_{\rm ls}$ 1.67$\pm$0.07 1.58$\pm$0.09
: The values of $r_{0}$ and $\gamma$ for the 2SLAQ LRG Survey and AAOmega LRGs. Note that $r_{b}=1.5 \hmpc$ for the 2SLAQ LRGs, while $r_{b}=1.0
\hmpc$ for AAOmega LRGs. Also note that due to improved implementation of Limber’s formula and more accurate binning, the values given here for $r_{0}$ and $\gamma$ for the 2SLAQ LRG Survey from Limber’s Formula, supersede those given by @Ross07.[]{data-label="tab:w_theta_values"}
A single power-law, of the form $$\xi(r) = \left ( \frac{r}{r_{0}} \right)^{-\gamma},
\label{eqn:pl_fit}$$ where $r_{0}$ is the correlation length and $\gamma$ the power-law slope, has traditionally been fitted for the 3-D correlation function for galaxies, $\xi$, and from which the relation, $$w(\theta) = A \, \theta^{1-\gamma}
\label{eqn:pl_fit_w_theta}$$ where $A$ is amplitude, can be derived for the angular correlation function (e.g. Peebles, 1980). However, as was also found by @Ross07 for the 2SLAQ LRG $w(\theta)$, here we find that a double power-law model is required to fit the present measurement. Following that work, we use Limber’s Formula [see @Phillipps78] to relate the 3-D correlation function to the our measured $w(\theta)$. A double power-law of the form $$\xi(r)= \left\{
\begin{array}{ll}
\left(r / r_{0, \rm ss}\right)^{-\gamma_{\rm ss}} & r \leqslant r_{\rm{b}}
\;\;\; \rm{and}\;\; \\
\left(r / r_{0, \rm ls}\right)^{-\gamma_{\rm ls}} & r > r_{\rm{b}}
\end{array}
\right.
\label{eq:xipowerlaw2}$$ where ‘ss’ and ‘ls’ stand for small scales and large scales respectively, is assumed and calculated from Limber’s formula. The calculated values for $r_{0}$ and $\gamma$ are given in Table \[tab:w\_theta\_values\], where we fit over the range $0.1' < \theta < 40.0'$ and note that $r_{b}=1.5 \hmpc$ for the 2SLAQ LRGs, while $r_{b}=1.0
\hmpc$ for AAOmega LRGs. We also note that due to improved implementation of Limber’s formula and more accurate binning, the values given here for $r_{0}$ and $\gamma$ for the 2SLAQ LRG Survey from Limber’s Formula, supersede those given by @Ross07.
From Table \[tab:w\_theta\_values\], we can see that the $w(\theta)$ measurement for the AAOmega high-redshift data is comparable to the $z=0.55$ data from the 2SLAQ LRG survey. At small scales, the observed AAOmega $w(\theta)$ slope is nearly equal to the 2SLAQ LRG measurement, while at large-scales, the AAOmega slope is slightly shallower than the 2SLAQ LRGs: $\gamma=1.58\pm0.09$ for AAOmega compared to $\gamma=1.67\pm0.07$ for 2SLAQ. However, given the associated errors, the two measurements are in very good agreement. We leave further analysis of the angular correlation function as reported here to Sawangwit et al. (2008, in prep.) who shall investigate the evidence for a double power-law feature in a much larger LRG sample.
Given the AAOmega LRG Pilot $N(z)$ (Figure \[fig:AAOmega\_Nofz\]) and using Limber’s Formula, the AAOmega $w(\theta)$ amplitude is expected to be 13% lower than the 2SLAQ LRG amplitude if there is no clustering evolution in comoving coordinates. Thus, in terms of the overall amplitude, this reinforces the impression given in Table 5 that AAOmega LRGs have a large-scale amplitude which is at least as high as the 2SLAQ LRGs. This finding is further backed up by measurements of the projected correlation function, $w_{p}(\sigma)$. We do not present our $w_{p}(\sigma)$ results here, but note that our best fitting (single) power-law to this data has an amplitude $r_{0}=9.0\pm0.9 \hmpc$ and slope $\gamma=1.73\pm0.08$ over the scales $1.0 < \sigma/ \hmpc < 40.0$ (where $\sigma$ is the separation across the line-of-sight).
Redshift-space Correlation Function, $\xi(s)$
---------------------------------------------
Using the spectroscopic redshift data we obtained in the COSMOS, S11 and d05 fields we now calculate the 3-D redshift-space correlation function, $\xis$. We use the minimum variance estimator suggested by @LS93 (proven to be an optimum estimator by @Kerscher00) where $$\begin{aligned}
\xi(s) &=& 1+\left( \frac{N_{rd}}{N} \right)^{2}
\frac{DD(s)}{RR(s)}
- 2 \left( \frac{N_{rd}}{N} \right)
\frac{DR(s)}{RR(s)}
\label{lseq}\end{aligned}$$ and $DD$, $DR$ and $RR$ are the number of data-data, data-random and random-random pairs at separation $s$ respectively. We use bin widths of $\delta\log(s / \hmpc)$=0.2 and the number density of random points was 20$\times$ that of the LRGs.
The random catalogue was made taking into account the angular incompleteness and the radial distribution of the objects in this Pilot. For each 2dF field we constructed a “quadrant bullseye” angular mask which consisted of 5 concentric rings divided into 4 quadrants. Using both the input catalogue and the 2dF instrument configuration positions, a completeness map was made in each of the 20 sectors. These completenesses then went into mimicking the angular selection function, from which a random catalogue was generated. Corrections for fibre collisions on small, $\lesssim 30$ arcseconds, scales were made by taking the ratio of the input catalogue $w(\theta)$ to the observed redshift catalogue $w(\theta)$, as described by @Ross07. The radial distribution was described by a high-order polynomial fit (shown as the red curve in Figure 3) to the AAOmega $N(z)$ for the 335 $19.8 < i <20.5$ selected LRGs given in Figure 3. We also note that for ease of modelling, we truncate the polynomial fit (and thus the random radial distribution) at redshifts of $z \leq 0.50$ and $z \geq 0.90$.
![The AAOmega LRG Pilot Redshift-Space Correlation Function $\xi(s)$. The (blue) triangles are the measurements from the $riz$-selected $19.8 < i_{\rm deV} < 20.5$ sample, which yielded 335 $Q{\rm op} \geq 3$ LRGs and the associated “Field-to-Field” errors. The dashed (red) line is the redshift-space correlation function from the 2SLAQ LRG Survey [@Ross07].[]{data-label="fig:AAOmega_xis"}](ps/xis_AAOmega_2plfit_2SLAQ_line.ps){height="6.5cm" width="6.5cm"}
Figure \[fig:AAOmega\_xis\] shows our estimate of the 3-D redshift-space correlation function, $\xi(s)$. Again, our error estimates are based on “field-to-field” errors. For $\xi(s)$, we use a double power-law model of the form given in equation \[eq:xipowerlaw2\], motivated by the fact that we expect the small-scale correlation function to be smoothed bt the effect of velocity dispersion (or “Fingers-of-God”) whereas at larger scales we expect the correlation function simply to be boosted due to infall, characterised by the parameter $\beta=\Omega^{0.6}/b$. We adopt the same procedure as for $w(\theta)$ and do a jackknife error analysis in order to estimate the errorbars on the best-fit double power-law model parameters. We find that, $s_{0, \rm ss}= 16.5\pm4.0 \hmpc$ with $\gamma_{\rm ss}=1.09\pm0.28$ on scales $s < 4.5 \hmpc$ and $s_{0, \rm ls}= 9.9\pm0.7 \hmpc$ with $\gamma_{\rm ls}=1.83\pm0.35$ on scales $s > 4.5 \hmpc$. The clustering strength for the $19.8 < i <20.5$, $riz$-selected AAOmega LRGs is again very comparable to the 2SLAQ LRG Survey, where $s_{\rm
ss}=17.3^{+2.5}_{-2.0} \hmpc$ and $\gamma_{\rm ss}=1.03\pm0.07$ on scales $s < 4.5 \hmpc$ and $s_{\rm ls}= 9.40\pm0.19 \hmpc$ and $\gamma_{\rm
ls}=2.02 \pm 0.07$ on scales $s > 4.5 \hmpc$.
Survey mean redshift $n / h^{3} {\rm Mpc^{-3}}$ Luminosity $\hmpc$ $\gamma$ Reference
------------------- --------------- ---------------------------- ------------------ -------------------------------- --------------- -----------
AAOmega $riz$ LRG 0.68 $\sim2\times10^{-4}$ $\gtrsim 2L^{*}$ $r_{0}=$10.2$\pm$0.7 1.58$\pm$0.09 1
$r_{0}=$9.0$\pm$0.9 1.73$\pm$0.08 2
$s_{0}=$9.9$\pm$0.7 1.83$\pm$0.35 3
2SLAQ LRG 0.55 $\sim2\times10^{-4}$ $\gtrsim 2L^{*}$ $s_{0}=$9.40$\pm$0.19 1.98$\pm$0.07 4, 5
$r_{0}=$7.45$\pm$0.35 1.72$\pm$0.06 4, 5
SDSS LRG 0.28 $9.7\times10^{-5}$ $\geq 3L^{*}$ $s_{0}=$11.85$\pm$0.23 1.91$\pm$0.07 6
$r_{0}=$9.80$\pm$0.20 1.94$\pm$0.02 6
MegaZ-LRG 0.63 $5.6\times10^{-5}$ $\gtrsim 3L^{*}$ $r_{0}=$9.3$\pm$0.3 1.94$\pm$0.02 7
COMBO-17 0.6 $4\times10^{-3}$ $\sim L^{*}$ $r_{0}=$5.39$^{+0.30}_{-0.28}$ 1.94$\pm$0.03 8
NDWFS $\sim$0.7 $\approx1\times10^{-3}$ $>1.6L^{*}$ $r_{0}=$6.4$\pm$1.5 2.09$\pm$0.02 9, 10
\[tab:r\_nought\_values\]
Using the model of @Kaiser87, we can find the parameter $\beta$ via $$\xi(s)= \xi(r) \left({ 1+\frac{2}{3}\beta + \frac{1}{5}\beta^{2}} \right).$$ We use our power-law fit for $\xi(r)$ and our large-scale power-law fit to $\xi(s)$ and find that the ratio $\xi(s) / \xi(r) = 1.3\pm0.3$ corresponding to a value of $\beta \simeq 0.4 $ at a scale of $8
\hmpc$. This is not inconsistent with the value $\beta = 0.45\pm0.05$ found for the 2SLAQ LRGs, though clearly the errorbar is large. Nevertheless, for a reasonable value of $\beta$, our values of $s_{0}= 9.9 \pm 0.7\hmpc$ and $r_{0}=9.0\pm0.9 \hmpc$ appear consistent. These high clustering amplitudes clearly suggest that at $z\simeq0.7$, LRGs remain very strongly clustered.
Discussion
==========
Clustering amplitudes and bias of LRGs at $z\simeq 0.7$
--------------------------------------------------------
Now that we have calculated the AAOmega LRG angular, projected, and 3-D redshift-space correlation functions we can use these measurements to infer the physical properties of LRGs. Before proceeding to determine typical LRG halo masses using simple ‘halo occupation’ models, we first compare the clustering amplitudes and biases of the AAOmega LRGs with other LRG results, taking into account the different redshift and luminosity ranges. For reference, a summary of results of space densities, luminosity limits and clustering amplitudes from the AAOmega LRG, 2SLAQ LRG, SDSS LRG, MegaZ-LRG, COMBO-17 and NDWFS surveys, is given in Table 6. We note, however, that direct comparisons between clustering results from surveys with different e.g. magnitude and colour selections can be complex.
We have found that a 2-power law fit is consistent with AAOmega $w(\theta)$ data. The slopes of the AAOmega power-law fits are both less than those for the 2SLAQ LRG Survey [@Ross07]. This could be due to evolution with redshift but the errors on the AAOmega $w(\theta)$ are too large for this difference to be significant. Certainly the large scale results from $\xi(s)$ are perfectly consistent with the two surveys having the same large-scale slope and amplitude (see Fig. \[fig:AAOmega\_xis\]).
We further note that from both the fitting of Limber’s formula to $w(\theta)$ and describing $w_{p}(\sigma)$ with a simple power-law, we find the real-space clustering amplitude of AAOmega LRGs is consistent with that from the SDSS LRG Survey [@Zehavi05a], though our errors are large. Using our $r_{0}$ estimate from $w_{p}(\sigma)$, (which has the smaller error and more closely matched power-law slope), we note that AAOmega LRGs have a slightly lower clustering amplitude than SDSS LRGs, $r_{0}=9.0\pm0.9 \hmpc$ versus $r_{0}=9.80\pm0.20 \hmpc$ respectively. However, this is not surprising since SDSS LRGs have a redder colour selection and higher luminosity, and this may explain their higher clustering amplitude.
To calculate the value of the linear bias, $b$, for the AAOmega LRGs, we use the integrated correlation function [@Croom05; @daAngela08], $$\xi_{20}(r) = \frac{3}{r_{\rm max}^{3}} \int^{r_{\rm max}}_{0} \xi(r) r^{2} dr$$ where we set $r_{\rm max}=20 \hmpc$ since this is a large enough scale for linear theory to apply and also, due to the $r^{2}$ weighting, small-scale redshift-space distortions should be negligible. We first calculate the integrated mass correlation function using the $\sigma_{8}=0.84$ normalised $\Lambda$CDM model for $P(k)$ from @Smith03 with $\Omm(z=0)=0.27$. We find $\xi^{\rm mass}_{20}=0.12$ at the 2SLAQ LRG mean redshift $z=0.55$ and $\xi^{\rm mass}_{20}=0.11$ at the AAOmega LRG mean redshift $z\simeq0.70$.
We then calculate the integrated galaxy correlation function assuming $r_{0}=7.45\pm0.35 \hmpc$ and hold $\gamma$ fixed at 1.72 for the 2SLAQ LRGs @Ross07 and $r_{0}=9.03\pm0.93 \hmpc$, $\gamma=1.73$ for AAOmega LRGs. We find that $b_{\rm 2SLAQ} = 1.90\pm0.08$ and $b_{\rm AAOmega } = 2.35\pm0.22$, where $b=(\xi_{20} / \xi_{\rm mass,
20})^{1/2}$. The value of $b_{\rm 2SLAQ} = 1.90\pm0.08$ is higher, but consistent with that found by @Ross07, who found $b_{\rm
2SLAQ} = 1.66\pm0.35$, from $z$-space distortion analysis, and we suggest the error presented here may be an underestimate since $\gamma$ is being held at a fixed value. The value of $b_{\rm AAOmega
} = 2.35\pm0.22$ is higher than for the 2SLAQ LRGs, but the large error on the AAOmega result means there may be no inconsistency here. However, our value of $b_{\rm AAOmega } = 2.35\pm0.22$ is even higher than that reported for the SDSS LRGs at lower redshifts, who report values of $b\approx1.8$ [@Padmanabhan07]. Although an increase in bias is expected due to the higher redshift of the AAOmega sample, the effect is larger than predicted especially taking into account the bluer AAOmega selection. But again the large error on $b_{\rm
AAOmega}$ renders this difference statistically insignificant.
To see what sort of consistency with 2SLAQ might be expected, we can predict the value of $b$ at redshift $z=0.7$ by utilising the values measured by 2SLAQ at lower redshift, $b(z=0.55)=1.66\pm0.35$, and the bias evolution model given by @Fry96 [@Croom96], $$b(z) = 1 + [b(0) - 1] G(\Omm(0), \Omlam(0), z).
\label{eqn:bias_model_1}$$
Here, $G(\Omm(0), \Omlam(0), z)$ is the linear growth rate of the density perturbations [@Peebles80; @Peebles84; @Carroll92]. There are many other bias models, but here we are following @Ross07 [and references therein] by making the simple assumptions that galaxies formed at early times and their subsequent clustering is governed purely by their discrete motion within the gravitational potential produced by the matter density perturbations. This model would be appropriate, for example, in a “high-peaks” biasing scenario where early-type galaxies formed at a single redshift and their co-moving space density then remained constant to the present day.
Thus, assuming a growth rate of $G(0.3,0.7,z)$, to relate $\xi_{\rm
mm}(z=0.55)$ to $\xi_{\rm mm}(z=0.7)$, we therefore expect $\xi_{\rm
gg}(z=0.7) = 0.94 \, \xi_{\rm gg}(z=0.55)$ from this model. From Table 6 the $r_{0}$ values between 2SLAQ and AAOmega LRGs are consistent, although the errors on the AAOmega $r_{0}$ measurement are big. But the errors on $\xi(s)$ are smaller, and even here, the $s_{0}$ values agree to within the errors (see also Figure \[fig:AAOmega\_xis\]). The consistency of the clustering results is expected, since the 0.7 magnitudes deeper $19.8 < i_{deV} <
20.5$ selection was based on experience from the 2SLAQ LRG Survey and primarily designed to select similarly highly-biased red galaxies at redshift $z\simeq 0.7$. We conclude that the LRG correlation function amplitudes are similar at redshifts $z\approx0.55$ and $z \approx 0.7$ and that there is still no inconsistency with the simple bias model where the comoving density of LRGs are assumed to be constant with redshift.
Predictions of halo occupation models
-------------------------------------
An alternative approach to interpreting our measured level of clustering is to use the halo occupation model, in which the galaxy field is taken to be a superposition of contributions from dark-matter haloes, weighted by the number of galaxies per halo, $N(M)$. This methodology is commonly reffered to as a ‘halo occupation distribution’, or HOD, model and was used recently by [@Phleps06] to model the projected correlations in the COMBO-17 survey. We apply exactly the same method as described in that paper to model our AAOmega data, specifically for our $w_{p}(\sigma)$ measurement. Again we adopt a standard matter power spectrum, with $\Omega_m=0.3$, $\Omega_b=0.045$, $h=0.73$, $\sigma_8=0.85$, and a scalar spectral index of 0.97. The occupation model is the simplest possible: $N(M) = (M/M_{\rm
min})^\alpha$ for $M>M_{\rm min}$. These two free parameters are reduced to one if the model is also required to match the number density of LRGs, which is approximately $0.0002 \, h^3 \, {\rm
Mpc}^{-3}$.
Realistic occupation models will be more complicated than this simple power-law form, but Phleps et al. argue that the results can be expressed quite robustly in terms of an effective halo mass – i.e. the average halo mass weighted by the number of galaxies. For our current data, the occupation parameters that best match the clustering measurements are $\alpha\simeq 0.7$ and $M_{\rm min}\simeq 2 \times
10^{13} h^{-1}M_\odot$. These imply an average halo mass for the AAOmega LRGs at $z\simeq 0.7$ of $M_{\rm eff}\simeq 7\times 10^{13}
h^{-1} M_\odot$. Reasonably enough for particularly rare and luminous galaxies such as those studied here, this mass is somewhat larger than the figure found by Phleps et al. for the COMBO-17 red-sequence galaxies at $z\simeq 0.6$, which was $M_{\rm eff}\simeq 1.6\times
10^{13} h^{-1}M_\odot$, using the same methodology. Our AAOmega figure for $M_{\rm eff}$ is in fact almost identical to the average mass deduced for $z=0$ red-sequence galaxies in SDSS. Of course, this coincidence does not imply any direct correspondence between these populations: the haloes that host our $z\simeq0.7$ LRGs may have become much more massive by the present.
@Blake07 calculate the LRG angular correlation function using the “MegaZ-LRG” galaxy database, which is a large photometric-redshift catalogue of luminous red galaxies extracted from the SDSS imaging data [@Collister07]. They then successfully model the observations using a HOD model with a “central” galaxy contribution and a “satellite” galaxy component. Noting that comparison of results are strongly dependent on the overall normalization of the power spectrum, $\sigma_{8}$, we compare our effective mass value for the AAOmega LRGs at $z\simeq 0.7$ of $M_{\rm
eff}\simeq 7\times 10^{13} h^{-1} M_\odot$ ($\sigma_{8}=0.85$) to that of the highest redshift bin by @Blake07 of $0.6 < z < 0.65$ and find their $M_{\rm eff} = 9.5 \pm 0.7 \times 10^{13} h^{-1} M_\odot$ ($\sigma_{8}=0.8$) to be $\sim 30\%$ larger than our effective mass estimate. However, after further analysis these authors have revised their $M_{\rm eff}$ estimates (C. Blake priv. comm) and we await comparisons to their new results.
@White07 and @Brown08 have used data from the 9 deg$^{2}$ Bo$\ddot{\rm{o}}$tes field, which has been imaged in the optical and infrared as part of the NOAO Deep Wide Field Survey [NDWFS; @JD99; @Brown08], and by the [*Spitzer*]{} IRAC Shallow Survey [@Eisenhardt04]. @White07 use the clustering of luminous red galaxies from these observations (and $N$-body simulations) to argue that about $\frac{1}{3}$ of the most luminous satellite galaxies appear to undergo merging or disruption within massive halos between $z\simeq0.9$ and 0.5. @Brown08 report a correlation length of $r_{0}=6.4\pm1.5 \hmpc$ for their brightest red galaxy sample, $M_{B} -5 \log h <-21.0$ (corresponding to $L>1.6L^{*}$ galaxies), across the redshift range $0.6 < z < 0.8$. These authors also calculate the bias for this sample to be $b=2.15\pm0.08$. Thus, although the NDWFS LRGs and AAOmega LRGs have different selections (e.g. different magnitude and redshift limits), evidence from both surveys suggest that redshift $z=0.7$ LRGs are highly-biased objects and thus extremely well-suited to LSS studies.
LRGs versus ELGs
----------------
[**Scale**]{}
--------------------------- ---------------------------- ------------------------------------ ---------------------------- ------------------------------------ -------------- ----------------
$k/h\, {\rm Mpc^{-1}}$ $P/ h^{-3}\,{\rm Mpc^{3}}$ $V_{\rm eff}/h^{-3} {\rm Gpc^{3}}$ $P/ h^{-3}\,{\rm Mpc^{3}}$ $V_{\rm eff}/h^{-3} {\rm Gpc^{3}}$ 167/123 nts. Equal no. nts.
0.02 6.7$\times 10^{4}$ 1.1 $1\times 10^{5}$ 1.9 1.7 1.3
0.05 2.7$\times 10^{4}$ 0.82 $4\times 10^{4}$ 1.4 1.7 1.3
0.15 6.7$\times 10^{4}$ 0.42 $1\times 10^{4}$ 0.61 1.5 1.1
\[tab:LRGs\_vs\_ELGs\_2\]
One of the key questions that the AAOmega LRG Pilot Survey wanted to address, was whether a “blue” or a “red” galaxy survey be the more advantageous when pursuing BAOs at high redshift. In the previous sections, we have presented the $N(z)$ and clustering amplitudes for $\bar{z}=0.68$ Luminous Red Galaxies. As such, our ‘Pilot’ observations suggest, a spectroscopic redshift survey strategy to pursue BAOs with AAOmega LRGs might consist of $\approx$1.5 hour exposures with
- [ $\simeq 100$ fibres placed on $gri$-selected $i<19.8$ LRGs with $z \simeq 0.55$ and]{}
- [ $\simeq 260$ fibres placed on $riz$-selected $19.8<i<20.5$ LRGs with $z \simeq 0.7$]{}
in order to obtain LRGs over 3000deg$^{2}$ which will give an $\sim 4 \times$ bigger effective volume than the original SDSS LRG Survey of 45,000 LRGs [@Eisenstein05]. We shall compare this strategy, with an alternate “Emission Line Galaxy” (ELG) survey, in the remainder of this section.
@Glazebrook07 select “blue” emission line galaxies (ELGs) using SDSS and [*GALEX*]{} Far ultra-violet (FUV) and Near ultra-violet (NUV) imaging [@Martin05], for the [*WiggleZ*]{} BAO Dark Energy Survey. By using the reported $N(z)$ in @Glazebrook07 [Figure 2] which has an average redshift of $z\simeq0.6\pm0.2$ as well as their estimate of the clustering amplitude, we can make a comparison with our data. The clustering amplitude reported initially by @Glazebrook07 is $s_{0}= 3.81
\pm 0.20 \hmpc$ (their Figure 3). However, it has recently been suggested that an improved [*GALEX*]{} ELG Selection for [*WiggleZ*]{} may give a higher ELG clustering amplitude of $r_{0}\approx 6
\hmpc$ (C. Blake priv. comm.) leading to $s_{0}\approx 9 \hmpc$ assuming $\beta(z\approx0.7)=0.8$ and applying equation 11. We use this higher value, along with the appropriate redshift distributions for ELGs (truncated at redshift $z<0.5$ due to the [*WiggleZ*]{} Survey plans to focus on $z>0.5$ galaxies only) and LRGs (from our Fig. \[fig:AAOmega\_Nofz\]) and assuming that bias is scale independent.
We can calculate the effective volume surveyed using [e.g. @Tegmark06]: $$V_{\rm eff} = \int \left[
\frac{ n({\bf r}) \, P_{g}(k)}
{1 + n({\bf r}) \, P_{g}(k)}
\right]^{2} dV.$$ where $n({\bf r})$ is the comoving number density of the sample, (in units of $h^{3}$ Mpc$^{-3}$) and $P_{g}(k)$ is the value of the galaxy Power Spectrum at wavenumber $k$ (with units of $h$ Mpc$^{-1}$). For the LRG Survey we assume $\approx$ redshifts are required with 100 fibres targeted on $i<19.8$, redshift $z\simeq0.55$ 2SLAQ LRGs with 90% completeness, to account for 5% redshift incompleteness and 5% stellar contamination, and 260 fibres on $19.8
< i < 20.5$ $z\simeq0.7$ AAOmega LRGs with 70% completeness (15% redshift incompleteness and 15% stellar contamination). For the ELG Survey, we assume 360 fibres targeted on ELGs, as described above, with 80% redshift completeness. Therefore, we see that [*(i)*]{} a 167 night LRG survey would have $\approx 1.7 \times$ the effective volume of a 123 night ELG survey as envisaged by Glazebrook et al. and [*(ii)*]{} for equal telescope time, an LRG survey will sample $\approx 1.3 \times$ the effective volume of an ELG Survey (see Table 6). The above results are approximately in line with those of @Parkinson07 who present “Figures of Merit” (FoM) calculations to judge the optimality of different survey designs for future galaxy redshift-based BAO experiments.
Conclusions
===========
We have reported on the AAOmega-AAT LRG Pilot observing run to establish the feasibility of a large spectroscopic survey aimed at detecting BAO and present some of the first results from the new AAOmega instrument. We have confirmed that AAOmega has a factor of approximately four in improved throughput in its red ($>5700$Å) arm as compared to the old 2dF spectrographs. Utilising this new sensitivity, we observed Luminous Red Galaxies (LRGs) selected using single epoch SDSS $riz$-photometry in 3 fields including the COSMOS field, the COMBO-17 S11 field and the previously observed 2SLAQ Survey field, d05. Our main conclusions are:
- [We detect 1270 objects in three fields, of which 587 are confirmed high-redshift LRGs. The mean redshift for each selection was $\bar{z}=0.578 \pm 0.006$ from the $gri$-band selection with $17.5 < i_{\rm deV} < 20.5$, $\bar{z}=0.681 \pm
0.005$ from the $riz$-band selection with $19.8 < i_{\rm deV} <
20.5$ and $\bar{z}=0.698\pm0.015$ from the $riz$-band selection with $19.5 < z < 20.2$. At $i<20.5$, 84% redshift completeness for LRGs was achieved in 1.67hr exposures in reasonable conditions.]{}
- [We have compared our AAOmega spectroscopic redshifts to spectroscopic and photometric redshifts obtained by the 2SLAQ LRG Survey and COMBO-17 respectively. We find excellent agreement with the 2SLAQ spectroscopic redshifts, but a suggestion that there is a systematic tendency of the photometric redshifts to underestimate the spectroscopic redshifts by $\overline{\Delta z}=0.026 \pm 0.005$.]{}
- [We find that a simple power-law model, for $\wp$, gives a best fit value of $r_{0} = 9.03 \pm 0.93$ for our $\bar{z}=0.68$ LRG sample, compared to $r_{0} = 9.80 \pm 0.20$ for the $-21.2 < M_{r} < -23.2$ SDSS LRG sample and $r_{0} = 7.30 \pm
0.34$ for the $\bar{z}=0.55$ 2SLAQ LRG sample. This confirms that high-redshift luminous red galaxies are very good large-scale structure tracers, similar to their lower redshift counterparts [@Zehavi05a; @Eisenstein05; @Ross07].]{}
- [We also find that, taking into account the large errors on the AAOmega LRG $r_{0}$ measurement, there is no inconsistency with the simple bias model where the comoving density of LRGs are assumed to be constant with redshift.]{}
- [Finally, this Pilot project shows that a large-scale AAOmega spectroscopic survey of highly biased $z \sim 0.7 $ LRGs over 3000deg$^{2}$, remains a very promising and competitive route in order to measure the baryon acoustic oscillations and use this scale-length to investigate the potential evolution of the equation of state parameter, $w$.]{}
acknowledgement {#acknowledgement .unnumbered}
===============
We thank C. Wolf for supplying the COMBO-17 photometric redshift catalogue data in the S11 field and U. Sawangwit for providing the Bruzual and Charlot models. We also thank R. Angulo, C.M. Baugh and R.M. Bielby for useful discussion. This work was supported by a PPARC PhD Studentship and by National Science Foundation grant AST-0607634 (N.P.R.) We warmly thank all the present and former staff of the Anglo-Australian Observatory for their work in building and operating the AAOmega facility. The AAOmega LRG Pilot is based on observations made with the Anglo-Australian Telescope and with the SDSS. Funding for the creation and distribution of the SDSS Archive has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Aeronautics and Space Administration, the National Science Foundation, the U.S. Department of Energy, the Japanese Monbukagakusho, and the Max Planck Society. The SDSS Web site is [http://www.sdss.org/]{}. The SDSS is managed by the Astrophysical Research Consortium (ARC) for the Participating Institutions. The Participating Institutions are The University of Chicago, Fermilab, the Institute for Advanced Study, the Japan Participation Group, The Johns Hopkins University, the Korean Scientist Group, Los Alamos National Laboratory, the Max-Planck-Institute for Astronomy (MPIA), the Max-Planck-Institute for Astrophysics (MPA), New Mexico State University, University of Pittsburgh, University of Portsmouth, Princeton University, the United States Naval Observatory, and the University of Washington.
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The AAOmega LRG Pilot Data
==========================
In Table A1 we present properties of the first 40 objects from the AAOmega Pilot Catalogue in Right Ascension order. The full dataset is published in its entirety in the electronic edition of the [*Monthly Notices of the Royal Astronomical Society*]{}.
Object ID$^{1}$ $\alpha$ $\delta$ $r$-fibre $^{2}$ T$^{3}$ X-cor$^{4}$ redshift $Q{\rm op}$ Field $u$ $g$ $r$ $i$ $z$ $u_{\rm err}$ $g_{\rm err}$ $r_{\rm err}$ $i_{\rm err}$ $z_{\rm err}$
----------------------------------------------- ----------- ---------- ------------------ --------- ------------- ---------- ------------- ------- --------- --------- --------- --------- --------- --------------- --------------- --------------- --------------- ---------------
J095618.80+021623.2 149.07835 2.27312 20.14 1 6.34 0.6716 3 cos 23.6300 23.2400 21.7760 20.5030 19.8820 1.0310 0.2800 0.1210 0.0730 0.1770
J095634.36+015804.9 149.14317 1.96803 19.48 1 5.51 0.5510 3 cos 22.9990 22.0510 20.5420 19.4830 18.8840 0.7910 0.1420 0.0600 0.0410 0.0910
J095637.67+015254.7 149.15698 1.88187 20.10 1 3.47 0.5686 3 cos 24.3040 22.6720 21.4140 20.2050 19.8920 1.2250 0.1650 0.0890 0.0530 0.1490
J095639.34+021709.2 149.16393 2.28590 20.27 1 4.91 0.5457 3 cos 23.4230 23.1880 21.2570 20.2020 19.4730 1.1030 0.3510 0.1080 0.0720 0.1670
J095647.19+022325.2 149.19666 2.39035 19.73 1 5.57 0.5357 3 cos 25.4570 22.6930 20.7560 19.7260 19.7420 1.4010 0.2240 0.0710 0.0480 0.2130
J095649.24+021838.6 149.20519 2.31073 19.10 1 5.00 0.4806 3 cos 23.1220 21.7150 20.0390 19.1010 18.6840 0.8970 0.0990 0.0400 0.0290 0.0840
J095649.42+023355.3 149.20592 2.56537 19.93 2 3.30 0.9373 1 cos 21.5160 26.4460 21.3550 19.9320 19.3920 0.5760 1.6790 0.3160 0.1550 0.3360
J095649.86+020743.4 149.20776 2.12875 20.33 2 3.06 0.9444 2 cos 22.5640 23.1070 21.2320 20.3290 19.4470 0.5350 0.2970 0.0920 0.0670 0.1380
J095654.01+020225.0 149.22507 2.04028 20.41 1 6.03 0.6589 3 cos 23.1800 23.3520 21.9510 20.6320 20.0930 0.6970 0.2810 0.1320 0.0660 0.1870
J095702.89+024208.9 149.26208 2.70250 19.78 1 3.49 0.6792 4 cos 22.2880 21.7660 20.7220 19.7830 19.4340 0.3290 0.0900 0.0620 0.0470 0.1110
J095702.94+014853.3 149.26228 1.81481 20.29 1 2.98 0.7001 2 cos 24.7150 24.6660 22.2380 20.9450 20.6940 1.5400 0.8110 0.2040 0.1040 0.3320
J095703.84+015133.8 149.26600 1.85940 20.48 1 3.90 0.6355 3 cos 24.8210 22.9440 22.1000 20.8850 20.5140 1.2970 0.2120 0.1840 0.1030 0.2660
J095705.34+024238.2 149.27227 2.71061 20.46 3 3.27 0.0225 2 cos 23.4330 22.8750 22.5620 20.7060 20.4770 1.0860 0.3040 0.4100 0.1370 0.3660
J095705.95+021834.5 149.27483 2.30959 20.14 1 3.47 0.8355 2 cos 26.6360 24.5210 22.0400 20.1440 19.9200 0.8420 1.3890 0.3450 0.1050 0.3750
J095707.64+023955.2 149.28184 2.66533 19.95 3 7.88 0.0010 3 cos 24.5950 22.7660 21.6200 19.9550 19.3470 1.5510 0.2390 0.1550 0.0610 0.1150
J095708.35+014747.4 149.28483 1.79652 20.16 2 2.75 0.9454 2 cos 22.0610 22.9460 21.9640 20.1640 19.5390 1.2060 1.0070 0.6780 0.2210 0.5190
J095709.77+020506.6 149.29071 2.08519 20.35 3 4.97 0.0004 3 cos 22.9240 22.7870 21.4720 20.2040 19.7110 0.6020 0.1890 0.0930 0.0510 0.1410
J095712.04+015554.2 149.30018 1.93172 20.01 1 3.40 0.4818 3 cos 25.9540 22.1380 21.0040 20.0080 19.8880 0.7680 0.1210 0.0780 0.0530 0.1750
J095712.53+021202.8 149.30223 2.20079 20.26 1 3.43 0.8407 3 cos 22.0080 21.9630 21.7840 20.5500 20.0970 0.4650 0.1580 0.2190 0.1200 0.3680
J095713.39+015241.0 149.30579 1.87807 19.82 1 6.31 0.6434 3 cos 24.6310 22.6460 20.9660 19.8160 19.2770 1.5990 0.2100 0.0840 0.0500 0.1140
J095713.89+015210.8 149.30789 1.86967 20.14 3 2.44 0.0449 2 cos 22.7680 22.4500 21.5830 20.1400 19.9110 1.0200 0.3280 0.2630 0.1210 0.3740
J095719.14+020154.6 149.32975 2.03184 19.85 1 3.78 0.5553 3 cos 23.0910 22.6420 20.7870 19.8540 19.7550 0.8480 0.2100 0.0660 0.0470 0.1960
J095724.21+013159.5 149.35090 1.53321 19.99 1 3.06 0.5992 2 cos 22.3180 22.9680 21.5120 19.9900 19.5990 1.1650 0.7730 0.3610 0.1420 0.3850
J095724.98+022905.3 149.35408 2.48483 18.66 1 5.44 0.4814 3 cos 26.7010 21.3450 19.5470 18.6610 18.2400 0.7670 0.1060 0.0340 0.0280 0.0630
J095728.41+014307.4 149.36841 1.71873 19.39 3 3.11 0.0336 1 cos 24.3030 22.4020 20.5450 19.3900 18.9600 2.4410 0.2450 0.0740 0.0430 0.1180
J095728.89+021721.2 149.37040 2.28924 20.24 3 4.21 0.0008 3 cos 25.0320 22.8120 21.4660 20.2770 19.7580 1.1850 0.1700 0.0960 0.0550 0.1390
J095731.70+020327.6 149.38210 2.05768 20.39 2 2.74 0.9269 1 cos 23.0820 22.8770 22.2150 20.3910 19.6020 1.0690 0.3210 0.2880 0.0930 0.2070
J095733.18+021546.5 149.38826 2.26293 20.26 1 4.90 0.6974 3 cos 26.0990 23.2870 21.6230 20.4430 19.6990 0.9610 0.4190 0.1720 0.0980 0.2120
J095733.33+013144.5 149.38888 1.52905 20.33 1 2.52 0.6524 2 cos 23.9740 23.1710 21.3590 20.4520 19.5900 1.3940 0.2940 0.1030 0.0690 0.1170
J095734.28+025024.9 149.39286 2.84025 19.90 1 6.10 0.5540 3 cos 23.4570 22.4010 20.8830 19.9010 19.0370 1.0000 0.1950 0.0840 0.0620 0.0940
J095737.86+014333.3 149.40775 1.72593 20.44 1 3.31 0.4784 1 cos 25.9890 22.8020 21.5670 20.2210 19.4930 1.4960 0.3870 0.2040 0.1020 0.2090
J095741.91+020033.7 149.42466 2.00938 19.85 1 9.80 0.6901 4 cos 22.5380 22.3760 20.9850 19.6760 19.2900 0.6310 0.2390 0.1160 0.0600 0.1570
J095742.07+025028.4 149.42533 2.84123 19.88 1 6.65 0.5144 3 cos 22.4440 22.1470 20.7750 19.8810 19.5570 0.3270 0.1150 0.0570 0.0460 0.1090
J095742.56+023452.1 149.42734 2.58114 19.42 1 11.34 0.7019 5 cos 22.5590 22.1410 20.7490 19.4190 18.7640 0.6400 0.1740 0.0820 0.0430 0.0810
J095742.58+024432.1 149.42745 2.74228 20.11 3 5.51 0.0008 3 cos 22.6610 25.3400 21.9470 20.1110 19.6740 0.9930 2.2170 0.4110 0.1420 0.3160
J095744.56+023835.8 149.43567 2.64330 20.07 1 6.84 0.7358 3 cos 22.8760 23.2550 21.5570 20.3540 19.5620 0.6980 0.3720 0.1420 0.0830 0.1410
J095744.60+012447.1 149.43585 1.41309 20.11 3 3.97 0.0337 1 cos 25.3800 22.8650 21.7010 20.1050 19.7220 3.5850 0.6350 0.3840 0.1410 0.3750
J095745.96+014240.1 149.44153 1.71115 20.13 3 3.05 0.0002 1 cos 24.3480 22.5120 20.7400 19.5390 18.7560 3.0220 0.3360 0.1120 0.0590 0.1220
J095746.79+030025.7 149.44499 3.00716 20.08 1 3.70 0.7130 3 cos 23.1230 23.2530 20.6030 19.6350 18.8280 2.1560 0.8570 0.1320 0.0920 0.1670
J095748.08+025642.0 149.45035 2.94503 19.85 3 8.03 0.0001 3 cos 23.1270 22.3340 21.0780 19.4360 18.4770 0.9510 0.1660 0.0920 0.0350 0.0540
\[tab:The\_AAOmega\_Pilot\_Catalogue\_Top40\]
: The first 40 objects from the AAOmega Pilot Catalogue in RA order. The table for the full sample is available online only. $^1$Using the SDSS nomenclature. $^2$$r$-band SDSS Fibre Magnitude. $^3$Model galaxy template to fit observed spectra. A value of 1 signifies the “early-type” template provided the best-fit, a value of 2 is for a $k+a$ Balmer absorption spectrum template and 3 indicates the M-star template. $^4$Cross-correlation co-efficient between the model and observed galaxy spectra.
[^1]: email: [email protected]
[^2]: “$Q{\rm op}$” represents an integer redshift quality flag assigned by visual inspection of the galaxy spectrum and the redshift cross-correlation function. A value of 3 or greater represents a $>95$% confidence that the redshift obtained from the spectrum is valid.
|
---
abstract: 'We develop a unified framework to classify topological defects in insulators and superconductors described by spatially modulated Bloch and Bogoliubov de Gennes Hamiltonians. We consider Hamiltonians $\mathcal{H}({\bf k},{\bf r})$ that vary slowly with adiabatic parameters ${\bf r}$ surrounding the defect and belong to any of the ten symmetry classes defined by time reversal symmetry and particle-hole symmetry. The topological classes for such defects are identified, and explicit formulas for the topological invariants are presented. We introduce a generalization of the bulk-boundary correspondence that relates the topological classes to defect Hamiltonians to the presence of protected gapless modes at the defect. Many examples of line and point defects in three dimensional systems will be discussed. These can host one dimensional chiral Dirac fermions, helical Dirac fermions, chiral Majorana fermions and helical Majorana fermions, as well as zero dimensional chiral and Majorana zero modes. This approach can also be used to classify temporal pumping cycles, such as the Thouless charge pump, as well as a fermion parity pump, which is related to the Ising non-Abelian statistics of defects that support Majorana zero modes.'
author:
- 'Jeffrey C.Y. Teo and C.L. Kane'
title: Topological Defects and Gapless Modes in Insulators and Superconductors
---
Introduction {#sec:introduction}
============
The classification of electronic phases according to topological invariants is a powerful tool for understanding and predicting the behavior of matter. This approach was pioneered by Thouless, et al.[@tknn](TKNN), who identified the integer topological invariant characterizing the two dimensional (2D) integer quantum Hall state. The TKNN invariant $n$ gives the Hall conductivity $\sigma_{xy}=ne^2/h$ and characterizes the Bloch Hamiltonian ${\cal H}({\bf k})$, defined as a function of ${\bf k}$ in the magnetic Brillouin zone. It may be expressed as the first Chern number associated with the Bloch wavefunctions of the occupied states. A fundamental consequence of this topological classification is the [*bulk-boundary correspondence*]{}, which relates the topological class of the bulk system to the number of gapless chiral fermion edge states on the sample boundary.
Recent interest in topological states[@qizhang10; @moore10; @hk10] has been stimulated by the realization that the combination of time reversal symmetry and the spin orbit interaction can lead to topological insulating electronic phases [@km05a; @km05b; @moorebalents07; @roy1; @fkm07; @qihugheszhang08] and by the prediction[@bhz06; @fukane07; @zhang09] and observation [@konig1; @konig2; @roth; @hsieh08; @hsieh09a; @roushan09; @xia09a; @hor09; @chen09; @hsieh09b; @park10; @alpichshev10; @hsieh09c] of these phases in real materials. A topological insulator is a two or three dimensional material with a bulk energy gap that has gapless modes on the edge or surface that are protected by time reversal symmetry. The bulk boundary correspondence relates these modes to a $\mathbb{Z}_2$ topological invariant characterizing time reversal invariant Bloch Hamiltonians. Signatures of these protected boundary modes have been observed in transport experiments on 2D HgCdTe quantum wells[@konig1; @konig2; @roth] and in photoemission and STM experiments on 3D crystals of Bi$_{1-x}$Sb$_x$[@hsieh08; @hsieh09a; @roushan09], Bi$_2$Se$_3$[@xia09a], Bi$_2$Te$_3$[@chen09; @hsieh09b; @alpichshev10] and Sb$_2$Te$_3$[@hsieh09c]. Topological insulator behavior has also been predicted in other classes of materials with strong spin orbit interactions [@shitade09; @pesin10; @chadov10; @lin10a; @lin10b; @lin10c; @yan10].
Superconductors, described within a Bogoliubov de Gennes (BdG) framework can similarly be classified topologically[@roy08; @schnyder08; @kitaev09; @qi09]. The Bloch-BdG Hamiltonian ${\cal H}_{BdG}({\bf k})$ has a structure similar to an ordinary Bloch Hamiltonian, except that it has an exact particle-hole symmetry that reflects the particle-hole redundancy inherent to the BdG theory. Topological superconductors are also characterized by gapless boundary modes. However, due to the particle-hole redundancy, the boundary excitations are Majorana fermions. The simplest model topological superconductor is a weakly paired spinless $p$ wave superconductor in 1D[@kitaev00], which has zero energy Majorana bound states at its ends. In 2D, a weakly paired $p_x+ip_y$ superconductor has a chiral Majorana edge state[@readgreen]. Sr$_2$RuO$_4$ is believed to exhibit a triplet $p_x+ip_y$ state[@mackenzie03]. The spin degeneracy, however, leads to a doubling of the Majorana edge states. Though undoubled topological superconductors remain to be discovered experimentally, superfluid $^3$He B is a related topological phase[@volovik03; @roy08; @schnyder08; @qi09; @volovik09] and is predicted to exhibit 2D gapless Majorana modes on its surface. Related ideas have also been used to topologically classify Fermi surfaces[@horava].
Topological insulators and superconductors fit together into an elegant mathematical framework that generalizes the above classifications[@schnyder08; @kitaev09]. The topological classification of a general Bloch or BdG theory is specified by the dimension $d$ and the 10 Altland Zirnbauer symmetry classes[@altland97] characterizing the presence or absence of particle-hole, time reversal and/or chiral symmetry. The topological classifications, given by $\mathbb{Z}$, $\mathbb{Z}_2$ or $0$ show a regular pattern as a function of symmetry class and $d$, and can be arranged into a [*periodic table*]{} of topological insulators and superconductors. Each non trivial entry in the table is predicted, via the bulk-boundary correspondence, to have gapless boundary states.
Topologically protected zero modes and gapless states can also occur at topological defects, and have deep implications in both field theory and condensed matter physics[@jackiwrebbi; @jackiwrossi; @ssh; @volovik03]. A simple example is the zero energy Majorana mode that occurs at a vortex in a $p_x+ip_y$ superconductor[@readgreen]. Similar Majorana bound states can be engineered using three dimensional heterostructures that combine ordinary superconductors and topological insulators[@fukane08], as well as semiconductor structures that combine superconductivity, magnetism and strong spin orbit interactions[@sau10; @alicea10; @lutchyn10; @oreg10]. Recently, we showed that the existence of a Majorana bound state at a point defect in a three dimensional Bogoliubov de Gennes theory is related to a $\mathbb{Z}_2$ topological invariant that characterizes a family of Bogoliubov de Gennes Hamiltonians ${\cal H}_{BdG}({\bf k},{\bf r})$ defined for $\bf r$ on a surface surrounding the defect[@teokane10]. This suggests that a more general formulation of topological defects and their corresponding gapless modes should be possible.
=3.2in
In this paper we develop a general theory of topological defects and their associated gapless modes in Bloch and Bloch-BdG theories in all symmetry classes. As in Ref. , we assume that far away from the defect the Hamiltonian varies slowly in real space, allowing us to consider adiabatic changes of the Hamiltonian as a function of the real space position ${\bf r}$. We thus seek to classify Hamiltonians ${\cal H}({\bf k},{\bf r})$, where ${\bf
k}$ is defined in a $d$ dimensional Brillouin zone (a torus $T^d$), and ${\bf r}$ is defined on a $D$ dimensional surface $S^D$ surrounding the defect. A similar approach can be used to classify cyclic temporal variations in the Hamiltonian, which define adiabatic pumping cycles. Hereafter we will drop the BdG subscript on the Hamiltonian with the understanding that the symmetry class dictates whether it is a Bloch or BdG Hamiltonian.
In Fig. \[figure1\] we illustrate the types of topological defects that can occur in $d=1$, $2$ or $3$. For $D=0$ we regard $S^0$ as two points ($\{-1,+1\}$). Our topological classification then classifies the [*difference*]{} of ${\cal H}({\bf
k},+1)$ and ${\cal H}({\bf k},-1)$. A non trivial difference corresponds to an interface between two topologically distinct phases. For $D=1$ the one parameter families of Hamiltonians describe line defects in $d=3$ and point defects in $d=2$. For $d=1$ it could correspond to an adiabatic temporal cycle $H({\bf k},t)$. Similarly for $D=2$, the two parameter family describes a point defect for $d=3$ or an adiabatic cycle for a point defects in $d=2$.
Classifying the $D$ parameter families of $d$ dimensional Bloch-BdG Hamiltonians subject to symmetries leads to a generalization of the periodic table discussed above. The original table corresponds to $D=0$. For $D>0$ we find that for a given symmetry class the topological classification ($\mathbb{Z}$, $\mathbb{Z}_2$ or $0$) depends only on $$\delta = d - D.
\label{delta}$$ Thus, all line defects with $\delta = 2$ have the same topological classification, irrespective of $d$, as do point defects with $\delta = 1$ and pumping cycles with $\delta = 0$. Though the classifications depend only on $\delta$, the [*formulas*]{} for the topological invariants depend on both $d$ and $D$.
This topological classification of ${\cal H}({\bf k},{\bf r})$ suggests a generalization of the bulk-boundary correspondence that relates the topological class of the Hamiltonian characterizing the defect to the structure of the protected modes associated with the defect. This has a structure reminiscent of a mathematical [*index theorem*]{}[@nakahara] that relates a topological index to an analytical index that counts the number of zero modes[@jackiwrebbi; @jackiwrossi; @volovik03; @goldstone; @weinberg; @witten; @witten82; @davis]. In this paper we will not attempt to [*prove*]{} the index theorem. Rather, we will observe that the topological classes for ${\cal H}({\bf k},{\bf r})$ coincide with the expected classes of gapless defect modes. In this regards the dependence of the classification on $\delta$ in is to be expected. For example, a point defect at the end of a one dimensional system ($\delta = 1-0$) has the same classification as a point defects in two dimensions ($\delta = 2-1$) and three dimensions ($\delta =
3-2$).
We will begin in section \[sec:periodictable\] by describing the generalized periodic table. We will start with a review of the Altland Zirnbauer symmetry classes[@altland97] and a summary of the properties of the table. In Appendix \[appendix:periodicities\] we will justify this generalization of the table by introducing a set of mathematical mappings that relate Hamiltonians in different dimensions and different symmetry classes. In addition to establishing that the classifications depend only on $\delta=d-D$, these mappings allow other features of the table, already present for $D=0$ to be easily understood, such as the pattern in which the classifications vary as a function of symmetry class as well as the Bott periodicity of the classes as a function of $d$.
In section \[sec:linedefects\] and \[sec:pointdefect\] we will outline the physical consequences of this theory by discussing a number of examples of line and point defects in different symmetry classes and dimensions. The simplest example is that of a line defect in a 3D system with no symmetries. In section \[sec:linechiraldirac\] we will show that the presence of a 1D [*chiral Dirac fermion*]{} mode (analogous to an integer quantum Hall edge state) on the defect is associated with an integer topological invariant that may be interpreted as the winding number of the “$\theta$" term that characterizes the magnetoelectric polarizability[@qihugheszhang08]. This description unifies a number of methods for “engineering" chiral Dirac fermions, which will be described in several illustrative examples.
Related topological invariants and illustrative examples will be presented in Sections \[sec:linechiralmajorana\]-\[sec:lineclassc\] for line defects in other symmetry classes that are associated with gapless 1D helical Dirac fermions, 1D chiral Majorana fermions and 1D helical Majorana fermions. In section \[sec:pointdefect\] we will consider point defects in 1D models with chiral symmetry such as the Jackiw Rebbi model[@jackiwrebbi] or the Su, Schrieffer, Heeger model[@ssh], and in superconductors without chiral symmetry that exhibit Majorana bound states or Majorana doublets. These will also be related to the early work of Jackiw and Rossi[@jackiwrossi] on Majorana modes at point defects in a model with chiral symmetry.
Finally, in Section \[sec:pump\] we will regard ${\bf r}$ as including a temporal variable, and apply the considerations in this paper to classify cyclic pumping processes. The Thouless charge pump[@thouless; @thoulessniu] corresponds to a non trivial cycle in a system with no symmetries and $\delta=0$ ($d=D=1$). A similar pumping scenario can be applied to superconductors and defines a [*fermion parity pump*]{}. This, in turn, is related to the non-Abelian statistics of Ising anyons, and provides a framework for understanding braidless operations on systems of three dimensional superconductors hosting Majorana fermion bound states. Details of several technical calculations can be found in the Appendices.
An interesting recent preprint by Freedman et al.[@freedman10], which appeared when this manuscript was in its final stages discusses some aspects of the classification of topological defects in connection with a rigorous theory of non-Abelian statistics in higher dimensions.
Periodic Table for defect classification {#sec:periodictable}
========================================
Table \[tab:periodic\] shows the generalized periodic table for the classification of topological defects in insulators and superconductors. It describes the equivalence classes of Hamiltonians ${\cal H}({\bf k},{\bf r})$, that can be continuously deformed into one another without closing the energy gap, subject to constraints of particle-hole and/or time reversal symmetry. These are mappings from a [*base space*]{} defined by $({\bf
k},{\bf r})$ to a [*classifying space*]{}, which characterizes the set of gapped Hamiltonians. In order to explain the table, we need to describe (i) the symmetry classes, (ii) the base space, (iii) the classifying space and (iv) the notion of stable equivalence. The repeating patterns in the table will be discussed in Section \[sec:properties\]. Much of this section is a review of material in Refs. . What is new is the extension to $D>0$.
----- ------ ------------ --------- ---------- ---------------- ---------------- ---------------- ---------------- ---------------- ---------------- ---------------- ----------------
$s$ $\Theta^2$ $\Xi^2$ $ \Pi^2$ $0$ $1$ $2$ $3$ $4$ $5$ $6$ $7$
0 A $0$ $0$ $0$ $\mathbb{Z}$ $0$ $\mathbb{Z}$ $0$ $\mathbb{Z}$ $0$ $\mathbb{Z}$ $0$
1 AIII $0$ $0$ $1$ $0$ $\mathbb{Z}$ $0$ $\mathbb{Z}$ $0$ $\mathbb{Z}$ $0$ $\mathbb{Z}$
0 AI $1$ $0$ $0$ $\mathbb{Z}$ $0$ $0$ $0$ $2\mathbb{Z}$ $0$ $\mathbb{Z}_2$ $\mathbb{Z}_2$
1 BDI $1$ $1$ $1$ $\mathbb{Z}_2$ $\mathbb{Z}$ $0$ $0$ $0$ $2\mathbb{Z}$ $0$ $\mathbb{Z}_2$
2 D $0$ $1$ $0$ $\mathbb{Z}_2$ $\mathbb{Z}_2$ $\mathbb{Z}$ $0$ $0$ $0$ $2\mathbb{Z}$ $0$
3 DIII $-1$ $1$ $1$ $0$ $\mathbb{Z}_2$ $\mathbb{Z}_2$ $\mathbb{Z}$ $0$ $0$ $0$ $2\mathbb{Z}$
4 AII $-1$ $0$ $0$ $2\mathbb{Z}$ $0$ $\mathbb{Z}_2$ $\mathbb{Z}_2$ $\mathbb{Z}$ $0$ $0$ $0$
5 CII $-1$ $-1$ $1$ $0$ $2\mathbb{Z}$ $0$ $\mathbb{Z}_2$ $\mathbb{Z}_2$ $\mathbb{Z}$ $0$ $0$
6 C $0$ $-1$ $0$ $0$ $0$ $2\mathbb{Z}$ $0$ $\mathbb{Z}_2$ $\mathbb{Z}_2$ $\mathbb{Z}$ $0$
7 CI $1$ $-1$ $1$ $0$ $0$ $0$ $2\mathbb{Z}$ $0$ $\mathbb{Z}_2$ $\mathbb{Z}_2$ $\mathbb{Z}$
----- ------ ------------ --------- ---------- ---------------- ---------------- ---------------- ---------------- ---------------- ---------------- ---------------- ----------------
: Periodic table for the classification of topological defects in insulators and superconductors. The rows correspond to the different Altland Zirnbauer (AZ) symmetry classes, while the columns distinguish different dimensionalities, which depend only on $\delta = d-D$. []{data-label="tab:periodic"}
Symmetry Classes {#sec:symmetryclasses}
----------------
The presence or absence of time reversal symmetry, particle-hole symmetry and/or chiral symmetry define the 10 Altland Zirnbauer symmetry classes[@altland97]. Time reversal symmetry implies that $${\cal H}({\bf k},{\bf r}) = \Theta {\cal H}(-{\bf k},{\bf r})
\Theta^{-1},
\label{trsym}$$ where the anti unitary time reversal operator may be written $\Theta = e^{i\pi S^y/\hbar}K$. $S^y$ is the spin and $K$ is complex conjugation. For spin 1/2 fermions, $\Theta^2 =
-1$, which leads to Kramers theorem. In the absence of a spin orbit interaction, the extra invariance of the Hamiltonian under rotations in spin space allows an additional time reversal operator $\Theta' = K$ to be defined, which satisfies $\Theta'^2 = +1$.
Particle-hole symmetry is expressed by $${\cal H}({\bf k},{\bf r}) = -\Xi {\cal H}(-{\bf k},{\bf r})
\Xi^{-1},
\label{phsym}$$ where $\Xi$ is the anti unitary particle-hole operator. Fundamentally, $\Xi^2 = +1$. However, as was the case for $\Theta$, the absence of spin orbit interactions introduces an additional particle hole symmetry, which can satisfy $\Xi^2=-1$.
Finally, chiral symmetry is expressed by a unitary operator $\Pi$, satisfying $${\cal H}({\bf k},{\bf r}) = -\Pi {\cal H}({\bf k},{\bf r})
\Pi^{-1}.
\label{chiralsym}$$ A theory with both particle-hole and time reversal symmetry automatically has a chiral symmetry $\Pi = e^{i\chi} \Theta \Xi$. The phase $\chi$ can be chosen so that $\Pi^2=1$.
Specifying $\Theta^2 = 0, \pm 1$, $\Xi^2 = 0, \pm 1$ and $\Pi^2 = 0, 1$ (here $0$ denotes the absence of symmetry) defines the 10 Altland Zirnbauer symmetry classes. They can be divided into two groups: 8 [*real*]{} classes that have anti unitary symmetries $\Theta$ and or $\Xi$ plus 2 [*complex*]{} classes that do not have anti unitary symmetries. Altland and Zirnbauer’s notation for these classes, which is based on Cartan’s classification of symmetric spaces, is shown in the left hand part of Table \[tab:periodic\].
=2.0in
To appreciate the mathematical structure of the 8 real symmetry classes it is helpful to picture them on an 8 hour “clock", as shown in Fig. \[clock\]. The $x$ and $y$ axes of the clock represent the values of $\Xi^2$ and $\Theta^2$. The “time" on the clock can be represented by an integer $s$ defined modulo 8. Kitaev[@kitaev09] used a slightly different notation to label the symmetry classes. In his formulation, class D is described by a real Clifford algebra with no constraints, and in the other classes Clifford algebra elements are constrained to anticommute with $q$ positive generators. The two formulations are related by $s = q+2$ mod $8$. The complex symmetry classes can similarly be indexed by an integer $s$ defined modulo 2. For all classes, the presence of chiral symmetry is associated with odd $s$.
Base space, Classifying space and stable equivalence {#sec:basespace}
----------------------------------------------------
The Hamiltonian is defined on a base space composed of momentum ${\bf k}$, defined in a $d$ dimensional Brillouin zone $T^d$ and real space degrees of freedom ${\bf r}$ in a sphere $S^D$ (or $S^{D-1}\times S^1$ for an adiabatic cycle). The total base space is therefore $T^d \times S^D$ (or $T^d \times S^{D-1} \times S^1$). As in Ref. , we will simplify the topological classification by treating the base space as a sphere $S^{d+D}$. The “strong" topological invariants that characterize the sphere will also characterize $T^d \times S^D$. However, there may be additional topological structure in $T^d \times S^D$ that is absent in $S^{d+D}$. These correspond to “weak" topological invariants. For $D=0$ these arise in layered structures. A weak topological insulator, for example, can be understood as a layered two dimensional topological insulator. There are similar layered quantum Hall states. For $D\ne 0$, then there will also be a “weak" invariant if the Hamiltonian ${\cal H}({\bf k},{\bf r}_0)$ for fixed ${\bf r}={\bf r}_0$ is topologically non trivial. As is the case for the classification of bulk phases $D=0$, we expect that the topologically protected gapless defect modes are associated with the strong topological invariants.
The set of Hamiltonians that preserve the energy gap separating positive and negative energy states can be simplified without losing any topological information. Consider the retraction of the original Hamiltonian ${\cal H}({\bf k},{\bf r})$ to a simpler Hamiltonian whose eigenvalue spectrum is “flattened", so that the positive and negative energy states all have the same energy $\pm E_0$. The flattened Hamiltonian is then specified the set of all $n$ eigenvectors (defining a $U(n)$ matrix) modulo unitary rotations within the $k$ conduction bands or the $n-k$ valence bands. The flattened Hamiltonian can thus be identified with a point in the Grassmanian manifold, $$G_{n,k} = U(n)/U(k)\times U(n-k).
\label{grassmanian}$$
It is useful to broaden the notion of topological equivalence to allow for the presence of extra trivial energy bands. Two families of Hamiltonians are [*stably equivalent*]{} if they can be deformed into one another after adding an arbitrary number of trivial bands. Thus, trivial insulators with different numbers of core energy levels are stably equivalent. Stable equivalence can be implemented by considering an expanded [*classifying space*]{} that includes an infinite number of extra conduction and valence bands, ${\cal C}_0 = U / U \times U \equiv \bigcup_{k=0}^\infty
G_{\infty,k}$.
With this notion of stable equivalence, the equivalence classes of Hamiltonians ${\cal H}({\bf k},{\bf r})$ can be formally added and subtracted. The addition of two classes, denoted $[{\cal H}_1] + [{\cal
H}_2]$ is formed by simply combining two independent Hamiltonians into a single Hamiltonian given by the matrix direct sum, $[{\cal H}_1
\oplus {\cal H}_2]$. Additive inverses are constructed through reversing conduction and valence bands, $[{\cal H}_1]-[{\cal H}_2] = [{\cal H}_1 \oplus -{\cal H}_2]$. $[{\cal H} \oplus -{\cal H}]$ is guaranteed to the be trivial class $[0]$. Because of this property, the stable equivalence classes form an Abelian group, which is the key element of K theory[@karoubi; @lawson; @atiyah94].
Symmetries impose constraints on the classifying space. For the symmetry classes with chiral symmetry, restricts $n=2k$ and the classifying space to a subset ${\cal C}_1 = U(\infty) \subset U/U \times U$. The anti unitary symmetries (\[trsym\],\[phsym\]) impose further constraints. At the special points where ${\bf k}$ and $-{\bf k}$ coincide, the allowed Hamiltonians are described by the 8 classifying spaces ${\cal R}_q$ of Real K theory.
Properties of the periodic table {#sec:properties}
--------------------------------
For a given symmetry class $s$, the topological classification of defects is given by the set of stable equivalence classes of maps from the base space $({\bf k},{\bf r}) \in S^{D+d}$ to the classifying space, subject to the symmetry constraints. These form the K group, which we denote as $K_{\mathbb{C}}(s;D,d)$ for the complex symmetry classes and $K_{\mathbb{R}}(s;D,d)$ for the real symmetry classes. These are listed in Table \[tab:periodic\].
Table \[tab:periodic\] exhibits many remarkable patterns. Many can be understood from the following basic periodicities, $$\begin{aligned}
K_{\mathbb{F}}(s;D,d + 1) &=& K_{\mathbb{F}}(s- 1;D,d) \label{period1},\\
K_{\mathbb{F}}(s;D + 1, d) &=& K_{\mathbb{F}}(s+ 1;D,d)\label{period2}.\end{aligned}$$ Here $s$ is understood to be defined modulo 2 for $\mathbb{F} = \mathbb{C}$ and modulo 8 for $\mathbb{F} = \mathbb{R}$. We will establish these identities mathematically in Appendix \[appendix:periodicities\]. The basic idea is to start with some Hamiltonian in some symmetry class $s$ and dimensionalities $D$ and $d$. It is then possible to explicitly construct two new Hamiltonians in one higher dimension which have either (i) $d \rightarrow d+1$ or (ii) $D \rightarrow D+1$. These new Hamiltonians belongs to new symmetry classes that are shifted by one “hour" on the symmetry clock and characterized by (i) $s \rightarrow
s+1$ or (ii) $s \rightarrow s-1$. We then go on to show that this construction defines a 1-1 correspondence between the equivalence classes of Hamiltonians with the new and old symmetry classes and dimensions, thereby establishing and .
The periodicities and have a number of consequences. The most important for our present purposes is they can be combined to give $$K_\mathbb{F}(s;D+1,d+1) = K_\mathbb{F}(s;D,d).$$ This $(1,1)$ periodicity shows that the dependence on the dimensions $d$ and $D$ only occurs via $\delta =
d-D$. Thus the dependence of the classifications on $D$ can be deduced from the table for $D=0$. This is one of our central results.
In addition, the periodicities (\[period1\],\[period2\]) explain other features of the table that are already present for $D=0$. In particular, the fact that $s$ is defined modulo 2 (8) for the complex (real) classes leads directly to the Bott periodicity of the dependence of the classifications on $d$: $$\begin{aligned}
K_{\mathbb{C}}(s;D,d+2) &=& K_{\mathbb{C}}(s;D,d), \\
K_{\mathbb{R}}(s;D,d+8) &=& K_{\mathbb{R}}(s;D,d).\end{aligned}$$ Moreover, (\[period1\],\[period2\]) shows that $K_a(s;D,d)$ depends only on $d-D-s$. This explains the diagonal pattern in Table \[tab:periodic\], in which the dependence of the classification on $d$ is repeated in successive symmetry classes. Thus, the entire table could be deduced from a single row.
Equations and do not explain the pattern of classifications within a single row. Since this is a well studied math problem there are many routes to the answer[@bott; @JMilnor; @lawson]. One approach is to notice that for $d=0$, $K_\mathbb{F}(s,D,0)$ is simply the $D$’th homotopy group of the appropriate classifying space which incorporates the symmetry constraints. For example, for class BDI ($s=1$, $\Xi^2 = +1$, $\Theta^2=+1$) the classifying space is the orthogonal group $O(\infty)$. Then, $K_{\mathbb{R}}(1,D,0) = \pi_D(O(\infty))$, which are well known. This implies $$K_{\mathbb{R}}(s;D,d) =\pi_{s+D-d-1}(O(\infty)).$$
Additional insight can be obtained by examining the interconnections between different elements of the table. For example, the structure within a column can be analyzed by considering the effect of “forgetting" symmetries. Hamiltonians belonging to the real chiral (non chiral) classes are automatically in complex class AIII (A). There are therefore K group homomorphisms that send any real entries in table \[tab:periodic\] to complex ones directly above. In particular, as detailed in Appendix \[appendix:representatives\] this distinguishes the $\mathbb{Z}$ and $2\mathbb{Z}$ entries, which indicate the possible values of Chern numbers (or $U(n)$ winding numbers) for even (or odd) $\delta$. In addition, the dimensional reduction arguments given in Refs. lead to a dimensional hierarchy, which helps to explain the pattern within a single row as a function of $d$.
Line defects {#sec:linedefects}
============
Line defects can occur at the edge of a 2D system ($\delta = 2-0$) or in a 3D system ($\delta = 3-1$). From Table \[tab:periodic\], it can be seen that there are five symmetry classes which can host non trivial line defects. These are expected to be associated with gapless fermion modes bound to the defect. Table \[linetab\] lists non trivial classes, along with the character of the associated gapless modes. In the following subsections we will discuss each of these cases, along with physical examples.
Symmetry Topological classes 1D Gapless Fermion modes
---------- --------------------- --------------------------
A $\mathbb{Z}$ Chiral Dirac
D $\mathbb{Z}$ Chiral Majorana
DIII $\mathbb{Z}_2$ Helical Majorana
AII $\mathbb{Z}_2$ Helical Dirac
C $2\mathbb{Z}$ Chiral Dirac
: Symmetry classes that support topologically non trivial line defects and their associated protected gapless modes. []{data-label="linetab"}
Class A: Chiral Dirac Fermion {#sec:linechiraldirac}
-----------------------------
### Topological Invariant {#sec:linechiraldiracinvariant}
A line defect in a generic 3D Bloch band theory with no symmetries is associated with an integer topological invariant. This determines the number of chiral Dirac fermion modes associated with the defect. Since ${\cal H}({\bf k},{\bf r})$ is defined on a compact 4 dimensional space, this invariant is naturally expressed as a second Chern number, $$n = {1\over {8\pi^2}} \int_{T^3\times S^1} {\rm Tr}[{\cal F}\wedge{\cal
F}],
\label{2ndchern}$$ where $${\cal F} = d{\cal A} + {\cal A} \wedge {\cal A}
\label{F(A)}$$ is the curvature form associated with the non-Abelian Berry’s connection ${\cal A}_{ij} = \langle u_i|d u_j\rangle$ characterizing the valence band eigenstates $|u_j({\bf k},s)\rangle$ defined on the loop $S^1$ parameterized by $s$.
It is instructive to rewrite this as an integral over $s$ of a quantity associated with the local band structure. To this end, it is useful to write ${\rm Tr}[{\cal F}\wedge{\cal F}] = d
{\cal Q}_3$, where the Chern Simons 3 form is, $${\cal Q}_3 = {\rm Tr}[{\cal A}\wedge d{\cal A} + {2\over 3} {\cal A}\wedge{\cal
A}\wedge{\cal A}].
\label{q3}$$ Now divide the integration volume into thin slices, $T^3 \times \Delta
S^1$, where $\Delta S^1$ is the interval between $s$ and $s+\Delta s$. In each slice, Stokes’ theorem may be used to write the integral as a surface integral over the surfaces of the slice at $s$ and $s+\Delta s$. In this manner, Eq. \[2ndchern\] may be written $$n = {1\over {2\pi}} \oint_{S^1} ds {d\over {ds}}\theta(s),
\label{n(theta)}$$ where $$\theta(s) = {1\over {4\pi}} \int_{T^3} {\cal Q}_3({\bf k},s).
\label{theta(s)}$$ Eq. \[theta(s)\] is precisely the Qi, Hughes, Zhang formula[@qihugheszhang08] for the “$\theta$" term that characterizes the magnetoelectric response of a band insulator. $\theta = 0$ for an ordinary time reversal invariant insulator, and $\theta = \pi$ in a strong topological insulator. If parity and time reversal symmetry are broken then $\theta$ can have any intermediate value. We thus conclude that the topological invariant associated with a line defect, which determines the number of chiral fermion branches is given by the winding number of $\theta$.
We now consider several examples of 3D line defects that are associated with chiral Dirac fermions.
### Dislocation in a 3D Integer quantum Hall state {#sec:qhedislocation}
A three dimensional integer quantum Hall state can be thought of as a layered version of the two dimensional integer quantum Hall state. This can be understood most simply by considering the extreme limit where the layers are completely decoupled 2D systems. A line dislocation, as shown in Fig. \[dislocation\] will then involve an edge of one of the planes and be associated with a chiral fermion edge state. Clearly, the chiral fermion mode will remain when the layers are coupled, provided the bulk gap remains finite. Here we wish to show how the topological invariant reflects this fact.
=2.5in
On a loop surrounding the dislocation parameterized by $s\in[0,1]$ we may consider a family of Hamiltonians $H({\bf k},s)$ given by the Hamiltonian of the original bulk crystal displaced by a distance $s{\bf B}$, where ${\bf
B}$ is a lattice vector equal to the Burgers vector of the defect. The corresponding Bloch wavefunctions will thus be given by, $$u_{m{\bf k},s}({\bf r}) = u^0_{m\bf k}({\bf r}- s {\bf B}),
\label{dislocationu}$$ where $u^0_{m\bf k}({\bf r})$ are Bloch functions for the original crystal. It then follows that the Berry’s connection is $${\cal A} = {\cal A}^0 + {\bf B}\cdot ({\bf k} - {\bf a}^p({\bf k})) ds,
\label{dislocationa}$$ where ${\cal A}^0_{mn}({\bf k}) =
\langle u^0_{m\bf k}|\nabla_{\bf k} |u^0_{n\bf k}\rangle\cdot d{\bf k}$ and $${\bf a}^p_{mn}({\bf k}) = \langle u^0_{m\bf k}| (\nabla_{\bf r} + {\bf k} ) |u^0_{n\bf
k}\rangle.$$ With this definition, ${\bf a}^p({\bf k})$ is a periodic function: ${\bf
a}^p({\bf k}+{\bf G}) = {\bf a}^p({\bf k})$ for any reciprocal lattice vector ${\bf G}$[@blount].
If the crystal is in a three dimensional quantum Hall state, then the non zero first Chern number is an obstruction to finding the globally continuous gauge necessary to evaluate . We therefore use , which can be evaluated by noting that $${\rm Tr}[{\cal F}\wedge{\cal F}] =
{\rm Tr}\left[{\bf B}\cdot\left(2 {\cal F}^0 \wedge d{\bf k} -
d [{\cal F}^0,{\bf a}^p]\right) \wedge ds \right].
\label{dislocationff}$$ Upon integrating ${\rm Tr}[{\cal F}\wedge{\cal F}]$ the total derivative term vanishes due to the periodicity of ${\bf a}^p$. Evaluating the integral is then straightforward. The integral over $s$ trivially gives $1$. We are then left with $$n = {1\over {2\pi}} {\bf B} \cdot {\bf G}_c,$$ where $${\bf G}_c = {1\over {2\pi}}\int_{T^3} d{\bf k} \wedge {\rm Tr}[{\cal F}^0].$$ ${\bf G}_c$ is a reciprocal lattice vector that corresponds to the triad of Chern numbers that characterize a 3D system. For instance, in a cubic system ${\bf G}_c = (2\pi/a)(n_x,n_y,n_z)$, where, for example $n_z = (2\pi)^{-1}\int
{\rm Tr}[{\cal F}^0_{xy}] dk_x\wedge dk_y$, for any value of $k_z$.
An equivalent formulation is to characterize the displaced crystal in terms of $\theta$. Though can not be used, and can be used to implicitly define $\theta$ up to an arbitrary additive constant, $$\theta(s)= s {\bf B}\cdot{\bf G}_c.$$
### Topological insulator heterostructures {#sec:linechiraldirachetero}
Another method for engineering chiral Dirac fermions is use heterostructures that combine topological insulators and magnetic materials. The simplest version is a topological insulator coated with a magnetic film that opens a time reversal symmetry breaking energy gap at the surface. A domain wall is then associated with a chiral fermion mode. In this section we will show how this structure, along with some variants on the theme, fits into our general framework. We first describe the structures qualitatively, and then analyze a model that describes them.
Fig. \[chiraldevice\] shows four possible configurations. Figs. \[chiraldevice\](a,b) involve a topological insulator with magnetic materials on the surface. The magnetic material could be either ferromagnetic or antiferromagnetic. We distinguish these two cases based on whether inversion symmetry is broken or not. Ferromagnetism does not violate inversion symmetry, while antiferromagnetism does (at least for inversion about the middle of a bond). This is relevant because $\theta$ , discussed above, is quantized unless [*both*]{} time reversal and inversion symmetries are violated. Of course, for a non centrosymmetric crystal inversion is already broken, so the distinction is unnecessary.
Fig. \[chiraldevice\](a) shows a topological insulator capped with antiferromagnetic insulators with $\theta = \pm \epsilon$ separated by a domain wall. Around the junction where the three regions meet $\theta$ cycles between $\pi$, $+\epsilon$ and $-\epsilon$. Of course this interface structure falls outside the adiabatic regime that is based on. However, it is natural to expect that the physics would not change if the interface was “smoothed out" with $\theta$ taking the shortest smooth path connecting its values on either side of the interface.
Fig. \[chiraldevice\](b) shows a similar device with ferromagnetic insulators, for which $\theta = 0$ or $\pi$. In this case the adiabatic assumption again breaks down, however, as emphasized in Ref. , the appropriate way to think about the surface is that $\theta$ connects $0$ and $\pi$ along a path that is determined by the sign of the induced gap, which in turn is related to the magnetization. In this sense, $\theta$ cycles by $2\pi$ around the junction.
In Figs. \[chiraldevice\](c,d) we consider topological insulators which have a weak magnetic instability. If in addition to time reversal, inversion symmetry is broken, then $\theta \sim \pi \pm \epsilon$. Recently Li, Wang, Qi and Zhang[@li10] have considered such materials in connection with a theory of a dynamical axion and suggested that certain magnetically doped topological insulators may exhibit this behavior. They referred to such materials as topological magnetic insulators. We prefer to call them [*magnetic topological insulators*]{} because as magnetic insulators they are topologically trivial. Rather, they are topological insulators to which magnetism is added. Irrespective of the name, such materials would be extremely interesting to study, and as we discuss below, may have important technological utility.
Fig. \[chiraldevice\](c) shows two antiferromagnetic topological insulators with $\theta = \pi \pm \epsilon$ separated by a domain wall, and Fig. \[chiraldevice\](d) shows a similar device with ferromagnetic topological insulators. They form an interface with an insulator, which could be vacuum. Under the same continuity assumptions as above the junction where the domain wall meets the surface will be associated with a chiral fermion mode. Like the structure in Fig. \[chiraldevice\](a), this may be interpreted as an edge state on a domain wall between the “half quantized" quantum Hall states of the topological insulator surfaces. However, an equally valid interpretation is that the domain wall itself forms a single two dimensional integer quantum Hall state with an edge state. Our framework for topologically classifying the line defects underlies the equivalence between these two points of view.
Mong, Essen and Moore [@mong10] have introduced a [*different*]{} kind of antferromagnetic topological insulator that relies on the symmetry of time reversal combined with a lattice translation. Due to the necessity of translation symmetry, however, such a phase is not robust to disorder. They found that chiral Dirac modes occur at certain step edges in such crystals. These chiral modes can also be understood in terms of the invariant . Note that these chiral modes survives in the presence of disorder even though the bulk state does not. Thus, the chiral mode, protected by the strong invariant , is more robust than the bulk state that gave rise to it.
If one imagines weakening the coupling between the two antiferromagnetic topological insulators (using our terminology, not that of Mong, et al.[@mong10]) and taking them apart, then at some point the chiral mode has to disappear. At that point, rather than taking the “shortest path" between $\pi \pm \epsilon$, $\theta$ takes a path that passes through $0$. At the transition between the “short path" and the “long path" regimes, the gap on the domain wall must go to zero, allowing the chiral mode to escape. This will have the character of a plateau transition in the 2D integer quantum Hall effect.
Structures involving magnetic topological insulators would be extremely interesting to study because with them it is possible to create chiral fermion states with a single material. Indeed, one can imagine scenarios where a magnetic memory, encoded in magnetic domains, could be read by measuring the electrical transport in the domain wall chiral fermions.
To model the chiral fermions in these structures we begin with the simple three dimensional model for trivial and topological insulators considered in Ref. , $${\cal H}_0 = v \mu_x \vec\sigma \cdot {\bf k} + (m+ \epsilon |{\bf k}|^2)
\mu_z.
\label{h0chiral}$$ Here $\vec\sigma$ represents spin, and $\mu_z$ describes an orbital degree of freedom. $m>0$ describes the trivial insulator and $m<0$ describes the topological insulator. An interface where $m$ changes sign is then associated with gapless surface states.
Next consider time reversal symmetry breaking perturbations, which could arise from exchange fields due to the presence of magnetic order. Two possibilities include $$\begin{aligned}
{\cal H}_{af} = h_{af} \mu_y \label{haf},\\
{\cal H}_{f} = \vec h_f \cdot \vec\sigma \label{hf}.\end{aligned}$$ Either $h_{af}$ or $h_{f,z}$ will introduce a gap in the surface states, but they have different physical content. ${\cal H}_0$ has an inversion symmetry given by ${\cal H}_0({\bf k}) = P {\cal H}_0(-{\bf k}) P$ with $P = \mu_z$. Clearly, ${\cal H}_{f}$ respects this inversion symmetry. ${\cal H}_{af}$ does not respect $P$, but does respect $P\Theta$. We therefore associate ${\cal H}_f$ with ferromagnetic order and ${\cal H}_{af}$ with antiferromagnetic order.
Within the adiabatic approximation, the topological invariant can be evaluated in the presence of either or . The antiferromagnetic perturbation is most straightforward to analyze because ${\cal H}_0 + H_{af}$ is a combination of 5 anticommuting Dirac matrices. On a circle surrounding the junction parameterized by $s$ it can be written in the general form $$H({\bf k},s) = {\bf h}({\bf k},s) \cdot \vec\gamma,
\label{hgamma}$$ where $\vec\gamma = (\mu_x\sigma_x,\mu_x\sigma_y,\mu_x\sigma_z, \mu_z,\mu_y)$ and ${\bf h}({\bf k},s) = (v {\bf k},m(s)+ \epsilon|{\bf k}|^2, h_{af}(s))$. For a model of this form, the second Chern number is given simply by the winding number of the unit vector $\hat{\bf d}({\bf k},s) = {\bf
h}/|{\bf h}|\in S^4$ as a function of ${\bf k}$ and $s$. This is most straightforward to evaluate in the limit $\epsilon\rightarrow 0$, where $\hat{\bf
d}$ is confined to the “equator" $(d_1,d_2,d_3,0,0)$ everywhere except near ${\bf k}\sim 0$ and $|{\bf k}|\gtrsim 1/\epsilon$. The winding number is determined by the behavior at ${\bf k} \sim 0$, and may be expressed by with $\theta$ given by $$e^{i\theta} = \frac{m + ih_{af}}{\sqrt{m^2 + h_{af}^2}}.$$
We therefore expect a topological line defect to occur at an intersection between planes where $m$ and $h_{af}$ change sign. The chiral fermion mode associated with this defect can seen explicitly if we solve a simple linear model, $m = f_z z$, $h_{af} = f_y y$. This model, which has the form of a harmonic oscillator, is solved in Appendix \[appendix:ZM\], and explicitly gives the chiral Dirac fermion mode with dispersion $$E(k_x) = v {\rm sgn}(f_z f_y) k_x.
\label{E(kx)}$$
Class D: chiral Majorana fermions {#sec:linechiralmajorana}
---------------------------------
### Topological invariant {#sec:linechiralmajoranainvariant}
A line defect in a superconductor without time reversal symmetry is characterized by an integer topological invariant that determines the number of associated chiral Majorana fermion modes. Since the BdG Hamiltonian characterizing a superconductor has the same structure as the Bloch Hamiltonian, we can analyze the problem by “forgetting" about the particle hole symmetry and treating the BdG Hamiltonian as if it was a Bloch Hamiltonian. The second Chern number, given by can be defined. It can be verified that any value of the Chern number is even under particle-hole symmetry, so that particle-hole symmetry does not rule out a non zero Chern number. We may follow the same steps as (\[2ndchern\]-\[theta(s)\]) to express the integer topological invariant as $$\tilde n = {1\over{8\pi^2}} \int_{T^3\times S^1} {\rm Tr}[\tilde{\cal F} \wedge
\tilde{\cal F}],
\label{tilden}$$ where $\tilde{\cal F}$ is the curvature form characterizing the BdG theory. As in , $\tilde n$ may be expressed as a winding number of $\tilde\theta$, which is expressed as an integral over the Brillouin zone of the Chern Simons 3 form. The difference between $n$ and $\tilde n$ is that $\tilde n$ characterizes a BdG Hamiltonian. If we considered the BdG Hamiltonian for a non superconducting insulator, then due to the doubling in the BdG equation, we would find $$\tilde n = 2 n.$$ In this case, the chiral Dirac fermion that occurs for a $2\pi$ ($n=1$) winding of $\theta$ corresponds to a $4\pi$ ($n=2$) winding of $\tilde\theta$. Superconductivity allows for the possibility of a $2\pi$ winding in $\tilde\theta$: a chiral Dirac fermion can be split into a pair of chiral Majorana fermions.
### Dislocation in a layered topological superconductor {#sec:linechiralmajoranalayered}
The simplest example to consider is a dislocation in a three dimensional superconductor. The discussion closely parallels Section \[sec:qhedislocation\], and we find $$\tilde n = {1\over{2\pi}} {\bf B} \cdot \tilde {\bf G}_c,$$ where ${\bf B}$ is the Burgers vector of the dislocation and $\tilde{\bf G}_c$ characterizes the triad of first Chern numbers characterizing the 3D BdG Hamiltonian. A 3D system consisting of layers of a 2D topological superconductor will be characterized by a non zero $\tilde{\bf G}_c$. Since, as a 3D superconductor, the layered structure is in the topologically trivial class, such a state could be referred to as a [*weak topological superconductor*]{}.
The simplest model system in this class is a stack of 2D $p_x+ip_y$ superconductors. A dislocation would then have $\tilde n = 1$ and a single chiral Majorana fermion branch. A possible physical realization of the weak topological superconductor state is Sr$_2$RuO$_4$, which may exhibit triplet $p_x+ip_y$ pairing. Since the spin up and spin down electrons make two copies of the spinless state, a dislocation will be associated with $\tilde n = 2$. Thus, we predict that there will be two chiral Majorana modes bound to the dislocation, which is the same as a single chiral Dirac fermion mode.
### Superconductor Heterostructures {#sec:linechiralmajoranahetero}
We now consider heterostructures with associated chiral Majorana modes. The simplest to consider is a BdG analog of the structures considered in Fig. \[mchiraldevice\]. These would involve, for example an interface between a 3D time reversal invariant topological superconductor with a magnetic material with a magnetic domain wall. The analysis of such a structure is similar to that in Eq. \[h0chiral\] if we replace the Pauli matrices describing the orbital degree of freedom $\vec \mu$ with Pauli matrices describing Nambu space $\vec\tau$. Protected chiral Majorana fermion modes of this sort on the surface of $^3$He-B with a magnetic domain wall have been recently discussed by Volovik[@volovik10].
In Ref. a different method for engineering chiral Majorana fermions was introduced by combining an interface between superconducting and magnetic regions on the surface of a topological insulator. To describe this requires the 8 band model introduced in Ref. , $$H = \tau_z \mu_x \vec\sigma\cdot {\bf k} + (m + \epsilon |{\bf k}|^2) \tau_z\mu_z + \Delta
\tau_x + h \mu_y.
\label{h0d}$$ (Here, for simplicity we consider only the antiferromagnetic term). The surface of the topological insulator occurs at a domain wall (say, in the x-y plane) where $m(z)$ changes sign. The superconducting order parameter $\Delta$ and magnetic perturbation $h$ both lead to an energy gap in the surface states. This Hamiltonian is straightforward to analyze because $[{\cal
H},\tau_x\mu_y]=0$, which allows the $8\times 8$ problem to be divided into two $4\times 4$ problems, which have superconducting/magnetic mass terms $\Delta
\pm h$. Near a defect where $\Delta = h$ the $\Delta + h$ gap never closes, while the $\Delta - h$ gap can be critical. $\Delta>h$ leads to a superconducting state, while $\Delta<h$ leads to a quantum Hall like state. There is a transition between the two at $\Delta =
h$.
An explicit model for the line defect can be formulated with $m(z) = f_z z$, $\Delta - h = f_y y$ and $\Delta + h = M$. The topological invariant can be evaluated using a method similar to , and the chiral Majorana states can be explicitly solved along the lines of .
Class AII: Helical Dirac fermions {#sec:linehelicaldirac}
---------------------------------
### Topological Invariant {#sec:linehelicaldiracinvariant}
Line defects for class AII are characterized by a $\mathbb{Z}_2$ topological invariant. To develop a formula for this invariant we follow the approach used in Ref. to describe the invariant characterizing the quantum spin Hall insulator.
As in the previous section, a line defect in three dimensions is associated with a four parameter space $({\bf k},{\bf r})\in T^3\times S^1$. Due to time reversal symmetry, the second Chern number that characterized the line defects in must be zero. Thus there is no obstruction to defining Bloch basis functions $|u({\bf k},{\bf r})\rangle$ continuously over the entire base space. However, the time reversal relation between $(-{\bf k},{\bf r})$ and $({\bf k},{\bf r})$ allows for an additional constraint, so that the state is specified by the degrees of freedom in [*half*]{} the Brillouin zone.
As in Ref. it is useful to define a matrix $$w_{mn}({\bf k},{\bf r}) = \langle u_m({\bf k},{\bf r}) | \Theta |u_n(-{\bf
k},{\bf r})\rangle.
\label{wmn(k,r)}$$ Because $|u_m({\bf k},{\bf r})\rangle$ and $|u_n(-{\bf k},{\bf r})\rangle$ are related by time reversal symmetry $w({\bf k},{\bf r})$ is a unitary matrix, that depends on the gauge choice for the basis functions. Locally it is possible to choose a basis in which $$w({\bf k},{\bf r}) = w_0,
\label{trconstraint}$$ where $w_0$ is independent of ${\bf k}$ and ${\bf r}$, so that states at $(\pm {\bf k},{\bf r})$ have a fixed relation. Since for ${\bf k}=0$ $w = -w^T$, $w_0$ must be antisymmetric. A natural choice is thus $w_0 = i \sigma_2 \otimes 1$.
The $\mathbb{Z}_2$ topological invariant is an obstruction to finding such a constrained basis globally. The constrained basis can be defined on two patches, but the basis functions on the two patches are necessarily related by a topologically non trivial transition function. In this sense, the $\mathbb{Z}_2$ invariant resembles the second Chern number in .
In Appendix \[appendix:defectsaiidiii\] we will generalize the argument developed in Ref. to show that the transition function relating the two patches defines the $\mathbb{Z}_2$ topological invariant, which may be written[@fukui09] $$\nu = {1\over{8\pi^2}}\left(\int_{{1\over 2}T^3 \times S^1} {\rm Tr}[{\cal F}\wedge{\cal
F}]- \int_{\partial{1\over 2}T^3 \times S^1} {\cal Q}_3 \right) \ {\rm mod} \
2,
\label{fukane}$$ where ${\cal F}$ and ${\cal Q}_3$ are expressed in terms of the Berry’s connection ${\cal A}$ using and . The integral is over half of the base space $(1/2)(T^3\times S^1)$, defined such that $({\bf k},{\bf r})$ and $(-{\bf k},{\bf r})$ are never both included. The second term is over the boundary of $(1/2)(T^3\times S^1)$, which is closed under $({\bf k},{\bf r})
\rightarrow (-{\bf k},{\bf r})$. Eq. \[fukane\] must be used with care because the Chern Simons form in the second term depends on the gauge. A different continuous gauge can give a different $\nu$, but due to , they must be related by an even integer. Thus, an odd number is distinct.
In addition to satisfying , it is essential to use a gauge in which at least ${\cal Q}_3$ is continuous on $\partial{1\over 2}T^3 \times S^1$ (though not necessarily on all of ${1\over 2}T^3\times S^1$). This continuous gauge can always be found if the base space is a sphere $S^4$. However for $T^3\times S^1$, the “weak" topological invariants can pose an obstruction to finding a continuous gauge. We will show how to work around this difficulty at the end of the following section.
### Dislocation in a weak topological insulator {#sec:linehelicaldiracdislocation}
Ran, Zhang and Vishwanath recently studied the problem of a line dislocation in a topological insulator[@ran09]. They found that an insulator with non trivial weak topological invariants can exhibit topologically protected helical modes at an appropriate line dislocation. In this section we will show that these protected modes are associated with a non trivial $\mathbb{Z}_2$ invariant in . In addition to providing an explicit example for this invariant, this formulation provides additional insight into why protected modes can exist in a weak topological insulator. As argued in Ref. , the weak topological invariants lose their meaning in the presence of disorder. The present considerations show that the helical modes associated with the dislocation are protected by the [*strong*]{} topological invariant associated with the line defect. Thus if we start with a perfect crystal and add disorder, then the helical modes remain, even though the crystal is no longer a weak topological insulator. The helical modes remain even if the disorder destroys the crystaline order, so that dislocations become ill defined, [*provided*]{} the mobility gap remains finite in the bulk crystal. In this case, the Hamiltonian has a non trivial winding around the line defect, even though the defect has no obvious structural origin. Thus, the weak topological insulator provides a [*route*]{} to realizing the topologically protected line defect. But once present, the line defect is more robust than the weak topological insulator.
To evaluate the $\mathbb{Z}_2$ invariant for a line dislocation we repeat the analysis in section \[sec:qhedislocation\]. Because of the subtlety with the application of we will first consider the simplest case of a dislocation in a weak topological insulator. Afterwards we will discuss the case of a crystal with both weak and strong invariants.
The Bloch functions on a circle surrounding a dislocation are described by , and the evaluation of ${\rm Tr}[{\cal F}\wedge {\cal F}]$ proceeds exactly as in (\[dislocationa\]-\[dislocationff\]). To evaluate the second term in we need the Chern Simons 3 form. One approach is to use and . However, this is not continuously defined on $\partial (1/2)(T^3\times S^1)$ because ${\cal A}$ has a term ${\bf B}\cdot {\bf k} ds$ that is discontinuous at the Brillouin zone boundary. An alternative is to write $${\cal Q}_3 ={\rm Tr}[ {\bf B}\cdot \left( 2 {\cal A}^0\wedge d{\bf k}
- [{\cal F}^0,{\bf a}^p]\right)\wedge ds].
\label{newq3}$$ From this clearly satisfies ${\rm Tr}[{\cal F}\wedge{\cal F}] = d {\cal Q}_3$, and it is defined continuously on $\partial (1/2)(T^3\times S^1)$ as long as ${\cal
A}^0$ is continuously defined on $\partial (1/2)T^3$. For a weak topological insulator this is always possible, provided $(1/2)T^3$ is defined appropriately. Eq. \[newq3\] differs from Eq. \[q3\] by a total derivative.
Combining , and , the terms involving ${\bf a}^p$ cancel because ${\bf
a}^p$ is globally defined. (Note that ${\bf a}^p$ is unchanged by a ${\bf k}$ dependent – but ${\bf r}$ independent – gauge transformation). This can not be said of the term involving ${\cal A}^0$, however, because in a weak topological insulator ${\cal A}^0$ is [*not*]{} globally defined on $(1/2)T^3$. Performing the trivial integral over $s$ we then find $$\nu = {1\over {2\pi}} {\bf B} \cdot {\bf G}_\nu \ {\rm mod} \ 2,
\label{bdotgnu}$$ where $${\bf G}_\nu = \int_{{1\over 2}T^3} {\rm Tr}[{\cal F}^0]\wedge d{\bf k}
- \int_{\partial{1\over 2}T^3} {\rm Tr}[{\cal A}^0]\wedge d{\bf k}.
\label{gnu}$$
The simplest case to consider is a weak topological insulator consisting of decoupled layers of 2D quantum spin Hall insulator stacked with a lattice constant $a$ in the $z$ direction. In this case ${\cal F}^0 = {\cal F}^0(k_x,k_y)$ is independent of $k_z$, so the $k_z$ integral can be performed trivially. This leads to ${\bf G}_\nu = (2\pi/a)\nu \hat {\bf z}$, where $$\nu = {i\over{2\pi}}\left[\int_{{1\over 2}T^2} {\rm Tr}[{\cal F}^0] -
\int_{\partial{1\over 2}T^2} {\rm Tr}[{\cal A}^0]\right]
\label{nu2D}$$ is the 2D $\mathbb{Z}_2$ topological invariant characterizing the individual layers.
Eq. \[gnu\] also applies to a more general 3D weak topological insulator. A weak topological insulator is characterized by a triad of $\mathbb{Z}_2$ invariants $(\nu_1\nu_2\nu_3)$ that define a mod 2 reciprocal lattice vector[@fkm07; @fukane07], $${\bf G}_\nu = \nu_1 {\bf b}_1+\nu_2{\bf b}_2 + \nu_3 {\bf b}_3,
\label{gnutopo}$$ where ${\bf b}_i$ are primitive reciprocal lattice vectors corresponding to primitive lattice vectors ${\bf a}_i$ (such that ${\bf a}_i\cdot{\bf b}_j =
2\pi \delta_{ij}$). The indices $\nu_i$ can be determined by evaluating the 2D invariant on the time reversal invariant plane ${\bf k} \cdot {\bf a}_i = \pi$.
To show that ${\bf G}_\nu$ in and are equivalent, consider ${\bf G}_\nu \cdot {\bf a}_1$ in . If we write ${\bf k} = x_1 {\bf b}_1 + x_2 {\bf b}_2 + x_3{\bf
b}_3$, then the integrals over $x_2$ and $x_3$ have the form of . Since this is quantized, it must be independent of $x_1$, and will be given by its value at $x_1 = 1/2$. This then gives ${\bf G}_\nu \cdot {\bf a}_1 = 2\pi
\nu_1$. A similar analysis of the other components establishes the equivalence. A non trivial value of Eq. \[bdotgnu\] is the same as the criterion for the existence of protected helical modes on a dislocation Ran, Zhang, Vishwanath[@ran09] derived using a different method.
Evaluating in a crystal that is [*both*]{} a strong topological insulator and a weak topological insulator (such as Bi$_{1-x}$Sb$_x$) is problematic because the 2D invariants evaluated on the planes $x_1=0$ and $x_1=1/2$ are necessarily [*different*]{} in a strong topological insulator. This arises because a non trivial strong topological invariant $\nu_0$ is an obstruction to continuously defining ${\cal A}^0$ on $\partial (1/2)T^3$, so can not be evaluated continuously between $x_1=0$ and $x_1 = 1/2$. From the point of view of the topological classification of the [*defect*]{} on $T^3 \times S^1$, $\nu_0$ is like a [*weak*]{} topological invariant because it a property of $T^3$ and is independent of the real space parameter $s$ in $S^1$. Thus this complication is a manifestation of the fact that topological classification of Hamiltonians on $T^3\times S^1$ has more structure than those on $S^4$. The problem is not with the existence of the invariant $\nu$ on $T^3 \times
S^1$, but rather with applying the formulas (\[fukane\],\[gnu\]). The problem can be circumvented with the following trick.
Consider an auxiliary Hamiltonian $\tilde {\cal H}({\bf k},{\bf r}) =
{\cal H}({\bf k},{\bf r}) \oplus {\cal H}_{STI}({\bf k})$, where ${\cal
H}_{STI}$ is a simple model Hamiltonian for a strong topological insulator like Eq. \[h0chiral\], which can be chosen such that it is a constant independent of ${\bf k}$ everywhere except in a small neighborhood close to ${\bf k} = 0$ where a band inversion occurs. Adding such a Hamiltonian that is independent of ${\bf r}$ will have no effect on the topologically protected modes associated with a line defect, so we expect the invariant $\nu$ to be the same for both ${\cal H}({\bf k},{\bf r})$ and $\tilde{\cal H}({\bf k},{\bf r})$. If ${\cal H}({\bf k},{\bf r})$ has a non trivial strong topological invariant $\nu_0=1$ then $\tilde{\cal H}({\bf k},{\bf r})$ will have $\nu_0 = 0$, so that Eq. \[gnu\] can be applied. ${\bf G}_\nu$ will then be given by the 2D invariant evaluated for $\tilde{\cal H}$, which will be independent of $x_1$. Since ${\cal H}_{STI}({\bf k})$ is ${\bf
k}$ independent everywhere except a neighborhood of ${\bf k}=0$, this will agree with the 2D invariant evaluated for ${\cal H}$ at $x_1 = 1/2$, but not $x_1=0$. It then follows that even in a strong topological insulator the invariant characterizing a line dislocation is given by , where ${\bf G}_\nu$ is given by in terms of the weak topological invariants.
### Heterostructure geometries {#sec:linehelicaldirachetero}
In principle it may be possible to realize 1D helical fermions in a 3D system that does not rely on a weak topological insulating state. It is possible to write down a 3D model, analogous to that has bound helical modes. However, it is not clear how to physically implement this model. This model will appear in a more physical context as a BdG theory in the following section.
Class DIII: helical Majorana fermions {#sec:linehelicalmajorana}
-------------------------------------
Line defects for class DIII are characterized by a $\mathbb{Z}_2$ topological invariant that signals the presence or absence of 1D helical Majorana fermion modes. As in Section \[sec:linechiralmajorana\], the BdG Hamiltonian has the same structure as a Bloch Hamiltonian, and the $\mathbb{Z}_2$ invariant can be deduced by “forgetting" the particle hole symmetry, and treating the problem as if it was a Bloch Hamiltonian in class AII.
There are several ways to realize helical Majorana fermions. The simplest is to consider the edge of a 2D time reversal invariant superconductor or superfluid, or equivalently a dislocation in a layered version of that 2D state. A second is to consider a topological line defect in a 3D class DIII topological superconductor or superfluid. Such line defects are well known in of $^3$He B[@grinevich88; @volovik03] and have recently been revisited in Ref. .
=1.5in
Here we will consider a different realization that uses topological insulators and superconductors. Consider a linear junction between two superconductors on the surface of a topological insulator as shown in Fig. \[hmdevice\]. In Ref. it was shown that when the phase difference between the superconductors is $\pi$ there are gapless helical Majorana modes that propagate along the junction. This can be described by an 8 band minimal model that describes a topological insulator surface with a superconducting proximity effect, $${\cal H} = v \tau_z \mu_x \vec\sigma\cdot {\bf k} + (m + \epsilon|{\bf
k}|^2) \tau_z \mu_z + \Delta_1 \tau_x.$$ Here $m$ is the mass describing the band inversion of a topological insulator, as in , and $\Delta_1$ is the real part of the superconducting gap parameter. This model has time reversal symmetry with $\Theta = i \sigma_y K$ and particle-hole symmetry with $\Xi = \sigma_y\tau_y K$. The imaginary part of the superconducting gap, $\Delta_2 \tau_y$ violates time reversal symmetry. A line junction along the $x$ direction with phase difference $\pi$ at the surface of a topological insulator corresponds to the intersection of planes where $m(z)$ and $\Delta_1(y)$ change sign.
The $\mathbb{Z}_2$ invariant characterizing such a line defect is straightforward to evaluate because $[{\cal H},\mu_y\tau_x]=0$. This extra symmetry allows a “spin Chern number" to be defined, $n_\sigma = (16\pi^2)^{-1}\int {\rm Tr}[\mu_y\tau_x {\cal
F}\wedge{\cal F}]$. Since the system decouples into two time reversed versions of , $n_\sigma = 1$. By repeating the formulation in Appendix \[appendix:defectsaiidiii\] of the $\mathbb{Z}_2$ invariant $\nu$, it is straightforward to show that this means $\nu = 1$.
The helical modes can be explicitly seen by solving the linear theory, $m = f_z z$, $\Delta_1 = f_y y$, which leads to the harmonic oscillator model studied in Appendix \[appendix:ZM\]. In the space of the two zero modes the Hamiltonian has the form $${\cal H} = v k_x \sigma_x,$$ and describes 1D helical Majorana fermions.
Class C: Chiral Dirac fermions {#sec:lineclassc}
------------------------------
We finally briefly consider line defects in class C. Class C can be realized when time reversal symmetry is broken in a superconductor without spin orbit interactions that has even parity singlet pairing. Line defects are characterized by an integer topological invariant that determines the number of chiral Majorana fermion modes associated with the line. As in class $D$, this may be evaluated by “forgetting" the particle-hole symmetry and evaluating the corresponding Chern number that would characterize class $A$. The $2\mathbb{Z}$ in Table \[linetab\] for this case, however, means that the Chern integer computed in this manner is necessarily [*even*]{}. This means that there will necessarily be an even number $2n$ of chiral Majorana fermion modes, which may equivalently viewed as $n$ chiral Dirac fermion modes.
An example of such a system would be a 2D $d_{x^2-y^2}+ id_{xy}$ superconductor[@laughlin], which exhibits chiral Dirac fermion edge states, or equivalently a dislocation in a 3D layered version of that state.
Point defects {#sec:pointdefect}
=============
Point defects can occur at the end of a 1D system ($\delta = 1-0$) or at topological defects in 2D ($\delta = 2-1$) or 3D ($\delta = 3-2$) systems. From the $\delta = 1$ column Table \[tab:periodic\], it can be seen that there are five symmetry classes that can have topologically non trivial point defects. These are expected to be associated with protected zero energy bound states. Table \[pointtab\] lists the non trivial classes, along with the character of the associated zero modes. In this section we will discuss each of these cases.
Symmetry Topological classes $E=0$ Bound States
---------- --------------------- --------------------------
AIII $\mathbb{Z}$ Chiral Dirac
BDI $\mathbb{Z}$ Chiral Majorana
D $\mathbb{Z}_2$ Majorana
DIII $\mathbb{Z}_2$ Majorana Kramers Doublet
(= Dirac)
CII $2\mathbb{Z}$ Chiral Majorana Kramers
Doublet (=Chiral Dirac)
: Symmetry classes supporting non trivial point topological defects and their associated $E=0$ modes. []{data-label="pointtab"}
Classes AIII, BDI and CII: chiral zero modes {#sec:pointchiral}
--------------------------------------------
### Topological invariant and zero modes {#sec:pointchiralinvariant}
Point defects in classes AIII, BDI and CII are characterized by integer topological invariants. The formula for this integer invariant can be formulated by exploiting the chiral symmetry in each class. In a basis where the chiral symmetry operator is $\Pi = \tau_z$, the Hamiltonian may be written, $${\cal H}({\bf k},{\bf r}) = \left(\begin{array}{cc} 0 & q({\bf k},{\bf r}) \\
q({\bf k},{\bf r})^\dagger & 0\end{array}\right).
\label{hchiral}$$ When the Hamiltonian has a flattened eigenvalue spectrum ${\cal H}^2 = 1$, $q({\bf k},{\bf r})$ is a unitary matrix. For a point defect in $d$ dimensions, the Hamiltonian as a function of $d$ momentum variables and $D=d-1$ position variables is characterized by the winding number associated with the homotopy $\pi_{2d-1}[U(n\rightarrow\infty)]=\mathbb{Z}$, which is given by $$n = {(d-1)!\over {(2d-1)! (2\pi i)^d}}\int_{T^d \times S^{d-1}} {\rm Tr}[(q d q^\dagger)^{2d-1}].
\label{nchiral}$$ For a Hamiltonian that is built from anticommuting Dirac matrices, ${\cal
H}({\bf k},{\bf r}) = \hat {\bf d}({\bf k},{\bf r}) \cdot \vec \gamma$, this invariant is given simply by the winding degree of the mapping $\hat {\bf d}({\bf k},{\bf r})$ from $T^{d}\times S^{d-1}$ to $S^{2d-1}$, which is expressed as an integral of the Jacobian, $$n = {(d-1)!\over{2\pi^d}}\int_{T^d\times S^{d-1}} d^d{\bf k} d^{d-1}{\bf r} {\partial\hat{\bf
d}({\bf k},{\bf r})\over \partial^d{\bf k}\partial^{d-1}{\bf r}}.
\label{windingdegree}$$
In class AIII there are no constraints on $q({\bf k},{\bf r})$ other than unitarity, so all possible values of $n$ are possible. There are additional constraints for the chiral classes with antiunitary symmetries. As shown in Appendix \[appendix:representatives\], this is simplest to see by analyzing the constraints on the winding degree discussed above. $n$ must be zero in classes CI and DIII. There is no constraint on $n$ in class BDI, while $n$ must be even in class CII.
The topological invariant is related to an index that characterizes the chirality of the zero modes, $$n = N_+-N_-,
\label{index}$$ where $N_{\pm}$ are the number of zero modes that are eigenstates of $\Pi$ with eigenvalue $\pm 1$. To see that these zero modes are indeed protected consider $N_+=n>0$ and $N_-=0$. Any term in the Hamiltonian that could shift any of the $N_+$ degenerate states would have to have a nonzero matrix element connecting states with the same chirality. Such terms are forbidden, though, by the chiral symmetry $\{{\cal H},\Pi\}=0$. In the superconducting classes BDI and CII the zero energy states are Majorana bound states. In class CII, however, since time reversal symmetry requires that $n$ must be even, the paired Majorana states can be regarded as zero energy Dirac fermion states. In the special case where ${\cal H}({\bf k},{\bf r})$ has the form of a massive Dirac Hamiltonian, by introducing a suitable regularization for $|{\bf k}|\rightarrow\infty$ the topological invariant (\[nchiral\],\[windingdegree\]) can be expressed in a simpler manner as a topological invariant characterizing the mass term. In the following subsections we consider this in the three specific cases $d = 1,2,3$.
### Solitons in d=1 {#sec:pointchirald=1}
The simplest topological zero mode occurs in the Jackiw Rebbi model[@jackiwrebbi], which is closely related to the Su Schrieffer Heeger model[@ssh]. Consider $${\cal H}(k,x) = v k \sigma_x + m \sigma_y.$$ Domain walls where $m(x)$ changes sign as a function of $x$ are associated with the well known zero energy soliton states.
To analyze the topological class requires a regularization for $|k|\rightarrow
\infty$. This can either be done with a lattice, as in the Su, Schrieffer, Heeger model or by adding a term $\epsilon k^2 \sigma_y$, as in so that $|k|\rightarrow\infty$ can be replaced by a single point. In either case, the invariant changes by $1$ when $m$ changes sign.
### Jackiw Rossi Model in d=2 {#sec:pointchirald=2}
Jackiw and Rossi introduced a two dimensional model that has protected zero modes[@jackiwrossi]. The Hamiltonian can be written $${\cal H}({\bf k},{\bf r}) = v \vec \gamma \cdot {\bf k}
+ \vec\Gamma\cdot\vec\phi({\bf r}),
\label{jackiwrossi}$$ where ${\bf k} = (k_x,k_y)$ and $(\gamma_1,\gamma_2)$ and $(\Gamma_1,\Gamma_2)$ are anticommuting Dirac matrices. They showed that the core of a vortex where $\phi = \phi_1 + i \phi_2$ winds by $2\pi n$ is associated with $n$ zero modes that are protected by the chiral symmetry. Viewed as a BdG Hamiltonian, these zero modes are Majorana bound states.
This can be interpreted as a Hamiltonian describing superconductivity in Dirac fermions. In this interpretation the Dirac matrices are expressed as $(\gamma_1,\gamma_2) = \tau_z(\sigma_x,\sigma_y)$ and $(\Gamma_1,\Gamma_2) = (\tau_x,\tau_y)$, where $\vec\sigma$ is a Pauli matrix describing spin and $\vec\tau$ describes particle-hole space. The superconducting pairing term is $\Delta = \phi_1 + i\phi_2$. In this interpretation a vortex violates the physical time reversal symmetry $\Theta = i\sigma_y K$. However, even in the presence of a vortex this model has a fictitious “time reversal symmetry" $\tilde\Theta = \sigma_x\tau_x K$ which satisfies $\tilde\Theta^2 = +1$. This symmetry would be violated by a finite chemical potential term $\mu \tau_z$. Combined with particle-hole symmetry $\Xi = \sigma_y\tau_y K$ ($\Xi^2 = +1$), $\tilde\Theta$ defines the BDI class with chiral symmetry $\Pi = \sigma_z\tau_z$.
Evaluating the topological invariant again requires a $|{\bf k}|\rightarrow\infty$ regularization. One possibility is to add $\epsilon |{\bf k}|^2 \tau_x$, so that $|{\bf
k}|\rightarrow \infty$ can be replaced by a single point. In this case the invariant can be determined by computing the winding degree of $\hat {\bf d}({\bf k},{\bf r})$ on $S^3$. In the limit $\epsilon\rightarrow 0$ the ${\bf k}$ integral can be performed, so that can be expressed as the winding number of the phase of $\phi_1+i\phi_2 = |\Delta|e^{i\varphi}$, $$n = {1\over{2\pi }} \int_{S^1} d\varphi.
\label{phiwinding}$$
### Hedgehogs in $d=3$ {#sec:pointchirald=3}
In Ref. we introduced a three dimensional model for Majorana bound states that can be interpreted as a theory of a vortex at the interface between a superconductor and a topological insulator. In the special case that the chemical potential is equal to zero, model has the same form as , except that now all of the vectors are three dimensional. In the topological insulator model we have $\vec\gamma = (\gamma_1,\gamma_2,\gamma_3) = \mu_x\tau_z \vec\sigma$ and $\vec\Gamma = (\Gamma_1,\Gamma_2,\Gamma_3) = (\mu_z\tau_z,\tau_x,\tau_y)$. $\vec\tau$ and $\vec\sigma$ are defined as before, while $\vec\mu$ describes a orbital degree of freedom. The chiral symmetry, $\Pi = \mu_y\tau_z$ is violated if a chemical potential term $\mu \tau_z$ is included.
Following the same steps that led to the invariant is given by the winding number of $\hat\phi = \vec\phi/|\vec\phi|$ on $S^2$, $$n = {1\over{4\pi}}\int_{S^2} \hat \phi \cdot (d\hat\phi \times
d\hat\phi).$$
Class D: Majorana bound states {#sec:pointmajorana}
------------------------------
### Topological invariant {#sec:pointmajoranainvariant}
Point defects in class D are characterized by a $\mathbb{Z}_2$ topological invariant that determines the presence or absence of a Majorana bound state associated with the defect. These include the well known end states in a 1D p wave superconductor and vortex states in a 2d $p_x+ip_y$ superconductor. In Ref. we considered such zero modes in a three dimensional BdG theory describing Majorana zero modes in topological insulator structures. Here we develop a unified description of all of these cases.
For a point defect in $d$ dimensions, the Hamiltonian depends on $d$ momentum variables and $D=d-1$ position variables. In appendix \[appendix:pointdefectclassd\] we show that the $\mathbb{Z}_2$ invariant is given by, $$\nu = {2\over{d!}}\left({i\over {2\pi}}\right)^d \int_{T^d\times S^{d-1}} {\cal Q}_{2d-1} \ {\rm mod} \ 2,
\label{muchernsimons}$$ where ${\cal Q}_{2d-1}$ is the Chern Simons form. The specific cases of interest are, $$\begin{aligned}
{\cal Q}_1 &=& {\rm Tr}[{\cal A}],\\
{\cal Q}_3 &=& {\rm Tr}[{\cal A} d {\cal A} + {2\over 3}{\cal A}^3], \label{cs3d}\\
{\cal Q}_5 &=& {\rm Tr}[{\cal A} (d{\cal A})^2 + {3\over 2}{\cal A}^3 d{\cal A} +
{3\over 5}{\cal A}^5].\end{aligned}$$
It is instructive to see that reduces to in the case in which a system also has particle-hole symmetry. In this case, as detailed in Appendix \[appendix:pointdefectclassd\] it is possible to choose a gauge in which ${\cal A} = q^\dagger dq/2$, so that ${\cal Q}_{2d-1} \propto
(qdq^\dagger)^{2d-1}$.
### End States in a 1D superconductor {#sec:pointmajorana1d}
The simplest example of a point defect in a superconductor occurs in Kitaev’s model[@kitaev00] of a one dimensional p wave superconductor. This is described by a simple 1D tight binding model for spinless electrons, which includes a nearest neighbor hopping term $t c_i^\dagger c_{i+1} + {\rm h.c.}$ and a nearest neighbor p wave pairing term $\Delta c_i c_{i+1} + {\rm h.c.}$. The Bogoliubov de Gennes Hamiltonian can then be written as $${\cal H}(k) = ( t \cos k - \mu )\tau_z + \Delta \sin k \tau_x.
\label{1DKitaev}$$ This model exhibits a weak pairing phase for $|\mu| < t$ and a strong pairing phase for $|\mu|>t$. The weak pairing phase will have zero energy Majorana states at its ends.
The topological invariant can be easily evaluated. We find ${\cal A} = d\theta/2$, where $\theta$ is the polar angle of ${\bf d}(k) = ( t\cos k - \mu,\Delta \sin k)$. It follows that for $|\mu|<t$, the topological invariant is $\nu = 1$ mod 2.
### Vortex in a 2D topological superconductor {#sec:pointmajorana2d}
In two dimensions, a Majorana bound state occurs at a vortex in a topological superconductor. This can be easily seen by considering the edge states of the topological superconductor in the presence of a hole[@readgreen]. Particle-hole symmetry requires that the quantized edge states come in pairs. When the flux is an odd multiple of $h/2e$, the edge states are quantized such that a zero mode is present. In this section we will evaluate the topological invariant associated with a loop surrounding the vortex[@thankroman].
We begin with the class D BdG Hamiltonian ${\cal H}^0_p(k_x,k_y)$ characterizing the topological superconductor when the superconducting phase is zero. We include the subscript $p$ to denote the first Chern number that classifies the topological superconductor. We can then introduce a nonzero superconducting phase by a gauge transformation, $${\cal H}_p({\bf k},\varphi) = e^{-i\varphi\tau_z/2} {\cal H}_p^0({\bf k})
e^{i\varphi\tau_z/2},
\label{phaserotation}$$ where $\tau_z$ operates in the Nambu particle-hole space. We now wish to evaluate for this Hamiltonian when phase $\varphi(s)$ winds around a vortex. There is, however, a difficulty because the Chern Simons formula requires a gauge that is continuous throughout the entire base space $T^2 \times S^1$. The nonzero Chern number $p$ characterizing ${\cal
H}_p^0({\bf k})$ is an obstruction to constructing such a gauge. A similar problem arose in Section \[sec:linehelicaldiracdislocation\], when we discussed a line dislocation in a weak topological superconductor. We can adapt the trick we used there to get around the present problem. We thus double the Hilbert space to include two copies of our Hamiltonian–one with Chern number $p$ and one with Chern number $-p$, $$\tilde{\cal H}^0({\bf k}) = \left(\begin{array}{cc} {\cal H}^0_p({\bf k}) & 0 \\ 0 & {\cal H}^0_{-p}({\bf k})
\end{array}\right).$$ We then put the vortex in only the $+p$ component, $$\tilde{\cal H}({\bf k},\varphi) = e^{-i \varphi q} \tilde{\cal
H}^0({\bf k}) e^{i \varphi q},$$ where $$q = {{1+\tau_z}\over 2} \left(\begin{array}{cc} 1 & 0 \\ 0 & 0
\end{array}\right).
\label{qequation}$$ We added an extra phase factor by replacing $\tau_z$ by $1+\tau_z$ in order to make $e^{i \varphi q}$ periodic under $\varphi\rightarrow\varphi+2\pi$.
Since the Chern number characterizing $\tilde {\cal H}^0({\bf k})$ is zero, there exists a continuous gauge, $$|\tilde u_i({\bf k},\varphi)\rangle = e^{i\varphi q} |\tilde
u^0_i({\bf k})\rangle,$$ which allows us to evaluate the Chern Simons integral. The Berry’s connection $\tilde{\cal A}_{ij} = \langle \tilde u_i |d\tilde u_j\rangle$ is given by $$\tilde{\cal A} = \tilde{\cal A}^0 + i Q d\varphi,$$ where $\tilde{\cal A}^0({\bf k})$ is the connection describing $\tilde {\cal
H}^0({\bf k})$ and $Q_{ij}({\bf k}) = \langle \tilde u^0_i({\bf k})|q|\tilde u^0_j({\bf k})\rangle$. Inserting this into and rearranging terms we find $${\cal Q}_3 = {\rm Tr}[ 2 Q \tilde{\cal F}^0 - d(Q \tilde{\cal A}^0) ] \wedge
d\varphi,$$ where $\tilde {\cal F}^0 = d\tilde{\cal A}^0 + \tilde{\cal A}^0\wedge\tilde{\cal A}^0$. Since the second term is a total derivative it can be discarded. For the first term there are two contributions from the $1$ and the $\tau_z$ in . Upon integrating over ${\bf k}$, the $\tau_z$ term can be shown to vanish as a consequence of particle-hole symmetry. The $1$ term simply projects out the Berry curvature of the original Hamiltonian ${\cal H}^0_p({\bf k})$, so that $${\cal Q}_3 = {\rm Tr}[{\cal F}^0]\wedge d\varphi.$$ It follows from that the $\mathbb{Z}_2$ invariant characterizing the vortex is $$\nu = p m \ {\rm mod} \ 2,$$ where $p$ is the Chern number characterizing the topological superconductor and $m$ is the phase winding number associated with the vortex.
It is also instructive to consider this invariant in the context of the simple two band model introduced by Read and Green[@readgreen]. This can be written as a simple tight binding model $${\cal H}^0(k_x,k_y) = (t (\cos k_x + \cos k_y) - \mu) \tau_z + \Delta (\sin k_x \tau_x +
\sin k_y \tau_y).
\label{readgreenh}$$ where the superconducting order parameter $\Delta$ is real. As in , this model exhibits weak and strong pairing phases for $|\mu|<t$ and $|\mu|>t$. These are distinguished by the Chern invariant, which in turn is related to the winding number on $S^2$ of the unit vector $\hat {\bf d}({\bf k})$, where $\vec d({\bf k})$ are the coefficients of $\vec\tau$ in . A nonzero superconducting phase is again introduced by rotating about $\tau_z$ as in . Here we wish to show that in this two band model the $\mathbb{Z}_2$ invariant $\nu$ can be understood from a geometrical point of view.
=3.3in
The $Z_2$ invariant characterizing a vortex can be understood in terms of the topology of the maps $\hat {\bf
d}(k_x,k_y,\phi)$. from $T^2 \times S^1$ to $S^2$. These maps were first classified by Pontrjagin[@pontrjagin], and have also appeared in other physical contexts[@jaykka; @kapitanski; @teokane10]. Without losing generality, we can reduce the torus $T^2$ to a sphere $S^2$, so the mappings are $S^2 \times S^1 \rightarrow S^2$. When for fixed $\phi$ $\hat {\bf d}(k_x,k_y,\phi)$ has an $S^2$ winding number of $\pm p$, the topological classification is $\mathbb{Z}_{2p}$. In the case of interest, $p=1$, so there are two classes.
This $\mathbb{Z}_2$ Pontrjagin invariant can be understood pictorially by considering inverse image paths in $({\bf k},\phi)$ space, which map to two specific points on $S^2$. These correspond to 1D curves in $S^2 \times S^1$. Fig. \[pontrjagin\] shows three examples of such curves. The inner sphere corresponds to $\phi=0$, while the outer sphere corresponds to $\phi=2\pi$. Since $p=1$, for every point on $S^2$ the inverse image path is a single curve connecting the inner and outer spheres. The key point is to examine the linking properties of these curves. The $\mathbb{Z}_2$ invariant describes the number of twists in a pair of inverse image paths, which is $1$ in (a), $2$ in (b) and $0$ in (c). The configuration in (b) can be continuously deformed into that in (c) by dragging the paths around the inner sphere. This can be verified by a simple demonstration using your belt. The twist in (a), however, can not be undone. The number of twists thus defines the $\mathbb{Z}_2$ Pontrjagin invariant.
### Superconductor Heterostructures {#sec:pointmajoranahetero}
Finally, in three dimensions, a non trivial point defect can occur at a superconductor heterostructure. An example is a vortex in the superconducting state at the interface between a superconductor and a topological insulator. As shown in Ref. , this can be described by the simple Hamiltonian, $${\cal H} = v \tau_z \mu_x \vec\sigma \cdot {\bf k} - \mu \tau_z + (m +
\epsilon|{\bf k}|^2 ) \tau_z \mu_z + \Delta_1 \tau_x + \Delta_2
\tau_y.$$ Here $m$ is a mass which distinguishes a topological insulator from a trivial insulator, and $\Delta = \Delta_1 + i\Delta_2$ is a superconducting order parameter. For $\mu = 0$, this Hamiltonian has the form of the three dimensional version of discussed in IV.A.4, where the mass term is characterized by the vector $\vec\phi = (m,\Delta_1,\Delta_2)$. A vortex in $\Delta$ at the interface where $m$ changes sign then corresponds to a hedgehog singularity in $\vec\phi$. From , it can be seen that the class BDI $\mathbb{Z}$ invariant is $n=1$. This then establishes that the class D $\mathbb{Z}$ invariant is $\nu =
1$. The $\mathbb{Z}_2$ survives when a nonzero chemical potential reduces the symmetry from class BDI to class D.
Class DIII: Majorana doublets {#sec:pointdiii}
-----------------------------
Point defects in class DIII are characterized by a $\mathbb{Z}_2$ topological invariant. These are associated with zero modes, but unlike class D, the zero modes are required by Kramers theorem to be doubly degenerate. The zero modes thus form a Majorana doublet, which is equivalent to a single Dirac fermion.
In Table \[tab:periodic\], Class DIII, $\delta = 1$ is an entry that is similar to Class AII, $\delta = 2$. The $\mathbb{Z}_2$ for DIII invariant bears a resemblance to the invariant for AII, which is a generalization of the $\mathbb{Z}_2$ invariant characterizing the 2D quantum spin Hall insulator. In Appendix \[appendix:pdDIII\] we will establish a formula that employs the same gauge constraint, $$w({\bf k},{\bf r}) = w_0,
\label{DIIIgc}$$ where $w_0$ is a constant independent of ${\bf k}$ and ${\bf r}$. $w({\bf k},{\bf r})$ relates the time reversed states at ${\bf k}$ and $-{\bf k}$ and is given by . [*Provided*]{} we choose a gauge that satisfies this constraint, the $\mathbb{Z}_2$ invariant is given by, $$\tilde\nu=\frac{1}{d!}\left(\frac{i}{2\pi}\right)^d
\int_{T^d\times S^{d-1}}\mathcal{Q}_{2d-1}\quad\mbox{mod 2}.
\label{diiiformula1}$$ This formula is almost identical to the formula for a point defect in class D, but they differ by an important factor of two. Due to the combination of time reversal and particle-hole symmetry the Chern Simons integral is guaranteed to be an integer, but the integer is not gauge invariant. When the time reversal constraint is satisfied, the parity $\tilde\nu$ is gauge invariant. It then follows that the class D invariant in , $\nu = 0$ mod 2.
In the special case $d=1$ there is a formula that does not rely on the gauge constraint, though it still requires a globally defined gauge. It is related to the similar “fixed point" formula for the invariant for the 2D quantum spin Hall insulator[@fukane06], and has recently been employed by Qi, Hughes and Zhang[@qhz10] to classify one dimensional time reversal invariant superconductors. In class DIII, it is possible to choose a basis in which the time reversal and particle hole operators are given by $\Theta = \tau_y K$ and $\Xi = \tau_x K$, so that the chiral operator is $\Pi = \tau_z$. In this basis, the Hamiltonian has the form , where $q({\bf k},{\bf r}) \rightarrow q(k)$ satisfies $q(-k) = - q(k)^T$. Thus, ${\rm Pf}[q(k)]$ is defined for the time reversal invariant points $k = 0$ and $k=\pi$. $q(k)$ is related to $w(k)$ because in a particular gauge it is possible to choose $w(k) = q(k)/\sqrt{|{\rm Det}[q(k)]|}$. The $\mathbb{Z}_2$ invariant is then given by, $$(-1)^{\tilde\nu} = {{\rm Pf}[q(\pi)]\over{{\rm Pf}[q(0)]}} {\sqrt{{\rm
Det}[q(0)]}\over\sqrt{{\rm Det}[q(\pi)]}},
\label{diiiformula2}$$ where the branch $\sqrt{{\rm Det}[q(k)]}$ is chosen continuously between $k=0$ and $k=\pi$. The equivalence of and for $d=1$ is demonstrated in Appendix \[appendix:fixedpointformulas\]. Unlike , however, the fixed point formula does not have a natural generalization for $d>1$.
Majorana doublets can occur at topological defects in time reversal invariant topological superconductors, or in Helium 3B. Here we consider a different configuration at a Josephson junction at the edge of a quantum spin Hall insulator (Fig. \[pijunction\]). When the phase difference across the Josephson junction is $\pi$, it was shown in Ref. that there is a level crossing in the Andreev bound states at the junction. This corresponds precisely to a Majorana doublet.
This can be described by a the simple continuum 1D theory introduced in Ref. . $$\label{1Djj}
{\cal H} = v k \tau_z \sigma_z + \Delta_1 \tau_x$$ Here $\sigma_z$ describes the spin of the quantum spin Hall edge state, and $\Delta_1$ is the real superconducting order parameter. This model has particle-hole symmetry $\Xi = \sigma_y\tau_y K$ and time reversal symmetry $\Theta = i \sigma_y K$ and is in class DIII. A $\pi$ junction corresponds to a domain wall where $\Delta_1$ changes sign. Following appendix \[appendix:ZM\], it is straightforward to see that this will involve a degenerate pair of zero modes indexed by the spin $\sigma_z$ and chirality $\tau_y$ constrained by $\tau_y\sigma_z=-1$.
=2.5in
The Hamiltonian should be viewed as an a low energy theory describing the edge of a 2D quantum spin Hall insulator. Nonetheless, we may describe a domain wall where $\Delta_1$ changes sign using an effective one dimensional theory by introducing a regularization replacing $\Delta_1$ by $\Delta_1 + \epsilon k^2$. This regularization will not effect the topological structure of a domain wall where $\Delta_1$ changes sign. A topologically equivalent lattice version of the theory then has the form, $${\cal H} = t \sin k \tau_z \sigma_z + (\Delta_1 + u(1-\cos k))\tau_x.
\label{hdiiilattice}$$ where we assume $|\Delta_1|< 2u$.
The topological invariant can be evaluated using either or . To use , note that has exactly the same form as two copies (distinguished by $\sigma_z=\pm 1$) of . The evaluation of then proceeds along the same lines. It is straightforward to check that in a basis where the time reversal constraint is satisfied (this fixes the relative phases of the $\sigma_z=\pm 1$ states), ${\cal A} = d\theta$, where $\theta$ is the polar angle of ${\bf d}(k) = (t\sin k, \Delta_1 + u(1-\cos k))$. It follows that a defect where $\Delta_1$ changes sign has $\tilde\nu = 1$.
To use , we transform to a basis in which $\Theta = \tau_y K$, $\Xi = \tau_x K$ and $\Pi = \tau_z$. This is accomplished by the unitary transformation $U = \exp[i (\pi/4) \sigma_y\tau_z] \exp[i (\pi/4) \tau_x]$. Then, ${\cal H}$ has the form of Eq. \[h0chiral\] with $q(k) = -i( t\sin k \sigma_z
+ (\Delta_1 + u (1-\cos k)) \sigma_y$. It follows that ${\rm det}[q(k)]$ is real and positive for all $k$. Moreover, ${\rm Pf}[q(0)]/\sqrt{{\rm det}[q(0)]} = {\rm sgn}[\Delta_1]$ while ${\rm Pf}[q(\pi)]/\sqrt{{\rm det}[q(\pi)]} =1$. Again, a defect where $\Delta_1$ changes sign has $\tilde\nu = 1$.
Adiabatic pumps {#sec:pump}
===============
In this section we will consider time dependent Hamiltonians ${\cal H}({\bf k},{\bf r},t)$, where in additional to having adiabatic spatial variation ${\bf r}$ there is a cyclic adiabatic temporal variation parameterized by $t$. We will focus on point like spatial defects, in which the dimensions of ${\bf k}$ and ${\bf
r}$ are related by $d-D = 1$.
Adiabatic cycles in which ${\cal
H}({\bf k},{\bf r},t=T) = {\cal H}({\bf k},{\bf r},t=0)$ can be classified topologically by considering $t$ to be an additional “spacelike" variable, defining $\tilde D = D + 1$. Such cycles will be classified by the $\delta = 0$ column of Table \[tab:periodic\]. Topologically non trivial cycles correspond to adiabatic pumps. Table \[pumptab\] shows the symmetry classes which host non trivial pumping cycles, along with the character of the adiabatic pump. There are two general cases. Classes A, AI and AII define a charge pump, where after one cycle an integer number of charges is transported towards or away from the point defect. Classes BDI and D define a fermion parity pump. We will discuss these two cases separately.
We note in passing that the $\delta = 0$ column of Table \[tab:periodic\] also applies to topological [*textures*]{}, for which $d=D$. For example a spatially dependent three dimensional band structure ${\cal H}({\bf k},{\bf r})$ can have topological textures analogous to Skyrmions in a 2D magnet. Such textures have recently been analyzed by Ran, Hosur and Vishwanath[@ran10b] for the case of class D, where they showed that the $\mathbb{Z}_2$ invariant characterizing the texture corresponds to the fermion parity associated with the texture. Thus, non trivial textures are fermions.
Symmetry Topological classes Adiabatic Pump
---------- --------------------- ------------------------
A $\mathbb{Z}$ Charge
AI $\mathbb{Z}$ Charge
BDI $\mathbb{Z}_2$ Fermion Parity
D $\mathbb{Z}_2$ Fermion Parity
AII $2\mathbb{Z}$ Charge Kramers Doublet
: Symmetry classes that support non trivial charge or fermion parity pumping cycles. []{data-label="pumptab"}
Class A, AI, AII: Thouless Charge Pumps {#sec:pumpcharge}
---------------------------------------
The integer topological invariant characterizing a pumping cycle in class A is simply the Chern number characterizing the Hamiltonian ${\cal H}({\bf
k},{\bf r},t)$[@thouless; @thoulessniu]. Imposing time reversal symmetry has only a minor effect on this. For $\Theta^2 = -1$ (Class AII), an odd Chern number violates time reversal symmetry, so that only even Chern numbers are allowed. This means that the pumping cycle can only pump Kramers pairs of electrons. For $\Theta^2 = +1$ (Class AI) all Chern numbers are consistent with time reversal symmetry.
The simplest charge pump is the 1D model introduced by Thouless[@thouless]. A continuum version of this model can be written in the form, $${\cal H}(k,t) = v k \sigma_z + (m_1(t)+ \epsilon k^2) \sigma_x + m_2(t)
\sigma_y.$$ When the masses undergo a cycle such that the phase of $m_1+im_2$ a single electron is transmitted down the wire. In this case, ${\cal
H}(k,t)$ has a non zero first Chern number. The change in the charge associated with a point in a 1D system is given by the difference in the Chern numbers associated with either side of the point. Thus, after a cycle a charge $e$ accumulates at the end of a Thouless pump.
A two dimensional version of the charge pump can be developed based on Laughlin’s argument[@laughlin81] for the integer quantum Hall effect. Consider a 2D $\nu=1$ integer quantum Hall state and change the magnetic flux threading a hole from 0 to $h/e$. In the process, a charge $e$ is pumped to the edge states surrounding the hole. This pumping process can be characterized by the second Chern number characterizing the 2D Hamiltonian ${\cal
H}(k_x,k_y,\theta,t)$, where $\theta$ parameterizes a circle surrounding the hole. A similar pump in 3D can be considered, and is characterized by the third Chern number.
Class D, BDI: Fermion Parity Pump {#sec:pumpparity}
---------------------------------
Adiabatic cycles of point defects in class D and BDI are characterized by a $\mathbb{Z}_2$ topological invariant. In this section we will argue that a non trivial pumping cycle transfers a unit of fermion parity to the point defect. This is intimately related to the Ising non-Abelian statistics associated with defects supporting Majorana bound states.
Like the point defect in class DIII ($\delta = 1$), the temporal pump ($\delta = 0$) in class D occupies an entry in Table \[tab:periodic\] similar to the line defect ($\delta = 2$) in class AII, so we expect a formula that is similar to the formula for the 2D quantum spin Hall insulator. This is indeed the case, though the situation is slightly more complicated. The Hamiltonian ${\cal H}({\bf k},{\bf r},t)$ is defined on a base space $T^d \times S^{d-1}\times S^1$. In appendix \[appendix:pumpinvariant\] we will show that the invariant can be written in a form that resembles , $$\label{FPPFK}
\nu=\frac{i^d}{d!(2\pi)^d}
\left[\int_{\mathcal{T}_{1/2}}\mbox{Tr}(\mathcal{F}^d) -
\oint_{\partial\mathcal{T}_{1/2}}\mathcal{Q}_{2d-1}\right]\quad\mbox{mod
2},$$ where $\mathcal{T}_{1/2}$ is half of the base manifold, say $k_1\in[0,\pi]$, and the Chern-Simons form $\mathcal{Q}_{2d-1}$ is generated by a continuous valence frame $u_v({\bf k},{\bf r},t)|_{k_1=0,\pi}$ that obeys certain particle-hole gauge constraint. This is more subtle than the time reversal gauge condition for line defects in AII and point defects in DIII. Unlike , we do not have a computational way of checking whether or not a given frame satisfies the constraint. Nevertheless, it can be defined, And in certain simple examples, the particle-hole constraint is automatically satisfied.
The origin of the difficulty is that unlike time reversal symmetry, particle hole symmetry connects the conduction and valence bands. The gauge constraint therefore involves both. Valence and conduction frames can be combined to form a unitary matrix, $$G_{{\bf k},{\bf r},t}=\left(\begin{array}{*{20}c}|&|\\
u_v({\bf k},{\bf r},t)&u_c({\bf k},{\bf r},t)\\
|&|\end{array}\right)\in U(2n).$$ The orthogonality of conduction and valence band states implies that $$\label{FPPgc}
G_{{\bf k},{\bf r},t}^\dagger\Xi G_{-{\bf k},{\bf r},t}=0.$$ In general, we call a frame $G:\partial\mathcal{T}_{1/2}\to U(2n)$ particle-hole trivial if it can continuously be deformed to a constant [*while satisfying throughout the deformation*]{}. The Chern Simons term in requires a gauge that is built from the valence band part of a particle-hole trivial frame.
Though the subtlety of the gauge condition makes a general computation of the invariant difficult, it is possible to understand the invariant in the context of specific models. Consider, a theory based on a point defect in the $d$ dimensional version of , $${\cal H}({\bf k},{\bf r},t) = v \vec\gamma\cdot{\bf k} +
\vec\Gamma\cdot \vec\phi({\bf r},t).$$ Here $\vec\Gamma$ and $\vec\gamma$ are $2^d\times 2^d$ Dirac matrices, and we suppose that for fixed $t$, the $d$ dimensional mass vector $\vec \phi({\bf r},t)$ has a point topological defect at ${\bf
r}_0(t)$. If Ref. we argued that adiabatic cycles for such point defects are classified by a Pontrjagin invariant similar to that discussed in section \[sec:pointmajorana2d\]. This may also be understood in terms of the rotation of the “orientation" of the defect. Near the defect, suppose $\vec\phi({\bf r},t) = O(t)\cdot ({\bf r}-{\bf
r}_0(t))$, where $O(t)$ is a time dependent $O(d)$ rotation. In the course of the cycle, the orientation of the topological defect, characterized by $O(t)$ goes through a cycle. Since for $d\ge 3$, $\pi_1(O(d)) =
\mathbb{Z}_2$, there are two classes of cycles. As shown in Ref. , the non trivial cycle, which corresponds to a $2\pi$ rotation changes the sign of the Majorana fermion wavefunction associated with the topological defect. We will argue below that this corresponds to a change in the local fermion parity in the vicinity of the defect. For $d=2$, $\pi_1(O(2)) = \mathbb{Z}$. However, the change in the sign of the Majorana bound state is given by the parity of the $O(2)$ winding number. In theories with more bands, it is only this parity that is topologically robust.
In $d=1$, the single $\Gamma$ matrix in the 2 band model does not allow for continuous rotations. Consider instead Kitaev’s model[@kitaev00] for a 1D topological superconductor with at time dependent phase, $${\cal H}(k,t) = (t \cos k - \mu )\tau_z + \Delta_1(t) \sin k \tau_x +
\Delta_2(t) \sin k \tau_y.$$ In this case it is possible to apply the formula because on the boundary $\partial {\cal T}$, which is $k = 0$ or $k=\pi$ the Hamiltonian is independent of $t$, so that the gauge condition is automatically satisfied. Moreover, the second term in involving the Chern Simons integral is equal to zero, so that the invariant is simply the integral of ${\cal F}(x,t)$ over ${\cal T}_{1/2}$. It is straightforward to check that this gives $\nu=1$.
=2.5in
In order to see why this corresponds to a pump for fermion parity, suppose a topological superconductor is broken in two places, as shown in Fig. \[fppump\]. At the ends where the superconductor is cut there will be Majorana bound states. The pair of bound states associated with each cut define two quantum states which differ by the [*parity*]{} of the number of electrons. If the two ends are weakly coupled by electron tunneling then the pair of states will split. Now consider advancing the phase of the central superconductor by $2\pi$. As shown in Refs. , the states interchange as depicted in Fig. \[fppump\]. The level crossing that occurs at $\pi$ phase difference is protected by the conservation of fermion parity. Thus, at the end of the cycle, one unit of fermion parity has been transmitted from one circled region to the other.
The pumping of fermion parity also applies to adiabatic cycles of point defects in higher dimensions, and is deeply connected with the Ising non-Abelian statistics associated with those defects[@teokane10].
Conclusion {#sec:conclusion}
==========
In this paper we developed a unified framework for classifying topological defects in insulators and superconductors by considering Bloch/BdG Hamiltonians that vary adiabatically with spatial (and/or temporal) parameters. This led to a generalization of the bulk-boundary correspondence, which identifies protected gapless fermion excitations with topological invariants characterizing the defect. This leads to a number of additional questions to be addressed in future work.
The generalized bulk-boundary correspondence has the flavor of a mathematical index theorem, which relates an analytic index that characterizes the zero modes of a system to a topological index. It would be interesting to see a more general formulation of this relation[@fukui10a; @fukui10b] that applies to the classes without chiral symmetry that have $\mathbb{Z}_2$ invariants, and goes beyond the adiabatic approximation we used in this paper. Though the structure of the gapless modes associated with defects make it clear that such states are robust in the presence of disorder and interactions, it would be desirable to have a more general formulation of the topological invariants characterizing a defect that can be applied to interacting and/or disordered problems.
An important lesson we have learned is that topologically protected modes can occur in a context somewhat more general than simply boundary modes. This expands the possibilities for engineering these states in physical systems. It is thus an important future direction to explore the possibilities for heterostructures that realize topologically protected modes. The simplest version of this would be to engineer protected chiral fermion modes using a magnetic topological insulator. The perfect electrical transport in such states could have far reaching implications at both the fundamental and practical level. In addition, it is worth considering the expanded possibilities for realizing Majorana bound states in superconductor heterostructures, which could have implications for quantum computing.
Finally, it will be interesting to generalize these topological considerations to describe inherently correlated states, such as the Laughlin state. Could a [*fractional*]{} quantum Hall edge state arise as a topological line defect in a 3D system? Understanding the topological invariants that would characterize such a defect would lead to a deeper understanding of topological states of matter.
We thank Claudio Chamon, Liang Fu, Takahiro Fukui and Roman Jackiw for helpful discussions. This work was supported by NSF grant DMR-0906175.
Periodicity in symmetry and dimension {#appendix:periodicities}
=====================================
In this appendix we will establish the relations (\[period1\],\[period2\]) between the $K$-groups in different position-momentum dimensions $(D,d)$ and different symmetry classes $s$. We will do so by starting with an arbitrary Hamiltonian in $K_{\mathbb{F}}(s;D,d)$ and then explicitly constructing new Hamiltonians in one higher position or momentum dimension, which have a symmetry either added or removed. The new Hamiltonians will then belong to $K_{\mathbb{F}}(s+1;D,d+1)$ or $K_{\mathbb{F}}(s-1;D+1,d)$. The first step is to identify the mappings and show they preserve the group structure. This defines group homomorphisms relating the $K$ groups. The next step is to show they are [*isomporphisms*]{} by showing that the maps have an inverse, up to homotopic equivalence.
Hamiltonian mappings {#appendix:hamiltonianmaps}
--------------------
There are two classes of mappings: those that add symmetries and those that remove symmetries. These need to be considered separately.
=3.0in
We consider first the symmetry removing mappings that send a Hamiltonian ${\cal H}_c$ with chiral symmetry to a Hamiltonian ${\cal H}_{nc}$ without chiral symmetry. Suppose $\{{\cal H}_c({\bf k},{\bf r}),\Pi\}=0$, where $\Pi$ is the chiral operator. Then define $${\cal H}_{nc}({\bf k},{\bf r},\theta) = \cos\theta {\cal H}_c({\bf k},{\bf r})
+ \sin\theta \Pi
\label{ctonc}$$ for $-\pi/2 \leq \theta \leq \pi/2$. This has the property that at $\theta = \pm
\pi/2$ the new Hamiltonian is $\pm\Pi$, independent of ${\bf k}$ and ${\bf r}$. Thus, at each of these points we may consider the base space $T^d \times S^D$ defined by ${\bf k}$ and ${\bf
r}$ to be contracted to a point. The new Hamiltonian is then defined on the [*suspension*]{} $\Sigma(T^d\times S^D)$ of the original base space (see fig.\[suspension1\]). If we treat the original base space as a $d+D$ dimensional sphere, then the suspension is a $d+D+1$ dimensional sphere.
Without loss of generality we assume ${\cal H}_c$ is flattened, so that ${\cal H}_c^2 = 1$. Since $\{{\cal H}_c,\Pi\}=0$ it follows that ${\cal H}_{nc}^2 = 1$ as well. The second term in violates the chiral symmetry. Thus, if ${\cal H}_c$ belongs to the complex class AIII (with no anti unitary symmetries), then ${\cal H}_{nc}$ belongs to class A. Eq. thus provides a mapping from class AIII to class A.
=3.0in
For the real classes, which have anti unitary symmetries, the second term will violate either particle-hole symmetry or time reversal symmetry, depending on whether $\theta$ is a momentum or position type variable (odd or even under $\Theta$ and $\Xi$). This will lead to a new non chiral symmetry class related to the original class by either a clockwise or counter clockwise turn on the symmetry clock (fig.\[clockarrow\]). To determine which it is, note that if we require $[\Theta,\Xi]=0$ then $(\Theta\Xi)^2 = \Theta^2 \Xi^2 =
(-1)^{(s-1)/2}$. The unitary chiral symmetry operator (satisfying $\Pi^2=1$) can then be written $$\Pi = i^{(s-1)/2} \Theta \Xi.$$ It follows that if $\theta$ is momentum like, then time reversal symmetry is violated when $s = 1\ {\rm mod}\ 4$, while particle hole is violated when $s =
3\ {\rm mod}\ 4$. This corresponds to corresponds to a clockwise rotation on the symmetry clock, $s \rightarrow s+1$. If $\theta$ is position like then $s \rightarrow s-1$.
We next build a chiral Hamiltonian from a non chiral one by adding a symmetry. This is accomplished by doubling the number of bands in a manner similar to the doubling employed in the Bogoliubov de Gennes description of a superconductor. We thus write $${\cal H}_c({\bf k},{\bf r},\theta) = \cos\theta {\cal H}_{nc}({\bf k},{\bf r})
\otimes \tau_z + \sin \theta \openone\otimes\tau_a,
\label{nctoc}$$ where $a = x$ or $y$. Here $\vec\tau$ are Pauli matrices that act on the doubled degree of freedom. As in , gives a new Hamiltonian defined on a base space that is the suspension of the original base space. If ${\cal H}_{nc}^2 = 1$ it follows that ${\cal H}_c^2=1$, so the energy gap is preserved. It is also clear that the new Hamiltonian has a chiral symmetry because it anticommutes with $\Pi = i \tau_z \tau_a$. Thus, if ${\cal H}_{nc}$ is in class A, then ${\cal H}_c$ is in class AIII.
For the real symmetry classes $a=x$ or $y$ must be chosen so that the second term in preserves the original anti unitary symmetry of ${\cal H}_{nc}$. This depends on the original anti unitary symmetry and whether $\theta$ is chosen to be a momentum or a position variable. For example, if ${\cal H}_{nc}$ has time reversal symmetry, $\Theta$, and $\theta$ is a momentum (position) variable, then we require $a=y$ ($a=x$). In this case, ${\cal H}_c$ has the additional particle-hole symmetry $\Xi = \tau_x \Theta$ ($\Xi = i\tau_y \Theta$) that satisfies $\Xi^2 =
\Theta^2$ ($\Xi^2 = -\Theta^2$). A similar analysis when ${\cal H}_{nc}$ has particle hole symmetry allows us to conclude that the symmetry class of ${\cal H}_c$ is given by a clockwise rotation on the symmetry clock, $s \rightarrow s+1$, when $\theta$ is a momentum variable. When $\theta$ is a position variable, $s \rightarrow s-1$ gives a counter clockwise rotation.
Equations and map a Hamiltonian into a new Hamiltonian in a different dimension and different symmetry class. It is clear that two Hamiltonians that are topologically equivalent will be mapped to topologically equivalent Hamiltonians, since the mapping can be done continuously on a smooth interpolation between the original Hamiltonians. Thus, (\[ctonc\],\[nctoc\]) define a mapping between equivalence classes of Hamiltonians. Moreover, since the direct sum of two Hamiltonians is mapped to the direct sum of the new Hamiltonians, the group property of the equivalence classes is preserved. (\[ctonc\],\[nctoc\]) thus define a $K$-group homomorphism, $$\begin{aligned}
K_{\mathbb{F}}(s;D,d)&\longrightarrow&K_{\mathbb{F}}(s+1;D,d+1)\label{arrowm},\\
K_{\mathbb{F}}(s;D,d)&\longrightarrow&K_{\mathbb{F}}(s-1;D+1,d)\label{arrowp},\end{aligned}$$ for $\mathbb{F}=\mathbb{R},\mathbb{C}$.
Invertibility {#appendix:invertivility}
-------------
In order to establish that (\[arrowm\],\[arrowp\]) are isomorphisms we need to show that there exists an inverse. This is [*not*]{} true of the Hamiltonian mappings. A general Hamiltonian cannot be built from a lower dimensional Hamiltonian using (\[ctonc\],\[nctoc\]). However, we will argue that it is possible to continuously deform any Hamiltonian into the form given by or . Thus, the mappings between equivalence classes have an inverse. To show this we will use a mathematical method borrowed from Morse theory[@JMilnor].
Without loss of generality we again consider [*flattened*]{} Hamiltonians having equal number of conduction and valence bands with energies $\pm 1$. Consider ${\cal H}({\bf k},{\bf r},\theta)$, where $\theta \in [-\pi/2,\pi/2]$ is either a position or momentum variable and ${\cal H}$ is independent of ${\bf k}$ and ${\bf r}$ at $\theta = \pm\pi/2$. We wish to show that ${\cal H}({\bf k},{\bf r},\theta)$ can be continuously deformed into the form or . To do so we define an artificial “action" $$\label{Haction}
S[\mathcal{H}({\bf k},{\bf r},\theta)]=\int d\theta d^d{\bf k}d^D{\bf
r}\mbox{Tr}\left(\partial_\theta\mathcal{H}\partial_\theta\mathcal{H}\right).$$ $S$ can be interpreted as a “height" function in the space of gapped symmetry preserving Hamiltonians. Given any Hamiltonian there is always a downhill direction. These downhill vectors can then be integrated into a deformation trajectory. Since the action is positive definite, it is bounded below. The deformation trajectory must end at a Hamiltonian that locally minimizes the action.
Under the flatness constraint $\mathcal{H}^2=1$, minimal Hamiltonians satisfy the Euler-Lagrange equation $$\label{HEL}
\partial^2_\theta\mathcal{H}+\mathcal{H}=0.$$ The solutions must be a linear combination of $\sin\theta$ and $\cos\theta$. The coefficient of $\sin\theta$ must be constant because the base space is compactified to points at $\theta=\pm\pi/2$. A minimal Hamiltonian thus has the form $$\mathcal{H}({\bf k},{\bf
r},\theta)=\cos\theta\mathcal{H}_1({\bf k},{\bf
r})+\sin\theta\mathcal{H}_0\label{Asol}.$$ The constraint $\mathcal{H}({\bf k},{\bf r},\theta)^2=1$ requires $$\mathcal{H}_0^2=\mathcal{H}_1({\bf k},{\bf
r})^2=1,\quad\{\mathcal{H}_0,\mathcal{H}_1({\bf k},{\bf
r})\}=0.
\label{pAsol}$$
If $\mathcal{H}({\bf k},{\bf r},\theta)$ is non-chiral, then eq. is already in the form of eq. with $\Pi=\mathcal{H}_0$ and $\mathcal{H}_c({\bf k},{\bf r}) = \mathcal{H}_1({\bf k},{\bf r})$. $\mathcal{H}_1$ automatically has chiral symmetry due to Eq.. This shows that and are invertible when $s$ is odd.
If $\mathcal{H}({\bf k},{\bf r},\theta)$ is chiral, then both $\mathcal{H}_0$ and $\mathcal{H}_1({\bf k},{\bf r})$ anticommute with the chiral symmetry operator $\Pi$. Rename $\mathcal{H}_0=\tau_a$ and $\Pi=i\tau_z \tau_a$, where $a = x$ ($a=y$) when $\theta$ is a position (momentum) variable. It follows that $\{\mathcal{H}_1,\tau_x\}=\{\mathcal{H}_1,\tau_y\}=0$, so we can write $$\mathcal{H}_1({\bf k},{\bf r})=h({\bf k},{\bf r})\otimes\tau_z.$$ Eq. thus takes the form of Eq. with ${\cal H}_{nc} = h$. Since $\tau_z$ anti-commutes with either $\Theta$ or $\Xi$, $h({\bf k},{\bf r})$ carries exactly one anti-unitary symmetry and is therefore non-chiral. This shows that (\[arrowm\]) and (\[arrowp\]) are invertible when $s$ is even.
Representative Hamiltonians, and classification by winding numbers {#appendix:representatives}
==================================================================
In this appendix we construct representative Hamiltonians for each of the symmetry classes that are built as linear combinations of Clifford algebra generators that can be represented as anticommuting Dirac matrices. This allows us to relate the integer topological invariants, corresponding to the $\mathbb{Z}$ and $2\mathbb{Z}$ entries in Table I, to the winding degree in maps between spheres. Similar construction for defectless bulk Hamiltonians can be found in Ref.\[\] by Ryu, [*et.al*]{}. In general, Hamiltonians do not have this specific form. However, since each topological class of Hamiltonians includes representatives of this form, it is always possible to smoothly deform ${\cal H}({\bf k},{\bf r})$ into this form.
The simplest example of this approach is the familiar case of a two dimensional Hamiltonian with no symmetries (class $A$). A topologically non trivial Hamiltonian can be represented as a $2\times 2$ matrix that can be expressed in terms of Pauli matrices as ${\cal H}({\bf k}) = {\bf h}({\bf k})\cdot\vec\sigma$. The Hamiltonian can then be associated with a unit vector $\hat {\bf d}({\bf
k}) = {\bf h}({\bf k})/|{\bf h}({\bf k})| \in S^2$. It is then well known that the Chern number characterizing ${\cal H}({\bf k})$ in two dimensions is related to the [*degree*]{}, or winding number, of the mapping from ${\bf k}$ to $S^2$. This approach also applies to higher Chern numbers characterizing Hamiltonians in even dimensions $d=2n$. In this case, a Hamiltonian that is a combination of $2n+1$ $2^n\times 2^n$ Dirac matrices, and can be associated with a unit vector $\hat{\bf
d} \in S^{2n}$.
For the complex chiral class AIII, the $U(n)$ winding number characterizing a family of Hamiltonians can similarly be expressed as a winding number on spheres. For example, in $d=1$, a chiral Hamiltonian can be written ${\cal H}(k) = h_x(k) \sigma_x +
h_y(k)\sigma_y$ (so $\{{\cal H},\sigma_z\}=0$), and is characterized by $\hat{\bf d}(k) \in S^1$. The integer topological invariant can then be expressed by the winding number of $\hat{\bf d}(k)$. Similar considerations apply to the integer invariants for chiral Hamiltonians in higher odd dimensions.
For the real symmmetry classes we introduce “position type" Dirac matrices $\Gamma_\mu$ and “momentum type" Dirac matrices $\gamma_i$. These satisfy $\{\Gamma_\mu,\Gamma_\nu\}=2\delta_{\mu\nu}$, $\{\gamma_i,\gamma_j\}=2\delta_{ij}$, $\{\Gamma_\mu,\gamma_j\}=0$ and are distinguished by their symmmetry under anti unitary symmetries. If there is time reversal symmetry we require $$[\Gamma_\mu,\Theta] = \{\gamma_i,\Theta\} = 0,$$ while with particle-hole symmetry, $$\{\Gamma_\mu,\Xi\} = [\gamma_i,\Xi] = 0.$$
For a Hamiltonian that is a combination of $p$ momentum like matrices $\gamma_{1,\ldots,p}$ and $q+1$ position like matrices $\Gamma_{0,\ldots,q}$, $$\mathcal{H}({\bf k},{\bf r})={\bf R}({\bf k},{\bf
r})\cdot\vec\Gamma+{\bf K}({\bf k},{\bf
r})\cdot\vec\gamma,
\label{Hmodel}$$ the coefficients must satisfy the involution $$\begin{aligned}
{\bf R}(-{\bf k},{\bf r})&=&{\bf R}({\bf k},{\bf r}),\label{Diracinvol1}\\
{\bf K}(-{\bf k},{\bf r})&=&-{\bf K}({\bf k},{\bf r})\label{Diracinvol2}.\end{aligned}$$ This can be characterized by a unit vector $$\hat{\bf d}({\bf k},{\bf r}) = {\frac{({\bf K},{\bf R})}{\sqrt{|{\bf
K}|^2 + |{\bf R}|^2}}} \in S^{p+q},$$ where $S^{p+q}$ is a $(p+q)$-sphere in which $p$ of the dimensions are odd under the involution (\[Diracinvol2\]).
The symmetry class $s$ of ${\cal H}({\bf k},{\bf r})$ is related to the indices $(p,q)$ characterizing the numbers of Dirac matrices by $$p-q = s \ {\rm mod} \ 8.
\label{p-q}$$ To see this, start with a Hamiltonian ${\cal H}_0 = R_0({\bf k},{\bf r})
\Gamma_0$ that involves a single $1\times 1$ position like “Dirac matrix" $\Gamma_0 = \openone$, so $(p,q)=(0,0)$. This clearly has time reversal symmetry, with $\Theta = K$, and corresponds to class AI with $s=0$. Next, generate Hamiltonians ${\cal H}_s$ with different symmetries $s$ by using the Hamiltonian mappings introduced in appendix \[appendix:hamiltonianmaps\]. Both the mappings and define a new Clifford algebra with one extra generator that is either position or momentum type. The mappings that correspond to clockwise rotations on the symmetry clock ($s\rightarrow s+1$) introduce an additional position like generator ($p
\rightarrow p+1$), while the mappings that correspond to counterclockwise rotations ($s\rightarrow s-1$) introduce an additional momentum like generator ($q \rightarrow q+1$). Eq. follows because this procedure can be repeated to generate Hamiltonians with any indices $(p,q)$. Some examples are listed in table \[exDiracMatrix\]
--- ------ ------------------ ------------------ ------------------ ---------- ---------- -------------------- ------------- ------------
s AZ $\Gamma_0$ $\vec\gamma$ $\Theta$ $\Xi$ $\Pi$
0 AI $K$
1 BDI $\sigma_z$ $\sigma_y$ $K$ $\sigma_xK$ $\sigma_x$
2 D $\sigma_z$ $\sigma_y$ $\sigma_x$ $\sigma_xK$
3 DIII $\tau_z\sigma_z$ $\tau_z\sigma_y$ $\tau_z\sigma_x$ $\tau_x$ $i\tau_y\sigma_xK$ $\sigma_xK$ $\tau_y$
4 AII $\tau_z\sigma_z$ $\tau_z\sigma_y$ $\tau_z\sigma_x$ $\tau_x$ $\tau_y$ $i\tau_y\sigma_xK$
--- ------ ------------------ ------------------ ------------------ ---------- ---------- -------------------- ------------- ------------
: Examples of Dirac matrices for $(p,q)=(s,0)$. []{data-label="exDiracMatrix"}
The integer topological invariants in Table \[tab:periodic\] (which occur when $s-\delta$ is even) can be related to the winding degree of the maps $\hat {\bf d}: S^{D+d} \rightarrow
S^{p+q}$. This can be non zero when the spheres have the same total dimensions. In light of , $(p,q)$ can always be chosen so that $d+D=p+q$. The anti unitary symmetries impose constraints on the possible values of these winding numbers, which depend on the relation between $\delta = d-D$ and $s=p-q$.
The involutions on $S^{d+D}$ and $S^{p+q}$ have opposite orientations when $\delta-s\equiv2$ or 6 mod 8, and therefore an involution preserving map $S^{d+D}\to S^{p+q}$ can have non-zero winding degree only when $\delta-s\equiv0$ or 4 mod 8. Symmetry gives a further constraint on the latter case. Consider a sphere map $S^{2}_{\theta,\phi}\to
S^{2}_{\vartheta,\varphi}$, where the involutions on the spheres send $(\theta,\phi)\mapsto(\theta,\phi+\pi)$ and $(\vartheta,\varphi)\mapsto(\vartheta,\varphi)$. In order for $\varphi(\theta,\phi)=\varphi(\theta,\phi+\pi)$, the winding number must be even. Together, these show $$\deg\in\left\{\begin{array}{*{20}c}\mathbb{Z},&\mbox{for
$\delta-s\equiv0$ (mod 8)}\\2\mathbb{Z},&\mbox{for $\delta-s\equiv4$ (mod
8)}\\0,&\mbox{otherwise}\hfill\end{array}\right.$$ This gives a topological understanding of the $\mathbb{Z}$’s and $2\mathbb{Z}$’s on the periodic table in terms of winding number, which can be identified with the more general analytic invariants, namely Chern numbers for non-chiral classes $$n=\frac{1}{\left(\frac{d+D}{2}\right)!}\left(\frac{i}{2\pi}\right)^{\frac{d+D}{2}}
\int_{T^d\times S^D}\mbox{Tr}\left(\mathcal{F}^{\frac{d+D}{2}}\right)$$ and winding numbers of the chiral flipping operator $q({\bf k},{\bf r})$ for chiral ones (See Eqs. \[hchiral\] and \[nchiral\])). $$n=\frac{\left(\frac{d+D-1}{2}\right)!}{(d+D)!(2\pi i)^{\frac{d+D+1}{2}}}
\int_{T^d\times S^D}\mbox{Tr}\left((qdq^\dagger)^{d+D}\right)$$
The $\mathbb{Z}_2$’s on the periodic table are not directly characterized by winding degree, but rather through dimensional reduction. Given a Hamiltonian $\mathcal{H}({\bf k},k_1,k_2,{\bf r})$ with $s\equiv\delta$ mod 8, its winding degree mod 2 determines the $\mathbb{Z}_2$-classification of its equatorial offspring $\mathcal{H}_{k_2=0}({\bf k},k_1,{\bf r})$ and $\mathcal{H}_{k_{1,2}=0}({\bf k},{\bf r})$. For example, topological insulators in two and three dimensions are equatorial restrictions of a four dimensional model $\hat{\bf d}:S^{4}\xrightarrow{=}S^{4}$ with unit winding number. Around the north pole, the Hamiltonian has the form $$\mathcal{H}({\bf k},k_4)=(m+\varepsilon
k^2)\mu_1+{\bf k}\cdot\mu_3\vec\sigma+k_4\mu_2$$ and on the equator $k_4=0$, this gives a three dimensional Dirac theory of mass $m$ around ${\bf k=0}$ that locally describes 3D topological insulators $\mbox{Bi}_2\mbox{Se}_3$ and Bi$_2$Te$_3$ around $\Gamma$.
Zero Modes in the Harmonic Oscillator Model {#appendix:ZM}
===========================================
We present exact solvable soliton states of Dirac-type defect Hamiltonians. These include zero modes at a point defect of a Hamiltonian in the chiral class AIII, and chiral modes along a line defect of a Hamiltonian in the non-chiral class A. We establish the connection between the two kinds of boundary modes through the Hamiltonian mapping .
A non-trivial chiral Hamiltonian isotropic around a point defect at ${\bf r}=0$ is a Dirac operator $$\mathcal{H}=-i\vec\gamma\cdot\nabla+{\bf
r}\cdot\vec\Gamma\label{app-chd}$$ where the chiral operator is $\Pi=i^d\prod_{j=1}^d\gamma_j\Gamma_j$, and its adiabatic limit $e^{-i{\bf k}\cdot{\bf r}}\mathcal{H}e^{i{\bf k}\cdot{\bf
r}}={\bf k}\cdot\vec\gamma+{\bf r}\cdot\vec\Gamma$ has unit winding degree on $S^{2d-1}=\{({\bf k},{\bf
r}):k^2+r^2=1\}$.$$\mathcal{H}^2=-\nabla^2+r^2-i\vec\gamma\cdot\vec\Gamma$$ and the spectrum is determined by the quantum numbers $n_j\geq0$ of the harmonic oscillator and the parities $\xi_j$ of the mutually commuting matrices $i\gamma_j\Gamma_j$, for $j=1,\ldots,d$. $$\mathcal{E}^2=\sum_{j=1}^d 2n_j+1 -\xi_j$$ The unique zero energy state $|\Psi_0\rangle$, indexed by $n_j=0$ and $\xi_j=1$, has positive chirality $\Pi=+1$, and is exponentially localized at the point defect as $\Psi_0({\bf r})\propto
e^{-\frac{1}{2}r^2}$.
Next we consider a non-chiral Hamiltonian isotropic along a line defect. $$\mathcal{H}(k_\|)=k_\|\Pi-i\vec\gamma\cdot\nabla+{\bf
r}\cdot\vec\Gamma\label{app-nhd}$$ where $k_\|$ is parallel to the defect line, ${\bf r}$ and $\nabla$ are normal position and derivative. Its adiabatic limit $e^{-i{\bf k}\cdot{\bf
r}}\mathcal{H}(k_\|)e^{i{\bf k}\cdot{\bf r}}=k_\|\Pi+{\bf
k}\cdot\vec\gamma+{\bf r}\cdot\vec\Gamma$ is related to that of (\[app-chd\]) by (1,1)-periodicity, and has unit winding degree on $S^{2d}=\{(k_\|,{\bf k},{\bf r}):k_\|^2+|{\bf k}|^2+|{\bf
r}|^2=1\}$. The zero mode $|\Psi_0\rangle$ of (\[app-chd\]) gives rise to a positive chiral mode, $\mathcal{H}(k_\|)|\Psi_0\rangle=k_\|\Pi|\Psi_0\rangle=+k_\||\Psi_0\rangle$.
\[engplot\]
The two examples verified bulk-boundary correspondence through identifying analytic information of the defect-bound solitons and the topology of slowly spatial modulated theories far away from the defect. The single zero mode of (\[app-chd\]) and spectral flow of (\[app-nhd\]) are equated to unit winding degree of an adiabatic limit. In general, bulk-boundary correspondence is mathematically summarized by index theorems that associate certain analytic and topological indices of Hamiltonians.[@jackiwrebbi; @jackiwrossi; @weinberg; @volovik03; @fukui10a; @fukui10b]
Invariant for Point Defects in Class D and BDI {#appendix:pointdefectclassd}
==============================================
We follow the derivation given in Ref.\[\], which was based on Qi, Hughes and Zhang’s formulation of the topological invariant characterizing a three dimensional topological insulator[@qihugheszhang08]. For a point defect in $d$ dimensions, the Hamiltonian ${\cal H}({\bf k},{\bf r})$ depends on $d$ momentum variables and $d-1$ position variables. We introduce a one parameter deformation $\widetilde{\cal H}(\lambda,{\bf
k},{\bf r})$ that connects $\widetilde{\cal H}({\bf k},{\bf r})$ at $\lambda =0$ to a constant Hamiltonian at $\lambda = 1$, while breaking particle-hole symmetry. The particle-hole symmetry can be restored by including a mirror image $\tilde {\cal H}(\lambda,{\bf k},{\bf r}) = - \Xi {\cal H}(-\lambda,{\bf
k},{\bf r})\Xi^{-1}$ for $-1<\lambda<0$. For $\lambda = \pm $, $({\bf k},{\bf r})$ can be replaced by a single point, so the $2d$ parameter space $(\lambda,{\bf k},{\bf r})$ is the suspension $\Sigma(T^d\times S^{d-1})$ of the original space. The Hamiltonian defined on this space is characterized by its $d$’th Chern character $$\nu = {\frac{1}{d!}} \left({\frac{i}{2\pi}}\right)^d \int_{\Sigma(T^d\times S^{d-1})} {\rm
Tr}[ {\cal F}^d ].$$ Due to particle-hole symmetry, the contributions from the two hemispheres $\lambda >0$, $\lambda<0$ are equal. Using the fact that the integrand is the derivative of the Chern Simons form, ${\rm Tr}[{\cal F}^d] = d{\cal Q}_{2d-1}$, we can therefore write $$\label{CSD}
\nu = \frac{2}{d!}\left(\frac{i}{2\pi}\right)^d \int_{T^d\times S^{d-1}} {\cal Q}_{2d-1}$$ As was the case in Refs.\[\], $\nu$ can be different for different deformations ${\cal
H}(\lambda,{\bf k},{\bf r})$. However, particle-hole symmetry requires the difference is an even integer. Thus, the parity of defines the $\mathbb{Z}_2$ invariant.
The Chern Simons form $\mathcal{Q}_{2d-1}$ can be expressed in terms of the connection $\mathcal{A}$ via the general formula $$\label{defCS}
\mathcal{Q}_{2d-1}=d\int_0^1dt\mbox{Tr}\left[\mathcal{A}(td\mathcal{A}+t^2\mathcal{A}^2)^{d-1}
\right]$$
In the addition of time reversal symmetry $\Theta^2=1$ or equivalently a chiral symmetry $\Pi=\Theta\Pi=\tau_z$, a valence frame of the BDI Hamiltonian can be chosen to be $$u({\bf k},{\bf
r})=\frac{1}{\sqrt{2}}\left(\begin{array}{*{20}c}q({\bf k},{\bf
r})\\-\openone\end{array}\right)$$ where $q$ is unitary and $\openone$ is the identity matrix. This corresponds the Berry connection $\mathcal{A}=u^\dagger du=\frac{1}{2}q^\dagger dq$ and Chern-Simons form $$\begin{aligned}
Q_{2d-1}&=&\frac{d}{2}\int_0^1dt\left(\frac{t}{2}\left(\frac{t}{2}-1\right)\right)^{d-1}\mbox{Tr}\left[(q^\dagger
dq)^{2d-1}\right]\nonumber\\&=&\frac{(-1)^d}{2}\frac{d!(d-1)!}{(2d-1)!}\mbox{Tr}\left[(q^\dagger
dq)^{2d-1}\right]\label{CS=wn/2}\end{aligned}$$ This equates the winding number of $q$ to the Chern Simons invariant .
Invariant for line defects in class AII and point defects in DIII {#appendix:defectsaiidiii}
=================================================================
We formulate a topological invariant that characterizes line defects in class AII in all dimensions that is analogous to the integral formula invariant characterizing the quantum spin Hall insulator introduced in Ref. . This can be applied to weak topological insulators in three dimensions with dislocation around a line defect. The invariant can be indirectly applied to strong topological insulators through decomposition into strong and weak components. As a consequence of the Hamiltonian mapping that identifies $(s=4,\delta=2)$ and $(s=3,\delta=1)$, this gives a new topological invariant that classified point defects in class DIII in all dimensions.
Line defects in class AII {#appendix:defectsaii}
-------------------------
The base space manifold is $\mathcal{T}^{2d-2}=T^d\times S^{d-2}$, where $T^d$ is the Brillouin zone and $S^{d-2}\times\mathbb{R}$ is a cylindrical neighborhood that wraps around the line defect in real space. Divide the base space into two pieces, $\mathcal{T}^{2d-2}_{1/2}$ and its time reversal counterpart (see fig.\[FK31\](a)). We will show the $\mathbb{Z}_2$-invariant $$\nu=\frac{i^{d-1}}{(d-1)!(2\pi)^{d-1}}
\left[\int_{\mathcal{T}^{2d-2}_{1/2}}\mbox{Tr}(\mathcal{F}^{d-1}) -\oint_{\partial\mathcal{T}^{2d-2}_{1/2}}\mathcal{Q}_{2d-3}\right]
\label{FKf}$$ topologically classifies line defects in AII, where the Chern-Simons form, defined by , is generated by the Berry connection $\mathcal{A}_{mn}=\langle u_m({\bf k},{\bf r})|du_n({\bf k},{\bf r})\rangle$, and the valence frame $u_m({\bf k},{\bf r})$ satisfies the gauge condition $$\label{FKgc}w_{mn}({\bf k},{\bf r})
=\langle u_m({\bf k},{\bf r})|\Theta u_n(-{\bf k},{\bf r})\rangle=
\mbox{constant}$$ on the boundary $({\bf k},{\bf r})\in\partial\mathcal{T}^{2d-2}_{1/2}$.
=3.0in
The non-triviality of the $\mathbb{Z}_2$-invariant is a topological obstruction to choosing a global continuous valence frame $|u_m({\bf k},{\bf r})\rangle$ that satisfies the gauge condition (\[FKgc\]) on the whole base space $\mathcal{T}^{2d-2}$. If there is no topological obstruction from the bulk[@ldtopobst], the gauge condition forces the valence frame to be singular at two points, depicted in fig.\[FK31\](b), related to each other by time reversal. One removes the singularity by picking another valence frame locally defined on two small balls enclosing the two singular points, denoted by $B$ in fig.\[FK31\](b). We therefore have two valence frames $|u_m^{A/B}({\bf k},{\bf r})\rangle$ defined on two patches of the base space, $A=\mathcal{T}^{2d-2}\backslash B$ and B, each obeying the gauge condition (\[FKgc\]).
The wavefunctions on the two patches translate into each other through transition function $$t^{AB}_{mn}({\bf k},{\bf r})
=\langle u_m^A({\bf k},{\bf r})|u_n^B({\bf k},{\bf r})\rangle\in U(k)$$ on the boundary $\partial B\approx S^{2d-3}\cup S^{2d-3}$. The function behavior on the two disjoint $(2d-3)$-spheres is related by time reversal. The topology is characterized by the winding of $t^{AB}:S^{2d-3}\to U(k)$ on one of the spheres. $$\begin{aligned}
\nu&=&\frac{(d-2)!}{(2d-3)!(2\pi i)^{d-1}}\oint_{S^{2d-3}}\mbox{Tr}\left[\left(t^{AB}d(t^{AB})^\dagger\right)^{2d-3}\right]
\quad\quad\label{windingFK}\\&=&\frac{(-1)^d}{(d-1)!(2\pi i)^{d-1}}\oint_{S^{2d-3}}\left(\mathcal{Q}_{2d-3}^A-\mathcal{Q}_{2d-3}^B\right)
\label{FKtf}\end{aligned}$$ The two integrals can be evaluated separately. Since $d\mathcal{Q}_{2d-3}=\mbox{Tr}(\mathcal{F}^{d-1})$, Stokes’ theorem tells us $$\begin{aligned}
\int_{A\cap \mathcal{T}^{2d-2}_{1/2}}\mbox{Tr}(\mathcal{F}^{d-1})
&=&\left(\oint_{\partial\mathcal{T}^{2d-2}_{1/2}}
-\oint_{S^{2d-3}}\right)\mathcal{Q}_{2d-3}^A\nonumber\\
\int_{B\cap \mathcal{T}^{2d-2}_{1/2}}\mbox{Tr}(\mathcal{F}^{d-1})
&=&\oint_{S^{2d-3}}\mathcal{Q}_{2d-3}^B\nonumber\end{aligned}$$ Combining these into eq.(\[FKtf\]) identifies the $\mathbb{Z}_2$-invariant (\[FKf\]) with the winding number of the transition function.
The curvature term in (\[FKf\]) is gauge invariant. Any gauge transformation on the boundary $\partial\mathcal{T}^{2d-2}_{1/2}$ respecting the gauge condition (\[FKgc\]) has even winding number and would alter the Chern-Simons integral by an even integer. The gauge condition is therefore essential to make the formula non-vacuous.
### Spin Chern number {#appendix:spinchernnumber}
A quantum spin Hall insulator is characterized by its spin Chern number $n_\sigma=(n_\uparrow-n_\downarrow)/2$. We generalize this to time reversal invariant line defects of all dimensions by equating it with Eq.(\[FKf\]). This applies in particular to a model we considered for a linear Josephson junction in section \[sec:linehelicalmajorana\].
A spin operator $S$ is a unitary operator, square to unity, commutes with the Hamiltonian, and anticommutes with the time reversal operator. The valence spin frame $$|u_m^\downarrow({\bf k},{\bf r})\rangle
=\Theta|u_m^\uparrow(-{\bf k},{\bf r})\rangle$$ automatically satisfies the time reversal gauge constraint (\[FKgc\]). It is straightforward to check that the curvature and Chern-Simons form can be split as direct sums according to spins. $$\begin{aligned}
\mathcal{F}({\bf k},{\bf r})&=&\mathcal{F}^\uparrow({\bf k},{\bf r})\oplus
\mathcal{F}^\uparrow(-{\bf k},{\bf r})^\ast\\
\mathcal{Q}_{2d-3}({\bf k},{\bf r})&=&\mathcal{Q}_{2d-3}^\uparrow({\bf k},{\bf r})\oplus
\mathcal{Q}_{2d-3}^\uparrow(-{\bf k},{\bf r})^\ast\end{aligned}$$ Again assuming that there is no lower dimensional “weak" topology, the $\uparrow$-frame can be defined everywhere on $\mathcal{T}^{2d-2}$ with a singularity at one point, say in $\mathcal{T}^{2d-2}_{1/2}$, and the $\downarrow$-frame is singular only at the time reversal of that point.
The curvature term of (\[FKf\]) splits into two terms $$\int_{\mathcal{T}^{2d-2}_{1/2}}\mbox{Tr}(\mathcal{F}^{d-1})
=\left[\int_{\mathcal{T}^{2d-2}_{1/2}}-\int_{\mathcal{T}^{2d-2}\backslash\mathcal{T}^{2d-2}_{1/2}}\right]
\mbox{Tr}(\mathcal{F}^{d-1}_\uparrow)$$ And the two spin components of the Chern-Simons term $\oint_{\partial\mathcal{T}^{2d-2}_{1/2}}\mathcal{Q}_{2d-3}$ add up into $$2\oint_{\partial\mathcal{T}^{2d-2}_{1/2}}\mathcal{Q}^\uparrow_{2d-3}
=-2\int_{\mathcal{T}^{2d-2}\backslash\mathcal{T}^{2d-2}_{1/2}}
\mbox{Tr}(\mathcal{F}^{d-1}_\uparrow)$$ by Stokes theorem.
Combining these two, we equate (\[FKf\]) to the spin Chern number $$n_\uparrow=\frac{i^{d-1}}{(d-1)!2\pi^{d-1}}
\int_{\mathcal{T}^{2d-2}}\mbox{Tr}(\mathcal{F}^{d-1}_\uparrow)$$ Time reversal requires $n_{tot}=n_\uparrow+n_\downarrow=0$ and therefore $n_\sigma=(n_\uparrow-n_\downarrow)/2=n_\uparrow$.
Point defects in class DIII {#appendix:pdDIII}
---------------------------
The base space manifold is $\mathcal{T}^{2d-1}=T^d\times S^{d-1}$. The Hamiltonian mapping relates a point defect Hamiltonian $\mathcal{H}({\bf k},{\bf r})$ in class DIII to a line defect Hamiltonian $\mathcal{H}({\bf k},{\bf r},\theta)=\cos\theta\mathcal{H}({\bf k},{\bf r})
+\sin\theta\Pi$ in class AII, where $\Pi=i\Theta\Xi$ is the chiral operator, $({\bf k},{\bf r},\theta)\in\Sigma\mathcal{T}^{2d-1}$ (see fig.\[susp2\](a)) and $\theta$ is odd under time reversal.
=2.5in
The line defect Hamiltonian $\mathcal{H}({\bf k},{\bf r},\theta)$ is topologically characterized by the generalization of (\[FKf\]), which was proven to be identical to the winding number (\[windingFK\]) of the transition function $t^{AB}$ (see fig.\[susp2\](b) for the definition of patches $A$ and $B$). We will utilize this to construct a topological invariant that characterizes point defects in class DIII.
Set $\Theta=\tau_yK$ and $\Xi=\tau_xK$ under an appropriate choice of basis. A [*canonical*]{} valence frame of $\mathcal{H}({\bf k},{\bf r},\theta)$ can be chosen to be $$\label{vfpDIII}u^B_+({\bf k},{\bf r},\theta)
=\left(\begin{array}{*{20}c}\sin\left(\frac{\pi}{4}-\frac{\theta}{2}\right)q({\bf k},{\bf r})\\
-\cos\left(\frac{\pi}{4}-\frac{\theta}{2}\right)\openone\end{array}\right)$$ where $q({\bf k},{\bf r})\in U(k)$ is from the canonical form of the chiral Hamiltonian $\mathcal{H}({\bf k},{\bf r})$ in eq., $\openone$ is the $k\times k$ identity matrix, and the valence frame is non-singular everywhere except at $\theta=-\pi/2$. There is a gauge transformation $u^B_+\to u^A=u^B_+t^{BA}$ everywhere except $\theta=\pm\pi/2$ such that the new frame $u^A$ satisfies the gauge condition (\[FKgc\]).[@ldtopobst2] A valence frame on patch B can be constructed by requiring $u^B_-(\theta)=\Theta u^B_+(-\theta)$ around $\theta=-\pi/2$.
The $\mathbb{Z}_2$-topology is characterized by the evenness or oddness of the winding number of $t^{AB}$ as in . This can be evaluated by the integral along the equator $\theta=0$ $$\label{wnpDIII}\tilde\nu=\frac{(d-1)!}{(2d-1)!(2\pi i)^d}\int_{T^d\times S^{d-1}}\mbox{Tr}\left[\left(t^{AB}d(t^{AB})^\dagger\right)^{2d-1}\right]$$ where $u^A=u^B_+t^{BA}$ is a solution to the gauge condition (\[FKgc\]), or equivalently $t^{BA}$ satisfies $$\label{FKgcq}q({\bf k},{\bf r})
=t^{BA}(-{\bf k},{\bf r})\sigma_yt^{BA}({\bf k},{\bf r})^T$$ where the constant in eq.(\[FKgc\]) is chosen to be $i\sigma_y$.
The winding number can also be expressed as a Chern-Simons integral. $$\tilde\nu=\frac{i^d}{d!(2\pi)^d}\int_{T^d\times S^{d-1}}
\left(\mathcal{Q}^B_{2d-1}-\mathcal{Q}^A_{2d-1}\right)\label{DIIICS1}$$ where $\mathcal{Q}^{A/B}$ are the Chern-Simons form generated by valence frames $u^{A/B}$. Restricted to $\theta=0$, gives $u^B({\bf k},{\bf r})=\frac{1}{\sqrt{2}}(q({\bf k},{\bf r}),-\openone)$. Following , the first term of equals half of the winding number of $q$, which is guaranteed to be zero by time reversal and particle-hole symmetries. And therefore point defects in DIII are classified by the Chern-Simons invariant $$\tilde\nu=\frac{1}{d!}\left(\frac{i}{2\pi}\right)^d\int_{T^d\times S^{d-1}}
\mathcal{Q}_{2d-1}\quad\mbox{mod 2}\label{DIIICS2}$$ where the Chern-Simons form is generated by a valence frame that satisfies the time reversal gauge constraint .
Note that the integrality of the Chern-Simons integral is a result of particle-hole symmetry. Forgetting time reversal symmetry, point defects in class D are classified by the Chern-Simons invariant $\nu=2\tilde\nu$ with a factor of 2. Time reversal symmetry requires the zero modes to form Kramers doublets, and therefore $\nu=2\tilde\nu$ must be even. A gauge transformation in general can alter $\tilde\nu$ by any integer. Thus, similar to the formula in class AII, the time reversal gauge constraint is essential so that is non-vacuous.
### Fixed points formula in 1D {#appendix:fixedpointformulas}
We here identify , or equivalently , to a fixed point invariant in 1 dimension. In ref.\[\], Qi, Hughes and Zhang showed that 1D TRI superconductors are $\mathbb{Z}_2$-classified by the topological invariant $$\label{QHZTRIS}
(-1)^{\tilde\nu}=\frac{\mbox{Pf}(q_{k=\pi})}{\mbox{Pf}(q_{k=0})}
\exp\left(\frac{1}{2}\int_0^\pi\mbox{Tr}(q_kdq_k^\dagger)\right)$$ under the basis $\Theta=\tau_yK$ and $\Xi=\tau_xK$, where $q_k$ is the chiral flipping operator in . Time reversal and particle-hole symmetry requires $q_k=-q_{-k}^T$. Hence the Pfaffians are well defined as $q_k$ is antisymmetric at the fixed points $k=0,\pi$.
Using the gauge condition , we can expressed the Pfaffians as $\mbox{Pf}(\Theta q_{k=0,\pi})=\det(t_{k=0,\pi})
\mbox{Pf}(\sigma_y)$, where $t_k^{BA}$ is abbreviated to $t_k$. $$\frac{\mbox{Pf}(q_{k=\pi})}
{\mbox{Pf}(q_{k=0})}=\exp\left(-\int_0^\pi
\mbox{Tr}\left(t_kdt_k^\dagger\right)\right)$$ Substitute into the Cartan form $\mbox{Tr}(q_kdq_k^\dagger)$ gives $$\mbox{Tr}\left(q_kdq_k^\dagger\right)
=\mbox{Tr}\left(t_{-k}dt_{-k}^\dagger\right)
+\mbox{Tr}\left(t_kdt_k^\dagger\right)$$ Combining these into , $$\begin{aligned}
(-1)^{\tilde\nu}&=&\exp\left(\frac{1}{2}\int_0^\pi
\mbox{Tr}\left(t_{-k}dt_{-k}^\dagger-t_kdt_k^\dagger\right)\right)\\
&=&\exp\left(-\frac{1}{2}\int_{-\pi}^\pi
\mbox{Tr}\left(t_kdt_k^\dagger\right)\right)\end{aligned}$$ which agrees .
Invariant for fermion parity pumps {#appendix:pumpinvariant}
==================================
In the appendix, we will show that a $\mathbb{Z}_2$-invariant under a particle-hole gauge constraint topologically classified fermion parity pumps in dimension $\delta=0$ and class D or BDI. (See sec.\[sec:pumpparity\] for the full statement.)
=2.5in
We will show using a construction similar to a reasoning in Moore and Balents \[\]. We consider a deformation of the Hamiltonian along with the base manifold, so that the boundary $\partial\mathcal{T}_{1/2}$ is deformed into a single point (see fig \[defjm\]). Let $s\in[0,1]$ be the deformation variable, and denote $\mathcal{T}_{1/2}^+(s)$ and $\partial\mathcal{T}_{1/2}(s)$ be the corresponding deformation slices. Chern invariant $$n=\frac{1}{d!}\left(\frac{i}{2\pi}\right)^d
\int_{\mathcal{T}^+_{1/2}(s=1)}\mbox{Tr}(\mathcal{F}^d)$$ integrally classifies Hamiltonians on the half manifold $\mathcal{T}^+_{1/2}(s=1)$. Particle-hole symmetry requires opposite Chern invariant on the other half. A different choice of deformation could only change the Chern invariant by an even integer.[@FPPapp2Z] Hence the Chern invariant modulo 2 defines a $\mathbb{Z}_2$-invariant.
The Chern integral can be further deformed and decomposed into $$\begin{aligned}
&&\int_{\mathcal{T}^+_{1/2}(s=0)}\mbox{Tr}(\mathcal{F}^d)
+\int_0^1ds\int_{\partial\mathcal{T}^+_{1/2}(s)}\mbox{Tr}(\mathcal{F}^d)
\nonumber\\&=&\int_{\mathcal{T}^+_{1/2}(s=0)}\mbox{Tr}(\mathcal{F}^d)
-\oint_{\partial\mathcal{T}^+_{1/2}(s=0)}\mathcal{Q}_{2d-1}\end{aligned}$$ where Stokes’ theorem is used and the negative sign is from a change of orientation of the boundary. This proves . The particle-hole gauge constraint is built-in since the $G_{{\bf k},{\bf r},t}(s)$ deforms into a constant at $s=1$ while respecting particle-hole symmetry at all $s$.
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Bulk topology is a lower dimenional topological obstruction coming purely from the momentum part $T^d$ of the base space $T^d\times S^{d-2}$. One could mathematically remove the bulk topological obstruction by adding a defectless Hamiltonian $\widetilde{\mathcal{H}}({\bf k},{\bf r})
=\mathcal{H}({\bf k},{\bf r})\oplus\mathcal{H}_0({\bf k})$.
Again, assume the bulk has trivial topology. Or otherwise remove the topological obstruction mathematically by adding a defectless Hamiltonian.
Deformation of Hamiltonians on $\partial\mathcal{T}_{1/2}(s)$, for $s\in[0,1]$ has dimension $\delta=(d-1)-(d-1+1+1)=-2$, and is classified by $2\mathbb{Z}$.
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abstract: 'We characterize a niobium-based superconducting quantum interference proximity transistor (Nb-SQUIPT) built upon a Nb-Cu-Nb SNS weak link. The Nb-SQUIPT and SNS devices are fabricated simultaneously in two separate lithography and deposition steps, relying on Ar ion cleaning of the Nb contact surfaces. The quality of the Nb-Cu interface is characterized by measuring the temperature-dependent equilibrium critical supercurrent of the SNS junction. In the Nb-SQUIPT device, we observe a maximum flux-to-current transfer function value of about $ 55~{\mathrm{nA/\Phi_0}}$ in the sub-gap regime of bias voltages. This results in suppression of power dissipation down to a few ${\mathrm{fW}}$. The device can implement a low-dissipation SQUIPT, improving by up to two orders of magnitude compared to a conventional device based on an Al-Cu-Al SNS junction and an Al tunnel probe (Al-SQUIPT).'
author:
- 'R. N. Jabdaraghi, J. T. Peltonen, D. S. Golubev, and J. P. Pekola'
title: Magnetometry with low resistance proximity Josephson junction
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Introduction
============
The superconducting quantum interference proximity transistor (SQUIPT) [@Giazotto2010] is a sensitive magnetometer based on the proximity effect [@Gennes1964; @Gennes1966; @Belzig1999; @Lambert1998; @Pannetier2000; @Buzdin2005]. The SQUIPT consists of a superconducting loop (${\mathrm{S_1}}$) interrupted by a normal metal (N) in clean contact with it while a superconducting probe (${\mathrm{S_2}}$) is tunnel coupled to the middle of the weak link.
![Schematic view of a Nb-SQUIPT device, consisting of a normal metal Cu embedded in a superconducting Nb loop while a superconducting Al probe is tunnel coupled to the middle of the Cu normal metal. The resistance of the contact between Nb and Cu is $R_0$, whereas $R$ denotes the resistance of the junction between Al and Cu. []{data-label="fig:theory"}](Figure1.pdf){width="100.00000%"}
The operation of the device relies on the magnetic field modulation of the DoS [@Belzig1999; @Belzig1996; @Gueron1996; @Sueur2008] in the proximized normal metal (N) [@Petrashov1994; @Petrashov1995; @Belzig2002] embedded between superconducting electrodes (${\mathrm{S_1}}$), resulting in the opening of a minigap [@Belzig1999; @Belzig1996; @Zhou1998] in the N part. As shown in Fig. \[fig:theory\], the superconducting loop and probe are Nb and Al, respectively. A large number of applications has been proposed for SQUIPTs, similar to SQUIDs [@Clarke2004; @Tinkham1996; @Likharev1986], including measurement of magnetic flux induced by atomic spins, single-photon detection and nanoelectronical measurements [@Foley2009; @Hao2005; @Hao2007]. The SQUIPT represents a sensitive interferometer with reduced power dissipation and potentially low flux noise in contrast to conventional SQUIDs [@Giazotto2010; @Giazotto2011]. So far, a number of SQUIPT devices with an aluminum tunnel probe (S-SQUIPT) [@Giazotto2010; @Meschke2011; @Najafi2014; @Ronzani2014; @Najafi2016; @Najafi2017; @Ligato2017; @Ronzani2017; @Ambrosio2015] have been reported due to the high quality native oxide of Al, yielding excellent tunnel barriers [@Ligato2017].
The flux sensitivity of a SQUIPT device can be improved by maximizing the proximity effect in the normal metal N, hence increasing the responsivity of the device to a change in magnetic field. In this respect, replacing the superconducting loop with a larger-gap superconductor such as vanadium [@Ligato2017] or niobium [@Najafi2016] is an evident direction to look for sensitive SQUIPTs. Recently, a device with normal-conducting probe (N-SQUIPT) has been shown to be a promising candidate for the implementation of SQUIPTs with very low dissipated power [@Ambrosio2015]. However, implementing low dissipation S-SQUIPTs has not been explored in detail up to now. Operating the S-SQUIPT at sub-gap bias voltages on the supercurrent branch allows to obtain extra-low power dissipation, reduced by up to two orders of magnitude compared to earlier devices [@Giazotto2010; @Najafi2014]. In this article, we present a detailed experimental demonstration of Nb-Cu-Nb weak links and a SQUIPT based on a Nb superconducting loop (Nb-SQUIPT), first considered in Ref. 25. The devices are realized using an etching-based two-step fabrication process. We first characterize Nb-Cu-Nb junctions through low temperature switching current measurement and then investigate a Nb-SQUIPT, obtained by attaching an Al tunnel probe to the SNS weak link. In this prototype magnetometer, we observe the maximum flux-to-current transfer function value of about $|\partial I/\partial \Phi|_{\mathrm{max}}\approx 55~ {\mathrm{nA/{\mathrm{\Phi_0}}}}$ at $T=80~{\mathrm{mK}}$, which has been measured close to zero bias voltage. Here, ${\mathrm{\Phi_0}}=h/(2e)$ is the superconducting flux quantum.
Experimental methods and fabrication
====================================
![False color scanning electron micrograph of a Nb-SQUIPT showing (a) an enlarged view of the NIS junction region, together with the measurement scheme under voltage bias. (b) Current-voltage (IV) characteristics of a SNS structure measured at different bath temperatures below 500 mK. Arrows indicate the corresponding switching $I_{\mathrm{sw}}$ and retrapping $I_{\mathrm{r}}$ currents. The inset shows a representative SEM image of a SNS weak link, consisting of a normal metal (Cu) wire embedded between two superconducting electrodes (Nb). $L_1$ is the distance between the superconducting electrodes, whereas $L_2$ is the total normal metal length.[]{data-label="fig:SEM"}](Figure2.pdf){width="100.00000%"}
![Current-voltage characteristics of a Nb-SQUIPT device measured at $T=78$ mK for two values of the external magnetic flux through the superconducting loop, ${\mathrm{\Phi}}=0$ (red) and ${\mathrm{\Phi}}=0.5 ~{\mathrm{\Phi}}_0$ (blue). The inset illustrates the extent of the flux modulation of the IV curve around the onset of the quasiparticle current at four selected magnetic flux values in the range $0<{\mathrm{\Phi}}<0.5~{\mathrm{\Phi}}_0$. (c) Current modulation $I ({\mathrm{\Phi}})$ for several values of bias voltage in the bias regime between 250 and 320 $\mu$V.[]{data-label="fig:IV"}](Figure3.pdf){width="100.00000%"}
![Flux modulation of the IV curve in the sub-gap region at two bath temperatures $T=80$ mK (blue) and $T=190$ mK (red). (b) $I ({\mathrm{\Phi}})$ at several bias voltages between -25 $\mu$V (blue) and 25 $\mu$V (red) at $T=190$ mK. (c) Temperature dependence of the maximum flux-to-current transfer function $|\partial I/\partial\Phi|_{\mathrm{max}}$ in the sub-gap region of bias voltages.[]{data-label="fig:subgap"}](Figure4.pdf){width="100.00000%"}
![Flux-to-current transfer function measured at two different bias voltages, in the supercurrent regime at $V=32~{\mathrm{\mu V}}$ and in the quasiparticle branch at $V=307~{\mathrm{\mu V}}$.[]{data-label="fig:res_flux"}](Figure5.pdf){width="100.00000%"}
![Maximum flux-to-current transfer function as a function of bias voltage, measured at $T=78$ mK.[]{data-label="fig:res_vbias"}](Figure6.pdf){width="100.00000%"}
Both Nb-SQUIPT devices and Nb-Cu-Nb junctions on the same chip are fabricated using a process with two separate electron beam lithography (EBL) steps as discussed in detail in Ref. 25. The first step accomplishes patterning of the Nb structures. We start with an oxidized 4 inch Si wafer with 200 nm sputter-deposited Nb. In the first lithography step, we write into a layer of spin-coated positive tone AR-P 6200.13 resist. The lithography is followed by wet development and 5 min reflow baking at $150 ^\circ {\mathrm{C}} $. In order to transfer the exposed pattern into the Nb film, reactive ion etching (RIE) is used with a mixture of ${\mathrm{SF_6}}$ and Ar. Nb etching is followed by the second EBL step using a conventional bilayer resist [@Dolan1977], consisting of a 200 nm thick polymethyl methacrylate (PMMA) layer on top of a 900 nm layer of copolymer. This step provides the NIS tunnel junction and the Cu wire, and importantly, the clean electrical contacts between Nb and Cu in the Nb-Cu-Nb weak link.
The creation of a transparent contact between Nb and Cu is achieved by exposing the chip to $in~situ$ Ar ion etching, prior to the Cu deposition in the same vacuum cycle. The Nb-Cu-Nb weak link and Nb-SQUIPT devices with an Al-AlOx-Cu Normal metal$-$Insulator$-$Superconductor (NIS) tunnel junction are simultaneously fabricated: After Ar etching, 25 nm of Al at an angle $\theta=27^\circ$ is deposited and oxidized for $1-5$ min at a pressure of $1-5$ mbar to form the tunnel barrier of the NIS probe. Next, approximately 25 nm Cu at $\theta=-14^\circ$ is evaporated to complete the NIS junction and to form the normal metal wire. Figure \[fig:SEM\][$\left(\mathrm{a}\right)\;$]{}shows a scanning electron microscope (SEM) image of a typical Nb-SQUIPT device with an enlarged view of the zone around the weak link. The measurements of the samples have been performed in a $^3$He/$^4$He dilution refrigerator [@Pekola1994] down to the base temperature of about 80 mK.
Results and discussion
======================
The relevant dimensions of a typical measured SNS structure \[see inset of Fig. \[fig:SEM\][$\left(\mathrm{b}\right)$]{}\] are the minimum Nb electrode separation $L_1=0.49~ \mu {\mathrm{m}}$, the full length $L_2=2.15~ \mu {\mathrm{m}}$ and minimum width $w=0.54~ \mu {\mathrm{m}}$ of the Cu wire. We measure typical low temperature resistance value of the Cu strip to be about $0.4~{\mathrm{\Omega}}$. In the diffusive SNS junction, the Thouless energy $E_{\mathrm{Th}}=\hbar D/L^2$ determines the energy scale for the proximity effect. Here $D$ and $L$ are the diffusion constant and the effective length of the Cu wire, respectively [@Giazotto2008; @Blum2004; @Dubos2008]. The effective length of the normal metal $L=1.2~{\mathrm{\mu m}}$ is derived from the estimated values of $E_{\mathrm{Th}}\simeq 4.5 ~{\mathrm{\mu eV}}$ and the measured Cu diffusion constant $D=0.01~{\mathrm{m^2s^{-1}}}$ [@Najafi2016]. The Nb-Cu-Nb sample considered here is in the long junction limit $L\approx16~ \xi_0$, where $\xi_0=\sqrt{\hbar D/{\mathrm{\Delta}}}$ is the coherence length and ${\mathrm{\Delta}}\approx1.2~{\mathrm{meV}}$ the superconducting energy gap of Nb.
The current-voltage characteristic of the Nb-Cu-Nb structure is shown in the main panel of Fig. \[fig:SEM\][$\left(\mathrm{b}\right)\;$]{}at different values of bath temperatures. The dc voltage $V_{\mathrm{SNS}}$ is measured across the structure biased by a current $I_{\mathrm{SNS}}$ in a four probe configuration as indicated in the inset of Fig. \[fig:SEM\][$\left(\mathrm{b}\right)$]{}. As shown by the IV curves, the hysteretic behavior from self-heating in the finite-voltage state is observable at $T<650~{\mathrm{mK}}$ and hence the magnitudes of switching $I_{\mathrm{sw}}$ and retrapping $I_{\mathrm{r}}$ current differ from each other. At the base temperature, we find them to be about $30~{\mathrm{\mu A}}$ and $6~{\mathrm{\mu A}}$, respectively.
An IV characteristic of the Nb-SQUIPT is illustrated in Fig. \[fig:IV\][$\left(\mathrm{a}\right)$]{}, measured at two different magnetic fields, ${\mathrm{\Phi}}=0$ (red) and ${\mathrm{\Phi}}=0.5 ~\Phi_0$ (blue), corresponding to maximum and minimum minigap opened in the normal weak link [@Sueur2008; @Zhou1998]. Furthermore, the flux-dependent onset of current on the quasiparticle branch is shown in the inset when $|V|$ exceeds the sum of the probe electrode gap ${\mathrm{\Delta}}_{\mathrm{Al}}/e$ and the minigap induced in the N part. In the diffusive regime of a metallic SNS junction, the minigap in the normal metal is of the order of the Thouless energy $E_{\mathrm{Th}}$ [@Pannetier2000]. Considering the bias voltage dependence of the differential conductance [@Meschke2011; @Najafi2014] and subtracting the contribution of an effective series resistance $R_{\mathrm{S}}=1$ k${{\mathrm{\Omega}}}$, arising from the two point measurement technique, the magnitudes of the superconducting probe electrode gap and minigap are ${\mathrm{\Delta}}_{\mathrm{Al}}=253~~\mu{\mathrm{eV}}$ and ${\mathrm{\Delta}}_{\mathrm{g}}=27~\mu{\mathrm{eV}}$, respectively. As a consequence of the minigap variations, the current $I ({\mathrm{\Phi}})$ is periodic as a function of the flux through the loop. Typical current modulations are displayed in Fig. \[fig:IV\][$\left(\mathrm{c}\right)\;$]{}for some selected values of bias voltages $V$ in the quasiparticle branch from $250~\mu {\mathrm{V}}$ to $320~\mu {\mathrm{V}}$. Figure \[fig:IV\][$\left(\mathrm{c}\right)\;$]{}indicates that the measured current modulation reaches a peak-to-peak amplitude as large as $\delta I=5.5~ {\mathrm{nA}}$ at the bias voltage $V=300~\mu {\mathrm{V}}$. We now consider the flux modulation in the low bias regime. Figure \[fig:subgap\][$\left(\mathrm{a}\right)\;$]{} demonstrates the extent of the flux modulation of the IV curve in the sub-gap regime close to zero bias at two different temperatures, $T=80$ mK (blue) and $T=190$ mK (red). At $T=190$ mK, almost full magnetic flux modulation of the IV is visible due to increased SNS weak link inductance in comparison to the loop inductance [@Ronzani2014]. We can furthermore observe current peak at the bias voltage $V=30$ $\mu$V with the maximum current $I_{\max}=3.6$ nA. It is caused by multiple Andreev reflections, which are visible due to relatively low tunnel resistance of the probe junction, $R_{{\mathrm{T}}}=1$ k${{\mathrm{\Omega}}}$. In this sub-gap regime, the current modulation as a function of magnetic flux $I ({\mathrm{\Phi}})$ is shown in Fig. \[fig:subgap\][$\left(\mathrm{b}\right)\;$]{}at few selected values of applied bias voltage. At $|V|\approx 30~\mu {\mathrm{V}}$, we find the maximum peak-to-peak amplitude as large as $\delta I\approx 2.5~ {\mathrm{nA}}$, corresponding to the maximum sensitivity $|\partial I/\partial {\mathrm{\Phi}}|_{\mathrm{max}}\approx 55~ {\mathrm{nA/\Phi_0}}$. The effect of temperature $T$ on the flux-to-current transfer function is displayed in Fig. \[fig:subgap\][$\left(\mathrm{c}\right)$]{}, showing the temperature dependence of the maximum sensitivity in the low bias regime. It decreases monotonously with increasing $T$. For comparison, the transfer functions close to the optimum bias values in the supercurrent and quasiparticle regions are plotted in Fig. \[fig:res\_flux\].
We now discuss the bias voltage dependence of the maximum sensitivity $|\partial I/\partial{\mathrm{\Phi}}|_{\mathrm{max}}$ at the base temperature $T\approx 80~ {\mathrm{mK}}$. Figure \[fig:res\_vbias\] indicates that the maximum current responsivity is obtained at $|V|\simeq30~\mu {\mathrm{V}}$, corresponding to $|\partial I/\partial{\mathrm{\Phi}}|_{\mathrm{max}}\approx 55~{\mathrm{nA/\Phi_0}}$. At this working point, the typical output current level is $I\approx 3~{\mathrm{nA}}$ implying an average total power dissipation for the Nb-SQUIPT to be below ${\mathrm{10^{-13}~W}}$. The value can be further reduced by simply increasing the resistance of the probing junction. This power is two orders of magnitude smaller than in our previous device with an Al superconducting probe [@Najafi2014]. The sub-gap operation of a SQUIPT is thus a possible choice for applications where very low dissipation is required.
![Subgap structure related to multiple Andreev reflections in the I-V curve taken at zero magnetic field and at $T=78$ mK. The contribution of a 1 k${{\mathrm{\Omega}}}$ series resistor has been subtracted. []{data-label="fig:res_MAR"}](Figure7.pdf){width="100.00000%"}
Let us now briefly discuss the mechanism behind the SQUIPT response to the magnetic field at low bias. Considering first an ideal case, we assume that the response comes from the Josepshson current of the probe junction, which is modulated by the field together with the minigap in the copper island. The expected value of the critical current may be estimated by means of the Ambegaokar-Baratoff formula [@AB] for an asymmetric junction with the gaps of the leads ${{\mathrm{\Delta}}}_{\rm g}({{\mathrm{\Phi}}})$ and ${{\mathrm{\Delta}}}_{\rm Al}$. With the junction resistance $R_{\mathrm{T}}=1$ k${{\mathrm{\Omega}}}$ we find a maximum critical current $I_{\mathrm{C}}=150$ nA.
Assuming now that the minigap is modulated as ${{\mathrm{\Delta}}}_{{\mathrm{g}}}({{\mathrm{\Phi}}}) \approx {{\mathrm{\Delta}}}_{{\mathrm{g}}}\left|\cos(\pi{{\mathrm{\Phi}}}/{{\mathrm{\Phi}}}_0)\right|$, we can estimate the maximum responsivity of the device as $\sim\pi I_{\mathrm{C}}/{{\mathrm{\Phi}}}_0\approx 500$ nA/${{\mathrm{\Phi}}}_0$, which is about 10 times higher than in the experiment. Unfortunately, in our sample we observe only a very small critical current $I_{\mathrm{C}}= 2.4$ nA, which, on top of that, is not modulated by flux, and this simple mechanism is not working. However, our analysis shows that improving the quality of the probe junction may potentially increase the sensitivity of the SQUIPT. We believe that the observed low-bias current modulation in our device comes from multiple Andreev reflections (MAR). The tunnel barrier of the probe junction is rather transparent (overlap size ${\mathrm{310~nm\times 510~ nm}}$, characteristic resistance ${\mathrm{\approx150~\Omega \mu m^2}}$) and MAR signatures are clearly visible in the I-V curve plotted in Fig. \[fig:res\_MAR\]. The observed pattern agrees well with the one expected for a junction between two different superconductors with the gaps ${{\mathrm{\Delta}}}_{\rm g}$ and ${{\mathrm{\Delta}}}_{\rm Al}$ [@Cuevas]. Namely, we find current peaks at $eV={{\mathrm{\Delta}}}_{\rm g}$, $eV={{\mathrm{\Delta}}}_{\rm Al}$, a current jump at $eV={{\mathrm{\Delta}}}_{\rm Al}+{{\mathrm{\Delta}}}_{\rm g}$, and a series of less visible features at voltages ${{\mathrm{\Delta}}}_{\rm Al}/2e$, ${{\mathrm{\Delta}}}_{\rm Al}/3e$ etc. In terms of responsivity at low bias, the MAR peak centered around ${{\mathrm{\Delta}}}_{\rm g}/e=27$ ${\rm \mu}$V is important. The position of this peak follows the magnetic field dependent minigap ${{\mathrm{\Delta}}}_{\rm g}({{\mathrm{\Phi}}})$, which results in the modulation of the current at fixed voltage shown in Fig. \[fig:subgap\](b). The peak is smeared by increasing temperature and eventually vanishes at $T>{{\mathrm{\Delta}}}_{\rm g}/k_{\mathrm{B}}\approx 300$ mK, see Fig. \[fig:subgap\](c).
Summary
=======
To summarize, we have investigated Nb-Cu-Nb weak links and, consequently, a Nb-SQUIPT at low bias voltages of the tunnel probe. The structures are based on a fabrication process with two independent lithography and deposition steps, relying on Ar ion cleaning of the Nb contact surfaces. The typical power dissipation in the fW range gives the opportunity of using the Nb-SQUIPT as a low dissipation magnetometer and, furthermore, in other detector applications such as ultrasensitive bolometers and calorimeters.\
**Acknowledgments**\
The work has been supported by the Academy of Finland (project numbers 284594 and 275167). We acknowledge Micronova Nanofabrication Centre of OtaNano research infrastructure for providing the processing facilities, and for the sputtered Nb films.
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abstract: 'We introduce the idea of Assur graphs, a concept originally developed and exclusively employed in the literature of the kinematics community. The paper translates the terminology, questions, methods and conjectures from the kinematics terminology for one degree of freedom linkages to the terminology of Assur graphs as graphs with special properties in rigidity theory. Exploiting recent works in combinatorial rigidity theory we provide mathematical characterizations of these graphs derived from ‘minimal’ linkages. With these characterizations, we confirm a series of conjectures posed by Offer Shai, and offer techniques and algorithms to be exploited further in future work.'
author:
- 'Brigitte Servatius[^1]'
- 'Offer Shai[^2]'
- 'Walter Whiteley [^3]'
bibliography:
- 'assurcombArX.bib'
title: Combinatorial Characterization of the Assur Graphs from Engineering
---
Introduction
============
Working in the theory of mechanical linkages, the concept of ‘Assur groups’ was developed by Leonid Assur (1878-1920), a professor at the Saint-Petersburg Polytechnical Institute. In 1914 he published a treatise (reprinted in [@Assur]) entitled [*Investigation of plane bar mechanisms with lower pairs from the viewpoint of their structure and classification*]{}. In the kinematics literature it is common to introduce ‘Assur groups’ (selected groups of links) as special minimal structures of links and joints with zero mobility, from which it is not possible to obtain a simpler substructure of the same mobility [@PK]. Initially Assur’s paper did not receive much attention, but in 1930 the well known kinematician I.I. Artobolevskiĭ, a member of the Russian academy of sciences, adopted Assur’s approach and employed it in his widely used book [@Arto]. From that time on Assur groups are widely employed in Russia and other eastern European countries, while their use in the west is not as common. However, from time to time Assur groups are reported in research papers for diverse applications such as: position analysis of mechanisms [@Mitsi]; finding dead-center positions of planar linkages [@PK] and others.
The mechanical engineering terminology for linkages (kinematics) and their standard counting techniques are introduced via an example in the next section. Central to Assur’s method is the decomposition of complex linkages into fundamental, minimal pieces whose analyses could then be merged to give an overall analysis. Many of these approaches for Assur groups were developed from a range of examples, analyzed geometrically and combinatorially, but never defined with mathematical rigor.
In parallel, rigidity of bar and joint structures as well as motions of related mechanisms have been studied for several centuries by structural engineers and mathematicians. Recently (since 1970) a focused development of a mathematical theory using combinatorial tools was successful in many applications. For example for planar graphs there is a simple geometric duality theory, which, if applied to mechanisms and frameworks yields a relation between statics and kinematics: any locked planar mechanism is dual to an unstable planar isostatic framework (determinate truss) [@Shai; @CW2].
The purpose of this paper is twofold. First, we want to draw together the vocabulary and questions of mechanical engineering with the rigidity theory terminologies of engineering and mathematics. Second, we want to apply the mathematical tools of rigidity theory, including the connections between statics and kinematics, to give precision and new insights into the decomposition and analysis of mechanical linkages.
The mathematical tools we need are briefly sketched with references provided in §2.3-2.7. Our main result is the description of Assur graphs (our term for Assur groups) in Engineering terms (§2.1,2.2) and its reformulation in mathematical terms. We show that our mathematical reformulation allows us in a natural way to embed Assur’s techniques in the theory of frameworks (§3) and bring the results back to linkages. In the process we veriy several conjectured characterizations presented by Offer Shai in his talk concerning the generation of Assur graphs and the decomposition of linkages into Assur graphs, at the 2006 Vienna Workshop on Rigidity and Flexibility §3.1. We also give algorithmic processes for decomposing general linkages into Assur graphs, as well as for generating all Assur graphs (§3.2-3.3).
In a second paper [@SSW2], we will apply the geometric theory of bar-and-joint framework rigidity in the plane to explore additional properties and characterizations of Assur graphs. This exploration includes singular (stressed) positions of the frameworks, explored using reciprocal diagrams, and the introduction of ‘drivers’, which appear in passing in the initial example in the next section.
Preliminaries
=============
In the first two sub-sections we present the mechanical engineering vocabulary, problems and approaches through an example. These offer the background and the motivation for the concepts of the paper, but do not yet give the formal mathematical definitions. In the remaining five sub-sections we give the framework basics needed to mathematically describe these approaches.
Linkages and Assur graphs
--------------------------
A linkage is a mechanism consisting of rigid bodies, the [*links*]{}, held together by joints. Since we only consider linkages in the plane, all our joints are pin joints, or pins. A complex linkage may be efficiently studied by decomposing it into simple pieces, the Assur groups. (Engineers use the term group to mean a specified set of links. In mathematics the word group is used for an algebraic structure, but most of the tools we will use come from graph theory, so the word graph seems more natural and we will use it as soon as we start our mathematical section). We introduce these ideas via an example.
Figure \[tracfig01a\] depicts an excavator attached with a linkage system. In the following, we illustrate how the schematic drawing of this system is constructed and how it is decomposed into Assur groups.
![The excavator with its kinematic system.\[tracfig01a\]](tracfig01a)
In order to get a uniform scheme, termed [*structural scheme*]{}, it is common to represent all the connections between the links by revolute joints as appears in Figure \[tracfig02b\]. Here joints $0_1$ and $0_3$ attach the excavator to the vehicle (fixed ground) and these special joints are marked with a small hatched triangle, and are called [*pinned joints*]{}. All other joints are called [*inner*]{} joints. A link which can be altered (e.g. by changing its length) is called a [*driving link*]{}. A driving link can be thought of as driving or changing the distance between its endpoints like the pistons in our excavator example, which may be modeled in the structural scheme by a rotation of an inserted link 1.
![The unified structural scheme of the kinematic system of the excavator.\[tracfig02b\]](tracfig02b "fig:")![The unified structural scheme of the kinematic system of the excavator.\[tracfig02b\]](tracfig03 "fig:")
Once the engineering system is represented in the structural scheme, to start the analysis, all the driving links are deleted and replaced by pinned joints (mathematically speaking the driving links are contracted and their endpoints identified). In the current example, links 1 and 4 are deleted and joints A and D are pinned. Then, the system is decomposed into three Assur groups, each consisting of two links, one inner joint and two pinned joints. In the literature the Assur groups of this type are referred to as dyads [@Norton]. The order of the decomposition is important. If an inner joint of a group, $G_1$, becomes a pinned joint in group $G_2$, then $G_1$ should precede $G_2$.
In our example (see Figure \[tracfig04\]), the unique order of decomposition is: Dyad$_{1} = \{2,3\}$; Dyad$_{2}=\{5,6\}$; Dyad$_{3}=\{7,8\}$.
![Assur group decomposition of the structural scheme of the kinematic system of the excavator.\[tracfig04\]](tracfig04)
Degree of freedom of a mechanism: Grübler’s equation
----------------------------------------------------
The degree of freedom (DOF) of a linkage is the number of independent coordinates or measurements required to define its position.
In mechanical engineering [@Norton], Grübler’s equation relates the (least number of internal) degrees of freedom $F$ of a linkage mechanism to the number $L$ of links and the number $J$ of joints in the mechanism. In the plane, if $J_i$ is the number of joints from which $i$ links emanate, $i\geq 2$ then $$\label{Grubler1}
F =3(L-1)-2\sum (i-1)J_i$$
In the example above $L=9$ because the fixed ground is considered a link, and $\sum (i-1)J_i= 11$, because the revolute joint $E$ is counted twice as it pins links 5, 6, and 7. By Grübler’s equation we get $F=3(9-1)-2\cdot11=2$, which is correct as there are two driving links (two distances being controled). If these two driving links are removed and their ends pinned (identified) as in the example analysis, we have only 7 links left and the number of revolute joints is now 9, so $F=0$. This is another indication that the drivers work independently.
Note that Grübler’s equation only gives a lower bound on the degree of freedom and there are many cases where the actual DOF is larger than the predicted one [@Norton]. If a linkage contains a sub-collection of links pinned in such a way among themselves that the Grübler count is negative for the sub-collection, then the predicted DOF for the linkage might be smaller than the actual DOF (see Figure \[miscount\](a)). This situation can be detected and corrected by combinatorial means as we will describe in Laman’s Theorem, see §2.5 Theorem \[OverviewTheorem\].
Counting techniques, however, cannot detect special geometries, e.g. parallelism or symmetry of links, which also might lead to a false Grübler DOF prediction. We will examine this type of geometric singularity of a combinatorially correct graph in [@SSW2].
![In some cases Grübler’s equation provides a false answer (a), due to an overcounted subgraph. In others (b) it correctly predicts a pinned isostatic framework (determinate truss). \[miscount\]](henny03)
Frameworks
----------
For a linkage in which all the links are bars, with revolute joints at the two endpoints of the bar, we can rewrite Grübler’s equation in terms of graph theory, by introducing a graph whose vertices, $V$, are the joints and whose edges, $E$, are the bars. With $V_i$ denoting the set of vertices of valence $i$, Grübler’s equation becomes $$F=3(|E|-1)-2\sum (i-1)J_i=3(|E|-1)-2\sum i|V_i| +2\sum |V_i|$$ $$=
3|E|-3-4|E|+2|V|=2|V|-3 -|E|.$$ So if the edges of a graph embedded in the plane are interpreted as rigid bars and the vertices as revolute joints, the graph needs to have at least $2|V|-3$ edges in order to have no internal degrees of freedom, an observation made already by Maxwell. The count $2|V|-3$ will be central to the rest of the paper.
By a [*framework*]{} we mean a graph $G = (V,E)$ together with a configuration ${\mathbf{p}}$ of $V$ into Euclidean space, for our purpose the Euclidean plane. We will always assume that the two ends of an edge (a bar) are distinct points). A motion of the framework is a displacement of the vertices which preserves the distance between adjacent vertices, and a framework is [*rigid*]{} if the only motions which it admits arise from congruences.
Let us assume that the location ${\mathbf{p}}_{i}$ of a vertex is a continuous function of time, so that we can differentiate with respect to time. If we consider the initial velocities, ${\mathbf{p}}'_{i}$, of the endpoints ${\mathbf{p}}_{i}$ of a single edge $(i,j)$ under a continuous motion of a framework, then, to avoid compressing or extending the edge, it must be true that the components of those velocities in the direction parallel to the edge are equal, i.e. $$\label{InfEqn}
({\mathbf{p}}_i - {\mathbf{p}}_j)\cdot({\mathbf{p}}'_i - {\mathbf{p}}'_j) = 0.$$ A function assigning vectors to each vertex of the framework such that equation \[InfEqn\] is satisfied at each edge is called an [*[first-order ]{}motion.*]{} If the only [first-order ]{}motions are [*trivial*]{}, that is, they arise from [first-order ]{}translations or [first-order ]{}rotations of ${\mathbb{R}}^2$, then we say that the framework is [first-order ]{}rigid in the plane. [First-order ]{}rigidity implies rigidity, see for example [@Connelly].
In our excavator example not all links are bars. In the structural scheme link 8 is modeled by a bar because it contains only two pins, while link 3, which contains 5 pins appears as a “body". We can replace such a body by a rigid subframework on these 5 vertices (or more). In general, any linkage consisting of rigid bodies held together by pin joints can be modeled as a framework by replacing the bodies with rigid frameworks.
The rigidity matrix
-------------------
Any graph $G$ can be considered a subgraph of the complete graph $K_n$ on the vertex set $V = \{1,\ldots, n\}$, where $n$ is large enough. Let ${\mathbf{p}}$ be a fixed [*configuration*]{} (embedding) of $V$ into ${\mathbb{R}}^2$.
Equation \[InfEqn\] defines a system of linear equations, indexed by the edges $(i,j)$, in the variables for the unknown velocities ${\mathbf{p}}'_i$. The matrix $R({\mathbf{p}})$ of this system is a real $n(n-1)/2$ by $2n$ matrix and is called the [*rigidity matrix*]{}. As an example, we write out coordinates of ${\mathbf{p}}$ and of the rigidity matrix $R({\mathbf{p}})$, in the case $n=4$. $${\mathbf{p}} = ({\mathbf{p}}_{1},{\mathbf{p}}_{2},{\mathbf{p}}_{3},{\mathbf{p}}_{4}) =
(p_{11},p_{12},p_{21},p_{22},p_{31},p_{32},p_{41},p_{42});$$ $$\left[\begin{array}{cccccccc}
_{p_{11}-p_{21}} & _{p_{12}-p_{22}} & _{p_{21}-p_{11}} & _{p_{22}-p_{12}} &
_{0} & _{0} & _{0} & _{0} \\
_{p_{11}-p_{31}} & _{p_{12}-p_{32}} & _{0} & _{0} &
_{p_{31}-p_{11}} & _{p_{32}-p_{12}} & _{0} & _{0} \\
_{p_{11}-p_{41}} & _{p_{12}-p_{42}} & _{0} & _{0} &
_{0} & _{0} & _{p_{41}-p_{11}} & _{p_{42}-p_{12}} \\
_{0} & _{0} & _{p_{21}-p_{31}} & _{p_{22}-p_{32}} &
_{p_{31}-p_{21}} & _{p_{32}-p_{22}} & _{0} & _{0} \\
_{0} & _{0} & _{p_{21}-p_{41}} & _{p_{22}-p_{42}} &
_{0} & _{0} & _{p_{41}-p_{21}} & _{p_{42}-p_{22}} \\
_{0} & _{0} & _{0} & _{0} &
_{p_{31}-p_{41}} & _{p_{32}-p_{42}} & _{p_{41}-p_{31}} & _{p_{42}-p_{32}}
\end{array}\right]$$
A framework $(V,E,{\mathbf{p}})$ is infinitesimally rigid (in dimension $2$) if and only if the submatrix of $R({\mathbf{p}})$ consisting of the rows corresponding to $E$ has rank $2n-3$. We say that the vertex set $V$ is in generic position if the determinant of any submatrix of $R({\mathbf{p}})$ is zero only if it is identically equal to zero in the variables ${\mathbf{p}}'_i$. For a generically embedded vertex set, linear dependence of the rows of $R({\mathbf{p}})$ is determined by the graph induced by the edge set under consideration. The rigidity properties of a graph are the same for any generic embedding. A graph $G$ on $n$ vertices is [*generically rigid*]{} if the rank $\rho$ of its rigidity matrix $R_G ({\mathbf{p}})$ is $2n-3$, where $R_G ({\mathbf{p}})$ is the submatrix of $R({\mathbf{p}})$ containing all rows corresponding to the edges of $G$, for a generic embedding of $V$. The (generic) DOF of $G$ is defined to be $ 2n-3-\rho$.
Results for the plane
---------------------
Linear dependence of the rows of the rigidity matrix defines a matroid on the set of rows and for generic configurations we speak about [*independent edge sets*]{} rather than independent rows of $R({\mathbf{p}})$. For a generic embedding of $n$ vertices into ${\mathbb{R}}^2$ we call the matroid on the complete graph obtained from $R({\mathbf{p}})$ the generic rigidity matroid in dimension $2$ on $n$ vertices, $\mathfrak{R}_2(n)$.
The following theorem characterizes $\mathfrak{R}_2(n)$.
\[OverviewTheorem\]
[*(Laman [@Laman])*]{} The independent sets of $\mathfrak{R}_2(n)$ are those sets of edges which satisfy Laman’s condition: $$\label{LamansIneq}
|F| \leq 2|V(F)| - 3
\mbox{~for all $F \subseteq E, F \not = \emptyset $};$$
Laman’s Theorem was proved in 1970 and it was this theorem that promoted the use of matroids to attack rigidity questions. There are many equivalent axiom systems known for matroids. These can be used to reveal structural properties of various types and their relationships. The fact that matroids are exactly those structures for which independent sets can be constructed greedily has important algorithmic conseqences.
From the count condition in the inequalities (\[LamansIneq\]) for independent edge sets it is straight forward to deduce count conditions for [*bases*]{} of $\mathfrak{R}_2(n)$ (edge sets inducing minimally rigid or isostatic graphs), as well as for minimally dependent sets, or [*circuits*]{}, which will play a fundamental role in our analysis and will be treated in the next section.
Independent sets of $\mathfrak{R}_2(n)$ may be constructed inductively [@TW]. Given an independent edge set $E$ in $\mathfrak{R}_2(n)$, we can extend $E$ by new edges provided that the inequalities \[LamansIneq\] are not violated. Starting with an independent set (e.g. a single edge):
1. We can attach a new vertex $v$ by two new edges $x=(v,u)$ and $y=(v,w)$ to the subgraph of $G$ induced by $E$ and $E\cup\{x,y\}$ is independent, see Figure \[extendfig01\]a. This is also called [*$2$-valent vertex addition*]{}.
2. Similarly, we can attach a new vertex $v$ by three new edges to the endpoints of an edge $e \in E$ plus any other vertex in the subgraph of $G$ induced by $E$, and ${E\setminus e}\cup\{x,y,z\}$ is independent, see Figure \[extendfig01\]b. This operation is called [*edge-split*]{}, because the new vertex $v$ is thought of as splitting the edge $e$.
![Building up an independent set of edges by: $2$-valent vertex (a) and edge-split (b)\[extendfig01\]](extendfig01)
These [*Henneberg techniques*]{} developed in [@TW] have become standard in rigidity theory, see also [@GSS], and when we resort to “the usual arguments” within some of the proofs to come, we have these standard proof techniques in mind. For further reference we state the following well known result.
Any independent set in $\mathfrak{R}_2(n)$ can be obtained from a single edge by a sequence of $2$-valent vertex additions and edge-splits.\[Henneberg\]
Figure \[henny01\] illustrates a sequence described in Theorem \[Henneberg\].
![A Henneberg sequence constructing an isostatic graph.\[henny01\]](henny02)
Rigidity circuits
-----------------
Minimally dependent sets, or circuits, in $\mathfrak{R}_2(n)$ are edge sets $C$ satisfying $|C|=2|V(C)|-2$ and every proper non-empty subset of $C$ satisfies inequality (\[LamansIneq\]). Note that these circuits, called [*rigidity circuits*]{}, always have an even number of edges. We will, as is commonly done, not distinguish between edge sets and the graphs they induce.
Similarly to the inductive constructions of independent sets (see Figure \[extendfig01\]), all circuits in $\mathfrak{R}_2(n)$ can be constructed from a tetrahedron (the complete graph on four vertices) by two simple operations, see [@Berg-Jordan], namely [*edge-split* ]{} as in Figure \[extendfig01\]b, and [*2-sum*]{}, where the 2-sum of two (disjoint) graphs is obtained by “gluing” the graphs along an edge and removing the glued edge, see Figure \[twosum\].
![2-sums taken along the lined up edge pairs combine circuits into a larger circuit. \[twosum\]](twosum)
Any circuit in $\mathfrak{R}_2(n)$ can be obtained from $K_4$ by a sequence edge-splits and 2-sums.\[BJ\]
Isostatic Pinned Framework
--------------------------
Given a framework associated with a linkage, we are interested in its internal motions, not the trivial ones, so following the mechanical engineers we pin the framework by prescribing, for example, the coordinates of the endpoints of an edge, or in general by fixing the position of the vertices of some rigid subgraph, see Figure \[pinsub01\]. We call these vertices with fixed positions [*pinned*]{}, the others [*inner*]{}. (Inner vertices are sometimes called [*free*]{} or [*unpinned*]{} in the literature.) Edges among pinned vertices are irrelevant to the analysis of a pinned framework. We will denote a pinned graph by $G(I,P;E)$, where $I$ is the set of inner vertices, $P$ is the set of pinned vertices, and $E$ is the set of edges, where each edge has at least one endpoint in $I$.
A pinned graph $G(I,P;E)$ is said to satisfy the [*Pinned Framework Conditions*]{} if $|E|=2|I|$ and for all subgraphs $G'(I', P'; E')$ the following conditions hold:
1. $|E'|\leq 2|I'|$ if $|P'| \geq 2$,
2. $|E'|\leq 2|I'| -1$ if $|P'|=1$ , and
3. $|E'|\leq 2|I'|-3$ if $P' = \emptyset$.
We call a pinned graph $G(I,P;E)$ [*pinned isostatic*]{} if $E=2|I|$ and $\widetilde{G}= G\cup K_{P}$ is rigid as an unpinned graph, where $K_{P}$ is a complete graph on a vertex set containing all pins (but no inner vertices). In other words, we “replace” the pinned vertex set by a complete graph containing the pins and call $G(I,P;E)$ isostatic, if choosing any basis in that replacement produces an (unpinned) isostatic graph.
A pinned graph $G(I,P;E)$ realized in the plane, with ${\mathbf{P}}$ for the pins, and ${\mathbf{p}}$ for all the vertices, is a [*pinned framework*]{}. A pinned framework is [*rigid*]{} if the matrix $R_{\widetilde{G}}$ has rank $2|I|$, where $R_{\widetilde{G}}$ is the rigidity matrix of $\widetilde{G}$ with the columns corresponding to the vertex set of $K_p$ removed, [*independent*]{} if the rows of $R_{\widetilde{G}}$ corresponding to $E$ are independent, and [*isostatic*]{}, if it is rigid and independent. The vertices $I$ of a pinned framework are in [*generic position*]{} if any submatrix of the rigidity matrix is zero only if it is identically equal to zero with the coordinates of the inner vertices as variables. The coordinates of the pins are prescribed constants.
Figure \[pinsub01\] shows an example of a pinned isostatic $G$ and a corresponding basis ${\widehat{G}}$ of $\widetilde{G}$.
![Framework (a) is pinned isostatic because Framework (b) is isostatic.\[pinsub01\]](pinsub01)
It is common in engineering to choose pins in advance and their placement ${\mathbf{P}}$ might not be generic, in fact not even in [*general position*]{} (no three points collinear), as it is sometimes necessary to have all pins on a line. The following result shows that this is not a problem.
\[PinThm\] Given a pinned graph $G(I,P;E)$, the following are equivalent:
\(i) There exists an isostatic realization of $G$.
\(ii) The Pinned Framework Conditions are satisfied.
\(iii) For all placements ${\mathbf{P}}$ of $P$ with at least two distinct locations and all generic positions of $I$ the resulting pinned framework is isostatic.
Let $\widetilde{G}= G\cup K_{P'}$ $P'\supseteqq P$, and $F$ a maximal independent edge set of $K_{P'}$. Then by Theorem \[OverviewTheorem\] we deduce that $E\cup F$ is isostatic if and only if the Pinned Framework Conditions are satisfied, so $(i)\Leftrightarrow (ii)$
In order to show $(ii)\Rightarrow (iii)$ we first show that we can extend $F$ to $F\cup E$ by a Henneberg sequence of 2-valent vertex additions and edge-splits (see Figure \[extendfig01\]). To this end we de-construct $\widetilde{G}$ first by removing inner vertices as follows. Assume that $|I|>1$. Since $|E|=2|I|$ and $I$ spans at most $2|I|-3$ edges, we must have at least three edges joining the set of inner vertices $I$ to the pinned vertices $P$. Therefore, if we sum over the valence of inner vertices, and denote the the set of edges with both endpoints in $I$ by $E_i$, the ones with one endpoint in $I$ by $E_p$, we obtain $\sum val(i) = 2|E_i|
+|E_p| = 2|I|+|E_i| \leq 4|I|-3$. So there will be some inner vertices of valence 2 or 3. If at this stage there is some vertex of degree $2$, we can just remove it, to create a smaller graph with the same isostatic count. If there is some vertex of degree $3$, then by the usual arguments [@GSS; @TW], it can be removed, and replaced by a new edge joining two of its neighbors, which were not yet joined in a remaining rigid subgraph (e.g. not both pinned), to create a smaller subgraph with the isostatic count. This produces a reverse sequence of smaller and smaller isostatic graphs until we have no inner vertices.
To obtain a realization, place $P$ in an arbitrary position ${\mathbf{P}}$ with at least two distinct vertices. Create an isostatic graph whose vertex set contains $P$, by, for example, ordering the vertices in $P$ with distinct positions arbitrarily, $P=\{ p_1, p_2, \ldots \}$, adding edges between consecutive vertices and attaching an extra vertex, ${p}_0$ by edges $(p_0,p_i)$. This graph is clearly isostatic, provided the point ${\mathbf{p}}_0$ is not placed on the line through ${\mathbf{p}}_i$ and ${\mathbf{p}}_{i+1}$ for any $i$, because it has the correct edge count and is rigid since it consists of a string of non-collinear triangles.
To complete the proof, we work back up the sequence of subgraphs we created in the de- construction process. We assume the current graph is realized as isostatic. When the next graph is created by adding a 2-valent vertex, then adding such a vertex in any position except on the line joining its two attachments gives a new isostatic realization.
When the next graph is created by an edge-split, note that at least one of the neighbors of the new vertex is inner, so this added inner vertex can be placed in a generic position ensuring that the three new attachments are not collinear. Therefore, by the usual arguments [@GSS; @TW], this insertion is also isostatic when placed along the line of the bar being removed, and therefore also when placed in any generic position. Since (iii) trivially implies (i), the proof is complete.
A pinned graph $G(I, P;E)$ satisfying the Pinned Framework Conditions must have at least two pins and in every isostatic realization of $G$ there must be at least two distinct pin locations. Placing all pins in the same location never yields an isostatic framework, but we can make an important observation about the DOF of such a “pin collapsed” framework.
\[contractThm\] Let $G(I, P; E)$ be a pinned graph satisfying the Pinned Framework Conditions. Identifying the pinned vertices to one vertex $p^{*}$ yields a graph $G^{*}(V,E)$, $V=I\bigcup \{p^{*}\}$ and the DOF of $G^{*}$ is one less than the number of rigidity circuits contained in $G^{*}$.
Since $|E|=2|I|=2|V|-2$, $G^{*}$ contains too many edges to be isostatic. If $G^{*}$ is rigid, it is overbraced by exactly one edge, hence contains exactly one rigidity circuit. If $G_p$ is not rigid, each of the rigidity circuits in $G^{*}$ must contain $p^{*}$. If two rigidity circuits intersected in a vertex other than $p^{*}$, the union of their edge sets, together with the pinned subgraph would violate the Pinned Subgraph Conditions. Therefore all circuits in $G^{*}$ have exactly the vertex $p$ in common. Removing exactly one edge from each circuit yields a basis for $G^{*}$ in $\mathfrak{R}(G^{*})$, establishing the desired connection between the DOF and the number of circuits.
Characterizations of Assur graphs
=================================
We start with two citations from the mechanical engineering literature as motivation for our combinatorial conditions. The following definition appears in [@YV]: “An Assur group is obtained from a kinematic chain of zero mobility by suppressing one or more links, at the condition that there is no simpler group inside". In [@SS] we find: “An element of an Assur group is a kinematic chain with free or unpaired joints on the links which when connected to a stationary link will have zero DOF. A basic rigid chain is a chain of zero DOF and whose subchains all have DOF greater than zero. In other words an element of an Assur group is a basic rigid chain with one of its links deleted".
These descriptions from the engineering literature are not definitions in the mathematical sense, but rather use ‘minimality’ informally, as in the original work of Assur. We are now ready to give a formal definition by confirming a series of equivalent combinatorial characterizations of Assur graphs. These statements are new, and (iii) and (iv) come from the conjectures offered by Offer Shai at the workshop.
Basic Characterization of Assur graphs
--------------------------------------
\[CharacterThm\] Assume $G= (I,P; E)$ is a pinned isostatic graph. Then the following are equivalent:
\(i) $G= (I,P; E)$ is minimal as a pinned isostatic graph: that is for all proper subsets of vertices $I'\cup P'$, $I' \cup P'$ induces a pinned subgraph $G' = (I'\cup P', E')$ with $|E'| \leq 2|I'| -1$.
\(ii) If the set $P$ is contracted to a single vertex $p^*$, inducing the unpinned graph $G^*$ with edge set $E$, then $G^*$ is a rigidity circuit.
\(iii) Either the graph has a single inner vertex of degree $2$ or each time we delete a vertex, the resulting pinned graph has a motion of all inner vertices (in generic position).
\(iv) Deletion of any edge from $G$ results in a pinned graph that has a motion of all inner vertices (in generic position).
\(i) implies (iv) If we delete an edge, there must be a motion by the count. If there is a set of inner vertices that are not moving, in generic position, then these vertices, and their edges to the pinned vertices, must form a proper isostatic pinned subgraph contradicting condition (i).
\(iv) implies (iii) Removing an edge with an endpoint of valence $2$ produces a graph with a pendant edge. This must be the only inner vertex, since any other inner vertices are not moving, contradicting (iv). Since removing a single edge results in a motion of all inner vertices, removing all edges incident with one particular vertex results in a framework with a motion on all the remaining vertices.
Conversely, (iii) implies (i) If the graph contains a minimal proper pinned subgraph, then removing any vertex outside of this subgraph will produce a motion at most in the vertices outside of the subgraph. This contradicts (iii).
\(i) is equivalent to (ii) If $G= (V,P; E)$ is a pinned isostatic graph, then identifying the vertices in $P$ to a single vertex $p*$ yields a graph $G^{*} = (V^{*}, E^{*})$ with $|E^{*}| = 2(|V|+1) = |V^{*}|-2$, so $G^{*}$ is dependent and if the minimality condition in (i) is satisfied, it must be a rigidity circuit. Conversely, if $G^{*}$ is a rigidity circuit, we can pick an arbitrary vertex of $G^{*}$ and call it $p^{*}$. Splitting $p^{*}$ into a vertex set $P$, $|P|\geq 2$, (where $P$ may have as many vertices as the valence of $p^{*}$ in $G^{*}$ allows) and specifying for each edge with endpoint $p^{*}$ a new endpoint from $P$ so that no isolated vertices are left, yields an isostatic pinned framework satisfying the minimality condition.
This theorem provides a rigorous mathematical definition: an [*Assur graph*]{} is a pinned graph satisfying one of the four equivalent conditions in Theorem \[CharacterThm\].
Condition (i) is a refinement of the Grübler count (\[Grubler1\]), in a form which is now necessary and sufficient.
Condition (ii) translates the minimality condition to minimal dependence in $\mathfrak{R}_2 (n)$ and thus serves as a purely combinatorial description of Assur graphs and may be checked by fast algorithms [@JacobsHendrickson; @pebblegame].
Conditions (iii) and (iv) are similar in nature. Condition (iii) provides the engineer with a quicker check for the Assur property for smaller graphs than (iv), since there are fewer vertices than edges to delete. However, condition (iv) tells the engineer that a driver inserted for an arbitrary edge will (generically) move all inner vertices. We will expand on this property in [@SSW2]
![Assur graphs\[assurwordfig01\]](assurwordfig01)
![Corresponding circuits for Assur graphs\[assurwordfig03\]](assurwordfig03)
Some examples of Assur graphs are drawn in Figure \[assurwordfig01\] and their corresponding rigidity circuits in Figure \[assurwordfig03\].
Decomposition of general isostatic frameworks
----------------------------------------------
We now show that a general isostatic framework can be decomposed into a partially ordered set of Assur graphs. The given framework can be re-assembled from these pieces by a basic linkage composition. Figure \[assurwordfig02\] shows isostatic pinned frameworks and Figure \[assurwordfig05\] indicates their decomposition.
![Decomposable - not Assur graphs\[assurwordfig02\]](assurwordfig02)
![The first step of a decomposition for isostatic frameworks in \[assurwordfig02\] - with identified subcircuit(s).\[assurwordfig04\]](assurwordfig04)
Given two linkages as pinned frameworks $H= (W,Q;F)$ and $G= (V,P; E)$ and an injective map $C: Q \rightarrow V\cup P$, the [*linkage composition*]{} $C(H,G)$ is the linkage obtained from $H$ and $W$ by identifying the pins $Q$ of $H$ with their images $C(Q)$.
Given two pinned isostatic graphs $H= (W,Q,F)$ , $G= (V,P; E)$, the composition $C(H,G)$ creates the new composite pinned graph: $C=(V\cup W, P, E\cup F)$ which is also isostatic.
By the counts, we have $|F|=2|W|$, and $|E|=2|V|$, so $|E\cup F| = 2|W \cup V|$. A similar analysis of the subgraphs confirms the isostatic status.
![Recomposing the pinned isostatic graphs in Figure \[assurwordfig02\] from their Assur components.\[assurwordfig05\]](assurwordfig05)
Under this operation, the Assur graphs will be the minimal, indecomposable graphs. We can show that every pinned isostatic graph $G$ is a unique composition of Assur graphs, which we will call the [*Assur components*]{} of $G$.
A pinned graph is isostatic if and only if it decomposes \[POTheorem\] into Assur components. The decomposition into Assur components is unique.
Take the isostatic pinned framework, and identify the ground pins. This is now a dependent graph. Using properties of $\mathfrak{R}_2 (n)$, see [@CMW], we can identify minimal dependent subgraphs - which, by Theorems \[contractThm\] and \[CharacterThm\], are Assur components after the pins are separated. These are the initial components. When all of these initial components are contracted in step two, we seek additional Assur components. We iterate the process until only the ground is left.
The decomposition process described in the proof of Theorem \[POTheorem\] naturally induces a partial order on the Assur components of an isostatic graph: component $A\leq B$ if $B$ occurs at a higher level, and $B$ has at least one vertex of $A$ as a pinned vertex. The algorithm for decomposing the graph guarantees that $A\leq B$ means that $B$ occurs at a later stage than $A$. This partial order can be represented in an [*Assur scheme*]{} as in Figure \[assurscheme\]. This partial order, with the identifications needed for linkage composition, can be used to re-assemble the graph from its Assur components.
![An isostatics pinned framework a) has a unique decomposition into Assur graphs b) which is represented by a partial order or Assur scheme c).\[assurscheme\]](AssurDecomp)
Replacing any edge in an isostatic framework produces a 1 DOF linkage. The decomposition described in Theorem \[POTheorem\] permits the analysis of this linkage in layers. In fact, we can place drivers in each Assur component to obtain linkages with several degrees of freedom and their complex behavior can be simply described by analyzing the individual Assur components. This process of adding drivers is studied in more detail in [@SSW2].
Generating Assur graphs
------------------------
We summarize inductive techniques to generate all Assur graphs. Engineers find such techniques of interest to generate basic building blocks for synthesizing new linkages.
![An Edge-Split takes an Assur graph with at least four vertices to an Assur graph\[assurwordfig06\]](assurwordfig06)
![2-sum of Assur graphs gives a new Assur graph with a removed pin.\[palo01Fig\]](palo01)
The dyad is the only Assur graph on three vertices. There is no Assur graph on four vertices. An Assur graph, whose corresponding rigidity circuit is $K_4$ is called a [*basic*]{} Assur graph.
To generate all Assur graphs (on five or more vertices) we use Theorem \[CharacterThm\](ii) together with Theorem \[BJ\] to generate all rigidity circuits. To get from a rigidity circuit $C$ to an Assur graph, we choose a vertex $p^*$ of $C$ and split it into two or more pins (as in the proof of Theorem \[CharacterThm\]). The choice of $p^*$, the splitting of $p^*$ into a set $P$ of pins ($2\leq |P|\leq val(p^*)$), and choosing for each edge incident to $p^*$ an endpoint from $P$ allows us to construct several Assur graphs from one rigidity circuit, see Figure \[PinRearrange\]. We say that $G(I, P; E)$ and $G'(I, P', E)$ are related by [*pin rearrangement*]{} if $G^* =G'^*$ (see Figure \[PinRearrange\]).
![Pin rearrangement (maintaining fact of at least two pins) \[PinRearrange\]](assurwordfig08)
The operations of edge-split and 2-sum, which were used to generate rigidity circuits inductively, can also be used directly on Assur graphs to generate new Assur graphs from old, see Figures \[assurwordfig06\] and \[palo01Fig\]. In particular, the operation of 2-sum may be of practical value if, for space reasons for example, a pinned vertex is to be eliminated, see Figure \[palo01Fig\].
All Assur graphs on 5 or more vertices can be obtained from basic Assur graphs by a sequence of edge-splits, pin-rearrangements and $2$-sums of smaller Assur graphs.
Since mechanical engineers might want to have additional tools readily available for generating Assur graphs, one can seek additional operations under which the class of Assur graphs is closed. Vertex-split (creating two vertices of degree at least three) is another operation which takes a rigidity circuit to a rigidity circuit, and therefore takes an Assur graph with at least three vertices to a larger Assur graph (Figure \[assurwordfig10\]). This is, in a specific sense, the dual operation to edge-split [@CMW]. More generally, [@CMW] explores a number of additional operations for generating larger circuits from smaller circuits. Each of these processes will take an Assur graph to a larger Assur graph.
![Vertex split taking an Assur graph to an Assur graph. \[assurwordfig10\]](assurwordfig10)
The inductive constructions for Assur graphs can be used to provide a visual [*certificate sequence*]{} for an Assur graph. If we are given a sequence of edge-splits and 2-sums starting from a dyad and ending with $G$, see Figure \[palo02Fig\], it is trivial to verify that $G$ is an Assur graph. We constructed such a sequence in the proof of Theorem \[PinThm\]. It is well known, see [@TW], that there are exponential algorithms to produce such a certificate. However, there are other fast algorithms, for example the so called pebble games, see [@JacobsHendrickson; @pebblegame] to detect all the rigidity properties of graphs that can be adapted to verify the Assur property.
![Certificate sequence for the final Assur graph.\[palo02Fig\]](palo02W)
Concluding comments
===================
The paper introduces, for the first time, the concept of Assur graphs, in the rigorous mathematical terminology of rigidity theory. This work paves a new channel for cooperation between the communities in kinematics and in rigidity theory. An example for such channel is the material appearing in §2.7 showing how to transform determinate trusses used by the kinematicians into isostatic frameworks, widely employed by the rigidity theory and structural engineering communities.
At this point, it is hard to predict all the practical applications that are to benefit from this new relation between the disciplines. Nevertheless, we anticipate practical results from the use of rigidity theory in mechanisms as introduced in the paper. Examples of such results, include using rigid circuits from rigidity theory to find the proper decomposition sequence of pinned isostatic framework into Assur components (Section 3.2) and generation of Assur graphs by applying two known operations to their corresponding rigidity circuits (Section 3.3).
It is expected that new opportunities will be opened up, for example, for mechanical engineers to comprehend topics in rigidity theory that are used today in many disciplines, including biology, communications and more. Mechanical engineers in the west may be motivated to use the Assur graphs (Assur groups) concept as it is widely applied in eastern Europe and Russia.
Decomposing a larger linkage into Assur graphs and analyzing these pieces one at a time is an effective way to analyze the overall motion, working in layers. This paper has given a precise mathematical foundation for the Assur method as well as a proof of its correctness and completeness.
We have followed standard engineering practice and developed the theory in the language of bar-and-joint frameworks, but of course there is no need to replace a larger link (rigid body) with an isostatic bar-and-joint sub-framework to apply the counting techniques. It is simply convenient to do so in order to streamline notation and graphics. All of our results may be reworded in terms of body and bar structures or body and pin frameworks. This presentation would be much closer to the original example in Figures 1 and 2 and the counts of §2.2.
To extend this type of analysis to 3D linkages, the lack of good characterization of isostatic bar-and-joint frameworks in $3$D is an initial obstacle. However, if we focus on body-and-bar or body-and-hinge structures (the analog of body-and-bar and body-and-pin frameworks in the plane) then the generic DOF of these structures is computable by analogous counting techniques [@WW2; @Wh], and all our combinatorial methods will carry over sucessfully.
[^1]: Mathematics Department, Worcester Polytechnic Institute. [email protected]
[^2]: Faculty of Engineering, Tel-Aviv University. [email protected]
[^3]: Department of Mathematics and Statistics, York University Toronto, ON, Canada. [email protected]. Work supported in part by a grant from NSERC Canada
|
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abstract: |
The presence of dark matter in the solar neighbourhood can be tested in the framework of stellar evolution theory by using the new results of helioseismology. If weakly interacting massive particles accumulate in the center of the Sun, they can provide an additional mechanism for transferring energy from the solar core. The presence of these particles produces a change in the local luminosity of the Sun of the order of $0.1 \%$, which is now within the reach of seismic solar experiments. We find that typical WIMPs with a mass of 60 GeV, annihilation cross-section of $10^{-32} \;
\mbox{cm}^3/\mbox{sec}$, and scalar scattering interaction between $10^{-36}\;\mbox{cm}^2$ and $10^{-41}\; \mbox{cm}^2$, the lower end of this range being that suggested by the DAMA direct detection experiment, modify the radial profile of the square of the sound speed in the solar core by $\sim 0.1 \%$. This level of change is strongly constrained by the most recent helioseismological results. Significantly more massive WIMPs have a much smaller effect on the solar core temperature profile. However future helioseismological experiments have the potential exploring most, if not all, of the WIMP parameter space accessible to current and planned direct detection experiments.
author:
- |
Ilídio P. Lopes, Joseph Silk and Steen H. Hansen\
Nuclear & Astrophysics Lab., 1 Keble Road, Oxford OX1 3RH, United Kingdom
date: Draft version
title: Helioseismology as a New Constraint on SUSY Dark Matter
---
== == == ==
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Sun: oscillations - Sun: interior - cosmology - dark matter
Introduction
============
The dynamical behaviour of various astronomical objects, from galaxies to galaxy clusters and to large-scale structure, in the observed universe can only be understood if the dominant component of the mean matter density is dark, amounting to $\Omega_m=0.3\pm 0.1.$ Constraints from primordial nucleosynthesis of the light elements provide a compelling measure of the mean baryon abundance, $\Omega_b=0.05\pm 0.02.$ The bulk of the dark matter is consequently non-baryonic, and the existence of particles that interact with ordinary matter on the scale of the weak force, so-called Weakly Interacting Massive Particles (WIMPs), provides one of the best-motivated candidates, arising from the lightest stable particle predicted by super-symmetry theories (SUSY), for resolving this problem.
One can show (Lee & Weinberg 1977) that if such a stable particle exists, its relic abundance is $\Omega_x h^{2} \simeq 3\times 10^{-27}
cm^3 s^{-1}/ \langle\sigma_a v\rangle$, where $\langle\sigma_a
v\rangle$ is the thermally averaged product of annihilation cross-section and relative velocity. Alternatively, if a new particle with weak-scale interactions exists, then its annihilation cross-section can be estimated to be $\langle\sigma_a v\rangle\sim
10^{-25} cm^3 s^{-1} $ (Jungman [*et.al.*]{} 1996). Indeed, this constitutes one of the strongest arguments for considering the WIMP to be one of the best particle candidates for the non-baryonic component of the dark matter.
These particles are predicted by super-symmetry theories (SUSY) to be thermally produced at the early stages of the universe (Ellis 2001). The best WIMP candidates are neutralinos, but other types of WIMPs (axions, axinos, gravitinos and Wimpzillas) are not excluded as an alternative to the constituents of the dark matter component (Roszkowski 2001). Nor are charged super-heavy particles with strong interactions (Albuquerque [*et.al.*]{} 2000).
It is usually assumed that the WIMPs are a thermal relic of the big bang. The resulting thermic relic particle should have a mass inferior to $340 TeV$, the unitary limit (Griest and Kamionkowski 1990). However, for SUSY models, this bound is typically 2 orders of magnitude stronger (Jungman [*et.al.*]{} 1996). The neutralino is the best candidate to be produced thermally, while CDM axinos, gravitinos and Wimpzillas are the best candidates to be produced by a non-thermal mechanism. For example, in the early Universe, vacuum fluctuations during or after inflation can produce a class of very weak super-massive particles with mass in the range of $10^{12} GeV$ up to $10^{13} GeV$ (Kuzmin & Tkachev 1999).
Many experiments around the world are currently engaged in searching for dark matter by assuming that it is dominated by the light neutralinos, as predicted by the minimal supersymmetric extension of the Standard Model (Ellis 2001). In particular, the DAMA collaboration (Bernabei [*et al.*]{} 2000) has reported evidence of an annual modulation of recoil energy, which was interpreted as being due to scattering of some cold dark matter particles with masses between 50 and 100 GeV, and spin-independent cross-sections on a proton of between $10^{-42} cm^2$ and $10^{-41} cm^{2}$.
Neutralinos might not constitute all the cold dark matter, but could be complemented by other particles such as axions and superheavy relics. Neutralinos are neutral Majorana particles, the lightest SUSY particles that are massive and stable (Jungman [*et.al.*]{} 1996). Neutralino properties as dark matter candidates are quite dependent on the neutralino type, such as the bino, the wino and the two neutral higgsinos (Rowszkowski 2001). The neutralino cross-section for elastic scattering on a nucleus is expected to be typically very small, roughly $10^{-42}{\rm cm}^2$. This is because the elastic cross-section is related to the cross-section of neutralino annihilation in the early Universe which must be only a fraction of the weak interaction strength in order to have $\Omega_x
h^2\sim 1$ (Bergström 2000). In this paper we adopt the neutralino as our WIMP test particle, unless stated otherwise.
The Sun traps WIMPs in the course of its lifetime, as we discuss below. If the Sun were to contain even a minute mass fraction of WIMPs, there could be a significant influence on its central thermal structure. Helioseismology provides a means for an independent test of the validity of this idea and can be used to discriminate between possible candidates for the Dark Matter ([Kaplan [*et al.*]{}]{} 1991). Indeed, helioseismology has become a mature discipline through the detection of more than three thousand solar acoustic modes with which it has been possible to constrain the internal structure of the Sun (Gough 1996). The large quantity of seismic data obtained by the ground-based network during the last 10 years, as well as the three seismic instruments on board the SOHO mission during the last 5 years, contributed to improve the accuracy of the frequency determination to a level of 1 part in $10^{5}$ (Bertello [*et al.*]{} 2000; Garcia [*et al.*]{} 2001). This has allowed us to obtain a solar model for which the equilibrium thermodynamical structure accurately reproduces the observed seismic data, the so-called seismic solar model.
In this paper, we discuss how WIMPs can modify the evolution of the Sun and ultimately the present internal structure, namely in the nuclear region. In particular, we are interested in WIMPs that interact with baryonic matter through a weak interaction with a spin-independent scattering cross-section with values in the range of the current experiments for the detection of dark matter, such as the DAMA collaboration, and other ongoing or future experiments. It follows from our analysis that the possible WIMP candidates proposed by the DAMA experiment are in disagreement with our current models.
The next section contains a description of the present status of the standard solar model, and the current view of the different difficulties in modelling the observed Sun. In Section 3, we present a theoretical description of WIMPs and how they interact with baryonic matter. In particular, we discuss qualitatively the type of WIMPs capable of modifying the evolution of the Sun. In Section 4, we discuss solar models that evolve in halos of dark matter, and we discuss how those WIMPs can modify the evolution of the Sun. We conclude with a discussion of our new results, stressing their pertinence to the current status of the dark matter problem and Cosmology.
THE PRESENT SUN: THE SOLAR STANDARD MODEL
=========================================
The Sun is a unique star for research because its proximity allows a superb quality of solar data, enabling precision measures of its luminosity, mass, radius and chemical composition. The Sun therefore naturally becomes a privileged target for testing stellar evolution theory. In recent years, different groups around the world have produced solar models in the framework of classical stellar evolution, taking into account the best known physics as well as all the available observational seismic data. This has led to determination of a well-established model for the Sun, the so-called standard solar model (Turck-Chiéze & Lopes 1993; Christensen-Dalsgaard [*et al.*]{} 1996; Brun, Turck-Chièze & Zahn 1999; Provost, Berthomieu & Morel 2000; Turck-Chiéze, Nghiem, Couvidat & Turcotte 2001, Turck-Chize S. [*et al.*]{} 2001a; Bahcall, Pinsonneault, Basu 2001), for which the acoustic modes are in very good agreement with observation. Furthermore, this model has established considerable consensus among the different research groups, concerning the predictions of the solar neutrino fluxes, and has unambiguously helped define the difference between the theoretical predictions and the experimental results.
The combined effort between helioseismology and stellar astrophysics has contributed to strongly constrain the internal structure of the present Sun. Indeed, among the different seismic diagnostics that the solar model is tested against, the radial profile of the square of the sound speed is known with a precision as high as $0.3\%$, and in particular in the nuclear region this precision goes up to $0.2\%$ (see Fig 1). This progress has been achieved through a detailed description of the microscopic physics, such as the nuclear reaction rates, the equation of state, the coefficient of opacities and the gravitational settling of chemical elements. More recently, the macroscopic mixing induced by differential rotation, seems to present a determining role in the evolution of the star. These physical processes take place in the thin shear layer, located between the radiative interior and the convection zone, the so-called tachocline (Elliott & Gough 1999).
The standard stellar evolution of the Sun assumes that the star is in hydrostatic equilibrium, is spherically symmetric and that the effects of rotation and magnetic fields are negligible. The present solar structure and evolution are computed starting from an initially homogeneous star of a given composition. The solar models used in this paper have been computed by using the CESAM code (Morel 1997). The main physical inputs are as follows. We used the solar composition of Grevesse, Noels & Sauval (1996), the OPAL96 tables of Iglesias & Rogers(1996), the equation of state of Rogers, Stenson & Iglesias (1996), the nuclear reaction rates of Adelberger et al. (1998), the Mitler (1977) prescription for the treatment of the nuclear screening rates, and the microscopic diffusion coefficients suggested by Michaud & Proffit (1993). The present structure of the Sun is obtained by evolving a initial star from the pre-main sequence, around 0.05 Gyr from the ZAMS, until its present age, 4.6 Gyr. The present solar model is obtained by choosing the initial Helium content, as well as the mixing length parameter of convection that best predicts the present solar luminosity and radius. The helioseismolgy requires that the present structure of the Sun, including the global quantities, should be determined with a very high precision. In particular, the calibration of the solar radius is done with a precision of $10^{-5}$.
THE EVOLUTION OF THE SUN IN THE PRESENCE OF DARK MATTER
=======================================================
In order to explain the rotation of spiral galaxies, it is necessary to assume that these are immersed in a halo of dark matter. In the present model, we assume the halo of the Milky Way to be an isothermal gas of WIMPs, the distribution of their velocities being Maxwellian with a mean velocity $\bar{v}\sim
240\; Km\;s^{-1}$. Furthermore, we assume that the local dark matter density near the Sun is about $0.3 \; GeV\;cm^{-3}$ (Jungman [*et.al.*]{} 1996). Typically, the local density by number is of the order of $3\;10^{-2}\; cm^{-3}$ for WIMPs with $m_{\rm x}\sim 10 {\rm GeV}$, and $3\;10^{-3}\;cm^{-3}$ for WIMPs of $m_{\rm x}\sim 100 \;{\rm GeV}$.
The WIMP scattering cross-section
---------------------------------
Any WIMP that crosses the interior of the Sun may interact with a nucleus and lose enough energy to be trapped by the star’s gravitational potential well. The WIMP interactions with the baryonic matter will depend upon their mass, $m_{\rm x}$ and on their scattering cross-section on the nucleons, $\sigma_p$. The elastic scattering cross-section of relic WIMPs scattering off a nucleus in the solar interior depends on the individual cross-sections of the WIMPs scattering off of quarks and gluons. For non-relativistic Majorana particles, such as the neutralino, these can be divided into two separate types. The coherent part described by an effective scalar coupling between the WIMP and the nucleus is proportional to the number of nucleons in the nucleus. The incoherent component of WIMP-nucleus cross section results from an axial current interaction of a WIMP with the constituent quarks, and couples the spin of the WIMP to the total spin of the nucleus. These two types of cross-sections have been computed exactly (Goodman & Witten 1985, Jungman [*et.al.*]{} 1996, and references therein), and in the case of scalar interaction it strongly depends on the particle physics model used. At this stage, it is worth noticing that in the case of scalar (or coherent) interactions, the total scattering cross-section is proportional to the mass of the nucleus, $ \sigma^{sc}\sim \mu_i^2
$, where $\mu_i=m_x m_i/(m_x+m_i)$ is the reduced mass, and in the case of axial (or incoherent) interactions, it can be shown that the cross-section, $ \sigma^{ax}\sim \mu_p^2 $, depends on the spin of the nucleus as $J(J+1)$. In the case of scalar interactions, because of the constructive interference between all nucleons in the nucleus, the cross-section rises rapidly with the atomic number and, as a consequence, WIMPs scatter much more efficiently on the heavier nuclei. Inside the Sun, elements heavier than hydrogen contribute substantially to the energy transfer and to the capture rate. Nevertheless, the Sun’s evolution on the main sequence is strongly dominated by the production of helium rather than heavier chemical elements. In such a scenario, the helium nucleus will be the dominant source of scalar scattering, mainly due to its high abundance, given that heavier nuclei such as iron, nitrogen, carbon and oxygen are sub-dominant species. However, in more detailed studies on the capture of WIMPs by the Sun, the contribution of heavier elements seems to be more significant than was previously thought (Gould 1987). In fact, the scattering on these species will increase the total number of WIMPs accreted by the Sun leading to a more significant change on the structure of the solar core. Consequently, by choosing to consider the scattering only for helium, we make a conservative approach to study the impact of WIMPs on stellar evolution and Helioseismology.
The essential particle physics parameters which enter the dark matter problem are the mass of the WIMP $m_x$ and the elastic scattering cross-section, $\sigma_p$, of the various nuclei which constitute the solar material, and $\langle\sigma_a v\rangle$ the thermal average of the annihilation cross-section times the relative velocity of the WIMPs at freeze-out. The annihilation cross section can be calculated for each model (Krauss [*et.al.*]{} 1985) however, since we lack certain theoretical guidance, we decided to concentrate on a typical simple case for WIMP-nucleus scalar scattering, as previously discussed. We relate the cross-section of the WIMP on nuclei to the WIMP-proton cross section $\sigma_p$ taken as a free parameter.
The Capture and Annihilation Rates of WIMPs in the Sun
------------------------------------------------------
The abundance of WIMPs in the Sun depends on the WIMPs accumulated in the Sun by capture from the Galactic halo and is depleted by annihilation. If $N_x$ is the number of WIMPs in the Sun, then the differential equation governing the time evolution of $N_x$ is (Gould [*et.al*]{} 1987) $$\begin{aligned}
\frac{dN_x}{dt}=\Gamma_c- C_a\;N_x^2,\end{aligned}$$ where $\Gamma_c$ is the rate of accretion of WIMPs onto the Sun. The determination of $\Gamma_c$ depends on the nature of the particle. If the halo density of WIMPs remains constant in time, $\Gamma_c$ is time-independent. The second term accounts for depletion of WIMPs, and is twice the annihilation rate in the Sun, $\Gamma_a=1/2\;C_a N_x^2$. The quantity $C_a$ depends on the WIMP total annihilation cross section times the relative velocity in the non-relativistic limit, $\langle\sigma_a v\rangle$, and the distribution of WIMPs in the Sun. The total annihilation cross-section of the new particle should possibly have a value that is consistent with the cosmological density of dark matter as specified, $\langle\sigma_a v\rangle \approx 3 \,
10^{-26} cm^3 s^{-1} \Omega_x^{-1} h^{-2} $. It follows that, not knowing the initial content of relic WIMPs, $\Omega_x$, we will consider different scenarios, such that $\langle\sigma_a v\rangle <
10^{-26} cm^3 s^{-1} $ (Jungman [*et.al.*]{} 1996).
Solving the previous first-order differential equation, the total number of WIMPs at time $t$, is given by $$\begin{aligned}
N_x(t)=\Gamma_c\;\tau\;\tanh{\left(\frac{t}{\tau}\right)},\end{aligned}$$ and the annihilation rate is given by $$\begin{aligned}
\Gamma_a(t)=\frac{1}{2}\;\Gamma_c\;\tanh^2{\left(\frac{t}{\tau}\right)},\end{aligned}$$ where $\tau=1/\sqrt{\Gamma_c \; C_a}$ is the time scale for capture and annihilation of WIMPs to equilibrate. As the age of the Sun is much larger than the equilibrium time-scale, $t_\odot/\tau \gg 1$, equilibrium is reached very rapidly, then $\Gamma_a=1/2\; \Gamma_c$. It follows that $$\begin{aligned}
\frac{t_\odot}{\tau}=322
\left(\frac{\Gamma_c}{s^{-1}}\right)^{1/2}\left(\frac{\langle\sigma_a
v \rangle}{cm^3
s^{-1}}\right)^{1/2}\left(\frac{m_x}{10GeV}\right)^{3/4},
\label{eq-ttau}\end{aligned}$$ where $t_\odot$ is the present age of the Sun (Jungman [*et.al.*]{} 1996).
The rate of accretion of WIMPs in the Sun was first calculated by Press and Spergel (1985). Even though the detailed computation of the accretion can be quite elaborate, the basic idea is simple (Srednicki [*et.al.*]{} 1987); if the WIMP elastically scatters from a nucleus with a velocity smaller than the escape velocity, $v_{esc}=\sqrt{2GM_\odot/R_\odot}$, then the WIMPs will be captured. The WIMPs have a typical velocity of the order of $300\;
km/s$, which is smaller than the escape velocity at the surface of the Sun $618\; km/s$, consequently, they are trapped quite efficiently in the solar interior. If we consider that all the WIMPs that hit the Sun lose enough energy to be trapped, the trapping rate is given by the product of the surface area of the Sun, about $6.1\;10^{22}\; cm^2$, and the flux of WIMPs, of about $n_{\rm x}\;\bar{v}$. It reads $
\Gamma_{\rm c}=4\pi R_\odot^2 n_{\rm x} \bar{v}$. Gravitational focussing effects enhance the previous trapping rate by $2GM_\odot/\bar{v}^2$. This occurs because of the enlargement of the Sun’s effective area due to the gravitational well and the requirement that the impact parameter be less than $R_\odot$. This yields a trapping rate of $\Gamma_{\rm c}=\pi(2GM_\odot R_\odot) n_{\rm
x}/\bar{v}$. Bouquet and Salati (1989) have generalized and refined the capture rate for main-sequence stars (Salati & Silk 1989) $$\begin{aligned}
\Gamma_c=10^{32}\; {\rm s^{-1}} \left(\frac{\rho_{\rm
x}}{1\;M_\odot {\rm pc}^{-3}}\right) \left(\frac{m_p}{m_x}\right)
\left(\frac{300\;{\rm km}{\rm s}^{-1}}{\bar{v}_x}\right) \nonumber
\\
\left[1+0.16\left(\frac{\bar{v}_x}{300\;{\rm km}{\rm
s}^{-1}}\right)^2 \left(\frac{M_\odot}{M}\right)
\left(\frac{R}{R_\odot}\right) \right] \nonumber
\\
\left(\frac{M}{M_\odot}\right) \left(\frac{R}{R_\odot}\right) {\rm
min}\left[1,\frac{\sigma_p}{\sigma_\odot}\frac{M}{M_\odot}
\left(\frac{R_\odot}{R}\right)^2\right].\end{aligned}$$ The total number of WIMPs accreted, $N_x$, can be computed taking into account that $t_\odot/\tau >> 1$, and is $$\begin{aligned}
N_x\simeq \Gamma_c\tau.\end{aligned}$$ The total number of WIMPs at present strongly depends on the ratio $
\sqrt{\langle\sigma_a v\rangle/\Gamma_c}$. If we consider $
\langle\sigma_a v\rangle$ to be constant, the capture is weak and proportional to the scattering cross-section and $N_x$ increases with $\Gamma_c$, for the case of low scattering cross-sections. On the contrary, for large scattering cross-sections, any WIMP which enters the star is captured, the captured flux saturates at the level of the incoming flux and the system reaches an equilibrium. Note that evaporation is unimportant for WIMPs in the mass range considered here, namely $m_x>10$ GeV (Gould 1987).
This concentration of WIMPs in the solar core has a marginal effect on the evolution of the Sun. Luckily, the precision presently obtained by seismic diagnostics allows to determine in certain cases the effect of WIMPs on the solar core. In fact, a more accurate expression for the capture should take into account the way that WIMPs scatter from the nucleus through scalar interactions and/or axial-vector interactions, as well as second-order effects (Jungman [*et.al.*]{} 1996). However, we choose to focus on a simplified picture in this first approach to the problem.
The energy transport by WIMPs in the solar interior
---------------------------------------------------
The energy transport by WIMPs in the solar interior is governed by three natural length scales: the mean free path of the WIMP, $l_x$, the inverse of the logarithmic temperature gradient, $\nabla \ln{T}$, and a typical geometric dimension of the system, such as the WIMP scale height, $r_x$. Bouquet and Salati (1989) showed that the WIMP distribution is approximatively gaussian and therefore $$\begin{aligned}
n_x(r)=n_0\;\exp{\left[-\frac{r^2}{r_x^2}\right]}\end{aligned}$$ where $n_0=N_x/\pi\sqrt{\pi} r_x^3$, and $r_x^2 = 3 T_c/(2 \pi m_x G \rho)$ which typically is $r_x \approx 10^{-2} \, R_\odot \, m_{100}^{-1/2}$ with $m_{100}$ the WIMP mass in units $100$ GeV.
There are two extreme regimes for the energy transport by WIMPs,
characterized by the values of the Knudsen number, $K$, defined as the ratio $l_x/r_x$. The Knudsen number strongly depends on the scattering cross-sections of the WIMPs on nuclei, and typically goes like $K
\approx 30 \, m_{100}^{3/2} / \sigma_{36}$ where $\sigma_{36}$ is the scattering cross section measured in units $10^{-36}$ cm$^2$.
In the case of small cross-sections, the WIMP mean free path is larger than the scale length $r_x$, $K > 1$, and two successive collisions are widely separated. This is the so-called large Knudsen number regime (or Knudsen regime). The transport of energy is non-local. Thus the WIMPs undergo few interactions on each orbit and therefore are not in thermal equilibrium with the nuclei. In such a case, it is convenient to define the average temperature of the WIMP core, $T_x$, as being the temperature of the star at $r_x$. At the center of the star, the WIMP temperature (or the averaged kinetic energy) is lower than the temperature of the nuclei, and in such conditions the net effect of the WIMP-nucleus collisions is to transfer energy from nuclear matter to WIMPs. Conversely, in the outskirts of the WIMP core, in certain cases the WIMPs give back the energy gained in the center of the star. This process is a very efficient mechanism at evacuating the energy from the nuclear region. In such a regime no precise analytic result exists. Nevertheless, an approximative solution has been found by Spergel and Press (1985). The luminosity carried by WIMPs can then be evaluated by $$\begin{aligned}
L_{\rm lp}(r)=32\sqrt{2\pi}\;\frac{\sigma_p\mu_p}{m_x+m_p}
\int_0^r n_p(r) \bar{T}(r) n_x(r)r^2dr\end{aligned}$$ where $\bar{T}$ reads $$\begin{aligned}
\bar{T}(r)=\sqrt{\frac{m_pT_x+m_x T(r)}{m_xm_p}}\left[T(r)-T_x\right].\end{aligned}$$
In the opposite limit, $K < 1$, the transport of energy is local and the energy transport proceeds by conduction, allowing for a much simpler treatment (Gilliland [*et.al.*]{} 1986). In this regime, the WIMP mean free path is much smaller than the dimension of the region where they are trapped. The WIMPs interact sufficiently often to be in local thermal equilibrium with the nuclei and therefore it is possible to define a temperature of the WIMP, which is equal to the temperature of the nuclei. $$\begin{aligned}
L_{\rm sp}(r)=-4\pi r^2 \;n(r)\;\sqrt{\frac{T(r)}{m_x}} l_{sp}(r)\;
\nabla T(r)\end{aligned}$$ where $l_{sp}(r)$ is the local mean free path of the WIMP, it reads $$\begin{aligned}
l_{\rm
sp}(r)=\left[\frac{\rho(r)}{m_u}\sum\sigma_i\frac{X_i(r)}{A_i}\right]^{-1}.\end{aligned}$$ The range of WIMP masses allowed by the cosmological model is quite large, even if we restrain ourselves to the WIMPs that are produced in a thermal scenario. Consequently, depending also on their scattering cross-section, the WIMPs can transport energy, not only in the conductive region and in the Knudsen regime, but also in an intermediate case. We have chosen to define the intermediate case as the one where the transport of energy is determined by the interpolation formula (Gilliland [*et.al.*]{} 1986) $$\begin{aligned}
L_x(r)=\frac{L_{lp}(r)L_{sp}(r)}{L_{lp}(r)+L_{sp}(r)}.\end{aligned}$$ This formula appropriately converges to the right approximation in each of the two regimes, accordingly to the value of $K$. The errors introduced are within the precision of our computation.
DISCUSSION AND CONCLUSIONS
==========================
[**log$_{\bf 10}$**]{}
The thermodynamical structure in the interior of the Sun, namely in the nuclear region, is presently known with a precision of much less than a few per cent. This level of accuracy in constraining the solar interior has been achieved by a systematic study of the differences between the acoustic spectrum obtained from helioseismology experiments and the theoretical spectrum. Presently, this difference is less than $10 \mu Hz$ for almost all of the 3000 modes that probe the interior of the Sun (Gough 1996). This level of precision in the description of the solar core allows us to discriminate physical processes that could not be discussed otherwise, in particular those that present a peculiar behaviour, as seems to be the case for WIMPs trapped in the solar core. In Fig. 1, we compare the square of the sound speed, $c_s^2$, of different solar models evolving within the presence of a halo of WIMPs and the solar standard model (Brun, Turck-Chièze & Morel 1998). The changes induced by the presence of WIMPs are concentrated in the inner core within 10% of the solar radius, typically seen in the profiles of the temperature, density and molecular weight.
Indeed, the WIMPs are thermalized within the solar core and are on Keplerian orbits around the solar center, interacting through elastic scattering with the solar nuclei, such as helium, and thereby providing an alternative mechanism of energy transport other than radiation. The result is a nearly flat temperature distribution, leading to an isothermal core. Consequently, the central temperature is reduced. This reduction of temperature has two main consequences: since central pressure support must be maintained, due to the hydrostatic equilibrium, the central density is increased in the solar models with WIMPs, and since less hydrogen is burnt at the centre of the Sun, the central helium abundance and the central molecular weight are smaller than in standard solar models. The increase of the central density and hydrogen partially offset the effect of lowering the central temperature in the central production of energy. In fact, this is the reason why minor changes are required to the initial helium abundance and the mixing-length parameter in order to produce a solar model of the Sun with the observed luminosity and solar radius. This readily leads to a balance between the temperature, and the molecular weight in the core, leading to the peculiar profile of the square of the sound speed, $c_s^2\propto T/\mu$. This seems to be the case for most of the solar models within WIMP halos. The balance between the temperature and molecular weight is critical for the profile in the center of the star, leading to some of the profiles presented in Fig. 1. In the same figure, we display the inversion of the sound speed obtained from the data of the three seismic experiments on board the SOHO satellite. It follows from our analysis that the presence of WIMPs in the solar core produces changes in the solar sound speed of the same order of magnitude as the difference between the sound speed of the standard solar model and the inverted sound speed. It is important to remark that the inversion of the sound speed still presents some uncertainty in the central region due to the lack of seismic data, mainly due to the small number of acoustic modes that reach the nuclear region. Furthermore, the inversions are not very reliable at the surface, above $98 \%$ of the solar radius, due to a poor description of the interaction of acoustic waves with the radiation field and the turbulent convection, namely, in the superadiabatic region (Lopes & Gough 2001). However, we can establish with certainty that the difference between the inverted sound speed and the theoretical sound speed is known with a precision of $0.3\%$ in the solar interior, within $95\% $ of the solar radius. The evolution of the Sun in a halo of WIMPs will increase the evacuation of energy from the solar core. The WIMPs will work as a ’cold bridge’ between the core and the more external layers of the Sun. The magnitude of the effect is proportional to the total number of WIMPs concentrated in the solar core. Nevertheless, even if some systematic effect is present in the inversion of the sound speed, the presence of WIMPs in the solar core leads to a quite different nucleosynthesis history from the solar standard model case, and from that to a peculiar radial profile of the sound speed. In this way, the effect of WIMPs in the solar core can be inferred on the basis of seismic diagnostics, such as the inversion of the square of the sound speed, among other possible techniques. The proposed method constitutes a new way to disentangle the contribution of different non-baryonic particles to the dark matter.
The luminosity in the core of the Sun is presently known with a precision of one part in $10^{-3}$. In the coming years, it is very likely that the new seismic data available from the SOHO experiments, will allow us to obtain a seismic model of the Sun with an accuracy of $10^{-5}$. In such conditions, the Sun can and should be used as an excellent probe for dark matter in our own galaxy. In Figs. 2 and 3, we compute the ratio of the WIMP luminosity against the Sun’s luminosity produced in the inner core of $5\%$ of the solar radius. A significative region of the $\sigma_{scat}-\langle\sigma_{a}v\rangle$ plot shows changes in the solar luminosity of the order of $10^{-3}$. This order of magnitude on the luminosity produced in the solar core can be tested through seismological data. In particular, we are interested in the lighter WIMPs, $m_x< 100GeV$, and the smallest scattering cross-section, $10^{-45}\; cm^2 $ up to $10^{-40}\;cm^2$. This range of parameters are presently being tested by the DAMA and CDMS experiments, among others, and are also well within the range of the parameters of future helioseismological experiments (see Fig. 3). It is interesting to note, that the X-ray Quantum Calorimeter (XQC) experiment (D. McCammon [ *et.al.*]{} 2001) may exclude scattering cross sections bigger than about $10^{-29}$ cm$^2$ for the mass range considered here, relevant for strongly interacting massive particles recently hypothesised as an alternative dark matter candidate to WIMPs and that penetrate neither subterranean laboratories nor the solar core (Wandelt [*et.al.*]{} 2000), hence being complementary to helioseismology in potentially excluding overlapping regions in parameter space.
If we believe in the simple model presented in this paper, DAMA candidate WIMPs with masses of 60 GeV and annihilation cross-section of the order of $10^{-32}cm^3/s$ cannot exist (see Fig. 1), otherwise their effect in the solar core should already have been identified by seismic diagnostics. However, we stress that in order to determine with certainty the range of masses and cross-sections for WIMPs that is in disagreement with the helioseismological results, a more careful analysis of the different regimes of energy transport by WIMPs should be done. Furthermore, the microscopic physics of this region of the star is not fully established, as there remains uncertainty in certain nuclear reaction rates, such as the p+p reaction and the dynamical screening in other nuclear interactions such as $^3He+^4He$ and $^4He+^4He$.
An important diagnostic of the core can be obtained from the seismology of gravity modes. Indeed, it is the low-degree internal gravity modes that are the most sensitive to the conditions in the core, the region where substantial deviations from the so-called standard solar model might occur. Solar models evolving in the presence of dark matter have a g-mode period spacing that is drastically different from that of other solar models, mainly due to the peculiar distribution of density on the nuclear region that occurs as a consequence of the energy transport enhanced by the WIMPs. Therefore, g-mode observations could provide a sensitive test to the radial distribution of density in the solar core, and ultimately to the presence of dark matter in the solar neighborhood. The observation of gravity modes by SOHO seismic experiments, such as GOLF (Turck-Chiéze [*et al.*]{} 2001b), could ultimately strongly constrain the physics of the solar core.
We have in this paper focused on a slightly simplified version of WIMP energy transport. There are several improvements which can be included in the analysis (Lopes [*et al.*]{} 2001), e.g. in the Knudsen regime the deviation from isotropy leads to a radius dependent luminosity suppression (Gould & Raffelt 1990b); in the conductive regime, the mass dependence of the thermal conductivity gives an additional factor $\sim 2 \sqrt{m_{100}}$ for scattering on helium (Gould & Raffelt 1990a). Also inclusion of scattering off other light elements might affect the results slightly. Finally, the transition region between non-local and conductive energy transport could be expressed in terms of the Knudsen number (Dearborn [*et al.*]{} 1991; Kaplan [*et al.*]{} 1991) instead of the intuitive interpolation formula used here, which would be useful for a detailed investigation of the transition region.
In conclusion, we have identified the range of WIMP masses, scattering cross-sections and annihilation cross-sections which reduce the luminosity in the core of the Sun. These effects on the solar structure are now within the range of effects capable of being probed by the diagnostic capabilities of helioseismology. We did not concern ourselves with particular particle physics models, but considered a generic case, for WIMPs which have a large range of masses, scattering cross-sections and annihilation cross-sections. The effect of WIMPs in the core of the Sun is of the same order of magnitude as the microscopic physics and dynamical processes that are now being discussed in the framework of stellar evolution in the light of the most recent results of helioseismology. Studies of the sun and of cosmology may have much to gain from each other.
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors thanks P. Morel for using the CESAM code and S. Turck-Chièze and R. Garcia for stimulating discussions and helioseismic analysis. The authors would like also to thank the referee D. Jungman for his valuable suggestions that lead to improve the original manuscript. SHH is supported by a Marie Curie Fellowship of the European Community under the contract HPMFCT-2000-00607. IPL is grateful for support by a grant from Fundação para a Ciência e Tecnologia.
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---
abstract: 'Our aim in this paper is to investigate the first Hochschild cohomology of [*admissible algebras*]{} which can be seen as a generalization of basic algebras. For this purpose, we study differential operators on an admissible algebra. Firstly, differential operators from a path algebra to its quotient algebra as an admissible algebra are discussed. Based on this discussion, the first cohomology with admissible algebras as coefficient modules is characterized, including their dimension formula. Besides, for planar quivers, the $k$-linear bases of the first cohomology of acyclic complete monomial algebras and acyclic truncated quiver algebras are constructed over the field $k$ of characteristic $0$.'
author:
- |
Fang Li and Dezhan Tan\
Department of Mathematics, Zhejiang University\
Hangzhou, Zhejiang 310027, China\
[email protected]; [email protected]
title: 'On the first Hochschild cohomology of admissible algebras [^1] '
---
**Keywords:** quiver, admissible algebra, differential operators, Hochschild cohomology.
**2010 Mathematics Subject Classifications**: 16E40, 16G20, 16W25, 16S32
[ Introduction ]{}
==================
The Hochschild cohomology of algebras is invariant under Morita equivalence. Hence it is enough to consider basic connected algebras when the algebras are Artinian. Let $\Gamma=(V,E)$ be a finite connected quiver where $V$ (resp. $E$) is the set of vertices (resp. arrows) in $\Gamma$. Let $k$ be an arbitrary field and $k\Gamma$ be the corresponding path algebra. Denote by $R$ the two-sided ideal of $k\Gamma$ generated by $E$. Recall that an ideal $I$ is called [*admissible*]{} if there exists $m\geq2$ such that $R^m\subseteq
I\subseteq R^2$ (See [@ASS]). According to the Gabriel theorem, a finite dimensional basic $k$-algebra over an algebraically closed field $k$ is in the form of $k\Gamma/I$ for a finite quiver $\Gamma$ and an admissible idea $I$.
An Artinian algebra is called a [*monomial algebra*]{} (see [@ARS]) if it is isomorphic to a quotient $k\Gamma/I$ of a path algebra $k\Gamma$ for a finite quiver $\Gamma$ and an idea $I$ of $k\Gamma$ generated by some paths in $\Gamma$. In particular, denote by $k^n\Gamma$ the ideal of $k\Gamma$ generated by all paths of length $n$. Then the monomial algebra $k\Gamma/k^n\Gamma$ is called the [*$n$-truncated quiver algebra*]{}.
The study of Hochschild cohomology of quiver related algebras started with the paper of Happel in 1989 [@Ha], who gave the dimensions of Hochschild cohomology of arbitrary orders of path algebras for acyclic quivers. Afterwards, there have been extensive studies on the Hochschild cohomology of quiver related algebras such as truncated quiver algebras, monomial algebras, schurian algebras and 2-nilpotent algebras [@Cibils][@Cib][@Ba][@Z][@L][@St][@Sa][@XHJ][@PS][@ACT][@Sa2]. In [@Ha], a minimal projective resolution of a finite dimensional algebra $A$ over its enveloping algebra is described in terms of the combinatorics when the field $k$ is an algebraically closed field. In these papers listed above, the authors use this kind of projective resolution or its improving version to compute the Hochschild cohomology.
In [@GL], the authors applied an explicit and combinatorial method to study $HH^1(k\Gamma)$. In this paper, we improve the method in [@GL] to the case of algebras with relations in order to study the $HH^1(k\Gamma/I)$ where $k\Gamma/I$ is an admissible algebra. This way does not depend on projective resolution and the requirement of $k$ being an algebraically closed field. Using this method, we can obtain some structural results which were not arisen by the classical method in the above listed papers.
If $I\subseteq R^2$ holds for a two-sided ideal $I$, we call $k\Gamma/I$ an [*admissible algebra*]{} (see Definition \[admissible algebra\]). So finite-dimensional basic algebras are always admissible algebras. We will give Proposition \[proposition 2.2\], which shows admissible algebras, including basic algebras, possess the similar characterization of monomial algebras and truncated quiver algebras, although it is not graded. From this point of view, admissible algebra is motivated to unify and generalize basic algebra and monomial algebra.
In the following, we always assume that $k\Gamma/I$ is an admissible algebra. This paper includes three sections except for the introduction. In Section 2, we introduce the basic definitions which are used in this paper. In particular, we define the notion of an *acyclic* admissible algebra, which can be thought as a generalization of the notion of an acyclic quiver. A sufficient and necessary condition is obtained for a linear operator from $k\Gamma$ to $k\Gamma/I$ to be a differential operator. Next, we give a standard basis of $\textsl{Diff}(k\Gamma,k\Gamma/I)$.
In Section 3, we investigate $H^1(k\Gamma,k\Gamma/I)$). In Eq.(\[3.23\]), a dimension formula of $H^1(k\Gamma,k\Gamma/I)$ is given for a finite dimensional admissible algebra. Moreover, in Theorem \[thm 3.7\], we construct a basis of $H^1(k\Gamma,k\Gamma/I)$ when $\Gamma$ is planar and $k\Gamma/I$ is an acyclic admissible algebra.
In Section 4, we characterize $HH^1(k\Gamma/I)$. In Eq.(\[5.28\]), we give the dimension formula of $HH^1(k\Gamma/I)$ for any finite dimensional admissible algebras $k\Gamma/I$. Moreover, we apply this method to complete monomial algebras and truncated quiver algebras. In Theorems \[tm 4.7\] and \[thm 5.3\], we construct $k$-linear bases of their first cohomology groups under certain conditions. The Hochschild cohomology of monomial algebras and truncated quiver algebras has been studied in [@Cib][@Z][@L][@XHJ][@S][@XHJ]. Our results in Section 4 can be seen as the generalization of those corresponding conclusions in the listed references above. In the same section, two examples of admissible algebras are given which are not monomial algebras. Their first Hochschild cohomology is characterized using our theory.
$k$-linear basis of $\emph{Diff}(k\Gamma,k\Gamma/I)$
=====================================================
We always assume $\Gamma=(V, E)$ is a finite connected quiver, where $V$ (resp. $E$) is the set of vertices (resp. arrows) in $\Gamma$. For a path $p$, denote its starting vertex by $t(p)$, called the **tail** of $p$, and the ending point by $h(p)$, called the **head** of $p$. For two paths $p$ and $q$, if $t(p)=t(q)$ and $h(p)=h(q)$, we say $p$ and $q$ to be [**parallel**]{}, denote as $p\parallel q$. Denote by $\mathscr{P}=\mathscr{P}_{\Gamma}$ the set of paths in a quiver $\Gamma$ including its vertices; denote by $\mathscr{P}_A$ the set of its acyclic paths. Trivially, $\Gamma$ is acyclic if and only if $\mathscr{P}_{\Gamma}\backslash
V=\mathscr{P}_A$. Throughout this paper, we always assume quivers are finite and connected.
\[admissible algebra\] Suppose $\Gamma=(V,E)$ is a quiver, $I$ is a two-sided ideal of $k\Gamma$, we call the quotient algebra $k\Gamma/I$ an **admissible algebra** if $I\subseteqq R^2$ where $R$ denotes the two-sided ideal of $k\Gamma$ generated by $E$.
\[proposition 2.2\] Suppose $k\Gamma/I$ is an admissible algebra, then there exists a subset $\mathscr{P}^{'}$ of $\mathscr{P}$ such that $V\cup E\subseteq \mathscr{P}^{'}$ and $\mathscr{Q}=\{\overline{x}|x\in\mathscr{P}^{'}\}$ forms a basis of $k\Gamma/I$ for $\overline{x}=x+I$
Let $X$ be a $k$-linear basis of $I$. Denotes by $\mathscr{P}_{\geq2}$ the set of all paths of length $\geq2$. Define $$T:=\{Y\subseteq k\Gamma: Y \text{ is linearly independent in}\ k\Gamma \text{ satisfying}\ X\subseteq Y\subseteq X\cup\mathscr{P}_{\geq2} \}.$$ $T$ becomes a partial set due to the order of inclusion between subsets of $k\Gamma$. It is easy to see $T\neq\emptyset$ and $T$ satisfies the upper bound condition of chains. So by the famous Zorn’s Lemma, $T$ has a maximal element, denoted by $Z$.
We claim that $Z$ is linearly equivalent to $\mathscr{P}_{\geq2}$. Otherwise, there exists $p\in\mathscr{P}_{\geq2}$ such that $p$ cannot be linearly expressed by $Z$, then $Z\cup\{p\}$ is linearly independent in $k\Gamma$, which contradicts to the maximal property of $Z$.
Since $Z$ is linearly equivalent to $\mathscr{P}_{\geq2}$, it follows that $V\cup E\cup Z$ is linearly equivalent to $\mathscr{P}=V\cup E\cup\mathscr{P}_{\geq2}$. By the definition of $T$, $Z\subseteq X\cup\mathscr{P}_{\geq2}$. And $I$ is generated by $X$. Hence $V\cup E\cup (Z\backslash X)$ forms a basis of the complement space of $I$ in $k\Gamma$. It means that $\mathscr Q =\{\bar x: x\in V\cup E\cup (Z\backslash X)\}$ forms a basis of $k\Gamma/I$. It is clear that $V\cup E\cup (Z\backslash X)\subseteq\mathscr{P}$ and it is the $\mathscr{P}^{'}$ we want.
When $I\subseteq R^2$ is finite dimensional, we have an explicit way to determine the $\mathscr{P}^{'}$. Concretely, suppose $\{x_1,x_2\cdots x_m\}$ is a basis of $I$. Then there exists a finite subset $\{p_1,p_2\cdots p_n\}$ of $\mathscr{P}$ such that $x_i$ can be expressed by the linear combinations of $p_j$. Suppose $x_i=\sum\limits_{j=1}^na_{ij}p_j$ for $i=1,2\cdots m$, then we obtain a $m\times n$ matrix $A=(a_{ij})$. We can transform the matrix $A$ into a row-ladder matrix $B=(b_{ij})$ through only row transformations. Suppose $b_{i,c(i)}$ is the first nonzero number of the $i$-th row of $B$. Since $B$ is a row-ladder matrix, we have $c_i\neq c_k$ for $i\neq k$. Then $\{x_1,x_2\cdots x_m\}\cup\{p_l|l\neq
c_1,c_2\cdots c_m\}$ is linearly equivalent to $\{p_1,p_2\cdots
p_n\}$. Hence $(\mathscr{P}\backslash\{p_1,p_2\cdots
p_n\})\cup\{p_l|l\neq c_1,c_2\cdots c_m\}$ is a basis of the complement space of $I$ in $k\Gamma$. Then the residue classes in $k\Gamma/I$ of all elements in this basis form a basis of $k\Gamma/I$.
On the other hand, in some special cases, e.g., when $k\Gamma/I$ is a monomial algebra, even if $I$ is not finite dimensional, the choice of $\mathscr{P}'$ is also given in the same way. If $k\Gamma/I$ is a monomial algebra and $I$ is generated by a set of paths of length $\geq2$, the set of paths that do not belong to $I$ is just the $\mathscr{P}'$ required.
Let $A$ be a $k$-algebra and $M$ an $A$-bimodule. A **differential operator** (or say, **derivation**) from $A$ into $M$ is a $k$-linear map $D: A\longrightarrow M$ such that $$\label{Leibnitz rule}
D(xy)=D(x)y+xD(y).$$ In particular, when $M=A$, this coincides with the differential operator of algebras.
\[lemma 2.2\] Suppose $D$ is a differential operator from $k\Gamma$ into $k\Gamma/I$. Then $D$ is determined by its action on the set $V$ of vertices of $\Gamma$ and the set $E$ of arrows of $\Gamma$.
\[2-4\] Let $\Gamma$ be a quiver. Denote $kV$ (resp. $kE$) by the linear space spanned by the set $V$ of the vertices of $\Gamma$ (resp. the set $E$ of the arrows of $\Gamma$). Assume we have a pair of linear maps $D_0: kV\longrightarrow k\Gamma/I$ and $D_1: kE\longrightarrow
k\Gamma/I$ satisfying that $$\label{12}
D_0(x)x+xD_0(x)=D_0(x),\ x\in V,$$ $$\label{13}
D_0(x)y+xD_0(y)=0,\ x,y\in V,\ x\neq y,$$ $$\label{14}
D_0(x)q+xD_1(q)=D_1(q),\ x\in V,\ q\in E,\ t(q)=x,$$ $$\label{15}
D_1(q)y+qD_0(y)=D_1(q),\ y\in V,\ q\in E,\ h(q)=y.$$ Then, the pair of linear maps $(D_0,D_1)$ can be uniquely extended to a differential operator $D: k\Gamma\longrightarrow\ k\Gamma/I$ satisfying that $$\label{16}
D(p):=\sum\limits_{i=1}^lp_1\cdots p_{i-1}D_1(p_i)\cdots p_l.$$ for any path $p=p_1p_2\cdots p_l, p_i\in E,1\leq i\leq l, l\geq2$.
One only need to prove that $D$ is indeed a differential operator. For this, we need to check Eq.(\[Leibnitz rule\]) in the next four cases:
\(a) $x, y\in V$; (b) $x\in V, y\in \mathscr{P}\backslash V$; (c) $x\in\mathscr{P}\backslash V, y\in V$; (d) $x,
y\in\mathscr{P}\backslash V$.
However, the checking process is routine, so we omit it here.
In the sequel, we always suppose $k\Gamma/I$ is an admissible algebra for the given ideal $I$ and the notations in Definition \[admissible algebra\] are used. From Definition \[admissible algebra\] and Proposition \[proposition 2.2\], there exists a basis of $k\Gamma/I$ which consists of residue classes of some paths including that of $V$ and $E$. Denote the fixed basis of $k\Gamma/I$ by $\mathscr{Q}$. Suppose $D:k\Gamma\longrightarrow
k\Gamma/I$ is a linear operator, then for any $p\in\mathscr{P}$, $D(p)$ is a unique combination of the basis $\mathscr{Q}$ of $k\Gamma/I$. Write this linear combination by $$D(p)=\sum_{\overline{q}\in\mathscr{Q}}c_{\overline{q}}^p\overline{q}.$$ where all $c_{\overline{q}}^p\in k$. We will use this notation throughout this paper. As convention, for the empty set $\emptyset$, we say $\sum_{\overline{q}\in\emptyset}c_{\overline{q}}^p\overline{q}=0$.
Suppose $q_1, q_2\in\mathscr{P}, q_1, q_2\notin I$ and $\overline{q_1}=\overline{q_2}$ in $k\Gamma/I$, then $t(q_1)=t(q_2),
h(q_1)=h(q_2)$, i.e., $q_1\parallel q_2$.
If $t(q_1)\neq t(q_2)$, then $\overline{q_1}=\overline{t(q_1)}\overline{q_1}=\overline{t(q_1)}\overline{q_2}=\overline{0}$, a contradiction, so $t(q_1)=t(q_2)$. Similarly, $h(q_1)=h(q_2)$.
According to the Lemma above, for $\overline{p}\in\mathscr{Q}$, we can define $t(\overline{p}):=t(q)$(resp.$h(\overline{p}):=h(q)$) for any path $q\in\mathscr{P}$ satisfying $\overline{q}=\overline{p}$ in $k\Gamma/I$. For a path $s\in\mathscr{P}$ and $\overline{p}\in\mathscr{Q}$, if $t(s)=t(\overline{p})$ and $h(s)=h(\overline{p})$, we say $s$ and $\overline{p}$ to be parallel, denoted as $s\parallel\overline{p}$.
Denote $$\mathscr{Q}_A:=\{\overline{p}\in\mathscr{Q}|t(\overline{p})\neq
h(\overline{p})\}\ \text{ and} \ \
\mathscr{Q}_C:=\{\overline{p}\in\mathscr{Q}|t(\overline{p})=
h(\overline{p})\}.$$ Moreover $k\mathscr{Q}_A$(resp.,$k\mathscr{Q}_C$) denotes the subspace of $k\Gamma/I$ generated by $\mathscr{Q}_A$(resp.,$\mathscr{Q}_C$). Clearly, as $k$-linear spaces, $k\Gamma/I=k\mathscr{Q}_A\oplus k\mathscr{Q}_C$.
Using the above notations, an admissible algebra $k\Gamma/I$ is called [**acyclic**]{} if $$\mathscr{Q}_C\backslash\{\overline{v}|v\in V\}=\emptyset.$$
It is easy to see from this definition that\
(i) The fact whether the given $k\Gamma/I$ is acyclic is independent with the choice of $\mathscr{Q}$.\
(ii) If the quiver $\Gamma$ is acyclic, then $k\Gamma/I$ is acyclic; the converse is not true in general.\
(iii) If $k\Gamma/I$ is acyclic, then it is finite dimensional; the converse is not true, e.g., $k\Gamma/k^n\Gamma$ if $\Gamma$ is a loop for $n\geq2$.
\[2-5\] Let $D: k\Gamma\longrightarrow k\Gamma/I$ be a $k$-linear operator.
(i) If $D$ is a differential operator, then
\(a) for $v\in V$, $$\label{19}
D(v)=\sum_{\overline{q}\in\mathscr{Q}, t(\overline{q})=v,
h(\overline{q})\neq v}c_{\overline{q}}^{v}\overline{q}+
\sum_{\overline{q}\in\mathscr{Q}, h(\overline{q})=v,
t(\overline{q})\neq v}c_{\overline{q}}^{v}\overline{q},$$
\(b) for $p\in E$, $$\label{20}
D(p)=\sum_{\substack{\overline{q}\in\mathscr{Q},
\\h(\overline{q})=t(p),t(\overline{q})\neq
t(p)}}c_q^{t(p)}\overline{qp}+\sum_{\substack{\overline{q}\in\mathscr{Q},
\\\overline{q}\parallel p}}c_{\overline{q}}^{p}\overline{q}+\sum_{\substack{\overline{q}\in\mathscr{Q},
\\t(\overline{q})=h(p),h(\overline{q})\neq h(p)}}c_{\overline{q}}^{h(p)}\overline{pq}$$ where the coefficients are subject to the following condition: for any path $\overline{q}\in \mathscr{Q}$ such that $t(\overline{q})\neq h(\overline{q})$, $$\label{21}
c_{\overline{q}}^{h(\overline{q})}+c_{\overline{q}}^{t(\overline{q})}=0.$$
(ii) Conversely, assume the linear map $D$ from $kV\oplus kE$ to $k\Gamma/I$ satisfies Eqs.(\[19\]), (\[20\]), (\[21\]), then $D$ can be uniquely extended linearly to a differential operator as Eq.(\[16\]).
\(i) For a given $v\in V$, since $vv=v$, we have $$D(v)=D(vv)=D(v)v+v D(v).$$ So by the direct computation, we can get $$\begin{aligned}
D(v)&=&\sum_{\overline{q}\in \mathscr{Q},
h(\overline{q})=v}c_{\overline{q}}^{v}\overline{q}+\sum_{\overline{q}\in
\mathscr{Q}, t(\overline{q})=v}c_{\overline{q}}^{v} \overline{q}\notag.\\\end{aligned}$$ Moreover, $$D(v)=D(v)v+v D(v)=(D(v)v+v D(v))v+v D(v)=D(v)v+v D(v)v+v D(v),$$ so we have $v D(v)v=0$. That means $\sum\limits_{\overline{q}\in
\mathscr{Q},
t(\overline{q})=h(\overline{q})=v}c_{\overline{q}}^{v}\overline{q}=0$. So we get Eq.(\[19\]).
Also, for a given $p\in E$, we have $$\begin{aligned}
D(p)&=&D(t(p)ph(p))\\
&=&D(t(p)){ph(p)}+t(p)D(p)h(p)+t(p)p D(h(p))\\
&=&D(t(p))p+\sum_{\substack{\overline{q}\in\mathscr{Q}\\\overline{q}\parallel p}}c_{\overline{q}}^p\overline{q}+p D(h(p))\\\end{aligned}$$ Since $t(p),h(p)\in V$, by Eq.(\[19\]), we can easily get Eq.(\[20\]).
Let $x,y\in V,\ x\neq y$. By (\[19\]), $$\begin{aligned}
D(xy)&=&D(x)y+xD(y)\\
&=&\sum\limits_{\overline{q}\in\mathscr{Q},t(\overline{q})=x,h(\overline{q})=y}c_{\overline{q}}^{x}\overline{q}+\sum\limits_{\overline{q}\in\mathscr{Q}, t(\overline{q})=x,h(\overline{q})=y}c_{\overline{q}}^{y}\overline{q}\\
&=&\sum\limits_{\overline{q}\in\mathscr{Q},
t(\overline{q})=x,h(\overline{q})=y}(c_{\overline{q}}^{x}+c_{\overline{q}}^{y})\overline{q}\end{aligned}$$ But, $D(xy)=0$ since $xy=0$. So, $\sum\limits_{\overline{q}\in\mathscr{Q},
t(\overline{q})=x,h(\overline{q})=y}(c_{\overline{q}}^{x}+c_{\overline{q}}^{y})\overline{q}=0$.
For a path $\overline{q}\in\mathscr{Q}$ such that $t(\overline{q})\neq h(\overline{q})$, substituting $x$ and $y$ respectively with $t(\overline{q})$ and $h(\overline{q})$, we get Eq.(\[21\]).\
(ii) We only need to verify the conditions of Lemma \[2-4\] are satisfied. Because the process is straightforward, we leave it to the readers.
Next, we apply Proposition \[2-5\] to display a standard basis of differential operators from $k\Gamma$ to $k\Gamma/I$, for any admissible algebra $k\Gamma/I$.
(**Differential operator** $D_{r,\overline{s}}$.) For a quiver $\Gamma=(V, E)$, let $r\in E$ and $s\in \mathscr{P}$ with $r\parallel s$. Define the $k$-linear operator $D_{r,\overline{s}}:{k}V\oplus{k}E\longrightarrow k\Gamma/I$ satisfying $$\label{29}
D_{r,\overline{s}}(p)=\begin{cases}
\overline{s}, &p=r~~\text{for}~~ p\in E,\\
0, &p\neq r~~\text{for}~~ p\in E\cup V, \end{cases}$$ Then, the conditions of Lemma(\[2-4\]) are satisfied and thus, $D_{r,\overline{s}}$ can be uniquely extended to a differential operator from ${k}\Gamma$ to $k\Gamma/I$, denoted still by $D_{r,\overline{s}}$ for convenience.
Eqs.(\[12\]), (\[13\]), Eq.(\[14\]) and Eq.(\[15\]) can be checked easily by the definition of $D_{r,\overline{s}}$.
For a given $s\in \mathscr{P}$, we have the corresponding inner differential operator: $$\label{2.13}
D_{\overline{s}}: {k}\Gamma\rightarrow k\Gamma/I,\
D_{\overline{s}}(q)=\overline{sq}-\overline{qs},\ \forall
q\in\mathscr{P}.$$
\[theorem 3.2\] Let $\Gamma=(V,E)$ be a quiver and $I$ be an ideal such that $k\Gamma/I$ is an admissible algebra. Then the set $$\mathfrak{B}\:=\mathfrak{B}_1\cup\mathfrak{B}_{2}$$ is a basis of the $k$-linear space of differential operators from $k\Gamma$ to $k\Gamma/I$, where $$\label{eq2.15}
\mathfrak{B}_1:=\{D_{\overline{s}}|\overline{s}\in\mathscr{Q}_A\},\ \ \
\mathfrak{B}_{2}:=\{D_{r,\overline{s}}|r\in
E,\overline{s}\in\mathscr{Q}, r\parallel\overline{s} \}.$$
We only need to verify that the operators in $\mathfrak{B}$ are linearly independent and any differential operators can be generated $k$-linearly by $\mathfrak{B}$.
**Step 1. $\mathfrak{B}$ is linearly independent.** Suppose there are $c_{\overline{p}},c_{r,\overline{s}}\in k$ such that $$\label{independent}
\sum_{\overline{p}\in\mathscr{Q}, h(\overline{p})\neq
t(\overline{p})}c_{\overline{p}}D_{\overline{p}}+\sum_{r\in
E,\overline{s}\in \mathscr{Q},r\parallel
\overline{s}}c_{r,\overline{s}}D_{r,\overline{s}}=0.$$
Then for any given $\overline{p_0}\in \mathscr{Q},
h(\overline{p_0})\neq t(\overline{p_0})$, by the definition of $D_{\overline{p}}$ and $D_{r,\overline{s}}$, we have
$$\begin{aligned}
0&=&\sum_{\overline{p}\in\mathscr{Q}, t(\overline{p})\neq
h(\overline{p})}c_{\overline{p}}D_{\overline{p}}(h(\overline{p_0}))+\sum_{r\in
E,\overline{s}\in
\mathscr{Q}, r\parallel \overline{s}}c_{r,\overline{s}}D_{r,\overline{s}}(h(\overline{p_0}))\\
&=&\sum_{\overline{p}\in\mathscr{Q}, t(\overline{p})\neq h(\overline{p})}c_{\overline{p}}(\overline{ph(p_0})-\overline{h(p_0)p})+0\\
&=&\sum_{\overline{p}\in\mathscr{Q},t(\overline{p})\neq
h(\overline{p})=h(\overline{p_0})}c_{\overline{p}}\overline{p}-\sum_{\overline{q}\in\mathscr{Q},h(\overline{q})\neq
t(\overline{q})=h(\overline{p_0})}c_{\overline{q}}\overline{q}.\end{aligned}$$
In the last formula above, $\overline{p}\ne \overline{q}$ always holds. Thus, their coefficients are all zero. In particular, $c_{\overline{p_0}}=0$ for any $\overline{p_0}\in\mathscr{Q}$ with $h(\overline{p_0})\neq t(\overline{p_0})$.
Thus, from (\[independent\]), we get that $$\sum_{r\in E,\overline{s}\in
\mathscr{Q}, r\parallel
\overline{s}}c_{r,\overline{s}}D_{r,\overline{s}}=0.$$ Further, for any given $r_0\in E, \overline{s}\in \mathscr{Q}$ with $\overline{s}\parallel r_0$, we have:
$\sum\limits_{r\in E,\overline{s}\in \mathscr{Q},r\parallel
\overline{s}}c_{r,\overline{s}}D_{r,\overline{s}}(r_0)=0~~~~\Longrightarrow~~~~\sum\limits_{\overline{s}\in
\mathscr{Q},r_0\parallel \overline{s}}c_{r_0,\overline{s}}\overline{s}=0$.\
It follows that $c_{r_0,\overline{s}}=0$ for any $r_0\in E,
\overline{s}\in \mathscr{Q}$ with $r\parallel \overline{s}$.
Hence, $\mathfrak{B}$ is $k$-linearly independent.
**Step 2. $\mathfrak{B}$ is the set of $\bf k$-linear generators.** Let $D: {k}\Gamma\rightarrow k\Gamma/I$ be any differential operator. Then for $v\in V$ and $p\in E$, by Eqs.(\[19\]), (\[20\]) and (\[21\]) we have $$\label{033}
D(v)=\sum_{\overline{q}\in\mathscr{Q}, h(\overline{q})\neq
t(\overline{q})=v}c_{\overline{q}}^{v}\overline{q}+
\sum_{\overline{q}\in\mathscr{Q},t(\overline{q})\neq
h(\overline{q})=v}c_{\overline{q}}^{v}\overline{q}.$$ $$\label{33}
D(p)=-\sum_{\substack{\overline{q}\in\mathscr{Q}\\t(\overline{q})\neq
h(\overline{q})=t(p)}}c_{\overline{q}}^{t(\overline{q})}\overline{qp}+\sum_{\substack{\overline{q}\in\mathscr{Q}\\\overline{q}\parallel
p}}c_{\overline{q}}^p\overline{q}+
\sum_{\substack{\overline{q}\in\mathscr{Q}\\h(\overline{q})\neq
t(\overline{q})=h(p)}}c_{\overline{q}}^{t(\overline{q})}\overline{pq}.$$ We claim that $D$ agrees with the differential operator $\overline{D}$ defined by the linear combination $$\label{III.23}
\overline{D}=-\sum_{\overline{s}\in\mathscr{Q}, t(\overline{s})\neq
h(\overline{s})}c_{\overline{s}}^{t(\overline{s})}D_{\overline{s}}+\sum_{r\in
E,\overline{s}\in\mathscr{Q}, \overline{s}\parallel
r}c_{\overline{s}}^rD_{r,\overline{s}},$$ where $c_{\overline{s}}^{t(\overline{s})}$ and $c_{\overline{s}}^r$ come from Eqs.(\[033\]) and (\[33\]). Any path in $\mathscr{P}$ is either a vertex or a product of arrows. Thus by the product rule of differential operators, to show $D=\overline{D}$, we only need to verify that $D(q)=\overline{D}(q)$ for each $q=v\in V$ and $q=p\in
E$. The verification is straightforward, so we omit it.
We call the set $\mathfrak{B}$ in Theorem \[theorem 3.2\] the **standard basis** of the $k$-linear space $\emph{Diff}(k\Gamma, k\Gamma/I)$ generated by all differential operators from $k\Gamma$ to $k\Gamma/I$.
From this theorem, we get $\textsl{Diff}(k\Gamma,k\Gamma/I)=\mathfrak{D}_1\oplus
\mathfrak{D}_{2}$, where $\mathfrak{D}_i$ is the $k$-linear space generated by $\mathfrak{B}_i$ for $i=1,2$ in (\[eq2.15\]).
For any $p\in E$, $D_{p,\overline{p}}\in\mathfrak{B}_{2}$ is called **arrow differential operator** from $k\Gamma$ to $k\Gamma/I$. Let $\mathfrak{B}_E:=\{D_{p,\overline{p}}|p\in E\}$ and $\mathfrak{D}_E:=k\mathfrak{B}_E$ is called the **space of arrow differential operators**.
When $r\in E$ is a loop of $\Gamma$, i.e., $t(r)=h(r)$, then $D_{r,\overline{t(r)}}\in\mathfrak{B}_{2}$.
$H^1(k\Gamma,k\Gamma/I)$ for an admissible algebra $k\Gamma/I$
==============================================================
\[proposition 4.1\] Let $q\in\mathscr{P}$ be such that $h(q)=t(q)=v_0$. We have $$\label{4.21}
D_{\overline{q}}=\sum\limits_{p\in E,
t(p)=v_0}D_{p,\overline{qp}}-\sum\limits_{r\in E,
h(r)=v_0}D_{r,\overline{rq}}.$$
Note that the both sides of (\[4.21\]) are $k$-linearly generated by differential operators. So, by the product formula of differential operators, we only need to verify that the both sides always agree when they act on the elements of $V$ and $E$. Since the computation is direct, we omit it here.
For $v\in V$, it is clear that $t(v)=h(v)=v$. From Proposition \[proposition 4.1\], we have $$D_{\overline{v}}=\sum\limits_{p\in E,
t(p)=v}D_{p,\overline{p}}-\sum\limits_{r\in E,
h(r)=v}D_{r,\overline{r}}.$$ We call $D_{\overline{v}}$ the **vertex differential operator** from $k\Gamma$ to $k\Gamma/I$. Let $\mathfrak{D}_{V}$ denote the linear space spanned by $\{D_{\overline{v}}|v\in V\}$, called the **space of vertex differential operators**. It is clear that $\mathfrak{D}_{V}$ is a subspace of $\mathfrak{D}_{E}$.
\[lemma 4.2\] Let $p\in \mathscr{P}$, then $\overline{p}$ is always in the $k$-subspace $k\{\overline{q}\in
\mathscr{Q}|\ \overline{q}\parallel p\}$ generated by $\overline{q}$ with $\overline{q}\parallel p$.
Suppose $\overline{p}=\sum\limits_{\overline{q}\in
\mathscr{Q}}c_{\overline{q}}\overline{q}$, then $\overline{t(p)ph(p)}=\sum\limits_{\overline{q}\in
\mathscr{Q}}c_{\overline{q}}\overline{t(p)qh(p)}=\sum\limits_{\substack{\overline{q}\in
\mathscr{Q}\\\overline{q}\parallel p}}c_{\overline{q}}\overline{q}.$
\[corollary 4.3\] Let $q\in\mathscr{P}$ be such that $h(q)=t(q)$. Then $D_{\overline{q}}\in k\mathfrak{B}_{2}=\mathfrak{D}_{2}$.
For $r\in E, r\parallel s\in \mathscr{P}$; by Lemma \[lemma 4.2\], suppose $\overline{s}=\sum\limits_{\substack{\overline{q}\in
\mathscr{Q}\\\overline{q}\parallel s}}c_{\overline{q}}\overline{q}$, and it is clear that $D_{r,\overline{s}}=\sum\limits_{{\substack{\overline{q}\in
\mathscr{Q}\\\overline{q}\parallel
s}}}c_{\overline{q}}D_{r,\overline{q}}$, then use Proposition \[proposition 4.1\].
\[remark 4.1\] For $\overline{q}\in\mathscr{Q}, t(\overline{q})=h(\overline{q})$, from Theorem \[theorem 3.2\] and Corollary \[corollary 4.3\], we know that $D_{\overline{q}}\in k\mathfrak{B}_{2}=\mathfrak{D}_{2}$, but not in $k\mathfrak{B}_1=\mathfrak D_{1}$. Denote $\mathfrak{D}_C:=k\{D_{\overline{q}}\ |\
\overline{q}\in\mathscr{Q},\ t(\overline{q})= h(\overline{q})\}$. Then $\mathfrak{D}_{C}\subseteq\mathfrak{D}_{2}$ and $\mathfrak{D}_{C}\cap \mathfrak{D}_{1}=0$.
Denote by $\textsl{Inn-Diff}(k\Gamma, k\Gamma/I)$ the linear space consisting of inner differential operators from $k\Gamma$ to $k\Gamma/I$. Then, $\textsl{Inn-Diff}(k\Gamma, k\Gamma/I)=\mathfrak{D}_{1}+\mathfrak{D}_{C}$. Thus, we have $$\begin{aligned}
H^1(k\Gamma, k\Gamma/I)&=&\textsl{Diff}(k\Gamma, k\Gamma/I)/\textsl{Inn-Diff}(k\Gamma, k\Gamma/I)\\
&=&(\mathfrak{D}_{1}+\mathfrak{D}_{2})/(\mathfrak{D}_{1}+\mathfrak{D}_{C})\\
&\cong&\mathfrak{D}_{2}/(\mathfrak{D}_{2}\cap \mathfrak{D}_{C})\\
&\cong &\mathfrak{D}_{2}/ \mathfrak{D}_{C}.\end{aligned}$$ Since the basis of $k\Gamma/I$ given in Proposition \[proposition 2.2\] contains the residue classes of $V$ and $E$, we can see that the center of $k\Gamma/I$ as $k\Gamma$-bimodule and the center of $k\Gamma/I$ as an algebra are the same, denoted by $Z(k\Gamma/I)$.
Let $k\Gamma/I$ be a finite dimensional admissible algebra, then $$\label{3.23}
dim_kH^1(k\Gamma,k\Gamma/I)=|\mathfrak{B}_2|+dim_kZ(k\Gamma/I)-|\mathscr{Q}_C|.$$
By the discussion above, $dim_kHH^1(k\Gamma,k\Gamma/I)=|\mathfrak{B}_2|-dim_k\mathfrak{D}_{C}$. And$$\begin{aligned}
\mathfrak{D}_1\oplus\mathfrak{D}_C=\textsl{Inn-Diff}(k\Gamma, k\Gamma/I)&\cong&(k\Gamma/I)/Z(k\Gamma/I)\\
&\cong&(k\mathscr{Q}_C\oplus k\mathscr{Q}_A)/Z(k\Gamma/I)\\
&\cong&k\mathscr{Q}_C/(Z(k\Gamma/I))\oplus k\mathscr{Q}_A\\
&\cong&k\mathscr{Q}_C/(Z(k\Gamma/I))\oplus \mathfrak{D}_1,\end{aligned}$$ where the first isomorphism is assured by Eq.(\[2.13\]), the second and fourth isomorphisms are trivial, the third is because of the facts that $Z(k\Gamma/I)\subseteq k\mathscr{Q}_C$ and $Z(k\Gamma/I)\cap k\mathscr{Q}_A=0$. So $\mathfrak{D}_C\cong k\mathscr{Q}_C/Z(k\Gamma/I)$ as $k$-linear spaces, it follows that $$dim_kH^1(k\Gamma,k\Gamma/I)=dim_k\mathfrak{D}_{2}-dim_k\mathfrak{D}_C=|\mathfrak{B}_2|+dim_kZ(k\Gamma/I)-|\mathscr{Q}_C|.$$
If $k\Gamma/I$ is acyclic, then $Z(k\Gamma/I)\cong k$ and $|\mathscr{Q}_C|=|V|$. Thus, we have
If $k\Gamma/I$ is an acyclic admissible algebra (in particular, if $\Gamma$ is an acyclic quiver), then $$dim_kH^1(k\Gamma,k\Gamma/I)=|\mathfrak{B}_2|+1-|V|.$$
On the other hand, when $\Gamma$ is a planar quiver and $k\Gamma/I$ is an acyclic admissible algebra, we can apply the approach of [@GL] to give a basis of $HH^1(k\Gamma,k\Gamma/I)$. A planar quiver is a quiver with a fixed embedding into the plane $\mathbb{R}^2$. The set $F$ of faces of a planar quiver $\Gamma$ is the set of connected component of $\mathbb{R}^2\backslash\Gamma$.
We will need the famous [*Euler formula*]{} on planar graph, see [@B][@GT], which states that for any finite connected planar graph (which can be thought as the underlying graph of a quiver $\Gamma$), we have $$\label{4.22}
|V|-|E|+|F|=2.$$
For each face of $\Gamma$, its boundary is called a **primitive cycle**. Let $\mathbbm{p}_0$ denote the boundary of the unique unbounded face $f_0$ of $\Gamma$. Let $\Gamma_{\mathbb{P}}$ denote the set of primitive cycles of $\Gamma$ and $\Gamma_{\mathbb{P}}^{-}:=\Gamma_{\mathbb{P}}\backslash\mathbbm{p}_0$. Then clearly, the set $\Gamma_{\mathbb{P}}$ of primitive cycles of $\Gamma$ is in bijection with the set $F$ of the faces of $\Gamma$. So $|F|=|\Gamma_{\mathbb{P}}|$.
For a face $f\in F$, denote $\mathbbm{p}_f$ the corresponding primitive cycle of $f$. Suppose $\mathbbm{p}_f$ is comprised of an ordered list arrows $p_1,\cdots,p_s\in E$, define an operator from $k\Gamma$ to $k\Gamma/I$ $$D_{\mathbbm{p}_f}:=\pm D_{p_1,\overline{p_1}}\pm\cdots\pm
D_{p_s,\overline{p_s}},$$ where a $\pm D_{p_i,\overline{p_i}}$ is $+D_{p_i,\overline{p_i}}$ if $p_i$ is in clockwise direction when viewed from the interior of the face of $\mathbbm{p}_f$ and is $-D_{p_i,\overline{p_i}}$ otherwise. We call $D_{\mathbbm{p}_f}$ a **face differential operator** from $k\Gamma$ to $k\Gamma/I$. Let $\mathfrak{D}_{\mathbbm{P}}$ denote the linear space spanned by $\{D_{\mathbbm{P}}|\mathbbm{p}\in\Gamma_{\mathbb{P}}\}$, called the **space of face differential operators**.
The next lemma is similar to Theorem 4.9 in [@GL].
\[lemma 4.5\] Let $\Gamma$ be a planar quiver with the ground field $k$ of characteristic 0, then
\(a) $dim\mathfrak{D}_V=|V|-1$;
\(b) $dim\mathfrak{D}_{\mathbb{P}}=|F|-1=|\Gamma_{\mathbb{P}}^{-}|$;
\(c) $\mathfrak{D}_V$ and $\mathfrak{D}_{\mathbb{P}}$ are linearly disjoint subspaces of $\mathfrak{D}_E$.
\(a) Denote $\gamma_0=|V|$. Since $\overline{e}=\sum\limits_{i=1}^{\gamma_0}\overline{v_i}$ is the identity of $k\Gamma/I$, which clearly lies in the center of $k\Gamma/I$, we have $$D_{\overline{e}}=\sum\limits_{i=1}^{\gamma_0}D_{\overline{v_i}}=0.$$ So $dim\mathfrak{D}_V\leq\gamma_0-1$. We next prove that $dim\mathfrak{D}_V\geq\gamma_0-1$. We may assume that $\gamma_0\geq2$.
We claim that any $\gamma_0-1$ elements of $\{D_{\overline{v_i}}|i=1,\cdots,\gamma_0\}$ is linearly independent. In fact, suppose $\sum\limits_{i=1}^{\gamma_0-1}a_iD_{\overline{v_i}}=0$, where $a_i\in k$, which means that $\sum\limits_{i=1}^{\gamma_0-1}a_i\overline{v_i}$ is in the center of $k\Gamma/I$. Since $\Gamma$ is connected, let the vertex $v_{\gamma_0}$ be connected to $v_i$ by an arrow $p$ for $i\neq\gamma_0$. We may assume that $t(p)=v_i$ and $h(p)=v_{\gamma_0}$. We have $$a_i\overline{p}=(\sum\limits_{i=1}^{\gamma_0-1}a_i\overline{v_i})\overline{p}=\overline{p}(\sum\limits_{i=1}^{\gamma_0-1}a_i\overline{v_i})=\overline{0}.$$ so $a_i=0$. Note that $\Gamma$ is connected, we can repeat this process to get $a_j=0$ for any $j$.
\(b) Let $|F|=\gamma_2$. Through simple observation of planar quiver, we can see that if $p\in E$ is in the boundary, then it is at most in the boundary of two primitive cycles. Note that if $p\in E$ is in the boundary of two primitive cycles $\mathbbm{P}_1$ and $\mathbbm{P}_2$, then the sign of $D_{p,\overline{p}}$ in $D_{\mathbbm{P}_1}$ and $D_{\mathbbm{P}_2}$ are opposite. If $p\in E$ is in the boundary of only one primitive cycle $\mathbbm{P}$, then $D_{p,\overline{p}}$ occurs twice in $D_{\mathbbm{P}}$ with opposite sign. Thus we have $$\sum\limits_{j=0}^{\gamma_2-1}D_{\mathbbm{P}_i}=0,$$ where $\mathbbm{P}_0$ denotes the primite cycle corresponding to $f_0$ as above. So $dim\mathfrak{D}_{\mathbb{P}}\leq|F|-1$.
We next prove that $dim\mathfrak{D}_{\mathbb{P}}\geq|F|-1$. We may assume that $|F|\geq2$. Suppose $$\sum\limits_{j=1}^{\gamma_2-1}b_jD_{\mathbbm{P}_j}=0,$$ where $b_j\in k$. If $p\in E$ is in the boundary of $\mathbbm{P}_0$ and $\mathbbm{P}_j$ for $j>0$, then $\overline{0}=\sum\limits_{j=1}^{\gamma_2-1}b_jD_{\mathbbm{P}_j}(p)=\pm b_j\overline{p}$. So we have $b_j=0$. This means that if $\mathbbm{P}_j$ and $\mathbbm{P}_0$ have a common $p\in E$ in their boundary, then $b_j=0$. Replace $\mathbbm{P}_0$ with $\mathbbm{P}_j$, and repeat this process. Since the quiver is connected, we can get $b_j=0$ for any $j>0$.
\(c) From [@GL] and Theorem \[theorem 3.2\], we know that $\mathfrak{B}^o_E:=\{D_{p,p}| p\in E\}$ and $\mathfrak{B}_E:=\{D_{p,\overline{p}}| p\in E\}$ are $k$-linearly independent sets in $\textsl{Diff}(k\Gamma)$ and $\textsl{Diff}(k\Gamma, k\Gamma/I)$ respectively. Based on this, $D_{\overline{v_i}}$ and $D_{\mathbbm{p}_f}$ can be linearly expressed by using $\mathfrak{B}_E$, as well as $D_{v_i}$ and $D_{\mathfrak{c}_f}$ by using $\mathfrak{B}^o_E$ in [@GL]. Under this correspondence, referring to Theorem 4.9 in [@GL] in the same process, we obtain that $\mathfrak{D}_V$ and $\mathfrak{D}_{\mathbb{P}}$ are linearly disjoint subspaces of $\mathfrak{D}_E$.
By this lemma, $\mathfrak{B}_{\mathbbm{P}}:=\{D_{\mathbbm{P}}|\mathbbm{p}\in\Gamma_{\mathbb{P}}^{-}\}$ is a basis of $\mathfrak{D}_{\mathbbm{P}}$.
\[thm 3.7\] Let $\Gamma$ be a planar quiver and $k\Gamma/I$ be an acyclic admissible algebra with the ground field $k$ of characteristic 0. Then the union set $$(\mathfrak{B}_{2}\backslash\mathfrak{B}_E)\cup\mathfrak{B}_{\mathbb{P}}$$ is a basis of $H^1(k\Gamma,k\Gamma/I)$.
By the Euler formula and Lemma \[lemma 4.5\], we can get $\mathfrak{D}_E=\mathfrak{D}_V\oplus\mathfrak{D}_{\mathbb{P}}$. Because $k\Gamma/I$ is acyclic, we have $\mathfrak{D}_{C}=\mathfrak{D}_{V}$, then $$\begin{aligned}
H^1(k\Gamma, k\Gamma/I)&\cong &\mathfrak{D}_{2}/
\mathfrak{D}_{C}\\
&\cong &(\mathfrak{D}_E\oplus
k\{\mathfrak{B}_{2}\backslash\mathfrak{B}_E\})/\mathfrak{D}_{V}\\
&\cong&\mathfrak{D}_{\mathbb{P}}\oplus k\{\mathfrak{B}_{2}\backslash\mathfrak{B}_E\}\\
&\cong&k\mathfrak{B}_{\mathbbm{P}}\oplus
k\{\mathfrak{B}_{2}\backslash\mathfrak{B}_E\}.\end{aligned}$$
$HH^1(k\Gamma/I)$ for an admissible algebra $k\Gamma/I$
========================================================
\[lemma 5.1\] A differential operator of $k\Gamma/I$ can induce naturally a differential operator from $k\Gamma$ to $k\Gamma/I$. Conversely, a differential operator $D$ from $k\Gamma$ to $k\Gamma/I$ satisfying $D(I)=\overline{0}$ can induce a differential operator of $k\Gamma/I$.
Denote $p$ the canonical map from $k\Gamma$ to $k\Gamma/I$. Given a differential operator $D$ of $k\Gamma/I$, we claim that the composition $Dp$ is a differential operator from $k\Gamma$ to $k\Gamma/I$. Note that the canonical map from $k\Gamma$ to $k\Gamma/I$ is an algebra homomorphism, it can be directly verified. The converse result can be shown directly, too.
For a differential operator $D$ from $k\Gamma$ to $k\Gamma/I$ satisfying $D(I)=\overline{0}$, we denote $\overline{D}$ the induced differential operator on $k\Gamma/I$. Write $$\mathfrak{F}(I):=\{D\ |\
D\in\textsl{Diff}(k\Gamma,k\Gamma/I),\ D(I)=\overline{0}\},\ \ \ \mathfrak{F}_i(I):=\{D\ |\
D\in\mathfrak{D}_i,\ D(I)=\overline{0}\}\ \text{for}\ i=1,2.$$ It is clear that $D_{\overline{s}}(I)=\overline{0}$ for $s\in
\mathscr{P}$. So $\mathfrak{F}_1(I)$=$\mathfrak{D}_1$ and $\mathfrak{F}(I)=\mathfrak{D}_1\oplus\mathfrak{F}_2(I)$.
\[lemma 5.2\] $\mathfrak{F}(I)\cong \textsl{Diff}(k\Gamma/I)$ as $k$-linear spaces.
The map from $\mathfrak{F}(I)$ to $\textsl{Diff}(k\Gamma/I)$ is as follows, $$\mathfrak{F}(I)\longrightarrow \textsl{Diff}(k\Gamma/I),\ D\longmapsto \overline{D}.$$ The proof of Lemma \[lemma 5.1\] assures the map from $\mathfrak{F}(I)$ to $\textsl{Diff}(k\Gamma/I)$ is surjective. As for the injectivity, suppose $D_1, D_2\in\mathfrak{F}(I)$ and $D_1\neq D_2$, so according to Lemma \[lemma 2.2\], there exists a path $p\in V\cup E$ such that $D_1(p)\neq D_2(p)$. Since $\overline{0}\neq\overline{p}\in k\Gamma/I$, $\overline{D}_1(\overline{p})\neq
\overline{D}_2(\overline{p})$.
By this lemma, we can think $\textsl{Diff}(k\Gamma/I)$ is a $k$-subspace of $\textsl{Diff}(k\Gamma,k\Gamma/I)$.
From Lemma \[lemma 5.2\], we have $$\label{4.29}
HH^1(k\Gamma/I)\cong\mathfrak{F}(I)/(\mathfrak{D}_1\oplus\mathfrak{D}_{C})\cong(\mathfrak{D}_1\oplus\mathfrak{F}_2(I))/(\mathfrak{D}_1\oplus\mathfrak{D}_{C})\cong\mathfrak{F}_2(I)/\mathfrak{D}_{C}$$ as linear spaces. This means that $HH^1(k\Gamma/I)$ can be embedded into $H^1(k\Gamma,k\Gamma/I)\cong\mathfrak{D}_2/\mathfrak{D}_{C}$. Moreover, we have the next proposition.
\[pro 4.3\] Suppose $k\Gamma/I$ is a finite dimensional admissible algebra, then $$\label{5.28}
dim_kHH^1(k\Gamma/I)=dim_k\mathfrak{F}_2(I)+dim_kZ(k\Gamma/I)-|\mathscr{Q}_C|.$$
Note that $k\{\mathscr{Q}_C\}/(Z(k\Gamma/I))\cong\mathfrak{D}_{C}$ as linear spaces. By Equation $\ref{5.28}$, we have $$dim_kHH^1(k\Gamma/I)=dim_k\mathfrak{F}_2(I)-dim_k\mathfrak{D}_{C}=dim_k\mathfrak{F}_2(I)+dim_kZ(k\Gamma/I)-|\mathscr{Q}_C|.$$
\[cor 4.4\] If $k\Gamma/I$ is an acyclic admissible algebra (in particular, if $\Gamma$ is an acyclic quiver), then $$dim_kHH^1(k\Gamma/I)=dim_k\mathfrak{F}_2(I)+1-|V|.$$
If $k\Gamma/I$ is an acyclic admissible algebra, we have a standard procedure to compute $dim_k\mathfrak{F}_2(I)$. First note that for a differential operator $D$ from $k\Gamma$ to $k\Gamma/I$, $D(I)=\overline{0}$ if and only if $D(r_i)=\overline{0}$ where $\{r_1,\cdots,r_i,\cdots r_n\}$ is a minimal set of generators of $I$. This property follows easily from the Leibnitz rule of differential operators. Since $k\Gamma/I$ is acyclic, $|\mathfrak{B}_2|$ is finite for $\mathfrak{B}_2$ as given in Theorem \[theorem 3.2\]. Suppose $\sum\limits_{D_{r,\overline{s}}\in\mathfrak{B}_2}c_{r,\overline{s}}D_{r,\overline{s}}(r_i)=\overline{0}$ for $i=1,\cdots,n$. This means that the coefficients $c_{r,\overline{s}}$ satisfy the system of these homogeneous linear equations. So $dim\mathfrak{F}_2(I)$ is equal to the dimension of the solution space of the system of homogeneous linear equations.
Now we give two examples of admissible algebras that are not monomial algebras nor truncated quiver algebras, and characterize their first Hochschild cohomology.
Let $\Gamma=(V,E)$ be the quiver
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and $I=<\alpha_1\alpha_2-\beta_1\beta_2>$.
In this case, $\mathfrak{B}_2=\{D_{\alpha_1,\overline{\alpha_1}},D_{\alpha_2,\overline{\alpha_2}},D_{\beta_1,\overline{\beta_1}},D_{\beta_2,\overline{\beta_2}}\}$. So we have $$dim_kH^1(k\Gamma,k\Gamma/I)=|\mathfrak{B}_2|+dim_kZ(k\Gamma/I)-|\mathscr{Q}_C|=4+1-4=1.$$ Suppose that $$(aD_{\alpha_1,\overline{\alpha_1}}+bD_{\alpha_2,\overline{\alpha_2}}+cD_{\beta_1,\overline{\beta_1}}+dD_{\beta_2,\overline{\beta_2}})
(\alpha_1\alpha_2-\beta_1\beta_2)=(a+b-c-d)\overline{\alpha\beta}=\overline{0},$$ then we get $a+b-c-d=0$. Hence $dim_k\mathfrak{F}_2(I)=3$ and $dim_kHH^1(k\Gamma/I)=0$.
Let $\Gamma$ be the quiver having one vertex with two loops, equivalently, $k\Gamma=k\langle x,y\rangle$. Suppose the ideal $I=\langle xy-yx\rangle$. Then $k\Gamma/I=k[x,y]$.
In this case, $\mathfrak{B}_1=\emptyset$ and $\mathfrak{B}_2=\{D_{x,x^my^n}, D_{y,x^my^n}|m.n\geq0\}$ are the basis of $\textsl{Diff}(k\langle x,y\rangle, k[x,y])$, where $x^my^n$ means the multiplication in $k[x,y]$.
Since $k[x,y]$ is commutative, we get that $$\textsl{Inn-Diff}(k\langle x,y\rangle, k[x,y])=0, \ \ \
H^1(k\langle x,y\rangle, k[x,y])=\textsl{Diff}(k\langle x,y\rangle, k[x,y]).$$ Moreover, note that $D_{x,x^my^n}(xy-yx)=0$ and $D_{y,x^my^n}(xy-yx)=0$. Thus we obtain the basis of $HH^1(k[x,y])$ to be $$\{D_{x,x^my^n},
D_{y,x^my^n}|m.n\geq0\}.$$
Similarly we can obtain the first Hochschild cohomology for $k[x_1,x_2,\cdots,x_n]$.
Assume $k\Gamma/I$ is a monomial algebra. The residue classes of paths that do not belong to $I$ form a basis of $k\Gamma/I$. For convenience, we also denote by $\mathscr{Q}$ the basis of $k\Gamma/I$ when $k\Gamma/I$ is a monomial algebra.
A monomial algebra $k\Gamma/I$ is called [**complete**]{} if for any parallel paths $p,p'$ in $\Gamma$, $p\in I$ implies $p'\in I$.
\[thm 4.5\] Suppose $k\Gamma/I$ is a complete monomial algebra with $I\subseteq R^2$. Then the following set $$\overline{\mathfrak{B}}\:=\overline{\mathfrak{B}}_1\cup\overline{\mathfrak{B}}_{2}$$ is a basis of $\textsl{Diff}(k\Gamma/I)$, where $$\overline{\mathfrak{B}}_1:=\{\overline{D}_{\overline{s}}\
|\overline{s}\in\mathscr{Q}, h(\overline{s})\neq t(\overline{s})\},
\ \ \ \overline{\mathfrak{B}}_{2}:=\{\overline{D}_{r,\overline{s}}\ |r\in
E,\overline{s}\in\mathscr{Q}, r\parallel \overline{s}\}.$$
Since $k\Gamma/I$ is complete, we have $D_{r,\overline{s}}(p)=\overline{0}$ for any $D_{r,\overline{s}}\in\mathfrak{B}_2$, where $p$ is any path in $I$. Then $\mathfrak{F}_2(I)=\mathfrak{D}_2$. It follows that $\textsl{Diff}(k\Gamma,k\Gamma/I)\cong \textsl{Diff}(k\Gamma/I)$ as $k$-linear spaces. Thus due to Theorem \[theorem 3.2\], the result follows.
Suppose $k\Gamma/I$ is an acyclic complete monomial algebra with $I\subseteq R^2$. Then $$dim_kHH^1(k\Gamma/I)=|\overline{\mathfrak{B}}_2|+1-|V|.$$
By the proof of Proposition \[thm 4.5\], $dim_k\mathfrak{F}_2(I)=dim_k\mathfrak{D}_2=dim_k\overline{\mathfrak{D}}_2=|\overline{\mathfrak{B}}_2|$. By Corollary \[cor 4.4\], we get the required result.
In [@Sa2], the author gave a characterization of the first Hochschild cohomology of an acyclic complete monomial algebra through a projective resolution. However, its $k$-linear basis has not been constructed, so far. Here, we want to reach this aim in our method.
\[tm 4.7\] Let $\Gamma$ be a planar quiver, $k\Gamma/I$ be an acyclic complete monomial algebra with $I\subseteq R^2$ over the field $k$ of characteristic 0. Then the union set $$(\overline{\mathfrak{B}}_{2}\backslash\overline{\mathfrak{B}}_E)\cup\overline{\mathfrak{B}}_{\mathbb{P}}$$ is a basis of $HH^1(k\Gamma/I)$, where $\overline{\mathfrak{B}}_E=\{\overline{D}_{p,\overline{p}}|p\in E\}$ and $\overline{\mathfrak{B}}_{\mathbb{P}}=\{\overline{D}_{\mathbbm{p}}|\mathbbm{p}\in\Gamma_{\mathbb{P}}^{-}\}$.
By Eq.(\[4.29\]) and $\mathfrak{F}_2(I)=\mathfrak{D}_2$, we have $HH^1(k\Gamma/I)\cong H^1(k\Gamma,k\Gamma/I)$ in this case. So from Theorem \[thm 3.7\], we can directly get this theorem.
For a truncated quiver algebra $k\Gamma/k^n\Gamma$ with $n\geq2$, we can give a standard basis of $\textsl{Diff}(k\Gamma/k^n\Gamma)$. $k\Gamma/k^n\Gamma$ has the basis formed by the residue classes of the paths of length $\leq n-1$, denoted also by $\mathscr{Q}$.
\[thm 4.4\] Let $\Gamma=(V,E)$ be a quiver and the field $k$ be of characteristic 0. A basis of $\textsl{Diff}(k\Gamma/k^n\Gamma)$ for any truncated quiver algebra $k\Gamma/k^n\Gamma$ with $n\geq 2$ is given by the set $$\overline{\mathfrak{B}}\:=\overline{\mathfrak{B}}_1\cup\overline{\mathfrak{B}}_{2}$$ where $$\overline{\mathfrak{B}}_1:=\{\overline{D}_{\overline{s}}\
|\overline{s}\in\mathscr{Q}, h(\overline{s})\neq t(\overline{s})\},\ \ \
\overline{\mathfrak{B}}_{2}:=\{\overline{D}_{r,\overline{s}}\ |r\in
E,\overline{s}\in\mathscr{Q}, s\notin V, r\parallel \overline{s}\}.$$
It is clear that $D_{\overline{s}}(k^n\Gamma)=\overline{0}$ for $\overline{s}\in\mathscr{Q}, h(\overline{s})\neq
t(\overline{s})$ and $D_{r,\overline{s}}(k^n\Gamma)=\overline{0}$ for $r\in E,\overline{s}\in\mathscr{Q}, s\notin V,
\overline{s}\parallel r$. Note that when $r$ is a loop of $\Gamma$, $D_{r,\overline{h(r)}}\in \textsl{Diff}(k\Gamma,k\Gamma/k^n\Gamma)$, but $D_{r,\overline{h(r)}}(r^n)=nr^{n-1}\neq \overline{0}$. Moreover, for all loops $r_1,\cdots,r_s$ of $\Gamma$ and $c_1,\cdots,c_s$ not all $0$, we claim that $\sum\limits
c_iD_{r_i,\overline{h(r_i)}}(k^n\Gamma)\neq \overline{0}$. Without loss of generality, we can assume $c_1\neq 0$. So we have $$\sum\limits
c_iD_{r_i,\overline{h(r_i)}}(r_1^n)=nc_1r_1^{n-1}\neq
\overline{0}.$$ Then by Theorem \[theorem 3.2\], the union set $$\{D_{\overline{s}}|\overline{s}\in\mathscr{Q}, h(\overline{s})\neq
t(\overline{s})\}\bigcup\{D_{r,\overline{s}}|r\in
E,\overline{s}\in\mathscr{Q},\ s\notin V, r\parallel \overline{s}\}$$ forms a basis of the linear space $\mathfrak{F}_2(k^n\Gamma)$ for $I=k^n\Gamma$. By Lemma \[lemma 5.2\], we have $$\mathfrak{F}_2(k^n\Gamma)\cong\textsl{Diff}(k\Gamma/k^n\Gamma).$$ Note the map from $\mathfrak{F}_2(k^n\Gamma)$ to $\textsl{Diff}(k\Gamma/k^n\Gamma)$ in Lemma \[lemma 5.2\], we can see that the union set $\overline{\mathfrak{B}}\:=\overline{\mathfrak{B}}_1\cup\overline{\mathfrak{B}}_{2}$ is a $k$-linear basis of $\textsl{Diff}(k\Gamma/k^n\Gamma)$.
Thus $\textsl{Diff}(k\Gamma/k^n\Gamma)=\overline{\mathfrak{D}}_1\oplus
\overline{\mathfrak{D}}_{2}$, where $\overline{\mathfrak{D}}_i$ is the $k$-linear space generated by $\overline{\mathfrak{B}}_i$ for $i=1,2$.
\[corollary 5.4\] Let $\Gamma=(V,E)$ be a quiver and the field $k$ be of characteristic 0. Then $$dim_kHH^1(k\Gamma/k^n\Gamma)=|\overline{\mathfrak{B}}_2|+dim_kZ(k\Gamma/k^n\Gamma)-|\mathscr{Q}_C|.$$
By the proof of Proposition \[thm 4.4\] and the definition of $\mathfrak{F}_2(I)$, we can see that $\{D_{r,\overline{s}}\ |\ r\in
E,\overline{s}\in\mathscr{Q},\ s\notin V, r\parallel\overline{s}
\}$ is a basis of $\mathfrak{F}_2(k^n\Gamma)$ for $I=k^n\Gamma$. And by Proposition \[thm 4.4\], $\overline{\mathfrak{B}}_{2}:=\{\overline{D}_{r,\overline{s}}\ |r\in
E,\overline{s}\in\mathscr{Q}, s\notin V, r\parallel \overline{s}\}$. Then by Proposition \[pro 4.3\] and the correspondence between $D_{r,\overline{s}}$ and $\overline{D}_{r,\overline{s}}$ for each pair $(r,\overline{s})$, we get the required result.
This corollary has indeed been given as Theorem 1 in [@L] and Theorem 2 in [@XHJ]. The method we obtain it here is different with that in [@L] and [@XHJ].
Moreover, when $k\Gamma/k^n\Gamma$ is acyclic, we can get a basis of $HH^1(k\Gamma/k^n\Gamma)$ as in Theorem \[tm 4.7\].
\[thm 5.3\] Let $\Gamma$ be a planar quiver, $k\Gamma/k^n\Gamma$ for $n\geq2$ be acyclic over the field $k$ of characteristic 0, then the union set $$(\overline{\mathfrak{B}}_{2}\backslash\overline{\mathfrak{B}}_E)\cup\overline{\mathfrak{B}}_{\mathbb{P}}$$ is a basis of $HH^1(k\Gamma/k^n\Gamma)$, where $\overline{\mathfrak{B}}_E=\{\overline{D}_{p,\overline{p}}|p\in E\}$ and $\overline{\mathfrak{B}}_{\mathbb{P}}=\{\overline{D}_{\mathbbm{p}}|\mathbbm{p}\in\Gamma_{\mathbb{P}}^{-}\}$.
Since $k\Gamma/I$ is acyclic, $\mathfrak{D}_{C}=\mathfrak{D}_{V}$. By Lemma \[lemma 5.2\], $\overline{\mathfrak{D}}_{C}\cong\mathfrak{D}_{C}$, $\overline{\mathfrak{D}}_{E}\cong\mathfrak{D}_{E}$, $\overline{\mathfrak{D}}_{\mathbb{P}}\cong\mathfrak{D}_{\mathbb{P}}$. So the result can be obtained in the same way as the proof of Theorem \[thm 3.7\].
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[^1]: Project supported by the National Natural Science Foundation of China (No.11271318, No.11171296 and No.J1210038) and the Specialized Research Fund for the Doctoral Program of Higher Education of China (No. 20110101110010)
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abstract: 'Three-body correlations for the ground-state decay of the lightest two-proton emitter $^{6}$Be are studied both theoretically and experimentally. Theoretical studies are performed in a three-body hyperspherical-harmonics cluster model. In the experimental studies, the ground state of $^{6}$Be was formed following the $\alpha $ decay of a $^{10}$C beam inelastically excited through interactions with Be and C targets. Excellent agreement between theory and experiment is obtained demonstrating the existence of complicated correlation patterns which can elucidate the structure of $^{6}$Be and, possibly, of the $A$=6 isobar.'
author:
- 'L. V. Grigorenko$^{1,2,3}$, T. D. Wiser$^{4}$, K. Mercurio$^{4}$, R. J. Charity$^{5}$, R. Shane$^{4},$ L. G. Sobotka$^{4,5}$, J. M. Elson$^{5}$, A. Wuosmaa$^{6}$, A. Banu$^{7}$, M. McCleskey$^{7}$, L. Trache$^{7}$, R. E. Tribble$^{7}$, and M. V. Zhukov$^{8}$'
title: 'Three-body decay of $^{6}$Be'
---
Introduction
============
Two-proton (2*p*) radioactivity was predicted by V.I. Goldansky in 1960 [@gol60] as an exclusively quantum-mechanical phenomenon. True three-body decay, in his terms, is a situation where the sequential emission of the particles is energetically prohibited and all the final-state fragments are emitted simultaneously. These energy conditions are illustrated for $^6$Be in Fig. \[fig:6be-spec\] which shows that the $^{5}$Li ground state (g.s.) is not fully accessible for sequential decay. Since the experimental discovery of $^{45}$Fe two-proton radioactivity in 2002 [@pfu02; @gio02], this field has made fast progress. New cases of 2*p* radioactivity were found for $^{54}$Zn [@bla05], $^{19}$Mg [@muk07], and, maybe, $^{48}$Ni [@dos05]. The 2*p* correlations were recently measured in $^{45}$Fe [@mie07], $^{19}$Mg [@muk07; @muk08], $^{16}$Ne [@muk08], and $^{10}$C [@Mercurio08]. Very interesting [@muk06], but, so far, controversial [@pec07] case is possibility of $2p$ emission off deformed isomeric state in $^{94}$Ag. All these decays exhibit complex correlation patterns. It is argued that studies of these patterns could provide important information about the structure of the decaying nuclei.
With this active research as the background, there is one case which has been unduly forgotten. The $^{6}$Be nucleus is the lightest true two-proton emitter in the sense of Goldansky. As this is expected to be the simplest case (smallest Coulomb interaction, expressed cluster structure with closed-shell core), a full understanding of its physics would provide a reliable basis for all future studies of 2*p* decay. However until now, theoretical work on $^{6}$Be was limited to predicting the energies and widths of its states. In addition, precise experimental data do not exist. The last experimental work dedicated to correlations in $^{6}$Be g.s. is about 20 years old.
There is one more aspect which makes the $^{6}$Be case of special interest. In the last decade, large efforts have been directed to studies of $^{6}$He with special emphasis to the understanding of the halo properties in this comparatively simple and accessible case. The associated literature comprises hundreds of titles. To deduce the correlations in the neutron halo, one has to excite (e.g., Coulex) or destroy (e.g., knockout reactions) this nucleus. Therefore, the interpretation of the experimental data is influenced by the need to clarify details of the reaction mechanism [^1]. However, information about mirror system can be obtained without all this “violence”. The isobaric analogue state in $^{6}$Be decays to the $\alpha$+$p$+$p$ channel all by itself, providing the differential data on correlations. This data can be used directly to elucidate the structure of $^{6}$Be without the need to deal with the details of the reaction mechanism. Thus an important opportunity exists for a better understanding of $^{6}$He properties through detailed studies of the $^{6}$Be. This has not been exploited previously.
![Energy levels and decay scheme for $^{6}$Be [@til02]. The $^{6}$Be g.s. is a true two-proton emitter in the sense of Goldansky: the sequential decay of this state is not possible as the lowest possible intermediate, $^{5}$Li g.s., is not energetically accessible.[]{data-label="fig:6be-spec"}](aa-01-6be-spec){width="33.00000%"}
In this work, we provide detailed theoretical calculations of the three-body decay characteristics of $^{6}$Be in a three-body cluster $\alpha $+$p$+$p$ model. We demonstrate that, in certain aspects, $^{6}$Be may be a preferable tool for studies of the $A$=6 isobar, especially considering the high sensitivity of observables to the details of the theoretical models. We then discuss previous experimental and theoretical works on $^6$Be. Subsequently, we report on an experiment where $^{6}$Be fragments are formed after the $\alpha
$-decay of $^{10}$C projectiles excited by inelastic scattering. These data cover the complete kinematic space accessible for three-body decay and the correlations are compared to the theoretical predictions.
The $\hbar=c=1$ system of units is used in this work. The following notations are used: $E_T$ is the system energy and $E_{3r}$ is the three-body resonance energy relative to the three-body $\alpha$+$p$+$p$ threshold.
Theoretical model
=================
The theoretical framework of this paper is largely the same as that developed for the two-proton radioactivity and three-body decay studies in Refs. [@gri02; @gri02a; @gri03; @gri03a; @gri07; @gri07a]. It is based on the three-body cluster model using the hyperspherical-harmonics (HH) method. The predictions obtained with this approach were found to be in very good agreement with experimental widths and momentum distributions [@mie07; @muk07; @muk08].
In this section, we sketch the necessary formalism emphasizing only the points which differ from previous treatments.
Hyperspherical harmonics method
-------------------------------
For narrow states, the time-dependent wavefunction (WF) in a finite domain can be parameterized as $$\Psi _{3}^{(+)}(\rho ,\Omega _{\rho },t)=e^{-\frac{\Gamma }{2}t-iEt}\;\Psi
_{3}^{(+)}(\rho ,\Omega _{\rho })\;.
\label{eq:decay-param}$$ The radial part of this WF can be obtained with good precision as a solution of the inhomogeneous system of equations
\[eq:source\] $$\begin{gathered}
\left( \hat{H}-E_{3r}\right) \Psi _{E}^{(+)}(\rho ,\Omega _{\rho })
=-i\,(\Gamma /2)\,\Psi _{\text{box}}(\rho ,\Omega _{\rho })\;, \\
\hat{H}
=\hat{T}+\hat{V}_{cp}(\mathbf{r}_{cn_{1}})+\hat{V}_{cp}(\mathbf{r}_{cp_{2}})
+\hat{V}_{pp} (\mathbf{r}_{p_{1}p_{2}})\;.\end{gathered}$$
Here $\Psi _{\text{box}}$ and $E_{3r}$ are the eigenfunction and the eigenvalue of the equation $$\left( \hat{H}-E\right) \Psi _{\text{box}}(\rho ,\Omega _{\rho })=0\;,
\label{eq:box}$$ solved with a “box” boundary condition at large $\rho $. The hyperspherical coordinates are defined via the Jacobi vectors
\[eq:Jacobi\] $$\begin{aligned}
\mathbf{X} &=&\mathbf{r}_{p_1}-\mathbf{r}_{p_2}\;,\quad
\mathbf{Y}=(\mathbf{r}_{p_1}+\mathbf{r}_{p_2})/2-\mathbf{r}_{c}\;, \\
\rho ^{2} &=&\frac{2}{3}\left( r_{cp_{1}}^{2}+r_{cp_{2}}^{2}\right)
+\frac{1}{6}\,r_{p_{1}p_{2}}^{2}=\frac{1}{2}\,X^{2}+\frac{4}{3}\,Y^{2}\;, \\
\Omega _{\rho } &=&\{\theta _{\rho },\Omega _{x},\Omega _{y}\}\;,\quad
\theta _{\rho }=\text{arctan}\left[ \sqrt{\frac{3}{8}}\frac{X}{Y}\right] \;.\end{aligned}$$
These Jacobi variables are given in “T” Jacobi system (see Fig. \[fig:jacobi\]). The hyperradial components $\chi _{K\gamma }^{(+)}(\rho
)$ of the WF equation \[eq:source\], possessing the pure outgoing asymptotics $$\Psi _{E}^{(+)}(\rho ,\Omega _{\rho })=\rho ^{-5/2}\sum_{K\gamma }^{K_{\max
}}\,\chi _{K\gamma }^{(+)}(\varkappa \rho )\,\mathcal{J}_{K\gamma
}^{JM}(\Omega _{\rho })\;,$$ are matched to approximate boundary conditions of the three-body Coulomb problem obtained in Ref. [@gri03c]. The radial components of this WF at large $\rho $ values can be represented as $$\chi _{K\gamma }^{(+)}(\varkappa \rho )\sim \, A_{KLl_{x}l_{y}} ^{JSS_{x}}
(\varkappa ) \;\tilde{\mathcal{H}}_{K \gamma}^{(+)}(\varkappa \rho )\,.
%
\label{eq:psi3plus-ass}$$ In general, the functions $\tilde{\mathcal{H}}_{K\gamma }^{(+)}$ are some linear combinations of Coulomb functions with the outgoing asymptotic $G+iF$. The functions $\mathcal{J}_{K\gamma
}^{JM}(\Omega _{\rho })$ are hyperspherical harmonics coupled with spin functions to total spin $J$. “Multyindex” $\gamma$ denote the complete set of quantum numbers except the principal quantum number $K$: $\gamma=\{L,S,l_x,l_y\}$. The value $K_{\max }$ truncates the hyperspherical expansion. The hypermoment $\varkappa $ is expressed via the energies of the subsystems $E_{x}$, $E_{y}$ or via the Jacobi momenta $k_{x}$, $k_{y}$ conjugate to Jacobi coordinates $X$, $Y$:
\[eq:Jmom\] $$\begin{aligned}
\mathbf{k}_{x} & = & \frac{1}{2}\left( \mathbf{k}_{p_{1}}
-\mathbf{k}_{p_{2}}\right), \\
\mathbf{k}_{y} & = & \frac{2}{3}\left( \mathbf{k}_{p_{1}}
+\mathbf{k}_{p_{2}}\right) -\frac{1}{3}\mathbf{k}_{c}, \\
\varkappa ^{2} & = & 2ME_T=2M(E_{x}+E_{y})=2k_{x}^{2}+\frac{3}{4}\,k_{y}^{2}\;,
\\
\Omega _{\varkappa } & = & \{\theta _{k},\Omega _{k_{x}},\Omega
_{k_{y}}\}\;,\quad \theta _{k}=\text{arctan}[E_{x}/E_{y}]\;.\end{aligned}$$
A more detailed picture of the “T” and “Y” Jacobi systems in coordinate and momentum spaces can be found in Fig. \[fig:jacobi\].
The set of coupled equations for the functions $\chi ^{(+)}$ has the form $$\begin{gathered}
\left[ \frac{d^{2}}{d\rho ^{2}} - \frac{\mathcal{L}(\mathcal{L}+1)}{\rho ^{2}}
+2M\left\{ E-V_{K\gamma ,K\gamma }(\rho )\right\} \right] \chi _{K\gamma
}^{(+)}(\rho )= \\
2M\sum_{K^{\prime }\gamma ^{\prime }}V_{K\gamma ,K^{\prime }\gamma ^{\prime
}}(\rho )\chi _{K^{\prime }\gamma ^{\prime }}^{(+)}(\rho )+i\,\Gamma M\,\chi
_{K\gamma }(\rho )\,, \label{shredl}\end{gathered}$$ where $\mathcal{L}=K+3/2$ is “effective angular momentum” and $V_{K\gamma
,K^{\prime }\gamma ^{\prime
}}(\rho )$ is “three-body potential” (matrix elements of the pairwise potentials); $$\begin{gathered}
V_{K\gamma ,K^{\prime }\gamma ^{\prime }}(\rho ) = \\
\int \!\!d\Omega _{\rho }\,\mathcal{J}_{K^{\prime }\gamma ^{\prime
}}^{JM\ast }(\Omega _{\rho
})\sum_{i<j}V_{ij}(\mathbf{r}_{ij})\,\mathcal{J}_{K\gamma }^{JM}(\Omega _{\rho
})\, \label{hhpot}\end{gathered}$$ and $$\Psi _{\text{box}}(\rho ,\Omega _{\rho }) =\rho ^{-5/2}\sum_{K\gamma }\chi
_{K\gamma }(\rho )\,\mathcal{J}_{K\gamma }^{JM}(\Omega _{\rho })\,.$$
![Independent “T” and “Y” Jacobi systems for the core+$N$+$N$ three-body system in coordinate and momentum spaces.[]{data-label="fig:jacobi"}](aa-02-coor-mom){width="48.00000%"}
Width and momentum distribution
-------------------------------
Equation \[eq:source\] is first solved with an arbitrary value of $\Gamma $ and then the width is found according to the “natural” definition as the flux $j$ through a hypersphere with large radius $\rho _{\max }$ divided by the internal normalization $N$ (“number of particles” inside the sphere): $$\begin{aligned}
\Gamma _{\text{nat}} & = & j/N\;, \label{eq:nat} \\
j & = & \int d\Omega _{\rho }\,\frac{d\,j(\rho _{\max },\Omega _{\rho })}{d
\,\Omega _{\rho }}\;,
\label{eq:tot-flux} \\
%
N & = & \sum_{K\gamma }N_{K\gamma }=\sum_{K\gamma }\int_{0}^{\rho _{\text{int}}}
\!d\rho \,\left\vert \chi _{K\gamma }^{(+)}(\rho )\right\vert ^{2}\;.
\label{eq:int-norm}\end{aligned}$$ The differential flux through the hypersphere $\rho _{\max }$ is defined as $$\begin{gathered}
\frac{dj(\rho _{\max },\Omega _{\rho })}{d\Omega _{\rho }}= \\
\left.\mathop{\rm Im}\Bigl[\;\Psi _{3}^{(+)\dagger }\,\rho ^{5/2}\,\frac{d}{M
d\rho }\,\rho ^{5/2}\,\Psi _{3}^{(+)}\Bigr]\right\vert _{\rho =\rho _{\max
}}\,. \label{eq:dif-flux}\end{gathered}$$ If, for sufficiently large $\rho $, the coefficients $A_{Ll_{x}l_{y}}^{KSS_{x}}$ in Eq. \[eq:psi3plus-ass\] become independent of $\rho $, then the coordinate distribution becomes identical to the momentum distribution, i.e., $$\frac{j(\rho _{\max },\Omega _{\rho })}{d\Omega _{\rho }}\,\rightarrow \frac{
dj(\Omega _{\varkappa })} {d\Omega _{\varkappa }}\;.
\label{eq:current-equiv}$$ Further discussions of the validity of this approximation (Eq. \[eq:current-equiv\]), and detailed expressions for the momentum distributions, can be found in Ref. [@gri03c].
Potentials
----------
The $NN$ potential is taken either as a simple $s$-wave single-Gaussian form BJ (from the book of Brown and Jackson [@bro76]) $$V(r)=V_{0}\exp (-r^{2}/r_{0}^{2})\;,$$ with $V_{0}=-31$ MeV and $r_{0}=1.8$ fm, or the realistic “soft-core” potential GPT (Gogny-Pires-de Tourreil [@gog70]).
The Coulomb potential of the homogeneously charged sphere $r_{\text{sph}}=1.852$ fm is used in the $\alpha $-$p$ channel. In addition for this channel, we use an $\ell$-dependent potential SBB (Sack-Biedenharn-Breit [@sac54]) $$V(r)=V_{c}^{(\ell)}\exp (-r^{2}/r_{0}^{2})+(\mathbf{\ell}\cdot\mathbf{s})\,
V_{\ell s}\exp(-r^{2}/r_{0}^{2})\;,$$ where $r_{0}=2.30$ fm, $V_{c}^{(0)}=50$ MeV, $V_{c}^{(1)}=-47.32$ MeV, $V_{c}^{(2)}=-23$ MeV, and $V_{ls}=-11.71$ MeV. Historically, a somewhat modified SBBM potential has been used in the calculations of $A$=6 isobars in order to better reproduce the binding energies (e.g., Ref. [@dan91]). Later it was realized that it is more consistent to provide the phenomenological binding-energy correction using an additional short-range three-body potential (see, e.g., the discussion in Ref. [@gri07]). In this work, we used a short-range three-body potential of the form $$V_{3}(\rho )=\delta _{K\gamma ,K^{\prime }\gamma ^{\prime
}}V_{3}^{(0)}/[1+\exp ((\rho -\rho _{0})/d_{3})]\;, \label{eq:pot3}$$ where $\rho _{0}=2.5$ fm and $d_{3}=0.4$ fm. This “short-range” three-body potential (note the small diffuseness) does not distort the interactions in the subbarrier region which was found to be important for consistent studies of decay properties.
Three sets of nuclear potential are employed in this work. They are denoted as P1 (SBB+BJ), P2 (SBB+GPT), and P3 (SBBM+GPT). The values of $V_{3}^{(0)}$ used with potential sets P1, P2, and P3 are $-11.14$ MeV, $-13.22$ MeV, and $-0.64$ MeV, respectively. Throughout this paper when the potential set is not specified, the results of the calculations with the P2 set are shown.
Reaction models {#sec:reac}
---------------
In general, different definitions for the width of a decaying state coincide only in the limit when the width is very small. For the ground state of $^{6}
$Be, the definition of Eq. \[eq:nat\] is not very precise, as this state is comparatively broad ($\Gamma =92\pm 6$ keV) and thus the internal normalization $N$ (Eq. \[eq:int-norm\]) is sensitive to the integration limit $\rho _{\text{int}}$. For reasonable values of $\rho _{\text{int}}$ ranging from $10-20$ fm, the uncertainty in the width ($\Gamma _{\text{nat}}$) is about $25\%$ \[see Sec. \[sec:com-modis\], Fig. \[fig:corel-rho-dep\](a)\]. This problem does not exist for narrow $2p$ emitters ($\Gamma <1$ eV) where the WFs $\chi _{K\gamma }^{(+)}$ are vanishingly small under the Coulomb barrier. The densities for the dominating components of the $^{6}$He and $^{6}$Be WFs are shown in Fig. \[fig:wfs\]. For $^{6}$Be, it is clear that the WF under the barrier is not negligible.
![Densities $|\protect\chi_{K\protect\gamma}(\protect\rho)|^2$ and $|%
\protect\chi^{(+)}_{K\protect\gamma}(\protect\rho)|^2$ for the largest components of the $^{6}$He and $^{6}$Be g.s. WFs.[]{data-label="fig:wfs"}](aa-03-wf-comp){width="45.00000%"}
For moderately broad states, there are alternative ways to derive the width. These involve either the study of the $3\rightarrow 3$ scattering or the study of a particular reaction. For technical reasons, the latter is preferable for our application. For example, in order to determine the population of $^{6}$Be in a charge-exchange reaction on $^{6}$Li at zero angle, Eq. \[eq:source\] can be reformulated as $$\begin{gathered}
\left( \hat{H}-E_{T}\right) \Psi _{^{6}\text{Be}}^{(+)}(\rho ,\Omega _{\rho
})= \\
\sum_{i}\tau _{i}^{-}\sum_{M}\sigma _{i}^{(M)}\;\Psi _{^{6}\text{Li}}^{JM} (\rho
,\Omega _{\rho })\;.
\label{eq:ste-6li}\end{gathered}$$ This notation is based on the fact that for angles close to zero, the transitions in charge-exchange reactions, in the limit of high energies, are provided by the Gamow-Teller operator. Although this reaction is different to the one studied experimentally in this work, it is sufficient for our computational purposes. Namely, we will demonstrate that for the $^{6}$Be g.s. population, the choice of the reaction mechanism is not very important (there are still some exclusive situations, which we will discuss elsewhere).
Using the source function of Eq. \[eq:ste-6li\], the cross section for the population of the three-body continuum is proportional to the outgoing flux of the three particles on a hypersphere of some large radius $\rho = \rho_{\max
}$: $$d\sigma (E_{T})/dE_{T}\sim j(\rho _{\max },\Omega _{\rho })\,.
%
\label{eq:cross}$$ Differentials of this flux on the hypersphere provide angular and energy distributions among the decay products at the given decay energy $E_{T}$ in analogy with Eqs. \[eq:dif-flux\] and \[eq:current-equiv\].
“Feshbach” reduction
--------------------
Although the HH calculations for $^{6}$Be can be performed with $K_{\max
}=22-26$, these basis sizes may not be sufficient to obtain good convergence for all observables. However, the basis size can be effectively increased using the adiabatic procedure based on the so-called Feshbach reduction (FR) [@gri07]. Feshbach reduction eliminates from the total WF $\Psi =\Psi
_{p}+\Psi _{q\text{,}}$ an arbitrary subspace $q$ using the Green’s function of this subspace: $$H_{p}=T_{p}+V_{p}-V_{pq}G_{q}V_{pq}\;.$$ In an adiabatic approximation, we can assume that the radial part of kinetic energy is small under the centrifugal barrier in the channels where this barrier is large and can be approximated as a constant. In this approximation, the FR procedure is reduced to the construction of effective three-body interactions $V_{K\gamma ,K^{\prime }\gamma ^{\prime }}^{\text{eff%
}}$ by the matrix operations $$\begin{gathered}
G_{K\gamma ,K^{\prime }\gamma ^{\prime }}^{-1} =(H-E)_{K\gamma ,K^{\prime
}\gamma ^{\prime }}= \\
V_{K\gamma ,K^{\prime }\gamma ^{\prime }} \\
+\left[ E_{f}-E+\frac{(K+3/2)(K+5/2)}{2M\rho ^{2}}\right] \delta _{K\gamma
,K^{\prime }\gamma ^{\prime }}\end{gathered}$$ where $$\begin{gathered}
V_{K\gamma ,K^{\prime }\gamma ^{\prime }}^{\text{eff}} = \\
V_{K\gamma ,K^{\prime }\gamma ^{\prime }}-\sum V_{K\gamma ,\bar{K}\bar{\gamma%
}}G_{\bar{K}\bar{\gamma},\bar{K}^{\prime }\bar{\gamma}^{\prime }}V_{\bar{K}%
^{\prime }\bar{\gamma}^{\prime },K^{\prime }\gamma ^{\prime }}\;.\end{gathered}$$ Summations over indexes with the bar are made for the eliminated channels. We typically eliminate the channels with $K>K_{FR}$, where $K_{FR}$ provides the sector of the hyperspherical basis where the calculations remains fully dynamical. We take the Feshbach energy $%
E_{f}$ in our calculations as $E_{f}\equiv E$.
There are two ways to control the reliability of the FR procedure. (i) The “soft” method is to vary $K_{FR}$ from the maximum attainable in the dynamic calculations downwards for fixed $K_{\max }$. The results, in principle, should coincide. (ii) The “safe” method is to take $K_{\max }$ in the range attainable for dynamic calculations and compare the “reduced” $K_{\max }\rightarrow K_{FR}$ calculations (with much smaller dynamic basis size $K_{FR}$) with completely dynamic calculations with $K_{\max }$. For $%
^{6}$Be, these considerations show that we can safely use $K_{FR}=14$. However, the even safer value of $K_{FR}=22$ is used in this work.
Ground state
============
There are several convergence characteristics that should be understood before reliable results on $^{6}$Be are obtained. The convergence character is quite different for all the observables of interest and also depend strongly on the interaction in the $p$-$p$ channel.
Convergence of energy and width
-------------------------------
Because of the problem mentioned in Sect. \[sec:reac\], we need to begin our studies with the energy dependence of the cross section. The convergence of the cross-section profile with increasing size of the basis is demonstrated in Fig. \[fig:sig-con-k\]. The main character of the convergence is clearly seen here; the centroid energy decreases, while the width grows significantly.
![(Color online) Energy profile of the $^{6}$Be g.s. populated in the charge-exchange reaction with $^{6}$Li. The results are shown as a function of the basis size $K_{\max}$ where $K_{FR}=22$. For $K_{\max}\leq K_{FR}$ and no Feshbach reduction is needed.[]{data-label="fig:sig-con-k"}](aa-04-fwhm-c){width="38.00000%"}
The cross section for the $^{6}$Be g.s. population, shown in Fig. \[fig:sig-con-k\], clearly has a profile close to a slightly asymmetric Lorentzian. Can the profile of this three-body resonance be described by appropriately-modified R-matrix type expressions? A curious result is obtained here, the resonance profile, shown in Fig. \[fig:sig-con-k\] by the solid curve, can be fit with amazing precision by the following expression: $$\sigma (E_{T})\sim \frac{\Gamma (E_{T})}{(E_{T}-E_{3r})^{2}+\Gamma
(E_{T})^{2}/4}\;, \label{eq:sig-prof}$$ $$\Gamma (E_{T})=\Gamma _{0}\left[ \alpha \left( \frac{E_{T}}{E_{3r}}\right)
^{2}+(1-\alpha )\left( \frac{E_{T}}{E_{3r}}\right) ^{4}\right] \;,
\label{eq:g-ot-e}$$ where $\Gamma _{0}=98$ keV and $\alpha =0.65$. Equation \[eq:sig-prof\] is the ordinary expression for the inelastic cross section of an isolated resonance. The parameterization of Eq. \[eq:g-ot-e\] was chosen because, for the single-channel penetration through the hyperspherical barrier with $K=0$, the energy dependence of the width can be inferred as $\Gamma (E_{T})\sim
E_{T}^{2}$. For $K=2$ one has $\Gamma (E_{T})\sim E_{T}^{4}$ (see, e.g., Ref. [@gol04]. It should be understood that the $K=0$ component is equivalent to a “phase volume” with the characteristic energy behavior of $%
\sim E^{2}$). The energy dependence of the width obtained by Eq. \[eq:g-ot-e\] almost coincides with the calculated dependence of this width in a reasonable energy range (see Fig. \[fig:pen-ot-e\] when one uses $\alpha$=0.63 and 0.52 for potential P2 and P3, respectively). If we take the actual calculated partial widths for the $K=0$ and $2$ components from Table \[tab:struc-t\], then the value of $\alpha $ can be estimated as $$\alpha =N_{K=0}/(N_{K=0}+N_{K=2})\approx 0.58 \; .$$ This is quite close to the value 0.65 obtained by a fit.
The existence of this simple approximation, despite the fact that there are Coulomb interactions and other numerous channels involved, may demonstrate that the dynamics of the $^{6}$Be g.s. decay is largely defined by the penetration through the hyperspherical barriers. Possibly, this is due to the comparatively large $^{6}$Be decay energy of $E_{3r}=1.371$ MeV. Simple estimates shows that the state is “sitting” somewhere straight on the top of the Coulomb barrier.
![(Color online) Dependence of $^{6}$Be g.s. width on the decay energy $E_T$. Predications are shown for the three potential set P1-P3. The dotted curves show the approximation of Eq. \[eq:g-ot-e\]. []{data-label="fig:pen-ot-e"}](aa-05-pen-ot-e){width="39.00000%"}
It was found that the value of $j(E_{T})$ for $^{6}$Be g.s. is not sensitive to the particular choice of the source in Eq. \[eq:ste-6li\], which is typically within the width of the line [^2]. This means that the width defined by the procedure of Eqs. \[eq:sig-prof\] and \[eq:g-ot-e\] is very reliable. We can fine tune the value $\rho _{\text{int}}$ in Eq. \[eq:int-norm\] so that the definition of the width in Eq. \[eq:nat\] coincides with the definition in Eq. [eq:sig-prof]{} and subsequently we can reliably use Eq. \[eq:nat\]. All of the potential sets P1-P3 needed $\rho _{\text{int}}\approx 12.5$ fm.
![(Color online) Convergency of the resonance energy $E_{3r}$ and the width $\Gamma$ for the $^{6}$Be g.s. as a function of the basis size $%
K_{\max}$.[]{data-label="fig:en-con-k"}](aa-06-en-con-k-c){width="44.00000%"}
The convergences of the predicted resonance energy and width as a function of the hyperspherical basis size are shown in Fig. \[fig:en-con-k\] for each of the potential sets. In all cases, our calculations are fully converged. The resonance energies are forced to approach the experimental value $E_{3r}=1.371$ MeV. This is achieved by fine tuning the phenomenological potential of Eq. \[eq:pot3\]; this is a necessary approach in order to provide reasonable predictions for the decay characteristics. We can see that while the calculations with P1 and P2 (SBB potential in the $\alpha $-$p$ channel) are in good agreement with each other and with the experimental value, the width obtained with P3 (SBBM potential) is far too large.
An expected feature observed here is the much slower convergence of the calculations with a realistic potential in the $NN$ channel. An important, but often disregarded fact, which one can see in Fig. \[fig:en-con-k\], is the much slower convergence of the width as compared to the energy. This means that, *in general, an energy convergence does not guarantee the convergence of other important characteristics*. As we will see in Sect. [sec:com-modis]{}, the situation with momentum distributions is even more complicated than it is for the widths.
The sensitivity of the width to a number of the other parameters in the calculations is demonstrated in Fig. \[fig:rho-int-max\]. Figure [fig:rho-int-max]{}(a) shows the sensitivity of the width defined by Eq. [eq:nat]{} to the size $\rho _{\text{int}}$ of the region where the internal normalization is calculated. The stability of the calculations to the dynamical range $\rho _{\max }$ is demonstrated in Fig. \[fig:rho-int-max\](b). To attain $1\%$ numerical precision in the width calculations, we need to go beyond 60 fm in the hyperradius $\rho $.
![(a) Sensitivity of the width as defined in Eq. \[eq:nat\] to the size of the “internal region” $\protect\rho_{\text{int}}$. The dot shows the value of $\protect\rho_{\text{int}}$ at which this width coincides with that defined via the cross-section profile Eqs. \[eq:ste-6li\], \[eq:cross\], \[eq:sig-prof\], and [eq:g-ot-e]{}. (b) Relative precision of the width as a function of the matching radius $\protect\rho_{\max}$.[]{data-label="fig:rho-int-max"}](aa-07-wid-ot-rint "fig:"){width="24.40000%"} ![(a) Sensitivity of the width as defined in Eq. \[eq:nat\] to the size of the “internal region” $\protect\rho_{\text{int}}$. The dot shows the value of $\protect\rho_{\text{int}}$ at which this width coincides with that defined via the cross-section profile Eqs. \[eq:ste-6li\], \[eq:cross\], \[eq:sig-prof\], and [eq:g-ot-e]{}. (b) Relative precision of the width as a function of the matching radius $\protect\rho_{\max}$.[]{data-label="fig:rho-int-max"}](aa-07-rel-prec-wid "fig:"){width="23.00000%"}
Features of the momentum distributions in $^{6}$Be {#sec:theory}
--------------------------------------------------
The correlations in the decay of $^{6}$Be include both the generic features of the $2p$ decays, as discussed earlier in Refs. [@gri03a; @gri03c; @gri07], and some peculiarities which we present in more detail now. For nuclear states with $J\leq 1/2$ (as is the case for $^{6}$Be g.s. decay), the three-body correlations can be completely described by 2 parameters. There are a total of 9 degrees of freedom for three particles in the final state. Of these, three describe the center-of-mass motion, three describe the Euler rotation of the decay plane (for $J\leq 1$ all its orientations are quantum-mechanically identical), and the three-body decay energy is fixed. Thus we are left with two parameters to describe the correlations. It is convenient to choose the energy distribution parameter $\varepsilon $ between any two of the particles and the angle $\theta _{k}$ between the Jacobi momenta: $$\varepsilon =E_{x}/E_{T}\quad ,\quad \cos (\theta _{k})=\frac{\mathbf{k}%
_{x}\cdot \mathbf{k_{y}}}{k_{x}\,k_{y}}. \label{eq:corel-param}$$These parameters can be constructed in any Jacobi system and for $^{6}$Be there are two “irreducible” Jacobi systems, called “T” and “Y” , see Fig. \[fig:jacobi\]. The distributions constructed in different Jacobi systems are just different representations of the same physical picture. However, different aspects of the correlations may be better revealed in a particular Jacobi system.
![(Color online) Complete correlation picture for $^{6}$Be g.s.decay, presented in “T” and “Y” Jacobi systems.[]{data-label="fig:corel-all"}](aa-08-corel-be-3d2c){width="49.00000%"}
Predictions for the complete correlation picture of $^{6}$Be g.s. decay are shown in Fig. \[fig:corel-all\] for both the “T” and “Y” Jacobi systems. Schematic figures are included in this figure to help in visualizing the correlations associated with different regions of the Jacobi plots. The main features of these distributions are:
1. The energy distribution in the “T” system has a double-humped profile which is an indication of the $[p^{2}]$ configuration dominance which was pointed out in very early papers on $^{6}$Be [@boc87; @dan87; @boc89]. This double-humped configuration is expressed more in coordinate space (see the internal region in Fig. \[fig:corel-dens\]) and only marginally “survives” in the asymptotic region. The internal peaks in Fig. [fig:corel-dens]{} have the special names of “diproton” (protons are close to each other) and “cigar” (protons are in-line with $\alpha $-particle) configurations [@dan91].
2. There are kinematical regions where the presence of particles is suppressed due to Coulomb repulsions. Strong suppression in the $\alpha $-$p$ channel in regions (b) and (d) and a smaller suppression in the $p$-$p$ channel in region (e) are predicted.
3. There are enhancements due to the $p$-$p$ final-state interaction in regions (a) and (f). The $^{5}$Li g.s. resonance in the $\alpha $-$p$ channel is not accessible for decay. However, some hint of its presence can be obtained from the enhancement in region (g). This is a “back-to-back” configuration, where protons fly in the opposite directions. However, the reason for the enhancement of such a configuration is not fully understood.
4. The angular dependence in the “T” system almost vanishes for regions (a) and (c) ($E_{x}/E_{T}\sim 0$ and $E_{x}/E_{T}\sim 1$). It is clear that in the limit $E_{x}/E_{T}\rightarrow 0$ and $E_{x}/E_{T}\rightarrow 1$ the dependence on the relative orientation of $\mathbf{k}_{x}$ and $\mathbf{k}%
_{y}$ should become degenerate. However at intermediate values of $%
E_{x}/E_{T}$, this dependence is very pronounced.
5. The total-energy distribution in the “Y” system (see Fig. [fig:corel-dep-pot]{} for the projected distributions) is almost a symmetric bell-shape. This is the energy distribution between the core and one of the protons and its symmetry reflects the symmetry between protons. In heavy two-proton emitters, this distribution becomes very narrow and almost completely symmetric.
![(Color online) Dependence of energy distribution between the proton (“T” system) in the decay of $^{6}$Be g.s. on the decay energy $E_T$.[]{data-label="fig:corel-en-dep"}](aa-09-corel-en-dep-c){width="34.00000%"}
The correlation predictions shown in Fig. \[fig:corel-all\] are obtained on resonance. The dependence the of energy correlation on the decay energy of $^{6}$Be is demonstrated in Fig. \[fig:corel-en-dep\]. The double-humped shape of this spectrum becomes less pronounced when the energy decreases. With smaller energy, the relative contribution of the $[s^{2}]$ configuration to the decay grows compared to the $[p^{2}]$ configuration. The latter has an additional centrifugal component to the barrier and its contribution to the width should be suppressed at low energies. The pure $%
[s^{2}]$ configuration should produce a featureless phase-volume energy distribution $$dj/dE_{x}\sim \sqrt{E_{x}(E_{T}-E_{x})}\;.$$
The sensitivity of the projected distributions to the choice of the potential set P1-P3 is demonstrated in Fig. \[fig:corel-dep-pot\]. The angular distribution in the “T” system and the energy distribution in “Y” systems are practically insensitive to this choice. The other projected distributions demonstrate sensitivity on the level of $10-15\%$. However, local differences in certain kinematical regions are much larger.
![Sensitivity of the energy and angular distributions in the decay of $^{6}$Be g.s. to the choice of the potential set. Results are shown for both the “T” and “Y” Jacobi systems.[]{data-label="fig:corel-dep-pot"}](aa-10-corel-dep-pot-1 "fig:"){width="34.00000%"} ![Sensitivity of the energy and angular distributions in the decay of $^{6}$Be g.s. to the choice of the potential set. Results are shown for both the “T” and “Y” Jacobi systems.[]{data-label="fig:corel-dep-pot"}](aa-10-corel-dep-pot-2 "fig:"){width="34.00000%"}
Figures \[fig:corel-all\] and \[fig:corel-dep-pot\] demonstrate what we call the “softness” of the $^{6}$Be system: minor variations in the conditions or computational details lead to a noticeable variations in the observable properties. Heavier $2p$ emitters appear to be much “stiffer” in this respect.
Convergence of the momentum distributions {#sec:com-modis}
-----------------------------------------
In our calculations there are two projected distributions which are practically insensitive to convergence issues (the angular distribution in the “T” system and the energy distribution in the “Y” system). The other two distributions (the angular distribution in the “Y” system and the energy distribution in the “T” system) demonstrate strong sensitivity. The convergence of the energy distributions are illustrated in Figs. \[fig:corel-con-k\] and \[fig:corel-rho-dep\].
The convergence of the energy distribution between protons has a very curious character, see Fig. \[fig:corel-con-k\]. From $K_{\max }=8$ to $K_{\max }=22$ this distribution is very stable \[several curves almost coincide, see Fig. \[fig:corel-en-dep\](a)\]. Then from $K_{\max }=24$ to $K_{\max }\sim 70$ the distribution changes qualitatively, and up to $K_{\max}\sim 100$ there is still a noticeable variation \[Fig. \[fig:corel-en-dep\](b)\]. Hopefully with $K_{\max }=110$, we have a well converged distribution. Calculations with small basis sizes (e.g., $K_{\max }\leq 70$) for $^{6}$Be should provide a qualitatively wrong energy distribution in the “T” system. Similarly for the angular distribution in the “Y” system.
![(Color online) Convergency of the “T” energy distribution in the decay of $^{6}$Be g.s. as a function of the basis size $K_{\max}$.[]{data-label="fig:corel-con-k"}](aa-11-con-en-dis-c){width="34.00000%"}
This “softness” of the $^{6}$Be system makes it a very complicated object to study. Minimum basis sizes which provide convergence for the energy and width are far from sufficient for calculations of momentum distributions. This is a feature which we probably do not face in heavier $2p$ emitters as the Coulomb interaction in the core-$p$ channel plays a more dominant role in the decay dynamics.
![(Color online) Dependence of energy distribution in the decay of $%
^{6}$Be g.s. on the maximal dynamic range of the calculation $\protect\rho%
_{\max}$. For the “T” Jacobi system, $E_x$ is energy between two protons and in the “Y” Jacobi system, $E_x$ is energy between core and one of the protons.[]{data-label="fig:corel-rho-dep"}](aa-12-corel-rho-dep-c){width="34.00000%"}
The radial convergence of the energy distributions is illustrated in Fig. \[fig:corel-rho-dep\]. Calculations with $\rho _{\max }<300$ fm are clearly insufficient to stabilize the distribution. However by $\rho
_{\max}=800$ fm, the distributions seem to be well converged. Could there be some noticeable modifications of the distributions due to further propagation in the long-range Coulomb field? This question was analyzed in Ref. [@gri03c] for $^{45}$Fe using the classical trajectory approach. The complete stabilization takes place in $^{45}$Fe at $\rho \sim (3-6)\times 10^{4}$ fm, with a major part of the effect originating at $\rho \lesssim 1\times 10^{4}$ fm. The decay energies of $^{6}$Be and $^{45}$Fe g.s. are similar and the Coulomb interaction is $\sim 12$ times weaker in $^{6}$Be. Therefore, the majority of the long-range effects should be taken into account in calculations with $\rho _{\max }\sim 1000$ fm. The $^{6}$Be calculations of this work were typically done with $\rho _{\max }=1200$ fm.
Structure of the $^{6}$He and $^{6}$Be g.s.
-------------------------------------------
From another point of view, one can benefit from the “softness” of $^{6}$Be system. The high sensitivity of the observables to the details of the model ingredients increase our ability to discriminate these features and hence improve our ability to elucidate the details of the nuclear structure.
{width="34.40000%"} {width="31.70000%"} {width="32.00000%"}
--------- ------- ------- ------- ------- ------- -------
$[l^2]$ P1 P2 P3 P1 P2 P3
$[s^2]$ 8.11 8.58 8.35 10.54 11.15 10.84
$[p^2]$ 90.91 90.30 90.37 87.98 87.18 87.17
$[d^2]$ 0.47 0.53 0.61 0.69 0.77 0.95
$[f^2]$ 0.41 0.43 0.50 0.60 0.65 0.77
--------- ------- ------- ------- ------- ------- -------
: Weights of the shell-model-like configurations $[l^2]$ in the $^{6}$He and $^{6}$Be g.s. WFs in percent for the Jacobi “Y” system. The normalizations of the $^{6}$Be components are found for integration radius $\protect\rho_{\text{int}}=12.5$ fm.[]{data-label="tab:struc-y"}
Detailed information about the $^{6}$He and $^{6}$Be g.s. WFs is provided in Table \[tab:struc-t\]. In general, there is high degree of isobaric symmetry between the $^{6}$He and $^{6}$Be WFs in the internal region. This is not true, however, for the $K=0$ component, which differs the most. The reason for this is shown in Fig. \[fig:wfs\] where the magnitude of the $K=0$ WF in asymptotic region is comparable to its magnitude in the nuclear interior. Hence the nuclear boundary is not defined for this component in $^6$Be. This is also seen in Table \[tab:struc-y\], which provides the information about the WF in approximate “shell model” terms. After looking at the radial behavior of the WF’s components in Fig. \[fig:wfs\], we find that the concept of isobaric symmetry is relevant here strictly speaking *only* for the most interior region of the WF ($\rho <4-5$ fm). Beyond this point the radial behavior in $^{6}$He and $^{6}$Be differ drastically.
The weights of the components in Tables \[tab:struc-t\] and \[tab:struc-y\] are in *very* good relative agreement for the different potential sets P1-P3. Evidently these major features of the structure are not that sensitive to the fine details of the interactions.
--------------------------------------------------------------------------------------------
value P1 P2 P3 Exp.
--------------------------------------- ------- ------- ------- ----------------------------
$\langle \rho \rangle $ (fm) 5.088 5.156 5.491
$\langle r_{NN} \rangle$ (fm) 4.482 4.502 4.884
$\langle r_{cN} \rangle$ (fm) 4.113 4.172 4.430
$\langle r_{N} \rangle$ (fm) 3.211 3.248 3.469
$\langle r_{c} \rangle$ (fm) 1.321 1.171 1.232
$r_{\text{mat}}$ (fm) 2.396 2.421 2.540 $2.30\pm 0.07$ [@ege01]
$2.48\pm 0.03$ [@oza01]
$r_{\text{ch}}$(fm) 2.103 2.012 2.048 $2.054\pm 0.014$ [@wan04]
$r_{\text{ch}}$(fm) 2.113 2.043 2.079 $2.068\pm
0.011$ [@mue07]
$B_{GT}(^6$He$\rightarrow^6$Li) 5.004 5.058 4.930 $4.745 \pm 0.009$ [@til02]
$\Delta E_{\text{coul}}$ (MeV) 2.351 2.302 2.111 2.344 [@til02]
$\Gamma (^6$Be$_{\text{g.s.}})$ (keV) 98 112 154 $92\pm 6$ [@til02]
--------------------------------------------------------------------------------------------
: Radial properties of the $^{6}$He g.s. WF and some observables obtained for $^{6}$He and $^{6}$Be g.s. with potentials P1, P2, P3.[]{data-label="tab:observ"}
It can be seen that the partial widths $\Gamma _{i}$ of the $^{6}$Be WF components in Table \[tab:struc-t\] are drastically different as compared to weights $N_{i}$ in the internal region. This is a reflection of complicated dynamics in $2p$ decays, the WFs are strongly “rearranged” in the subbarrier region and by the long-range Coulomb pairwise fields. The $^{6}$He and $^{6}$Be WF correlation densities are shown in Fig. \[fig:corel-dens\]. The WFs are nearly identical in the internal region, while in the asymptotic region for $^{6}$Be we can clearly see how this “rearrangement” is taking place. Comparing different potential sets P1-P3 in Table \[tab:struc-t\], we see that P1 and P2 calculations are almost identical, while the major partial width in P3 differs strongly. We conclude that the decay dynamics is mainly defined by core-$p$ interaction.
Geometric properties of the $^{6}$He g.s. WF and several observables obtained for $^{6}$He and $^{6}$Be g.s. are shown in Table \[tab:observ\]. The root mean square values are given for $\rho$, $r_{NN}$ (distance between valence nucleons), $r_{cN}$ (distance between nucleon and core), $r_{N}$ (distance between valence nucleon and c.m.), $r_{cN}$ (distance between core and c.m.). The differences between these geometric characteristics for P1 and P2 are typically around $1\%$. In the case of P3, the differences are significantly larger. The $B_{GT}$ values obtained with P1-P3 also agree within $1.5\%$, but all differ more from the experimental value. Here, the “experimental” $B_{GT}$ value is obtained using the $^{6}$He lifetime $\tau _{1/2}=806.7\pm 1.5$ ms [@til02], and the $\beta $-decay constants of $ft(0^{+}\rightarrow
0^{+})=3072.40$ s and $\lambda =1.268$. It has already been discussed in the literature that the $4-7\%$ disagreement here could be connected with both the WF quality and the renormalization of the weak constant [@dan91]. Therefore, we give no definite conclusion about quality of the models here.
The next most precisely known characteristic for $^{6}$He is its charge radius. Recent studies have defined $r_{\text{ch}}$ with increasing precision [@wan04; @mue07]. The relative uncertainty of this value is now about $0.5\%$, while variations in the calculated value are around $4\%$ for P1-P3. However, comparison of this value with those theoretically calculated is not completely model independent. The theoretically calculated charge radius of $^{6}$He is noticeably sensitive to the neutron charge radius. The latter is inferred theoretically, rather than measured experimentally. This means that there exists considerable systematic uncertainty in the determination of the charge radii. According to our estimates, this uncertainty can be as large as $2\%$. This fact somewhat relieves the constrains on the WF connected with this observable. One can see in Table \[tab:observ\] that the P2 and P3 calculations, containing realistic $NN$ potentials can be regarded as consistent with the experiment.
The matter radius of $^{6}$He is defined in the cluster model using the matter radius of the $\alpha $-particle. The value $r_{\text{mat}}(\alpha
)=1.464$ fm is derived from the charge radius $r_{\text{ch}}(\alpha )=1.671$ using the neutron and proton charge radii; $r_{\text{ch}}^{2}(n)=-0.1161$ fm$^{2}$, $r_{\text{ch}}(p)=0.875$ fm. The experimental data on matter radii have large a systematic uncertainty. This is probably the reason for the controversial signal obtained in different experiments (see two examples in Table \[tab:observ\]). This observable so far does not seem to have discriminative power for theoretical models.
The Coulomb shift $\Delta E_{\text{coul}}$ and $^{6}$Be g.s. width obtained with P1 and P2 are in a good agreement with experiment. Some overestimation of the width in the three-body cluster model can be expected due to the admixture of different configurations in $^{6}$Be WF. The weight of such admixtures can be estimated as $6-14\%$, based on the P1 and P2 widths. However, the Coulomb shift and width obtained with P3 are clearly not acceptable. Our overall feeling is that the cumulative information on $^{6}$He and $^{6}$Be g.s. is sufficient to choose P2 as the only acceptable potential.
Theoretical discussion
======================
As we have already mentioned, most of the attention in the studies of the $A$=6 isobar has been paid to $^{6}$He. Even in the studies of $^{6}$Be, there are only few works which studied it’s width. In addition, there has been are only limited studies of the $^{6}$Be g.s. decay correlations. The first consistent calculations of the $^{6}$Be three-body decay width were performed in Ref. [@dan93] using the integral formalism. In papers [@gri00b; @gri01], the quantum-mechanical formalism for two-proton radioactivity and Coulombic three-body decay studies was developed. In these papers, the integral formalism was criticized in application to the decays of systems with strong three-body Coulomb interactions and a more preferable way to calculate widths was proposed \[see, Eq. \[eq:nat\]\]. The value $\Gamma =90$ keV was obtained in Ref. [@gri00b] with the P1 potential ($K_{\max }=20$), which as we can see in Fig. \[fig:en-con-k\], is reasonably well converged.
In our approach, the effects of antisymmetrization are taken into account in a simplified way. However, there are studies that treated the $^{6}$Be decay as a 6-body problem. In RGM calculations [@cso94], the $^{6}$Be width of $\Gamma =160$ keV for $E_{3r}=1.52$ MeV was found using the complex scaling method. Scaling this value to the experimental $2p$ decay energy with the help of Fig. \[fig:pen-ot-e\] we obtain $\Gamma =125$ keV which is considerably larger than the experimental value. An interesting algebraic method was developed for studies of $^{6}$Be decay in Ref. [@vas01]. Here, the hyperspherical decomposition is used for the WF both in the internal region (6-body HHs) and in the asymptotic region (three-body cluster HHs). A calculated width of $\Gamma =72$ keV was obtained for $E_{3r}=1.172$ MeV which scales to $\Gamma =110$ keV at the experimental $2p$ decay energy. In addition, we can expect a $10-15\%$ reduction due to the absence of the $S=1$ component in these calculations. This component is important in the internal region, but does not contribute to the width significantly. In addition, we can also expect roughly a factor of 2 increase due to the small basis size ($K_{\max }=10$) used in the asymptotic region in Ref. [@vas01]. According to Fig. \[fig:en-con-k\], with $K_{\max
}=10$ we can expect only $60\%$ of the width, at most. It seem that Ref. [@vas01] is more a concept demonstration, rather than a realistic calculation. Therefore at the present moment, it is not possible to draw any conclusions about importance of the 6-body effects in calculations of the $^{6}$Be decay properties.
The width of the $^{6}$Be g.s. was calculated in Ref. [@des06] via a method analogous to ours (hyperspherical harmonics), but having certain technical differences. An approximate treatment of the $3\rightarrow 3$ scattering is introduced in this work and the width is extracted from the energy behavior of the phase shifts. The width obtained was $\Gamma =65$ keV for $E_{3r}=1.26 $ MeV which scales to $\Gamma =84$ keV at the experimental $2p$ decay energy. It can be found in Ref. [@des06] that the calculation does not seem converged. If we extrapolate from $\Gamma =84~$keV using the convergence curves for P2, P3, then the value $\Gamma =110$ keV is obtained, which is is a good agreement with our P2 result.
An important result of the present work is the clear demonstration that any approach purporting to give satisfactory description of the $^{6}$Be g.s.decay properties should have a certain “dynamic range” both in radial and functional spaces (see Table \[tab:range\]). It can be found that not all of these conditions are satisfied in these other works dedicated to $^{6}$Be.
Our calculations demonstrate a noticeable sensitivity of the observables in the decay of $^{6}$Be g.s. to the ingredients of the model. Table \[tab:observ\] demonstrates that this sensitivity is enhanced in $^{6}$Be compared to $^{6}$He. Typical variations of the observables for $^{6}$He are $0.5-4\%$, while in $^{6}$Be there is about a $60\%$ difference in between the widths calculated with P1 and P3. The tunneling process can be seen as a kind of a “quantum amplifier”, which drastically emphasizes minor features in the structure. For that reason, it is possible that the indirect probe of $^{6}$Be decay is a more sensitive tool for determining the halo properties of $^{6}$He than direct investigations of $^{6}$He itself. We are referring to precision measurements of the correlations in $^{6}$Be decay which are discriminative with respect to the fine details of the momentum distributions. In the experimental studies presented in this work, the quality of the data is approaching fulfilment of such a high precision request.
value $E_{3r}$ $\Gamma$ distributions
---------------------- ---------- ---------- ---------------
$\rho_{\max}$ (fm) 20 60 300
$K_{\max}$ (SBB+BJ) 16 30 80
$K_{\max}$ (SBB+GPT) 40 70 110
: Minimal dynamical ranges of calculations required to provide reasonably converged different observables for $^{6}$Be. Different basis sizes are required for simplistic BJ and realistic GPT potentials in the $p$-$p$ channel.[]{data-label="tab:range"}
Existing experimental knowledge about $^{6}$Be
==============================================
Very precise results about the energy and width of the $^{6}$Be g.s. were obtained in the early studies: $E_{T}=1371(5)$ keV, $\Gamma=89(6)$ keV [@wha66]. The current value of the width is only slightly different $\Gamma =92(6)$ keV [@til02].
![Experimental energy distributions between protons in the decay of $^{6}$Be measured in (a) Ref. [@gee77] and (b) Ref. [@boc89]. The theoretical prediction (P1) is provided only to guide the eye, as now we have no idea about the required experimental corrections.[]{data-label="fig:exp-com-1"}](aa-14-exp-old){width="37.00000%"}
The first measurements of $^{6}$Be decay correlations were made in Ref. [@gee77], see Fig. \[fig:exp-com-1\](a). They determined the energy spectrum of $\alpha $-particles reconstructed in the $^{6}$Be c.m. frame. For $^{6}$Be g.s. events, this spectrum is the same as the correlation spectrum between two protons. The authors could not fit the data using simplistic decay scenarios (phase volume, diproton decay, simultaneous emission of $p$-wave protons) and concluded: “...no incoherent sum of the processes considered here will fit the data. Perhaps a full three-body computation is necessary to understand the energy spectrum.”
This ground-state decay, as well as decays of the $2^{+}$, $T=0$ states of the $A$=6 isobar, was further investigated in the series of works by the Kurchatov Institute group [@boc87; @boc89; @boc92 and Refs. therein], see Fig. \[fig:exp-com-1\](b). They developed a method of analyzing the $p$-$p$ correlations in the framework of a three-body partial-wave decomposition and applied this to the three-body decays of light nuclei [@dan87; @boc89]. In particular, the first kinematically complete study of $^{6}$Be proved the existence of three-particle $p$+$p$+$\alpha $ correlations with $S(p$-$p)=1$ and $S(p$-$p)=0$ [@boc89; @boc92] which matched the three-body components found theoretically in the $p$-shell structure of $^{6}$Be [@dan91]. One of the important result for $^{6}$Be g.s. was the realization that $S(p$-$%
p)=0$ and $S(p$-$p)=1$ components of the WF should produce very different correlation patterns. The presence of an “admixture” of $S(p$-$p)=1$ component to the WF was demonstrated by an experiment performed with special kinematics. In these works, the concept of “democratic decay” was coined. This describes the specific decay mode for three-body systems, when the events are not highly focused in narrow kinematical regions, but are distributed broadly (“democracy” among different kinematical regions). “Democratic decay” is now a popular term for this class of phenomena, but the correlations in $^{6}$Be decay have never been studied since that time. The spectra shown in Figs. \[fig:exp-com-1\] (a) and (b) are not in complete agreement with each other. Furthermore, there are large statistical uncertainties and the geometry of experiments may cause cuts in kinematical space which make comparison the theory difficult. It is clear a modern experiment on $^{6}$Be decay was needed.
Experiment
==========
Experimental Method
-------------------
The Texas A&M University K500 cyclotron facility was used to produce a 200 pnA beam of $^{10}$B at $E/A=15.0$ MeV. This primary beam impinged on a hydrogen gas cell held at a pressure of 2 atmospheres and kept at liquid-nitrogen temperature. A secondary beam of $E/A=10.7$ MeV $^{10}$C was produced through the $^{10}$B$(p,n)^{10}$C reaction and separated from other reaction products using the MARS spectrometer [@Tribble89]. This secondary beam, with intensity of $2\times 10^{5}$ s$^{-1}$, purity of $99.5\%$, an energy spread of $3\%$, and a spot size of $3.5\times 3.5$ mm was inelastically excited due to interactions with 14.1 mg/cm$^{2}$ Be and 13.4 mg/cm$^{2}$ C targets. Ground-state $^{6}$Be fragments were created from the $\alpha $ decay of these excited $^{10}$C particles. Following the decay of the $^{6}$Be g.s. fragment, the final exit channel is $2p$+$2\alpha $.
The four decay products were detected in an array of four Si $E$-$\Delta E$ telescopes located in a plane 14 cm downstream of the target. The telescopes, part of the HIRA array [@Wallace07], consisted of a 65 $\mu $m thick, single-sided Si-strip $\Delta E$ detector followed by a 1.5 mm thick, double-sided Si strip $E$ detector. All Si detectors were $6.4 \times
6.4$ cm in area with their position-sensitive faces divided into 32 strips. The telescopes were positioned in a square arrangement with each telescope offset from its neighbor to produce a small, central, square hole through which the unscattered beam passed. With this arrangement, the angular range from $\theta =1.3$ to $7.7^{\circ }$ was covered. More details of the experimental arrange can be found in Ref. [@Mercurio08].
Monte Carlo Simulations
-----------------------
Monte Carlo simulations of the experiment were performed in order to determine the experiment bias and to understand the effects of the gates applied to remove unwanted $2p$+$2\alpha$ events. The simulations included the $\alpha$ decay of the parent $^{10}$C fragments and the correlations between the $^{6}$Be decay products are sampled according to the theory of Sect. \[sec:theory\]. The effects of energy loss and small-angle scattering of all the decay products were considered following Refs. [@Ziegler85; @Anne88].
Simulated events were passed through a detector filter and the effects of the position and energy resolution of the detector were added. The “detected” simulated events were subsequently analyzed in the same manner as the experimental data. The velocity, excitation-energy, and angular distributions of the parent $^{10}$C states were chosen such that the secondary distributions that passed the detector filter were consistent with the experimental results. Similar simulations for other decay modes were found to reproduce the experimental resolution [@Mercurio08].
Event Selection
---------------
Apart from $\alpha $-$^{6}$Be g.s. decay, these are many other $^{10}$C decay modes that lead to the $2p$+$2\alpha$ exit channel and thus the detected events must be suitably gated to remove these unwanted decays. Of particular importance is the rejection of the large yield of decays where the $^{10}$C fragments undergoes two-proton decay (either sequential through $^{9}$B or prompt) leading to the creation of an $^{8}$Be g.s. [@Mercurio08]. These events can readily be identified from the correlations between the two $\alpha $ particles. The distribution of relative energy ($E_{rel}^{\alpha \alpha }$) between the two $\alpha $ particles contains a strong, narrow peak corresponding to $^{8}$Be g.s. decay [@Mercurio08]. This peak has a FWHM of 38 keV and sits on a negligible background [@Mercurio08] thus allowing for a clean rejection of these events with the gate $E_{rel}^{\alpha \alpha }<0.2$ MeV. Our Monte Carlo simulations suggests this gate has essentially no significant effect on true $\alpha $-$^{6}$Be g.s. decays with only $0.01\%$ of detected events being rejected.
The remaining events have contributions from $\alpha $-$^{6}$Be g.s. and $p$-$^{9}$B ($E^{\ast }$=2.43 MeV) decays [@Mercurio08]. The latter $^{9}$B excited state does not decay through $^{8}$Be g.s. but undergoes a three-body decay like the $^{6}$Be ground state. For both of these decays modes, there is a difficultly is trying to find the intermediate state (either $^{6}$Be or $^{9}$B) as there are two possible ways to construct this fragment from the detected $2p$+$2\alpha $ exit channel. Let us concentrate on the $^{6}$Be g.s. fragments first where we must determine which of the two detected $\alpha $ particles was the one initially emitted from the $^{10}$C parent and which was produced in the decay of $^{6}$Be. To this end, the $^{6}$Be excitation energy for the two ways of constructing the $^{6}$Be fragment are determined and ordered according to their maximum and minimum values; $E^{\ast }(^{6}$Be$)_{\max }$ and $E^{\ast
}(^{6}$Be$)_{\min}$. A two dimensional plot of these two excitation energies is shown in Fig. \[fig:Be6\_prep\]. A prominent ridge centered around $E^{\ast }(^{6}$Be$)_{\min }=0$ corresponding to $^{6}$Be g.s. decay is clearly visible. For those events in this ridge structure, the identification of which $\alpha $ particles was produced in $^{6}$Be decay is clearly the one associated with $E^{\ast }(^{6}$Be$)_{\min }$ when $E^{\ast
}(^{6}$Be$)_{\max }\gg E^{\ast }(^{6}$Be$)_{\min }$. However when $E^{\ast
}(^{6}$Be$)_{\max }\sim E^{\ast }(^{6}$Be$)_{\min }$ the Monte Carlo simulations indicate that misidentifications will occur. These simulations suggests that for $E^{\ast }(^{6}$Be$)_{\max }-E^{\ast }(^{6}$Be$)_{\min
}=0.5$ MeV, the probability of misidentifying the $\alpha $ particles is $0.03\%$. This condition is indicated in Fig. \[fig:Be6\_prep\] by the dashed line and only events above this line were used in the subsequent analysis of the experimental data. One can see from Fig. \[fig:Be6\_prep\] that this condition does not significantly cut into the ridge structure and the Monte Carlo simulations suggests we lose $4.7\%$ of the remaining $\alpha
$-$^{6}$Be g.s. events with this gate.
The remaining ridge structure still sits on a background. Part of this background can be traced to $^{10}$C$\rightarrow p$+$^{9}$B($E^{\ast }=2.43$ MeV) decays. These events can be identified from $E^{\ast }(^{9}$B$)_{\max }$ and $E^{\ast }(^{9}$B$)_{\min }$ information in a manner similar to the $\alpha
$-$^{6}$Be g.s. events. A ridge structure also is evident in this case and it also sits on an non-negligible background, which in turn has contributions from $\alpha $-$^{6}$Be g.s. decay. Although one cannot completely separate all $p$-$^{9}$B and $\alpha $-$^{6}$Be events, we do reject events in the $E^{\ast }(^{9}$B$)_{\min }$ ridge structure. This results in a slightly diminished yield of true $\alpha $-$^{6}$Be g.s.events, but more importantly, it reduces the relative background under the $^{6}$Be ridge structure shown in Fig. \[fig:Be6\_prep\]. The Monte Carlo simulations suggests only $2.7\%$ of the remaining true $\alpha $-$^{6}$Be g.s. events were rejected by this condition.
The distribution of $E_{T}$ for the final selection of events is shown in Fig. \[fig:ET6Be\] by the data points. The FWHM width of the peak associated with $^{6}$Be g.s. is 220 keV which is larger than the intrinsic value of $\Gamma=92$ keV due to detector resolution. The solid curve indicates the simulated distribution after a smooth background contribution (dashed curve) is added. This simulated distribution reproduces the experimental results quite well confirming that the Monte Carlo simulations correctly model the experimental resolution. Figure \[fig:ET6Be\] also shows the gate $G6$ used to select $^{6}$Be g.s. fragments and the two gates, $G_{B1}$ and $G_{B2}$ which, when combined, were used to estimate the background in the $G6$ gate. In all subsequent results, this background has been subtracted.
The excitation-energy distribution of $^{10}$C fragments associated with the selected events is shown in Fig. \[fig:ExDist\]. There is localized strength around $E^{\ast }$($^{10}$C$)=7$ MeV and a continuous distribution up to approximately 15 MeV. Thus many $^{10}$C excited states are contributing to the detected $^{6}$Be g.s. yield.
Comparison of theory and experiment
===================================
Comparisons of experimental and predicted correlations in both the “T” and “Y” Jacobi systems are shown in Fig. \[fig:lego\]. The experimental results \[Figs. \[fig:lego\](a) and \[fig:lego\](c)\] has been background subtracted and, for the predicted distributions \[Figs. \[fig:lego\](b) and \[fig:lego\](d)\], the effects of the detector resolution and bias has been incorporated via the Monte Carlo simulations. In this and subsequent plots, the simulated results has been normalized to the same number of counts as for the experiment data. In determining the Jacobi coordinates, there are two ways of choosing the order of the proton. For the experimental events, Jacobi coordinates were determined for both of these ways and thus each event contributes two counts to the spectra. For “T” system, this forces the $\cos \left( \theta _{k}\right) $ distribution to be symmetrized around $\cos \left( \theta _{k}\right) $=0. General overall agreement between theory and experiment is found, although statistical fluctuations are the limiting factor for the experimental data.
To allow for a more detailed comparison, we compare projections of the correlations on both the $E_{x}/E_{T}$ and $\cos \left( \theta _{k}\right) $ axes in Fig \[fig:all2\]. The experimental data are indicated by the data points while the dashed and solid curves show the predictions before and after the simulated bias of the experimental apparatus is included. Interestingly, the “soft” observables (energy distribution in the “T” system and the angular distribution in the “Y” system) which have the most sensitivity to the ingredients of the theoretical calculations and its numerical implementation also have the largest bias induced by the detector apparatus. The other projected distributions (angular distribution in “T” and energy distribution in “Y”) are practically unaffected the detector response.
The same comparison of theory and data for all three potentials P1-P3 is shown in Fig. \[fig:all4\]. All three sets of predictions reproduce the experimental data reasonable well. To highlight more details of the correlations, we show the $\cos \left( \theta _{k}\right) $ distributions gated on three equal region of $E_{x}/E_{T}$ in Fig. \[fig:cos\] for the “T” and “Y” Jacobi systems. Reasonable agreement between the experiment (data points) and the three calculations (curves) is also found, although the P1 and P2 calculation are somewhat better. To quantify this, we determine the $\chi ^{2}$ per degree of freedom ($\chi ^{2}/\nu $) of the theoretical fit to the two-dimension data of Fig. \[fig:lego\]. These values are listed in Table \[Tbl:chi\] for both the T and Y systems. For a good fit were need $\chi ^{2}/\nu \sim $1 and clearly both P1 and P2 satisfy this criteria. Again we find the P3 calculation is somewhat worse.
potential “T” “Y”
----------- ------ ------
P1 1.29 1.25
P2 1.17 1.14
P3 1.58 1.45
: $\protect\chi^2$ per degree of freedom for fits to the complete correlations data in the “T” and “Y” system with the three assumed potentials[]{data-label="Tbl:chi"}
Conclusions
===========
The first detailed studies of the correlations from the decay of $^{6}$Be g.s. are performed both experimentally and theoretically. We have found that certain correlations (namely, energy correlation between two protons and angular correlations in “Y” Jacobi system) are quite sensitive to the details of structure and interactions. We demonstrated that relative sensitivity of correlation patterns to the details of the interactions is higher in the decay of $^{6}$Be compared to the corresponding sensitivity of typical observables in $^{6}$He. We argue that further highly detailed studies of correlations in the decay of $^{6}$Be could provide a better access to the properties of $A$=6 isobar (and thus to halo properties of $^{6}$He nucleus) than the direct studies of $^{6}$He halo properties.
Experimentally $^{6}$Be fragments are produced from the $\alpha $ decay of $^{10}$C excited states formed by inelastically scattering a $^{10}$C beam off of Be and C targets. The $\alpha $+$2p$ decay products as well as the initially emitted $\alpha $ particle were detected in a Si array with good position and energy resolution. The experimentally measured correlations between $^{6}$Be g.s. decay products and the theoretical predicts were found to be in good agreement.
Acknowledgements
================
This work was supported by the U.S. Department of Energy, Division of Nuclear Physics under grants DE-FG02-87ER-40316, DE-FG02-93ER40773, and DE-FG02-04ER413. L.V.G. acknowledge the support from Russian Foundation for Basic Research grants RFBR 08-02-00892, RFBR 08-02-00089-a, and Russian Ministry of Industry and Science grant NS-3004.2008.2.
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----- --------------------------------------------- ------- ------- ------- ------- ------- ------- ------- ------- -------
$i$ $K \quad L \quad S \quad l_x \quad l_y$ P1 P2 P3 P1 P2 P3 P1 P2 P3
1 $0 \quad\; 0 \quad\; 0 \quad\; 0 \quad\; 0$ 4.32 4.65 4.27 6.72 7.24 6.65 50.44 50.77 41.03
2 $2 \quad\; 0 \quad\; 0 \quad\; 0 \quad\; 0$ 78.36 80.73 79.40 75.71 77.49 75.28 33.48 33.74 41.52
3 $2 \quad\; 1 \quad\; 1 \quad\; 1 \quad\; 1$ 14.19 11.28 12.02 13.09 10.60 11.44 3.89 3.31 6.15
4 $4 \quad\; 0 \quad\; 0 \quad\; 0 \quad\; 0$ 0.03 0.04 0.02 0.10 0.14 0.07 2.03 2.11 2.25
5 $4 \quad\; 0 \quad\; 0 \quad\; 2 \quad\; 2$ 0.48 0.50 0.58 0.44 0.45 0.53 6.10 6.48 4.97
6 $6 \quad\; 0 \quad\; 0 \quad\; 0 \quad\; 0$ 0.01 0.02 0.01 0.02 0.03 0.01 1.63 1.26 1.49
7 $6 \quad\; 0 \quad\; 0 \quad\; 2 \quad\; 2$ 1.13 1.18 1.56 1.56 1.60 2.32 0.67 0.73 0.78
8 $6 \quad\; 1 \quad\; 1 \quad\; 3 \quad\; 3$ 0.57 0.54 0.75 0.79 0.75 1.18 0.08 0.06 0.09
9 $8 \quad\; 0 \quad\; 0 \quad\; 0 \quad\; 0$ 0.28 0.31 0.37 0.47 0.51 0.66 0.85 0.69 0.85
10 $8 \quad\; 0 \quad\; 0 \quad\; 2 \quad\; 2$ 0.17 0.17 0.25 0.28 0.28 0.46 0.08 0.11 0.10
11 $8 \quad\; 0 \quad\; 0 \quad\; 4 \quad\; 4$ 0.03 0.03 0.04 0.05 0.05 0.08 0.37 0.40 0.32
----- --------------------------------------------- ------- ------- ------- ------- ------- ------- ------- ------- -------
[^1]: Evident exception is, of cause, the $\beta $-decay of $^{6}$He and $\beta $-delayed particle emission. These reactions exploit very “reliable” weak probe, providing important, but naturally limited information about this system.
[^2]: There are, however, some very special situations which we can not discuss in this work.
|
---
abstract: 'We report magnetization and specific heat measurements in the 2D frustrated spin-1/2 Heisenberg antiferromagnet Cs$_2$CuCl$_4$ at temperatures down to 0.05 K and high magnetic fields up to 11.5T applied along $a$, $b$ and $c$-axes. The low-field susceptibility $\chi (T)\simeq M/B$ shows a broad maximum around 2.8K characteristic of short-range antiferromagnetic correlations and the overall temperature dependence is well described by high temperature series expansion calculations for the partially frustrated triangular lattice with $J$=4.46K and $J''/J$=1/3. At much lower temperatures ($\leq 0.4$ K) and in in-plane field (along $b$ and $c$ -axes) several new intermediate-field ordered phases are observed in-between the low-field incommensurate spiral and the high-field saturated ferromagnetic state. The ground state energy extracted from the magnetization curve shows strong zero-point quantum fluctuations in the ground state at low and intermediate fields.'
author:
- 'Y. Tokiwa$^1$'
- 'T. Radu$^1$'
- 'R. Coldea$^2$'
- 'H. Wilhelm$^1$'
- 'Z. Tylczynski$^3$'
- 'F. Steglich$^1$'
title: 'Magnetic phase transitions in the two-dimensional frustrated quantum antiferromagnet Cs$_2$CuCl$_4$'
---
Introduction
============
Cs$_2$CuCl$_4$ is a quasi-2D Heisenberg antiferromagnet with $S$=1/2 Cu$^{2+}$ spins arranged in a triangular lattice with spatially-anisotropic couplings.[@Coldea01] The weak interlayer couplings stabilize magnetic order at temperatures below 0.62 K into an incommensurate spin spiral. The ordering wavevector is largely renormalized from the classical large-$S$ value and this is attributed to the presence of strong quantum fluctuations enhanced by the low spin, geometric frustrations and low dimensionality.[@renormalization; @Veillette] The purpose of the thermodynamic measurements reported here is to probe the phase diagrams in applied magnetic field and see how the ground state spin order evolves from the low-field region, dominated by strong quantum fluctuations, up to the saturated ferromagnetic phase, where quantum fluctuations are entirely suppressed by the field. Intermediate fields are particularly interesting as the combination of (still) strong quantum fluctuations, potentially degenerate states due to frustration and an effective “cancelling” of small anisotropies by the applied field may stabilize non-trivial forms of magnetic order.
The Hamiltonian of Cs$_2$CuCl$_4$ has been determined from measurements of the magnon dispersion in the saturated ferromagnetic phase.[@Coldea02] The exchanges form a triangular lattice with spatially-anisotropic couplings as shown in Fig. 1(b) with exchanges $J=0.374(5)$ meV (4.34 K) along $b$, $J^{\prime}=0.34(3)J$ along the zig-zag bonds in the $bc$ plane, and weak interlayer couplings $J^{\prime\prime}=0.045(5) J$ along $a$. In addition there is also a small Dzyaloshinskii-Moriya interaction $D=0.053(5) J$, which creates an easy-plane anisotropy in the ($bc$) plane (for details see Ref. ). Neutron diffraction measurements have shown rather different behavior depending on the field direction with respect to the easy-plane. For perpendicular fields (along $a$-axis) incommensurate cone order with spins precessing around the field axis is stable up to ferromagnetic saturation, however for fields applied along the $c$-axis (in-plane) the incommensurate order is suppressed by rather low fields, 2.1 T compared to the saturation field of 8.0 T along this axis.[@Coldea01] The purpose of the present magnetization and specific heat measurements is to explore in detail the phase diagram in this region of intermediate to high fields. From anomalies in the thermodynamic quantities we observe that for in-plane field several phases occur in-between the low-field spiral and the saturated ferromagnetic states. From the magnetization curve we extract the work required to fully saturate the spins and from this we derive the total ground state energy in magnetic field and the component due to zero-point quantum fluctuations.
Experimental details
====================
DC-magnetization of a high-quality single crystal of Cs$_2$CuCl$_4$ grown from solution was measured at temperatures down to 0.05K and high fields up to 11.5T using a high-resolution capacitive Faraday magnetometer[@Sak94]. A commercial superconducting quantum interference device magnetometer (Quantum Design MPMS) was used to measure the magnetization from 2K to 300 K. The specific heat measurements were carried out at temperatures down to 0.05 K in magnetic fields up to 11.5 T using the compensated quasi-adiabatic heat pulse method [@Wil04].
Measurements and results
========================
Temperature-dependence of susceptibility
----------------------------------------
 Temperature dependence of the susceptibility $\chi \simeq M/B$ of Cs$_2$CuCl$_4$ along the three crystallographic axes. Labels indicate magnetic long range order (LRO), short-range order (SRO) and paramagnetic (PM). (b) Susceptibility divided by the $g$-factor squared compared to calculations for a 2D anisotropic triangular lattice (see inset) with $J^{\prime}/J=1/3$ and $J=4.46$K (thick solid line), and non-interacting 1D chains with $J^{\prime}=0$ and $J=4.34$K (thick dashed line).](ChiT_v2.eps){height="10cm"}
 Magnetization curves of Cs$_2$CuCl$_4$ measured at the base temperature for the field applied along the three crystallographic axes. The curves for $B\parallel b$ and $c$ are shifted by 0.3 and 0.6$\mu_{\rm B}/$Cu, respectively. (b) Susceptibility ${dM}/{dB}$ vs field. Vertical arrows indicate anomalies associated with phase transitions (see text).](M_v2.eps){width="8cm"}
We first discuss the temperature-dependence of the magnetic susceptibility at low field and compare with theoretical predictions for an anisotropic triangular lattice as appropriate for Cs$_2$CuCl$_4$. Figure \[chi\](a) shows the measured susceptibility $\chi\simeq M/B$ in a field of 0.1 T. A Curie-Weiss local-moment behavior is observed at high temperatures and a broad maximum, characteristic of short-range antiferromagnetic correlations, occurs around $T_{\rm max}=2.8(1)$ K, in agreement with earlier data.[@Carlin] Upon further cooling the $b$- and $c$-axes susceptibilities show a clear kink at $T_{\rm N}$=0.62K, indicating the transition to long-range magnetic order. No clear anomaly at $T_{\rm N}$ is observed for $B\parallel a$. This is because the magnetic structure has ordered moments spiralling in a plane which makes a very small angle ($\sim 17\deg$) with the $bc$ plane.[@Coldea96] In this case the near out-of-plane ($a$-axis) susceptibility is much less sensitive to the onset of magnetic order compared to the in-plane susceptibility (along $b$ and $c$). Fitting the high-temperature data ($T\geq 20$ K) to a Curie-Weiss form $\chi(T)=C/(T+\Theta)$ with $C=N_{\rm{A}}g^2\mu^2_BS(S+1)/3k_B$ gives $\Theta=4.0 \pm $0.2 K and $g$-factors $g_a$=2.27, $g_b$=2.11 and $g_c$=2.36 for the $a$-, $b$- and $c$-axes, respectively. The $g$-factors are in good agreement with the values obtained by low-temperature ESR measurements $g$=(2.20, 2.08, 2.30)[@Bailleul94]. When the susceptibility is scaled by the determined $g$-values, $\chi/g^2$, the data along all three crystallographic directions overlap within experimental accuracy onto a common curve for temperatures above the peak, indicating that the small anisotropy term in the Hamiltonian (estimated to $\sim$5% $J$) are only relevant at much lower temperatures. In the temperature range $T \geq T_{max}$ we compare the data with high-temperature series expansion calculations[@Weihong04] for a 2D spin-1/2 Hamiltonian on an anisotropic triangular lattice (see Fig. \[chi\](b) inset). Very good agreement is found for exchange couplings $J^{\prime}/J=1/3$ and $J$=4.46K (0.384 meV) (solid line in Fig. \[chi\](b)). In contrast, the data departs significantly from the expected Bonner-Fisher curve for one-dimensional chains ($J^{\prime}=0$ and $J=4.34$ K).[@Weihong04]
Magnetization curve and ground-state energy
-------------------------------------------
Figure \[M\] shows the magnetization $M(B)$ and its derivative $\chi=dM/dB$ as a function of applied field at a base temperature of 0.05 K for the $a$- and $b$-axes and 0.07 K for the $c$-axis. For all three axes the magnetization increases linearly at low field but has a clear overall convex shape and saturates above a critical field $B_{\rm sat}$=8.44(2), 8.89(2) and 8.00(2) T along the $a$-, $b$- and $c$-axis, respectively. When normalized by the $g$-values the saturation fields are the same within 2% for the three directions, the difference being the same order of magnitude as the relative strength 5% of the anisotropy terms in the Hamiltonian.[@Coldea02] The saturation magnetizations $M_{\rm
sat}/g=\langle S_z \rangle$ are obtained to be only 1-2.5% below the full spin value of 1/2, which might be due to experimental uncertainties in the absolute units conversion or a slight overestimate of the $g$-values by this amount. Including such a small uncertainty in the $g$-values has only a small effect on the normalized susceptibility $\chi/g^2$ in Fig. \[chi\](b) and does not change significantly the results of the comparison with the series expansion calculation for the anisotropic triangular lattice.
Reduced magnetization, (b) susceptibility, and (c) ground-state energy, vs reduced field. Black (solid), blue (dashed), green (dash-dotted) and red (dotted) lines show experimental data for $B\parallel$a, semiclassical mean-field prediction, linear spin-wave theory including 1st order quantum corrections and Bethe-ansatz prediction for 1D chains $J^{\prime}=0$, respectively.](EnergyAnal4_v2.eps){width="8cm"}
Before analyzing in detail the various transitions in field identified by anomalies in the susceptibility $\chi=dM/dB$ (vertical arrows in Fig. \[M\]b) we briefly discuss how the ground-state energy varies with the applied field, as this gives important information about the effects of quantum fluctuations. The ground state energy is obtained by direct integration of the magnetization curve, i.e. $$E(B)=E(0)-\int_{0}^{B} M(B) dB$$ where the energy (per spin) above the saturation field takes the classical value $E(B>B_{\rm sat})=J(0)S^2-g \mu_B BS$ because the ferromagnetic state is an exact eigenstate of the Hamiltonian with no fluctuations.[@Coldea02] Here $J(0)=\frac{1}{2}\sum_{\delta}J_{\delta}$ is the sum of all exchange interactions equal to $J+2J^{\prime}$ for the main Hamiltonian in Cs$_2$CuCl$_4$ \[see Fig. \[chi\]b) inset\]. Figure \[EnergyAnal\] shows comparisons between the experimental data for $B$$\parallel$$a$ (black solid lines, similar results obtained using $b$- or $c$-axis data), a mean field calculation (blue dashed lines), a linear spin-wave theory(LSWT) with 1st order quantum correction (green dash-dotted lines) and Bethe-ansatz prediction for 1D chains $J^{\prime}=0$ (red dotted lines). In magnetic field, a cone structure is predicted by the classical mean field calculation [@Veillette] $E_{cl}(B<B_{\rm
sat})=S^2\left[J(Q)\cos^{2}\theta+J(0)\sin^2\theta\right]-g\mu_B B
S \sin\theta$ (blue dashed lines) where $Q$ is the classical ordering wavevector $Q=\cos^{-1}[-J^{\prime}/(2J)]$, $\theta=\sin^{-1}(B/B_{\rm sat})$, the saturation field is $g\mu_BB_{\rm sat}=2S[J(0)-J(Q)]$ and $J(Q)=J\cos(2Q)+2J^{\prime}\cos(Q)$. Here we use $J=0.374$ meV and $J^{\prime}/J=0.34$ for the main Hamiltonian in Cs$_2$CuCl$_4$. The experimentally-determined ground state energy (black solid line) is lower than the classical value (blue dashed line) due to zero-point quantum fluctuations. The energy difference in zero field is 85% of the expected classical energy $E_{cl}(B=0)=J(Q)S^2$, indicating rather strong quantum fluctuations in the ground state. The strongly non-linear (convex) shape of the magnetization curve compared to the classically-expected linear form $M/M_{\rm sat}(B<B_{\rm
sat})=B/B_{\rm sat}$ \[see Fig. \[EnergyAnal\](a)\] is a direct indication of the importance of zero-point quantum fluctuations. Including 1st order quantum correction to the classical result in a linear spin-wave approach gives[@Veillette] $E_{LSWT+1/S}=S(S+1)\left[J(Q)\cos^2\theta+
J(0)\sin^2\theta\right]-g\mu_BB(S+1/2)\sin\theta+\langle
\omega_{\bm{k}} \rangle /2$ where $\langle \omega_{\bm{k}} \rangle
$ is the average magnon energy in the 2D Brillouin zone of the triangular lattice. This improves the agreement with the data (green dash-dotted lines). Particularly at high fields it captures better the divergence of the susceptibility \[see Fig. \[EnergyAnal\](b)\] at the transition to saturation. The saturation field is underestimated slightly because we have here neglected the weak inter-layer couplings $J^{\prime\prime}$=4.5% $J$ and the DM interaction $D$=5.3%J, both of which increase the field required to ferromagnetically-align the spins. It is also illuminating to contrast the data with a model of non-interacting chains ($J^{\prime}=0$, red dotted lines). This would largely (by 48%) underestimate the observed saturation field and would predict a rather different functional form for the magnetization $M_{1D}(B<B_{\rm sat})=M_{\rm
sat}\frac{2}{\pi}\sin^{-1}\left(1-\frac{\pi}{2} +\frac{\pi
J}{g\mu_B B} \right)^{-1}, B_{\rm sat}=2J/g\mu_B$ and susceptibility ${\partial M_{1D}}/{\partial B}$ compared to the data, indicating that the 2D frustrated couplings are important.
![\[C\_b\] Specific heat as a function of temperature in magnetic fields along $b$-axis. Specific heat data in fields 4, 5 and 6T are shifted upwards by 0.5, 1.0 and 1.5 J/mol$\cdot$K$^2$, respectively.](C_b4_v2.eps){width="8cm"}
Phase diagrams in in-plane field
--------------------------------
When the magnetic field is applied along the $a$-axis perpendicular to the plane of the zero-field spiral the ordered spins cant towards the field axis and at the same time maintain a spiral rotation in the $bc$ plane thus forming a cone. The cone angle closes continuously at the transition to saturation and as expected in this case the susceptibility $dM/dB$ observes a sharp peak followed by a sudden drop as the field crosses the cone to saturated ferromagnet transition, see Fig. \[M\](b). However for fields applied along the $b$- and $c$-axes several additional anomalies are present in the magnetization curve apart from the sharp drop in susceptibility upon reaching saturation, indicating several different phases stabilized at intermediate field.
Before discussing in detail the experimental phase diagrams we note that for all field directions the magnetization increases in field up to saturation and no intermediate-field plateaus are observed, in contrast to the isostructural material Cs$_2$CuBr$_4$, where a narrow plateau phase occurs for in-plane field when the magnetization is near $1/3^{\rm rd}$ of saturation.[@Ono04] Such a plateau phase is expected for the fully-frustrated ($J^{\prime}/J=1$) triangular antiferromagnet and originates from the formation of the gapped collinear up-up-down state in field. The absence of a plateau in Cs$_2$CuCl$_4$ is probably related to the weaker frustration ($J^{\prime}/J$=0.34(3)) compared to Cs$_2$CuBr$_4$($J^{\prime}/J\sim 0.5$) [@Ono04].
A difference in the phase diagrams in field applied along the $a$-axis and in the $bc$ plane in Cs$_2$CuCl$_4$ is expected based on the presence of small DM terms in the spin Hamiltonian, which create a weak easy-plane anisotropy in the $bc$ plane.[@Coldea02] Semi-classical calculations which take this anisotropy into account predict two phases below saturation:[@Veillette] a distorted spiral at low field separated by a spin-flop like transition from a cone at intermediate field. The data in Fig. \[M\](b) however observe more complex behavior with several different intermediate-field ordered phases. Also early neutron scattering measurements did not observe the characteristic incommensurate magnetic Bragg peaks expected for a cone structure at $B>$2.1 T $\parallel$$c$-axis [@Coldea01], suggesting that the magnetic structure at intermediate field may be quite different from the classical prediction and may be stabilized by quantum fluctuations beyond the mean-field level. To map out the extent of the various phases in in-plane field we have made a detailed survey of the $B-T$ phase diagram using both temperature and field scans in magnetization and specific heat and the resulting phase diagrams are shown in Fig. \[B-T\]. Below we describe in detail the signature of those transitions in specific heat and magnetization data, first for field along the $b$-axis, then $c$-axis.
Magnetization and differential susceptibility $dM/dB$ in field along $b$ are shown in Fig. \[M\]. $dM/dB$ shows a sharp peak at $B$=2.76T and an additional small peak at $B$=8.57T and those two anomalies indicate two new phases at base temperature below the saturation field and above the spiral phase. To probe the extent in temperature of those phases we show in Fig. \[C\_b\] specific heat measurements as a function of temperature at constant magnetic field. At 3 T two peaks are clearly observed indicating two successive phase transitions upon cooling from high temperatures. The lower critical temperature increases rapidly with increasing field and gradually approaches the upper transition at 5 T and the two peaks appear to merge at 6 T.
![\[M(T)b\] Magnetization normalized by applied field $M/B$ (thick solid lines, left axis) and its derivative $d(M/B)/dT$ (thin solid lines, right axis) as a function of temperature for $B\parallel b$. Vertical arrows indicate anomalies.](AnormChiB3_v2.eps){width="8.5cm"}
![\[torque\] Raw capacitance data as a function of temperature in magnetic field of 4 and 6 T applied along $b$-axis. Data are vertically shifted for clarity. Filled and open arrows indicate anomalies. Thick (thin) solid lines correspond to measurements with (without) gradient field. Inset shows the temperature derivative of capacitance data in gradient field 10T/m. The curves are shifted vertically for clarity.](TorqueBall2_v2.eps){width="8cm"}
![\[AnormM(B)C\] Expanded plots of magnetization and susceptibility $\chi=dM/dB$ as a function of field along $c$-axis. Data are identical to those from Fig. 2(a) and (b).](AnormMBC_2_v2.eps){width="8cm"}
![\[C\_c\] Specific heat as a function of temperature in magnetic fields along $c$-axis. Specific heat data in fields 4, 5 and 6T are shifted upwards by 0.5, 1.0 and 1.5 J/mol$\cdot$K$^2$, respectively.](C_c3_v2.eps){width="7.5cm"}
![\[M(T)c\] Magnetization normalized by applied field $M/B$ (thick solid lines, left axis) and its derivative $d(M/B)/dT$ (thin solid lines, right axis) as a function of temperature for $B\parallel c$. Vertical arrows indicate anomalies.](AnormChiC4_v3.eps){width="8cm"}
Complementary magnetization data vs. temperature for field along $b$ is shown in Fig. \[M(T)b\]. At 3 and 4 T two anomalies are observed also in $M/B(T)$ and its derivative $d(M/B)/dT$, at essentially the same temperatures as the peaks in specific heat, indicating that those two anomalies are associated with magnetic phase transitions. The anomalies appear as kinks in $M/B(T)$ and steps in $d(M/B)/dT$. At 5 T however the scaled magnetization $M/B(T)$ only observes a clear anomaly at the lower of the two critical temperatures observed in specific heat. At 6 T no anomaly is visible in $M/B(T)$, but only the derivative $d(M/B)/dT$ shows a kink. The missing anomalies can however be seen in the raw capacitance data, plotted in Fig. \[torque\], which also contain information not only on the (longitudinal) magnetization but also the transverse spin components. At 4T the capacitance in both zero and non-zero gradient field show two successive transitions indicated by solid and open arrows. Those are in good agreement with the peaks observed in specific heat. Although there is no anomaly visible in the magnetization at 6T, the raw capacitance shows an anomaly (see derivative of $d{\cal C}/dT$ in inset of Fig. \[torque\])) at the same temperature as the peak in specific heat. The capacitance in non-zero gradient field contains information on the torque of the sample in addition to the magnetization, while that in zero gradient field does not depend on magnetization but only on the torque. The torque contribution is subtracted by measuring the capacitance in zero gradient field (details of measurement technique are described in Ref. \[5\]). The fact that there is no anomaly in magnetization implies that subtraction of torque effect cancels out the anomaly in the raw data. Therefore only the torque (transverse magnetization) has an anomaly and the longitudinal magnetization has no anomaly at the critical temperatures for these missing anomalies.
For field along $c$ it has been reported from neutron scattering study that the spiral phase at zero field is suppressed by magnetic field of 1.4T and above this field ordered spins form an incommensurate elliptical structure with elongation along the field direction [@Coldea01]. The elliptical phase is suppressed at 2.1T where the intensity of incommensurate magnetic Bragg peaks vanishes and the properties of the phase above 2.1T are still unknown. As shown in Fig. \[M\](b), the suppression of the spiral phase is clearly seen as a step in magnetization (a sharp peak in $dM/dB$) at 1.40T. In Fig. \[AnormM(B)C\] the magnetization at 0.07K and its derivative $dM/dB$ are expanded in order to show the four anomalies above the spiral phase. In Fig. \[AnormM(B)C\](a), $M(B)$ shows a small step (a peak in $\chi (B)$) at 2.05T, corresponding to the suppression of the elliptical phase. As indicated by an open arrow in Fig. \[AnormM(B)C\](a), a step in susceptibility at 2.18T is clearly seen, indicating possibly a new phase which may exist only in a very small range of fields from 2.05 to 2.18T. The next transition occurs at 3.67T (for increasing field) shown in Fig. \[AnormM(B)C\](b). $M(B)$ has a step accompanied by a hysteresis, indicative of a first order transition. Figure \[AnormM(B)C\](c) shows another transition at 7.09T with a clear hysteresis.
Fig. \[C\_c\] shows specific heat in magnetic fields along the $c$-axis. At 3T only one transition is observed upon cooling, whereas at 4 and 5T two successive transitions are observed. The lower temperature transition is very sharp, related to the first order behavior (hysteresis) on this transition line also observed in magnetization data $M(B)$ at 3.67T shown in Fig. \[AnormM(B)C\](b). The lower temperature transition shifts to higher temperatures with increasing field and almost merges with the upper transition at 6T.
Fig. \[M(T)c\] shows complementary magnetization data vs. temperature. At 3 T $M/B(T)$ and $d(M/B)/dT$ show a kink and a step at 0.35K, respectively. At 4T the position of the kink (step in $d(M/B)/dT$) is shifted to slightly higher temperature and another step-like anomaly (a negative peak in $d(M/B)/dT$) appears at lower temperatures 0.22K. At 5T this lower temperature step shifts to higher temperatures and the upper temperature anomaly (kink) can not be seen in $M/B(T)$ but is manifested as a kink in $d(M/B)/dT$ at 0.38K. Again the anomaly is missing in $M/B(T)$, but the raw capacitance data (not shown) exhibits an anomaly at 0.38K in good agreement with the specific heat result. As shown in the top panel of Fig. \[M(T)c\], $M/B(T)$ and $d(M/B)/dT$ have two anomalies at 6T. Note that due to the first order character of the lower temperature transition the anomalies of $M/B$ ($d(M/B)/dT$) indicated by open arrows in Fig. \[M(T)c\] are steps (peaks) rather than kinks (steps).
![\[B-T\] $B-T$ phase diagrams of Cs$_2$CuCl$_4$ for $B\parallel$ $b$- and $c$-axis. Data points of open circles (magnetization), squares (specific heat) and triangles (neutrons [@Coldea01]) connected by solid lines indicate phase boundaries. Solid circles show positions of the maximum in the temperature dependence of the magnetization and indicate a cross-over from paramagnetic to short-range order(SRO). “E” on the phase diagram for $B\parallel c$-axis denotes the elliptical phase. [@Coldea01]](Phase3_v2.eps){width="8cm"}
The phase diagrams for $B\parallel b$- and $c$-axis constructed using the anomalies discussed above are shown in Fig. \[B-T\]. The new data agree with and complement earlier low-field neutron diffraction results (open triangles).[@Coldea01] Apart from the phase transition boundaries identified above we have also marked the cross-over line between paramagnetic and antiferromagnetic short-range ordered(SRO) region, determined by the location of the peak in the temperature dependence of the magnetization such as in Fig. \[chi\](a). The peak position $T_{\rm{max}}$ decreases with increasing field and disappears above $B_c$, indicating suppression of antiferromagnetic correlations by magnetic field. For the field along $b$ and $c$-axis the phase diagrams are much more complicated than that for $B\parallel a$ which shows only one cone phase up to saturation field [@Coldea02; @Rad05]. For $B\parallel b$ three new phases appear above the spiral phase. Two of these phases occupy small areas of the $B-T$ phase diagram. For $B\parallel c$ four new phases are observed in addition to the spiral and elliptical phases.
We note that the absence of an observable anomaly in the temperature dependence of the magnetization upon crossing the phase transitions near certain fields (6 T along $b$ and 5 T along $c$) is consistent with Ehrenfest relation and is related to the fact that the transition boundary $T_c(B)$ is near flat around those points. The relation between the shape of the phase boundary and the anomaly in $M(T)$ was discussed by T. Tayama, [*et. al.*]{}[@Tay01] and is $$\Delta\left(\frac{d M}{d
T}\right)=-\frac{dT_c}{dB}\Delta\left(\frac{C}{T}\right)
\label{Ehrenfest}$$ where $\Delta(X)$ is the discontinuity of quantity $X$, $C$ is the specific heat and $T_c$ is the field-dependent critical temperature of second order phase transition. This shows that the discontinuity in $dM/dT$ vanishes when $dT_c/dB=0$, i.e. when the phase boundary is flat in field. This is indeed the case for 6T $\parallel b$ and at 5T $\parallel c$ \[see Fig. \[B-T\]\], and here only a kink and no discontinuity is seen in $dM/dT$.
Conclusions
===========
We have studied the magnetic phase diagrams of Cs$_2$CuCl$_4$ by measuring magnetization and specific heat at low temperatures and high magnetic fields. The low-field susceptibility in the temperature range from below the broad maximum to the Curie-Weiss region is well-described by high-order series expansion calculations for the partially frustrated triangular lattice with $J'/J$=1/3 and $J$=0.385meV. The extracted ground state energy in zero field obtained directly from integrating the magnetization curve is nearly a factor of 2 lower compared to the classical mean-field result. This indicates strong zero-point quantum fluctuations in the ground state, captured in part by including quantum fluctuations to order $1/S$ in a linear spin-wave approach. The obtained $B-T$ phase diagrams for in-plane field ($B
\parallel b$ and $c$-axis) show several new intermediate-field phases. The difference between the phase diagrams for $B\parallel
a, b$ and $c$ can not be explained by a semi-classical calculation for the main Hamiltonian in Cs$_2$CuCl$_4$, of a frustrated 2D Heisenberg model on an anisotropic triangular lattice with small DM terms. Further neutron scattering experiments are needed to clarify the magnetic properties of these new phases.
We would like to thank V. Yushankhai, D. Kovrizhin, D.A. Tennant, M. Y. Veillette and J.T. Chalker for fruitful discussions, Z. Weihong for sending the series data, P. Gegenwart and T. Lühmann for technical support.
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---
abstract: 'An appealing mechanism for inducing multiferroicity in materials is the generation of electric polarization by a spatially varying magnetization that is coupled to the lattice through the spin-orbit interaction. Here we describe the reciprocal effect, in which a time-dependent electric polarization induces magnetization even in materials with no existing spin structure. We develop a formalism for this dynamical multiferroic effect in the case for which the polarization derives from optical phonons, and compute the strength of the phonon Zeeman effect, which is the solid-state equivalent of the well-established vibrational Zeeman effect in molecules, using density functional theory. We further show that a recently observed behavior – the resonant excitation of a magnon by optically driven phonons – is described by the formalism. Finally, we discuss examples of scenarios that are not driven by lattice dynamics and interpret the excitation of Dzyaloshinskii-Moriya-type electromagnons and the inverse Faraday effect from the viewpoint of dynamical multiferroicity.'
author:
- 'Dominik M. Juraschek'
- Michael Fechner
- 'Alexander V. Balatsky'
- 'Nicola A. Spaldin'
bibliography:
- 'Nicola.bib'
title: Dynamical Multiferroicity
---
\[Introduction\] Introduction
=============================
Multiferroic materials, with their simultaneous magnetism and ferroelectricity, are of considerable fundamental and technological interest because of their cross-coupled magnetic and electric external fields and internal order parameters. Particularly intriguing are the class of multiferroics in which a non-centrosymmetric ordering of the magnetic moments induces directly a ferroelectric polarization. Prototypical examples are chromium chrysoberyl, Cr$_2$BeO$_4$ [@Newnham_et_al:1978], in which the effect was first proposed, and terbium manganite, TbMnO$_3$, which provided the first modern example and detailed characterization [@Kimura_et_al_Nature:2003; @kenzelmann:2005]. The mechanism responsible for the induced ferroelectric polarization in these materials is the Dzyaloshinskii-Moriya interaction [@Dzyaloshinskii:1957; @Dzyaloshinskii:1958; @Moriya1:1960; @Moriya:1960] as first proposed by Katsura *et al.* [@katsura:2005]: Electric polarization, $\mathbf{P} \sim \mathbf{M} \times (\nabla_\mathbf{r} \times \mathbf{M})$, emerges in spatial spin textures as a consequence of the lowest order coupling between polarization and magnetization, $\mathbf{P}\cdot( \mathbf{M} \times ( \nabla_{\mathbf r} \times \mathbf{M}))$, in the usual Ginzburg-Landau picture [@katsura:2005; @sergienko:2006; @Mostovoy:2006; @Cheong/Mostovoy:2007].
From a symmetry point of view a reciprocal effect, in which magnetization is induced by ferroelectric polarization, must also exist. The time-reversal and spatial-inversion properties of magnetization and polarization – $\mathbf{M}$ is symmetric under space inversion and anti-symmetric under time reversal, whereas $\mathbf{P}$ is symmetric under time reversal and anti-symmetric under space inversion – indicate that the product $\mathbf{M} \cdot (\mathbf{P} \times \partial_t \mathbf{P})$ couples $\mathbf{M}$ and $\mathbf{P}$ at the same lowest order as $\mathbf{P}\cdot( \mathbf{M} \times ( \nabla_{\mathbf r} \times \mathbf{M}))$. It follows that a magnetization $$\label{eq:dynmult}
\mathbf{M} \sim \mathbf{P} \times \partial_t \mathbf{P}$$ develops in the presence of an appropriate dynamical polarization. This mechanism of induced magnetization by time-dependent polarization provides a dynamical analogy of the usual static multiferroicity. While the generation of polarization by spatial spin textures is now well established both experimentally and in first-principles calculations (see for example Refs. [@yamasaki:2007; @malashevich:2008; @kajimoto:2009; @walker:2011]), discussions of the reciprocal dynamical effect are scarce, perhaps surprisingly so, since the idea is rooted in the classical induction of a magnetic field by a circulating current. One notable example is the proposal by Dzyaloshinskii and Mills [@dzyaloshinskii:2009] that it is the cause of the observed paramagnetism and specific heat increase in a ferroelectric insulator [@lashley:2007]; this analysis has been largely unrecognized to date.
The purpose of this paper is to develop the formalism of this dynamical multiferroic effect and to describe observable effects that it leads to. We begin by deriving a formalism for magnetic moments induced by phonons that allows computation of their magnitudes using first-principles electronic structure calculations. This allows us to provide a general derivation of the phonon Zeeman effect as the solid-state equivalent to the well-established vibrational Zeeman effect in molecules and to analyze the limits under which the effect will be experimentally observable. Next, we give an explanation for a recently observed behaviour, the resonant magnon excitation of a magnon using optically driven phonons reported by Nova *et al* [@nova:2017], in terms of dynamical multiferroicity. Finally we discuss the connection of dynamical multiferroicity to effects that are not related to lattice dynamics, specifically Dzyaloshinskii-Moriya-type electromagnons and the inverse Faraday effect.
Our investigation is particularly timely in light of recent progress in the ability to manipulate materials in the time domain to generate nonequilibrium states of matter with properties that differ from or are inaccessible in the static limit. Examples include the ultrafast structural phase transition in (La,Ca)MnO$_3$ driven by melting of orbital order [@Beaud_et_al:2009] and the report of enhanced superconducting transition temperature in the high-$T_\text{c}$ cuprate yttrium barium copper oxide [@Hu_et_al:2014; @mankowsky:2014; @Kaiser_et_al:2014]. In addition, a number of theoretical proposals exist, for example predictions within Floquet theory of emerging topological states [@lindner:2011], and of temporal control of spin-currents [@sato:2016]. Our work in turn contributes to this growing field of “dynamical materials design” by providing an additional mechanism – time-dependence of polarization – through which novel states can be dynamically generated.
Formalism for dynamical multiferroicity {#formalism-for-dynamical-multiferroicity .unnumbered}
=======================================
We begin by reviewing the duality between the time dependence of $\mathbf{P}$ and spatial gradient of $\mathbf{M}$ that we discussed in the introduction and that is illustrated in Fig. \[fig:magpolduality\]. A spatially varying magnetic structure induces a ferroelectric polarization; if the gradient is zero, the polarization vanishes. Both, the sense of the spin spiral and the direction of polarization, persist statically in one of two degenerate ground states: If the sense of the spin spiral is reversed, the polarization inverts. In the case of the dynamical multiferroic effect, a temporally varying polarization induces a magnetization; if the time derivative is zero, the magnetization vanishes. Here, both, the sense of the rotating polarization (i.e. the curl of $\mathbf{P}\times\partial_t\mathbf{P}$) and the direction of the magnetization, persist *dynamically* in one of two degenerate ground states of the dynamical system: If the sense of the rotating polarization is inverted, the magnetization reverses. A quasi-static state of the magnetization is realized when the rotation of the polarization is steady. The induced magnetization can then couple to lattice and magnetic degrees of freedom of the system, as we will show in various examples. We emphasize that magnetism can be induced by a time-varying polarization in a previously non-magnetic system, just as a spatially varying magnetization can induce a polarization in a system that is previously nonpolar.
For the general case of a magnetization induced by two perpendicular dynamical polarizations that are oscillating sinusoidally with frequencies $\omega_1$ and $\omega_2$ and a relative phase shift of $\varphi$ we write the time-dependent polarization as $$\label{eq:polarization}
\mathbf{P}(t) = \left( \begin{array}{c} P_1(t) \\
P_2(t) \\
0
\end{array}\right)
=
\left( \begin{array}{c} A_1 \sin(\omega_1 t + \varphi) \\
A_2 \sin(\omega_2 t) \\
0
\end{array}
\right),$$ where one of the components can, in principle, also be static ($\omega_1$ or $\omega_2$ equal to zero). Evaluating Eq. (\[eq:dynmult\]) with this polarization we obtain $$\begin{aligned}
\label{eq:magneticmoment}
\mathbf{M}(t) & \sim & \bigg[ \frac{\omega_{+}}{2} \sin\big(\omega_{-}t+\varphi\big) \nonumber \\
& & - \frac{\omega_{-}}{2}\sin\big(\omega_{+}t+\varphi\big) \bigg] A_1 A_2 \hat{z}.\end{aligned}$$ A time-varying magnetization is induced that is oriented perpendicular to the spatial orientations of both polarizations. The magnetization consists of a superposition of a large-amplitude oscillation with the difference frequency, $\omega_{-}=\omega_1-\omega_2$, and a small-amplitude oscillation with the sum frequency, $\omega_{+}=\omega_1+\omega_2$. If the frequencies are equal, $\omega_1=\omega_2$, the induced magnetization is static.
![ \[fig:magpolduality\] **Duality of magnetization and polarization.**\
**(a)** A spatially varying magnetization induces a polarization, as is well known for example in multiferroic TbMnO$_3$ [@Kimura_et_al_Nature:2003; @kenzelmann:2005; @katsura:2005; @Mostovoy:2006]. **(b)** A temporally varying polarization induces a magnetization as shown in this work. ](magpolduality.pdf)
Dynamical multiferroicity mediated by phonons {#dynamical-multiferroicity-mediated-by-phonons .unnumbered}
=============================================
The polar nature of optical phonons means that they can induce time-dependent polarization, and in turn dynamical multiferroicity, in materials. We write the polarization, [$\mathbf{P}$]{}, caused by the displacing atoms in a normal-mode vibration as the Born effective charges, [$\mathbf{Z}^{\ast}$]{}, of the ions multiplied by the normal coordinates, [$\mathbf{Q}$]{}, of the phonon modes, ${\ensuremath{\mathbf{P}}}={\ensuremath{\mathbf{Z}^{\ast}}}{\ensuremath{\mathbf{Q}}}$. (Note that another approach is to use a Berry phase calculation; this was shown to lead to similar results by Ceresoli and Tosatti [@ceresoli2:2002; @ceresoli:2002].) The connection to the usual relation between magnetic moment and angular momentum [@einsteindehaas:1915] is revealed by rewriting Eq. \[eq:dynmult\] to obtain $$\label{eq:einsteindehaas}
\mathbf{M} = \gamma \mathbf{Q} \times \mathbf{\dot{Q}} = \gamma \mathbf{L},$$ where $\gamma$ is a gyromagnetic ratio and $\mathbf{L}={\ensuremath{\mathbf{Q}}}\times\dot{{\ensuremath{\mathbf{Q}}}}$ an angular momentum. Optical phonons with perpendicular polarity induce circular motions of the ions (ionic loops) whose individual magnetic moments combine to produce an effective macroscopic magnetic moment, as depicted in Fig. \[fig:phononmagneticmoment\]. The direction of $\mathbf{M}$ is determined by the sense of the ionic loop. The magnetic moment of phonons is the solid-state equivalent of the vibrational magnetic moment of molecules [@wick:1948].
For a many-body system, such as a phonon, the simple tensorial relation of Eq. (\[eq:einsteindehaas\]) has generally to be extended. The total magnetic moment per unit cell given by the sum of the moments caused by the circular motion of each ion [@eshbach:1952]: $$\begin{aligned}
\label{eq:magmom_atomic}
\mathbf{M} = \sum\limits_{i} \mathbf{m}_{i} = \sum\limits_{i} \gamma_{i} \mathbf{L}_{i}.\end{aligned}$$ Here $\mathbf{m}_{i}$ is the magnetic moment of ion $i$, $\mathbf{L}_{i}$ its angular momentum and $\gamma_{i}$ its gyromagnetic ratio, and the sum is over all ions in one unit cell. The angular momentum results from the motion of the ion along the eigenvectors of all contributing phonon modes [@huttner:1978]: $$\begin{aligned}
\label{eq:angular_atomic_detail}
\mathbf{L}_i = \sum\limits_{\alpha{},\beta{}} \mathbf{Q}_{i\alpha}\times\mathbf{\dot{Q}}_{i\beta}
= \sum\limits_{\alpha{},\beta{}} Q_{\alpha}\dot{Q}_{\beta} \mathbf{q}_{i\alpha} \times
\mathbf{q}_{i\beta},\end{aligned}$$ where we wrote the displacement vector of ion $i$ corresponding to mode $\alpha$, $\mathbf{Q}_{i\alpha}$, in terms of a product of the normal mode coordinate amplitude, $Q_{\alpha}$, with the unit eigenvector, $\mathbf{q}_{i\alpha}$: $\mathbf{Q}_{i\alpha}=Q_{\alpha}\mathbf{q}_{i\alpha}$. Indices $\alpha{}$ and $\beta{}$ run over all contributing phonon modes. The gyromagnetic ratio tensor is the effective charge to mass ratio of the ion: $$\label{eq:gyro_atomic}
\gamma_{i}=\frac{e\mathbf{Z}^{\ast}_{i}}{2 M_{i}},$$ where $\mathbf{Z}^{\ast}_{i}$ is the Born effective charge tensor of ion $i$ and $M_{i}$ its mass.
We will now rewrite the magnetic moment of Eq. (\[eq:magmom\_atomic\]) in terms of the phononic system. We insert the expression for the angular momentum of Eq. (\[eq:angular\_atomic\_detail\]) in Eq. (\[eq:magmom\_atomic\]) to obtain: $$\begin{aligned}
\label{eq:magmom_atomic_detail}
\mathbf{M} & = & \sum\limits_{i} \gamma_i
\sum\limits_{\alpha{},\beta{}} Q_{\alpha}\dot{Q}_{\beta}
\mathbf{q}_{i\alpha} \times \mathbf{q}_{i\beta} \nonumber\\
& = & \sum\limits_{\alpha<\beta}
\big( Q_\alpha \dot{Q}_\beta - Q_{\beta} \dot{Q}_\alpha \big)
\sum\limits_{i} \gamma_{i} \mathbf{q}_{i\alpha} \times \mathbf{q}_{i\beta}.\end{aligned}$$ Now we write the difference of the normal mode coordinates as an angular momentum, analogously to Eq. (\[eq:angular\_atomic\_detail\]): $$\begin{aligned}
\label{eq:angular_phononic_detail}
\big( Q_\alpha \dot{Q}_\beta - Q_{\beta} \dot{Q}_\alpha \big)
= \mathbf{Q}_{\alpha\beta} \times \dot{\mathbf{Q}}_{\alpha\beta}
= \mathbf{L}_{\alpha\beta},\end{aligned}$$ where $\mathbf{Q}_{\alpha\beta}$ contains the normal coordinates of the modes $\alpha$ and $\beta$ in the basis of their symmetric representation. The remaining part of [$\mathbf{M}$]{} resembles a gyromagnetic ratio, therefore we write Eq. (\[eq:magmom\_atomic\_detail\]) as: $$\begin{aligned}
\label{eq:magmom_phononic}
\mathbf{M} = \sum\limits_{\alpha<\beta} \gamma_{\alpha\beta} \mathbf{L}_{\alpha\beta} = \sum\limits_{\alpha<\beta}
\mathbf{m}_{\alpha\beta},\end{aligned}$$ where $$\begin{aligned}
\label{eq:gyro_phononic}
\gamma_{\alpha\beta} = \sum_{i} \gamma_{i} \mathbf{q}_{i\alpha} \times \mathbf{q}_{i\beta}\end{aligned}$$ is the gyromagnetic ratio vector and $\mathbf{L}_{\alpha\beta}$ the angular momentum of a system of phonons. The induced magnetic moments, $\mathbf{m}_{\alpha\beta}$, are generated by the ionic loops caused by pairs of phonon modes, $\alpha$ and $\beta$. For only two contributing phonon modes Eq. (\[eq:magmom\_phononic\]) reduces to the simple tensorial form $$\begin{aligned}
\label{eq:magmom_phononic_twomodes}
\mathbf{M} = \mathbf{m}_{12} = \gamma_{12} \mathbf{L}_{12} = \gamma_{12} \mathbf{Q}_{12} \times \dot{\mathbf{Q}}_{12}.\end{aligned}$$ We note that all quantities in this section, particularly the Born effective charge tensors, $\mathbf{Z}^{\ast}_{i}$, and the phonon eigenvectors, $\mathbf{q}_{i\alpha}$, can be calculated from first principles using standard density functional theory methods.
![\[fig:phononmagneticmoment\] **Magnetic moments from ionic loops.** Schematic motion of ions in a diatomic A$^{+}$B$^{-}$ material driven by perpendicular optical phonons. The circular motions of the ions create local magnetic moments, $\mathbf{m}_{\text{A}}$ and $\mathbf{m}_{\text{B}}$. The area covered by the ionic loop of the lighter ion (here B$^-$) is larger, and therefore its local magnetic moment overcompensates that of the heavier ion. This leads to a macroscopic magnetic moment, [$\mathbf{M}$]{}. ](circularphonons.pdf)
Phonon Zeeman effect {#phonon-zeeman-effect .unnumbered}
--------------------
Next we discuss the Zeeman splitting of degenerate phonon modes, building on the early work extending the well-established vibrational Zeeman effect in molecules [@moss:1973] to solids by Anastassakis [@anastassakis:1972] and Rebane [@Rebane:1983], as well as a derivation for cubic perovskites by Ceresoli [@ceresoli:2002] and a phenomenological analysis by Dzyaloshinskii and Mills [@dzyaloshinskii:2009]. Here we derive a general formalism for the phonon Zeeman effect that is amenable to computation using density functional theory via the quantities defined in the previous section.
Consider two degenerate phonon modes ($\omega_1=\omega_2=\omega_0$) polarized along perpendicular axes and shifted in phase by $\varphi \neq 0$. Eq. (\[eq:magneticmoment\]) then reduces to $$\begin{aligned}
\label{eq:magmom_Zeeman}
\mathbf{M}(0) \sim \omega_0 \sin(\varphi) A_1 A_2 \hat{z}.\end{aligned}$$ We see that the magnetization induced by the atomic motions is static and its magnitude depends only on the amplitude of the lattice vibrations, on their frequency, and on the phase shift between the two sinusoidal fields, reaching a maximum at $\varphi=\pi/2$. If an external magnetic field, **B**, is applied to the system, this induced magnetization interacts with it via the usual Zeeman coupling, with the result that the degeneracy of the phonon modes is lifted.
The Lagrangian describing the interaction of this magnetic moment with an externally applied magnetic field is: $$\begin{aligned}
\label{eq:lagrangianSUPP}
\mathcal{L}(\mathbf{Q},\mathbf{\dot{Q}}) = \frac{1}{2} |\mathbf{\dot{Q}}|^2
- \frac{\omega^2}{2} |\mathbf{Q}|^2
+ \mathbf{B} \cdot \mathbf{M},\end{aligned}$$ where $\mathbf{B}$ is the external magnetic field, $\mathbf{Q}=\mathbf{Q}_{12}=(Q_1,Q_2,0)$ contains the normal coordinates of two degenerate phonon modes and $\mathbf{M}$ is their magnetic moment as in Eq. (\[eq:magmom\_phononic\_twomodes\]). We can write the Lagrangian component wise as: $$\begin{aligned}
\label{eq:lagrangian_detailed}
\mathcal{L}(Q_1,Q_2,\dot{Q}_1,\dot{Q}_2) & = & \frac{1}{2} \dot{Q}_{1}^{2} + \frac{1}{2} \dot{Q}_{2}^{2} -
\frac{\omega^2}{2} Q_{1}^{2} - \frac{\omega^2}{2} Q_{2}^{2} \nonumber\\
& & + \gamma B_z\big( Q_{1}\dot{Q}_{2} - Q_{2}\dot{Q}_{1} \big),\end{aligned}$$ where $\gamma=\gamma_{12}$ is the gyromagnetic ratio as derived in Eqs. (\[eq:gyro\_phononic\],\[eq:magmom\_phononic\_twomodes\]), and $B_z$ is the $z$ component of the magnetic field. After a Fourier transformation, $$\begin{aligned}
Q_{\alpha} & \rightarrow Q_{\alpha} = Q_{\alpha\omega}{\ensuremath{\text{e}^{i\omega t}}} \\
Q_{\alpha}^2 & \rightarrow Q_{\alpha\omega}Q_{\alpha\omega}^{\ast} \\
Q_{\alpha}\dot{Q}_{\beta} & \rightarrow \frac{1}{2}\left(Q_{\alpha\omega}\dot{Q}_{\beta\omega}^{\ast}+
Q_{\alpha\omega}^{\ast}\dot{Q}_{\beta\omega}\right),\end{aligned}$$ the Lagrangian becomes $$\mathcal{L}(\mathbf{Q_{\omega}},\mathbf{Q_{\omega}^{\ast}}) = \mathbf{Q_{\omega}} \mathbf{A}
\mathbf{Q_{\omega}^{\ast}},$$ where $\mathbf{Q_{\omega}}=(Q_{1\omega},Q_{2\omega},0)$ contains the normal mode coordinates in Fourier space, and $$\mathbf{A} = \left( \begin{array}{cc} \omega^2-\omega^2_0 & 2i\gamma B_z\omega \\ -2i\gamma B_z\omega &
\omega^2-\omega^2_0 \end{array} \right).$$ We solve the determinant for the zone centre ($\omega\rightarrow\omega_0$) and obtain a splitting of the degenerate phonon modes: $$\label{eq:phononZeemansplitting}
\omega = {\ensuremath{\omega_{0}}}\sqrt{1\pm\frac{2\gamma B}{{\ensuremath{\omega_{0}}}}} \approx {\ensuremath{\omega_{0}}}\pm \gamma B_z .$$ The splitting is sketched in Fig. \[fig:phononzeemansplitting\] for a magnetic moment aligned parallel and antiparallel to an external field, corresponding to a left- and right-handed sense of the phonons. Notably the phonon Zeeman effect is independent of the magnitude of the phonon magnetic moment and hence the amplitude of the lattice vibrations. It is therefore present in optical modes excited, for example thermally or due to zero-point fluctuations [@dzyaloshinskii:2009; @riseborough:2010], and does not require intense optical pumping.
We estimate the magnitude of the effect for the tetragonal phase of strontium titanate, SrTiO$_3$, using density functional theory. (Please see the “Computational methods for phonon calculations” section for details of the calculation.) Our calculated low frequency degenerate E$_\text{g}$ phonon modes lie at [$1\,\text{THz}$]{}, and their gyromagnetic ratio as calculated with Eq. (\[eq:gyro\_phononic\]) is $\gamma=0.9\times10^{7}\,\text{T}^{-1}\text{s}^{-1}$. (Note that the g-factor obtained by Ceresoli [@ceresoli2:2002] for cubic SrTiO$_3$ using the Berry phase approach has a similar value.) It follows that to obtain a relative splitting of the eigenfrequency of $2\gamma B/{\ensuremath{\omega_{0}}}\approx 10^{-3}$ one needs a magnetic field of the order of $B={\ensuremath{55\,\text{T}}}$; at this field strength, our splitting is four orders of magnitude larger than the value estimated by Dzyaloshinskii and Kats for the splitting of *acoustic* phonons [@dzyaloshinskii:2011]. While the splitting for SrTiO$_3$ is small, the effect will be significantly enhanced in materials with high Born effective charges, light ions, and low-frequency optical phonon modes. Promising candidates are ABO$_3$ compounds in which a heavy A/B-site ion ensures low frequency phonons, while the light B/A-site ion and the oxygens make up a large percentage of the motion in the normal mode. A further approach to increasing the magnitude of the effect is to identify materials with dynamically varying Born effective charges which will lead to an additional contribution to the angular momentum proportional to $(\partial_t\mathbf{Z}^\ast)\mathbf{Q}$.
![\[fig:phononzeemansplitting\] **Zeeman splitting of degenerate phonon modes.** Sketch of the phonon Zeeman splitting as a function of the external magnetic field $\mathbf{B}$. The phonons with their magnetic moment $\mathbf{M}$ aligned parallel to the external magnetic field $\mathbf{B}$ has a lower energy than the phonons with their moment aligned antiparallel to the field. ](phononZeemansplitting.pdf)
 
Resonant magnon excitation by\
optically driven phonons {#resonant-magnon-excitation-by-optically-driven-phonons .unnumbered}
------------------------------
A recently observed example of the dynamical multiferroic effect is the resonant excitation of magnons through the frequency-dependent magnetic moment of a system of phonons. Consider two nondegenerate phonon modes $(\omega_1\neq\omega_2)$ with perpendicular polarity. Setting $\varphi=0$ without loss of generality, Eq. (\[eq:magneticmoment\]) then reduces to $$\begin{aligned}
\label{eq:magmom_magnon}
\mathbf{M}(t) \sim \bigg[ \frac{\omega_{+}}{2} \sin(\omega_{-}t) - \frac{\omega_{-}}{2}\sin(\omega_{+}t) \bigg]
A_1 A_2 \hat{z}.\end{aligned}$$ We obtain an induced magnetization that varies in time as a superposition of a large-amplitude oscillation with the difference frequency, $\omega_{-}$, and a small-amplitude oscillation with the sum frequency, $\omega_{+}$.
Such a situation was recently realized experimentally by Nova *et al.* by applying an intense linearly polarized terahertz pulse along the \[110\] direction in the $a$-$b$ plane of orthorhombic perovskite-structure erbium ferrite, ErFeO$_3$ [@nova:2017]. The pulse drives the high-frequency IR-active phonon modes B$_{3\text{u}}$ and B$_{2\text{u}}$ with polarization along the $a$ and $b$ lattice vectors. Due to the $Pbnm$ symmetry of ErFeO$_3$, these modes have slightly different eigenfrequencies (17.0 and [$16.2\,\text{THz}$]{}). Intriguingly, the spontaneous excitation of a magnon at [$0.75\,\text{THz}$]{}, close to the difference frequency, was observed. Our analysis above indicates that this behavior is a manifestation of the dynamical multiferroic effect: The large-amplitude part of the induced magnetic moment, $\mathbf{M}$, oscillates with the difference frequency $\omega_{-}={\ensuremath{0.8\,\text{THz}}}$, sufficiently close to the [$0.75\,\text{THz}$]{} magnon of ErFeO$_3$ [@koshizuka:1980], which is thus resonantly excited by $\mathbf{M}$.
Indeed a dynamical simulation of the evolution of the phonons with parameters computed from density functional theory confirms the behaviour predicted phenomenologically in Eq. (\[eq:magmom\_magnon\]). (See the “Computational methods for phonon calculations” section for details of the calculation.) In Fig. \[fig:timeevolution\] **a** we show the induced magnetic moment as calculated by Eq. (\[eq:magmom\_phononic\]) and in Fig. \[fig:timeevolution\] **b** the normal mode coordinate of the high-frequency IR-active B$_{3\text{u}}$ and B$_{2\text{u}}$ modes following an excitation with an ultrashort terahertz pulse. The small-amplitude oscillation of **M** with the sum frequency is negligible and only the large-amplitude oscillation with the difference frequency contributes. The induced magnetic moment peaks at the order of magnitude of the nuclear magneton with $|\mathbf{M}|\approx 0.1\mu_{\text{N}}$ per unit cell. It decays together with the IR modes over the timescale of a few picoseconds; experimentally the induced magnon was found to survive for at least [$35\,\text{ps}$]{} [@nova:2017].
While the phonon Zeeman effect is independent of the magnitude of the phonon magnetic moment, the resonant effect discussed here is quadratic in the amplitude of the dynamical polarization and therefore dependent on the intensity of the exciting terahertz pulse. At high pulse intensities an anharmonic coupling of the excited IR phonons to Raman-active phonons was shown to be relevant in ErFeO$_3$ [@juraschek:2017]. These modes do not contribute to the gyromagnetic ratio however, and we can therefore neglect the effect of nonlinear phononics in the above analysis.
Computational methods for phonon calculations {#computational-methods-for-phonon-calculations .unnumbered}
---------------------------------------------
We calculated the phonon eigenfrequencies and eigenvectors, and the Born effective charges of SrTiO$_3$ and ErFeO$_3$ from first-principles using the density functional theory formalism as implemented in the Vienna ab-initio simulation package (VASP) [@kresse:1996; @kresse2:1996] and the frozen-phonon method as implemented in the phonopy package [@phonopy]. We used the default VASP PAW pseudopotentials with Er 4$f$ electrons treated as core states. We converged the Hellmann-Feynman forces to [$10^{-5}\,\text{eV/{\ensuremath{\text{\AA}}}{}}$]{} using a plane-wave energy cut-off of [$700\,\text{eV}$]{} and a 7$\times$7$\times$5 $k$-point mesh to sample the Brillouin zone for SrTiO$_3$ and [$850\,\text{eV}$]{}, 6$\times$6$\times$4 for ErFeO$_3$. For the exchange-correlation functional we chose the PBEsol [@PBEsol] form of the generalized gradient approximation (GGA) and imposed a Hubbard correction of $U={\ensuremath{3.7\,\text{eV}}}$ and a Hund’s exchange of $J={\ensuremath{0.7\,\text{eV}}}$ on the Fe $3d$ states in ErFeO$_3$. Our fully relaxed structures with lattice constants $a={\ensuremath{5.51\,\text{{\ensuremath{\text{\AA}}}}}}$ and $c={\ensuremath{7.77\,\text{{\ensuremath{\text{\AA}}}}}}$ for SrTiO$_3$ and $a={\ensuremath{5.19\,\text{{\ensuremath{\text{\AA}}}}}}$, $b={\ensuremath{5.56\,\text{{\ensuremath{\text{\AA}}}}}}$, and $c={\ensuremath{7.52\,\text{{\ensuremath{\text{\AA}}}}}}$ for ErFeO$_3$ fit reasonably well to the experimental values of Refs. [@eibschutz:1965; @kiat:1996], as do our calculated phonon eigenfrequencies [@fleury:1968; @galzerani:1982; @subbarao:1970; @koshizuka:1980; @nova:2017]. Our calculated values for the highest IR phonon frequencies are [$16.52\,\text{THz}$]{} for the B$_{3\text{u}}$ and [$15.95\,\text{THz}$]{} for the B$_{2\text{u}}$ mode in ErFeO$_3$. For the time evolution of IR phonons after a pulsed optical excitation, we obtain the time-dependent normal mode coordinates, $Q$, by numerically solving the dynamical equations of motion: $$\label{eq:EOM}
\ddot{Q}_{\alpha} + \kappa_{\alpha} \dot{Q}_{\alpha} + \omega^{2}_{\alpha} Q_{\alpha} = F,$$ where $\kappa_{\alpha}$ is the friction coefficient and $\omega_{\alpha}$ the eigenfrequency of mode $\alpha$. The periodic driving force $F$ models the terahertz pulse in a realistic fashion with gaussian shape with an amplitude of 10MVcm$^{-1}$, and a finite width in both time (fwhm=[$130\,\text{fs}$]{}) and frequency (peak frequency [$19.5\,\text{THz}$]{}, fwhm=[$6.5\,\text{THz}$]{}) [@nova:2017].
Beyond lattice dynamics {#beyond-lattice-dynamics .unnumbered}
=======================
In the previous two examples we showed how the magnetic moment arising from a system of phonons couples lattice and magnetic degrees of freedom. In the following we discuss examples in which the time-dependent polarization is not caused by lattice dynamics.
Dzyaloshinskii-Moriya-type electromagnons {#dzyaloshinskii-moriya-type-electromagnons .unnumbered}
-----------------------------------------
The existence of electromagnons – that is spin waves excited by a.c. electric fields – was demonstrated ten years ago in multiferroic materials with an incommensurate (cycloidal or helicoidal) magnetic structure [@pimenov:2006]. In the original report, the interaction was shown to be mediated through conventional Heisenberg coupling of spins, leading to a phonon-magnon hybridization that makes the magnon electroactive (and is therefore called an electromagnon) [@pimenov:2006; @senff:2007; @sushkov:2008; @kida:2008; @valdesaguilar:2009; @pimenov:2009; @lee:2009; @takahashi:2012]. A second mechanism for generating electromagnons, in which electroactive excitation of the spin spiral occurs via the Dzyaloshinskii-Moriya interaction, has also been identified [@katsura:2007; @pimenov:2008; @shuvaev:2010; @takahashi:2012; @takahashi:2013; @shuvaev:2013]. We show in the following that the description of these electromagnons from the viewpoint of dynamical multiferroicity is equivalent to the previously introduced formalism via the inverse effect of the Dzyaloshinskii-Moriya interaction [@katsura:2007].
The standard description of a Dzyaloshinskii-Moriya-type electromagnon in a helical magnet is through the coupling of the spin degrees of freedom of the cycloid, $\mathbf{S}_m$, with a uniform lattice displacement, $\mathbf{u}$. The ferroelectric polarization is given by $\mathbf{P}_\text{FE}\propto\mathbf{e}_{ij}\times(\mathbf{S}_i\times\mathbf{S}_j)$, where $\mathbf{e}_{ij}$ is the vector connecting the sites $i$ and $j$. The component $n$ of the lattice displacement that couples to the rotation of the spin plane is parallel to its rotation axis, $\mathbf{u}^n~||~(\mathbf{S}_i\times\mathbf{S}_{j})$. An a.c. electric field, $\mathbf{E}~||~\mathbf{u}^n$, then induces a magnetization, $\mathbf{m}~||~\mathbf{e}_{ij}$, and excites an electromagnon when it matches the frequency of the eigenmode of the spin cycloid [@katsura:2007]. Specifically in TbMnO$_3$ with a magnetic field applied along $b$, the spin cycloid lies in the $a$-$b$ plane. Since $(\mathbf{S}_i\times\mathbf{S}_j)~||~c$ and $\mathbf{e}_{ij}~||~b$, the resulting $\mathbf{P}_\text{FE}~||~a$. The a.c. electric field component of sub-terahertz radiation, $\mathbf{E}~||~c$, then induces a magnetization, $\mathbf{m}~||~b$, and excites a Dzyaloshinskii-Moriya-type electromagnon when it matches the eigenfrequency of the spin cycloid at [$0.63\,\text{THz}$]{} [@shuvaev:2010]. To disentangle this clutter of alignments, we illustrate the symmetry in Fig. \[fig:abcycloid\]. Note that a cooperative contribution of the ferroelectric polarization from symmetric magnetostriction has been reported [@mochizuki:2009; @mochizuki2:2010], which, however, leaves our symmetry analysis here unaffected.
This situation is exactly consistent with the formalism of dynamical multiferroicity when the two perpendicular polarizations in Eq. (\[eq:polarization\]) are given by $\mathbf{P}=(P_1(0),0,P_2(t))$, where $P_1(0)\equiv\mathbf{P}_\text{FE}$ ($\omega_1=0$, $\varphi=\pi/2$) and $P_2(t)\equiv\mathbf{E}(t)$ ($\omega_2=\omega_0$). Eq. (\[eq:magneticmoment\]) then reduces to $$\begin{aligned}
\label{eq:magmom_electromagnon}
\mathbf{M}(t) \sim \omega_0 \cos(\omega_0 t) A_1 A_2 \hat{y}.\end{aligned}$$ The formalism predicts an induced magnetization that oscillates with the frequency of the terahertz radiation, $\omega_0$, identical to the magnetization described above within the inverse Dzyaloshinskii-Moriya formalism, $\mathbf{M}(t)\equiv\mathbf{m}$.
![ \[fig:abcycloid\] **Dzyaloshinskii-Moriya-type electromagnon symmetry in TbMnO$_3$.** Spin cycloid oriented in the $a$-$b$ plane with ferroelectric polarization $\mathbf{P}_\text{FE}$ along $a$. To excite a Dzyaloshinskii-Moriya electromagnon, the a.c. electric field of the sub-terahertz radiation **E** has to be aligned along $c$ and the induced magnetization **m** is along $b$. ](abcycloid.pdf)
Inverse Faraday effect {#inverse-faraday-effect .unnumbered}
----------------------
Finally, we show that the inverse Faraday effect, which describes the generation of a static magnetization in a material when irradiated with circularly polarized light, can also be interpreted from the viewpoint of dynamical multiferroicity. The inverse Faraday effect was first demonstrated in the 1960’s [@pitaevskii:1961; @vanderziel:1965; @pershan:1966] and experienced a massive revival in the last decade, after it was shown that it can be used to control magnetization non-thermally with photomagnetic pulses [@kimel:2005]. The standard description of the effect is $$\label{eq:IFE}
\mathbf{M}(\omega) = \chi(\omega) \mathbf{E}(\omega) \times \mathbf{E^\ast}(\omega),$$ where $\mathbf{M}$ is the induced magnetization, $\chi$ the magneto-optic susceptibility and $\mathbf{E}$ the electric field component of the circularly polarized light. The electric field induces perpendicular time-varying polarizations in the material and the connection to dynamical multiferroicity is revealed by rewriting Eq. (\[eq:IFE\]) in the time domain: $$\begin{aligned}
\label{eq:IFE_DynMult}
\mathbf{M}(t) & = & \chi(t) \mathbf{E}(t) \times \partial_{t}\mathbf{E}(t) \nonumber\\
& \sim & \mathbf{P} \times \partial_{t}\mathbf{P}.\end{aligned}$$ The perpendicular time-dependent polarizations induced by the circularly polarized light both oscillate with the frequency of the light $(\omega_1=\omega_2=\omega_0)$ and are shifted in phase by $\varphi=\pi/2$. Eq. (\[eq:magneticmoment\]) therefore reproduces the induced static magnetization of the inverse Faraday effect: $$\begin{aligned}
\label{eq:magmom_IFE}
\mathbf{M}(0) \sim \omega_0 A_1 A_2 \hat{z}.\end{aligned}$$ Note that the investigation of the inverse Faraday effect has been extended in recent years by microscopic theories [@battiato:2014] that also describe ultrafast timescales [@reid:2010; @popova:2011], as well as the case of linearly polarized light [@ali:2010]; these extensions are beyond the scope of the analogy provided here.
Discussion {#discussion .unnumbered}
==========
In summary, we have introduced the concept of dynamical multiferroicity and shown that it provides a straightforward unifying interpretation of four diverse phenomena, summarized in Table \[tab:orderofexcitations\]: i) the Zeeman splitting of phonon spectra in a magnetic field, ii) resonant magnon scattering, iii) electromagnon coupling, and iv) the inverse faraday effect. In the case of i) and ii), in which the multiferroicity is mediated by lattice dynamics, we showed that the dynamical multiferroic coupling can be calculated quantitatively using standard density functional theory methods, and we performed such calculations for representative materials in each case.
Since the term *nonlinear* is used in both optics (for example second-harmonic generation) and phononics (for example cubic and higher-order phonon-phonon interactions), we also clarify the order (linear or higher order and in which parameter) of the excitations for the four cases. First, in the cases of SrTiO$_3$, ErFeO$_3$ and DyFeO$_3$, the materials are centrosymmetric, so a nonlinear second-harmonic optical response to the terahertz pulse is not possible. In the case of TbMnO$_3$, second-harmonic generation occurs at higher energies than the sub-THz radiation used in the example (see for example Ref. [@matsubara:2015]). In addition, while cubic and higher-order non-linear phononic coupling has indeed been demonstrated in some of these materials (see for example, Ref. [@juraschek:2017] for a detailed analysis in the case of ErFeO$_3$), the phenomena described here are not driven by non-linear phononic effects. The phonon Zeeman effect in SrTiO$_3$ is independent of the amplitude of the degenerate phonons and requires no external excitation and so is zeroth-order in the polarization amplitude. The magnon excitation in ErFeO$_3$ depends on the quadratic term in the phonon amplitude and so is nonlinear in the amplitude of the driven time-dependent polarization, $P(t)$. As for the Dzyaloshinskii-Moriya-type electromagnon excitation, the induced magnetization is *linear* in $P(t)$ (with the perpendicular ferroelectric polarization being static), while the inverse Faraday effect is nonlinear in $P(t)$ due to the excitation with circularly polarized light.
The phonon Zeeman splitting that we analyze here is distinct from previously discussed interactions between phonons and magnetism in *magnetic* materials. These include the phonon angular momentum arising from spin-phonon interaction [@zhang:2014], the splitting of acoustic phonons due to the spin-orbit interaction [@liu:2017], the splitting of nondegenerate Raman-active phonon modes accompanying a ferroelectric to paraelectric phase transition [@rovillain:2011] and the magnetic-field dependence of the phonon frequencies in Ce compounds [@schaack:1975; @schaack:1976; @schaack:1977]. In contrast, the phonon Zeeman effect does not require magnetic ions. The report of a splitting of degenerate phonons due to the interaction with magnetoexcitons in graphene [@remi:2014] is a fascinating phenomenon, but also a different mechanism from the phonon Zeeman effect. It is further distinct from the phonon Hall effect proposed for acoustic phonons as an analogue of the anomalous Hall effect [@strohm:2005; @sheng:2006].
With the increased availability of intense THz sources of radiation, we anticipate that additional manifestations of dynamical multiferroicity will be revealed over the next years. Reciprocally, we expect that the effect will be used to engineer new behaviors that are not accessible in the static domain. In this context, we point out that dynamical multiferroicity provides a unit-cell analogue of an electric motor, in which the time-dependent polarization acts as a nanoscale electromagnetic coil to generate magnetic fields. This analogy might open a pathway to unforeseen technological applications.
We thank G. Aeppli, A. Cavalleri, M. Fiebig, T. F. Nova, and A. Scaramucci for useful discussions. This work was supported by the ETH Zürich, by Dr. Max Rössler and the Walter Haefner Foundation through the ETH Zürich Foundation, by US DoE E3B7 and by the ERC Advanced Grant program numbers 291151 and DM-321031. Calculations were performed at the Swiss National Supercomputing Centre (CSCS) supported by the project IDs s624 and p504.
|
---
abstract: 'We analyze X-ray spectra of heavily obscured ($N_H > 10^{24} {\ifmmode{\rm\,cm^{-2}}\else{${\rm\,cm^{-2}}$}\fi}$) active galaxies obtained with , concentrating on the iron K$\alpha$ fluorescence line. We measure very large equivalent widths in most cases, up to 5 keV in the most extreme example. The geometry of an obscuring torus of material near the active galactic nucleus (AGN) determines the Fe emission, which we model as a function of torus opening angle, viewing angle, and optical depth. The starburst/AGN composite galaxies in this sample require small opening angles. Starburst/AGN composite galaxies in general therefore present few direct lines of sight to their central engines. These composite galaxies are common, and their large covering fractions and heavy obscuration effectively hide their intrinsically bright X-ray continua. While few distant obscured AGNs have been identified, we propose to exploit their signature large Fe K$\alpha$ equivalent widths to find more examples in X-ray surveys.'
author:
- 'N. A. Levenson, J. H. Krolik, P. T. Życki, T. M. Heckman, K. A. Weaver, H. Awaki, and Y. Terashima'
title: 'Extreme X-ray Iron Lines in Active Galactic Nuclei'
---
Iron Lines in the X-ray Spectra of AGN
======================================
The strongest line in the X-ray spectrum of an active galactic nucleus (AGN) at moderate energies ($4\lesssim E \lesssim 10$ keV) is due to iron fluorescence, particularly “neutral” Fe K$\alpha$ at 6.4 keV, from Fe less ionized than . If both the continuum and the Fe-emitting region are viewed directly (the “Type 1" view), the equivalent width (EW) is small, typically less than 200 eV. As [@Kro87] pointed out, the EW can increase greatly if the fluorescing material is exposed to a stronger continuum than the observer detects. In this case (the “Type 2" view), an obscuring “torus” of material near the active nucleus blocks direct views of the central engine along the line of sight. X-ray observations of Seyfert galaxies generally support this unification scenario. Seyfert 1s typically exhibit EW $\approx 150$ eV [@Nan94], while the EWs in Seyfert 2s reported hitherto tend to be larger and distributed more broadly, ranging from about 100 eV to 1 keV [@Tur97].
Seen from the “Type 2" view, the K$\alpha$ EW depends strongly on the torus geometry and total column density. Previous theoretical calculations [@Awa91; @Ghi94; @Kro94] have concentrated on producing EW $\lesssim 1$ keV, consistent with earlier observations. They indicated, however, that still larger EWs might result from Compton-thick tori (i.e., $N_H > 10^{24}{\ifmmode{\rm\,cm^{-2}}\else{${\rm\,cm^{-2}}$}\fi}$ toward the nucleus) with special geometries.
Motivated by this suggestion and the recent discovery in X-ray Observatory () observations of several much stronger lines [e.g., @Sam01], in this work, we examine the Fe K$\alpha$ properties of active galaxies that have previously been identified as Compton thick. Our sample, listed in Table \[tab:ew\], comprises all such galaxies has observed for which published results, archival data, or proprietary data are available. Most of these are classified as Seyfert 2 galaxies. NGC 4945 lacks the requisite optical emission line signature of Seyfert galaxies, but X-rays reveal its active nucleus [@Iwa93]. M51 is exceptional in this group for its low luminosity, and the source identified in the Deep Field South, CDF-S 202, is exceptional for its high luminosity, but they both fulfill the broad selection criteria. Although this sample is not complete and the data are heterogeneous, these measurements suggest that extremely large Fe equivalent widths are common in such heavily obscured AGN.
Observations and Spectral Fitting\[sec:fit\]
============================================
All of the sample members were observed with the Advanced CCD Imaging Spectrometer (ACIS), and the Circinus galaxy and Mrk 3 were observed with the transmission gratings in place. See @Wei00, G. Garmire et al., in preparation, and @Can00 for more information on , ACIS, and the transmission gratings, respectively.
@Sak00 analyze the spectra of Mrk 3. In order to examine the Fe lines in detail, we obtained these data from the Data Archive at the X-ray Center and extracted the High-Energy Transmission Grating Spectrometer spectrum following the method of @Yaq01, applying updated calibrations. Using XSPEC [@Arn96], we fit the spectrum from 4–8 keV, avoiding the complication of many emission lines at softer energies while providing enough high-energy coverage to fit the H-like Fe and neutral Fe K$\beta$ near 7 keV. We modeled the continuum as a strongly-absorbed power law, adding an unabsorbed power law constrained to have the same photon index to represent the scattered contribution. These data do not strongly constrain the intrinsic column density, but the EW is relatively insensitive to this uncertainty. (These data are in fact consistent with Compton thin obscuration; $N_H = 6.9 (+2.4, -3.9) \times 10^{23} {\ifmmode{\rm\,cm^{-2}}\else{${\rm\,cm^{-2}}$}\fi}$.) Three Gaussians, which represent the Fe lines, are significant, with the neutral K$\alpha$ EW$= 0.71$ keV.
M51, NGC 4945, NGC 5135, and NGC 7130, were all observed with ’s back-illuminated S3 ACIS detector. We obtained the data on NGC 4945 and M51 from the archive. See Madejski et al., in preparation, for more complete analysis of the observation of NGC 4945. @Ter01 present results from the observation of M 51, but we reanalyze the data here in order to utilize updated calibrations. The other two data sets are part of our program on starburst/AGN composite galaxies. We reprocessed all data from original Level 1 event files and applied the latest gain corrections, from 2001 July 31. We examined the lightcurves and excluded times during the NGC 4945 observation where flares were significant. We detected no significant flares in the other data. In each case, we extracted the spectrum from a circular region of radius about $2\arcsec$ and measured the local background in nearby source-free regions. Except in the case of NGC 4945, we did not bin the data, in order to retain the highest-energy photons. While the very soft X-ray spectra require thermal emission components, above 4 keV only the power-law continuum and Fe emission lines are significant. A single unabsorbed continuum component is required, which we interpret as the reprocessed contribution. The best-fitting continuum in each of these four spectra is relatively flat, consistent with pure Compton reflection.
Table \[tab:ew\] summarizes these results, listing the measured central energy and the equivalent width of the neutral Fe K$\alpha$ line for each sample member. We also record the continuum photon index, $\Gamma$, of the fits. The intrinsic continuum is strongly reprocessed in these obscured galaxies. While the simple power laws we apply are reasonable over the restricted energy ranges of the model fits, we caution that they are not appropriate descriptions over large energy ranges (e.g., for $E = 0.5$–$30$ keV). References in Table \[tab:ew\] include the identification of the sample members’ large column densities and published measurements, which we adopt for Circinus, NGC 1068, CDF-S 202, and IRAS 09104+4109. In these last two cases, @Nor02 and @Iwa01 apply reflection models, and the tabulated values of $\Gamma$ represent the intrinsic source spectra. Thus, these two are steeper than the directly-measured continua of the other highly obscured examples, although their observed continua are similarly flat.
Fe Line Models\[sec:model\]
===========================
The Fe K$\alpha$ EWs of the Compton thick AGNs in this sample tend to be extremely large, as expected when the intrinsic AGN continuum is fully hidden from direct view. We model the geometric dependence of continuum reprocessing with Monte Carlo simulations, which @Kro94 describe more completely. Although we introduce no new physics in these calculations, the current observations demonstrate that more extreme configurations are relevant and must be evaluated. We consider a uniform torus of neutral material having a square cross-section of Thomson optical depth, $\tau$, in each direction, and opening angle $\theta$ (Figure \[fig:cartoon\]). We consider all viewing angles, $i$, including Type 1 AGN orientations, where $i < \theta$. We adopt intrinsic $\Gamma = 1.9$ and solar Fe abundance in the basic model and discuss additional variations below. We do not consider any scattered or direct contribution from ionized material that may be present outside the torus cavity or near the central source, but these simplifications do not significantly affect the EW.
The EW rises smoothly with decreasing $\theta$ and increasing $i$ for constant $\tau$, as Figure \[fig:tau4\] illustrates for $\tau = 4$. For $i \leq \theta$, the EW we calculate falls below $\approx 100$ eV because we model only the torus contribution; the accretion disk, visible when $i < \theta$, would add to the EW in that case.
Figure \[fig:iang65\] shows the variation of EW in the $\tau-\theta$ plane for $i = 65^\circ$. For fixed $\tau$, the EW is greatest when $\theta$ is smallest, but EW does not increase monotonically with $\tau$. To achieve EW $> 2$ keV, $\tau > 1$ is required; the greatest EWs are found for $\tau \simeq 4$.
The case of $\theta = 10^\circ$ (Figure \[fig:theta10\]) demonstrates that extremely large EWs may arise over a wide range of viewing angles and optical depths for sufficiently small torus opening angle. While EW increases with $i$ for a given $\tau$, moderate values of $\tau$ and large values of $i$ produce the largest EWs, which some of these Compton-thick AGNs require.
These trends may be understood qualitatively as the interaction of several processes. Smaller opening angle means the torus obscures a larger solid angle around the nucleus, hence more Fe K-edge photons are captured and generate K$\alpha$ photons. Viewing angles closer to the equatorial plane see larger equivalent widths in the K$\alpha$ line because the continuum is most severely attenuated along that direction. Large, but not too large, optical depth promotes K$\alpha$ production by absorbing more hard X-rays: the continuum at K$\alpha$ is suppressed, while photons well above the Fe K-edge are absorbed and trigger fluorescence. Compton optical depths more than $\simeq 4$ diminish the equivalent width because the optical depth from the points of K$\alpha$ production to the surface is too large for the fluorescence photons to escape.
Variations in spectral shape, torus cross-section, and Fe abundance create changes in the EW that are only on the order of typical measurement errors. A shallower slope tends to increase the EW, because relatively more high-energy photons penetrate deeper into the torus, and the Fe-fluorescing region is then closer to the observer. Adopting $\Gamma = 1.7$, for example, the EW changes by less than 5% for $i \le 65^\circ$, and up to 30% at $i=75^\circ$ for $\theta = 30^\circ$ and $\tau = 2 $.
Previous calculations have emphasized the effect of Fe abundance in order to account for large EWs, but abundance variations are only a secondary effect. With twice-solar abundances, for example, the EW increases up to 60% in the edge-on geometry. In contrast, purely geometric (and physically plausible) variations can alter the EW by an order of magnitude.
The Fe line luminosity is correlated with the intrinsic AGN luminosity. For a given torus geometry, the K$\alpha$ luminosity is relatively insensitive to viewing geometry, typically varying by factors of 5 over all viewing angles, with departures to factors of 10 only in a small minority of cases.
We constrain the geometries of the obscuring regions of the galaxies with the largest EWs with these models. M51 requires $\theta \le 20^\circ$; for $\theta = 10^\circ$, a range of optical depth $\tau \ge 2$ and viewing angle $i \ge 65^\circ$ are consistent. NGC 5135 also favors $\theta \le 20^\circ$, allowing a range of $i \ge 35^\circ$, depending on $\tau$. We note that ionized Fe lines are not detected in these two galaxies. Their large neutral covering fractions suggest that even if an ionized region is present, the neutral material may completely block it from our view.
Significance\[sec:signif\]
==========================
Larger EWs require more extreme geometries, with torus opening angle $\theta \lesssim 30^\circ$ for EW$\gtrsim 2$ keV. In this sample, the galaxies that exhibit the largest EWs have concentrated circumnuclear starbursts, while in the only galaxy certainly lacking a starburst, Mrk 3, the K$\alpha$ EW is relatively small. This may be causal—the mechanical energy of a starburst, input as stellar winds and supernovae, may inflate the torus so that it covers a greater solid angle around the nucleus [e.g., @Wad02].
The compact starbursts of NGC 5135 and NGC 7130 are evident from both their vacuum ultraviolet spectra, which show absorption features formed in the winds and photospheres of massive stars, and from their optical spectra where the high-order Balmer series and He I lines are observed in absorption [@Gon98]. The nuclear spectrum of M51 reveals high-order Balmer lines in absorption, also characteristic of a young stellar population [@Hec80]. The 100-pc scale starburst of NGC 4945 is detected in Pa$\alpha$ images [@Mar00], while a 200-pc ring of H$\alpha$ and Br$\gamma$ is evidence of the young stars around the nucleus of Circinus [@Elm98; @Mai98]. The unpolarized nuclear spectrum of IRAS 09104+4109 exhibits a broad feature around $\lambda 4686$, attributable to Wolf-Rayet stars [@Tra00]. NGC 1068’s starburst is large, and this extended (kpc-scale) star-formation region [@Tel88] therefore does not shape the immediate environment of the central engine. Instead, the AGN significantly affects the conditions of the nuclear region, as the prominent ionized Fe lines illustrate in this case [@Uen94].
Most recent investigations of the significance of starbursts in active galaxies have concentrated on Type 2 AGNs [e.g., @Gon01] for the practical reason that the starburst signatures are more evident when the glare of the central engine is blocked. Our explanation for the exceptionally large K$\alpha$ EWs seen in starburst/AGN composites—very small opening angles—implies that any particular AGN with a starburst is more likely to be seen as Type 2, so few genuine starburst/Type 1 AGNs exist.
Starburst/Type 2 AGN composite galaxies are common. Approximately half of all Seyfert 2s contain circumnuclear starbursts [@Gon01]. The composite galaxies are also preferentially more obscured than their “pure” counterparts, which lack starbursts [@LWH01j]. The starbursts themselves are responsible for some of this obscuration. With typical star-formation rates per unit area $SFR \approx 50$–$100 M_\sun {\rm \, kpc^{-2} \, yr^{-1}}$ in the central 100 pc of composite galaxies [@Gon98], the correlation of @Ken98 implies the mean gas column density in the star-forming regions is about $10^{24} {\ifmmode{\rm\,cm^{-2}}\else{${\rm\,cm^{-2}}$}\fi}$. The causal connection of large column density and star formation may then work both ways to enhance EW: strong starbursts arise where a large reservoir of material is available, and the starburst may also inflate the torus. Thus, in a significant fraction of AGNs, accretion occurs behind large obscuring column densities that cover most lines of sight.
In order to reproduce the observed spectrum of the cosmic X-ray background, synthesis models [@Set89; @Com95] require a large population of obscured AGNs. An outstanding problem with these models has been the small number of obscured luminous AGNs—Type 2 quasars—that are actually identified. Most of these are radio-loud sources, which represent only a minority of AGNs [@Urr95]. Broad-band X-ray surveys are not generally an effective method for finding Type 2 quasars because even the X-ray continuum may not penetrate the enshrouding gas. For example, CDF-S 202 is the only high-luminosity obscured AGN reported in the 1-Ms exposure of the Deep Field South, and only one has been identified in the 185-ks observation of the Lynx field [@Ste02].
This sample suggests that prominent Fe K$\alpha$ (EW $> 1$ keV) is a common feature of very obscured AGNs, and the characteristic large EW can be exploited to find more of them. Specifically, we propose searching for Type 2 quasars in continuum-subtracted [*narrow-band*]{} X-ray images. At high redshift, in particular, where the host galaxy contribution is negligible, the Fe line contains nearly all the X-ray flux. Within a narrow energy range, $\Delta E \approx 300$ eV, some objects might become significant detections that would be undetectable when the background or sensitivity constraint of a wider band is included.
The Fe K$\alpha$ line is a valuable probe of buried AGNs and the material that hides them. Isolating this line emission both spectrally and spatially from additional diluting sources, these observations suggest that large EWs are common in obscured AGNs. The most extreme EWs require small torus opening angles and arise in the starburst/AGN composite galaxies.
We thank Tahir Yaqoob for essential assistance with and software for the grating spectra and thank David Strickland for providing a reduced spectrum of the NGC 4945 nucleus.
Arnaud, K. A. 1996, in ASP Conf. Ser. 101, Astronomical Data Analysis Software and Systems V, ed. G. Jacoby & J. Barnes (San Francisco: ASP), 17 Awaki, H., Koyama, K., Inoue, H., & Halpern, J. P. 1991, , 43, 195 Awaki, H., Koyama, K., Kunieda, H., & Tawara, Y. 1990, , 346, 544 Canizares, C. R., et al. 2000, , 539, L41 Comastri, A., Setti, G., Zamorani, G., & Hasinger, G. 1995, , 296, 1 Elmouttie, M., Koribalski, B., Gordon, S., Taylor, K., Houghton, S., Lavezzi, T., Haynes, R., & Jones, K. 1998, , 297, 49 Franceschini, A., Bassani, L., Cappi, M., Granato, G. L., Malaguti, G., Palazzi, E., & Persic, M. 2000, , 353, 910 Ghisellini, G., Haardt, F., & Matt, G. 1994, , 267, 743 Gonz[á]{}lez Delgado, R. M., Heckman, T. M., & Leitherer, C. 2001, , 546, 845 Gonz[á]{}lez Delgado, R. M., Heckman, T., Leitherer, C., Meurer, G., Krolik, J., Wilson, A. S., Kinney, A., & Koratkar, A. 1998, , 505, 174 Heckman, T. M., Crane, P. C., & Balick, B. 1980, , 40, 295 Iwasawa, K., Fabian, A. C., & Ettori, S. 2001, , 321, L15 Iwasawa, K., Koyama, K., Awaki, H., Kunieda, H., Makishima, K., Tsuru, T., Ohashi, T., & Nakai, N. 1993, , 409, 155 Kennicutt, R. C. 1998, , 498, 541 Koyama, K., Inoue, H., Tanaka, Y., Awaki, H., Takano, S., Ohashi, T., & Matsuoka, M. 1989, , 41, 731 Krolik, J. H., & Kallman, T. R. 1987, , 320, L5 Krolik, J. H., Madau, P., & Życki, P. T. 1994, , 420, L57 Levenson, N. A., Weaver, K. A., & Heckman, T. M. 2001, , 550, 230 Maiolino, R., Krabbe, A., Thatte, N., & Genzel, R. 1998, , 493, 650 Makishima, K., Ohashi, T., Kondo, H., Palumbo, G. G. C., & Trinchieri, G. 1990, , 365, 159 Marconi, A., Oliva, E., van der Werf, P. P., Maiolino, R., Schreier, E. J., Macchetto, F., & Moorwood, A. F. M. 2000, , 357, 24 Matt, G., et al. 1996, , 281, L69 Nandra, K., & Pounds, K. A. 1994, , 268, 405 Norman, C., et al. 2002, , 571, 218 Risaliti, G., Maiolino, R., & Salvati, M. 1999, , 522, 157 Sako, M., Kahn, S. M., Paerels, F., & Liedahl, D. A. 2000, , 543, L115 Sambruna, R. M., Netzer, H., Kaspi, S., Brandt, W. N., Chartas, G., Garmire, G. P., Nousek, J. A., & Weaver, K. A. 2001, , 546, L13 Setti, G., & Woltjer, L. 1989, , 224, L21 Stern, D., et al. 2002, , 568, 71 Telesco, C. M., & Decher, R. 1988, , 334, 573 Terashima, Y., & Wilson, A. S. 2001, , 560, 139 Tran, H. D., Cohen, M. H., & Villar-Martin, M. 2000, , 120, 562 Turner, T. J., George, I. M., Nandra, K., & Mushotzky, R. F. 1997, , 113, 23 Ueno, S., Mushotzky, R. F., Koyama, K., Iwasawa, K., Awaki, H., & Hayashi, I. 1994, , 46, L71 Urry, C. M., & Padovani, P. 1995, , 107, 803 Wada, K., & Norman, C. A. 2002, , 566, L21 Weisskopf, M. C., Tananbaum, H. D., Van Speybroeck, L. P., & O’Dell, S. L. 2000, , 4012, 2 Yaqoob, T., George, I. M., Nandra, K., Turner, T. J., Serlemitsos, P. J., & Mushotzky, R. F. 2001, , 546, 759 Young, A. J., Wilson, A. S., & Shopbell, P. L. 2001, , 556, 6
tab1
![\[fig:cartoon\] Cartoon of model geometry. The square torus cross-section is shaded, and the solid line marks the torus symmetry axis. The torus opening angle, $\theta$, and the viewing angle, $i$, are indicated along dashed and dot-dashed lines, respectively. ](f1.eps){width="6in"}
![\[fig:tau4\] EW as a function of $\theta$ and $i$, for $\tau = 4 $ and $\Gamma = 1.9$. The image is scaled linearly from 0 (white) to 10 keV (black). The dotted contour is at 100 eV, and the solid contours are scaled linearly in steps of 1 keV, beginning at 1 keV. Both $\theta$ and $i$ are indicated in degrees. ](f2.eps){width="4in"}
![\[fig:iang65\] EW as a function of $\tau$ and $\theta$, for $i = 65^\circ$. The image and contour scales are the same as in Fig. \[fig:tau4\]. ](f3.eps){width="4in"}
![\[fig:theta10\] EW as a function of $\tau$ and $i$, for $\theta = 10^\circ$. The image and contour scales are the same as in Fig. \[fig:tau4\]. ](f4.eps){width="4in"}
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[**Noisy kink in microtubules**]{}
H.C. Rosu$^{a,c}$ [^1], J.A. Tuszyński$^{b}$ [^2] and A. González$^{a}$ [^3] $^{a}$ [*Instituto de Física de la Universidad de Guanajuato, Apdo Postal E-143, Léon, Gto, México*]{}
$^{b}$ [*Department of Physics, University of Alberta, Edmonton, AB, T6G 2J1, Canada*]{}
$^{c}$ [*Institute of Gravitation and Space Sciences, Magurele-Bucharest, Romania*]{}
ABSTRACT. We study the power spectrum of a class of noise effects generated by means of a digital-like disorder in the traveling variable of the conjectured Ginzburg-Landau-Montroll kink excitations moving along the walls of the microtubules. We have found a 1/f$^{\alpha}$ noise with $\alpha \in$ (1.82-2.04) on the time scales we have considered.
PACS: 87.15.-v, 72.70+m, 87.10.+e
One can find extensive, descriptive presentations of microtubules (MTs) in many biological papers. Here, we shall give only elementary definitions as follows. They are ubiquitous protein polymers in eukaryotic cells belonging to the category of [*biological filaments*]{} and making the most part of the cytoskeleton. They are hollow cylinders 25 nm in outer diameter and 17 nm in the inner one, with lengths ranging from nm to mm in some neuronal cells. The walls of the cylinders are usually made of 13 (the seventh Fibonacci number) protofilaments laterally associated. The surface structure of MT walls is very interesting [@mand]. Structurally, MTs are quasi one-dimensional chains of tubulin polar dimers (negative $\alpha$ and positive $\beta$ monomers, each of 4 nm in length) undergoing conformational changes induced by the guanosine triphosphate to diphosphate (GTP-GDP) hydrolysis. The whole assembly process of a MT is due to the hydrolyzation of GTP. The cation Mg$^{++}$ is essential to increase the affinity of tubulin for binding GTP and thus for generating MTs. Moreover, a unique dynamical property is the so-called [*dynamical instability*]{} [@-1] which is a random growth and shrinkage of the more active plus ends of MTs. It has raised much interest in recent years [@-2].
An energy-transfer mechanism in MTs by means of Ginzburg-Landau-Montroll (GLM) kinklike protofilament excitations has been discussed by Satarić, Tuszyński and Zakula [@1] in 1993. Also sine-Gordon (SG) solitons have been discussed by Chou, Zhang and Maggiora in 1994 [@czm]. We recall that various types of solitons have found interesting applications in biological physics (Davydov’s model [@dav], DNA/RNA [@dna]). Usually, in order to get nonlinear differential equations one performs a continuum limit for some lattice models in which discreteness effects are neglected. Some authors have shown that such effects might be important and suggested various ways of including them in continuous differential equations [@discr]. The digital disorder we shall comment on and use next is just a possible way to incorporate discreteness in a moving solitonic pattern.
In 1993, Rosu and Canessa [@2] introduced a digital-like disorder in the Davydov $\beta$-kink leading to the 1/f$^{\alpha}$ noise, with $\alpha\approx1$, in the dynamics of that kink, and also commented on the multifractal features of the dynamics of the $\beta$-kink. The procedure is as follows. In the traveling variable $\xi=x-v_K t$ of the slowly moving kink (the acoustic Lorentz factor $\gamma _L \approx 1$) one puts $x_i=x_0+\Delta x_i$, where $x_0$ is the position of the center of mass of the kink, or its central position along the chain, and $\Delta x$ are small random displacements around $x_0$ (say, in the interval $\pm 1$). In the calculations, one can fix $x_0=0$. For each random position $\Delta x_i$ chosen from a uniform distribution, one calculates by means of a fast-Fourier-transform algorithm the [*noise power spectrum*]{} of the time series of the signal (considered to be the kink) in order to get the time correlations of the fluctuations of the signal, i.e., $$S_{K}(f)\propto\frac{1}{\tau}\left\langle|\int_0^{\tau}K(\xi)e^{2\pi ift}dt|^2
\right\rangle ~,
\eqno(1)$$ where $0<t<\tau\approx 1/f$ and K($\xi$) is the kink function, and the brackets stand for averaging over ensembles. This approach to noise effects has been taken from the literature on the self-organized criticality (SOC) paradigm, see, e.g., [@vm]. On the other hand, the standard treatment of noise effects when they originate in thermal fluctuations is by means of the Langevin equation method. Valls and Lust [@vl] have studied the effect of thermal noise on the front propagation in the GL case. They have found a crossover between constant-velocity propagation at early times and diffusive behavior at late times.
The purpose of the present work is to investigate the noise produced by the same type of disorder as in [@2] in the case of the GLM kink conjectured in MTs.
The main assumption in [@1] is that the assembly of tubulin dimers/dipoles (${\cal D}_n$) form a quasi one-dimensional ferrodistortive system for which the double-well on-site potential model is a standard framework $$V({\cal D}_{n})=-\frac{1}{2}A {\cal D}_{n}^{2}+\frac{1}{4}B{\cal D}_{n}^{4}~.
\eqno(2)$$ The variable ${\cal D}$ has been identified in [@1] with the amount of $\beta$-state distorsion vertically projected (the $\beta$-state is defined as having the mobile electron within the $\beta$-monomer). Moreover, in [@1] a GL hamiltonian/free-energy with intrinsic electric field and dissipation effects included led to the dimer Euler-Lagrange dimensionless equation of motion (EOM) in the traveling coordinate of the anharmonic oscillator form (with linear friction) $$%-\frac{Ma^2V_s^2}{\gamma _{L}^2}
|A|{\cal D}^{''}-\gamma \alpha v_K{\cal D}^{'}
-F({\cal D})=0~,
\eqno(3)$$ where $\gamma$ is the friction coefficient, $\alpha=|A|\gamma ^2 _L/Mv_{sound}^2$ and $F({\cal D})=A{\cal D}-B{\cal D} ^3+qE$, with $q$ denoting the effective charge of a single dimer of mass M, and $E$ the magnitude of the intrinsic electric field. This EOM is known to have a unique kink solution given by the formula [@3] $$K (\xi)=a+\frac{b-a}{1+\exp (\beta \xi)}\equiv a+
\frac{\beta}{\sqrt{2}}(1-\tanh(\beta\xi /2))~,
\eqno(4)$$ where $K={\cal D}/\sqrt{|A|/B}$ is a rescaled dipole variable, $\beta=(b-a)/\sqrt{2}$, whereas $a$ and $b$ are two of the solutions of the cubic equation $$F(K)\equiv(K -a)(K -b)(K -c)\equiv K^3 -K -\sigma=0~,
\eqno(5)$$ where $\sigma =q\frac{\sqrt{B/|A|}}{|A|}E$ and $u_0$ units are used, where $u_0=\sqrt{|A|/B}\approx 1.4\cdot 10^{-11}$ m is the amplitude of the dimer displacement (shift of the double-well potential). Notice that the GLM kink is thin. Its width is $w_K=\frac{1}{\beta}\approx 0.7 u_0$.
As we said, the type of digital disorder we consider here is very close to the ideas of the SOC paradigm that we understand in the broad sense of [*both*]{} spatial and temporal scaling of the dynamical state of the system [@soc]. The spatial scaling (self-similarity) is of the (multi)fractal type while the temporal scaling leads to $1/f^{\alpha}$ noises.
A strong motivation for dealing with digital dynamics is the possibility of generating broken symmetries and therefore of having various types of dynamical phase transitions [@ch]. Thus, digital disorder, though might look a rather ad-hoc approach, focuses on both self-organized properties of MTs, i.e., to driven steady states with long-range spatio-temporal correlations, and to (dynamical) phase transitions, since digital dynamics allows for symmetry breaking.
Our results are displayed in Figures 1 and 2 and show that the noise introduced by the digital disorder in the GLM kink variable is practically of the 1/f$^2$ (Brownian) type on the time scales we have considered. At low frequencies there is the known cross-over to a white noise due to the finite system size, which is moving to lower frequencies as the size of the system is increased [@fj]. On the other hand, the deviation from the power law at high frequencies is an artifact due to the so-called aliasing [@hjj]. The Brownian noise we have obtained is not unexpected since it is a common occurence in SOC models in any finite dimension [@cc], and only mean-field calculations reproduced the $1/f$ noise [@tb].
Finally, we would like to mention that if polarization does not exactly follow the displacement one needs a system of two coupled partial differential equations leading to two coupled traveling kink waves. Of course, the nonlinear models seem to be too simple-minded for the MT complexity. Nevertheless they provide guides for further insight and perhaps some partial answers. [**Acknowledgments**]{}
This work was partially supported by CONACyT (Mexico) and by NSERC (Canada).
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K.C. Chou, C.T. Zhang, G.M. Maggiora, Biopolymers [**34**]{} (1994) 143.
A.S. Davydov, [*Solitons in Molecular Systems*]{} (Reidel, Dordrecht, 1985).
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4ex
[\
Double logarithmic plot of the power spectrum of the digital noise perturbing the motion of the GLM kink moving along MTs at constant v$_K$ = 2m/s. The fitted slopes are as follows: (a) -2.032 (b) -2.0175 (c) -2.0164 for the time scales corresponding to that of the dimer, ten times bigger, and hundred times bigger, respectively. The errors in the slopes are at the level of 0.0003 for each case.]{}
2ex
4ex
[\
The same plot as in Fig. 1 for v$_K$ = 100m/s. The fitted slopes are (a) -2.0083 (b) -1.8641 (c) -1.8205 (d) -2.0000 for temporal scales of 10$^{-2}$, 10$^{-1}$, 10$^{0}$, and 10$^{1}$ times that of the tubulin dimer, respectively. The level of the errors is the same as in Fig. 1.]{}
[^1]: Electronic mail: [email protected]
[^2]: Electronic-mail:[email protected]
[^3]: Electronic mail: [email protected]
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[**Bethe subalgebras in affine Birman–Murakami–Wenzl algebras and flat connections for $q$-KZ equations.**]{}6
[**A.P.Isaev$^{*}$, A.N.Kirillov$^{**}$ and V.O.Tarasov$^{***}$**]{}
$^*$ Bogoliubov Laboratory of Theoretical Physics, JINR,\
141980, Dubna, Moscow region,\
and ITPM, M.V.Lomonosov Moscow State University, Russia\
E-mail: [email protected]
$^{**}$ Research Institute of Mathematical Sciences, RIMS,\
Kyoto University, Sakyo-ku, 606-8502, Japan\
[*URL: http://www.kurims.kyoto-u.ac.jp/-.05cm$\tilde{\quad}$-.17cm kirillov* ]{}\
and\
The Kavli Institute for the Physics and Mathematics of the Universe ( IPMU ),\
5-1-5 Kashiwanoha, Kashiwa, 277-8583, Japan\
and\
Department of Mathematics, National Research University Higher School of Economics,\
117312, Moscow, Vavilova str. 7, Russia
$^{***}$Department of Mathematical Sciences,\
Indiana University – Purdue University Indianapolis\
402 North Blackford St, Indianapolis, IN 46202-3216, USA\
and\
St. Petersburg Branch of Steklov Mathematical Institute Fontanka 27,\
St. Petersburg, 191023, Russia
*Dedicated to Professor Rodney Baxter on the occasion of his 75th Birthday.*
[**Abstract.**]{} Commutative sets of Jucys–Murphy elements for affine braid groups of $A^{(1)},B^{(1)},C^{(1)},D^{(1)}$ types were defined. Construction of $R$-matrix representations of the affine braid group of type $C^{(1)}$ and its distinguish commutative subgroup generated by the $C^{(1)}$-type Jucys–Murphy elements are given. We describe a general method to produce flat connections for the two-boundary quantum Knizhnik- Zamolodchikov equations as necessary conditions for Sklyanin’s type transfer matrix associated with the two-boundary multicomponent Zamolodchikov algebra to be invariant under the action of the $C^{(1)}$-type Jucys–Murphy elements. We specify our general construction to the case of the Birman–Murakami–Wenzl algebras ($BMW$ algebras for short). As an application we suggest a baxterization of the Dunkl–Cherednik elements $Y'$s in the double affine Hecke algebra of type $A$.
[**Mathematics Subject Classification (2010)**]{}. 81R50, 16T25, 20C08.
[**Key words**]{}. $C^{(1)}$-type affine braid group, Jucys–Murphy subgroup, Yang Baxter equations of types $A$ and $C$, Baxterization, affine Hecke and Birman–Murakami-Wenzl algebras, Bethe subalgebras, Gaudin models. Flat connections and two-boundary Knizhnik–Zamolodchikov equations.
Introduction
============
0
The quantum Knizhnik–Zamolodchikov equation (q-KZ equation for shot) is a system of difference equations which has been introduced by F.Smirnov [@Sm], [@Sm2], during the study of form factors of integrable models, and independently, by I. Frenkel and N.Reshetikhin, [@FreRe] during the study of the representation theory of quantum affine algebras. Since that time the literature that enter into the treatment of qKZ equations, their generalizations and applications, are enormous. We mention here only a few:
$\bullet$ [@JMN], which is concerned to the study of correlation functions of integrable systems;
$\bullet$ [@Ch], which is devoted to applications to the representation theory of affine Hecke algebras;
$\bullet$ [@KZ], [@Z], [@PZ], which are concerned to the study of variety applications to Algebraic Combinatorics and Algebraic Geometry of certain class of solutions to (boundary) q-KZ equations.
$\bullet$ [@RSV], devoted to the study of Jackson integral solutions of the boundary quantum Knizhnik-Zamolodchikov equation(s) with applications to the representation theory of quantum affine algebra $U_{q}(\widehat{{\mathfrak{sl}}(2)})$.
In the present paper we describe a general method for construction of [*two-boundary*]{} quantum KZ equations associated with affine Birman–Murakami–Wenzel algebras (BMW algebras) [@BirWen], [@Mur00], [@We], [@DRV], and give several examples to illustrate our method. The underlying idea of our construction is to describe relations/equations among the generators of the multicomponent two-boundary Zamolodchikov algebras [@GoshZam] which that the natural action of the distinguish commutative subgroup of the affine braid group $B_n(C^{(1)})$ of type $C^{(1)}$ generated by the Jucys–Murphy elements $\{ JM_{i} \},~i=1,2,\ldots, n,$ the “monodromy matrix” associated with the Zamolodchikov algebras in question, see Sections 2, 3 and 4 for details. For example, in Section 2 we describe [*distinguish*]{} commutative subgroups in the (non-twisted) affine braid groups of classical types. The generators of these distinguish subgroups will be called [*universal Jucys– Murphy elements*]{}, or $JM$-elements for short. Note that the well-known $JM$-elements in the group ring of the symmetric group [@Ju0], or (affine) Hecke [@GiNi], [@Doi], Birman–Murakami-Wenzl [@IsOg] and cyclotomic Hecke (and cyclotomic BMW) algebras, are images of the universal $JM$-elements. The main objective of our paper is to construct [*Baxterization*]{} of the $JM$-elements in the affine Birman–Murakami–Wenzl algebras of type $C^{(1)}$, i.e. to construct mutually commuting family of elements $JM_i(x) \in BMW(C^{1)}) \otimes \mathbb{Q}(x)$ depending on spectral parameter $x$, such that $JM_i(0)=JM_i,~ \forall i$.\
Now let us say few words about the content of our paper.\
As it was mentioned, in we recall definitions of [*distinguish*]{} commutative subgroups in the affine braid groups of classical types. Since the generators of these commutative subgroups are the major origin of the Jucys–Murphy elements in a big variety of algebras, we include the definitions and proofs of universal $JM$-elements basic properties.
We want to that in all known cases, such as the group ring of the symmetric groups, (affine, cyclotomic) Hecke, Brauer, $BMW$ algebras, the corresponding $JM$-elements come from the distinguish commutative subgroup in the corresponding (affine) braid group of classical type. In fact, birational representations of affine braid group associated with semisimple Lie algebras, give rise to the well-known and widely used integrable systems such as Heisenberg chains and Gaudin models, [@IsOg2], [@IK], Painlevé equations, [@No] and the literature quoted therein.\
In we describe a way how to construct $R$-matrix representations of the affine braid group $B_n(C^{(1)})$ of type $C^{(1)}$, and use these constructions to define the corresponding [*quantum $qKZ$ equations*]{} and two sets of [*flat connections*]{} associated with the former.\
contains one of our main results concerning of construction of [*flat connections*]{} based on the study of two-boundary (multi-component) Zamolodchikov algebras. Namely, $qKZ$ equations are making their appearance to ensure that the two boundary Zamolodchikov algebra in question is invariant under the action of the distinguish commutative subgroup in the corresponding affine braid group. In we present our main construction, namely that of [*flat connections*]{} for quantum Knizhnik–Zamolodchikov equations derived from the study of two boundary Zamolodchikov algebra and the $B_n(C^{(1)})$ universal Jucys–Murphy elements.
In we specify our general constructions presented in Section \[sec4\] to the case of affine $BMW$ algebras, and construct flat connections for the algebra $BMW(C^{(1)})$. To pass from general construction to the case of the affine Birman–Murakami–Wenzl algebras of type $C^{(1)}$, we rely on the use of embedding the braid group $B_n(C^{(1)})$ into the algebra $BMW(C^{(1)})$.
In we construct [*baxterized Jucys–Murphy elements*]{} in the affine $BMW$ algebras. Our approach is based on Sklyanin’s transfer matrix method [^1], [@Skl],[@Skly]. The key to apply the Sklyanin transfer matrix method to construction of [*baxterized $JM$-elements $\bar{y}_n(x;\vec{z}_{(n)})$*]{}, see (\[xxz55\]), in the fact that the family of algebras $\{ BMW_n(C) \}_{n \ge 1}$ can be provided with the [*Markov trace*]{}, namely, there exists a unique homomorphism $$Tr_{n+1} : BMW_{n+1}(C) \longrightarrow BMW_n(C),~~\forall n \ge 1$$ which satisfies a set of “good” properties, stated in Proposition \[sec6\].2 (cf [@Jo], [@Jones], [@IsOg2], [@Co2]). Let’s point out here on another important fact that the Jucys–Murphy element $y_n(x)$ satisfies the reflection equation (\[reflH\]). We also introduce a family of mutually commuting elements $\tau_n(x;;\vec{z}_(n)) \in BMW_n(C)$, the so-called [*dressing $JM$-operators*]{} which are an analogue of the Sklyanin transfer matrices [@Skl], and the coefficients in the expansion of $\tau_n(x;;\vec{z}_{(n)})$ over the variable $x$ (for the homogeneous case $z_i =1, \forall$ ) are the Hamiltonians for the open Birman–Murakami–Wenzl chain models with nontrivial boundary conditions, see e.g. [@IsOg2], and example at the end of Section \[sec6\].1. is devoted to construction of the Bethe subalgebras in the affine $BMW_n(C)$ algebras and a factorizibility property of the corresponding $qKZ$ connections. We will show that the flat connections ${\sf A}_{i}'(z)$, see (\[Bax05\]), are images under the map (\[Zam10K\]) of certain elements ${\sf J}_i \in B_n(C)$ which under the special limit (\[Zam77\]) one can deduce the $BMW$ analog (\[qkz03\]) of the Cherednik’s connections have been introduced in [@Ch] for Hecke algebras. As an application, in Section \[sec6\] we construct a [*baxterization*]{} of the type $A$ Dunkl–Cherednik elements $Y_i \in DAHA$, which have been in-depth studied in [@Ch].
Affine braid groups of type $A^{(1)},B^{(1)},C^{(1)},D^{(1)}$ and Jucys–Murphy elements\[ABG\]
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0
First consider affine braid group $B_n(C^{(1)})$ with generators $\{ T_0, \dots , T_n \}$ subject to defining relations T\_i T\_[i+1]{} T\_i = T\_[i+1]{} T\_[i]{} T\_[i+1]{} , i = 1, …, n-2 ,
[c]{} T\_1 T\_0 T\_1 T\_0 = T\_0 T\_1 T\_0 T\_1 ,\
\[0.2cm\] T\_[n-1]{} T\_n T\_[n-1]{} T\_n = T\_n T\_[n-1]{} T\_n T\_[n-1]{} ,
where $T_0,T_n$ — two affine generators. Let $||m_{ij}||$ be symmetric matrix with integer coefficients $m_{ij} \geq 2$. The structure relations (\[Affbg\]), (\[Affbg2\]) of the group $B_n(C^{(1)})$ can be written as $\underbrace{T_i \, T_{j} \, T_i \cdots}_{m_{ij}}
= \underbrace{T_{j} \, T_{i} \, T_{j} \cdots}_{m_{ji}}$ and correspond to the Coxeter graph of the type $C^{(1)}$
(17,4.5)
(2,2) (1.5,2.5)[$T_0$]{} (4.5,2) (4.5,2.5)[$T_1$]{} (2.2,1.9)[(1,0)[2.1]{}]{} (2.2,2.1)[(1,0)[2.1]{}]{} (4.7,2)[(1,0)[2]{}]{} (7,2)[$. \; . \; . \; . \; . \; . \; .$]{}
(10.5,2) (10.5,2.5)[$T_{n-2}$]{} (10.7,2)[(1,0)[2]{}]{} (13,2) (12.7,2.5)[$T_{n-1}$]{} (15.5,2) (15.5,2.5)[$T_n$]{} (13.2,1.9)[(1,0)[2.1]{}]{} (13.2,2.1)[(1,0)[2.1]{}]{}
where the number of lines between nodes $i$ and $j$ is equal to $(m_{ij}-2)$. Note that for the group $B_n(C^{(1)})$ defined by (\[Affbg\]), (\[Affbg2\]) we have two automorphisms $\rho_1$ and $\rho_2$: \_1(T\_i) = T\_i\^[-1]{} , \_2(T\_i) = T\_[n-i]{} .
The well known statement is:
[**Proposition \[ABG\].1.**]{}
*The affine braid group $B_n(C^{(1)})$ contains the commutative subgroups which are generated by the following sets of elements*
[l]{} J\_i= ( T\_[i-1]{}\^[-1]{} T\_[1]{}\^[-1]{}) (T\_0 T\_n) (T\_[n-1]{} T\_[i]{}) , i=1,…,n ,\
\[0.2cm\] \_i= (T\_[i-1]{} T\_[1]{}) (T\_0 T\_[n]{}) (T\_[n-1]{}\^[-1]{} T\_[i]{}\^[-1]{}) , i=1, …, n ,\
\[0.2cm\] (Jucys-Murphy elements) a\_[i]{}:= (T\_[i-1]{} T\_[1]{}) T\_0 (T\_[1]{} T\_[i-1]{}) , i=1,…,n ,\
\[0.2cm\] (Jucys-Murphy elements) b\_[i]{} := (T\_[i]{} T\_[n-1]{}) T\_n (T\_[n-1]{} T\_[i]{}) i=1, …, n .
[**Proof.**]{} The proof of commutativity of the elements $a_i$ is straightforward and follows from the fact that $[a_i, \, T_j]=0$ for $i > j$. The commutativity of the elements $b_i$ follows from the commutativity of elements $a_i$ since we have $b_{n-i+1} = \rho_2(a_i)$, where the automorphism $\rho_2$ is defined in (\[autom\]).
Now we prove the commutativity of the elements $J_i$ (it will be important for our consideration below). We introduce the element X = \_[k=0]{}\^[n]{} T\_k = T\_0 T\_[n]{} . For this element we have the following identities
[c]{} X T\_i = T\_[i+1]{} X , (i=1,…, n-2) ,\
\[0.2cm\] T\_1 X\^2 = T\_1 T\_0 T\_1 T\_0 (T\_2 T\_1) (T\_3 T\_2) (T\_[n-1]{} T\_[n-2]{}) T\_n T\_[n-1]{} T\_n = X\^2 T\_[n-1]{} ,
where in the proof of these identities we have used (\[Affbg\]), (\[Affbg2\]). With the help of the operator $X$ (\[XX01\]) the element $\overline{J}_k$ (\[jucys1\]) can be written as $$\overline{J}_k =
T_{k-1} \cdots T_1 \cdot X
\cdot T^{-1}_{n-1} \cdots T_k^{-1} =
T_{k-1} \cdots T_2 \cdot X
\cdot T_{n}^{-1} T^{-1}_{n-1} \cdots T_k^{-1} \; .$$ Let $k > r$. Then by using (\[Affbg\]), (\[Affbg2\]) and (\[XX02\]) we have $$\begin{array}{c}
\overline{J}_k \; \overline{J}_r = (T_{k-1} \cdots T_1 \cdot X
\cdot T^{-1}_{n-1} \cdots T_k^{-1}) \cdot
(T_{r-1} \cdots T_1 \cdot X
\cdot T^{-1}_{n-1} \cdots T_r^{-1}) = \\ [0.2cm]
= (T_{k-1} \cdots T_1) \cdot X
\cdot (T_{r-1} \cdots T_1) \cdot (T^{-1}_{n-1} \cdots T_k^{-1}) \cdot
X \cdot (T^{-1}_{n-1} \cdots T_r^{-1}) = \\ [0.2cm]
= (T_{k-1} \cdots T_1) \cdot (T_{r} \cdots T_2) \cdot X
\cdot X \cdot
(T^{-1}_{n-2} \cdots T_{k-1}^{-1}) \cdot (T^{-1}_{n-1} \cdots T_r^{-1}) =
\\ [0.2cm]
= (T_{r-1} \cdots T_1) \cdot (T_{k-1} \cdots T_1) \cdot X^2
\cdot (T^{-1}_{n-1} \cdots T_r^{-1}) \cdot
(T^{-1}_{n-1} \cdots T_{k}^{-1}) = \\ [0.2cm]
= (T_{r-1} \cdots T_1) \cdot (T_{k-1} \cdots T_2) \cdot X^2
\cdot T_{n-1} \cdot (T^{-1}_{n-1} \cdots T_r^{-1}) \cdot
(T^{-1}_{n-1} \cdots T_{k}^{-1}) = \\ [0.2cm]
= (T_{r-1} \cdots T_1) \cdot X \cdot (T_{k-2} \cdots T_1)
\cdot (T^{-1}_{n-1} \cdots T_{r+1}^{-1}) \cdot X
\cdot (T^{-1}_{n-1} \cdots T_{k}^{-1}) =
\\ [0.2cm]
= (T_{r-1} \cdots T_1) \cdot X
\cdot (T^{-1}_{n-1} \cdots T_{r}^{-1}) \cdot (T_{k-1} \cdots T_1)
\cdot X \cdot (T^{-1}_{n-1} \cdots T_{k}^{-1})
= \overline{J}_r \; \overline{J}_k \; ,
\end{array}$$ where to obtain the last line we use the identity $(k > r)$ $$\begin{array}{c}
(T_{k-2} \cdots T_1)
\cdot (T^{-1}_{n-1} \cdots T_{r+1}^{-1}) =
(T_{k-2} \cdots T_r) \cdot (T_{r-1} \cdots T_1) \cdot
(T^{-1}_{n-1} \cdots T_{k}^{-1}) (T^{-1}_{k-1} \cdots T_{r+1}^{-1})
= \\ [0.2cm]
= (T^{-1}_{n-1} \cdots T_{k}^{-1}) (T_{k-2} \cdots T_r)
\cdot (T^{-1}_{k-1} \cdots T_{r+1}^{-1}) (T_{r-1} \cdots T_1) = \\ [0.2cm]
= (T^{-1}_{n-1} \cdots T_{k}^{-1} T_{k-1}^{-1})
(T_{k-1} T_{k-2} \cdots T_r)
\cdot (T^{-1}_{k-1} \cdots T_{r+1}^{-1}) (T_{r-1} \cdots T_1) = \\ [0.2cm]
= (T^{-1}_{n-1} \cdots T_{k-1}^{-1})
(T^{-1}_{k-2} \cdots T_{r}^{-1}) \cdot
(T_{k-1} \cdots T_r)
(T_{r-1} \cdots T_1) =
(T^{-1}_{n-1} \cdots T_{r}^{-1}) \cdot (T_{k-1} \cdots T_1) \; .
\end{array}$$
The commutativity of the elements $J_i$ follows from the commutativity of the elements $\overline{J}_i$ since we have $\rho_1(\rho_2(\overline{J}_{n-i+1})) = J_i^{-1}$, where automorphisms $\rho_1$ and $\rho_2$ are defined in (\[autom\]).
The quotient of the group $B_n(C^{(1)})$ by additional relations $T_i^2 =1$ $(\forall i)$ is called Coxeter group of the type $C^{(1)}$. This group is denoted as $W_n(C^{(1)})$. At the end of this Section we present the explicit realization of $W_n(C^{(1)})$ which we use below. Introduce the set of spectral parameters $(z_1 , \dots , z_n)$, $z_i \in \mathbb{C}$. Now we define a representation ${\sf s}: \; T_i \to s_i$ of $B_{n}$:
[c]{} s\_i : (z\_1, …, z\_i, z\_[i+1]{} , …, z\_n) (z\_1, …, z\_[i+1]{}, z\_[i]{} , …, z\_n) (i=1,…,n-1) ,\
\[0.2cm\] s\_0 : (z\_1,z\_2, …, z\_n) ((z\_1),z\_2, …, z\_n) ,\
\[0.2cm\] s\_n : (z\_1, …, z\_[n-1]{} , z\_n) (z\_1, …, z\_[n-1]{} , |(z\_n) ) ,
where $\sigma$, $\bar{\sigma}$ are two involutive mappings $\mathbb{C} \to \mathbb{C}$ such that $(\sigma)^2=1$, $(\bar{\sigma})^2 =1$. We specify these involutions in next Sections. From (\[Affbg1\]) one can check that operators $s_0,s_i,s_n$ satisfy (\[Affbg\]), (\[Affbg2\]) and moreover we have $s_0^2 = s_n^2 =s_i^2=1$. Thus, equations (\[Affbg1\]) give the representation of the Coxeter group $W_n(C^{(1)})$. For special choices of $\sigma$ and $\bar{\sigma}$, namely $\sigma(z) = 1-z$ and $\bar{\sigma}(z) =-z$, the representation (\[Affbg1\]) have been used in [@Ch],[@Stok].
[**Remark 1.**]{} Denote by $B_n(C)$ the subgroup of the affine braid group $B_n(C^{(1)})$ generated by elements $T_i$ $(i=0,\dots, n-1)$ with defining relations given in (\[Affbg\]) and in first line of (\[Affbg2\]). The group $B_n(C)$ is associated to the Coxeter graph of $C$-type
(17,4.5)
(2,2) (1.5,2.5)[$T_0$]{} (4.5,2) (4.5,2.5)[$T_1$]{} (2.2,1.9)[(1,0)[2.1]{}]{} (2.2,2.1)[(1,0)[2.1]{}]{} (4.7,2)[(1,0)[2]{}]{} (7,2)[$. \; . \; . \; . \; . \; . \; .$]{}
(10.5,2) (10.5,2.5)[$T_{n-2}$]{} (10.7,2)[(1,0)[2.1]{}]{} (13,2) (12.7,2.5)[$T_{n-1}$]{}
Consider the homomorphism (projection) $\rho$: $B_n(C^{(1)}) \to B_n(C)$ such that $\rho(T_i) = T_i$ $(i=0,\dots, n-1)$ and $\rho(T_n) =1$. It is clear that under this projection we have $a_i = \rho(\overline{J}_i)$ and it means that the commutativity of $a_i$ follows from the commutativity of $\overline{J}_i$. The elements $a_i$ given in (\[jucys1\]) generate the commutative set in the subgroup $B_n(C) \subset B_n(C^{(1)})$.
[**Remark 2.**]{} Denote by $B_n(A^{(1)})$ the affine braid group which corresponds to the affine $A$-type Coxeter graph
(17,3) (1.9,1.1) (1.7,0.5)[$T'_1$]{} (2,1.1)[(1,0)[1]{}]{} (3.1,1.1) (2.9,0.5)[$T'_2$]{} (3.2,1.1)[(1,0)[1]{}]{} (4.3,1.1) (4.1,0.5)[$T'_3$]{} (4.4,1.1)[(1,0)[1]{}]{} (6,1.1)[. . . . . . . . . .]{} (10,1.1)[(1,0)[1]{}]{} (11.1,1.1) (10.7,0.5)[. . .]{} (11.2,1.1)[(1,0)[1]{}]{} (12.3,1.1) (12,0.5)[$T'_{n-1}$]{}
(2,1.1)[(4,1)[5]{}]{} (7.25,2.4)[(4,-1)[5]{}]{} (7.18,2.4) (7.5,2.5)[$T'_n$]{}
We call group $B_n(A^{(1)})$ $(n>2)$ a periodic $A$-type braid group. This group is generated by invertible elements $T'_i$ $(i=1,\dots,n)$ and according to its Coxeter graph we have the defining relations T’\_i T’\_[i+1]{} T’\_i = T’\_[i+1]{} T’\_[i]{} T’\_[i+1]{} , i = 1, …, n ,\
\[0.2cm\] where we impose the periodic conditions $T'_{i +n}=T'_i$.
Note that the group $B_n(A^{(1)})$ possesses automorphisms \_3( T’\_i) = T’\_[i+1]{} , \_4( T’\_i) = T’\_[n-i+1]{} , \_5( T’\_i) = T\_[i]{}\^[-1]{} . Define the extension $\bar{B}_n(A^{(1)})$ of the group $B_n(A^{(1)})$ by adding an additional generator $\bar{X}$ with defining relations (cf. (\[XX02\])) |[X]{} T’\_i = T’\_[i+1]{} |[X]{} (i=1,…, n) T’\_1 |[X]{}\^2 = |[X]{}\^2 T’\_[n-1]{} . Namely, we add operator $\bar{X}$ which serves the automorphism $\rho_3$: $\rho_3(T'_i) = \bar{X} \, T'_i \bar{X}^{-1}$ in (\[perBn1\]). Then for the group $\bar{B}_n(A^{(1)})$ one can construct the following commuting sets of elements
[c]{} J’\_k = T\^[-1]{}\_[k-1]{} T\^[-1]{}\_1 |[X]{} T’\_[n-1]{} T’\_k (k=1,…,n) ,\
\[0.2cm\] |[J]{}’\_k = \_5(J’\_k) = T’\_[k-1]{} T’\_1 |[X]{} T\^[-1]{}\_[n-1]{} T\_k\^[-1]{} (k=1,…,n) ,
where we have defined $\rho_5(\bar{X}) = \bar{X}$ (this is compatible with (\[perBn2\])).
Now we introduce the element $\bar{T}_n$ in $B_n(C^{(1)})$ as following |[T]{}\_n := X\^[-1]{} T\_1 X = X T\_[n-1]{} X\^[-1]{} B\_n(C\^[(1)]{}) , where $X$ is given in (\[XX01\]). The element (\[XX03\]) satisfies periodic braid relations $$\bar{T}_n \, T_{n-1} \, \bar{T}_n = T_{n-1} \, \bar{T}_n \, T_{n-1} \; , \;\;\;
\bar{T}_n \, T_{1} \, \bar{T}_n = T_{1} \, \bar{T}_n \, T_{1} \; ,$$ where we have used (\[XX02\]). Thus, we have the homomorphic maps (embeddings) $\rho'$: $B_n(A^{(1)}) \to B_n(C^{(1)})$ and $\rho^{\prime \prime}$: $\bar{B}_n(A^{(1)}) \to B_n(C^{(1)})$ such that $$\begin{array}{c}
\rho'(T'_i) = T_i \;\;\;\; (i=1,\dots,n-1) \; , \;\;\;\;\;
\rho'(T'_n) = \bar{T}_n \; , \\ [0.2cm]
\rho^{\prime \prime}(T'_i) = T_i \;\;\;\;
(i=1,\dots,n-1) \; , \;\;\;\;\;
\rho^{\prime \prime}(T'_n) = \bar{T}_n
\; , \;\;\; \rho^{\prime \prime}(\bar{X}) = X \; .
\end{array}$$ It means that $B_n(A^{(1)})$ and $\bar{B}_n(A^{(1)})$ are subgroups in $B_n(C^{(1)})$ with generators $(T_1,\dots,T_{n-1},\bar{T}_n)$ and $(T_1,\dots,T_{n-1},\bar{T}_n, X)$, respectively.
[**Remark 3.**]{} Consider the braid group $B_{n+1}(B^{(1)})$ which is associated to the graph
(17,4.5)
(2,3.2) (2,0.8) (0.9,3.4)[$T_0$]{} (0.9,0.2)[$T_{-1}$]{} (4.5,2) (4.5,2.5)[$T_1$]{} (2.2,3.1)[(2,-1)[2.1]{}]{} (2.2,0.9)[(2,1)[2.1]{}]{} (4.7,2)[(1,0)[2]{}]{} (7,2)[$. \; . \; . \; . \; . \; . \; .$]{}
(10.5,2) (10.5,2.5)[$T_{n-2}$]{} (10.7,2)[(1,0)[2.1]{}]{} (13,2) (12.7,2.5)[$T_{n-1}$]{} (15.5,2) (15.5,2.5)[$T_n$]{} (13.2,1.9)[(1,0)[2.1]{}]{} (13.2,2.1)[(1,0)[2.1]{}]{}
The defining relations for this group are
[c]{} T\_i T\_[i+1]{} T\_i = T\_[i+1]{} T\_[i]{} T\_[i+1]{} , i = 0,1, …, n-1 ,\
\[0.2cm\] T\_[-1]{} T\_1 T\_[-1]{} = T\_1 T\_[-1]{} T\_1 , T\_[-1]{} T\_0 = T\_0 T\_[-1]{} ,\
\[0.2cm\] T\_[n-1]{} T\_n T\_[n-1]{} T\_n = T\_n T\_[n-1]{} T\_n T\_[n-1]{} .
Introduce the element \_0 = T\_[-1]{} T\_0 , which in view of (\[Affbd\]) satisfies relation \_0 T\_1 \_0 T\_1 = T\_1 \_0 T\_1 \_0 So, $B_{n}(C^{(1)})$ is a subgroup in $B_{n+1}(B^{(1)})$ and we have the homomorphism (embedding) $\tilde{\rho}$: $B_{n}(C^{(1)})\to B_{n+1}(B^{(1)})$ which is defined by the map : T\_0 \_0 , T\_i T\_i (i=1,…,n) . Thus, according to the Proposition \[ABG\].1 we have the following commuting sets for the group $B_{n+1}(B^{(1)})$
[l]{} \_i= (\_[k=i-1]{}\^1 T\_[k]{}\^[-1]{} ) ( \_[k=n-1]{}\^i T\_[k]{} ) (i=1,…,n) ,\
\[0.2cm\] \_i= ( \_[k=i-1]{}\^1 T\_[k]{} ) ( \_[k=n-1]{}\^i T\_[k]{}\^[-1]{} ) (i=1, …, n) ,
where $\tilde{X}=\tilde{T}_0 \, T_1 \cdots T_n$ is the image of the element $X \in B_{n+1}(C^{(1)})$ presented in (\[XX01\]).
[**Remark 4.**]{} The braid group $B_{n+2}(D^{(1)})$ which is associated with the graph
(17,4.5)
(2,3.2) (2,0.8) (0.9,3.4)[$T_0$]{} (0.9,0.2)[$T_{-1}$]{} (4.5,2) (4.5,2.5)[$T_1$]{} (2.2,3.1)[(2,-1)[2.1]{}]{} (2.2,0.9)[(2,1)[2.1]{}]{} (4.7,2)[(1,0)[2]{}]{} (7,2)[$. \; . \; . \; . \; . \; . \; .$]{}
(10.5,2) (9.6,2.5)[$T_{n-2}$]{} (10.7,2)[(1,0)[2.1]{}]{} (13,2) (11.9,2.5)[$T_{n-1}$]{} (13.2,2.1)[(2,1)[2.1]{}]{} (13.2,1.9)[(2,-1)[2.1]{}]{} (15.9,3.4)[$T_n$]{} (15.9,0.2)[$T_{n+1}$]{} (15.5,3.2) (15.5,0.8)
has defining relations
[c]{} T\_i T\_[i+1]{} T\_i = T\_[i+1]{} T\_[i]{} T\_[i+1]{} , i = 0,1, …, n ,\
\[0.2cm\] T\_[-1]{} T\_1 T\_[-1]{} = T\_1 T\_[-1]{} T\_1 , T\_[-1]{} T\_0 = T\_0 T\_[-1]{} ,\
\[0.2cm\] T\_[n-1]{} T\_[n+1]{} T\_[n-1]{} = T\_[n+1]{} T\_[n-1]{} T\_[n+1]{} , T\_[n]{} T\_[n+1]{} = T\_[n+1]{} T\_[n]{} .
Note that the element $\tilde{T}_n = T_n \, T_{n+1}$ obeys relations $$\tilde{T}_n \, T_{n-1} \, \tilde{T}_n \, T_{n-1} =
T_{n-1} \, \tilde{T}_n \, T_{n-1} \, \tilde{T}_n \; .$$ Thus the elements $(T_{-1},T_0,T_1,\dots,T_{n-1},\tilde{T}_n)$ generate the subgroup $B_{n+1}(B^{(1)})$ in $B_{n+2}(D^{(1)})$ and we have the homomorphism (embedding) $\rho_0:$ $B_{n+1}(B^{(1)}) \to B_{n+2}(D^{(1)})$ such that \_0: T\_i T\_i (i=-1,0,1,…,n-1) , \_0: T\_n \_n . Define the element (cf. (\[XX01\])) $$X^{\prime \prime} = \tilde{T}_0 \, T_1 \cdots T_{n-1} \,
\tilde{T}_n \; ,$$ where $\tilde{T}_0$ is defined as in (\[tilT0\]). Then we again have two sets of commuting elements (cf. (\[jucys1\]), (\[jucys11\]))
[l]{} J\_i\^ = (\_[k=i-1]{}\^1 T\_[k]{}\^[-1]{} ) X\^ ( \_[k=n-1]{}\^i T\_[k]{} ) (i=1,…,n) ,\
\[0.2cm\] \_i\^= ( \_[k=i-1]{}\^1 T\_[k]{} ) X\^ ( \_[k=n-1]{}\^i T\_[k]{}\^[-1]{} ) (i=1, …, n) .
Finally we stress that the quotient of the group $B_{n+2}(D^{(1)})$ with respect to the relations $T_0 = T_{-1}$ (or $T_n = T_{n+1}$) is isomorphic to the braid group $B_{n+2}(D)$ associated to the Coxeter graph of classical $D$-type. The commutative elements in this case are given by the same formulas as in (\[jucys11\]), where instead of $\tilde{X}$ we have to substitute element $X(D) = T_0^2 T_1 \cdots T_{n-1} \tilde{T}_n$ (or $X(D) = \tilde{T}_0 \, T_1 \cdots T_{n-1} T_n^2$).
General picture
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#### 1. Affine root systems and affine Weyl groups (see [@C1 Section 1]).
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Let $R_n$ be a root system of type $A_n,B_n,\allowbreak\ldots,\allowbreak F_n,G_n$. We will write $R$ also for the type of the root system. Let $\alpha_1,\allowbreak\ldots,\allowbreak\alpha_n\in R_n$ be simple roots, $\omega^\vee_1,\allowbreak\ldots,\allowbreak \omega^\vee_n$ — fundamental coweights, $(\omega^\vee_i,\alpha_i)=\delta_i^j$, $\theta$ — the maximal root. The Dynkin diagram of the affine root system $R_n^{(1)}\!$ is obtained by adding the root $-\,\theta$ to the simple roots $\alpha_1,\allowbreak\ldots,\allowbreak\alpha_n$. The affine simple root is $\alpha_0=[-\,\theta,1]$ in the notation of [@C1].
For $\alpha\in R_n$, denote $\alpha^\vee=2\alpha/(\alpha,\alpha)$. Let $Q^\vee\!=\bigoplus_{i=1}^n\mathbb{Z}\alpha^\vee_i$ be the coroot lattice, $P^\vee\!=\bigoplus_{i=1}^n \mathbb{Z}\,\omega^\vee_i$ be the coweight lattice, and $P^\vee_+\!=\bigoplus_{i=1}^n \mathbb{Z}_{\ge0}\omega^\vee_i$
Let $s_\alpha$ be the reflection corresponding to a root $\alpha\in R_n^{(1)}\!$, and $s_i=s_{\alpha_i}$. The Weyl group $W$ of type $R_n$ is generated by the reflections $s_1,\allowbreak\ldots,\allowbreak s_n$.
The affine Weyl group $W^{(a)}$ of type $R_n^{(1)}\!$ is generated by the reflections $s_0,s_1,\allowbreak\ldots,\allowbreak s_n$ and is isomorphic to the semidirect product $W\ltimes Q^\vee$, with $s_0=\theta^\vee s_\theta$. Here we identify $W$ and $Q^\vee\!$ with the respective subgroups of $W^a\!$.
The extended affine Weyl group $W^{(b)}$ of type $R_n^{(1)}\!$ is the semidirect product $\,\widetilde{W}=W\ltimes P^\vee$. It is also isomorphic to the semidirect product $\Pi\ltimes W^a$, where $\Pi=P^\vee/Q^\vee$. The elements of the subgroup $\Pi\subset\widetilde{W}$ “permute” the reflections $s_0,\allowbreak\ldots,\allowbreak s_n$ — for any $i$ and $\pi\in\Pi$, $\,\pi s_i\pi^{-1}=s_j$ for some $j=\pi[i]$.
Define the length on $\widetilde{W}$ by $\ell(s_i)=1$ and $\ell(\pi)=0$ for $\pi\in\Pi$. Then for $b,b'\!\in P^\vee_+\!\subset\widetilde{W}$, \[ellb\] (b+b’)=(b)+(b’), see [@C1 Proposition 1.4].
The affine braid group $B(R_n^{(1)})$ is generated by the elements $S_0,\allowbreak\ldots,\allowbreak S_n$ subject to the same braid relations as $s_0,\allowbreak\ldots,\allowbreak s_n$ (we use $S_i$ to keep distinction from the generators $T_i$ in Section \[ABG\].) The extended affine braid group $\widetilde{B}(R_n^{(1)})$ is the semidirect product $\Pi\ltimes B(R_n^{(1)})$ — for any $i$ and $\pi\in\Pi$, $\,\pi S_i\pi^{-1}=S_{\pi[i]}$, (cf. with relations (i), (ii) in [@C1 Definition 3.1]).
For $\widetilde{w}\in\widetilde{W}$ with a reduced decomposition $\widetilde{w}=\pi s_{i_1}\dots s_{i_k}$, $\pi\in\Pi$, $k=\ell(\widetilde{w})$, the element $S_{\widetilde{w}}=\pi S_{i_1}\dots S_{i_k}\in\widetilde{W}$ does not depend on the reduced decomposition, and $S_{\widetilde{w}\widetilde{w}'}=S_{\widetilde{w}}S_{\widetilde{w}'}$ provided $\ell(\widetilde{w}\widetilde{w}')=\ell(\widetilde{w})+\ell(\widetilde{w}')$, $\,\widetilde{w},\widetilde{w}'\in\widetilde{W}$. Hence, the elements $S_b$, $b\in P^\vee_+\!\subset\widetilde{W}$ generate a commutative subgroup of $\widetilde{W}$ because $S_bS_{b'}=S_{b+b'}=S_{b'}S_b$ for any $b,b'\!\in P^\vee_+\!\subset\widetilde{W}$, see . (Cf. with [@C1 formula (3.8)].)
For fundamental coweights $\omega^\vee_1,\allowbreak\ldots,\allowbreak \omega^\vee_n$, set \[Yi\] Y\_i=S\_[\^\_i]{},i=1,…,n. The elements $Y_1,\allowbreak\ldots,\allowbreak Y_n\in\widetilde{W}$ pairwise commute.
#### 2. Groups $\widehat{B}(C_n^{(1)})$ and $\widetilde{B}(C_n^{(1)})$.
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The group $\widehat{B}(C_n^{(1)})$ is generated by the elements $T_0,\allowbreak\ldots,\allowbreak T_n\in B(C_n^{(1)})$, see (\[Affbg\]), (\[Affbg2\]) and by the element $U$ with relations \[UT\] UT\_iU\^[-1]{}=T\_[n-i]{},i=0,…,n. In other words, $UGU^{-1}=\rho_2(G)$ for any $G\in B(C_n^{(1)})$, where $\rho_2$ is given by formula (\[autom\]). The element $U^2$ is central.
Set $I_i\,=\,J_1\dots J_i$, $i=1,\allowbreak\ldots,\allowbreak n$, where $J_1,\allowbreak\ldots,\allowbreak J_n$ are given by (\[jucys1\]). Also, \[Ii\] I\_i=(X T\_[n-1]{}…T\_i)\^[i]{},i=1,…,n, where $X=T_0\dots T_n$, see (\[XX01\]). Let \[Zn\] Z=T\_0…T\_[n-1]{}T\_0…T\_[n-2]{}…T\_0T\_1T\_0U. The element $Z$ commutes with $T_1,\allowbreak\ldots,\allowbreak T_{n-1}$ and $X$, and hence by , commutes with $I_1,\allowbreak\ldots,\allowbreak I_n$. Moreover, $Z^2=I_n U^2$. One more nice formula \[Io\] I\_i= X\^[i]{}T\_[n-i]{}…T\_1T\_[n-i+1]{}…T\_2…T\_[n-1]{}…T\_i.
The group $\widetilde{B}(C_n^{(1)})$ is the quotient of $\widehat{B}(C_n^{(1)})$ by relation $U^2=1$. The identification is $S_i=T_i$, $i=0,\allowbreak\ldots,\allowbreak n$, and $\Pi=\{1,U\}$. Also, $Y_i=I_i$, $i=1,\allowbreak\ldots,\allowbreak n-1$, and $Y_n=Z$.
#### 3. Groups $B(B_n^{(1)})$ and $\widetilde{B}(B_n^{(1)})$.
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The group $\widetilde{B}(B_n^{(1)})$ is the quotient of $B(C_n^{(1)})$ by relation $T_0^2=1$. The identification is $S_i=T_i$, $i=1,\allowbreak\ldots,\allowbreak n$, $S_0=T_0T_1T_0$, and $\Pi=\{1,T_0\}$. Thus $S_0S_1=S_1S_0$ and $S_0S_2S_0=S_2S_0S_2$. Also $Y_i=I_i$, $i=1,\allowbreak\ldots,\allowbreak n$. The commutative subgroup in $B(B_n^{(1)})$ is generated by the products $J_1J_i=I_1\,I_i\,I_{i-1}^{-1}$, $i=1,\allowbreak\ldots,\allowbreak n$. Here $I_0=1$.
The relation with elements (\[jucys11\]), (\[jucys5\]) is explained farther.
#### 4. Groups $B(D_n^{(1)})$ and $\widetilde{B}(D_n^{(1)})$.
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The groups $B(D_n^{(1)})$ and $\widetilde{B}(D_n^{(1)})$ are subquotients of $\widetilde{B}(C_n^{(1)})$. Let $\widetilde{B}'(C_n^{(1)})$ be the quotient of $\widetilde{B}(C_n^{(1)})$ by relations $T_0^2=1$, $T_n^2=1$. (Recall that $U^2=1$ in $\widetilde{B}(C_n^{(1)})$.) The subgroup $B(D_n^{(1)})\subset\widetilde{B}'(C_n^{(1)})$ is generated by $S_0=T_0T_1T_0$, $S_n=T_nT_{n-1}T_n$, and $S_i=T_i$, $i=1,\allowbreak\ldots,\allowbreak n-1$.
Let $\Pi_n=\{1,T_0U,(T_0U)^2,(T_0U)^3\}=
\{1,T_0U,T_0T_n,T_nU\}$ if $n$ is odd, and $\Pi_n=\{1,T_0T_n,U,T_0T_nU\}$ if $n$ is even. The subgroup $\widetilde{B}(D_n^{(1)})\subset\widetilde{B}'(C_n^{(1)})$ is generated by $B(D_n^{(1)})$ and $\Pi=\Pi_n$.
Also $Y_i=I_i$, $i=1,\allowbreak\ldots,\allowbreak n-2$, $\,Y_{n-1}=I_{n-1}Z^{-1}$ and $Y_n=Z$. The subgroup $\Pi_n$ can be recovered from the requirement that $I_1$ and $Z$ belong to the subgroup generated by $B(D_n^{(1)})$ and $\Pi=\Pi_n$.
The commutative subgroup in $B(D_n^{(1)})$ is generated by the products $J_1J_i=I_1\,I_i\,I_{i-1}^{-1}$, $i=1,\allowbreak\ldots,\allowbreak n$. Here $I_0=1$.
#### 5. Recursive definition of $I_1,\allowbreak\ldots,\allowbreak I_n$.
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Let $T'_0,\allowbreak\ldots,\allowbreak T'_{n-1}$ denote the generators of $B(C_{n-1}^{(1)})$, and similarly for $I'_1,\allowbreak\ldots,\allowbreak I'_{n-1}$. There is an embedding \[mu\] :B(C\_[n-1]{}\^[(1)]{})B(C\_n\^[(1)]{}), $$\mu(T'_i)\,=\,T_i\,,\quad i=0,\allowbreak\ldots,\allowbreak n-2\,,\qquad
\mu(T'_{n-1})\,=\,T_{n-1}T_nT_{n-1}\,.$$ Then \[muI\] (I’\_i)=I\_i,i=1,…,n-1. This suggests a proof that the elements $I_1,\allowbreak\ldots,\allowbreak I_{n-1}, Z$ pairwise commute. Since $Z^2=I_n U^2$, by induction it suffices to prove only that $Z$ commutes with $I_1,\allowbreak\ldots,\allowbreak I_{n-1}$. This follows from the fact that $Z$ commutes with $T_1,\allowbreak\ldots,\allowbreak T_{n-1}$ and $X$, and formula . The recursive definition of $I_1,\allowbreak\ldots,\allowbreak I_n$ is reminiscent of the construction of Gelfand-Zetlin subalgebras, though $I_n$ is not central.
#### 6. Relation with elements (\[jucys11\]), (\[jucys5\]).
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To get elements (\[jucys11\]) compose the embedding $\mu$ with automorphisms (\[autom\]) for $B(C_{n-1}^{(1)})$ and $B(C_n^{(1)})$: $\lambda=\rho_2\circ\mu\circ\rho'_2$, \[la\] (T’\_i)=T\_[i+1]{},i=1,…,n-1,(T’\_0)=T\_1T\_0T\_1. The elements of the image $\lambda\bigl(B(C_{n-1}^{(1)})\bigr)$ commute with $T_0$. The next formulae define one more embedding $\tilde\lambda:B(C_{n-1}^{(1)})\to B(C_n^{(1)})$: \[lat\] (T’\_i)=T\_[i+1]{},i=1,…,n-1,(T’\_0)=T\_0T\_1T\_0T\_1. Taking the quotient by the relation $T_0^2=1$ projects $B(C_n^{(1)})$ into $\widetilde{B}(B_n^{(1)})$ and formulae into \[latb\] (T’\_i)=S\_[i+1]{},i=1,…,n-1,(T’\_0)=S\_0S\_1. (Recall that $S_0=T_0T_1T_0$ and $S_i=T_i$, $i=1,\allowbreak\ldots,\allowbreak n$.) Formulae coincide with the embedding $\tilde\rho$ in (\[mapT\]) up to relabeling of generators.
To get elements (\[jucys5\]), the game is similar. First take an embedding $B(C_{n-2}^{(1)})\to B(C_n^{(1)})$, \[Cn-2\] T”\_iT\_[i+1]{},i=1,…,n-3,T”\_0T\_0T\_1T\_0T\_1,T”\_[n-2]{}T\_[n-1]{}T\_nT\_[n-1]{}T\_n, where $T''_0,\allowbreak\ldots,\allowbreak T''_{n-2}$ are the generators of $B(C_{n-2}^{(1)})$, and then the quotient by the relations $T_0^2=1$, $T_n^2=1$. Then formulae induce an embedding $B(C_{n-2}^{(1)})\to\widetilde{B}(B_n^{(1)})$, \[Bn-2\] T”\_iS\_[i+1]{},i=1,…,n-3,T”\_0S\_0S\_1,T”\_[n-2]{}S\_[n-1]{}S\_n. Recall that $S_0=T_0T_1T_0$, $S_n=T_nT_{n-1}T_n$, and $S_i=T_i$, $i=1,\allowbreak\ldots,\allowbreak n-1$. Formulae coincide with the embedding $\rho_0$ in (\[mapT2\]) up to relabeling of generators.
#### 7. One more automorphism of $B(C_n^{(1)})$.
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Consider the element $Z$, see . In addition to commutativity ZT\_i=T\_iZ, i=1,…,n-1,ZX=XZ, we also have ZT\_1…T\_n=T\_0…T\_[n-1]{}Z,ZT\_1…T\_[n-1]{}=T\_1…T\_[n-1]{}Z, that is, $ZT_0^{-1}X=XT_n^{-1}Z$ and $ZT_0^{-1}XT_n^{-1}=T_0^{-1}XT_n^{-1}Z$. Consider an automorphism \[phi\] :(C\_n\^[(1)]{})(C\_n\^[(1)]{}),(G)=ZGZ\^[-1]{}. Then $\varphi(T_i)\,=\,T_i$, $i=1,\allowbreak\ldots,\allowbreak n-1$, \[phiT0\] (T\_0)=T\_0T\_1…T\_[n-1]{}T\_nT\_[n-1]{}\^[-1]{}…T\_0\^[-1]{}= XT\_nX\^[-1]{}. \[phiTn\] (T\_n)=T\_[n-1]{}\^[-1]{}…T\_1\^[-1]{}T\_0T\_1…T\_[n-1]{}= T\_nX\^[-1]{}T\_0XT\_n\^[-1]{}=J\_nT\_n\^[-1]{}, Notice that $Z$ commutes with $I_1,\allowbreak\ldots,\allowbreak I_n$ given by , that is, $\varphi(I_i)=I_i$, $i=1,\allowbreak\ldots,\allowbreak n$.
The subgroup $B(C_n^{(1)})$ is invariant under the automorphism $\varphi$.
$R$-matrix representation of $B_n(C^{(1)})$.\[sec3\]
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Define an $R$-operator acting in the tensor product $V \otimes V$ of two $N$-dimensional vector spaces $V$ R(x,y) (\_[k\_1]{} \_[k\_2]{}) = (\_[i\_1]{} \_[i\_2]{}) R\^[i\_1i\_2]{}\_[k\_1k\_2]{}(x,y) . Here vectors $\{ \vec{e}_1,\dots,\vec{e}_N \}$ form a basis in $V$ and components $R^{i_1i_2}_{k_1k_2}(x,y)$ are functions of two spectral parameters $x$ and $y$. Let operator $R$ satisfies Yang-Baxter equation: R\_[12]{}(x,y) R\_[13]{}(x,z) R\_[23]{}(y,z) = R\_[23]{}(y,z) R\_[13]{}(x,z) R\_[1 2]{}(x,y) (V V V) , where we have used the standard matrix notations [@FRT]. Now we introduce two matrices $||K^i_j|| \in {\rm Mat}(V)$ and $||\overline{K}^i_j|| \in {\rm Mat}(V)$ with elements which are operators acting in the spaces $\widetilde{V}$ and $\widetilde{V}'$, respectively. In other words we have two operators $K \in {\rm End}(V \otimes \widetilde{V})$ and $\overline{K} \in {\rm End}(V \otimes \widetilde{V}')$. Let these operators be solutions of the equation R\_[12]{}(x,y) K\_1(x) R\_[21]{}(y,|[x]{}) K\_[2]{}(y) = K\_[2]{}(y) R\_[12]{}(x,|[y]{}) K\_1(x) R\_[21]{}(|[y]{},|[x]{}) ( V V) , which is called reflection equation and equation (cf. (\[Zam06\])) R\_[12]{}(x,y) \_2(y) R\_[21]{}(,x) \_[1]{}(x) = \_[1]{}(x) R\_[12]{}(,y) K\_2(y) R\_[21]{}(,) (V V ’) , which is called dual reflection equation. We explain this terminology and the meaning of the equations (\[Zam04\]), (\[Zam06\]) and (\[Zam13\]) in the next Section. In equations (\[Zam06\]) and (\[Zam13\]) we have used notations |[x]{} = (x) , = |(x) , where $\sigma$ and $\bar{\sigma}$ are the same involutive mappings $\mathbb{C} \to \mathbb{C}$ which were introduced in (\[Affbg1\]).
Using operator $R(x,y)$, which is defined in (\[Zam02\]) and (\[Zam04\]), we introduce the set of $R$-operators $R_{k,k+1}(x,y)$ $(k=1,\dots,n-1)$ which act in the space $V^{\otimes n}$ R\_[k,k+1]{}(x,y) = I\^[(k-1)]{} R(x,y) I\^[(n-k-1)]{} . For us it will be also convenient to introduce operators \_k(x,y) \_[k,k+1]{}(x,y) = I\^[(k-1)]{} P R(x) I\^[(n-k-1)]{} (k=1,…,n-1) , R\_[k,r]{}(x,y) = P\_[r,k+1]{} (I\^[(k-1)]{} R(x) I\^[(n-k-1)]{}) P\_[r,k+1]{} , where $P$ is a permutation operator in $V \otimes V$ $$P \cdot (v_1 \otimes v_2) = (v_2 \otimes v_1) \;\;\;\;\; \forall v_1,v_2 \in V \; ,$$ and $P_{r,k}= P_{k,r}$ is the permutation operator in $V^{\otimes n}$ such that $$P_{r,k} (v_1 \otimes \cdots \otimes v_k \otimes \cdots \otimes v_r \otimes \cdots \otimes v_{n}) =
(v_1 \otimes \cdots \otimes v_r \otimes \cdots \otimes v_k \otimes \cdots \otimes v_{n}) \; .$$ In terms of operators (\[Zam08\]) equations (\[Zam04\]), (\[Zam06\]) and (\[Zam13\]) can be written in the form \_[k]{}(x,y) \_[k+1]{}(x,z) \_[k]{}(y,z) = \_[k+1]{}(y,z) \_[k]{}(x,z) \_[k+1]{}(x,y) , \_[12]{}(x,y) K\_1(x) \_[12]{}(y,|[x]{}) K\_[1]{}(y) = K\_[1]{}(y) \_[12]{}(x,|[y]{}) K\_1(x) \_[12]{}(|[y]{},|[x]{}) , \_[12]{}(x,y) \_2(y) \_[12]{}(,x) \_[2]{}(x) = \_[2]{}(x) \_[12]{}(,y) \_2(y) \_[12]{}(,) , where $$\begin{array}{c}
K_k(x) = I^{\otimes (k-1)} \otimes K(x) \otimes I^{\otimes n-k-1} \; , \;\;\;
\overline{K}_{k}(x) = I^{\otimes (k-1)} \otimes \overline{K}(x)
\otimes I^{\otimes n-k-1} \;\;\;\;\;\; (k =1,\dots,n-1) \; .
\end{array}$$
Introduce the set of spectral parameters $\{ z_1 , \dots , z_n \}$. By using the group of the elements $s_i$ (see (\[Affbg1\])) and matrices $\hat{R}_{k}(z_k,z_{k+1})$, $K_1(z_1)$, $\overline{K}_n(z_n)$ we construct the representation $\rho$ of the affine group $B_n(C^{(1)})$ in $\tilde{V} \otimes V^{\otimes n} \otimes \tilde{V}'$
[c]{} (T\_i) = s\_i \_[i]{}(z\_i,z\_[i+1]{}) (i=1,…,n-1) , (T\_0) = K\_1(z\_1) s\_0 , (T\_n) = \_n(z\_n) s\_n .
One can directly check that $\rho(T_i)$ $(i=0,\dots,n)$ satisfy defining relations in (\[Affbg\]), (\[Affbg2\]) if $\hat{R}_{k}(z_k,z_{k+1})$ and $K_1(z_1)$, $\overline{K}_n(z_n)$ satisfy relations (\[Zam09\]), (\[Zam10\]), (\[Zam10a\]).
Further we will use the operator $D_{z_k}$ such that for any wave function $\Psi(z_1, \dots , z_n)$ and any operator $f(z_1, \dots , z_n)$ we have
[c]{} D\_[z\_k]{} f(z\_1, …, z\_k, …, z\_n) = f(z\_1, …, , …, z\_n) D\_[z\_k]{} ,\
\[0.2cm\] D\_[z\_k]{} (z\_1, …, z\_k, …, z\_n) = (z\_1, …, , …, z\_n) ,
where $\tilde{\bar{z_k}} = \bar{\sigma} (\sigma (z_k))$. We note that the operator $D_{z_k}$ in (\[Zam57\]) can be written in the representation (\[Affbg1\]) as D\_[z\_k]{} = (s\_[k-1]{} s\_1) (s\_0 s\_n) (s\_[n-1]{} s\_k) = [s]{}(J\_k) , where elements $J_k$ were introduced in (\[jucys1\]).
[**Theorem \[sec3\].1.**]{}
*The images of the commutative elements (\[jucys1\]) are operators in $\tilde{V} \otimes V^{\otimes n} \otimes \tilde{V}'$*
[c]{} (J\_i)=[A]{}\_i = \_[k-1]{}\^[-1]{}(z\_[k-1]{},z\_k) \_1\^[-1]{}(z\_1,z\_k) K\_1(z\_k) \_[1]{}(z\_1,|[z]{}\_k) \_[k-1]{}(z\_[k-1]{},|[z]{}\_k)\
\[0.2cm\] \_[k]{}(z\_[k+1]{},|[z]{}\_k) \_[n-1]{}(z\_[n]{},|[z]{}\_k) \_n(|[z]{}\_k) D\_[z\_k]{} \_[n-1]{}(z\_k,z\_n) \_[k]{}(z\_k,z\_[k+1]{}) ,\
\[0.3cm\] (\_i)=\_i = \_[k-1]{}(z\_k,z\_[k-1]{}) \_1(z\_k,z\_1) K\_1(z\_k) \_[1]{}(z\_1,|[z]{}\_k) \_[k-1]{}(z\_[k-1]{},|[z]{}\_k)\
\[0.2cm\] \_[k]{}(z\_[k+1]{},|[z]{}\_k) \_[n-1]{}(z\_[n]{},|[z]{}\_k) \_n(|[z]{}\_k) D\_[z\_k]{} \_[n-1]{}\^[-1]{}(z\_n,z\_k) \_[k]{}\^[-1]{}(z\_[k+1]{},z\_k) ,
form two sets of flat connections for quantum Knizhnik-Zamolodchikov equations
[c]{} [A]{}\_k(z\_1, …, z\_k, …, z\_n) (z\_1, …, z\_k, …, z\_n) = (z\_1, …, z\_k, …, z\_n) ,\
\[0.2cm\] \_k(z\_1, …, z\_k, …, z\_n) (z\_1, …, z\_k, …, z\_n) = (z\_1, …, z\_k, …, z\_n) ,
where functions $\Psi, \overline{\Psi} \in
\tilde{V} \otimes V^{\otimes n} \otimes \tilde{V}'$.
\
[**Proof.**]{} Formulas (\[Zam05ww\]) are obtained by direct calculations. The flatness of the connections (\[Zam16w\]) $$[{\sf A}_k , \, {\sf A}_j] = 0 =
[ \overline{\sf A}_k , \, \overline{\sf A}_j] \; ,$$ follows from the Proposition \[ABG\].1.
Flat connections for quantum Knizhnik-Zamolodchikov equations. Approach with Zamolodchikov algebra.\[sec4\]
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Zamolodchikov algebra.
----------------------
Introduce a set of operators $A^i(z)$ $(i=1,2,\dots,N)$ which act in the complex vector space ${\cal H}$. Each operator $A^i(z)$ is a function of the spectral parameter $z$. The operators $A^i(z)$ are generators of the algebra ${\cal Z}$ with quadratic defining relations (see e.g. [@GoshZam] and references therein) A\^[i\_1]{}(x) A\^[i\_2]{}(y) = R\^[i\_1i\_2]{}\_[k\_1k\_2]{}(x,y) A\^[k\_2]{}(y) A\^[k\_1]{}(x) . where $R^{i_1i_2}_{k_1k_2}(x,y) \in \mathbb{C}$ are functions of the spectral parameters $x$ and $y$ and also are components of an $R$-operator acting in the tensor product $V \otimes V$ of two $N$-dimensional vector spaces $V$ (see (\[Zam02\])). The algebra ${\cal Z}$ is called Zamolodchikov algebra. Relations (\[Zam01\]) can be written in concise matrix notations [@FRT] as following A\^[1 ]{}(x) A\^[2 ]{}(y) = R\_[1 2]{}(x,y) A\^[2 ]{}(y) A\^[1 ]{}(x) . Consider the product $A^{i_1}(x) A^{i_2}(y) A^{i_3}(z)$ of three operators and reorder it with the help of (\[Zam01\]) as following $$A^{i_1}(x) A^{i_2}(y) A^{i_3}(z) \;\; \to \;\; A^{k_3}(z) A^{k_2}(y) A^{k_1}(x) \; ,$$ in two different ways in accordance with the arrangement of brackets (A\^[i\_1]{}(x) A\^[i\_2]{}(y)) A\^[i\_3]{}(z) = A\^[i\_1]{}(x) (A\^[i\_2]{}(y) A\^[i\_3]{}(z)) . As a result we obtain the self-consistence condition for the matrix $R(x,y)$ in the form of the Yang-Baxter equation (\[Zam04\]). The solutions $R(x,y)$ of the equation (\[Zam04\]) define Zamolodchikov algebra (\[Zam01\]).
Now we extend (see [@GoshZam]) the algebra ${\cal Z}$ by adding new ”boundary" operators $B^\alpha$ ($\alpha =1,2, \dots,M$) which act in ${\cal H}$ and obey relations
[c]{} A\^[i]{}(x) B\^= K\^[i ]{}\_[k ]{}(x) A\^[k]{}( |[x]{} ) B\^ A\^[1 ]{}(x) B = K\_[1]{}(x) A\^[1 ]{}( |[x]{} ) B ,\
\[0.2cm\] |[x]{} = (x) ,
where $\bar{x}$ is a reflected spectral parameter and $\sigma$ – involutive operation $\mathbb{C} \to \mathbb{C}$ such that $\sigma^2 =1$. E.g., for rational and trigonometric $R$-matrices (pay attention to the special dependence of spectral parameters) R(x,y)=R(x-y) , R(x,y)=R(x/y) , one can take $\sigma = \sigma_a$ and $\sigma = \sigma^{tri}_b$, respectively, where \_a (x) = a - x , \^[tri]{}\_b(x) = b/x , and $a,b \in \mathbb{C}$ are parameters which specify involutions $\sigma$, $\sigma^{tri}$. Matrix $K$ with components $K^{i \alpha}_{k \beta}(x)$ acts in the space $V \otimes \widetilde{V}$, where $\widetilde{V}$ is $M$-dimensional vector space. This matrix is called reflection matrix and describes a reflection of particles from right boundary [@GoshZam]. For simplicity, in the second formula in (\[Zam05\]) and below, we omit indices $\alpha,\beta,\dots$ related to the space $\widetilde{V}$.
In the same way as in (\[lexy\]), one can consider two different ways for the reordering of special product of 3 generators (including $B^\alpha$): $$[A^{i_1}(x) \; A^{i_2}(y)] \; B = A^{i_1}(x) \; [A^{i_2}(y) \; B ]
\;\;\;\; \longrightarrow \;\;\;\;
A^{k_1}(\bar{x}) \; A^{k_2}(\bar{y}) \; B \; .$$ As a result, in addition to the Yang-Baxter equation (\[Zam04\]), we obtain new consistence condition for the reflection matrix $K$ in the form of the reflection equation (\[Zam06\]).
Now besides the “right” boundary operators $B^\alpha$ with relations (\[Zam05\]), we also introduce the “left” boundary operators $\overline{B}_{\alpha'}$ $(\alpha' = 1,\dots,M')$ with relations
[c]{} \_[’]{} A\^[i]{}(x) = \^[i’]{}\_[k ’]{}(x) \_[’]{} A\^[k]{}( ) A\^[1 ]{}(x) = \_[1]{}(x) A\^[1 ]{}( ) ,\
\[0.2cm\] = |(x) ,
where $\bar{\sigma}$ is another involutive operation in $\mathbb{C}$: $\bar{\sigma}^2=1$ (e.g., one can define $\bar{\sigma}$ as in (\[Zam05s\]) but with another parameters $a,b$). Equations (\[Zam12\]) and operator $\overline{K}(x) \in {\rm End}(V \otimes \tilde{V}')$, where $\widetilde{V}'$ is $M'$-dimensional vector space, describe the reflection of particles from the left boundary. Two different ways for the reordering of the product of 3 generators (including $\overline{B}$): $$\overline{B} \; A^{i_1}(x) \; A^{i_2}(y) \;\;\;\; \longrightarrow \;\;\;\;
\overline{B} \; A^{k_1}(\tilde{x}) \; A^{k_2}(\tilde{y}) \; ,$$ give additional consistence condition in the form of the dual reflection equation (\[Zam13\]).
Note that applying defining relations (\[Zam03\]), (\[Zam05\]) and (\[Zam12\]) twice, we deduce three unitary relations for matrices $R$, $\overline{K}$ and $K$ R\_[12]{}(x,y) R\_[21]{}(y,x) = I I , K\_[1]{}(x) K\_1(|[x]{}) = I , \_[1]{}(x) \_1() = I ’ , where $I$, $\widetilde{I}$ and $\widetilde{I}'$ – unite operators in $V$, $\widetilde{V}$ and $\widetilde{V}'$, correspondingly.
In physics the matrices $R$ and $K$, $\overline{K}$ which satisfy equations (\[Zam04\]), (\[Zam06\]), (\[Zam13\]) and (\[Zam07\]) describe the factorizable scattering on a half line [@GoshZam], [@Cher1], or define the integrable spin chains with nontrivial boundary conditions [@Skl]. Note that if matrices $R$, $K$ and $\overline{K}$ satisfy unitarity conditions (\[Zam07\]), then for the representation (\[Zam10b\]) we have $(\rho(T_i))^2 = I$, where $I$ is the unit operator in $\tilde{V} \otimes V^{\otimes n} \otimes \tilde{V}'$. Thus, in this case the equations (\[Zam10b\]) define the representation of the Coxeter group $W_n(C^{(1)})$.
Flat connections for quantum Knizhnik-Zamolodchikov equations.
--------------------------------------------------------------
Consider the boundary Zamolodchikov algebra ${\cal Z}_{LR}$ with generators $\{ A^i(x), \; B^\alpha, \; \overline{B}_{\beta'} \}$. Namely, the algebra ${\cal Z}_{LR}$ includes the generators $A^i(x)$ of the Zamolodchikov algebra ${\cal Z}$ and both left and right boundary operators $B^\alpha$ and $\overline{B}_{\beta'}$. Consider the special element in ${\cal Z}_{LR}$: \[\^\_[’]{}\]\^[i\_n …i\_1]{}(z\_n, …, z\_k, …, z\_1) = \_[’]{} A\^[i\_n]{}(z\_n) A\^[i\_k]{}(z\_k) A\^[i\_1]{}(z\_1) B\^ , and push the $k$-th operator $A^{i_k}(z_k)$, in the ordered product $\bigl( A^{i_n}(z_n) \cdots A^{i_1}(z_1)\bigr)$ in the right hand side of (\[Zam14\]), with the help of equations (\[Zam01\]) to the right. Then we reflect this operator from the right boundary operator $B^\alpha$ with the help of (\[Zam05\]), and push the reflected operator $A_{(k)}(\bar{z}_k)$ backward to the left with the help of (\[Zam01\]) up to the left boundary operator $\overline{B}_{\beta'}$. Then we reflect the operator $A_{(k)}(\bar{z}_k)$ from this boundary operator and finally place the operator $A_{(k)}(\tilde{\bar{z}}_k)$ on its initial $k$-th position in the ordered product $A_{(n)}(z_n) \cdots A_{(2)}(z_2) \; A_{(1)}(z_1) $. As a result we obtain the equation
[c]{} (\^\_[’]{})\^[i\_1 …i\_n]{}(z\_n, …, z\_k, …, z\_1) = \[[A]{}\_k(z\_1, …, z\_k, …, z\_n)\]\^[i\_1 …i\_n; ’]{}\_[j\_1 …j\_n; ’ ]{} (\^\_[’]{})\^[j\_1 …j\_n]{} (z\_n, …, \_k, …, z\_1) ,\
\[0.2cm\] \_k=| ((z\_k)) ,
where involutions $\sigma$ and $\bar{\sigma}$ were introduced in (\[Zam05\]) and (\[Zam12\]) while the matrix $$\begin{array}{c}
[{\cal A}_k(z_1, \dots , z_k, \dots , z_n)]_{12 \dots n} =
K_{k}(z_k; \, \vec{z}_{(1,k-1)}) \cdot \overline{K}_{k}(\bar{z}_k; \, \vec{z}_{(k+1,n)}) \; , \\ [0.3cm]
\vec{z}_{(1,k-1)}=(z_1,\dots,z_{k-1}) \; ,
\;\;\; \vec{z}_{(k+1,n)}=(z_{k+1},\dots,z_{n}) \; ,
\end{array}$$ is defined by means of dressed reflection matrices
[c]{} K\_[k]{}(x;\_[(1,k-1)]{}) = R\_[k,k-1]{}(x,z\_[k-1]{}) R\_[k1]{}(x,z\_1) K\_k(x) R\_[k1]{}(z\_1,|[x]{}) R\_[k,k-1]{}(z\_[k-1]{},|[x]{}) =\
\[0.2cm\] = \_[k-1]{}\^[-1]{}(z\_[k-1]{},x) \_2\^[-1]{}(z\_2,x) \_1\^[-1]{}(z\_1,x) K\_1(x) \_[1]{}(z\_1,|[x]{}) \_2(z\_2,|[x]{}) \_[k-1]{}(z\_[k-1]{},|[x]{}) =\
\[0.2cm\] = \_[k-1]{}\^[-1]{}(z\_[k-1]{},x) K\_[k-1]{}(x;\_[(k-2)]{}) \_[k-1]{}(z\_[k-1]{},|[x]{}) ,
[c]{} \_[k]{}(|[x]{};\_[(k+1,n)]{}) = R\_[k+1,k]{}(z\_[k+1]{},|[x]{}) R\_[nk]{}(z\_[n]{},|[x]{}) \_k(|[x]{}) R\_[kn]{}(,z\_n) R\_[k,k+1]{}(,z\_[k+1]{}) =\
\[0.2cm\] \_[k]{}(z\_[k+1]{},|[x]{}) \_[n-1]{}(z\_[n]{},|[x]{}) \_n(|[x]{}) \_[n-1]{}(,z\_n) \_[k]{}(,z\_[k+1]{}) =\
\[0.2cm\] = \_[k]{}(z\_[k+1]{},|[x]{}) \_[k+1]{}(|[x]{};\_[(k+2,n)]{}) \_[k]{}(,z\_[k+1]{}) .
To write expression (\[Zam55\]) for the matrix $K_{k}(x;\vec{z}_{(1,k-1)})$ we take into account the unitarity condition for the $R$-operator $\hat{R}_{k}(x,z) =\hat{R}_{k}^{-1}(z,x)$.
For rational and trigonometric $R$-matrices (\[RatTri\]) the involutions $\sigma$ and $\bar{\sigma}$ could be defined as in (\[Zam05s\]) $${\rm rational \;\; case}: \;\; \sigma = \sigma_a \; , \;\;\; \bar{\sigma} = \sigma_{a'} \;\; ; \;\;\;
{\rm trigonometric \;\; case}: \;\; \sigma = \sigma^{tri}_b \; , \;\;\;
\bar{\sigma} = \sigma^{tri}_{b'} \; ; \;\;\;$$ and we respectively obtain = \_[a’]{}( \_a ( x)) = (a’-a) + x , = \^[tri]{}\_[b’]{} ( \^[tri]{}\_b ( x)) = x , i.e., for the rational case the spectral parameter $\tilde{\bar{x}}$ is a shift of $x$ by a constant $(a'-a)$, while for the trigonometric case the parameter $\tilde{\bar{x}}$ is a multiplication of $x$ by a constant $b'/b$. In view of this, for rational and trigonometric cases the operator $D_z$ (\[Zam57\]), (\[Zam57D\]) can be considered as finite difference derivatives. Note that $\bar{\sigma} \sigma \neq \sigma \bar{\sigma} $.
One can write eqs. (\[Zam15\]) in the form of quantum Knizhnik-Zamolodchikov equations (see (\[Zam16w\]): \_k(z\_1, …, z\_k, …, z\_n) (z\_1, …, z\_k, …, z\_n) = (z\_1, …, z\_k, …, z\_n) , where we interpret $\Psi$ (\[Zam14\]) as a wave function and introduce connections
[c]{} [A]{}\_k(z\_1, …, z\_k, …, z\_n) = [A]{}\_k(z\_1, …, z\_k, …, z\_n) D\_[z\_k]{} = K\_[k]{}(z\_k; \_[(1,k-1)]{}) \_[k]{}(|[z]{}\_k; \_[(k+1,n)]{}) D\_[z\_k]{} =\
\[0.3cm\] = K\_[k]{}(z\_k; \_[(1,k-1)]{}) \_[k]{}(|[z]{}\_k; \_[(k+1,n)]{}) .
In the right hand side of (\[Zam05f\]) we use the dressed reflection matrix (\[Zam55\]) for $x=z_k$ which can be written in the representations (\[Affbg1\]) and (\[Zam10b\]) as the following
[c]{} K\_[k]{}(z\_k; \_[(1,k-1)]{}) = \_[k-1]{}\^[-1]{}(z\_[k-1]{},z\_k) \_1\^[-1]{}(z\_1,z\_k) K\_1(z\_k) \_[1]{}(z\_1,|[z]{}\_k) \_[k-1]{}(z\_[k-1]{},|[z]{}\_k) =\
\[0.2cm\] = (T\_[k-1]{}\^[-1]{} T\_1\^[-1]{} T\_0 T\_1 T\_[k-1]{}) (s\_[k-1]{} s\_1 s\_0 s\_1 s\_[k-1]{}) = (|[a]{}\_k) (a\_k) ,
where $\bar{a}_k = T_{k-1}^{-1} \cdots T_1^{-1} T_0 T_1 \cdots T_{k-1}$ and elements $a_k$ were defined in (\[jucys1\]). Besides this we also define new dressed reflection matrix
[c]{} \_[k]{}(|[x]{}; \_[(k+1,n)]{}) = \_[k]{}(|[x]{}; \_[(k+1,n)]{}) D\_[x]{} = \_[k]{}(z\_[k+1]{},|[x]{}) \_[k+1]{}(|[x]{}; \_[(k+2,n)]{}) \_[k]{}(x,z\_[k+1]{}) =\
\[0.3cm\] = \_[k]{}(z\_[k+1]{},|[x]{}) \_[n-1]{}(z\_[n]{},|[x]{}) \_n(|[x]{}) D\_[x]{} \_[n-1]{}(x,z\_n) \_[k]{}(x,z\_[k+1]{}) ,
which includes the finite difference operator $D_{x}$ (\[Zam57\]). In the representations (\[Affbg1\]) and (\[Zam10b\]), for $x=z_k$, the matrix (\[Zam05K\]) can be written as the following \_[k]{}(|[z]{}\_k; \_[(k+1,n)]{}) = (s\_[k-1]{} s\_1 s\_0 s\_1 s\_[k-1]{}) ( T\_[k]{} T\_[n-1]{} T\_n T\_[n-1]{} T\_[k]{}) = [**s**]{}(a\_k) (b\_k) , where $a_k$ and $b_k$ were defined in (\[jucys1\]). To obtain relations (\[Zam05d\]) and (\[Zam05dd\]) we have used formulas (\[Zam57D\]) and $$\begin{array}{c}
\bar{z}_k = (s_{k-1} \cdots s_1 \, s_0 \, s_1 \cdots s_{k-1}) \, z_k \,
(s_{k-1} \cdots s_1 \, s_0 \, s_1 \cdots s_{k-1}) \; .
\end{array}$$ Finally, using (\[Zam05d\]) and (\[Zam05dd\]) one can write connections (\[Zam05f\]) in the form \_k(z\_1, …, z\_n) = (|[a]{}\_k) (b\_k) = (J\_k) .
Applying equation (\[Zam16\]) twice (for two different indices $k$ and $r$) we deduce the consistency condition $$[{\sf A}_k , \; {\sf A}_r] \;\; \Psi(z_1, \dots , z_n) = 0 \; ,$$ and our conjecture is that the connections $A_k$, explicitly given in (\[Zam05f\]) and (\[Zam05ff\]), are flat: \[[A]{}\_k , \_r\] = 0 . One can prove this identity directly by using the fact that connections $A_k$ (\[Zam05ff\]) are the images of the commuting elements $J_k \in B_n(C^{(1)})$ (see Proposition \[ABG\].1). Note that commutativity (\[Zam17\]) of connections $A_k$ (\[Zam05f\]), where matrix $K_{k}(z_k; \, \vec{z}_{(1,k-1)})$ is taken in the form (\[Zam05d\]), is valid even for the case when $R$-matrix is not satisfies unitarity condition. So, we have proved the following statement:
[**Theorem \[sec4\].1.**]{} [*Connections ${\sf A}_k$ which were defined in (\[Zam05f\]), (\[Zam05d\]), (\[Zam05K\]) are flat (\[Zam17\]) for any matrices $R$, $K$ and $\overline{K}$ satisfying eqs. (\[Zam09\]),(\[Zam10\]) and (\[Zam10a\]) and any involutive operations $\sigma,\bar{\sigma}$.*]{}
[**Remark 1.**]{} One can think about boundary operators $B^\alpha$ and $\overline{B}_{\alpha'}$ in (\[Zam05\]), (\[Zam12\]) and (\[Zam14\]) as about boundary states $| B^\alpha \rangle \in {\cal H}$ and $\langle \overline{B}_{\alpha'} | \in {\cal H}^*$ with the same conditions as in (\[Zam05\]), (\[Zam12\]). In this case the operator (\[Zam14\]) is represented as the matrix element \[\^\_[’]{}\]\^[i\_n …i\_1]{}(z\_n, …, z\_2, z\_1) = \_[’]{} | A\^[i\_n]{}(z\_n) A\^[i\_2]{}(z\_2) A\^[i\_1]{}(z\_1) | B\^ , and the equation (\[Zam16\]), with the wave function $\Psi$ which is given in (\[Zam18\]), is nothing but the quantum Knizhnik-Zamolodchikov (q-KZ) equations for the system with nontrivial boundary conditions. One can put $\tilde{V} = \tilde{V}'$, $\beta' = \alpha$ in (\[Zam18\]) and sum over $\alpha$. As a result we obtain the following form of the solution of q-KZ equation \^[i\_n …i\_1]{}(z\_n, …, z\_2, z\_1) = [Tr]{}\_[H]{} ( A\^[i\_n]{}(z\_n) A\^[i\_2]{}(z\_2) A\^[i\_1]{}(z\_1) ) , where $\rho = | B^\alpha \rangle \langle \overline{B}_\alpha |$ can be considered as a density matrix.
[**Remark 2.**]{} For systems with periodic boundary conditions one can deduce q-KZ equations by using the same method as was used above for the systems with nontrivial boundary conditions and open boundaries. Consider the function (\[Zam19\]) with any operator $\rho$ and require that this operator satisfies commutation relations with generators $A^i(x)$: A\^i(x) = Q\^i\_j(x) A\^i() , = | ((x)) . Here functions $Q^i_j(x)$ are components of a numerical matrix. Taking into account (\[Zam20\]) we obtain the following periodicity condition for the wave function (\[Zam19\])
[c]{} \^[i\_n …i\_1]{}(z\_n, …, z\_2, z\_1) =\
\[0.2cm\] = [Tr]{}\_[H]{} ( A\^[i\_n]{}(z\_n) A\^[i\_3]{}(z\_3) A\^[i\_2]{}(z\_2) Q\^[i\_1]{}\_[j\_1]{}(z\_1) A\^[j\_1]{}(\_1) ) = Q\^[i\_1]{}\_[j\_1]{}(z\_1) \^[j\_1 i\_n …i\_2]{}( \_1, z\_n, …, z\_2) .
The associativity equation $A^{i_1}(x)(A^{i_2}(y)\; \rho) = (A^{i_1}(x)A^{i_2}(y)\;) \rho$ requires consistency condition for matrix $Q^i_j(x)$ R\_[12]{}(z\_1,z\_2) Q\_1(z\_1) Q\_2(z\_2) = Q\_1(z\_1) Q\_2(z\_2) R\_[12]{}(\_1,\_2) . We also require the condition $$R_{12}(\tilde{\bar{z}}_1,\tilde{\bar{z}}_2) = R_{12}(z_1,z_2) \;\;\;\; \Leftrightarrow \;\;\;\;
D_{z_1} \, D_{z_2} \, R_{12}(z_1,z_2) = R_{12}(z_1,z_2) \, D_{z_1} \, D_{z_2} \; ,$$ which is obtained automatically for the rational and trigonometric cases, when involutions $\bar{\sigma}$, $\sigma$ are fixed as in (\[Zam05ss\]). In this case equation (\[Zam21\]) is written as $$(D_{z_1} \, Q_1(z_1)) \, (D_{z_2} \, Q_2(z_2)) \, R_{12}(z_1,z_2) =
R_{12}(z_1,z_2) \, (Q_1(z_1) \, D_{z_1}) \, (Q_2(z_2) \, D_{z_2}) \; .$$ Now we again pick up the generator $A^{i_k}(z_k)$ in the right hand side of (\[Zam19\]) push this generator to the right with the help of (\[Zam01\]), then use relation (\[Zam20\]) and cyclic property of the trace and finally place the operator $A^{i_k}(\tilde{\bar{z}}_k)$ on its initial $k$-th position. As a result we obtain equation (z\_n, …, z\_2, z\_1) = [A]{}\_k(\_[(1,n)]{}) (z\_n, …, z\_2, z\_1) , where $\vec{z}_{(1,n)}=(z_1,\dots,z_n)$ and ${\sf A}_k(\vec{z}_{(1,n)})$ is the flat connection for q-KZ equation in the periodic case [@FreRe]:
[cl]{} [A]{}\_k(\_[(1,n)]{}) = & R\_[k,k-1]{}(z\_k,z\_[k-1]{}) R\_[k,2]{}(z\_k,z\_[2]{}) R\_[k,1]{}(z\_k,z\_[1]{}) Q\_k(z\_k) D\_[z\_k]{}\
\[0.3cm\] & R\_[kn]{}(z\_k,z\_n) R\_[k,n-1]{}(z\_k,z\_[n-1]{}) R\_[k,k+1]{}(z\_k,z\_[k+1]{}) .
Here the finite difference operator $D_{z_k}$ is the same as in (\[Zam57\]). Using for the periodic braid group elements $T_i$ the same $R$-matrix representation (\[Zam10b\]) we write connection (\[Zam22f\]) as (cf. (\[perBn3\]))
[c]{} [A]{}\_k(\_[(1,n)]{}) = (T\_[k-1]{} T\_1) X ( T\^[-1]{}\_[n-1]{} T\_k\^[-1]{}) ,\
\[0.3cm\] X : = Q\_1(z\_1) D\_[z\_1]{} \_1 \_[n-1]{} ,
where $\hat{\sf s}_k = P_{k,k+1} \, {\sf s}_k$ and we have used unitarity conditions $T_i^2=1$. We have (for simplicity we write $T_i$ instead of $\rho(T_i)$)
[c]{} T\_i \_[i+1]{} \_i = \_[i+1]{} \_i T\_[i+1]{} , T\_[i+1]{} \_i \_[i+1]{} = \_i \_[i+1]{} T\_i ,\
\[0.2cm\] X T\_i = T\_[i+1]{} X , (i=1,…, n-2) ,\
\[0.2cm\] T\_1 X\^2 = T\_1 Q\_1 D\_[z\_1]{} Q\_2 D\_[z\_2]{} (\_1 \_[n-1]{})\^2 = Q\_1 D\_[z\_1]{} Q\_2 D\_[z\_2]{} T\_1 (\_1 \_[n-1]{})\^2 =\
\[0.2cm\] = Q\_1 D\_[z\_1]{} Q\_2 D\_[z\_2]{} T\_1 (\_2 \_1) (\_3 \_2) (\_[n-1]{} \_[n-2]{}) =\
\[0.2cm\] = Q\_1 D\_[z\_1]{} Q\_2 D\_[z\_2]{} (\_2 \_1) (\_[n-1]{} \_[n-2]{}) T\_[n-1]{} = X\^2 T\_[n-1]{} .
One can check that the element T\_n := X\^[-1]{} T\_1 X = X T\_[n-1]{} X\^[-1]{} , satisfies periodic braid relations $$T_n \, T_{n-1} \, T_n = T_{n-1} \, T_n \, T_{n-1} \; , \;\;\;
T_n \, T_{1} \, T_n = T_{1} \, T_n \, T_{1} \; .$$ Let $T_1$ be unitary operator $T_1^2 =1$. In this case the connection (\[Zam22Ap\]) satisfies the periodicity condition $${\sf A}_k(\vec{z}_{(1,n)}) =
T_{k-1} \cdots T_1 \cdot X
\cdot T^{-1}_{n-1} \cdots T_k^{-1} =
T_{k-1} \cdots T_2 \cdot X
\cdot T_{n}^{-1} T^{-1}_{n-1} \cdots T_k^{-1} \; .$$
[**Proposition \[sec4\].2**]{} [@FreRe]. [ *For the periodic chain the connections (\[Zam22Ap\])are flat, i.e. satisfy (\[Zam17\]).*]{}\
[**Proof.**]{} The proof is the same as the proof of the Proposition \[ABG\].1 in Section [**\[ABG\]**]{}.
[**Remark 3.**]{} Consider operator $T_{V{\cal V}}(x) \in {\rm End}(V \otimes {\cal V})$ which satisfies the intertwining relations
[c]{} [R]{}\_[[V]{}[V]{}’]{}(x,y) T\_[1 [V]{}]{}(x) T\_[1 [V]{}’]{}(y) = T\_[1 [V]{}’]{}(y) T\_[1 [V]{}]{}(x) \_[[V]{}[V]{}’]{}(x,y) (V ’)
R\_[12]{}\^[-1]{}(x,y) T\_[1 [V]{}]{}(x) T\_[2 [V]{}]{}(y) = T\_[2[V]{}]{}(y) T\_[1 [V]{}]{}(x) R\_[12]{}\^[-1]{}(x,y) (V V ) , where we denote by ${\cal V}'$ the second copy of the vector space ${\cal V}$, the numbers $1,2$ numerate vector spaces $V$, and the matrix $R_{12}(x,y)\in {\rm End}(V \otimes V)$, as well as the matrix ${\cal R}(x,y) \in {\rm End}({\cal V} \otimes {\cal V}')$, satisfy the Yang-Baxter equation (\[Zam04\]). Consider the transfer-matrix (z\_1, …, z\_n) = [Tr]{}\_[V]{} ( T\_[nV]{}(z\_n) T\_[2V]{}(z\_2) T\_[1V]{}(z\_1) \_[V]{} ) , where the operator $\rho_{\cal V} \in {\rm End}({\cal V})$ is such that \_[[V]{}[V]{}’]{}(x,y) \_[V]{} \_[V’]{} = \_[V]{} \_[V’]{} [R]{}\_[[V]{}[V]{}’]{}(x,y) . Then we have\
[**Proposition \[sec4\].3.**]{} [*Transfer-matrices $\tau(z_1, \dots , z_n)$ and $\tau(z_1', \dots , z_n')$, defined in (\[Zam24\]), are commutative generating functions \[ (z\_1, …, z\_n) , (z\_1’, …, z\_n’)\] = 0 , if parameters $(z_1, \dots , z_n)$, $(z_1', \dots , z_n')$ and the matrix ${\cal R}(x,y)$ are such that (z\_n,z\_n’) = [R]{}(z\_k,z\_k’) k=1,2,…,n-1 .* ]{} [**Proof.**]{} Let ${\cal V}'$ be the second copy of the space ${\cal V}$. Then we have $$\begin{array}{c}
\tau(z_1, \dots , z_n) \; \tau(z_1', \dots , z_n') =
{\rm Tr}_{{\cal V}{\cal V}'} \Bigl( T_{n\cal V}(z_n) T_{n\cal V '}(z_n') \cdots
T_{1\cal V}(z_1) T_{1\cal V'}(z_1') \; \rho_{\cal V} \rho_{\cal V'} \Bigr) = \\ [0.3cm]
=
{\rm Tr}_{{\cal V}{\cal V}'} \Bigl( {\cal R}^{-1}_{\cal V V'}(z_n,z_n') \cdot
T_{n\cal V '}(z_n') T_{n\cal V}(z_n) \cdots
T_{1\cal V'}(z_1') T_{1\cal V}(z_1) \cdot
{\cal R}_{\cal V V'}(z_1,z_1') \; \rho_{\cal V'} \rho_{\cal V} \Bigr) = \\ [0.3cm]
= {\rm Tr}_{{\cal V}{\cal V}'} \Bigl(T_{n\cal V '}(z_n') T_{n\cal V}(z_n) \cdots
T_{1\cal V'}(z_1') T_{1\cal V}(z_1) \; \rho_{\cal V'} \rho_{\cal V} \Bigr) =
\tau(z_1', \dots , z_n') \; \tau(z_1, \dots , z_n) \; ,
\end{array}$$ where ${\rm Tr}_{{\cal V}{\cal V}'}={\rm Tr}_{{\cal V}}{\rm Tr}_{{\cal V}'}$ and we have used relations (\[Zam23\]), (\[Zam25\]).
Note that for the rational (or trigonometric) $R$-matrices, when we have $R(x,y) = R(x-y)$ (or $R(x,y) = R(x/y)$), relation (\[Zam25r\]) is fulfilled for the choice $z_k - z_k' = x - y$ (or $z_k / z_k' = x/y$) for all $k$, where $x$ and $y$ are two fixed parameters. For example, in the trigonometric case the commutative transfer-matrix can be taken in the form $\tau(x; z_1, \dots , z_n) = \tau(x \, z_1, \dots , x \, z_n)$ and commutativity condition (\[Zam25rt\]) is written as $$[\tau(x; z_1, \dots , z_n), \; \tau(y; z_1, \dots , z_n)] = 0 \; .$$
Now, in addition to the relation (\[Zam25\]), we require that the operator $\rho_{\cal V}$ satisfies (cf. (\[Zam20\])):
[c]{} T\_[1V]{}(x) \_[V]{} Q\_1 = \_[V]{} Q\_1 T\_[1V]{}() , Q (V)
where for the invertible matrix $Q$ we have (cf. \[Zam21\]) $$R_{12}(x,y) Q_1 Q_2 = Q_1 Q_2 R_{12}(\tilde{\bar{x}},\tilde{\bar{y}}) \; .$$ Equation (\[Zam26\]) serves twisted periodic boundary conditions of the type (\[percond\]) for the transfer-matrix (\[Zam24\]).
At the end of this Section we formulate the following statement.
[**Proposition \[sec4\].4.**]{} [*Flat connections (\[Zam22f\]) commute with the transfer-matrix (\[Zam24\]) \_k(z\_1,…,z\_n) (z\_1, …, z\_n) = (z\_1, …, z\_n) \_k(z\_1,…,z\_n) .* ]{} [**Proof.**]{} Take the operator $T_{k\cal V}(z_k)$ (in the right hand side of (\[Zam24\])) and use the same procedure as in Remark 2. for the cyclic moving of $T_{k\cal V}(z_k)$. After direct calculations with the use of the relations (\[Zam26\]), (\[Zam23a\]) and identity $$\tau(z_1, \dots , \tilde{\bar{z}}_k, \dots , z_n) =
D_{z_k} \; \tau(z_1, \dots , z_k, \dots , z_n) \; D_{z_k}^{-1} \; ,$$ we deduce relation (\[Zam33\]).
[**Consequence.**]{} By using the statement of the Proposition \[sec4\].4 we deduce the following result. Let $\Psi(z_n,\dots,z_1)$ be any solution of the periodic quantum Knizhnik-Zamolodchikov equation (\[Zam22\]). Then, the vector $$\Psi'(z_n,\dots,z_1) = \tau(z_1, \dots , z_n) \cdot \Psi(z_n,\dots,z_1) \; ,$$ is also a solution of the periodic quantum Knizhnik-Zamolodchikov equation (\[Zam22\]).
Flat connections for Birman–Murakami–Wenzl algebra.\[aBMW\]
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#### 1. Birman–Murakami–Wenzl algebra. Definition and basic relations.
${}$
The [*Birman–Murakami–Wenzl algebra*]{} $BMW_n(q,\nu)$ was defined in [@BirWen], [@Mur00] and [@Mur0]. It is generated over $\mathbb{C}$ by invertible elements ${\sf T}_1,\dots,{\sf T}_{n-1}$ with the following defining relations \_i [T]{}\_[i+1]{} [T]{}\_i=[T]{}\_[i+1]{} [T]{}\_i [T]{}\_[i+1]{} ,\_i [T]{}\_[j]{}=[T]{}\_j [T]{}\_[i]{} |i-j|>1 , \_i [T]{}\_i=[T]{}\_i\_i=\_i , \_i [T]{}\_[i-1]{}\^\_i=\^[-]{}\_i , \_i [T]{}\_[i+1]{}\^\_i=\^[-]{}\_i =1 , where \_i:=1- . Here $q$ and $\nu$ are complex parameters of the algebra which we assume generic in the sequel; in particular, the definition (\[bmw3\]) makes sense, the denominator in the right hand side does not vanish. Note that the algebra $BMW_n(q,\nu)$ with defining relations (\[bmw01\])–(\[bmw2\]) possesses the automorphism $\rho_2({\sf T}_i) = {\sf T}_{n-i}$ (cf. (\[autom\])).
.2cm The quotient of the algebra $BMW_n(q,\nu)$ by the ideal generated by the elements $\kappa_1,\dots,\kappa_n$ (in fact, this ideal is generated by any one of these elements, say, $\kappa_1$) is isomorphic to the Hecke algebra $H_n(q)$. It is also well-known that the braid group ${\cal{B}}_n$ (of type $A$) embeds in the $BMW_n$ algebra ${\cal{B}}_n \hookrightarrow BMW_n$. We shall often omit the parameters in the notation for the algebras and write simply $BMW_n$ and $H_n$.
.2cm Let \[lammu\]== .The following relations can be derived from (\[bmw01\])–(\[bmw2\]): $$\lb{bmw4}\kappa_i^2 = \mu \, \kappa_i$$ then, with $\varepsilon =\pm 1$, $$\begin{aligned}
\lb{bmw5}\kappa_i\, {\sf T}_{i+\varepsilon}\, {\sf T}_i\!\! &=&\!\! {\sf T}_{i+\varepsilon}\, {\sf T}_i \,\kappa_{i+\varepsilon} \;
,\\[.5em]
\lb{bmw8}\kappa_i\,\kappa_{i+\varepsilon}\,\kappa_i\!\! &=&\! \! \kappa_{i}\; ,\\[.5em]
\lb{bmw9}\bigl({\sf T}_i-\!(q -q^{-1})\bigr)\kappa_{i+\varepsilon}\bigl({\sf T}_i-\!(q -q^{-1})\bigr)\!\! &=&\!\!
\bigl( {\sf T}_{i+\varepsilon}-\!(q -q^{-1})\bigr)\kappa_{i}\bigl({\sf T}_{i+\varepsilon} -\!(q -q^{-1})\bigr)\, ,\\[.5em]
\lb{bmw8a}{\sf T}_{i+\varepsilon}\,\kappa_i\, {\sf T}_{i +\varepsilon}\!\! &=&\!\! {\sf T}^{-1}_i\,\kappa_{i+\varepsilon}\, {\sf T}^{-1}_i\;
,\end{aligned}$$ and $$\begin{aligned}
\lb{bmw6}\kappa_i\, {\sf T}_{i+\varepsilon}\, {\sf T}_{i}&=&\kappa_i\,\kappa_{i+\varepsilon}\; ,\\[.5em]
\lb{bmw7}\kappa_i\, {\sf T}^{-1}_{i+\varepsilon}\, {\sf T}^{-1}_{i}&=&\kappa_i\,\kappa_{i+\varepsilon}\; ,\\[.5em]
\lb{bmw8b}\kappa_{i+\varepsilon}\,\kappa_i\,\bigl({\sf T}_{i+\varepsilon}-(q -q^{-1})\bigr) &=&
\kappa_{i+\varepsilon}\,\bigl( {\sf T}_i-(q -q^{-1})\bigr)\; ,\end{aligned}$$ together with their images under the anti-automorphism $\rho_a$ of the algebra $BMW_n$ defined on the generators by \_a([T]{}\_i)=[T]{}\_i , \_a([T]{}\_i [T]{}\_k)=[T]{}\_k [T]{}\_i , \_a([T]{}\_i [T]{}\_j [T]{}\_k)= [T]{}\_k [T]{}\_j [T]{}\_i , ….
#### 2. Baxterized elements.
${}$
The [*baxterized elements*]{} $T_i(u,v)$ are defined by T\_i(u,v):=[T]{}\_i++ \_i T\_i(u/v) , see [@CGX], [@Isa], [@Jones] and [@Mur]. They are rational functions in complex variables $u$ and $v$ which are called [*spectral variables*]{}. The elements $T_i(u,v)$ depend on the ratio of the spectral parameters; for us it is more convenient to have both spectral variables in the notation (\[a00\]) for the baxterized element. However for brevity we shall denote sometimes the baxterized elements by $T_i(u/v) \equiv T_i(u,v)$ (with one argument only).
.2cm The baxterized elements satisfy the braid relation of the form T\_i(u\_[2]{},u\_[3]{})T\_[i+1]{}(u\_1,u\_[3]{})T\_i(u\_1,u\_[2]{})= T\_[i+1]{}(u\_1,u\_[2]{})T\_i(u\_1,u\_[3]{})T\_[i+1]{}(u\_[2]{},u\_[3]{}) . The inverses of the baxterized elements are given by T\_i(v,u)\^[-1]{}=T\_i(u,v) f(u,v) ,where f(u,v)==f(v,u) .
#### 3. Jucys–Murphy elements.
${}$
The Jucys–Murphy elements of the algebra $B\!M\!W_n$ are defined by y\_1=1 , y\_[k+1]{}=[T]{}\_k…[T]{}\_2 [T]{}\_1\^2 [T]{}\_2…[T]{}\_k , k=1, …,n-1 .The elements $y_1,\dots,y_n$ pairwise commute and satisfy the identities \_j y\_[j+1]{} y\_j=y\_j y\_[j+1]{}\_j=\^2\_j .
The Jucys–Murphy elements were originally used for constructing idempotents for the symmetric groups in [@Ju], [@Mu]. Analogues of the Jucys–Murphy elements can be defined for a number of important algebras related to the symmetric group rings (e.g., the Hecke and Brauer algebras); they turn out to generate maximal commutative subalgebras in these rings (see [@IMO], [@IsOg3], [@OPdA], [@OV] and references therein). The commutative subalgebra, generated by the Jucys–Murphy elements $y_1,\dots,y_n$, of the generic algebra $B\!M\!W_n$ is maximal as well; it follows from the results in [@IsOg],[@LeRa].
#### 4. Affine BMW algebras of type $C$
(see, e.g., [@IsOg] and references therein).
${}$
Affine Birman-Murakami-Wenzl algebras $BMW_{n}(C)$ of type $C$ are extensions of the algebras $BMW_{n}$. The algebra $BMW_{n}(C)$ is generated by the elements $\{{\sf T}_1,\dots,{\sf T}_{n-1}\}$ with relations (\[bmw01\]), (\[bmw1\]), (\[bmw2\]), (\[bmw3\]) and the affine element ${\sf T}_0 = y_1 \neq 1$ which satisfies
[c]{} [T]{}\_1 [T]{}\_0 [T]{}\_1 [T]{}\_0 = [T]{}\_0 [T]{}\_1 [T]{}\_0 [T]{}\_1 , \[[T]{}\_k, [T]{}\_0\]=0 k > 1 ,\
\[0.2cm\] \_1 \_0 [T]{}\_1 [T]{}\_0 [T]{}\_1 = \_1 = [T]{}\_1 [T]{}\_0 [T]{}\_1 [T]{}\_0 \_1 ,
\_1 \_0\^k \_1 = \^[(k)]{} \_1 , k=1,2,3,… , where $\hat{z}$, $\hat{z}^{(k)}$ are central elements. Initially, the affine version of the Brauer algebras (which are the special limit $q \to 1$ of $BMW_{n}(C)$), was introduced by M. Nazarov [@Nazar]. Note that the central elements $\{ \hat{z}^{(k)} \}$ generate an infinite dimensional abelian subalgebra in $BMW_{n+1}(C)$ and we denote this subalgebra as $BMW_{0}(C)$.
Consider the set of affine elements (cf. with elements $a_i$ in (\[jucys1\])) $$y_1 = {\sf T}_0 \; , \;\;\;
y_{k+1} = {\sf T}_k \, y_k \, {\sf T}_k \; , \;\;\; k=1,2,\dots ,n-1 \; .$$ The elements $y_k$ $(k=1,2,\dots,n)$ generate a commutative subalgebra $Y_{n}$ in $BMW_{n}(C)$.
[**Proposition \[aBMW\].1**]{} [@IsOg2],[@IMO2] [*For the affine BMW algebra the element L\_j(u)= , c=- q\^[-1]{} \^[-1]{} , is the baxterized solution of the reflection equation T\_[j]{}(u,v) L\_[j]{}(u) T\_[j]{}(v,|[u]{} ) L\_[j]{}(v) = L\_[j]{}(v) T\_[j]{}(u,|[v]{} ) L\_[j]{}(u) T\_[j]{}(u,v) , (j=1,…,n-1) , where $\bar{u} = 1/(c \, u)$.*]{}\
[**Proof.**]{} The formula (\[rea14\]) is checked by direct calculations.
Since we have $T_j\bigl(u,v \bigr) = T_j\bigl(\bar{v},\bar{u} \bigr)$, the equation (\[rea14\]) is a realization of the reflection equation (\[Zam10\]) if we identify $$L_j(v) \to K_j(v) \; , \;\;\;
T_j\bigl(u,v \bigr) \to \hat{R}_j(u,v)
\; , \;\;\; \bar{x} = \sigma(x) = \frac{1}{c \, x} \; .$$
#### 5. Affine BMW algebras of type $C^{(1)}$.
${}$
The algebra $BMW_{n}(C^{(1)})$ is generated by the elements $\{T_0,T_1, \dots,T_n \}$ and is associated to the Coxeter graph (\[C1\]) of type $C^{(1)}$. The algebra $BMW_{n}(C^{(1)})$ is an extension of the affine algebra $BMW_{n}(C)$ (we add new generator $T_n$). We require that the algebra $BMW_{n}(C^{(1)})$ possesses the automorphism $\rho_2$ which is defined in (\[autom\]). Thus, applying automorphism $\rho_2$ to the relations (\[bmw02\]), we obtain relations for the affine element $T_n$ in the form
[c]{} [T]{}\_[n-1]{} [T]{}\_n [T]{}\_[n-1]{} [T]{}\_n = [T]{}\_n [T]{}\_[n-1]{} [T]{}\_n [T]{}\_[n-1]{} , \[[T]{}\_k, [T]{}\_n\]=0 k < n-1 ,\
\[0.2cm\] \_[n-1]{} \_n [T]{}\_[n-1]{} [T]{}\_n [T]{}\_[n-1]{} = ’ \_[n-1]{} = [T]{}\_[n-1]{} [T]{}\_n [T]{}\_[n-1]{} [T]{}\_n \_[n-1]{} ,\
\[0.2cm\] \_[n-1]{} \_n\^k \_[n-1]{} = \^[(k)]{} \_[n-1]{} , k=1,2,3,… .
where $\hat{z}' = \rho_2(\hat{z})$, $\hat{z}^{\prime(k)} = \rho_2(\hat{z}^{(k)})$ (as well as $\hat{z}$, $\hat{z}^{(k)}$) are the central elements in the algebra $BMW_{n}(C^{(1)})$.
Consider the set of affine elements (cf. with elements $b_i$ in (\[jucys1\])) $$\bar{y}_n = {\sf T}_n = \rho_2({\sf T}_0) \; , \;\;\;
\bar{y}_{k-1} = {\sf T}_{k-1} \, \bar{y}_k \, {\sf T}_{k-1} = \rho_2(y_{n-k+2}) \; , \;\;\;
k=2,\dots ,n \; .$$ The elements $\bar{y}_k$ $(k=1,\dots,n)$ generate a commutative subalgebra $\bar{Y}_{n}$ in $BMW_{n}(C^{(1)})$.
Since the element (\[rea15\]) is a solution of the reflection elution (\[rea14\]), the element |[L]{}\_j(u)= = \_2(L\_[n-j+1]{}) BMW\_[n]{}(C\^[(1)]{}) , c’ =- q\^[-1]{} \^[-1]{} = \_2(c) , is the baxterized solution of the dual reflection equation which is obtained as the image of (\[rea14\]) under the automorphism $\rho_2$ T\_[j]{}(u,v) |[L]{}\_[j+1]{}(u) T\_[j]{}(v, ) |[L]{}\_[j+1]{}(v) = |[L]{}\_[j+1]{}(v) T\_[j]{}(u, ) |[L]{}\_[j+1]{}(u) T\_[j]{}(u,v) , (j=1,…,n-1) , where $\tilde{u} = 1/(u \, c^{\prime})$. Taking into account relations (\[aa\]) and identities $$T_j(\tilde{v},u) = T_j(\tilde{u},v) \; , \;\;\;
\bar{L}_{j}(\tilde{u}) = \frac{1}{c'} \; \bar{L}_{j}(u)^{-1}$$ we write (\[rea14d\]) in the form T\_[j]{}(v,u) |[L]{}\_[j+1]{}() T\_[j]{}(,u ) |[L]{}\_[j+1]{}() = |[L]{}\_[j+1]{}() T\_[j]{}(,v ) |[L]{}\_[j+1]{}() T\_[j]{}(,) , (j=1,…,n-1) . The equation (\[rea14dd\]) can be represented as the reflection equation (\[Zam10a\]) if we identify $$\bar{L}_j(\tilde{v}) \to K_j(v) \; , \;\;\;
T_j\bigl(u,v \bigr) \to \hat{R}_j(u,v)
\; , \;\;\; \tilde{x} = \bar{\sigma}(x) = \frac{1}{c' \, x} \; .$$
#### 6. Embedding of the braid group $B_n(C^{(1)})$ into the algebra $BMW_{n}(C^{(1)})$.
${}$
Let $\{z_1,\dots,z_n \}$ be a set of spectral parameters. Consider the Weyl group generated by the operators $s_i$ (see (\[Affbg1\])) and the elements $$T_i(z_i,z_{i+1}) \; , \;\; L_1(z_1) \; , \;\; \bar{L}_n(z_n) \;\; \in \;\; BMW_{n}(C^{(1)}) \; .$$ Then we have the following statement.
[**Proposition \[aBMW\].2**]{} The map $\rho_{b}$ of the affine braid group $B_n(C^{(1)})$ into $BMW_{n}(C^{(1)})$ defined as (cf. (\[Zam10b\]))
[c]{} \_[b]{}(T\_i) = s\_i T\_[i]{}(z\_i,z\_[i+1]{}) (i=1,…,n-1) , \_[b]{}(T\_0) = L\_1(z\_1) s\_0 , \_[b]{}(T\_n) = s\_n \_n(z\_n) ,
is the representation of $B_n(C^{(1)})$.\
[**Proof.**]{} One can directly check that $\rho_{b}(T_i)$ $(i=0,\dots,n)$ satisfy defining relations in (\[Affbg\]), (\[Affbg2\]) if $T_{k}(z_k,z_{k+1})$, $L_1(z_1)$ and $\overline{L}_n(z_n)$ satisfy relations (\[ybe0\]), (\[rea14\]) and (\[rea14d\]), respectively.
[**Corollary.**]{} The map $\rho_{c}$ of the affine braid group $B_n(C)$ into $BMW_{n}(C)$
[c]{} \_[c]{}(T\_i) = s\_i T\_[i]{}(z\_i,z\_[i+1]{}) (i=1,…,n-1) , \_[c]{}(T\_0) = L\_1(z\_1) s\_0 ,
is the representation of $B_n(C)$.
#### 7. Flat connections for the algebra $BMW_{n}(C^{(1)})$.
${}$
Flat connections for the algebra $BMW_{n}(C^{(1)})$ are defined as images $\rho_b(J_i)$ and $\rho_b(\overline{J}_i)$ of the elements $J_i$ and $\overline{J}_i$ (see (\[jucys1\])) which form the commuting sets of elements in affine braid group $B_n(C^{(1)})$. The explicit formulas are (cf. (\[Zam05f\])) \_k(z\_1,…,z\_n) = \_b(J\_k) = K\_k(z\_k ; z\_1, …, z\_[k-1]{}) \_k(|[z]{}\_k ; z\_[k+1]{}, …, z\_[n]{}) , where (cf. (\[Zam05d\]), (\[Zam05K\]))
[c]{} K\_[k]{}(z\_k; \_[(1,k-1)]{}) = T\_[k-1]{}\^[-1]{}(z\_[k-1]{},z\_k) T\_1\^[-1]{}(z\_1,z\_k) L\_1(z\_k) T\_[1]{}(z\_1,|[z]{}\_k) T\_[k-1]{}(z\_[k-1]{},|[z]{}\_k) =\
\[0.2cm\] = \_b(T\_[k-1]{}\^[-1]{} T\_1\^[-1]{} T\_0 T\_1 T\_[k-1]{}) (s\_[k-1]{} s\_1 s\_0 s\_1 s\_[k-1]{}) = \_b(|[a]{}\_k) (a\_k) ,
[c]{} \_[k]{}(|[z]{}\_k; \_[(k+1,n)]{}) =\
\[0.3cm\] = T\_[k]{}(z\_[k+1]{},|[z]{}\_k) T\_[n-1]{}(z\_[n]{},|[z]{}\_k) \_n(|[z]{}\_k) D\_[z\_k]{} T\_[n-1]{}(z\_k ,z\_n) T\_[k]{}(z\_k ,z\_[k+1]{}) = [s]{}(a\_k) \_b(b\_k) ,
where the finite difference operator $D_{z_k}$ was defined in (\[Zam57\]) with $$\tilde{\bar{z}} = \bar{\sigma}(\sigma(z)) = \bar{\sigma}\Bigl(\frac{1}{c\, z}\Bigr)
= \frac{c}{c'} \; z \; .$$ We stress that $L_j(u) = D_u$ and $\bar{L}_{j+1}(u) = D_u^{-1}$ (as well as $L_j(u) = 1$ and $\bar{L}_{j+1}(u) = 1$) are solutions of the reflection equations (\[rea14\]) and (\[rea14d\]), respectively. For example, we can substitute solution $\bar{L}_{j+1}(u) = 1$ into (\[Zam05Kb\]) and reduce the flat connection (\[Zam77\]) into the form \_k(z\_1,…,z\_n) = K\_k(z\_k ; z\_1, …, z\_[k-1]{}) \_k’(|[z]{}\_k ; z\_[k+1]{}, …, z\_[n]{}) , where $$\begin{array}{c}
\overline{\sf K}'_{k}(\bar{z}_k; \, \vec{z}_{(k+1,n)}) =
T_{k}(z_{k+1},\bar{z}_k) \cdots T_{n-1}(z_{n},\bar{z}_k)
\cdot D_{z_k} \; \cdot
T_{n-1}(z_k ,z_n) \cdots T_{k}(z_k ,z_{k+1}) \; ,
\end{array}$$
#### 8. Braid–Hecke algebra ${\cal{BH}}_n(q,\nu)$.
${}$
The [*Braid–Hecke algebra*]{} ${\cal{BH}}_n(q,\nu)$, as far as we know, was introduced in [@Co1],[@Co2]. It is generated over $\mathbb{C}$ by the invertible [*braid type generators*]{}, $${\sf T_1}, \cdots, {\sf T_{n-1}},$$ subject the following defining relations
\_i [T]{}\_[i+1]{} [T]{}\_i=[T]{}\_[i+1]{} [T]{}\_i [T]{}\_[i+1]{} ,\_i [T]{}\_[j]{}=[T]{}\_j [T]{}\_[i]{} |i-j|>1 , \_i [T]{}\_i=[T]{}\_i\_i=\_i , \_[i 1]{}[T]{}\_[i]{} \_[i 1]{} - \_[i]{}\_[i 1]{} = [T]{}\_[i]{} [T]{}\_[i 1]{} \_[i]{} - \_[ i 1]{}\_[i]{}, \_i\_[i 1]{}\_[i]{}- \_[i]{} = \_[i 1]{} \_[i]{} \_[i 1]{} - \_[i 1]{},
where \_i:=1- . Here $q$ and $\nu$ are complex parameters of the algebra which we assume generic in the sequel; in particular, the definition (\[bmw3a\]) makes sense, the denominator in the right hand side does not vanish. Note that the algebra $BMW_n(q,\nu)$ is the quotient of the algebra ${\cal{BH}}_n(1,\nu)$ by the two-sided ideal generated by the [*tangle relations*]{} \_i\_[i 1]{}\_[i]{}- \_[i]{} = 0, \_[i 1]{} \_[i]{} \_[i 1]{} - \_[i 1]{} = 0.
It is easy to see that $$({\sf T}_i -q)({\sf T}_{i}+q^{-1})({\sf T}_{i} - \nu)=0,~~\kappa_{i}^2=
{\frac{(q- \nu)(q^{-1} + \nu)}{\nu (q-q^{-1})}}~~\kappa_{i}.$$ It follows from (\[bmv33\]), that the elements $\{ \kappa_1,\ldots, \kappa_{n-1} \}$ generate the Hecke algebra ${\cal{H}}_n(p)$ corresponding to a parameter $p$ such that $$p+p^{-1} = {\frac{ (q- \nu)(q^{-1} + \nu)}{\nu (q-q^{-1})}}.$$ Note that the algebra ${\cal{BH}}_n(q,\nu)$ with defining relations (\[bmw012\])–(\[bmv33\]) possesses the automorphism $\rho_2({\sf T}_i) = {\sf T}_{n-i}$ (cf. (\[autom\])). It is well-known that $\dim (BMW_n) = (2n-1) !!$. As for the algebra ${\cal{BH}}_n(q,\nu)$, it is known [@Co2] that it has finite dimension, but as far as we know, the exact value of its dimension is still unknown.\
$\bullet$ The baxterized elements $ \{T_i(u,v),~i=1,\ldots,n-1 \}$ are defined by (cf (\[ybe0\]), ($z:=u/v))$, $$(\nu+q~z)~ T_i(z) = q~z~{\sf T}_i+\nu~{\sf T}_i^{-1}+(q-q^{-1})~\frac{z(q+\nu)}{z-1}~
\equiv (\nu+q~z)~T_i(u,v).$$ $\bullet$ The Jucys–Murphy elements $\{ y_i,~i=1,\ldots,n-1 \}$ of the algebra ${\cal{BH}}_n(q,\nu)$ are defined by (\[jme\]). The $JM$ elements $y_2, ldots, y_{n-1}$ pairwise commute and satisfy the identities $$\kappa_j~y_{j+1}~y_{j} = y_{j}~y_{j+1}~\kappa_{j},~~~1 \le j < n-1.$$ $\bullet$ Affine braid-Hecke algebra ${\cal{BH}}_n(C)$ of type $C$ is an extension of the algebra ${\cal{BH}}_n(q,\nu)$ by the the affine element ${\sf T_{0}}=y_{0} \not= 1$, subject to the set of “crossing relations” (\[bmw02\]) and (\[bmw02z\]). One can check that the set of elements $$y_1= {\sf T}_{0},~~ y_{k+1}:= {\sf T}_{k}~y_k~{\sf T}_{k},~~~k=1,\ldots,n-1$$ generate a commutative subalgebra in ${\cal{BH}}_n(C)$.
$\bullet$ (Markov trace, cf [@Co2]) The family of algebras $\{ {\cal{BH}}_n(q,\nu) \hookrightarrow {\cal{BH}}_{n+1}(q,\nu) \}_{n \ge 1}$ can be provided with (unique) set of homomorphisms $$Tr_{n+1} : {\cal{BH}}_{n+1}(q,\nu) \longrightarrow {\cal{BH}}_n(q,\nu)$$ which satisfy the conditions stated in Proposition \[sec6\].2.
, the all properties of the algebra ${\cal{BH}}_n(q,\nu)$ stated in the item ${\bf 8}$ allow the use of the methods developed in Sections \[aBMW\] and \[sec6\], to construct families of [*commutative subalgebras*]{} in the algebra ${\cal{BH}}_n(q,\nu)$, as well as ${\cal{BH}}_n(q,\nu)$-valued flat connections and associated $qKZ$ equations. Details will appear.
.2cm
Sklyanin’s transfer-matrices for affine BMW algebra.\[sec6\]
============================================================
0
In this Section, to simplify formulas we make the redefinition of all spectral parameters $z \to c^{-1/2} z$. In this case the baxterized element (\[a00\]) does not changed (since it depends on the ratio of spectral parameters) and statement of the Proposition [**\[aBMW\].1**]{} reads as the following. For the affine BMW algebra $BMW_{n}(C^{(1)})$ the element L\_j(c\^[-1/2]{} u)= y\_j(u) , y\_j(u) y\_j(u\^[-1]{}) = c\^[-1]{} , where $c=- \nu \, q^{-1} \, \hat{z}^{-1}$, is the baxterized solution of the reflection equation T\_[j]{}(u/v) y\_[j]{}(u) T\_[j]{}(v u) y\_[j]{}(v) = y\_[j]{}(v) T\_[j]{}(v u ) y\_[j]{}(u) T\_[j]{}(u/v) , (j=1,…,n-1) .
Sklyanin’s transfer-matrix elements for the algebra $BMW_n(C)$
--------------------------------------------------------------
In this Subsection we generalize to the BMW algebra case results obtained in [@Isa07],[@IK] for the Hecke algebra case.
[**Definition \[sec6\].1**]{} Let $\vec{z}_{(k)}=(z_1,\dots,z_{k})$ be $k$ parameters and $y_1(x) \in BMW_{n}(C^{(1)})$ is any [*local*]{} (i.e., $[y_1(x), T_k]=0,$ $\forall k > 1$) solution of the reflection equation (\[rea14z\]) with $j=1$: T\_1 (x /z ) y\_[1]{}(x) T\_1(x z) y\_[1]{}(z) = y\_[1]{}(z) T\_1(x z) y\_[1]{}(x) T\_1 ( x /z ) , where solution $y_1(z)$ is given in (\[rea15z\]) for $j=1$. Define the elements (cf. (\[Zam05db\]))
[c]{} y\_[k]{}(x;\_[(k-1)]{}) = T\_[k-1]{}() T\_2() T\_1() y\_1(x) T\_[1]{}( x z\_1) T\_2( x z\_2) T\_[k-1]{}( x z\_[k-1]{}) =\
\[0.2cm\] = T\_[k-1]{}() y\_[k-1]{}(x;\_[(k-2)]{}) T\_[k-1]{}( x z\_[k-1]{}) ,
which we call “baxterized” Jucys–Murphy elements.
[**Proposition \[sec6\].1**]{} [*The “baxterized” Jucys–Murphy element (\[xxz55\]) is a solution of the reflection equation T\_k (x /z ) y\_[k]{}(x;\_[(k-1)]{}) T\_k( x z) y\_[k]{}(z;\_[(k-1)]{}) = y\_[k]{}(z;\_[(k-1)]{}) T\_k(x z) y\_[k]{}(x;\_[(k-1)]{}) T\_k ( x /z ) ,* ]{} [**Proof.**]{} The case $k=1$ of the equation (\[reflH\]) corresponds to our assumption that $y_1(x)$ satisfies the equation (\[reflH1\]). The general case follows by induction using the definition (\[xxz55\]) of elements $y_{k}(x;\vec{z}_{(k-1)}).$
For example, in the case of the affine BMW algebra $BMW_{n}(C^{(1)})$, one can use the local solution (\[rea15z\]) for $j=1$ (recall that $y_1 = {\sf T}_0$): y\_1(x)= [ c\^[- 1/2]{} x - y\_1 c\^[1/2]{} x y\_1 - 1]{} = [ c\^[- 1/2]{} x - [T]{}\_0 c\^[1/2]{} x [T]{}\_0 - 1]{} , y\_1(1) = - c\^[- 1/2]{} .
Further we consider only one-boundary affine BMW algebra $BMW_{n}(C)$ of type $C$ which is obtained as the projection ${\sf T}_n=1$ from the two-boundary affine BMW algebra $BMW_{n}(C^{(1)})$ (see paragraph 4 in Section [**\[aBMW\]**]{}).
Consider the following inclusions of the subalgebras $BMW_{1}(C) \subset BMW_{2}(C) \subset \dots \subset BMW_{n+1}(C)$: $$\{{\sf T}_0; {\sf T}_1, \dots ,{\sf T}_{k-1}\} \in BMW_{k}(C) \subset BMW_{k+1}(C) \ni
\{{\sf T}_0; {\sf T}_1, \dots ,{\sf T}_{k-1},{\sf T}_k \} \; .$$ For the subalgebras $BMW_{k+1}(C)$ we introduce linear mapping (quantum trace) $${\rm Tr}_{(k+1)}: \;\; BMW_{k+1}(C) \to BMW_{k}(C) \; , \;\;\;\; (k=1,2,\dots,n) \; ,$$ which is defined by the formula \_[k+1]{} X\_[k+1]{} \_[k+1]{} = Tr\_[([k+1]{})]{}(X\_[k+1]{}) \_[k+1]{} , X\_[k+1]{} BMW\_[k+1]{}(C) . [**Proposition \[sec6\].2**]{}
*For the map $Tr_{(k+1)}$: $BMW_{k+1}(C) \to BMW_{k}(C)$ we have the following properties $(\forall X_{k},X_k' \; \in \; BMW_{k}(C) \; ,
\;\; \forall Y_{k+1} \; \in \; BMW_{k+1}(C))$*
[c]{} Tr\_[(k+1)]{}([T]{}\_[k]{})=1 , Tr\_[(k+1)]{}([T]{}\_[k]{}\^[-1]{})=\^2 , \_[(k+1)]{} ( X\_k ) = X\_k ,\
Tr\_[(k+1)]{}(\_k)= , \_[(1)]{} ([T]{}\_0\^k)= \^[(k)]{} ,
Tr\_[(k+1)]{}([T]{}\_[k]{} X\_[k]{} [T]{}\_[k]{}\^[-1]{}) = Tr\_[(k)]{}(X\_[k]{}) = Tr\_[(k+1)]{}([T]{}\_[k]{}\^[-1]{} X\_[k]{} [T]{}\_[k]{}) , Tr\_[(k+1)]{}([T]{}\_[k]{} X\_[k]{} \_[k]{}) = Tr\_[(k+1)]{}(\_[k]{} X\_[k]{} [T]{}\_[k]{}) ,
[c]{} [Tr]{}\_[(k+1)]{} ( X\_k Y\_[k+1]{} X’\_k ) = X\_k \_[(k+1)]{}(Y\_[k+1]{}) X\_k’ ,\
[Tr]{}\_[(k)]{} [Tr]{}\_[(k+1)]{} ( [T]{}\_k Y\_[k+1]{} ) = [Tr]{}\_[(k)]{} [Tr]{}\_[(k+1)]{} ( Y\_[k+1]{} \_k) .
[**Proof.**]{} Eqs. (\[qtr1\]) follow from (\[bmw2\]), (\[bmw4\]), (\[bmw8\]) and (\[bmw02z\]). Using (\[bmw6\]), (\[bmw7\]) and (\[bmw8\]) we have $$\begin{array}{c}
\frac{1}{\nu} \, Tr_{(k+1)}({\sf T}_{k} \, X_{k} \, {\sf T}_{k}^{-1}) \kappa_{k+1} =
\kappa_{k+1} \, {\sf T}_{k} \, {\sf T}_{k+1} \, X_{k} \, {\sf T}_{k+1}^{-1} \, {\sf T}_{k}^{-1} \, \kappa_{k+1}=
\\ [0.2cm]
= \kappa_{k+1} \, \kappa_{k} \, X_{k} \, \kappa_{k} \, \kappa_{k+1}
= \frac{1}{\nu} \, Tr_{(k)}(X_{k}) \,
\kappa_{k+1} \, \kappa_{k} \, \kappa_{k+1} =
\frac{1}{\nu} \, Tr_{(k)}(X_{k}) \,
\kappa_{k+1} \; ,
\end{array}$$ which is equivalent to the first equality in (\[qtr2\]) (second equality in (\[qtr2\]) can be proved analogously). Eq. (\[qtr3\]) can be proved in the following way $$\begin{array}{c}
\kappa_{k+1} \, {\sf T}_{k} \, X_{k} \, \kappa_{k} \, \kappa_{k+1}=
\kappa_{k+1} \, \kappa_{k} \, {\sf T}_{k+1}^{-1} \, X_{k} \,
\kappa_{k} \, \kappa_{k+1}=
\kappa_{k+1} \, \kappa_{k} \, X_{k} \, {\sf T}_{k+1}^{-1}
\, \kappa_{k} \, \kappa_{k+1}
= \\ [0.2cm]
= \kappa_{k+1} \, \kappa_{k} \, X_{k} \, {\sf T}_{k} \, \kappa_{k+1}\; .
\end{array}$$ The first eq. in (\[map\]) is evident and the proof of second eq. in (\[map\]) is the following. First of all for any $Y_{k+1}' \in BMW_{k+1}(C)$ we have $$\kappa_{k+2} \kappa_k \kappa_{k+1} \, Y_{k+1}' \, \kappa_{k+1} \kappa_k \kappa_{k+2}
= \frac{1}{\nu} \kappa_{k+2} \kappa_k \kappa_{k+1} {\rm Tr}_{(k+1)}(Y_{k+1}')
\, \kappa_k \kappa_{k+2} =$$ $$= \frac{1}{\nu} \kappa_{k+2} \kappa_k {\rm Tr}_{(k+1)}(Y_{k+1}')
\, \kappa_k = \frac{1}{\nu^2} \kappa_{k+2} \kappa_k
{\rm Tr}_{(k)} {\rm Tr}_{(k+1)}(Y_{k+1}') \; .$$ Then, using this equation and relations (\[bmw6\]), (\[bmw7\]) we obtain $$\begin{array}{c}
\frac{1}{\nu^2} \kappa_{k+2} \kappa_k
{\rm Tr}_{(k)} {\rm Tr}_{(k+1)}(Y_{k+1} \, {\sf T}_k) =
\kappa_{k+2} \kappa_k \kappa_{k+1} \, Y_{k+1} {\sf T}_k \, \kappa_{k+1} \kappa_k \kappa_{k+2}
= \kappa_{k+2} \kappa_k \kappa_{k+1} \, Y_{k+1}
{\sf T}_{k+1}^{-1} \, \kappa_{k+2} \kappa_k = \\[0.2cm]
= \kappa_{k+2} \kappa_k \kappa_{k+1} \, Y_{k+1}
{\sf T}_{k+2} \, \kappa_{k+1} \, \kappa_{k+2} \kappa_k =
\kappa_{k+2} \kappa_k \kappa_{k+1} \, {\sf T}_{k+2} Y_{k+1}
\, \kappa_{k+1} \, \kappa_{k+2} \kappa_k = \\[0.2cm]
= \kappa_{k+2} \kappa_k \kappa_{k+1} \, {\sf T}_{k} Y_{k+1}
\, \kappa_{k+1} \, \kappa_{k+2} \kappa_k = \frac{1}{\nu^2} \kappa_{k+2} \kappa_k
{\rm Tr}_{(k)} {\rm Tr}_{(k+1)}({\sf T}_k \, Y_{k+1}) \; .
\end{array}$$
Below we use the following identities for baxterized elements (\[a00\]):
[c]{} T\_n(x) T\_n(y)= T\_n(x y) + 1 + \_n\
\[0.2cm\] T\_n(x) = T\_n(y) + + \_n ,
Note that identity (\[aa\]) is a consequence of the first relation in (\[bax1\]) if we substitute there $y=x^{-1}$ and take into account $(1-xy) \; T_n(x \, y) \; \stackrel{\; y \to x^{-1}}{\longrightarrow} \;
(q-q^{-1})$.
Using the properties (\[map\]) of the map ${\rm Tr}_{_{(n+1)}}$ and relations (\[bax1\]), one can prove the Lemma.
[**Lemma \[sec6\].1**]{}
*For all $X_k \in BMW_{k}(C)$ and all spectral parameters $x$ and $z$ the following identity is true:*
[c]{} [Tr]{}\_[\_[(k+1)]{}]{} ( T\_k(x) X\_k T\_k(z) ) = \_[\_[(k+1)]{}]{} ( T\_k X\_k \_k ) +\
\[0.2cm\] + [Tr]{}\_[\_[(k)]{}]{} (X\_k) - X\_k ,
where $T_k(x)$ and $T_k(z)$ are Baxterized elements (\[a00\]).
\
[**Proof.**]{} Direct calculations with the help of properties (\[qtr1\]) – (\[map\]).
From eq. (\[mapp1\]), for $x \, z = q^2 \nu^{-2}$, we obtain the “crossing-unitarity relation” \_[\_[(k+1)]{}]{} ( T\_k(x) X\_k T\_k ( q\^2 \^[-2]{}/ x) ) = \_[\_[(k)]{}]{} (X\_k) , where $F(x) =\frac{ (x \nu + q)^2}{(x \nu + q^3)(x\nu + q^{-1})}$. Note that identity (\[mapp2\]) was obtained in [@IsOg2] for slightly different definition of the baxterized elements (\[a00\]).
[**Proposition \[sec6\].3**]{} (see also [@IsOg2], [@Isa07]). [*Let $y_{k}(x) \in BMW_{k}(C)$ be any solution of the RE (\[reflH\]). The operators \_[k-1]{}(x) = \_[\_[(k)]{}]{} ( y\_[k]{}(x) ) BMW\_[k-1]{}(C) , form a commutative family of operators =0 (x,z) , in the subalgebra $BMW_{k-1}(C) \subset BMW_{n}(C)$ $(k < n)$.* ]{}
[**Proof.**]{} Using properties (\[qtr2\]), (\[map\]) and relations (\[mapp2\]), (\[reflH\]) we find $$\begin{array}{c}
\tau_{k-1}(x) \, \tau_{k-1}(z) = Tr_{_{(k)}} \left( y_{k}(x) \, \tau_{k-1}(z) \right) = \\ [0.2cm]
= F(x \, z) \,
\Tr _{_{(k)}} \left( y_{k}(x) \, \Tr _{_{(k+1)}}
\Bigl( T_k (x \, z) \, y_{k}(z)
T_k(q^2 (\nu^2 xz)^{-1}) \Bigr)
\right)= \\ [0.2cm]
= F(x \, z) \,
\Tr _{_{(k)}} \Tr _{_{(k+1)}} \Bigl( T_k(x/z) \, y_{k}(x) \, T_k (xz) \, y_{k}(z)
\, T_k^{-1}(x/z) \, T_k (q^2 (\nu^2 xz)^{-1})
\Bigr)= \\ [0.2cm]
= F(x \, z) \,
\Tr _{_{(k)}} \Tr _{_{(k+1)}} \left( y_{k}(z) \, T_k (xz) \, y_{k}(x)
\, T_k (q^2 (\nu^2 xz)^{-1}) \right)= \\ [0.2cm]
=
\Tr _{_{(k)}}\left( y_{k}(z) \, \tau_{k-1}(x) \right) = \tau_{k-1}(z) \, \tau_{k-1}(x) \; ,
\end{array}$$ where $F(x)$ was defined in (\[mapp2\]).\
Now we consider the operators (\[tau11\]) \_n(x;\_[(n)]{}) = \_[\_[(n+1)]{}]{} ( y\_[n+1]{}(x;\_[(n)]{})) BMW\_[n]{}(C) , where solution $y_{n+1}(x) \in BMW_{n+1}(C)$ of the reflection equation is taken in the form (\[xxz55\]). We stress that the elements (\[bethe02\]) are nothing but the analogs of Sklyanin’s transfer-matrices [@Skl] and the coefficients in the expansion of $\tau_n(x;\vec{z}_{(n)})$ over the variable $x$ (for the homogeneous case $z_k=1$) are the Hamiltonians for the open Birman-Murakami-Wenzel chain models with nontrivial boundary conditions.
For example let us redefine all baxterized elements in (\[a00\]) T\_i(x) \_i(x) = (1-x) T\_i(x) = (1-x) ([T]{}\_i + \_i ) + (q-q\^[-1]{})x , such that the new elements $\tilde{T}_i(x)$ satisfies conditions $$\left. \tilde{T}_i(x) \right|_{x=1} = (q-q^{-1}) \;\; , \;\;\;\;\;
\left. \partial_x \, \tilde{T}_i(x) \right|_{x=1} = -{\sf T}_i
- \frac{(q-q^{-1})}{1+ \nu^{-1}q} \, \kappa_i + (q-q^{-1}) \; .$$ \_i(u/v) \_i(v/u) = . Now we respectively redefine the Sklyanin’s transfer-matrix element (\[bethe02\]) as the following \_n(x;\_[(n)]{}) = \_[i=1]{}\^n ( (1-x/z\_i) (1-x z\_i ) ) \_[\_[(n+1)]{}]{} ( y\_[n+1]{}(x;\_[(n)]{})) = \_[\_[(n+1)]{}]{} ( \_[n+1]{}(x;\_[(n)]{})) , where $\tilde{y}_{n+1}(x;\vec{z}_{(n)})$ is given by (\[xxz55\]) with substitution $T_i(x) \to \tilde{T}_i(x)$ and $k \to n+1$. I.e., we have
[c]{} \_[k]{}(x;\_[(k-1)]{}) = \_[k-1]{}() \_2() \_1() y\_1(x) \_[1]{}( x z\_1) \_2( x z\_2) \_[k-1]{}( x z\_[k-1]{}) =\
\[0.2cm\] = \_[k-1]{}() \_[k-1]{}(x;\_[(k-2)]{}) \_[k-1]{}( x z\_[k-1]{}) .
Using “unitarity condition” (\[bax4\]) we represent baxterized Jucys-Murphy elements (\[xxz55w\]) in the form
[c]{} \_[k]{}(x;\_[(k-1)]{}) = ( \_[i=1]{}\^[k-1]{} ) \_[k]{}’(x;\_[(k-1)]{}) ,\
\[0.3cm\] \_[k]{}’(x;\_[(k-1)]{}) \_[k-1]{}\^[-1]{}() \_1\^[-1]{}() y\_1(x) \_[1]{}( x z\_1) \_2( x z\_2) \_[k-1]{}( x z\_[k-1]{}) .
We will use this form below.
Then, for the homogeneous case $z_i=1$ $(\forall i)$, we consider the coefficient $$\frac{c^{1/2} (q-q^{-1})^{1-2n}}{2\nu \mu } \Bigl(
\left. \partial_x \, \tilde{\tau}_n(x;z_i=1)\; \right|_{x=1} \Bigr) =
\sum_{i=1}^{n-1} \Bigl( {\sf T}_i
+ \frac{(q-q^{-1})}{1+ \nu^{-1}q} \, \kappa_i \Bigr)
+ \frac{(q-q^{-1})}{2} \frac{c {\sf T}_0^2 -1}{(c^{1/2} {\sf T}_0 -1)^2} + {\rm constant} \; ,$$ in the expansion of the generating function $\tilde{\tau}_n(x;z_i=1)$ for commutative elements. This coefficient gives (up to an additional constant) the element $${\cal H} = \frac{(q-q^{-1})}{2} \frac{c {\sf T}_0^2 -1}{(c^{1/2} {\sf T}_0 -1)^2} +
\sum_{i=1}^{n-1} \Bigl( {\sf T}_i
+ \frac{(q-q^{-1})}{1+ \nu^{-1}q} \, \kappa_i \Bigr) \;\; \in \;\;
BMW_n(C) \; ,$$ being the local Hamiltonian for the open BMW chain model with nontrivial boundary condition for the first site of the chain.
Consider the expansion of $\tau_n(x;\vec{z}_{(n)})$ over $x$ for the inhomogeneous case: \_n(x;\_[(n)]{}) = \_[k=-]{}\^\_k(\_[(n)]{}) x\^k BMW\_[n]{}(C) . According to the Proposition \[sec6\].3, for fixed parameters $\vec{z}_{(n)}=(z_{1},\dots,z_n)$, the elements $\Phi_k(\vec{z}_{(n)})$ generate a commutative subalgebra $\hat{\cal B}_n(\vec{z}_{(n)}) \subset BMW_{n}(C) $. These elements are interpreted as Hamiltonians for the inhomogeneous open Hecke chain models. Following [@MTV] we call the subalgebras $\hat{\cal B}_n(\vec{z}_{(n)})$ as Bethe subalgebras of the affine algebra $BMW_{n}(C)$.
Bethe subalgebras for affine BMW algebra and q-KZ connections
-------------------------------------------------------------
In this Section and below we will use the normalized baxterized elements (\[bax3\]): $\widetilde{T}_k(x) = (1-x) T_k(x)$. Consider the transfer-matrix operator (\[bethe02z\]) and fix the spectral parameter as $x=z_k$, where $1 \leq k \leq n$ (analogous results can be obtained if instead we take $x=z_k^{-1}$). In view of relation $\left. T_k(x/z_k)\right|_{x=z_k} = (q-q^{-1})$ we deduce for the transfer-matrix operator (\[bethe02z\]) $$\begin{array}{c}
B_k(\vec{z}) = \frac{1}{(q-q^{-1})} \tau_{n}(z_k ;\vec{z}_{(n)}) =
\Tr _{_{\!\! (n+1)}} \! \Bigl( \!
\underline{\tilde{T}_{n}(\frac{z_k}{z_{n}}) \cdots \tilde{T}_{k+1}(\frac{z_k}{z_{k+1}})}
\tilde{T}_{k-1}(\frac{z_k}{z_{k-1}}) \cdots \tilde{T}_1(\frac{z_k}{z_1}) \, y_1(z_k) \cdot \\ [0.3cm]
\cdot
\tilde{T}_{1}(z_k z_1) \cdots \tilde{T}_{k-1} (z_k z_{k-1}) \tilde{T}_k(z_k^2)
\tilde{T}_{k+1} (z_k z_{k+1}) \cdots \tilde{T}_{n}(z_k z_{n})\! \Bigr) = \\ [0.3cm]
\Tr _{_{\!\! (n+1)}} \! \Bigl( \!
\tilde{T}_{k-1}(\frac{z_k}{z_{k-1}}) \cdots \tilde{T}_1(\frac{z_k}{z_1}) \, y_1(z_k) \cdot \\ [0.3cm]
\cdot \tilde{T}_{1}(z_k z_1) \cdots \tilde{T}_{k-1} (z_k z_{k-1}) \cdot
\underline{\tilde{T}_{n}(\frac{z_k}{z_{n}}) \cdots \tilde{T}_{k+1}(\frac{z_k}{z_{k+1}})} \tilde{T}_k(z_k^2)
\tilde{T}_{k+1} (z_k z_{k+1}) \cdots \tilde{T}_{n}(z_k z_{n})\! \Bigr) =
\end{array}$$
[c]{} \_[\_[ (n+1)]{}]{} ( \_[k-1]{}() \_1() y\_1(z\_k) \_[1]{}(z\_k z\_1) \_[k-1]{} (z\_k z\_[k-1]{})\
\[0.3cm\] \_[k]{}(z\_k z\_[k+1]{}) \_[n-1]{}(z\_k z\_[n]{}) \_n(z\_k\^2) \_[n-1]{}() \_[k+1]{} () \_[k]{} () ) .
Now we use relations (\[qtr1\]) to obtain $$\Tr _{_{\!\! (n+1)}} \! \Bigl( \! \tilde{T}_n(z^2) \! \Bigr) =
\frac{(q^2 -z^2 \nu^2)(z^2 \nu +q^{-1})}{(z^2\nu +q)}
\equiv N(z^2) \; .$$ Then for (\[qkz01\]) we deduce
[c]{} B\_k() = N(z\_k\^2) ( T\_[k-1]{}() T\_1() y\_1(z\_k) T\_[1]{}(z\_k z\_1) T\_[k-1]{} (z\_k z\_[k-1]{}) )\
\[0.3cm\] ( T\_[k]{}(z\_k z\_[k+1]{}) T\_[n-1]{}(z\_k z\_[n]{}) T\_[n-1]{}() T\_[k]{} () ) =\
\[0.3cm\] = N(z\_k\^2) \_k (z\_k, \_[(1,k-1)]{}) \_k (z\_k, \_[(k+1,n)]{}) = N(z\_k\^2) ( \_[i=1]{}\^[k-1]{} ) [A]{}’\_k() ,
where ’\_k() = \_[k]{}’(z\_k;\_[(k-1)]{}) \_k (z\_k, \_[(k+1,n)]{}) , $$\begin{array}{c}
\overline{y}_k (z_k, \vec{z}_{(k+1,n)}) =
\! \tilde{T}_{k}(z_k z_{k+1}) \cdots
\tilde{T}_{n-1}(z_k z_{n}) \cdot \tilde{T}_{n-1}(\frac{z_k}{z_{n}})
\cdots \tilde{T}_{k+1} (\frac{z_k}{z_{k+2}}) \tilde{T}_{k} (\frac{z_k}{z_{k+1}}) \; ,
\end{array}$$ and elements $\tilde{y}_k (x, \vec{z}_{(1,k-1)})$, $\tilde{y}_k'(x, \vec{z}_{(1,k-1)})$ were defined in (\[xxz55w\]), (\[xxz55y\]).
Operators (\[qkz02\]) are equal to the transfer-matrix operator $\tau_{n}(x ;\vec{z}_{(n)})$ evaluated at the points $x=z_k$. Thus, by definition the operators $\{ B_1(\vec{z}\,) , \dots , B_n(\vec{z}\,) \}$ form a commutative set of elements in the algebra $BMW_{n}(C)$: \[ B\_k() , B\_r() \] = 0 (k,r = 1, …, n) . Thus, operators $\{ B_1(\vec{z}\,) , \dots , B_n(\vec{z}\,) \}$ for fixed parameters $\{ z_1, \dots, z_n \}$ can be considered as generators of the Bethe subalgebra in $BMW_{n}(C)$.
The validity of the identities (\[Bax06\]) can be shown in different way. For this, in view of (\[qkz02\]), we need to prove the commutativity of the set of elements ${\sf A}'_k(\vec{z}\,) \; \in \; BMW_{n}(C)$ which can be interpreted as analogs of flat connections (\[Zam77\]) for quantum Knizhnik-Zamolodchikov equations. Taking into account (\[Zam10D\]) we see that the map $\tilde{\rho}_{c}$: $B_n(C) \; \to \; BMW_{n}(C)$:
[c]{} \_[c]{}(T\_i) = s\_i \_[i]{}(z\_i,z\_[i+1]{}) (i=1,…,n-1) , \_[c]{}(T\_0) = y\_1(z\_1) s\_0 ,
where $s_0$ is defined in (\[Affbg1\]) with $\sigma(z_1) = 1/z_1$ , is the representation of $B_n(C)$. Then we have the following statement.
[**Proposition \[sec6\].4**]{} [*The flat connections ${\sf A}_i'(\vec{z}\,)$ (\[Bax05\]) are images $\tilde{\rho}_{c}({\sf J}_i)$ of the pairwise commuting elements $${\sf J}_i = (T_{i-1}^{-1} \cdots T_1^{-1} T_0 T_1 \cdots T_{i-1})
(T_i \cdots T_{n-1} \cdot T_{n-1} \cdots T_i) \in B_n(C) \; ,$$ which are obtained by the projection $T_n \to 1$ from the elements $J_i \in B_n(C^{(1)})$ given in (\[jucys1\])*]{}.\
[**Proof.**]{} The formula ${\sf A}_i'(\vec{z}\,) =
\tilde{\rho}_{c}({\sf J}_i)$ can be checked directly with the use of definition (\[Bax05\]) of ${\sf A}_i'(\vec{z}\,)$ and formulas (\[Zam10K\]) for the map $\tilde{\rho}_{c}$.
[**Remark.**]{} Using the special limit in (\[Zam77\]), one can deduce the BMW analog of the Cherednik’s connections A\_k() = T\_[k-1]{}( ) T\_1() y\_1\^[-1]{} [T]{}\_1\^[-1]{} \_[n-1]{}\^[-1]{} D\_[z\_k]{} T\_[n-1]{}() T\_[k]{} ( ) BMW\_n(C) , which were presented for the Hecke algebra case in [@Ch] (see there eq. (4.12) in Section 4.2). The finite difference operator $D_{z_k}$ is given in (\[Zam57\]) with $\tilde{\bar{z}}= \frac{c}{c'} z$. In (\[qkz03\]) we have to take into account that Cherednik’s affine elements $Y_k$ are related to ours by $Y_k = y_k^{-1}$.
To rewrite our expression (\[Zam77\]) to the Cherednik’s one (\[qkz03\]) we need to convert the factor L\_1(z\_k) T\_[1]{}( c z\_k z\_1) T\_[k-1]{} (c z\_k z\_[k-1]{}) T\_[k]{}(c z\_k z\_[k+1]{}) T\_[n-1]{}(c z\_k z\_[n]{}) |[L]{}\_n( ) , entered into the expression (\[Zam77\]) to the factor $y_1^{-1} {\sf T}_1^{-1} \cdots {\sf T}_{n-1}^{-1}$. It can be done if we first make in (\[Zam77\]) the redefinition of all spectral parameters $z_r \to t z_r$ and then consider the limit $t \to \infty$. To do this we note that only the product (\[qkz05\]) in (\[Zam77\]) will be dependent on $t$, where we have to use limits $$\lim_{t \to \infty} T_r(t^2 c z_k z_r) =
{\sf T}_r - (q - q^{-1}) + (q - q^{-1}) \kappa_r = {\sf T}_r^{-1} \; ,$$ $$\lim_{t \to \infty} L_1(t z_k) = \frac{1}{c} \, y_1^{-1}
= \frac{1}{c} \, {\sf T}_0^{-1}\; ,
\;\;\;\;
\lim_{t \to \infty} \bar{L}_n \Bigl( \frac{1}{t c z_k} \Bigr)
= \bar{y}_n = {\sf T}_n \; .$$ Here we used the expressions for baxterized elements (\[a00\]), (\[rea15\]) and (\[rea151\]). Thus, the limit of the factor (\[qkz05\]) is $$y_1^{-1} \cdot {\sf T}_1^{-1} \cdots {\sf T}_{n-1}^{-1} \cdot \bar{y}_n =
{\sf T}_0^{-1} \cdot {\sf T}_1^{-1} \cdots {\sf T}_{n-1}^{-1}
\cdot {\sf T}_n \equiv {\sf X} \; ,$$ and for the limit of the connection (\[Zam77\]) we obtain expression $$A_k'(\vec{z}) =
T_{k-1}\Bigl( \frac{ z_k}{z_{k-1}}\Bigr)
\cdots T_1\Bigl(\frac{ z_k}{z_1}\Bigr) \cdot
{\sf X} \, D_{z_k} \cdot
T_{n-1}\Bigl(\frac{z_k}{z_{n}}\Bigr)
\cdots T_{k+1} \Bigl(\frac{z_k}{z_{k+2}}\Bigr) T_{k} \Bigl( \frac{z_k}{z_{k+1}}
\! \Bigr) \; \in \; BMW_n(C^{(1)})\; ,$$ which is a generalization of (\[qkz03\]). The projection ${\sf T}_n \to 1$ for connection $A_k'(\vec{z})$ gives the BMW analog of the Cherednik’s connection (\[qkz03\]).
[**Acknowledgment.**]{} The work of API was supported by RSCF grant 14-11-00598.
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[^1]: Naive replacement of generators $T_i$ in $(\ref{jucys1})$ by its baxterization $T_i(u/v)$ defined in (\[a00\]), leads to the set of elements in the $BMW$ algebra, which do not commute in general
|
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abstract: 'A standard form of analysis for linguistic typology is the universal implication. These implications state facts about the range of extant languages, such as “if objects come after verbs, then adjectives come after nouns.” Such implications are typically discovered by painstaking hand analysis over a small sample of languages. We propose a computational model for assisting at this process. Our model is able to discover both well-known implications as well as some novel implications that deserve further study. Moreover, through a careful application of hierarchical analysis, we are able to cope with the well-known sampling problem: languages are not independent.'
author:
- |
Hal Daumé III\
School of Computing\
University of Utah\
[[email protected]]{} Lyle Campbell\
Department of Linguistics\
University of Utah\
[[email protected]]{}
title: A Bayesian Model for Discovering Typological Implications
---
Introduction
============
Linguistic typology aims to distinguish between logically possible languages and actually observed languages. A fundamental building block for such an understanding is the *universal implication* [@greenberg63universals]. These are short statements that restrict the space of languages in a concrete way (for instance “object-verb ordering implies adjective-noun ordering”); , and provide excellent introductions to linguistic typology. We present a statistical model for automatically discovering such implications from a large typological database [@wals].
Analyses of universal implications are typically performed by linguists, inspecting an array of $30$-$100$ languages and a few pairs of features. Looking at all pairs of features (typically several hundred) is virtually impossible by hand. Moreover, it is insufficient to simply look at counts. For instance, results presented in the form “verb precedes object implies prepositions in 16/19 languages” are nonconclusive. While compelling, this is not enough evidence to decide if this is a statistically well-founded implication. For one, maybe $99\%$ of languages have prepositions: then the fact that we’ve achieved a rate of $84\%$ actually seems really bad. Moreover, if the $16$ languages are highly related historically or areally (geographically), and the other $3$ are not, then we may have only learned something about geography.
In this work, we propose a statistical model that deals cleanly with these difficulties. By building a computational model, it is possible to apply it to a very large typological database and search over many thousands of pairs of features. Our model hinges on two novel components: a statistical noise model a hierarchical inference over language families. To our knowledge, there is no prior work directly in this area. The closest work is represented by the books *Possible and Probable Languages* [@newmeyer05probable] and *Language Classification by Numbers* [@mcmahon05language], but the focus of these books is on automatically discovering phylogenetic trees for languages based on Indo-European cognate sets [@dyen92indoeuropean].
Data
====
\[sec:data\]
\[fig:world\]
------------------ --------------------- --------------------- ------------------- --------------------- -------------- -------------------
[**Numeral**]{} [**Glottalized**]{} [**Number of**]{}
[**Language**]{} [**Classifiers**]{} [**Rel/N Order**]{} [**O/V Order**]{} [**Consonants**]{} [**Tone**]{} [**Genders**]{}
English Absent NRel VO None None Three
Hindi Absent RelN OV None None Two
Mandarin Obligatory RelN VO None Complex None
Russian Absent NRel VO None None Three
Tukang Besi Absent ? Either Implosives None Three
Zulu Absent NRel VO Ejectives Simple Five+
------------------ --------------------- --------------------- ------------------- --------------------- -------------- -------------------
\[tab:example-features\]
The database on which we perform our analysis is the *World Atlas of Language Structures* [@wals]. This database contains information about $2150$ languages (sampled from across the world; Figure \[fig:world\] depicts the locations of languages). There are $139$ *features* in this database, broken down into categories such as “Nominal Categories,” “Simple Clauses,” “Phonology,” “Word Order,” etc. The database is *sparse*: for many language/feature pairs, the feature value is unknown. In fact, only about $16\%$ of all possible language/feature pairs are known. A sample of five languages and six features from the database are shown in Table \[tab:example-features\].
Importantly, the density of samples is not random. For certain languages (eg., English, Chinese, Russian), nearly all features are known, whereas other languages (eg., Asturian, Omagua, Frisian) that have fewer than five feature values known. Furthermore, some features are known for many languages. This is due to the fact that certain features take less effort to identify than others. Identifying, for instance, if a language has a particular set of phonological features (such as glottalized consonants) requires only listening to speakers. Other features, such as determining the order of relative clauses and nouns require understanding much more of the language.
Models
======
\[sec:models\]
In this section, we propose two models for automatically uncovering universal implications from noisy, sparse data. First, note that even well attested implications are not always exceptionless. A common example is that verbs preceding objects (“VO”) implies adjectives following nouns (“NA”). This implication (VO $\supset$ NA) has one glaring exception: English. This is one particular form of noise. Another source of noise stems from transcription. WALS contains data about languages documented by field linguists as early as the 1900s. Much of this older data was collected before there was significant agreement in documentation style. Different field linguists often had different dimensions along which they segmented language features into classes. This leads to noise in the properties of individual languages.
Another difficulty stems from the *sampling problem.* This is a well-documented issue (see, eg., [@croft03typology]) stemming from the fact that any set of languages is not sampled uniformly from the space of all probable languages. Politically interesting languages (eg., Indo-European) and typologically unusual languages (eg., Dyirbal) are better documented than others. Moreover, languages are not independent: German and Dutch are more similar than German and Hindi due to history and geography.
The first model, <span style="font-variant:small-caps;">Flat</span>, treats each language as independent. It is thus susceptible to sampling problems. For instance, the WALS database contains a half dozen versions of German. The <span style="font-variant:small-caps;">Flat</span> model considers these versions of German just as statistically independent as, say, German and Hindi. To cope with this problem, we then augment the <span style="font-variant:small-caps;">Flat</span> model into a <span style="font-variant:small-caps;">Hier</span>archical model that takes advantage of known hierarchies in linguistic phylogenetics. The <span style="font-variant:small-caps;">Hier</span> model explicitly models the fact that individual languages are *not* independent and exhibit strong familial dependencies. In both models, we initially restrict our attention to pairs of features. We will describe our models as if all features are binary. We expand any multi-valued feature with $K$ values into $K$ binary features in a “one versus rest” manner.
The <span style="font-variant:small-caps;">Flat</span> Model
------------------------------------------------------------
\[sec:models:flat\]
In the <span style="font-variant:small-caps;">Flat</span> model, we consider a $2 \times N$ matrix of feature values. The $N$ corresponds to the number of languages, while the $2$ corresponds to the two features currently under consideration (eg., object/verb order and noun/adjective order). The order of the two features is important: $f_1$ implies $f_2$ is logically different from $f_2$ implies $f_1$. Some of the entries in the matrix will be unknown. We may safely remove all languages from consideration for which *both* are unknown, but we do *not* remove languages for which only one is unknown. We do so because our model needs to capture the fact that if $f_2$ is *always* true, then $f_1
\supset f_2$ is uninteresting.
The statistical model is set up as follows. There is a single variable (we will denote this variable “$m$”) corresponding to whether the implication holds. Thus, $m=1$ means that $f_1$ implies $f_2$ and $m=0$ means that it does not. Independent of $m$, we specify two feature priors, $\pi_1$ and $\pi_2$ for $f_1$ and $f_2$ respectively. $\pi_1$ specifies the prior probability that $f_1$ will be true, and $\pi_2$ specifies the prior probability that $f_2$ will be true. One can then put the model together naïvely as follows. If $m=0$ (i.e., the implication does not hold), then the entire data matrix is generated by choosing values for $f_1$ (resp., $f_2$) independently according to the prior probability $\pi_1$ (resp., $\pi_2$). On the other hand, if $m=1$ (i.e., the implication *does* hold), then the first column of the data matrix is generated by choosing values for $f_1$ independently by $\pi_1$, but the second column is generated differently. In particular, if for a particular language, we have that $f_1$ is true, then the fact that the implication holds means that $f_2$ *must* be true. On the other hand, if $f_1$ is false for a particular language, then we may generate $f_2$ according to the prior probability $\pi_2$. Thus, having $m=1$ means that the model is significantly more constrained. In equations:
$$\begin{aligned}
p(f_1 \| \pi_1) &= \pi_1^{f_1} (1-\pi_1)^{1-f_1} \\
p(f_2 \| f_1, \pi_2, m) &=
\brack{ f_2 & m = f_1 = 1 \\
\pi_2^{f_2} (1-\pi_2)^{1-f_2} & \text{otherwise} }
$$
The problem with this naïve model is that it does not take into account the fact that there is “noise” in the data. (By noise, we refer either to mis-annotations, or to “strange” languages like English.) To account for this, we introduce a simple noise model. There are several options for parameterizing the noise, depending on what independence assumptions we wish to make. One could simply specify a noise rate for the entire data set. One could alternatively specify a language-specific noise rate. Or one could specify a feature-specific noise rate. We opt for a blend between the first and second option. We assume an underlying noise rate for the entire data set, but that, conditioned on this underlying rate, there is a language-specific noise level. We believe this to be an appropriate noise model because it models the fact that the majority of information for a single language is from a single source. Thus, if there is an error in the database, it is more likely that other errors will be for the same languages.
In order to model this statistically, we assume that there are latent variables $e_{1,n}$ and $e_{2,n}$ for each language $n$. If $e_{1,n}=1$, then the first feature for language $n$ is wrong. Similarly, if $e_{2,n}=1$, then the second feature for language $n$ is wrong. Given this model, the probabilities are exactly as in the naïve model, with the exception that instead of using $f_1$ (resp., $f_2$), we use the exclusive-or[^1] $f_1 \otimes e_1$ (resp., $f_2 \otimes e_2$) so that the feature values are flipped whenever the noise model suggests an error.
\[fig:flat-gm\]
The graphical model for the <span style="font-variant:small-caps;">Flat</span> model is shown in Figure \[fig:flat-gm\]. Circular nodes denote random variables and arrows denote conditional dependencies. The rectangular plate denotes the fact that the elements contained within it are replicated $N$ times ($N$ is the number of languages). In this model, there are four “root” nodes: the implication value $m$; the two feature prior probabilities $\pi_1$ and $\pi_2$; and the language-specific error rate $\ep$. On all of these nodes we place Bayesian priors. Since $m$ is a binary random variable, we place a Bernoulli prior on it. The $\pi$s are Bernoulli random variables, so they are given independent Beta priors. Finally, the noise rate $\ep$ is also given a Beta prior. For the two Beta parameters governing the error rate (i.e., $a_\ep$ and $b_\ep$) we set these by hand so that the mean expected error rate is $5\%$ and the probability of the error rate being between $0\%$ and $10\%$ is $50\%$ (this number is based on an expert opinion of the noise-rate in the data). For the rest of the parameters we use uniform priors.
The <span style="font-variant:small-caps;">Hier</span> Model
------------------------------------------------------------
\[sec:models:hier\]
A significant difficulty in working with any large typological database is that the languages will be sampled *non*uniformly. In our case, this means that implications that seem true in the <span style="font-variant:small-caps;">Flat</span> model may only be true for, say, Indo-European, and the remaining languages are considered noise. While this may be interesting in its own right, we are more interested in discovering implications that are truly universal.
We model this using a hierarchical Bayesian model. In essence, we take the <span style="font-variant:small-caps;">Flat</span> model and build a notion of language relatedness into it. In particular, we enforce a hierarchy on the $m$ implication variables. For simplicity, suppose that our “hierarchy” of languages is nearly flat. Of the $N$ languages, half of them are Indo-European and the other half are Austronesian. We will use a nearly identical model to the <span style="font-variant:small-caps;">Flat</span> model, but instead of having a single $m$ variable, we have three: one for IE, one for Austronesian and one for “all languages.”
For a general tree, we assign one implication variable for each node (including the root and leaves). The goal of the inference is to infer the value of the $m$ variable corresponding to the root of the tree.
All that is left to specify the full <span style="font-variant:small-caps;">Hier</span> model is to specify the probability distribution of the $m$ random variables. We do this as follows. We place a zero mean Gaussian prior with (unknown) variance $\si^2$ on the root $m$. Then, for a non-root node, we use a Gaussian with mean equal to the “$m$” value of the parent and tied variance $\si^2$. In our three-node example, this means that the root is distributed $\Nor(0,\si^2)$ and each child is distributed $\Nor(m_{\text{root}},\si^2)$, where $m_\text{root}$ is the random variable corresponding to the root. Finally, the leaves (corresponding to the languages themselves) are distributed *logistic-binomial*. Thus, the $m$ random variable corresponding to a leaf (language) is distributed $\Bin(s(m_{\text{par}}))$, where $m_\text{par}$ is the $m$ value for the parent (internal) node and $s$ is the sigmoid function $s(x) = [1 + exp(-x)]^{-1}$.
The intuition behind this model is that the $m$ value at each node in the tree (where a node is either “all languages” or a specific language family or an individual language) specifies the extent to which the implication under consideration holds for that node. A large positive $m$ means that the implication is very likely to hold. A large negative value means it is very likely to not hold. The normal distributions across edges in the tree indicate that we expect the $m$ values not to change too much across the tree. At the leaves (i.e., individual languages), the logistic-binomial simply transforms the real-valued $m$s into the range $[0,1]$ so as to make an appropriate input to the binomial distribution.
Statistical Inference
=====================
In this section, we describe how we use Markov chain Monte Carlo methods to perform inference in the statistical models described in the previous section; provide an excellent introduction to MCMC techniques. The key idea behind MCMC techniques is to approximate intractable expectations by drawing random samples from the probability distribution of interest. The expectation can then be approximated by an empirical expectation over these sample.
For the <span style="font-variant:small-caps;">Flat</span> model, we use a combination of Gibbs sampling with rejection sampling as a subroutine. Essentially, all sampling steps are standard Gibbs steps, except for sampling the error rates $e$. The Gibbs step is not available analytically for these. Hence, we use rejection sampling (drawing from the Beta prior and accepting according to the posterior).
The sampling procedure for the <span style="font-variant:small-caps;">Hier</span> model is only slightly more complicated. Instead of performing a simple Gibbs sample for $m$ in Step (4), we first sample the $m$ values for the internal nodes using simple Gibbs updates. For the leaf nodes, we use rejection sampling. For this rejection, we draw proposal values from the Gaussian specified by the parent $m$, and compute acceptance probabilities.
In all cases, we run the outer Gibbs sampler for $1000$ iterations and each rejection sampler for $20$ iterations. We compute the marginal values for the $m$ implication variables by averaging the sampled values after dropping $200$ “burn-in” iterations.
Data Preprocessing and Search
=============================
After extracting the raw data from the WALS electronic database [@wals][^2], we perform a minor amount of preprocessing. Essentially, we have manually removed certain feature values from the database because they are underrepresented. For instance, the “Glottalized Consonants” feature has eight possible values (one for “none” and seven for different varieties of glottalized consonants). We reduce this to simply two values “has” or “has not.” $313$ languages have no glottalized consonants and $139$ have some variety of glottalized consonant. We have done something similar with approximately twenty of the features.
For the <span style="font-variant:small-caps;">Hier</span> model, we obtain the hierarchy in one of two ways. The first hierarchy we use is the “linguistic hierarchy” specified as part of the WALS data. This hierarchy divides languages into families and subfamilies. This leads to a tree with the leaves at depth four. The root has $38$ immediate children (corresponding to the major families), and there are a total of $314$ internal nodes. The second hierarchy we use is an areal hierarchy obtained by clustering languages according to their latitude and longitude. For the clustering we first cluster all the languages into $6$ “macro-clusters.” We then cluster each macro-cluster individually into $25$ “micro-clusters.” These micro-clusters then have the languages at their leaves. This yields a tree with $31$ internal nodes.
Given the database (which contains approximately $140$ features), performing a raw search even over all possible *pairs* of features would lead to over $19,000$ computations. In order to reduce this space to a more manageable number, we filter:
There must be at least $250$ languages for which *both* features are known.
There must be at least $15$ languages for which both feature values hold simultaneously.
Whenever $f_1$ is true, at least half of the languages also have $f_2$ true.
Performing all these filtration steps reduces the number of pairs under consideration to $3442$. While this remains a computationally expensive procedure, we were able to perform all the implication computations for these $3442$ possible pairs in about a week on a single modern machine (in Matlab).
Results
=======
The task of discovering universal implications is, at its heart, a data-mining task. As such, it is difficult to evaluate, since we often do not know the correct answers! If our model only found well-documented implications, this would be interesting but useless from the perspective of aiding linguists focus their energies on new, plausible implications. In this section, we present the results of our method, together with both a quantitative and qualitative evaluation.
Quantitative Evaluation
-----------------------
In this section, we perform a quantitative evaluation of the results based on *predictive power.* That is, one generally would prefer a system that finds implications that hold with high probability across the data. The word “generally” is important: this quality is neither necessary nor sufficient for the model to be good. For instance, finding $1000$ implications of the form $A_1 \supset X, A_2
\supset X, \dots, A_{1000} \supset X$ is completely uninteresting if $X$ is true in $99\%$ of the cases. Similarly, suppose that a model can find $1000$ implications of the form $X \supset A_1, \dots, X
\supset A_{1000}$, but $X$ is only true in five languages. In both of these cases, according to a “predictive power” measure, these would be ideal systems. But they are both somewhat uninteresting.
Despite these difficulties with a predictive power-based evaluation, we feel that it is a good way to understand the relative merits of our different models. Thus, we compare the following systems: <span style="font-variant:small-caps;">Flat</span> (our proposed flat model), <span style="font-variant:small-caps;">LingHier</span> (our model using the phylogenetic hierarchy), <span style="font-variant:small-caps;">DistHier</span> (our model using the areal hierarchy) and <span style="font-variant:small-caps;">Random</span> (a model that ranks implications—that meet the three qualifications from the previous section—randomly).
The models are scored as follows. We take the entire WALS data set and “hide” a random $10\%$ of the entries. We then perform full inference and ask the inferred model to predict the missing values. The accuracy of the model is the accuracy of its predictions. To obtain a sense of the quality of the ranking, we perform this computation on the top $k$ ranked implications provided by each model; $k \in \{ 2,4,8,\dots,512,1024 \}$.
The results of this quantitative evaluation are shown in Figure \[fig:quant\] (on a log-scale for the x-axis). The two best-performing models are the two hierarchical models. The flat model does significantly worse and the random model does terribly. The vertical lines are a standard deviation over $100$ folds of the experiment (hiding a different $10\%$ each time). The difference between the two hierarchical models is typically not statistically significant. At the top of the ranking, the model based on phylogenetic information performs marginally better; at the bottom of the ranking, the order flips. Comparing the hierarchical models to the flat model, we see that adequately modeling the *a priori* similarity between languages is quite important.
Cross-model Comparison
----------------------
The results in the previous section support the conclusion that the two hierarchical models are doing something significantly different (and better) than the flat model. This clearly must be the case. The results, however, do not say whether the two hierarchies are substantially different. Moreover, are the results that they produce substantially different. The answer to these two questions is “yes.”
We first address the issue of tree similarity. We consider all pairs of languages which are at distance $0$ in the areal tree (i.e., have the same parent). We then look at the mean tree-distance between those languages in the phylogenetic tree. We do this for all distances in the areal tree (because of its construction, there are only three: $0$, $2$ and $4$). The mean distances in the phylogenetic tree corresponding to these three distances in the areal tree are: $2.9$, $3.5$ and $4.0$, respectively. This means that languages that are “nearby” in the areal tree are quite often very far apart in the phylogenetic tree.
To answer the issue of whether the results obtained by the two trees are similar, we employ Kendall’s $\tau$ statistic. Given two ordered lists, the $\tau$ statistic computes how correlated they are. $\tau$ is always between $0$ and $1$, with $1$ indicating identical ordering and $0$ indicated completely reversed ordering. The results are as follows. Comparing <span style="font-variant:small-caps;">Flat</span> to <span style="font-variant:small-caps;">LingHier</span> yield $\tau =
0.4144$, a very low correlation. Between <span style="font-variant:small-caps;">Flat</span> and <span style="font-variant:small-caps;">DistHier</span>, $\tau = 0.5213$, also very low. These two are as expected. Finally, between <span style="font-variant:small-caps;">LingHier</span> and <span style="font-variant:small-caps;">DistHier</span>, we obtain $\tau=0.5369$, a very low correlation, considering that both perform well predictively.
Qualitative Analysis
--------------------
For the purpose of a qualitative analysis, we reproduce the top $30$ implications discovered by the <span style="font-variant:small-caps;">LingHier</span> model in Table \[tab:qualitative\] (see the final page).[^3] Each implication is numbered, then the actual implication is presented. For instance, \#7 says that any language that has adjectives preceding their governing nouns also has numerals preceding their nouns. We additionally provide an “analysis” of many of these discovered implications. Many of them (eg., \#7) are well known in the typological literature. These are simply numbered according to well-known references. For instance our \#7 is implication \#18 from Greenberg, reproduced by . Those that reference Hawkins (eg., \#11) are based on implications described by ; those that reference Lehmann are references to the principles decided by in Ch 4 & 8.
Some of the implications our model discovers are obtained by composition of well-known implications. For instance, our \#3 (namely, OV $\supset$ Genitive-Noun) can be obtained by combining Greenberg \#4 (OV $\supset$ Postpositions) and Greenberg \#2a (Postpositions $\supset$ Genitive-Noun). It is quite encouraging that $14$ of our top $21$ discovered implications are well-known in the literature (and this, not even considering the tautalogically true implications)! This strongly suggests that our model is doing something reasonable and that there is true structure in the data.
In addition to many of the known implications found by our model, there are many that are “unknown.” Space precludes attempting explanations of them all, so we focus on a few. Some are easy. Consider \#8 (Strongly suffixing $\supset$ Tense-aspect suffixes): this is quite plausible—if you have a language that tends to have suffixes, it will probably have suffixes for tense/aspect. Similarly, \#10 states that languages with verb morphology for questions lack question particles; again, this can be easily explained by an appeal to economy.
Some of the discovered implications require a more involved explanation. One such example is \#20: labial-velars implies no uvulars.[^4] It turns out that labial-velars are most common in Africa just north of the equator, which is also a place that has very few uvulars (there are a handful of other examples, mostly in Papua New Guinea). While this implication has not been previously investigated, it makes some sense: if a language has one form of rare consonant, it is unlikely to have another.
As another example, consider \#28: Obligatory suffix pronouns implies no possessive affixes. This means is that in languages (like English) for which pro-drop is impossible, possession is not marked morphologically on the head noun (like English, “book” appears the same regarless of if it is “his book” or “the book”). This also makes sense: if you cannot drop pronouns, then one usually will mark possession on the pronoun, not the head noun. Thus, you do not need marking on the head noun.
Finally, consider \#25: High and mid front vowels (i.e., /u/, etc.) implies large vowel inventory ($\geq 7$ vowels). This is supported by typological evidence that high and mid front vowels are the “last” vowels to be added to a language’s repertoire. Thus, in order to get them, you must also have many other types of vowels already, leading to a large vowel inventory.
Not all examples admit a simple explanation and are worthy of further thought. Some of which (like the ones predicated on “SV”) may just be peculiarities of the annotation style: the subject verb order changes frequently between transitive and intransitive usages in many languages, and the annotation reflects just one. Some others are bizzarre: why not having fricatives should mean that you don’t have tones (\#27) is not a priori clear.
[**\#**]{} [**Implicant**]{} [**Implicand**]{} [**Analysis**]{}
------------ ----------------------------- ---------------------------------- --------------------------------------------
1 Postpositions Genitive-Noun Greenberg \#2a
2 OV Postpositions Greenberg \#4
3 OV Genitive-Noun Greenberg \#4 + Greenberg \#2a
4 Genitive-Noun Postpositions Greenberg \#2a (converse)
5 Postpositions OV Greenberg \#2b (converse)
6 SV Genitive-Noun ???
7 Adjective-Noun Numeral-Noun Greenberg \#18
8 Strongly suffixing Tense-aspect suffixes Clear explanation
9 VO Noun-Relative Clause Lehmann
10 Interrogative verb morph No question particle Appeal to economy
11 Numeral-Noun Demonstrative-Noun Hawkins XVI (for postpositional languages)
12 Prepositions VO Greenberg \#3 (converse)
13 Adjective-Noun Demonstrative-Noun Greenberg \#18
14 Noun-Adjective Postpositions Lehmann
15 SV Postpositions ???
16 VO Prepositions Greenberg \#3
17 Initial subordinator word Prepositions Operator-operand principle (Lehmann)
18 Strong prefixing Prepositions Greenberg \#27b
19 Little affixation Noun-Adjective ???
20 Labial-velars No uvular consonants See text
21 Negative word No pronominal possessive affixes See text
22 Strong prefixing VO Lehmann
23 Subordinating suffix Strongly suffixing ???
24 Final subordinator word Postpositions Operator-operand principle (Lehmann)
25 High and mid front vowels Large vowel inventories See text
26 Plural prefix Noun-Genitive ???
27 No fricatives No tones ???
28 Obligatory subject pronouns No pronominal possessive affixes See text
29 Demonstrative-Noun Tense-aspect suffixes Operator-operand principle (Lehmann)
30 Prepositions Noun-Relative clause Lehmann, Hawkins
\[tab:qualitative\]
Multi-conditional Implications
------------------------------
Many implications in the literature have multiple implicants. For instance, much research has gone into looking at which implications hold, considering only “VO” languages, or considering only languages with prepositions. It is straightforward to modify our model so that it searches over triples of features, conditioning on two and predicting the third. Space precludes an in-depth discussion of these results, but we present the top examples in Table \[tab:multiconditional\] (after removing the tautalogically true examples, which are more numerous in this case, as well as examples that are directly obtainable from Table \[tab:qualitative\]). It is encouraging that in the top $1000$ multi-conditional implications found, the most frequently used were “OV” ($176$ times) “Postpositions” ($157$ times) and “Adjective-Noun” ($89$ times). This result agrees with intuition.
[**Implicants**]{} [**Implicand**]{}
------------------------- -- -------------------
Postpositions
Adjective-Noun
Posessive prefixes
Tense-aspect suffixes
Case suffixes
Plural suffix
Adjective-Noun
Genitive-Noun
High cons/vowel ratio
No front-rounded vowels
Negative affix
Genitive-Noun
No front-rounded vowels
Labial velars
Subordinating suffix
Tense-aspect suffixes
No case affixes
Prepositions
Strongly suffixing
Plural suffix
: Top implications discovered by the <span style="font-variant:small-caps;">LingHier</span> multi-conditional model.[]{data-label="tab:multiconditional"}
Discussion
==========
We have presented a Bayesian model for discovering universal linguistic implications from a typological database. Our model is able to account for noise in a linguistically plausible manner. Our hierarchical models deal with the sampling issue in a unique way, by using prior knowledge about language families to “group” related languages. Quantitatively, the hierarchical information turns out to be quite useful, regardless of whether it is phylogenetically- or areally-based. Qualitatively, our model can recover many well-known implications as well as many more potential implications that can be the object of future linguistic study. We believe that our model is sufficiently general that it could be applied to many different typological databases — we attempted not to “overfit” it to WALS. Our hope is that the automatic discovery of such implications not only aid typologically-inclined linguists, but also other groups. For instance, well-attested universal implications have the potential to reduce the amount of data field linguists need to collect. They have also been used computationally to aid in the learning of unsupervised part of speech taggers [@schone01univerals]. Many extensions are possible to this model; for instance attempting to uncover typologically hierarchies and other higher-order structures. We have made the full output of all models available at <http://hal3.name/WALS>.
#### Acknowledgments.
We are grateful to Yee Whye Teh, Eric Xing and three anonymous reviewers for their feedback on this work.
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[^1]: The exclusive-or of $a$ and $b$, written $a \otimes b$, is true exactly when either $a$ or $b$ is true but not both.
[^2]: This is nontrivial—we are currently exploring the possibility of freely sharing these data.
[^3]: In truth, our model discovers several tautalogical implications that we have removed by hand before presentation. These are examples like “SVO $\supset$ VO” or “No unusual consonants $\supset$ no glottalized consonants.” It is, of course, good that our model discovers these, since they are obviously true. However, to save space, we have withheld them from presentation here. The $30$th implication presented here is actually the $83$rd in our full list.
[^4]: Labial-velars and uvulars are rare consonants (order 100 languages). Labial-velars are joined sounds like /kp/ and /gb/ (to English speakers, sounding like chicken noises); uvulars sounds are made in the back of the throat, like snoring.
|
-1cm 15.5cm 0.7cm -1.6cm
To appear in [*J. Phys. G*]{}
[**QUARK FAMILY DISCRIMINATION AND\
FLAVOUR-CHANGING NEUTRAL CURRENTS IN THE\
$\mbox{SU(3)}_C \otimes \mbox{SU(3)}_L
\otimes \mbox{U(1)}$ MODEL WITH RIGHT-HANDED NEUTRINOS**]{}\
and [**Vo Thanh Van**]{}\
[*Institute of Physics, National Centre for Natural Science and Technology,\
P.O.Box 429, Bo Ho, Hanoi 10000, Vietnam*]{}\
Abstract\
Contributions of flavour-changing neutral currents in the 3 3 1 model with right-handed neutrinos to mass difference of the neutral meson system $\Delta m_P (P = K, D, B)$ are calculated. Using the Fritzsch anzats on quark mixing, we show that the third family should be different from the first two. We obtain a lower bound on mass of the new heavy neutral gauge boson as 1.02 TeV.
PACS number(s): 12.15.-y, 13.15.J\
[**I. Introduction**]{}\
The standard model (SM) has been very successful in explaing high energy phenomena. However there still remain some important questions we should understand. It is especially necessary to answer the quark family problem and hierarchy puzzle. In addition, the SuperKamiokande atmospheric neutrino data [@superk] provided an evidence for neutrino oscillation and consequently non-zero neutrino mass. It is known that, neutrinos are massless in the SM, therefore the SuperKamiokande result calls firstly for the SM extension.
Among the possible extensions, the models based on the $\mbox{SU(3)}_C \otimes \mbox{SU(3)}_L\otimes \mbox{U(1)}_N$ (3 3 1) gauge group [@ppf; @flt] have some interesting properties such as: first, it can explain why family number $N$ is equal to three. Second, one quark family is treated differently from the other two, and this gives some indication as to why the top quark is unbalancing heavy. Third, the Peccei-Quinn symmetry, necessary to solve the strong-CP problem, follows naturally from particle content in these models [@pal].
There are two main versions of 3 3 1 models: the minimal [@ppf; @dng] in which all lepton components ($\nu, l, (l_L)^c$) belong to the lepton triplet and a variant, in which right-handed neutrinos (r.h.neutrinos) are included i.e. ($\nu, l, (\nu)_L^c$) (hereafter we call it the model with r.h. neutrinos).
The fact that one quark family is treated differently from the other two, leads to the flavour-changing neutral currents (FCNC’s), which give a contribution to the mass difference of the neural meson systems at the tree level. The effect in the minimal model was considered in [@liu]. In this paper we shall consider the FCNC’s effect in the model with r.h. neutrinos.
[**II. The model** ]{}\
Let us briefly recapitulate the basic elements of the model. Fermions are in triplet $$f^{a}_L = \left( \begin{array}{c}
\nu^a_L\\ e^a_L\\ (\nu^c_L)^a
\end{array} \right) \sim (1, 3, -1/3), e^a_R\sim (1, 1, -1),
\label{l}$$ where a = 1, 2, 3 is the family index.
Two of the three quark families transform identically and one family (it does not matter which one) transforms in a different representation of the gauge group $\mbox{SU(3)}_C \otimes \mbox{SU(3)}_L\otimes
\mbox{U(1)}_N$: $$Q_{iL} = \left( \begin{array}{c}
d_{iL}\\-u_{iL}\\ D_{iL}\\
\end{array} \right) \sim (3, \bar{3}, 0),
\label{q}$$ $$u_{iR}\sim (3, 1, 2/3), d_{iR}\sim (3, 1, -1/3),
D_{iR}\sim (3, 1, -1/3),\ i=1,2,$$ $$Q_{3L} = \left( \begin{array}{c}
u_{3L}\\ d_{3L}\\ T_{L}
\end{array} \right) \sim (3, 3, 1/3),$$ $$u_{3R}\sim (3, 1, 2/3), d_{3R} \sim (3, 1, -1/3),
T_{R}\sim (3, 1, 2/3),$$ where $D$ and $T$ are exotic quarks with electric charges $-\frac{1}{3}$ and $\frac{2}{3}$, respectively.
Fermion mass generation and symmetry breaking can be achieved with just three $SU(3)_{L}$ triplets $$\chi = \left( \begin{array}{c}
\chi^o\\ \chi^-\\ \chi^{,o}\\
\end{array} \right) \sim (1, 3, -1/3),
\rho = \left( \begin{array}{c}
\rho^+\\ \rho^o\\ \rho^{,+}\\
\end{array} \right) \sim (1, 3, 2/3),\
\eta = \left( \begin{array}{c}
\eta^o\\ \eta^-\\ \eta^{,o}\\
\end{array} \right) \sim (1, 3, -1/3).
\label{h2}$$ All the Yukawa terms of quarks are given $$\begin{aligned}
{\cal L}_{Yuk}^{\chi}&=&\lambda_1\bar{Q}_{3L}T_{R}\chi +
\lambda_{2ij}\bar{Q}_{iL}d^{'}_{jR}\chi^{*} +
\mbox{h.c.}\nonumber\\
&=&\lambda_1(\bar{u}_{3L}\chi^o+\bar{d}_{3L}\chi^-
+\bar{T}_L\chi^{,o})T_R
+\lambda_{2ij}(\bar{d}_{iL}\chi^{o*}-\bar{u}_{iL}\chi^++
\bar{D}_{iL}\chi^{,o*})D_{jR} + \mbox{h.c.}\nonumber\\
{\cal L}_{Yuk}^{\eta}&=&\lambda_{3a}\bar{Q}_{3L}
u_{aR}\eta+\lambda_{4ia}\bar{Q}_{iL}
d_{aR}\eta^{*}+\mbox{h.c.}\nonumber\\
&=&\lambda_{3a}(\bar{u}_{3L}\eta^o+
\bar{d}_{3L}\eta^-+\bar{T}_L\eta^{,o})
u_{aR}+\lambda_{4ia}(\bar{d}_{iL}
\eta^{o*}-\bar{u}_{iL}\eta^+
+\bar{D}_{iL}\eta^{,o*})d_{aR}+\mbox{h.c.}\nonumber\\
{\cal L}_{Yuk}^{\rho}&=&\lambda_{1a}
\bar{Q}_{3L}d_{aR}\rho +
\lambda_{2ia}\bar{Q}_{iL}u_{aR}\rho^{*}\nonumber\\
&=&\lambda_{1a}(\bar{u}_{3L}\rho^++\bar{d}_{3L}\rho^o+
\bar{T}_L\rho^{,+})d_{aR}+
\lambda_{2ia}(\bar{D}_{iL}\rho^-
-\bar{u}_{iL}\rho^{o*}\nonumber\\
& &+\bar{D}_{iL}\rho^{,-})u_{aR}
+\mbox{h.c.}.
\label{yukawa}\end{aligned}$$ In this model the Higgs triplets in Eq. (\[h2\]) should develop VEVs as follow: $$\langle\chi \rangle^T = (0, 0, \omega/\sqrt{2}),\
\langle\rho \rangle^T = (0, u/\sqrt{2}, 0),\
\langle\eta \rangle^T = (v/\sqrt{2}, 0, 0).$$
The new complex gauge bosons in this model are $\sqrt{2} X^0_\mu = W^4_\mu - i W^5_\mu,
\sqrt{2} Y^+_\mu = W^6_\mu - i W^7_\mu$. Both these bosons carry lepton number two, hence they are called bileptons. In [@li] the first constraints on masses of the bileptons in this model are derived by considering $S, T$ parameters: $213\ {\rm GeV} \leq M_{Y^+} \leq 234\
{\rm GeV},\ 230\ {\rm GeV} \leq M_{X^0} \leq 251$ GeV.
The physical neutral gauge bosons are mixtures of $Z, Z'$: $$\begin{aligned}
Z^1 &=&Z\cos\phi - Z'\sin\phi,\nonumber\\
Z^2 &=&Z\sin\phi + Z'\cos\phi,\end{aligned}$$ where the photon field $A_\mu$ and $Z,Z'$ are given by: $$\begin{aligned}
A_\mu &=& s_W W_{\mu}^3 + c_W\left(-\frac{t_W}{\sqrt{3}}
\ W^8_{\mu} +\sqrt{1-\frac{t^2_W}{3}}\
B_{\mu}\right),\nonumber\\
Z_\mu &=& c_W W^3_{\mu} - s_W\left(-\frac{t_W}{\sqrt{3}}\
W^8_{\mu}+ \sqrt{1-\frac{t_W^2}{3}}\ B_{\mu}\right), \\
\label{apstat}
Z'_\mu &=& \sqrt{1-\frac{t_W^2}{3}}\ W^8_{\mu}+
\frac{t_W}{\sqrt{3}}\ B_{\mu}\nonumber.\end{aligned}$$ Here $s_W$ stands for $ \sin \theta_W$. The mixing angle $\phi$ is defined by $$\tan^2\phi =\frac{m_{Z}^2-m^2_{Z^1}}{M_{Z^2}^2-m_{Z}^2},
\label{tphi}$$ where $m_{Z^1}$ and $M_{Z^2}$ are the [*physical*]{} mass eigenvalues.
The interactions among fermions and $Z_1, Z_2$ are given as follows: $$\begin{aligned}
{\cal L}^{NC}&=&\frac{g}{2c_W}\left\{\bar{f}\gamma^{\mu}
[a_{1L}(f)(1-\gamma_5) + a_{1R}(f)(1+\gamma_5)]f
Z^1_{\mu}\right.\nonumber\\
& &+ \left.\bar{f}\gamma^{\mu}
[a_{2L}(f)(1-\gamma_5) + a_{2R}(f)(1+\gamma_5)]f
Z^2_{\mu}\right\}.
\label{nc}\end{aligned}$$ where $$\begin{aligned}
a_{1L,R}(f) &=&\cos\phi\ [T^3(f_{L,R})-s_W^2 Q(f)]\nonumber\\
& &- c_W^2\left[\frac{3N(f_{L,R})}{(3-4s_W^2)^{1/2}}
-\frac{(3-4s_W^2)^{1/2}}{2c^2_W}Y(f_{L,R})\right]
\sin\phi,\nonumber\\
a_{2L,R}(f)&=& c_W^2\left[\frac{3N(f_{L,R})}{(3-4s_W^2)^{1/2}}
-\frac{(3-4s_W^2)^{1/2}}{2c^2_W}Y(f_{L,R})\right]
\cos\phi\nonumber\\
& &+ \sin\phi\ [T^3(f_{L,R})-s_W^2 Q(f)].
\label{vaz}\end{aligned}$$ Here $T^3(f)$ and $Q(f)$ are, respectively, the third component of the weak isospin and the charge of the fermion $f$.
[**III. Flavour-changing neutral currents and mass difference of the neutral meson systems**]{}\
Due to the fact that one family of left-handed quarks is treated differently from the other two, the N charges for left-handed quarks are different too (see Eq. (\[q\])). Therefore flavour-changing neutral currents $Z_1, Z_2$ occur through a mismatch between weak and mass eigenstates.
Let us diagolize mass matrices by three biunitary transformations $$\begin{aligned}
U'_L & = & V_L^U U_L,\ U'_R = V_R^U U_R,\nonumber\\
D'_L & = & V_L^D D_L,\ D'_R = V_R^D D_R,
\label{tran}\end{aligned}$$ where $U \equiv (u, c, t)^T, \ D \equiv (d,s,b)^T$.\
The usual Cabibbo-Kobayashi-Maskawa matrix is given by $$V_{CKM} = V_L^{U+} V_L^D.
\label{vckm}$$
Using unitarity of the $V^D$ and $V^U$ matrices, we get flavour-changing neutral interactions $$\begin{aligned}
{\cal L}^{NC}_{ds}&=&\frac{g c_W}{2 \sqrt{3-4 s_W^2}}
\left[V^{D*}_{Lid}
V^D_{Lis}\right] \bar{d}_L \gamma_\mu s_L
\left(\cos \phi Z_2^\mu - \sin \phi Z_1^\mu \right ),
\nonumber\\
{\cal L}^{NC}_{uc}&=&\frac{g c_W}{2 \sqrt{3-4 s_W^2}}
\left[V^{U*}_{Liu} V^U_{Lic}\right] \bar{u}_L \gamma_\mu c_L
\left(\cos \phi Z_2^\mu - \sin \phi Z_1^\mu \right ),\\
{\cal L}^{NC}_{db}&=&\frac{g c_W}{2 \sqrt{3-4 s_W^2}}
\left[V^{D*}_{Lid} V^D_{Lib}\right] \bar{d}_L \gamma_\mu b_L
\left(\cos \phi Z_2^\mu - \sin \phi Z_1^\mu \right )\nonumber,
\label{fcnc}\end{aligned}$$ where $i$ denotes the number of "different” quark family i.e. the $ \mbox{SU(3)}_L $ quark triplet.
For the neutral kaon system, we get then effective Lagrangian $${\cal L}^{\Delta S=2}_{eff} = \frac{\sqrt{2} G_F\ c_W^4
\cos^2 \phi}{(3-4 s_W^2)}\left[V^{D*}_{Lid}
V^D_{Lis}\right]^2| \bar{d}_L \gamma^\mu s_L|^2
\left( \frac{m^2_{Z_1}}{M^2_{Z_2}} + \tan^2 \phi \right).
\label{eff}$$ Similar expressions can be easily written out for $D^0 -
\bar{D}^0$ and $B^0 - \bar{B}^0$ systems. From (\[eff\]) it is straightforward to get the mass difference $$\Delta m_P = \frac{4 G_F\ c_W^4\
\cos^2 \phi}{3 \sqrt{2}
(3-4 s_W^2)}\left[V^{D*}_{Lid}
V^D_{Li\alpha}\right]^2 \left(
\frac{m^2_{Z_1}}{M^2_{Z_2}} +
\tan^2 \phi \right) f^2_P B_P m_P,
\label{masdif}$$ where $\alpha = s$ for the $K_L - K_S$ and $\alpha = b$ for the $B^0 - \bar{B}^0$ mixing systems. The $D^0 - \bar{D}^0$ mass difference is given by the expression for the $K^0$ system with replace of $V^D$ by $V^U$. The $Z - Z'$ mixing angle $\phi$ was bounded and to be [@flt; @dng]: $ |\phi| \leq 10^{-3} $, hence if $M_{Z_2}$ is in order of one hundred TeV, the $Z - Z'$ mixing has to be taken into account.
In the usual case, the $Z-Z'$ mixing is constrained to be very small, it can be safely neglected. Therefore FCNC’s occur only via $Z_2$ couplings. For the shothand hereafter we rename $Z_1$ to be $Z$ and $Z_2$ to be $Z'$.
Since it is generally recognized that the most stringent limit from $\Delta m_K$, we shall mainly discuss this quantity. We use the experimental values [@caso] $$\begin{aligned}
\Delta m_K & = & ( 3.489 \pm 0.009)\times 10^{- 12}\
{\rm MeV}, \hspace*{1cm} m_K \simeq 498\ {\rm MeV}
\label{data}\end{aligned}$$ and $$\begin{aligned}
\sqrt{B_K} f_K & = & 135 \pm 19 \ {\rm MeV}.
\label{fb}\end{aligned}$$
Following the idea of Gaillard and Lee [@gali], it is reasonable to expect that $Z'$ exchange contributes a $\Delta m$ no larger than observed values. Substituting (\[data\]) and (\[fb\]) into (\[masdif\]) we get $$\begin{aligned}
M_{Z'}& >& 2.63 \times 10^5\ \eta_{Z'} \left[Re(V^{D*}_{Lid}
V^D_{Lis})^2\right]^{1/2} \ {\rm GeV}.
\label{gh}\end{aligned}$$ where $\eta_{Z'} \approx 0.55$ is the leading order QCD corrections [@gwi].
Let us call $ \Delta m_K^{min}, \Delta m_K^{rhn}$ contributions to $\Delta m$ from the $Z'$ in the minimal 3 3 1 model and in the model with r.h. neutrinos, respectively. We have then $$R \equiv \frac{ \Delta m_K^{min}}{ \Delta m_K^{rhn}}
= \frac{2 (3 - 4 s_W^2) }{3 ( 1 - 4 s^2_W)} = 19.7,
\label{rel}$$ for [@caso] $s_W^2 = 0.2312$. Because of the denominator, the relation is highly sensitive to the value of the Weinberg angle. It is easy to see that a limit for the $Z'$ following from Eq (\[gh\]) in the model with r.h.neutrinos is approximately 4.4 times smaller than that in the minimal version.
From the present experimental data we cannot get the constraint on $V^{U,D}_{Lij}$. These matrix elemetns are only constrainted by (\[vckm\]). However, it would seem more natural, if Higgs scalars are associated with fermion generations, to have the choice of nondiagonal elements depends on the fields to which the Higgs scalars couple. By this way, the simple Fritzsch [@hf] scheme gives us $$V^D_{ij} \approx \left( \frac{m_i}{m_j} \right)^{1/2},
\hspace*{1cm} i < j.
\label{hfr}$$
Combining (\[gh\]) and (\[hfr\]) we get the following bounds on $M_{Z'}$: $$\begin{aligned}
M_{Z'}& \geq & 38\ {\rm TeV}, \hspace*{0.3cm} {\rm if\ the\
first\ or\ the \ second\ quark\ family\ is\
different\ (\ in\ triplet)}
\nonumber\\
M_{Z'}& \geq& 1.02 \ {\rm TeV}, \hspace*{0.3cm}
{\rm if\ the\ third\ quark\ family\ is\ different}
\label{thu}\end{aligned}$$ From (\[thu\]) we see that to keep relatively low bounds on $M_{Z'}$ the third family should be the one that is different from the other two i.e. is in triplet.
Our numerical estimation is based on the fact that all the phases of the matrix elements equal to zero. The inclusion of complex phases would induce to a reduction in the mass limit. However the hierarchical picture should not be modified.
[**IV. Summary**]{}\
We have studied the FCNC’s in the 3 3 1 model with r.h. neutrinos arisen from the family discrimination in this model. This gives a reason to conclude that the third family should be treated differently from the first two. In this sense, the $\Delta m_K$ gives us the lower bound on $M_{Z'}$ as 1.02 TeV. It is to be mentioned that our conclusion is similar with that in the minimal version [@dpp]: $$\begin{aligned}
M_{Z'}& \geq & 315\ {\rm TeV}, \hspace*{0.3cm} {\rm if\ the\
first\ or\ the \ second\ quark\ family\ is\
different\ (\ in\ triplet)}
\nonumber\\
M_{Z'}& \geq& 10 \ {\rm TeV}, \hspace*{0.3cm}
{\rm if\ the\ third\ quark\ family\ is\ different}
\nonumber\end{aligned}$$
It is interesting to note that in the Fritzsch anzats, the limits for $M_{Z'}$ following from $\Delta m_B$ are independent of the family choice.
We emphasize that the FCNC’s in the minimal model are larger than those in the considered version due to the factor $\frac{1}{\left(1 - 4 s_W^2\right)}$. Therefore the lower bounds on $M_{Z'}$ are smaller accordingly.
In both versions of the 3 3 1 models, the third quark family should be different from the first two, and this gives us some indication of why the top quark is so heavy.
[**Acknowledgements**]{}
This work has been initialed when the first author (H.N.L) was at the Department of Physics, Chuo University, Tokyo. He expresses sincere gratitude to Professor T. Inami for helpful discussions and warm hospitality. This work was supported in part by Research Programme under grant $N^0$ : QT 98.04 and KT - 04.1.2.
[99]{} Y. Fukuda [*et al*]{}., Phys. Lett. [**B433**]{}, 9 (1998); Phys. Rev. Lett. [**81**]{}, 1562 (1998). F. Pisano and V. Pleitez, Phys. Rev. D[**46**]{}, 410 (1992); P. H. Frampton, Phys. Rev. Lett. [**69**]{}, 2889 (1992); R. Foot, O.F. Hernandez, F. Pisano, and V. Pleitez, Phys. Rev. D[**47**]{}, 4158 (1993). R. Foot, H. N. Long, and Tuan A. Tran, Phys. Rev. D[**50**]{}, R34 (1994); H. N. Long, Phys. Rev. D [**54**]{}, 4691 (1996). P. B. Pal, Phys. Rev. D[**52**]{}, 1659 (1995). Daniel Ng, Phys. Rev. D[**49**]{}, 4805 (1994). J. T. Liu, Phys. Rev. D[**50**]{}, 542 (1994); D. G. Dumm, F. Pisano, and V. Pleitez, Mod. Phys. Lett. [**A9**]{}, 1609 (1994); T. H. Lee and D. S. Hwang, Int’l . J. Mod. Phys. [**A12**]{}, 4411 (1997). H. N. Long and T. Inami, hep-ph/9902475 (1999) to be published. C. Caso [*et al*]{}, Particle Data Group, Eur. Phys. J. [**C3**]{} 1, (1998). M. K. Gaillard and B. W. Lee, Phys. Rev. D[**10**]{}, 897 (1974). F. J. Gilman and M. B. Wise, Phys. Rev. D[**27**]{}, 1128 (1983). H. Fritzsch, Phys. Lett. [**B73**]{}, 317 (1978); Nucl. Phys. [**B155**]{}, 189 (1979). D. G. Dumm [*et al*]{} in \[6\].
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---
author:
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Nikhilesh Dasgupta and Neena Gupta\
\
\
\
title: Nice derivations over principal ideal domains
---
|
---
abstract: 'We show that in $N=2$ supergravity, with a special quaternionic manifold of (quaternionic) dimension $h_1+1$ and in the presence of $h_2$ vector multiplets, a $h_2+1$ dimensional abelian algebra, intersecting the $2h_1+3$ dimensional Heisenberg algebra of quaternionic isometries, can be gauged provided the $h_2+1$ symplectic charge–vectors $V_I$, have vanishing symplectic invariant scalar product $V_I\times V_J=0$. For compactifications on Calabi–Yau three–folds with Hodge numbers $(h_1,h_2)$ such condition generalizes the half–flatness condition as used in the recent literature. We also discuss non–abelian extensions of the above gaugings and their consistency conditions.'
---
\
[ ]{}\
\
$\S$[*Dipartimento di Fisica, Politecnico di Torino\
C.so Duca degli Abruzzi, 24, I-10129 Torino\
Istituto Nazionale di Fisica Nucleare, Sezione di Torino, Italy*]{}\
[E-mail: [email protected], [email protected]]{}\
$\sharp$[*CERN, Physics Department, CH 1211 Geneva 23, Switzerland.*]{}\
[*INFN, Laboratori Nucleari di Frascati, Italy*]{}\
[E-mail: [email protected]]{}\
$\natural$[*DESY, Theory Group\
Notkestrasse 85, Lab. 2a D-22603 Hamburg, Germany\
II. Institut für Theoretische Physik,\
Luruper Chaussee 149, D-22761 Hamburg, Germany*]{}\
[E-mail: [email protected]]{}
6 mm
The Heisenberg algebra
======================
It is well known [@cfg; @fs] that the moduli space of a Calabi–Yau compactification of Type II string theory is a product of a special quaternionic manifold ${\Scr M}_{SQ}$ of quaternionic dimension $h_1+1$ and a special Kähler manifold ${\Scr M}_{SK}$ of complex dimension $h_2$ where $h_1=h_{(2,1)},\, h_2=h_{(1,1)}$ for Type IIA and the reverse for Type IIB.\
The special quaternionic geometry has some general properties [@fs; @dwvp1; @dwvp2], i.e. the $2h_1+3$ coordinates which describe $2h_1+2$ R–R scalar fields and the $a$ scalar field dual to the antisymmetric tensor field $B_{\mu\nu}$ [@dsv], parametrize, in the “solvable description” of the manifold [@adfft], a Heisenberg algebra of the form: $$\begin{aligned}
\label{Heis}
\left[X^\Lambda,\, Y_\Sigma\right]&=&\delta^\Lambda{}_\Sigma\, {\Scr Z}\,\,\,;\,\,\,\,\,
\Lambda=0,\dots, h_1\,,\nonumber\\
\left[X^\Lambda,\, X^\Sigma\right]&=&\left[Y_\Lambda,\,
Y_\Sigma\right]=\left[X^\Lambda,\,{\Scr
Z}\right]=\left[Y_\Lambda,\, {\Scr Z}\right]=0\,.\end{aligned}$$ In Calabi–Yau compactifications the generators $X^\Lambda,\,Y_\Sigma$ in (\[Heis\]) are parametrized by the RR real scalars which in Type IIA come from the internal components of the complex 3–form $A_{(3)}$ [@bcf; @bghl; @lm]: $$\begin{aligned}
\{\tilde{\zeta}_\Lambda,\,\zeta^\Lambda\}&\rightarrow
\{A_{ijk},\,A_{i\bar{j}\bar{k}},\,A_{\bar{i}\bar{j}\bar{k}},\,A_{\bar{i}jk},\,\}\,.\end{aligned}$$ while Type IIB they originate from the 2–form and 4–form cohomology: $$\begin{aligned}
\{\tilde{\zeta}_\Lambda,\,\zeta^\Lambda\}&\rightarrow
\{C,\,C_{\bar{i}j \,\bar {l}k},\,C_0,\,C_{i\bar{j}}\}\,,\end{aligned}$$ where $C$ is the dual of $C_{\mu\nu}$.
The universal hypermultiplet contains, besides the dilaton and the $a$ field which parametrizes the generator ${\Scr Z}$ in (\[Heis\]),he $\Lambda=0$ component of the above coordinates, namely $\{{\rm
Re}A_{ijk},\,{\rm Im}A_{ijk}\}$ in Type IIA and $\{C_0,\, C\}$ in Type IIB. In each case such multiplets parametrize ${\Scr
M}_U={\rm SU}(1,2)/{\rm U}(2)\subset {\Scr M}_{SQ}$. Under the group of motions generated by the Heisenberg algebra the scalar fields $\tilde{\zeta}_\Lambda,\,\zeta^\Lambda$ transform as follows [@fs]: $$\begin{aligned}
\delta\zeta^\Lambda&=&u^\Lambda\,\nonumber\\
\delta\tilde{\zeta}_\Lambda&=& v_\Lambda\,\nonumber\\
\delta a &=&
w+u^\Lambda\tilde{\zeta}_\Lambda-v_\Lambda\zeta^\Lambda\,.
\label{transf}\end{aligned}$$ Noting that $\delta(\tilde{\zeta}_\Lambda\zeta^\Lambda)=u^\Lambda\tilde{\zeta}_\Lambda+v_\Lambda\zeta^\Lambda$ we may redefine $a$ in such a way that one of the two scalar–dependent terms in $\delta a$ is eliminated.
The gaugings
============
Let us define a gauge algebra through the following infinitesimal field transformations: $$\begin{aligned}
\delta A_\mu^I&=&\partial_\mu \lambda^I\,,\nonumber\\
\delta \zeta^\Lambda&=&a_I{}^\Lambda\, \lambda^I\,,\nonumber\\
\delta \tilde{\zeta}_\Lambda&=&b_{I\Lambda}\, \lambda^I\,,\nonumber\\
\delta a &=&c_I\,
\lambda^I+(a_I{}^\Lambda\,\tilde{\zeta}_\Lambda-b_{I\Lambda}\,\zeta^\Lambda)\,\lambda^I\,,\end{aligned}$$ where $I=0,\dots , h_2$, $h_2$ being the number of vector multiplets. Note that no relation exists between $h_1,\,h_2$ so that the above algebra is not in general contained in the Heisenberg algebra.
The covariant derivatives read: $$\begin{aligned}
D_\mu \zeta^\Lambda&=&\partial_\mu\zeta^\Lambda-a_I{}^\Lambda\,
A^I_\mu\,,\nonumber\\
D_\mu
\tilde{\zeta}_\Lambda&=&\partial_\mu\tilde{\zeta}_\Lambda-b_{I\Lambda}\,
A^I_\mu\,,\nonumber\\
D_\mu a&=&\partial_\mu
a-(a_I{}^\Lambda\,\tilde{\zeta}_\Lambda-b_{I\Lambda}\,\zeta^\Lambda)\,
A^I_\mu-c_I\,A^I_\mu\,.\end{aligned}$$ One can verify that: $$\begin{aligned}
\delta (D_\mu \zeta^\Lambda)&=&\delta(D_\mu
\tilde{\zeta}_\Lambda)=0\,,\nonumber\\ \delta(D_\mu
a)&=&(a_I{}^\Lambda\,D_\mu\tilde{\zeta}_\Lambda-b_{I\Lambda}\,D_\mu\zeta^\Lambda)\,\lambda^I\,,\end{aligned}$$ where in order to derive the last equation we required requires the following condition: $$\begin{aligned}
c_{IJ}&\equiv &b_{I\Lambda}\, a_J{}^\Lambda-b_{J\Lambda}\,
a_I{}^\Lambda\,=0\,,\label{cocy}\end{aligned}$$ which we shall characterize in the sequel as a “cocycle” condition of the Lie algebra. If we consider $\{
a_J{}^\Lambda,\,b_{I\Lambda}\}$ to be the $2\,h_1+2$ components of a symplectic vector $V_I$, condition (\[cocy\]) can be rephrased as the vanishing of the symplectic scalar product $V_I\times
V_J=0$. Such condition is also equivalent to the closure of the abelian gauge algebra whose generators $\{T_I\}$ are: $$\begin{aligned}
\label{embe1}
T_I&=&b_{I\Lambda}\, X^\Lambda+a_I{}^\Lambda\, Y_\Lambda+c_I\,
{\Scr Z}\,\,\,;\,\,\,\,\, \left[T_I,\,T_J\right]=0\,.\end{aligned}$$
Gauging of special quaternionic $\sigma$–model
==============================================
There is an elegant way of writing the RR scalars in the quaternionic manifold in terms of the symplectic section: $$\begin{aligned}
Z&=&\left(\matrix{\zeta^\Lambda\cr
\tilde{\zeta}_\Lambda}\right)\,,\end{aligned}$$ and the symplectic (symmetric) matrix ${\Scr M}$ [@cdf]: $$\begin{aligned}
{\Scr M}&=&\left(\matrix{\bfone & -{\rm Re}({\Scr N})\cr 0 &
\bfone }\right)\left(\matrix{{\rm Im}({\Scr N}) & 0\cr 0 & {\rm
Im}({\Scr N})^{-1} }\right)\left(\matrix{\bfone &0\cr -{\rm
Re}({\Scr N})& \bfone}\right)\,.\end{aligned}$$ Indeed the kinetic term [@fs] is given by: $$\begin{aligned}
&&K_{a\bar{b}}\, \partial_\mu z^a\partial^\mu
\bar{z}^{\bar{b}}-\frac{1}{4\,\phi^2}\,(\partial\phi)^2-
\frac{1}{4\,\phi^2}\,(\partial a -Z\times
\partial Z)^2-\frac{1}{2\,\phi}\,\partial_\mu Z\, {\Scr
M}\,\partial^\mu Z\,.\end{aligned}$$ Note that invariance under the Heisenberg algebra with symplectic parameters: $$\begin{aligned}
\Theta&=&\left(\matrix{u^\Lambda\cr v_\Sigma}\right)\,,\end{aligned}$$ is manifest sice $$\begin{aligned}
\delta a&=&w+\Theta\times Z\,\,;\,\,\,\delta
Z=\Theta\,\,\Rightarrow \,\,da-Z\times dZ \,\,\mbox{invariant}\,.\end{aligned}$$ The gauging of the non linear $\sigma$–model goes as follows. We consider an abelian $h_2+1$ dimensional gauge group whose embedding in the Heisenberg algebra is described by $h_2+1$ symplectic charge-vectors $$\begin{aligned}
V_I&=&\left(\matrix{a_I{}^\Lambda\cr b_{I\,\Lambda}}\right)\,,\end{aligned}$$ and whose connection $U$ is expressed in terms of the $h_2+1$ vector fields $A^I_\mu$ as follows: $$\begin{aligned}
U&=&A^I\,V_I\,,\\
\delta U&=& d\Theta\,,\end{aligned}$$ $\Theta$ being now $\Theta=\lambda^I\,V_I$, where $\lambda^I$ are the gauge parameters. The covariant derivative of $Z$ is then $$\begin{aligned}
DZ&=&dZ-U\,,\end{aligned}$$ and the covariant derivative of $a$ reads $$\begin{aligned}
Da&=&da-U\times Z\,,\end{aligned}$$ since $\delta a =\Theta\times Z$.
If we transform $Da$ we obtain $$\begin{aligned}
\delta(Da)&=&\Theta\times dZ-U\times \Theta=\Theta\times (dZ+U)\,,\end{aligned}$$ which is not $\Theta\times DZ$ since $DZ=dZ-U$. Therefore closure implies $\Theta\times U=0$ which is equivalent to condition (\[cocy\]) since: $$\begin{aligned}
\Theta\times U&=&2\,a_{[I}{}^\Lambda\,
b_{J]\,\Lambda}\,\lambda^I\, A^J\,.\end{aligned}$$ If $\Theta\times U=0$ we can write the RR sector of the gauged Lagrangian $$\begin{aligned}
&&- \frac{1}{4\,\phi^2}\,(D a -Z\times D
Z)^2-\frac{1}{2\,\phi}\,D_\mu Z\, {\Scr M}\,D^\mu Z\,.\end{aligned}$$ Upon addition of the minimal coupling $\partial a -c_I\, A^I$ to the covariant derivative of $a$ the vector boson mass matrix $M^2_{IJ}$ will read: $$\begin{aligned}
M^2_{IJ}&=&\frac{1}{2\,\phi^2}\,(c_I-2\,Z\times
V_I)\,(c_J-2\,Z\times V_J) +\frac{1}{\phi}\,V_I\,{\Scr M}V_J\,.\end{aligned}$$
Non–abelian gauging
===================
Let us now see what are the requirements which have to be satisfied in order to embed a non–abelian gauge algebra in the Heisenberg algebra.\
Quite generally we introduce a non–abelian gauge algebra defined by:$$\label{gaugealg}
[T_I,\,T_J]=f^K_{\phantom {T}IJ}T_K\,.$$ Using the embedded expression for the gauge algebra generators given in equation (\[embe1\]), the embedding condition $$\label{embe2} f^K_{\phantom {T}IJ}T_K=c_{IJ}{\Scr
Z}\,,$$ implies the following relations: $$\begin{aligned}
\label{cond1}
f^K_{\phantom {T}IJ}c_K&=&c_{IJ}\,, \\
\label{cond2}f^K_{\phantom {T}IJ}b_{K\Lambda}&=&0\,,\\
\label{cond3}f^K_{\phantom {T}IJ}a_K{}^{\Lambda}&=&0\,.\end{aligned}$$ In terms of the Lie algebra cohomology equation (\[cond1\]) means that $c_{IJ}$ is a non trivial cocycle of the gauge algebra, while (\[cond2\]) and (\[cond3\]) imply that $b_{I\Lambda}$ and $a_I{}^{\Lambda}$ are coboundaries. When $c_{IJ}=0$ the cohomology is trivial and we are in the case of the abelian gauge algebra discussed in the previous section. Since the algebra (\[embe2\]) contains a central charge it is non-semisimple and according to a theorem of Lie algebra cohomology we may have a non trivial cocycle $c_I$ in the adjoint representation of the algebra (this would be impossible if the gauge algebra were semisimple since in that case the only non trivial cocycle should be in the trivial representation of the algebra). In fact a solution of conditions (\[cond1\]),(\[cond2\]),(\[cond3\]) may be found as follows. We first consider the case in which $h_2+1=2 h_1+3$, so that the number of vector matches the dimension of the Heisenberg algebra. The gauge generators $T_I$ decompose in the following way: $$\begin{aligned}
\{T_I\}&=&\{T_\Lambda,\,T^\Lambda,\,T_0\}\,.\end{aligned}$$ The charge matrices are chosen to be $$\begin{aligned}
b_{0\Lambda}&=&b_{\Sigma\Lambda}=0\,\,;\,\,\,b^\Sigma{}_\Lambda=b_\Lambda\,\delta^\Sigma{}_\Lambda\,,\nonumber\\
a_{0}{}^{\Lambda}&=&a^{\Sigma\Lambda}=0\,\,;\,\,\,a_\Sigma{}^\Lambda=a^\Lambda\,\delta_\Sigma{}^\Lambda\,,\\\end{aligned}$$ The cocycle condition (\[cond1\]) becomes $$\begin{aligned}
c_0\,f_{\Lambda}^0{}^\Sigma&=&(b_\Lambda
a^\Lambda)\,\delta_\Lambda{}^\Sigma\,,\end{aligned}$$ with no summation over the index $\Lambda$. Conditions (\[cond2\]), (\[cond3\]) are manifestly verified. If $h_2>2\,(h_1+1)$ we may apply the above construction to $2\,
h_1+3$ vectors while the remaining $h_2-2\,(h_1+1)$ vectors stay spectators. Viceversa, if $h_2<2\,(h_1+1)$ we can select a Heisenberg subalgebra with $\bar{h}_1=\frac{h_2}{2}-1$ and apply to it the construction described above.
Gauging and half–flatness
=========================
Let us consider Calabi–Yau compactifications on a *half–flat* manifold [@halfflat]. For the kind of manifolds considered in [@gm; @glmw], in the absence of fluxes, we have the following couplings in Type IIA and IIB theories $$\begin{aligned}
\mbox{IIA}&&a_I{}^\Lambda=0\,\,\,;\,\,\,\,b_{I\Lambda=0}=\epsilon_I\,;\,\,\,\,(\mbox{0
otherwise})\,,\\
\mbox{IIB}&&a_I{}^\Lambda=0\,\,\,;\,\,\,\,b_{I=0\Lambda}=\epsilon_\Lambda\,;\,\,\,\,(\mbox{0
otherwise})\,,\end{aligned}$$ and the cocycle condition (\[cocy\])is identically satisfied (recall that, according to our notations, in Type IIA $I=0,\dots,
h_{1,1}$ and $\Lambda=0,\dots, h_{2,1}$ while in Type IIB $I=0,\dots, h_{2,1}$, $\Lambda=0,\dots, h_{1,1}$). Note that we use the same symbols to denote the charges $a_I{}^\Lambda,\,b_{I\Lambda}$ in Type IIA and IIB theories although they are described in the two cases by different matrices with different dimensions. If we turn on a NS 3–form flux in Type IIA theory we have $a_{I=0}{}^\Lambda=p^\Lambda\neq 0$ and $b_{I=0\Lambda}=q_\Lambda\neq 0$, and then, on a half–flat manifold we should also have a non vanishing $a_{I}{}^\Lambda$ since the cocycle condition requires: $$\begin{aligned}
a_{I}{}^\Lambda \, b_{J\Lambda}&=&a_{J}{}^\Lambda \,
b_{I\Lambda}\Rightarrow q_0\,a_I{}^0=p^0\, \epsilon_I\,.\end{aligned}$$ On the Type IIB side, if we turn on an electric NS 3–form flux we get a covariant derivative of the type [@m]: $$\begin{aligned}
D_\mu\tilde{\zeta}_0=\partial_\mu\tilde{\zeta}_0 -q_{I}\,
A^I_\mu\,,\end{aligned}$$ where $q_I$ is the *electric* flux $b_{I\,\Lambda=0}$. If Type IIB background is half–flat [@gm] we also have $b_{I=0\,\Lambda}=\epsilon_{\Lambda}\neq 0$. In this case, as expected, $b_{0\,0}=\epsilon_0=q_0$. For the *magnetic* NS 3–form flux the correspondence is non–local.
The abelian gauging of the Heisenberg algebra discussed in the previous sections therefore generalizes the results on flux–compactifications on half–flat manifolds as discussed in the literature [@gm; @glmw], to arbitrary values of $I,\,\Lambda$. Consistency always requires in the “dual theories” the cocycle condition to be satisfied: $$\begin{aligned}
a_{[I}{}^\Lambda\,
b_{J]\Lambda}&=&0\,\,\,\,;\,\,\,\,\,(I,J=0,\dots,h_2\,;\,\,\Lambda=0,\dots,
h_1)\,.\nonumber\end{aligned}$$ Mirror symmetry on the other hand implies $$\begin{aligned}
b^{(B)}{}_{I\,\Lambda}&=&(b^{(A) T})_{
I\Lambda}\,\,\,;\,\,\,\,(I=0,\dots,h_{2,1}\,;\,\,\Lambda=0,\dots,
h_{1,1})\,,\nonumber\end{aligned}$$ In Type IIA theory we can interpret the parameters $a_I{}^\Lambda,\,b_{I\,\Lambda}$ of the gauging in terms of the following deformation of the Calabi–Yau cohomology [@glmw; @gm]: $$\begin{aligned}
d\alpha_\Lambda&=&b_{i\Lambda}\,
\omega^i\,\,;\,\,\,\,d\beta^\Lambda=a_i{}^\Lambda\,
\omega^i\,,\nonumber\\
d\omega_i&=&a_i{}^\Lambda\,\alpha_\Lambda-b_{i\Lambda}\,\beta^\Lambda\,,\nonumber\\
\omega_i&\in& H^{(1,1)}\,\,;\,\,\,\,\omega^i\in
H^{(2,2)}\,\,;\,\,\,\,i=1\dots, h_{1,1}\,.\label{eqs1}\end{aligned}$$ in the presence of a non trivial NS flux: $$\begin{aligned}
\hat{H}_{(3)}&=&dB_{(2)}+d(b^i\,\omega_i)-a_{0}{}^{\Lambda}\,
\alpha_\Lambda+b_{0\Lambda}\beta^\Lambda\,.\end{aligned}$$ Integrability on the cohomology side gives: $$\begin{aligned}
d\omega^i&=&0\,\,\,;\,\,\,\,\,d^2\omega_i=-(a_i{}^\Lambda\,b_{j\Lambda}-a_{j}{}^\Lambda\,b_{i\Lambda})\,
\omega^j=0\,,\label{cocy1}\end{aligned}$$ while on the NS flux it implies $$\begin{aligned}
d\hat{H}_{(3)}&=&0\,\,\,\Rightarrow\,\,\,\,\,(a_0{}^\Lambda\,b_{j\Lambda}-a_{j}{}^\Lambda\,b_{0\Lambda})\,
\omega^j=0\,.\label{cocy2}\end{aligned}$$ Conditions (\[cocy1\]),(\[cocy2\])are equivalent to the cocycle condition (\[cocy\]).
One can show that the definition of the $a_I{}^\Lambda,\,b_{I\,\Lambda}$ given in (\[eqs1\]) is consistent. Indeed, for instance, on one hand we can write: $$\begin{aligned}
\int d\alpha_\Lambda\wedge \omega_i&=&-\int \alpha_\Lambda\wedge
(a^\Sigma{}_{j}\,\alpha_\Sigma-b_{j\Sigma}\,\beta^\Sigma)=b_{j\Lambda}\,,\end{aligned}$$ while on the other hand we have: $$\begin{aligned}
b_{i\Lambda}\,\int \omega^i\wedge \omega_j&=&b_{j\Lambda}\,.\end{aligned}$$ By performing a compactification on such *half–flat* Calabi–Yau in the presence of a NS flux we have $$\begin{aligned}
d\hat{A}&=& dA_0\,,\nonumber\\
d\hat{B}_{(2)}&=& dB_{(2)}+db^i\wedge \omega_i-b^i
\,(a_i{}^\Lambda\,\alpha_\Lambda-b_{i\Lambda}\,\beta^\Lambda)\,,
\nonumber\\
d\hat{C}_{(3)}&=& d\tilde{A}^i\wedge \omega_i-(\zeta^\Lambda\,
b_{i\Lambda}-\tilde{\zeta}_\Lambda\,
a_i{}^\Lambda)\,\omega^i-(d\zeta^\Lambda-a_i{}^\Lambda\,\tilde{A}^i)\wedge
\alpha_\Lambda+\nonumber\\&&(d\tilde{\zeta}_\Lambda-b_{i\Lambda}\,\tilde{A}^i)\wedge
\beta^\Lambda\,,\nonumber\\
\hat{F}_{(4)}&=&d\hat{C}_{(3)}+\hat{H}_{(3)}\wedge
A^0=dA^i\wedge\omega_i-b^i\,dA^0\wedge
\omega_i-(d\zeta^\Lambda-a_I{}^\Lambda\,
A^I)\wedge\alpha_\Lambda+\nonumber\\&&(d\tilde{\zeta}_\Lambda-b_{I\,\Lambda}\,
A^I)\wedge\beta_\Lambda-(\zeta^\Lambda\,
b_{i\Lambda}-\tilde{\zeta}_\Lambda\,
a_i{}^\Lambda)\,\omega^i+dB_{(2)}\wedge A^0\,,\\
A^i&=&\tilde{A}^i+b^i\,A^0\,.\nonumber\end{aligned}$$ So we obtain the correct gauging of the Ramond isometries. The covariant derivative of the scalar field $a$ dual to $B_{\mu\nu}$ is obtained from the topological term in the IIA ten–dimensional action as in [@glmw].
Conclusions
===========
In this note we have studied the gauging of the Heisenberg algebra which is common to all special quaternionic manifolds, and proved that, for an abelian gauge algebra, it requires a vanishing cocycle condition (i.e. that a certain Lie algebra cocycle be trivial). This gauge algebra, as it appears in Calabi–Yau compactification with fluxes or/and half–flat manifolds, corresponds to the gauging of isometries acting on RR scalars and the (dual of) the NS 2–form. The symplectic structure exhibited by the RR scalars embedded in a special quaternionic manifold suggests the general form of the gauging and a mirror relation when switching to the Heisenberg algebra of the mirror theory. It is suggestive that, if this is done, new couplings are predicted that do not usually appear in the perturbative formulation of Type IIA and Type IIB theories. The general gauging of the Heisenberg algebra also induces a scalar potential which, in some particular cases, has been studied in [@gm] and [@glmw], and whose general properties are under investigation.
Acknowledgements
================
R.D. and M.T. would like to thank the Physics Department of CERN, where part of this work was done, for its kind hospitality.
Work supported in part by the European Community’s Human Potential Program under contract HPRN-CT-2000-00131 Quantum Space-Time, in which R. D’A. is associated to Torino University. The work of S.F. has been supported in part by European Community’s Human Potential Program under contract HPRN-CT-2000-00131 Quantum Space-Time, in association with INFN Frascati National Laboratories and by D.O.E. grant DE-FG03-91ER40662, Task C. The work of S.V. has been supported by DFG – The German Science Foundation, DAAD – the German Academic Exchange Service.
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---
abstract: 'Conventional quantum mechanics describes a pre- and post-selected system in terms of virtual (Feynman) paths via which the final state can be reached. In the absence of probabilities, a weak measurement (WM) determines the probability amplitudes for the paths involved. The weak values (VW) can be identified with these amplitudes, or their linear combinations. This allows us to explain the “unusual” properties of the VW, and avoid the “paradoxes” often associated with the WM.'
author:
- 'D. Sokolovski$^{a,b}$'
date:
-
-
title: 'Weak measurements measure probability amplitudes (and very little else)'
---
0.5cm
introduction
============
Evers since inception in 1988 [@Ah] the subject of the so-called quantum weak values (WV) remained a controversial topic (for early critique see [@C1], [@C2]). In more recent developments, the authors of [@CRAP] put forward a controversial [@COMM], [@DS1] of generalising the WV to classical theories, while Steinberg [@Stein] suggested that weak measurements (WM) can be used for probing certain “surreal” elements of quantum physics. Recently, the authors of [@Nature] have demonstrated experimentally how WM can be used to (indirectly) measure the system’s wave function. One might feel that a clarification of what actually happens in a WM is in order, and the purpose of this paper is to provide one based on the concepts conventionally used in quantum theory.
The history of WV goes back to Feynman, who used mean value of a functional, averaged with the [*probability amplitudes*]{}, to illustrate certain aspects of quantum motion [@FEYN1]. Feynman averages naturally arise, for example, in an attempt to measure time spent by a tunnelling particle in the barrier [@SB]. The WM, designed to perturb the measured system is little as possible, were later studied in terms of Krauss operators and the POVM’s, and found applications in the analysis of continuous measurements [@Mensky1], [@Caves], [@Mensky2]. The subject gained in popularity when the authors of [@Ah] pointed out some “unusual” properties of the VW. Subsequent attempts to better understand these properties were made, for example, in the analysis of the “complex probabilities” in [@Hoff]. In a recent review of the practical aspects of VW [@Rev] the authors characterised the VW as [*“complex numbers that one can assign to the powers of a quantum observable operator $\hat{A}$ using two states, and initial state $|i\ra$...., and a final state $|f\ra$...”*]{} This still leaves open the original question posed by the authors of [@Ah]: what, if anything, the WV tell us about the intermediate state of a pre- and post-selected system?
We will answer it in the following way: in the case of intermediate measurements on a pre- and post-selected system one must consider the system’s histories referring to at least three different moments of time. Such histories, in general, interfere, and are conventionally characterised by probability amplitudes [@FEYN]. A WM destroys coherence between the histories only slightly and, in the absence of probabilities, measures the corresponding probability amplitudes or, more generally, various combinations of its real and imaginary parts. We will show that this simple observation allows one to avoid the notions of “anomalous” weak values [@Ah], [@CRAP], quantum system “being at two different places at the same time” [@Ahbook], “photons disembodied from its polarisation” [@CAT],[@CAT2], or violation of Einstein’s causality in classically forbidden transitions [@NIM]. For consistency, we will need to reproduce some of the known results, and we will try to do it in the briefest possible manner in the following Sections.
Paths, amplitudes, and meters
=============================
Following [@Ah] we consider a system in an $N$-dimensional Hilbert space with a Hamiltonian $\hat{H}$. We also consider an arbitrary operator ${\hat{S}}$, with the eigenvalues $S_i$ and the eigenstates $|i\ra$, $i=1,2,,..,N$. At $t=0$ the system is prepared (pre-selected) in a state $|\psi\ra$ and at $t=T$ we check if the system is (post-select the system) in another state $|\phi\ra$. If it is, we will keep the results of all other measurements we may make halfway into the transition, at $t=T/2$. The probability amplitude for a successful post-selecton is then $A^{\phi \gets \psi}=\la \phi|\exp(-i\hat{H}T)|\psi\ra$. Inserting the unity $\sum_{i}|i\ra\la i|=1$ at $t=T/2$ we have $$\begin{aligned}
\label{a3}
A^{\phi \gets \psi}=\sum_{i=1}^NA^{\phi \gets \psi}_i, {\quad}{\quad}{\quad}{\quad}{\quad}{\quad}{\\ \nonumber}A^{\phi \gets \psi}_i\equiv \la \phi|\exp(-i\hat{H}T/2)|i\ra\la i|exp(-i\hat{H}T/2)|\psi\ra\end{aligned}$$ This can be seen as a variant of the most basic quantum mechanical problem [@FEYN]: a system may reach the final state from the initial state via $N$ paths or path (see Fig. 1). The paths are determined by the nature of the quantity ${\hat{S}}$, and their amplitudes depend on ${\hat{S}}$, as well as on the initial and final states $|\psi\ra$ and $|\phi\ra$. The paths may be either interfering or exclusive alternatives [@FEYN], depending on what is done at $t=T/2$. If nothing is done, $N$ [*virtual*]{} paths form a single route, and their amplitudes should be added as in Eq.(\[a3\]) [@FEYN]. The probability to arrive in $\phi$ is then given by $P^{\phi \gets \psi}=|\sum_{i=1}^NA^{\phi \gets \psi}_i|^2$. Alternatively, an external meter can destroy interference between the paths. If the destruction is complete, the paths become [*real*]{} and can be equipped with probabilities $|A^{\phi \gets \psi}_i|^2$. The probability of a successful post-selection is now given by $P^{\phi \gets \psi}=\sum_{i=1}^N|A^{\phi \gets \psi}_i|^2$.
![(Color online) A system in a $5$ dimensional Hilbert space can reach the final state $|\phi\ra$ via five virtual paths $\{i\}$ with probability amplitudes $ A^{\phi \gets \psi}_i$. An accurate measurement of an operator ${\hat{S}}=\sum_{i=1}^2|i\ra\la i|-\sum_{i=3}^5|i\ra\la i|$ with degenerate eigenvalues of $1$ and $-1$ creates two real pathways $I=\{1+2\}$ and $II=\{3+4+5\}$, travelled with the probabilities $\omega_{I}$ and $\omega_{II}$ given by Eq.(\[a7aaa\]). A WM of ${\hat{S}}$ determines the difference relative amplitudes for the virtual paths $I$ and $II$ in Eq.(\[a9a\]), $\alpha_{I}-\alpha_{II}$. []{data-label="fig:3"}]({FIG1.pdf}){width="8cm" height="5cm"}
Von Neumann measurements with post-selection
============================================
To see how interference between the paths shown in Fig. 1 can be destroyed, we employ a von Neumann pointer with a position $f$ and momentum $\lambda$, briefly coupled to the system around $t=T/2$ via an interaction Hamiltonian $-i\delta(t-T/2)\partial_f {\hat{S}}$ (we use $\hbar=1$). The meter is prepared in a state $|M\ra$, such that $G(f)\equiv \la f|M\ra$ is a real function which peaks around the origin $f=0$ with a width $\Delta f$, $$\begin{aligned}
\label{a4c}
G(f)=\la f| M\ra=(\Delta f)^{-1/2}G_0(f/\Delta f)\end{aligned}$$ where $G_0(f)=G_0(-f)$, $G_0(f)_{|f|\to \infty}\to 0$ and $\int G_0^2(f)df=1$. After a successful post-selection, the meter is in a pure state $|M'\ra$ (the result is well known, see, for example, [@Ah]) $$\begin{aligned}
\label{a4a}
G'(f)=\la f| M'\ra= \sum_{i=1}^NA^{\phi\gets\psi}_iG(f-S_i).\end{aligned}$$ In the momentum space, the meter’s final state is given by $$\begin{aligned}
\label{a4b}
G'(\lambda)=\la {{\lambda }}| M'\ra= G(\lambda) \sum_{i=1}^NA^{\phi\gets\psi}_i\exp(-i\lambda S_i),\end{aligned}$$ where $G'(f)=(2\pi)^{-1/2}\int G(\lambda)\exp(i\lambda f)d\lambda$. Repeating the experiment many times we can evaluate the mean pointer position or the momentum after the measurement, $$\begin{aligned}
\label{a4}
\la f\ra_{{\hat{S}}}= \int f |G(f)|^2 df/ \int |G(f)|^2 df,\end{aligned}$$ and $$\begin{aligned}
\label{a4e}
\la \lambda \ra_{{\hat{S}}}= \int \lambda |G({{\lambda }})|^2 d{{\lambda }}/ \int |G({{\lambda }})|^2 d{{\lambda }}.\end{aligned}$$ So what can be learnt about the condition of a pre- and post-selected system at $t=T/2$? It is convenient to write the operator ${\hat{S}}$ as a sum of projectors on its eigenstates, $$\begin{aligned}
\label{a5}
{\hat{S}}=\sum_{i=1}^N S_i\hat{P}_i, {\quad}\hat{P}_i\equiv |i\ra\la i|.\end{aligned}$$ and consider the measurement of a $\hat{P}_i$ for various values of $\Delta f$.
Accurate (strong) measurements
==============================
Consider first an accurate (strong) measurement of a $\hat{P}_i$. Since $\Delta f$ determines the uncertainty in the initial setting of the pointer, an accurate measurement would require $\Delta f\to 0$. If so, we easily find that $$\begin{aligned}
\label{a6}
\la f\ra_i^{strong}= |A^{\phi\gets\psi}_i|^2/|\sum_{i'=1}^N|A^{\phi\gets\psi}_{i'}|^2\equiv \omega_i.\end{aligned}$$ Thus, an accurate meter completely destroys the coherence between the paths in Fig.1. Moreover, the measured mean value of the projector $\hat{P}_i$ gives the [*relative frequency*]{} with which the real path passing through the $i$-th state is travelled if the experiment is repeated many times. It is a simple matter to verify that for an arbitrary operator ${\hat{S}}$ with non-degenerate eigenvalues, $S_i\ne S_j$, the mean value of the pointer position gives the weighted sum of its eigenvalues, $$\begin{aligned}
\label{a7}
\la f\ra^{strong}_{{\hat{S}}}= \sum_{i=1}^N\omega_iS_i.\end{aligned}$$ This has an obvious classical meaning: if the value of the quantity ${\hat{S}}$ on the $i$-th path is $S_i$, and the $i$-th path is travelled with the probability $\omega_i$, then the average value over many trials is given by by the sum (\[a7\]). If $K$ and $(N-K)$ eigenvalues of the measured ${\hat{S}}$ are degenerate, e.g., $S_{1}=...=S_{K}\equiv S_I$, $S_{K+1}=...=S_{N}\equiv S_{II}$, Eq.(\[a4a\]) shows that the interference between the paths within each group of eigenvalues are not destroyed by a strong measurement (SM) of ${\hat{S}}$ (see Fig. 1). Rather, in accordance with the Uncertainty Principle [@FEYN],[@DS2], they are combined into two real routes, with amplitudes, $$\begin{aligned}
\label{a7aa}
A^{\phi\gets\psi}_{I} =\sum_{i=1}^KA^{\phi\gets\psi}_{i},{\quad}\text{and} {\quad}A^{\phi\gets\psi}_{II} =\sum_{i=K+1}^NA^{\phi\gets\psi}_{i},{\quad}\end{aligned}$$ which are travelled with the probabilities $$\begin{aligned}
\label{a7aaa}
\omega_{I}=|A^{\phi\gets\psi}_{I}|^2/(|A^{\phi\gets\psi}_{I}|^2+|A^{\phi\gets\psi}_{II}|^2), {\\ \nonumber}\omega_{II}=|A^{\phi\gets\psi}_{I|}|^2/(|A^{\phi\gets\psi}_{I}|^2+|A^{\phi\gets\psi}_{II}|^2).\end{aligned}$$ Accordingly, we have $$\begin{aligned}
\label{a7a}
\la f\ra^{strong}_{{\hat{S}}}= \omega_{I} S_I+ \omega_{II} S_{II}.\end{aligned}$$ Equation (\[a7a\]) is easily generalised to the case where there are more than two groups of degenerate eigenvalues. Thus, a strong measurement measures probabilities (and their linear combinations) for the real paths created by the meter according to the simple rules (\[a7\]) and (\[a7a\]). Importantly, different choices of the operator ${\hat{S}}$ lead to different sets of the real paths and, therefore, to different statistical ensembles which may have nothing more in common than their parent quantum system [@DS2]. We will return to this point in Sections IX and X.
Inaccurate (weak) measurements
==============================
Suppose next that we want to know something about the quantity ${\hat{S}}$ at $t=T/2$, provided all paths are allowed to interfere. The Uncertainty Principle may warn us against such an attempt [@DS1], [@DS2], but we want to proceed anyway, by making the measurement highly inaccurate, or weak. We send $\Delta f \to \infty$, so that the r.h.s of Eq.(\[a4a\]) becomes $G(f)= const\times \sum_{i=1}^NA^{\phi\gets\psi} +\delta$, where $\delta$ is a small correction which would, hopefully, tell us something about the system when we measure $\la f \ra_{{\hat{S}}}$. But what precisely? With the interference almost untouched, we have no probabilities for the virtual paths, yet there are always probability amplitudes. We would risk a guess: $\la f \ra_{i}$ will tell us something about the [ probability amplitude]{} to travel the $i$-th path. As before, we will proceed by trying to measure the projectors $\hat{P}_i$ in the limit of vanishing accuracy, $\Delta f \to \infty$, and determine the mean pointer reading $\la f\ra_{i}$. Evaluating, to the first order of $\partial_fG$, the average in Eq.(\[a4\]) we have $$\begin{aligned}
\label{a8}
\la f\ra^{weak}_i= {\text{Re } }\alpha_i,\end{aligned}$$ where $$\begin{aligned}
\label{a8a}
\alpha_i\equiv
\frac{A^{\phi\gets\psi}_i}{\sum_{i'=1}^NA^{\phi\gets\psi}_{i'}}.
\end{aligned}$$ We note that Eq.(\[a8\]) has the same structure as Eq.(\[a6\]), but with probabilities replaced with the real parts of the corresponding [*relative probability amplitudes*]{} $\alpha_i$. Similarly, the mean reading of a weak meter set to measure an arbitrary operator ${\hat{S}}$ is just the weighted sum of its eigenvalues, $$\begin{aligned}
\label{a9}
\la f\ra^{weak}_{{\hat{S}}}= \sum_{i=1}^NS_i {\text{Re } }\alpha_i.\end{aligned}$$ For $\hat{H}=0$, this can be re-written in an equivalent form, used, for example, in [@Ah], $$\begin{aligned}
\label{a9}
\la f\ra^{weak}_{{\hat{S}}}= \sum \la \phi|i\ra \hat{A}\la i|\psi\ra/\la\phi|\psi\ra=\frac{\la \phi|\hat{A}|\psi\ra}{\la\phi|\psi\ra}.\end{aligned}$$ where Since there are no [*a priori*]{} restrictions on the sign of ${\text{Re } }\alpha_i$, Eq.(\[a9\]) has no simple probabilistic interpretation, [@DS3] and $\la f\ra^{weak}_{{\hat{S}}}$ may take any real value at all. We note that Eq.(\[a9\]) is valid whether or not the eigenvalues of ${\hat{S}}$ are degenerate. For example, for an operator with two sets of degenerate eigenvalues, $S_{I}$ and $S_{II}$, discussed in Sect. IV, Eq.(\[a9\]) gives $$\begin{aligned}
\label{a9a}
\la f\ra^{weak}_{{\hat{S}}}= S_I\text{Re } \alpha_I+S_{II}\text{Re }\alpha_{II},{\quad}{\\ \nonumber}\alpha_{I,II}=A^{\phi \gets \psi}_{I,II}/(A^{\phi \gets \psi}_{I}+A^{\phi \gets \psi}_{II}).\end{aligned}$$ To measure the imaginary parts of the $\alpha$’s, we follow [@Ah], and look at the mean momentum acquired by the pointer given by Eq.(\[a4e\]). For the projector $\hat{P}_i$, whose only non-zero eigenvalue is $S_i=1$, Eq.(\[a4b\]) reduces to $$\begin{aligned}
\label{a10}
G'(\lambda)= G(\lambda) [\sum_{j\ne i}A^{\phi\gets\psi}_{j}+A^{\phi\gets\psi}_i\exp(-i\lambda )].\end{aligned}$$ With $G(f)$ broad in the co-ordinate space, $G(\lambda)$ is narrow. Thus, $\exp(-i\lambda )\approx 1-i \lambda$, and using Eq.(\[a4b\]), for the projector $\hat{P}_i$ and, for an arbitrary quantity ${\hat{S}}$ we have $$\begin{aligned}
\label{a11}
\la \lambda \ra^{weak}_{i}= 2\int \lambda^2 |G(\lambda)|^2 d{{\lambda }}\times {\text{Im } }\alpha_i, \end{aligned}$$ and $$\begin{aligned}
\label{a12}
\la \lambda \ra^{weak}_{{\hat{S}}}= 2\int \lambda^2 |G(\lambda)|^2 d{{\lambda }}\times \sum_{i=1}^N S_i{\text{Im } }\alpha_i, \end{aligned}$$ respectively. Therefore, a weak von Neumann meter can be used to completely determine the complex valued amplitudes $\alpha_i= {\text{Re } }\alpha_i+i{\text{Im } }\alpha_i$. If we cannot measure the projectors $\hat{P}_i$ directly, but can do so for a set of N operators ${\hat{S}}^j$ such that the matrix $S^j_i$ is non-degenerate, the $2N$ measured mean readings, $\la f \ra^{weak}_{{\hat{S}}^j}$ and $\la {{\lambda }}\ra^{weak}_{{\hat{S}}^j}$ can be used to reconstruct all amplitudes $\alpha_i$. Thus, an inaccurate meter does nor create new real pathways, and measures instead the [probability amplitudes]{} for the virtual paths shown in Fig.1. Confusing probabilities with amplitudes may lead to “paradoxes”, but it would be wrong to attribute them the quantum theory, as will be discussed below.
Tomography of a transition
==========================
Since a weak meter perturbs a transition only slightly, several WM can be performed sequentially, or even at the same time [@AhM]. To check this we can use the close relation between the accuracy and the perturbation incurred in a measurement. Let us measure operators ${\hat{S}}^{(j)}$, $j=1,2,..,J$, which may or may not commute, at the same time $t=T/2$, and with all pointers prepared in the same state (\[a4c\]). The coupling Hamiltonian is now $\hat{H}_{int}=-i\delta(t-T/2)\sum_{j=1}^J \partial_{f_j},{\hat{S}}^{(j)}$ and changing the variables $f'_j=f_j/\Delta f$, we may rewrite it as $-i(\Delta{f})^{-1}\delta(t-T/2)\sum_{j=1}^J \partial_{f'_j}{\hat{S}}^{(j)}$. Determination of all meter positions and momenta gives the results $\overline{f}\equiv(f_1,f_2,...,f_N)$, and $\overline{{{\lambda }}}\equiv({{\lambda }}_1,{{\lambda }}_2,...,{{\lambda }}_N)$, respectively. Making all measurements weak by sending $\Delta f \to \infty$, for the evolution operator over the time of interaction with the meters we obtain $$\begin{aligned}
\label{d1}
\exp(-i \hat{H}_{int}/\Delta f)\approx 1-\frac{1}{\Delta f}\sum_{j=1}^J \partial_{f'_j}{\hat{S}}^{(j)}.\end{aligned}$$ Using (\[d1\]) to evolve the initial state $(\Delta f)^{K/2}\prod_j G_0(f'_j)|\psi\ra $, projecting the result on the final state $|\phi\ra$, and returning to the original variables $f_j$, we find the (yet unnormalised) final state of the pointers, $$\begin{aligned}
\label{d2}
G'(\overline{f})=\prod_j\la f_j| M'\ra= {\quad}{\quad}{\quad}{\quad}{\quad}{\quad}{\quad}{\quad}{\quad}{\\ \nonumber}\prod_{j=1}^K G(f_j)\left [\la \phi|\psi\ra- \sum_{j=1}^K\frac{\partial_{f_j}G(f_j)}{G(f_j)}]\la \phi|{\hat{S}}^{(j)}|\psi\ra\right ].\end{aligned}$$ Evaluating, to the first order of $\partial_{f_j}G(f_j)$, the mean pointer positions gives then the same result as a WM of the operator ${\hat{S}}^{(j)}$ alone in Eq.(\[a9\]) ($d\overline{f}=\prod_i df_i$), $$\begin{aligned}
\label{d3}
\nonumber
\int f_j \rho_f(\overline{f})d\overline{f}/\int \rho_f(\overline{f})d\overline{f}=
\sum_{i=1}^NS^{(j)}_i {\text{Re } }\alpha^{(j)}_i,
$$ where $\rho(\overline{f})\equiv |G'(\overline{f})|^2$, and $\alpha^{(j)}_i$ is the relative amplitude for the $i$-path, defined in the representation in which operator ${\hat{S}}^{(j)}$ is diagonal with the eigenvalues $S^{(j)}_i$. Similarly, for the mean momentum of the $j$-th meter we recover the result (\[a12\]), $$\begin{aligned}
\label{d4}
\nonumber
\int {{\lambda }}_j \rho_{{\lambda }}(\overline{{{\lambda }}})d\overline{{{\lambda }}}/\int\rho_{{\lambda }}(\overline{{{\lambda }}})d\overline{{{\lambda }}}
{\quad}{\quad}{\quad}{\quad}{\quad}{\quad}\\
= 2\int \lambda^2 |G(\lambda)|^2 d{{\lambda }}\times \sum_{i=1}^N S^{(j)}_i{\text{Im } }\alpha^{(j)}_i,{\quad}{\quad}\end{aligned}$$ where $\rho_{{\lambda }}(\overline{{{\lambda }}})\equiv |G'(\overline{{{\lambda }}})|^2$, and $G'(\overline{f})$ $=(2\pi)^{-K/2}\int d\overline{{{\lambda }}}_K$ $ G'((\overline{{{\lambda }}}) \exp(i\sum_j {{\lambda }}_jf_j)$. One way to use simultaneous WM is to employ, at the same time, $N$ pairs of meters, each pair measuring one of the orthogonal projectors in the same representation. $\hat{P}_{j}=|j\ra \la j|$. Determining all mean positions and mean momenta, we will then have a complete set of the path amplitudes $\alpha_i$ in Eq.(\[a8a\]). We can repeat the experiments using different basis, thus performing a complete “tomography” of the transition between the states $|\psi\ra$ and $|\phi\ra$. One can also simultaneously measure the amplitudes for different basis’ (provided, of course, that the joint effect of all the meters involved on the studied transition is small). The knowledge of all relative amplitudes $\alpha_i$ allows one to predict the results of any strong measurement without actually making it. In the simplest case where none of the eigenvalues of ${\hat{S}}$ are degenerate, the ratios of the frequencies $\omega$ in Eq.(\[a7\]) are readily expressed in terms of the $\alpha$’s, $\nu_i\equiv\omega_i/\omega_N=|\alpha_i|^2/|\alpha_N|^2$, $i=1,2,...,N-1$. The first $N-1$ probabilities $\omega_i$ are then found by solving $N-1$ linear equations, $$\begin{aligned}
\label{d5}
\sum_{i=1}^N (1+\delta_{ij}/\nu_j)\omega_i=1, {\quad}j=1,2,...,N-1,\end{aligned}$$ and the remaining $\omega_N$ is just $1-\sum_{i=1}^{N-1}\omega_i$. The case of degenerate eigenvalues can be treated in a similar manner, and we will not go into details here. There is no conflict with the Uncertainty Principle: virtual amplitudes for all possible paths are potentially present in the unperturbed transition, just as the projections on all possible basis are potentially present in he state $|\psi\ra$ describing the system at a given time. Nothing of the above is unusual. From the first-order perturbation theory it is well known that a response of a quantum system to a small perturbation contains information about both the moduli and phases of the amplitudes involved (see, for instance, [@DSC]).
Interpretation of the weak values
=================================
As was discussed in Sect. IV, a strong mean value of the projector $\hat{P}_i$ gives us the frequency with which the $i$-th real route, created by the meter, would be travelled if the experiment is repeated many times. A weak mean value of $\hat{P}_i$ in Eq.(\[a11\]), on the other hand, simply tells us what the real or imaginary part of the amplitude $\alpha_i$ is. The same applies to any operator ${\hat{S}}$. The quantity $\la f\ra^{weak}_{{\hat{S}}}$ is just a weighted sum of the real parts of $\alpha_i$. It is a simple matter to show that for any $|\psi\ra$, and $N$ complex quantities $z_1$, $z_2$, ...$z_N$ adding to unity, $\sum_i z_i=1$, one can always find $|\phi\ra$ such that $\alpha_i= z_i$. Indeed, equating $\alpha_i$ to $z_1$ gives us a set of linear equations. $$\begin{aligned}
\label{a10a}
\sum_{i=1}^N(z_j-\delta_{ij})A^{\phi\gets\psi}_i=0, {\quad}j=1,..,N\end{aligned}$$ which can always be solved for $A^{\phi\gets\psi}_i=\la\phi |i \ra \la i|\psi \ra $ and, therefore for $\phi$, provided all $z_i$ add up to unity. By the same token, one can always find a pre- and post-selected system such that the measured weak value (\[a9\]) will be [*any*]{} real number, large or small, positive or negative. In Ref.[@DS1] we linked this to the Uncertainty Principle, which forbids dividing interfering alternatives into sets which have individual physical significance. Now “any” means that there would be values well outside the spectrum of the operator ${\hat{S}}$, and this is not surprising. There will also be values inside the spectrum, but this is not a rule. Finally, there must be a value exactly the same as the one we would obtain in the strong measurement, $\la f\ra^{weak}_{{\hat{S}}}=\la f\ra^{strong}_{{\hat{S}}}$, but it has no special significance. (It is easy to check that this would be the case for a system post-selected in $|\phi\ra = \exp(-i\hat{H}T)|\psi\ra$). It is our central argument that one shouldn’t look for an interpretation more meaningful than the one given above. For example, it would be unwise to interpret $\la f\ra^{weak}_{{\hat{S}}}$ as the mean value of ${\hat{S}}$ under the conditions where the interference is not destroyed. Such an interpretation would lead to an intriguing but wrong conclusion that quantally, and for reasons unknown, any physical quantity can take a huge value, and complicate the issue, rather than simplify it. Thus, anyone measuring weakly, say, the mass of an electron, and finding a value of $10^{10}$ kg, (let alone $-10^{10}$ kg) would need to account for the absence of a massive gravitational field such a mass would produce. Not even the innocent looking $\la f\ra^{weak}_{{\hat{S}}}=\la f\ra^{strong}_{{\hat{S}}}$ can be interpreted in the same way as $\la f\ra^{strong}_{{\hat{S}}}$ itself. Like all other weak values, it is obtained under different physical conditions, where the meter measures amplitudes rather than probabilities. Recognising $\la f\ra^{weak}_{{\hat{S}}}$ as nothing more than a weighted combination of probability amplitudes, which can take any value by their very nature, returns the discussion to the realm of the reasonable. Next we will illustrate this by looking at some of the most often discussed weak measurement “paradoxes”, simplifying the narrative where possible.
How can a measurement of a spin $1/2$ give a result $100$?
==========================================================
Following [@Ah] we consider a weak measurement of the $z$-component of a spin-$1/2$, ${\hat{S}}=\sigma_z=|1\ra\la1|-|2\ra\la2|$. The spin is pre- and post-selected in the states $$\begin{aligned}
\label{b1}
\psi=(|1\ra+|2\ra)/\sqrt{2}, {\quad}\phi=(|1\ra+b|2\ra)/\sqrt{1+|b|^2},\end{aligned}$$ and we put $\hat{H}=0$, so that nothing happens before and after the WM is made. There are, therefore, two virtual paths, with the relative probability amplitudes of $\alpha_1=1/(1+b^*)$ and $\alpha_1=b^*/(1+b^*)$, respectively. Let us choose $b$ to be real, $b=-99/101$. It is easy to see that the mean reading of the weak meter will equal $100$, $$\begin{aligned}
\label{b2}
\la f\ra^{weak}_{\sigma_z}=100.\end{aligned}$$ What can we infer from this simple and verifiable result? The authors of [@Ah] conclude that [*“... the usual measuring procedure for preselected and post-selected ensembles of quantum systems gives unusual results. Under some natural conditions of weakness of the measurement, its result consistently defines a new kind of value for a quantum variable, which we call the weak value.”*]{} Recalling that ${\hat{S}}=\sigma_z$ has eigenvalues $s_1=1$ and $s_2=-1$, and consulting with Eq.(\[a9\]), we see, however, that the “new kind of quantum variable”, in this case, is just the difference between the real parts of the corresponding amplitudes, which indeed equals $100$, $Re \alpha_1-Re \alpha_2 = Re(1-b^*)/(1+b^*)=100$. There is nothing “unusual” about this result.
The “quantum Cheshire cat”
==========================
The authors of [@CAT] consider a system consisting of two parts, each described by a variable taking two possible values. The particle (the cat) can be found in the state $|L\ra$ (on the left), or in the state $|R\ra$ (on the right). The particle carries a spin (or the cat carries a smile) whose projections on the $z$-axis may take values $\pm 1$. Strong measurement (SM) of the operator ${\hat{\Pi}}_R=|R\ra \la R|$ establishes whether the cat is on the right, without asking questions about the spin. Strong measurement of the projectors ${\hat{\sigma}}^+_R= {\hat{\Pi}}_R|+\ra \la +|$ and ${\hat{\sigma}}^-_R= \Pi_R|-\ra \la -|$ checks whether the cat is on the right, with its spin up and down, respectively. The operator ${\hat{\sigma}}_{L,R}=({\hat{\sigma}}^+_{L,R}-{\hat{\sigma}}^-_{L,R})$ indicates, according to [@CAT], the presence of angular momentum at the corresponding location. The authors of [@CAT] show that there exist initial and final states such that an intermediate SM of ${\hat{\Pi}}_R$ at $t=T/2$ give (it is also assumed that $\hat{H}=0$) $$\begin{aligned}
\label{b2a}
\la f\ra^{strong}_{{\hat{\Pi}}_R}=0,
$$ while a SM of ${\hat{\sigma}}_{R}$ registers the presence on the right of the particles with the spin pointing up and down, with the probabilities $\omega_3$ and $\omega_4$, respectively. Concluding from (\[b2a\]) that the particle (the photon in the optical realisation of the experiment) must always pass through the left arm of the apparatus, yet seeing the angular momentum (polarisation) in the right arm, the authors of [@CAT] suggest that they [*“... know with certainty that the photon went through the left arm, yet find angular momentum in the right arm”.* ]{} They continue to assert that [*“... physical properties can be disembodied from the objects they belong to.”*]{}. We might argue against these conclusions by pointing out that in a SM of $\Pi_R$ a particle, which is indeed on the left, still carries its spin in a state $a_+|+\ra+a_-|-\ra$, $a_\pm=\la R|\la \pm|\psi\ra$, polarised along a direction other then the $z$-axis. Thus, the smile has not completely left the cat, but just moved to some other part, perhaps to the cat’s back. Similarly, $\omega_3+\omega_4$ gives the total probability for the particle with a known $z$-projection of its spin to pass through the right arm, so the smile there is not entirely without a cat. It can also be argued differently. In the chosen representation, there are four virtual paths connecting the two states: $\{1\}$ with the cat on the left and the spin up, $\{2\}$ with the cat on the left and spin down, $\{3\}$ with the cat on the right and the spin up, and $\{4\}$, with the cat on the right and the spin down. There are also four relative amplitudes $\alpha_i$, $i=1,2,3,4$. Consulting with Sect. IV we note that a SM of ${\hat{\Pi}}_R$ creates two real paths, $\{1+2\}$ comprising the paths $\{1\}$ and $\{2\}$, and $\{3+4\}$, consisting of the paths $\{3\}$ and $\{4\}$. Since the second real path is not travelled, we know that $\alpha_4=-\alpha_3$. A SM of the operator ${\hat{\sigma}}_R$ creates three real paths, $\{1+2\}$ corresponding to its doubly degenerate eigenvalue $0$, and $\{3\}$ and $\{4\}$, corresponding to the eigenvalues $1$ and $-1$, respectively. The last two are travelled with equal probabilities $\omega_3$ and $\omega_4$, Thus, whatever the interpretation of the results reported in [@CAT], they correspond to two completely different statistical ensembles, only one of which can be “shaped” out of the original quantum system at any given time. Hence no paradox. Familiar with this type of reasoning, the authors of [@CAT] agree that if the SM of ${\hat{\Pi}}_R$ and ${\hat{\sigma}}_{R}$, are carried out at the same time the “paradox” evaporates, destroyed by the mutual disturbance affecting both measurements. To avoid the disturbance, they introduce the left-side projector ${\hat{\Pi}}_L=1-{\hat{\Pi}}_R$, and perform simultaneous weak measurements of the four operators involved. It is the suggested use of VM which is of interest for us. The results of the measurements are $$\begin{aligned}
\label{z2}
\la f\ra^{weak}_{{\hat{\Pi}}_R}=\alpha_3+\alpha_4=0,{\\ \nonumber}\la f\ra^{weak}_{{\hat{\Pi}}_L}=\alpha_1+\alpha_2=1,{\\ \nonumber}\la f\ra^{weak}_{{\hat{\sigma}}_R}=\alpha_3-\alpha_4=1,{\\ \nonumber}\la f\ra^{weak}_{{\hat{\sigma}}_L}=\alpha_1-\alpha_2=0, \end{aligned}$$ and it is also found that $\la {{\lambda }}\ra^{weak}_{{\hat{\Pi}}_R}=\la {{\lambda }}\ra^{weak}_{{\hat{\Pi}}_L}=\la {{\lambda }}\ra^{weak}_{{\hat{\sigma}}_R}=\la {{\lambda }}\ra^{weak}_{{\hat{\sigma}}_L}=0$. It is claimed then in [@CAT] that the weak values (\[z2\]) tell us that [*“the photon is in the left arm, while the angular momentum is in the right arm”*]{}. But this claim is unwarranted. All that we may learn from (\[z2\]) is that the relative amplitudes $\alpha_i$ are real, and $\alpha_1=\alpha_2=\alpha_3=-\alpha_4=1/2$. In accordance with standard quantum mechanics, this allows us to predict the results of the SM in Eq.(\[b2\]), should they be made (for example, from (\[z2\]) we know that $\la f\ra^{strong}_{{\hat{\sigma}}^\pm_R}=\omega_{3,4}=1/4 $). It by no means serves as a proof that the results of the SM of ${\hat{\Pi}}_R$ and ${\hat{\sigma}}_R$, discussed above, in “exist”, any sense, simultaneously.
Weak measurements and counterfactual statements
===============================================
The “ quantum Cheshire cat paradox”, as well as other counterfactual quantum “paradoxes” [@Ah3], [@AhH], [@DS4] are easily dismissed if one is content to see a quantum system as a “toolbox” from which one may assemble different classical statistical ensembles by performing different measurements at different times [@DS2]. As yet another example of this kind, consider the “3-box” case [@AhM], [@Ah3], where a three-state system can reach the final state via three routes defined by intermediate projections on three states $|i\ra$, $i=1,2,3$. The initial and final states are chosen so that $\alpha_1=\alpha_3=1$ and $\alpha_2=-1$. A strong measurement of the projector $\hat{P}_1=|1\ra\la1|$ creates two real pathways, one consisting of the route $1$, and the second one comprising routes $\{2\}$ and $\{3\}$, with the special property that it is never travelled since $\alpha_2+\alpha_3=0$. Thus, at $t=T/2$, the system is always found in the state $|1\ra$. Similarly, a strong measurement of $\hat{P}_3=|3\ra\la3|$ creates two [*different*]{} pathways, with the property that the intermediate state of the system is now $|3\ra$. Since the pathways are different, arguing that the system is [ in two different states at the same time]{} is no more meaningful than to use the same piece of plasticine to make first a ball, then a cube, and later argue that [ a body can be round and rectangular at the same time]{}. As in the previous Section, the authors of [@AhH] suggest that the use of WM allows one to test the two occurrences simultaneously - [*“to test - to some extent - assertions that have been otherwise regarded as counterfactual.”*]{} Again, the question is to which extent? Let us measure $\hat{P}_1$ and $\hat{P}_2$ weakly, and obtain the mean pointer readings of $1$ in both cases. We have, therefore, learnt that the relative amplitudes $\alpha_1$ and $\alpha_3$ in Eq. (\[a8\]) both have values of 1. Also, since $\sum_i\alpha_i=1$, we conclude that $\alpha_2=-1$. But we already know that, since we have prepared the transition in this particular way. As in the previous Section, we have measured the three amplitudes, and found them consistent with what would happen in two different experiments involving strong measurements of $\hat{P}_1$ and $\hat{P}_2$. This is, of course, in line with the rule elementary quantum mechanics provides for assigning probabilities to exclusive alternatives [@FEYN]. Again, we have found no evidence that the two strongly measured properties are possessed by the system at the same time, just that the system can be manipulated to give the discussed results in different circumstances.
Nature’s own “weak measurement”
===============================
The WM interference mechanism is by no means unique to the von Neumann measurements performed on pre- and post-selected systems, and our conclusions apply in those cases as well. To demonstrate this we consider the case of apparently “superluminal” wave packet tunnelling analysed in detail in Ref. [@ANN]. A particle of a mass $\mu$ described by a Gaussian wave packet with a coordinate width $\Delta x$ and mean momentum $p$, $\psi(x,t=0)=(\Delta x)^{-1/2}G_0(x/\Delta x)\exp(ipx)$, $G_0(x)=(2/\pi)^{1/4}\exp(-x^2)$ is incident on a potential barrier placed sufficiently far to the right, whose transmission amplitude is $T(p)$. The energy of the particle is, $E(p)=p^2/\mu$, its free velocity is $v=p/\mu$, and we will ignore the spreading of the wave packet during the time of experiment (the case of spreading is analysed, e.g., in Ref. [@ANN]). As $t\to \infty$, the transmitted part of the wave function is $\psi^T(x,t)=\int T(k)G(k-p)\exp[ik(x-ct)]dk$, where $G(x)=(2pi)^{-1/2}\int G(k)\exp(ikx)dk$. Rewriting the Fourier transform as a convolution in the coordinate space we obtain an equation, similar to (\[a4\]) $$\begin{aligned}
\label{y1}
\psi^T(x,t)=\exp(ipx-vpt/2)\int G(x-vt-x')A(x')dx',{\quad}{\quad}$$ where $$\begin{aligned}
\label{y2}
A(x)=(2\pi)^{-1/2}\exp(-ipx)\int T(k)\exp(ikx)dk.{\quad}\end{aligned}$$ Apart from an overall phase factor, Eq.(\[y1\]) has the same form as (\[a4a\]), with the discreet variable $S_i$ is replaced by a continuous $x'$. The transmitted wave packet is now built from the envelopes $G(x-vt -x')$, shifted forwards relative to the freely propagating envelope $G(x-vt)$ if $x'>0$, or backwards of $x'<0$. The amplitude $A(x')$ for each shift is determined by the barrier and the incident mean momentum. We have, therefore, a “quantum measurement” of the spacial shift $x'$ with which the transmitted particle emerges from the barrier, performed to the accuracy determined by the wave packet’s width $\Delta x$. Notably, since the particle’s own position plays the role of the pointer, the “measurement” requires no external meter, and “is made” every time a wave packet is sent towards the barrier. Our aim is to compare the mean position $\la x\ra$ of a tunnelled particle with that of freely propagating one at some large $t$, and learn something about how long it takes for a particle to tunnel. To ensure that the particle does not go over the barrier, we must choose $E(p)$ below the barrier height, and $\Delta x$ very large, thus making the “measurement” weak. Proceeding as in Sect.V we easily find [@ANN] $$\begin{aligned}
\label{y3}
\delta x\equiv \la x\ra-vt=\int dx' x' {\text{Re } }\alpha(x'), {\\ \nonumber}\alpha(x)\equiv A(x)/\int dx' A(x')dx'\end{aligned}$$ while the change in the mean momentum $\la k \ra$ of the transmitted particle, obtained because higher momenta tunnel more easily, is given by [@ANN] $$\begin{aligned}
\label{y4}
\delta k \equiv \la k \ra-p=2\int k^2 |G(k)|^2 dk \int dx' x' {\text{Im } }\alpha(x').{\quad}\end{aligned}$$ Both quantities are readily expressed in terms of $T(p)=|T(p)|\exp[i\Phi(p)]$, $$\begin{aligned}
\label{y5}
\delta x =\partial_p \Phi(p),{\quad}\delta k =2\int k^2 |G(k)|^2 dk{\\ \nonumber}\times \partial_p \log |T(p)|.\end{aligned}$$ For a broad rectangular barrier of a width $d$ and a hight $V$ we have [@ANN] $T(p)\sim \exp[(q-ip)d]$ with $q\equiv \sqrt{2\mu(V-E(p)}>0$, resulting in $\delta x \sim -d <0$. At this point we note that since a rectangular barrier does not support bound states, $T(k)$ cannot have poles in the upper half of the complex $k$-plane, and $a(x<0)\equiv 0$ [@ANN]. Thus, $\delta x$ is one of those weak values which lie outside the domain where the measured quantity, in our case the shift, normally has its values. So what have we learnt about the delay a particle experiences during a tunnelling transmission? Firstly, as always is the case with the WM, the simple mathematics is correct, and, secondly, the effect can be observed (see, for example [@NIM]). Interpretation of the result (\[y3\]) is a different matter. Having found, on average, the tunnelled particle ahead of the freely propagating one, one is tempted to conclude that the former has spent in the barrier a time, shorter by $\delta \tau = \delta x/v$. If so, the time its has spent in the barrier is $\tau=d/v-\delta \tau$, known also as the “phase time” [@REV]. But $\tau$ is almost zero for a broad rectangular barrier, and this, perhaps, means that the particle’s speed under the barrier exceeds the speed of light $c$. A long standing discussion of whether a particle in a classically forbidden region may cheat Einstein’s relativity can be found, for example, in Refs. [@REV]. We recognise all this as yet another WM “paradox” entirely of one’s own making. The notions of “time spent in the barrier” and “under-barrier velocity”, as used above, are vague, and have no clear definition in quantum theory. In fact, we have decomposed the transmission amplitude which determines the transition of the particle from one side of the barrier to the other into a set of sub-amplitudes \[note that $\int dx A(x) = (2\pi)^{1/2}T(p)$\]. We then learned something about the real and imaginary parts of their weighted sum $\int dx' x' \alpha(x')$, and that is all.
Summary and discussion
======================
In summary, the phenomenon of weak measurements has a simple explanation within elementary quantum mechanics. A quantum system at a given time is fully characterised by a vector in its Hilbert space, $|\psi\ra$ or, more precisely, by the complex amplitudes $\la i|\psi\ra$, yielding its wave function in a representation corresponding to the basis $\{|i\ra\}$. In a similar way, a system making a transition between the states $|\psi\ra$ and $|\phi\ra$ is characterised by virtual (Feynman) paths connecting the two states, and the probability amplitudes ascribed to them. In case of a single intermediate measurement, these amplitudes are given by $\la \phi|i\ra\la i|\psi\ra$, in the eigen basis $\{|i\ra\}$ of the measured quantity. An inaccurate weak measurement does not destroy interference between the paths and, in the absence of probabilities, determines the values of the relative amplitudes, $\alpha_i=\la \phi|i\ra\la i|\psi\ra/\sum_{i'}'\la \phi|i'\ra\la i'|\psi\ra$ or, more generally, of their linear combinations. One can be forgiven for thinking that recently the WM have been granted more importance than they probably deserve [@BBC], [@DMAIL]. The possible over-interpretation of the WM results began with the original paper [@Ah], whose authors were vague on the nature of the weak values , qualifying them simply as a “new kind of quantum variable”. They also lead the reader to believe that their result was “unusual”, while the measured quantity is just the difference between the amplitudes $\alpha_i$, which naturally become large in the case of an improbable transition, $\sum_{i'}'\la \phi|i'\ra\la i'|\psi\ra\to 0$. The same vagueness is responsible for the notion that the WM can help resolve counterfactual “paradoxes”, since different weak values can be observed simultaneously [@CAT], [@AhH]. In reality the WM only allow one to reconstruct the amplitudes $\alpha_i$, which can later be used to predict which happens to a system under different incompatible conditions, and by no means suggest that these conditions are, in some sense, realised at the same time. Another example of over-interpretation of WM results is the identification of the “phase time” with the mean time a tunnelling particle spends in the barrier. This leads to a conflict with Einstein’s causality, which is, however, easily resolved by realising that all that is measured is only a particular integral involving the sub-amplitudes into which the transmission amplitude is partitioned. Finally, while certain amount of publicity given to a subject can be beneficial (notably, to the present author, who would otherwise have to find a different topic), one would like to avoid references to “surreal elements of quantum mechanics” [@Stein], “electrons with disembodied charge” [@CAT], a “weird quantum phenomenon known as the ’Cheshire Cat’ effect” [@DMAIL], or “violation of relativistic causality” [@NIM], unless absolutely necessary. Conventional quantum mechanics provides a way around these exotic suggestions, by offering a simple, albeit by far less intriguing explanation of the WM phenomena. Our discussion may appear unbalanced if we do not consider also what good can, and has been, achieved with the help of the WM scheme. It is well known that quantum perturbation theory results contain information about the phase of the relevant amplitudes and wave functions. Consider, for example a spin-1/2 in a state $|\psi\ra =a|1\ra+b|2\ra$. Projection on the state $|1\ra$ succeeds with the probability $P=|a|^2$, and repeating it we can determine both $|a|$ and $|b|=\sqrt{1- |a|^2}$. We may then add a small magnetic field along the $x$-axis, so that $|\psi\ra$ evolves into $(1-i\gamma \sigma_x)|\psi\ra$. The probability for a successful projection is now $P+\delta P$, where $\lim_{\gamma \to 0}\delta P/\gamma$ is easily found to be $2|a||b|\sin \Delta \varphi$, and $\Delta \varphi$ is the phase difference between $a$ and $b$. Thus, response of the system to a small perturbation allows one determine, in an indirect way, its wave function.
The WM scheme offers a similar possibility. Let us post-select the spin in a known state $|\phi\ra$, so that the two states are connected by two virtual paths with the relative amplitudes $\alpha_i=\la \phi|i\ra\la i|\psi\ra/\la\phi|\psi\ra$, ($\hat{H}=0$). Employing a weakly coupled meter, one can determine the real and imaginary parts of $\alpha_i$, as discussed in Sects. V an IV. Then, since $\la \phi|i\ra$ are known, one determines the values of $a=\la 1|\psi\ra$ and $b=\la 2|\psi\ra$ up to an unknown constant $\la\phi|\psi\ra$, and with them the all of the state $|\psi\ra$.
This was the strategy followed by the authors of [@Nature] in the more interesting case of infinite number of dimensions , who showed that also in this case the wave function can, in principle, be extracted from the amplitudes $\alpha_i$. \[Note, however, a mistake in Eq.(6) of [@Nature]: the first factor in the numerator depends on $a$, and should appear inside the sum in Eq.(7), making the suggested connection between the wave function and the WV of the projector much less direct. Rather, the said WV comes out proportional to the corresponding relative amplitude $\alpha$, which is one of the main points we make in this paper.\]
It should not then come as a surprise that quantities such as the gradient of its phase, representing local velocity in Bohm’s version of quantum mechanics [@Bohm],[@Holl], can also be “measured” indirectly in a WM scheme. A measurement of its photonic equivalent, the Pointing vector was reported in [@Scie], whose results were further interpreted in [@NJP]. Both experiments represent considerable technological advances, but do not, we argue, benefit from the exotic notions mentioned at the beginning of this Section. In their context a “weak measurement” should mean only “a perturbative scheme where the observed mean value of an additional degree of freedom can be expressed in terms of the amplitudes on the virtual paths connecting the initial and final states of the studied system”. Which is also the precise meaning of the title given to this paper.
Acknowledgements
================
Support of the Project Grupos Consolidados UPV/EHU del Gobierno Vasco (IT-472-10) and the MINECO Grant No. FIS2012-36673-C03-01, as well as useful discussions with Prof. E. Akhmatskaya are gratefully acknowledged.
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abstract: |
In this paper, for foliations with spin leaves, we compute the spectral action for sub-Dirac operators.\
[**Keywords:**]{}sub-Dirac operators; spectral action ; Seely-dewitt coefficients\
author:
- 'Yong Wang\'
title: '[**The spectral action for sub-Dirac operators**]{} '
---
=14.2cm =21.3cm =-0.30in =-0.30in
\[section\] \[section\] \[section\] \[section\] \[section\]
Introduction
============
Connes’spectral action principle (\[Co\]) in noncommutative geometry states that the physical action depends only on the spectrum. We assume that space-time is a product of a continuous manifold and a finite space. The spectral action is defined as the trace of an arbitrary function of the Dirac operator for the bosonic part and a Dirac type action of the fermionic part including all their interactions. In \[CC1\], Chamseddine and Connes computed the Spectral action for Dirac operators on spin manifolds and the Chamseddine-Connes spectral action comprises the Einsiein-Hilbert action of general relativity and the bosonic part of the action of the standard model of particle physics. In \[HPS\], Hanisch, Pfäffle and Stephan derived a formula for the gravitional part of the spectral action for Dirac operators on $4$-dimensional spin manifolds with totally anti-symmetric torsion. They also deduced the Lagrangian for the Standard Model of particle Physics in the presence of torsion from the Chamseddine-Connes spectral action. In \[CC2\], Chamseddine and Connes studied the spectral action for spin manifolds with boundary and generalized this action to noncommutative spaces which are products of a spin manifold and a finite space. In \[EILS\],\[ILS\], the spectral actions for the noncommutative torus and $SU_q(2)$ are computed explicitly.\
In this paper, we consider a compact foliation $M$ with spin leaves. We don’t assume that $M$ is spin, so we have no Dirac operators on $M$, then we can not derive the physical action from the Chamseddine-Connes spectral action for Dirac operators. In \[LZ\], in order to prove the Connes’ vanishing theorem for foliations with spin leaves, Liu and Zhang introduced sub-Dirac operators instead of Dirac operators. The sub-Dirac operator is a first order formally self adjoint elliptic differential operator. So we have a commutative spectral triple and we compute the spectral action for sub-Dirac operators.\
This paper is organized as follows: In Section 2, we review the sub-Dirac operator and compute the spectral action for sub-Dirac operators. In Section 3, we compute the spectral action for sub-Dirac operators for the Standard Model.In Section 4, we compute the spectral action for sub-Dirac operators for foliations with boundary.\
The spectral action for sub-Dirac operators
============================================
Let $(M,F)$ be a closed foliation and $g^F$ be a metric on $F$. Let $g^{TM}$ be a metric on $TM$ which restricted to $g^F$ on $F$. Let $F^\perp$ be the orthogonal complement of $F$ in $TM$ with respect to $g^{TM}$. Then we have the following orthogonal splitting, $$TM=F\oplus F^\perp;~~g^{TM}=g^F\oplus g^{F^\perp},\eqno(2.1)$$ where $g^{F^\perp}$ is the restriction of $g^{TM}$ to $F^\perp$. Let $P,P^\perp$ be the orthogonal projection from $TM$ to $F$,$F^\perp$ respectively. Let $\nabla^{TM}$ be the Levi-Civita connection of $g^{TM}$ and $\nabla^F$ (resp. $\nabla^{F^\perp})$ be the restriction of $\nabla^{TM}$ to $F$ (resp. $F^\perp$). That is, $$\nabla^F=P\nabla^{TM}P,~~\nabla^{F^\perp}=P^\perp\nabla^{TM}P^\perp.\eqno(2.2)$$ We assume that $F$ is oriented, spin and carries a fixed spin structure. We also assume that $F^\perp$ is oriented and that both $2p={\rm dim}F$ and $q={\rm dim}F^{\perp}$ are even.\
Let $S(F)$ be the bundle of spinors associated to $(F,g^F)$. For any $X\in \Gamma(F),$ denote by $c(X)$ the Clifford action of $X$ on $S(F)$. Since ${\rm dim}F$ is even, we have the splitting $S(F)=S_+(F)\oplus S_-(F)$ and $c(X)$ exchanges $S_+(F)$ and $S_-(F)$.\
Let $\wedge(F^{\perp,\star})$ be the exterior algebra bundle of $F^{\perp}$. Then $\wedge(F^{\perp,\star})$ carries a canonically induced metric $g^{\wedge(F^{\perp,\star})}$ from $g^{F^\perp}$. For any $U\in \Gamma(F^\perp)$, let $U^*\in \Gamma(F^{\perp,*})$ be the corresponding dual of $U$ with respect to $g^{F^\perp}$. Now for $U\in \Gamma(F^\perp)$, set $$c(U)=U^*\wedge-i_U,~~\widehat{c}(U)=U^*\wedge+i_U,\eqno(2.3)$$ where $U^*\wedge$ and $i_U$ are the exterior and inner multiplication. Let $h_1.\cdots,h_q$ be an oriented local orthonormal basis of $F^\perp$. Then $\tau=(-\sqrt{-1})^{\frac{q(q+1)}{2}}c(h_1)\cdots c(h_q)$ and $\tau
^2=1$. Now the $+1$ and $-1$ eigenspaces of $\tau$ give a splitting $\wedge(F^{\perp,\star})=\wedge_+(F^{\perp,\star})\oplus
\wedge_-(F^{\perp,\star}).$ Let $S(F)\widehat{\otimes}\wedge(F^{\perp,\star})$ be the ${\bf Z}_2$ graded tensor product of $S(F)$ and $\wedge(F^{\perp,\star})$. For $X\in \Gamma(F),~U\in \Gamma(F^\perp)$, the operators $c(X),~c(U),~\widehat{c}(U)$ extend naturally to $S(F)\widehat{\otimes}\wedge(F^{\perp,\star})$ and they are anticommute. The connections $\nabla^F,~\nabla^{F^\perp}$ lift to $S(F)$ and $\wedge(F^{\perp,\star})$ naturally. We write them $\nabla^{S(F)}$ and $\nabla^{\wedge(F^{\perp,\star})}$. Then $S(F)\widehat{\otimes}\wedge(F^{\perp,\star})$ carries the induced tensor product connection $\nabla^{S(F)\widehat{\otimes}\wedge(F^{\perp,\star})}$.\
Let $S\in \Omega(T^*M)\otimes \Gamma({\rm End}(TM))$ be defined by $$\nabla^{TM}=\nabla^{F}+\nabla^{F^\perp}+S.\eqno(2.4)$$ Then for any $X\in \Gamma(TM)$, $S(X)$ exchanges $\Gamma(F)$ and $\Gamma(F^\perp)$ and is skew-adjoint with respect to $g^{TM}$. Let $V$ be a complex vector bundle with the metric connection $\nabla^V$. Then $S(F)\widehat{\otimes}\wedge(F^{\perp,\star})\otimes V$ carries the induced tensor product connection $\nabla^{S(F)\widehat{\otimes}\wedge(F^{\perp,\star})\otimes V}$. Let $\{f_i\}_{i=1}^{2p}$ be an oriented orthonormal basis of $F$. Let$$\widetilde{\nabla}=\nabla^{S(F)\widehat{\otimes}\wedge(F^{\perp,\star})}+
\frac{1}{2}\sum_{j=1}^{2p}\sum_{s=1}^q<S(.)f_j,h_s>c(f_j)c(h_s)$$ $$\widetilde{\nabla}^{F,V}=\widetilde{\nabla}
\otimes {\rm Id}_V+{\rm Id}_{S(F)\widehat{\otimes}\wedge(F^{\perp,\star})}\otimes
\nabla^V.\eqno(2.5)$$ Since the vector bundle $F^\perp$ might well be non-spin, Liu and Zhang \[LZ\] introduced the following sub-Dirac operator:\
[**Definition 2.1**]{} Let $D_{F,V}$ be the operator mapping from $\Gamma({S(F)\widehat{\otimes}\wedge(F^{\perp,\star})\otimes
V})$ to itself defined by $$D_{F,V}=\sum_{i=1}^{2p}c(f_i)\widetilde{\nabla}^{F,V}_{f_i}
+\sum_{s=1}^{q}c(h_s)\widetilde{\nabla}^{F,V}_{h_s}.\eqno(2.6)$$
Let $\triangle^{F,V}$ be the Bochner Laplacian defined by $$\triangle^{F,V}:=-\sum_{i=1}^{2p}(\widetilde{\nabla}^{F,V}_{f_i})^2
-\sum_{s=1}^{q}(\widetilde{\nabla}^{F,V}_{h_s})^2+
\widetilde{\nabla}^{F,V}_{\sum_{i=1}^{2p}\nabla^{TM}_{f_i}f_i} +
\widetilde{\nabla}^{F,V}_{\sum_{s=1}^{q}\nabla^{TM}_{h_s}h_s}.
\eqno(2.7)$$ Let $r_M$ be the scalar curvature of the metric $g^{TM}$. Let $R^{F^\bot}$ and $R^V$ be curvature of $F^\bot$ and $V$. Then we have the following Lichnerowicz formula for $D_{F,V}$.\
[**Theorem 2.2(\[LZ\])**]{} [*The following identity holds*]{} $$D^2_{F,V}=\triangle^{F,V}+\frac{1}{2}\sum_{i,j=1}^{2p}c(f_i)c(f_j)R^V(f_i,f_j)$$ $$+\sum_{i=1}^{2p}\sum_{s=1}^qc(f_i)c(h_s)R^V(f_i,h_s)
+\frac{1}{2}\sum_{s,t=1}^{q}c(h_s)c(h_t)R^V(h_s,h_t)$$ $$+\frac{r_M}{4}+\frac{1}{4}\sum_{i=1}^{2p}\sum_{r,s,t=1}^q\left<R^{F^\bot}(f_i,h_r)h_t,h_s\right>
c(f_i)c(h_r)\widehat{c}(h_s)\widehat{c}(h_t)$$ $$+\frac{1}{8}\sum_{i,j=1}^{2p}\sum_{s,t=1}^q\left<R^{F^\bot}(f_i,f_j)h_t,h_s\right>
c(f_i)c(f_j)\widehat{c}(h_s)\widehat{c}(h_t)$$ $$+\frac{1}{8}\sum_{s,t,r,l=1}^q\left<R^{F^\bot}(h_r,h_l)h_t,h_s\right>
c(h_r)c(h_l)\widehat{c}(h_s)\widehat{c}(h_t).\eqno(2.8)$$\
When $V$ is a complex line bundle, we write $D_{F}$ instead of $D_{F,E}$. For the sub-Dirac operator $D_{F}$ we will calculate the bosonic part of the spectral action. It is defined to be the number of eigenvalues of $D_{F}$ in the interval $[-\wedge,\wedge]$ with $\wedge\in {\bf R}^+$. As in \[CC1\], it is expressed as $$I={\rm tr}\widehat{F}\left(\frac{D^2_F}{\wedge^2}\right).$$ Here tr denotes the operator trace in the $L^2$ completion of $\Gamma (S(F)\widehat{\otimes}\wedge(F^{\perp,\star}))$, and $\widehat{F}:{\bf R}^+\rightarrow {\bf R}^+$ is a cut-off function with support in the interval $[0,1]$ which is constant near the origin. Let ${\rm dim}M=m$. By Theorem 2.2, we have the heat trace asymptotics for $t\rightarrow 0$, $${\rm tr}(e^{-tD_F^2})\sim \sum_{n\geq
0}t^{n-\frac{m}{2}}a_{2n}(D_F^2).$$ One uses the Seely-deWitt coefficients $a_{2n}(D_F^2)$ and $t=\wedge^{-2}$ to obtain an asymptotics for the spectral action when ${\rm dim} M=4$ \[CC1\] $$I={\rm tr}\widehat{F}\left(\frac{D^2_F}{\wedge^2}\right)\sim
\wedge^4F_4a_0(D^2_F)+\wedge^2F_2a_2(D^2_F)+\wedge^0F_0a_4(D^2_F)~~{\rm
as} ~~\wedge\rightarrow \infty \eqno(2.9)$$ with the first three moments of the cut-off function which are given by $F_4=\int_0^{\infty}s\widehat{F}(s)ds,$\
$F_2=\int_0^{\infty}\widehat{F}(s)ds$ and $F_0=\widehat{F}(0)$. Let $$-E=\frac{r_M}{4}+W=\frac{r_M}{4}+\frac{1}{4}\sum_{i=1}^{2p}\sum_{r,s,t=1}^q\left<R^{F^\bot}(f_i,h_r)h_t,h_s\right>
c(f_i)c(h_r)\widehat{c}(h_s)\widehat{c}(h_t)$$ $$+\frac{1}{8}\sum_{i,j=1}^{2p}\sum_{s,t=1}^q\left<R^{F^\bot}(f_i,f_j)h_t,h_s\right>
c(f_i)c(f_j)\widehat{c}(h_s)\widehat{c}(h_t)$$ $$+\frac{1}{8}\sum_{s,t,r,l=1}^q\left<R^{F^\bot}(h_r,h_l)h_t,h_s\right>
c(h_r)c(h_l)\widehat{c}(h_s)\widehat{c}(h_t),\eqno(2.10)$$ and $$\Omega_{ij}=\widetilde{\nabla}_{e_i}\widetilde{\nabla}_{e_j}-\widetilde{\nabla}_{e_j}\widetilde{\nabla}_{e_i}
-\widetilde{\nabla}_{[e_i,e_j]},\eqno(2.11)$$ where $e_i$ is $f_i$ or $h_s$. We use \[G, Thm 4.1.6\] to obtain the first three coefficients of the heat trace asymptotics: $$a_0(D_F)=(4\pi)^{-\frac{m}{2}}\int_M{\rm tr}(\rm
Id)dvol,\eqno(2.12)$$ $$a_2(D_F)=(4\pi)^{-\frac{m}{2}}\int_M{\rm
tr}[(r_M+6E)/6]dvol,\eqno(2.13)$$ $$a_4(D_F)=\frac{(4\pi)^{-\frac{m}{2}}}{360}\int_M{\rm
tr}[-12R_{ijij,kk}+5R_{ijij}R_{klkl}$$ $$-2R_{ijik}R_{ljlk}+2R_{ijkl}R_{ijkl}-60R_{ijij}E+180E^2+60E_{,kk}+30\Omega_{ij}
\Omega_{ij}]dvol.\eqno(2.14)$$
Since ${\rm
dim}[S(F)\widehat{\otimes}\wedge(F^{\perp,\star})]=2^{p+q}$ and $m=2p+q$, then we have $a_0(D_F)=\frac{1}{2^p\pi^{p+\frac{q}{2}}}\int_Mdvol.$ By Clifford relations and cyclicity of the trace and the trace of the odd degree operator being zero, we have $${\rm tr}(c(f_i))=0;~{\rm tr}(c(f_i)c(f_j))=0 ~{\rm for} ~i\neq j;$$ $$~{\rm tr}(c(h_r)c(h_l)\widehat{c}(h_s)\widehat{c}(h_t))=0, ~{\rm for}
~r\neq l.\eqno(2.15)$$ and $${\rm
tr}E=-2^{p+q}\cdot\frac{r_M}{4},~~a_2(D_F)=-\frac{1}{12\cdot2^p\pi^{p+\frac{q}{2}}}\int_Mr_Mdvol.\eqno(2.16)$$ Let $I_1,I_2, I_3$ denote respectively the last three terms in (2.10). By (2.15), we have $${\rm tr}(E^2)={\rm
tr}(\frac{r_M^2}{16}+W^2)={\rm
tr}(\frac{r_M^2}{16}+I_1^2+I_2^2+I_3^2).\eqno(2.17)$$ $${\rm
tr}(I_1^2)=\frac{1}{16}\sum_{i,i'=1}^{2p}\sum_{r,r',s,s',t,t'=1}^q\left<R^{F^\bot}(f_i,h_r)h_t,h_s\right>
\left<R^{F^\bot}(f_{i'},h_{r'})h_{t'},h_{s'}\right>$$ $$\cdot{\rm
tr}[c(f_i)c(h_r)\widehat{c}(h_s)\widehat{c}(h_t)
c(f_{i'})c(h_{r'})\widehat{c}(h_{s'})\widehat{c}(h_{t'})]\eqno(2.18)$$ Similar to (2.15), we have $${\rm
tr}[c(f_i)c(h_r)\widehat{c}(h_s)\widehat{c}(h_t)
c(f_{i'})c(h_{r'})\widehat{c}(h_{s'})\widehat{c}(h_{t'})]$$ $$=-\delta_i^{i'}\delta_r^{r'}2^p{\rm
tr}_{\wedge(F^{\perp,\star})}[\widehat{c}(h_s)\widehat{c}(h_t)\widehat{c}(h_{s'})\widehat{c}(h_{t'})]\eqno(2.19)$$ Since $t\neq s,~t'\neq s'$. we get $${\rm
tr}_{\wedge(F^{\perp,\star})}[\widehat{c}(h_s)\widehat{c}(h_t)\widehat{c}(h_{s'})\widehat{c}(h_{t'})]
=(\delta_t^{s'}\delta_s^{t'}-\delta_t^{t'}\delta_s^{s'})2^q\eqno(2.20)$$ By (2.19) and (2.20), we have $${\rm
tr}(I_1^2)=\frac{2^{p+q}}{8}\sum_{i=1}^{2p}\sum_{r,s,t=1}^q\left<R^{F^\bot}(f_i,h_r)h_t,h_s\right>^2.\eqno(2.21)$$ Similarly we have $${\rm
tr}(I_2^2)=\frac{2^{p+q}}{16}\sum_{i,j=1}^{2p}\sum_{s,t=1}^q\left<R^{F^\bot}(f_i,f_j)h_t,h_s\right>^2;\eqno(2.22)$$ $${\rm
tr}(I_3^2)=\frac{2^{p+q}}{16}
\sum_{s,t,r,l=1}^q\left<R^{F^\bot}(h_r,h_l)h_t,h_s\right>^2.\eqno(2.23)$$ So we get $${\rm tr
E^2}=\frac{2^{p+q}}{16}r_M^2+\frac{2^{p+q}}{16}||R^{F^\bot}||^2,\eqno(2.24)$$ where $$||R^{F^\bot}||^2=2\sum_{i=1}^{2p}\sum_{r,s,t=1}^q\left<R^{F^\bot}(f_i,h_r)h_t,h_s\right>^2$$ $$+\sum_{i,j=1}^{2p}\sum_{s,t=1}^q\left<R^{F^\bot}(f_i,f_j)h_t,h_s\right>^2+
\sum_{s,t,r,l=1}^q\left<R^{F^\bot}(h_r,h_l)h_t,h_s\right>^2.\eqno(2.25)$$ Nextly we compute ${\rm tr}[\Omega_{ij} \Omega_{ij}]$ in a local coordinate, so we can assume that $M$ is spin and $\widetilde{\nabla}$ is the standard twisted connection on the twisted spinors bundle $S(TM)\otimes S(F^\bot)$. Then $$\Omega_{ij}=R^{S(TM)}(e_i,e_j)\otimes {\rm Id}_{S(F^\bot)}+{\rm
Id}_{S(TM)}\otimes R^{S(F^\bot)}(e_i,e_j)$$ $$=-\frac{1}{4}R^M_{ijkl}c(e_k)c(e_l)
\otimes {\rm Id}_{S(F^\bot)}-\frac{1}{4}{\rm Id}_{S(TM)}\otimes
\left<R^{F^\bot}(e_i,e_j)h_s,h_t\right>c(h_s)c(h_t).\eqno(2.26)$$ Similar to the computations of ${\rm tr E^2}$, we get $${\rm
tr}[\Omega_{ij}
\Omega_{ij}]=-\frac{2^{p+q}}{8}(R_{ijkl}^2+||R^{F^\bot}||^2)\eqno(2.27)$$ By the divergence theorem and (2.24) and (2.27), we have $$a_4(D_F^2)=\frac{1}{360\cdot2^p\pi^{p+\frac{q}{2}}}\int_M\left(\frac{5}{4}r_M^2-2R_{ijik}R_{ljlk}-\frac{7}{4}
R_{ijkl}^2+\frac{15}{2}||R^{F^\bot}||^2\right)dvol.\eqno(2.28)$$
The spectral action for the Standard Model associated to sub-Dirac operators
============================================================================
In this section, we let $m=4$. We consider the product space $\cal{H}$ of the $L^2$ completion of $\Gamma (S(F)\widehat{\otimes}\wedge(F^{\perp,\star}))$ and a finite dimensional Hilbert space $ {\cal H}_f$ (called internal Hilbert space). The specific particle model is encoded in $ {\cal
H}_f$. On the bundle $S(F)\widehat{\otimes}\wedge(F^{\perp,\star})\otimes{\cal H}_f$ one considers a connection $\widetilde{\nabla}^{F,{\cal H}_f}$ in (2.5) and $\nabla^{{\cal H}_f}$ is a covariant derivative in the trivial bundle ${{\cal H}_f}$ induced gauge fields. The associated Dirac operator to $\widetilde{\nabla}^{F,{\cal H}_f}$ is called $D^f_F$. The generalized Dirac operator of the Standard Model $D_{F,\Phi}$ contains the Higgs boson, Yukawa couplings, neutrino masses and the CKM-matrix encoded in a field $\Phi$ of endomorphisms of ${{\cal
H}_f}$. We define $D_{F,\Phi}$ for sections $\psi\otimes \chi\in
{\cal H}$ as $$D_{F,\Phi}(\psi\otimes
\chi)=D^f_F(\psi\otimes
\chi)+\gamma_5\psi\otimes\Phi\chi,\eqno(3.1)$$ where $\gamma_5=e_0e_1e_2e_3$ is the volume element. We choose the same $\Phi$ as $\Phi$ in \[CC1\]. The bosonic part of the Lagrangian of the Standard Model is obtained by replacing $D_F$ by $D_{F,\Phi}$ in (2.9). In (2.8), we write $D^2_{F,{\cal H}_f}=\triangle^{F,{\cal
H}_f}+W_1$. Then direct computations show $$D_{F,\Phi}^2=\triangle^{F,{\cal H}_f}-E_\Phi,\eqno(3.2)$$ where the potential is defined as $$E_\Phi(\psi\otimes
\chi)=-W_1(\psi\otimes
\chi)+\sum_{i=1}^4\gamma_5c(e_i)\cdot\psi\otimes[\nabla_{e_i}^{H_f},\Phi]\chi-\psi\otimes\Phi^2\chi.\eqno(3.3)$$ We denote the trace on ${\cal H}$ and on ${\cal H}_f$ as Tr and ${\rm tr}_f$. From (3.3), we have $${\rm Tr}(E_\Phi)={\rm dim}{\cal
H}_f\cdot 2^{p+q-2}r_M-2^{p+q}{\rm tr}_f(\Phi^2).\eqno(3.4)$$For Seely-deWitt coefficient $a_4(D_{F,\Phi}^2)$ we also need to calculate $$(E_\Phi)^2(\psi\otimes
\chi)=W_1^2(\psi\otimes
\chi)+\sum_{i,j=1}^4\gamma_5c(e_i)\gamma_5c(e_j)\cdot\psi\otimes
[\nabla_{e_i}^{H_f},\Phi][\nabla_{e_j}^{H_f},\Phi]\chi$$ $$+\psi\otimes\Phi^4\chi-2E\psi\otimes\Phi^2\chi+
\frac{1}{2}\sum_{i,j=1}^{2p}c(f_i)c(f_j)\psi\otimes [\Phi^2R^{{\cal
H}_f}(f_i,f_j)+R^{{\cal H}_f}(f_i,f_j)\Phi^2]\chi$$ $$+\sum_{i=1}^{2p}\sum_{s=1}^qc(f_i)c(h_s)\psi\otimes [\Phi^2R^{{\cal H}_f}(f_i,h_s)
+R^{{\cal H}_f}(f_i,h_s)\Phi^2]\chi$$ $$+\frac{1}{2}\sum_{s,t=1}^{q}c(h_s)c(h_t)\psi\otimes [\Phi^2R^{{\cal
H}_f}(h_s,h_t)+R^{{\cal H}_f}(h_s,h_t)\Phi^2]\chi$$ $$-\sum_{i=1}^4\gamma_5c(e_i)\psi\otimes(\Phi^2[\nabla_{e_i}^{H_f},\Phi]+[\nabla_{e_i}^{H_f},\Phi]\Phi^2)\chi$$ $$+\sum_{i=1}^4(E\gamma_5c(e_i)\psi+\gamma_5c(e_i)E\psi)\otimes
[\nabla_{e_i}^{H_f},\Phi]\chi$$ $$-\frac{1}{2}\sum_{i,j,k=1}^4\gamma_5c(e_i)c(e_j)c(e_k)\psi\otimes[\nabla_{e_i}^{H_f},\Phi]
R^{{\cal H}_f}(e_j,e_k)\chi$$ $$-
\frac{1}{2}\sum_{i,j,k=1}^4c(e_j)c(e_k)\gamma_5c(e_i)\psi\otimes
R^{{\cal H}_f}(e_j,e_k)[\nabla_{e_i}^{H_f},\Phi] \chi.\eqno(3.5)$$ By Clifford relations and cyclicity of the trace and the trace of the odd degree operator being zero, only the first four summands on the right-hand side contribute to the trace of $(E_\Phi)^2$. By direct computations, we get $${\rm Tr}(E_\Phi^2)={\rm dim}{\cal
H}_f\frac{2^{p+q}}{16}(r_M^2+||R^{F^\bot}||^2)-2^{p+q-1}\sum_{i,j=1}^4{\rm
tr}_f(\Omega^f_{ij}\Omega^f_{ij})$$ $$+2^{p+q-1}r_M{\rm
tr}_f(\Phi^2)+2^{p+q}{\rm tr}_f(\Phi^4)+2^{p+q}\sum_{i=1}^4{\rm
tr}_f([\nabla_{e_i}^{H_f},\Phi]^2).\eqno(3.6)$$ By (2.27), we have $${\rm Tr}(\widetilde{\Omega}^f_{ij}\widetilde{\Omega}^f_{ij})
=-{\rm dim}{\cal H}_f\cdot
\frac{2^{p+q}}{8}(R_{ijkl}^2+||R^{F^\bot}||^2)+2^{p+q}{\rm
tr}_f(\Omega^f_{ij}\Omega^f_{ij}).\eqno(3.7)$$ We choose the finite space ${\cal H}_f$ according to the construction of the noncommutative Standard Model \[CC1\], ${\rm dim}{\cal
H}_f=96$ and $\nabla^{{\cal H}_f}$ is the appropriate covariant derivative associated to the Standard Model gauge group $U(1)_Y\times SU(2)_\omega\times SU(3)_c.$ We know that (for related notations see \[HPS\], \[IKS\]), $${\rm
tr}_f(\Omega^f_{ij}\Omega^f_{ij})=\frac{48}{5}g_3^2||G||^2+\frac{48}{5}g_2^2||F_1||^2+16g_1^2||B||^2,\eqno(3.8)$$ $${\rm
tr}_f([\nabla_{e_i}^{H_f},\Phi]^2)=4a|D_\nu\varphi|^2,~~{\rm
tr}_f(\Phi^2)=4a|\phi|^2+2c,~~{\rm
tr}_f(\Phi^4)=4b|\phi|^4+8e|\phi|^2+2d.\eqno(3.9)$$ Then we get $$a_0(D_{F,\Phi})=\frac{96}{2^p\pi^{p+\frac{q}{2}}}\int_Mdvol,\eqno(3.10)$$ $$a_2(D_{F,\Phi})=\frac{1}{2^p\pi^{p+\frac{q}{2}}}\int_M(40r_M-4a|\phi|^2-2c)dvol,\eqno(3.11)$$ $$a_4(D_{F,\Phi})=\frac{1}{360\cdot 2^p\pi^{p+\frac{q}{2}}}\int_M
\left\{4000r_M^2-192R_{ijik}R_{ljlk}-168R^2_{ijkl}+120ar_M|\varphi|^2\right.$$ $$+60cr_M+720||R^{F^\bot}||^2-576g_3^2||G||^2
-576g_2^2||F_1||^2-960g_1^2||B||^2$$ $$\left.+720b|\varphi|^4+1440e|\varphi|^2+360d+720|D_\nu\varphi|^2\right\}dvol.\eqno(3.12)dvol$$ In presence of the Standard Model fields we obtain essentially one new term (apart from the usual suspects) $$I_{\rm new}=\frac{2}{
2^p\pi^{p+\frac{q}{2}}}\int_M||R^{F^\bot}||^2dvol.\eqno(3.13)$$
The spectral action for foliations with boundary
=================================================
In this section, we let $M$ be a foliation with boundary $\partial M$. Let $\psi\in \Gamma(S(F)\widehat{\otimes}\wedge(F^{\perp,\star})$, We impose the Dirichlet boundary conditions $\psi|_{\partial
M}=0$. With the Dirichlet boundary conditions, we have the heat trace asymptotics for $t\rightarrow 0$ \[BG\], $${\rm tr}(e^{-tD_F^2})\sim \sum_{n\geq
0}t^{\frac{n-m}{2}}a_{n}(D_F^2).$$ One uses the Seely-deWitt coefficients $a_{n}(D_F^2)$ and $t=\wedge^{-2}$ to obtain an asymptotics for the spectral action when ${\rm dim} M=4$ \[ILV (18)\] $$I={\rm tr}\widehat{F}\left(\frac{D^2_F}{\wedge^2}\right)\sim
\wedge^4F_4a_0(D^2_F)+\wedge^3F_3a_1(D^2_F)$$ $$+\wedge^2F_2a_2(D^2_F)+\wedge
F_1a_3(D^2_F)+\wedge^0F_0a_4(D^2_F)~~{\rm as} ~~\wedge\rightarrow
\infty \eqno(4.1)$$ where $F_k:=\frac{1}{\Gamma(\frac{k}{2})}\int_0^{\infty}\widehat{F}(s)s^{\frac{k}{2}-1}ds.$ Let $N=e_m$ be the inward pointing unit normal vector on $\partial
M$ and $e_i, 1\leq i\leq m-1$ be the orthonormal frame on $T(\partial M)$. Let $ L_{ab}=(\nabla_{e_a}e_b,N)$ be the second fundamental form and indices $\{a,b,\cdots\}$ range from $1$ through $m-1$. We use \[BG, Thm 1.1\] to obtain the first five coefficients of the heat trace asymptotics: $$a_0(D_F)=(4\pi)^{-\frac{m}{2}}\int_M{\rm tr}(\rm
Id)dvol_M,\eqno(4.2)$$ $$a_1(D_F)=-4^{-1}(4\pi)^{-\frac{(m-1)}{2}}\int_{\partial M}{\rm tr}(\rm
Id)dvol_{\partial M},\eqno(4.3)$$ $$a_2(D_F)=(4\pi)^{-\frac{m}{2}}6^{-1}\{\int_M{\rm
tr}(r_M+6E)dvol_M+2\int_{\partial M}{\rm tr}(L_{aa})dvol_{\partial
M}\},\eqno(4.4)$$ $$a_3(D_F)=-4^{-1}(4\pi)^{-\frac{(m-1)}{2}}96^{-1}\{\int_{\partial M}{\rm
tr}(96E+16r_M$$ $$+8R_{aNaN}+7L_{aa}L_{bb}-10L_{ab}L_{ab})dvol_{\partial
M})\},\eqno(4.5)$$ $$a_4(D_F)=\frac{(4\pi)^{-\frac{m}{2}}}{360}\{
\int_M{\rm
tr}[-12R_{ijij,kk}+5R_{ijij}R_{klkl}$$ $$-2R_{ijik}R_{ljlk}+2R_{ijkl}R_{ijkl}-60R_{ijij}E+180E^2+60E_{,kk}+30\Omega_{ij}
\Omega_{ij}]dvol_M$$ $$+\int_{\partial M}{\rm
tr}(-120E;_N-18r_M;_N+120EL_{aa}+20r_ML_{aa}+4R_{aNaN}L_{bb}-12R_{aNbN}L_{ab}$$ $$+4R_{abcd}L_{ac}+24L_{aa;bb}
+40/21L_{aa}L_{bb}L_{cc}-88/7L_{ab}L_{ab}L_{cc}+320/21L_{ab}L_{bc}L_{ac})dvol_{\partial
M}\}.\eqno(4.6)$$ By (2.16) and (2.28) and the divergence theorem for manifolds with boundary, we get $$a_0(D_F)=\frac{1}{2^p\pi^{p+\frac{q}{2}}}\int_Mdvol_M,\eqno(4.7)$$ $$a_1(D_F)=-4^{-1}(4\pi)^{-\frac{(m-1)}{2}}2^{p+q}\int_{\partial M}dvol_{\partial M},\eqno(4.8)$$ $$a_2(D_F)=\frac{1}{12\cdot
2^p\pi^{p+\frac{q}{2}}}(-\int_Mr_Mdvol_M+4\int_{\partial
M}L_{aa}dvol_{\partial M}),\eqno(4.9)$$ $$a_3(D_F)=-4^{-1}(4\pi)^{-\frac{(m-1)}{2}}96^{-1}2^{p+q}\{\int_{\partial
M}(-8r_M$$ $$+8R_{aNaN}+7L_{aa}L_{bb}-10L_{ab}L_{ab})dvol_{\partial
M}\},\eqno(4.10)$$ $$a_4(D_F)=\frac{(4\pi)^{-\frac{m}{2}}}{360}2^{p+q}\{
\int_M\left(\frac{5}{4}r_M^2-2R_{ijik}R_{ljlk}-\frac{7}{4}
R_{ijkl}^2+\frac{15}{2}||R^{F^\bot}||^2\right)dvol_M$$ $$+
\int_{\partial M}{\rm
tr}(-51r_{M;N}-10r_ML_{aa}+4R_{aNaN}L_{bb}-12R_{aNbN}L_{ab}$$ $$+4R_{abcd}L_{ac}+24L_{aa;bb}
+40/21L_{aa}L_{bb}L_{cc}-88/7L_{ab}L_{ab}L_{cc}+320/21L_{ab}L_{bc}L_{ac})dvol_{\partial
M}\}.\eqno(4.11)$$\
[**Acknowledgement.**]{} This work was supported by NSFC No.10801027 and Fok Ying Tong Education Foundation No. 121003.\
[**References**]{}\
\[BG\] Branson, T. P.; Gilkey, P. B. The asymptotics of the Laplacian on a manifold with boundary, Comm. Partial Differential Equations 15 (1990), no. 2, 245-272.\
\[CC1\] Chamseddine, A. H.; Connes, A., The spectral action principle. Comm. Math. Phys. 186 (1997), no. 3, 731-750.\
\[CC2\] Chamseddine, A. H.; Connes, A. Noncommutative geometric spaces with boundary: spectral action. J. Geom. Phys. 61 (2011), no. 1, 317-332.\
\[Co\] Connes, A., Gravity coupled with matter and the foundation of non-commutative geometry. Comm. Math. Phys. 182 (1996), no. 1, 155-176.\
\[EILS\] Essouabri, D.; Iochum, B.; Levy, C.; Sitarz, A., Spectral action on noncommutative torus. J. Noncommut. Geom. 2 (2008), no. 1, 53-123.\
\[Gi\]Gilkey, P. B., [*Invariance theory, the heat equation, and the Atiyah-Singer index theorem.*]{} Mathematics Lecture Series, 11. Publish or Perish, 1984\
\[HPS\]Hanisch, F.; Pfäffle, F.; Stephan, C. A., The spectral action for Dirac operators with skew-symmetric torsion. Comm. Math. Phys. 300 (2010), no. 3, 877-888.\
\[IKS\]Iochum, B.; Kastler, D.; Sch¨¹cker, T., On the universal Chamseddine-Connes action. I. Details of the action computation. J. Math. Phys. 38 (1997), no. 10, 4929-4950.\
\[ILS\] Iochum, B.; Levy, C.; Sitarz, A., Spectral action on ${\rm SU}_q(2)$. Comm. Math. Phys. 289 (2009), no. 1, 107-155.\
\[ILV\]Iochum, B.; Levy, C.; Vassilevich, D., Spectral action for torsion with and without boundaries, arXiv:1008.3630.\
\[LZ\] Liu, K.; Zhang, W., Adiabatic limits and foliations, Contemp. Math.,2001,195-208.\
\
E-mail: [*[email protected]*]{}\
|
---
abstract: |
Let $G$ be a graph of order $n$ and let $q\left( G\right) $ be that largest eigenvalue of the signless Laplacian of $G.$ In this note it is shown that if $k\geq2$, $n>5k^{2},$ and $q\left( G\right) \geq n+2k-2,$ then $G$ contains a cycle of length $l$ for each $l\in\left\{ 3,4,\ldots,2k+2\right\} .$ This bound on $q\left( G\right) $ is asymptotically tight, as the graph $K_{k}\vee\overline{K}_{n-k}$ contains no cycles longer than $2k$ and $$q\left( K_{k}\vee\overline{K}_{n-k}\right) >n+2k-2-\frac{2k\left(
k-1\right) }{n+2k-3}.$$ The main result of this note gives an asymptotic solution to a recent conjecture about the maximum $q\left( G\right) $ of a graph $G$ with forbidden cycles. The proof of the main result and the tools used therein could serve as a guidance to the proof of the full conjecture.
**AMS classification:** *15A42, 05C50*
**Keywords:** *signless Laplacian; maximum eigenvalue; forbidden cycles; spectral extremal problems.*
author:
- 'Vladimir Nikiforov[^1]'
title: 'An asymptotically tight bound on the $Q$-index of graphs with forbidden cycles'
---
Introduction
============
Given a graph $G,$ the $Q$-index of $G$ is the largest eigenvalue $q\left(
G\right) $ of its signless Laplacian $Q\left( G\right) $. In this note we give an asymptotically tight upper bound on $q\left( G\right) $ of a graph $G$ of a given order, with no cycle of specified length. Let us start by recalling a general problem in spectral extremal graph theory:
*How large can* $q\left( G\right) $ *be if* $G$ *is a graph of order* $n,$ *with no subgraph isomorphic to some forbidden graph* $F?\medskip$
This problem has been solved for several classes of forbidden subgraphs; in particular, in [@FNP13] it has been solved for forbidden cycles $C_{4}$ and $C_{5}.$ In addition, it seems a folklore result that $q\left( G\right)
>n$ implies the existence of $C_{3},$ and this bound is exact in view of the star of order $n$. For longer cycles, a general conjecture has been stated in [@FNP13], which we reiterate next to clarify the contribution of the present note.
Let $S_{n,k}$ be the graph obtained by joining each vertex of a complete graph of order $k$ to each vertex of an independent set of order $n-k;$ in other words, $S_{n,k}=K_{k}\vee\overline{K}_{n-k}.$ Also, let $S_{n,k}^{+}$ be the graph obtained by adding an edge to $S_{n,k}.$
\[con1\] Let $k\geq2$ and let $G$ be a graph of sufficiently large order $n.$ If $G$ has no $C_{2k+1},$ then $q\left( G\right) <q\left(
S_{n,k}\right) ,$ unless $G=S_{n,k}.$ If $G$ has no $C_{2k+2},$ then $q\left( G\right) <q\left( S_{n,k}^{+}\right) ,$ unless $G=S_{n,k}^{+}.$
Conjecture \[con1\] seems difficult, but not hopeless. It is very likely that it will be solved completely in the next couple of years. Thus, one of the goals of this note is to make some suggestions for such a solution and to emphasize the relevance of some supporting results.
The starting point of our work is the observation that both $q\left(
S_{n,k}\right) $ and $q\left( S_{n,k}^{+}\right) $ are very close to $n+2k-2$ whenever $n$ is large. In fact, the difference between these values is $\Omega\left( 1/n\right) ,$ as can be seen from the following proposition.
\[pro1\]If $k\geq2$ and $n>5k^{2},$ then $$n+2k-2-\frac{2\left( k^{2}-k\right) }{n+2k+2}>q\left( S_{n,k}^{+}\right)
>q\left( S_{n,k}\right) >n+2k-2-\frac{2\left( k^{2}-k\right) }{n+2k-3}.$$
These bounds prompt a weaker, yet asymptotically tight version of Conjecture \[con1\], which we shall prove in this note.
\[th1\]Let $k\geq2,$ $n>6k^{2},$ and let $G$ be a graph of order $n.$ If $q\left( G\right) \geq n+2k-2,$ then $G$ contains cycles of length $2k+1$ and $2k+2.$
Before going further, let us note a corollary of Theorem \[th1\].
Let $k\geq2,$ $n>6k^{2},$ and let $G$ be a graph of order $n.$ If $q\left(
G\right) \geq n+2k-2,$ then $G$ contains a cycle of length $l$ for each $l\in\left\{ 3,4,\ldots,2k+2\right\} .$
Indeed, if $l\geq5,$ the conclusion follows immediately from Theorem \[th1\]. For $l\in\left\{ 3,4\right\} ,$ recall the bound $$q\left( G\right) \leq\max\left\{ d_{u}+d_{v}:\left\{ u,v\right\} \in
E\left( G\right) \right\} .$$ In view of $q\left( G\right) \geq n+2,$ there must be an edge $\left\{
u,v\right\} $ belonging to two triangles; hence $G$ contains both $C_{3}$ and $C_{4}.\medskip$
Even though Theorem \[th1\] is weaker than Conjecture \[con1\], our proof is not too short. To emphasize its structure, we have extracted a few important points into separate statements, which we give next.
\[le1\]If $G$ is a graph with no $P_{2k+1},$ then for each component $H$ of $G,$ either $v\left( H\right) =2k$ or $e\left( H\right) \leq\left(
k-1\right) v\left( H\right) .$
Write $K_{2k}+v$ for the graph obtained by joining a vertex $v$ to a single vertex of the complete graph $K_{2k}$.
\[le2\]Let $v$ be a vertex of a graph $G$ of order $n.$ If $G$ contains no $P_{2k+1}$ with both endvertices different from $v,$ then$$2e\left( G\right) -d_{v}\leq\left( 2k-1\right) \left( n-1\right) ,$$ unless $G$ is a union of several copies of $K_{2k}$ and one $K_{2k}+v.$
We also need bounds on $q\left( G\right) $ for some special classes of graphs. Since the known upper bounds did not work in these cases, we came up with a few technical results giving the required bounds.
\[le3\]Let the integers $k,p,m,$ and $n$ satisfy $$k\geq2,\text{ \ }m\geq1,\text{ \ }p\geq0,\text{ \ }n=2kp+m,\text{ \ \ }n\geq6k+13.$$ Let $H$ be a graph of order $m$ and let $F$ be the union of $p$ disjoint graphs of order $2k,$ which are also disjoint from $H$. Let $G$ be the graph obtained by taking $F\cup H$ and joining some vertices of $F$ to a single vertex $w$ of $H$. If$$q\left( H\right) \leq m+2k-2+\frac{6pk}{n+3}, \label{co1}$$ then $q\left( G\right) \leq n+2k-2,$ with equality holding if and only if equality holds in (\[co1\]).
The reason for Lemma \[le3\] being so technical is that it must support the proof of the following two quite different corollaries.
\[cor1\]Let $k,p,$ and $n$ be integers such that $k\geq2$ and $n=2\left(
p+1\right) k+2.$ Let $$G=K_{1}\vee\left( \left( pK_{2k}\right) \cup K_{2k+1}\right) .$$ If $n\geq6k+13,$ then $q\left( G\right) <n+2k-2.$
Given a graph $G$ and $u\in V\left( G\right) ,$ write $G-u$ for the graph obtained by removing the vertex $u.$
\[cor2\]Let $k\geq2,$ $G$ be a graph of order $n,$ and $w\in V\left(
G\right) .$ Suppose that for each component $C$ of $G-w,$ either $v\left(
C\right) =2k$ or $e\left( C\right) \leq\left( k-1\right) v\left(
C\right) .$ If $n\geq6k+13,$ then $q\left( G\right) <n+2k-2.$
In the next section we outline some notation and results needed in our proofs. The proofs themselves are given in Section \[pf\].
Notation and supporting results
===============================
For graph notation and concepts undefined here, we refer the reader to [@Bol98]. For introductory and reference material on the signless Laplacian see the survey of Cvetković [@C10] and its references. In particular, let $G$ be a graph, and $X$ and $Y$ be disjoint sets of vertices of $G.$ We write:
- $V\left( G\right) $ for the set of vertices of $G,$ $E\left( G\right) $ for the set of edges of $G,$ and $e\left( G\right) $ for $\left\vert
E\left( G\right) \right\vert $;
- $G\left[ X\right] $ for the graph induced by $X,$ and $e\left( X\right)
$ for $e\left( G\left[ X\right] \right) ;$
- $e\left( X,Y\right) $ for the number of edges joining vertices in $X$ to vertices in $Y;$
- $\Gamma_{u}$ for the set of neighbors of a vertex $u,$ and $d_{u}$ for $\left\vert \Gamma_{u}\right\vert .\medskip$
We write $P_{k},$ $C_{k},$ and $K_{k}$ for the path, cycle, and complete graph of order $k.\medskip$
Given a graph $G$ and a vertex $u\in V\left( G\right) ,$ note that $$\sum_{v\in\Gamma_{u}}d_{v}=2e\left( \Gamma_{u}\right) +e\left( \Gamma
_{u},V\left( G\right) \backslash\Gamma_{u}\right) .$$ Below we shall use this fact without reference.
Some useful theorems
--------------------
Here we state several known results, all of which are used in the proof of Theorem \[th1\]. We start with two classical theorems of Erdős and Gallai [@ErGa59].
\[EGp\]Let $k\geq1.$ If $G$ is a graph of order $n,$ with no $P_{k+2},$ then $e\left( G\right) \leq kn/2,$ with equality holding if and only if $G$ is a union of disjoint copies of $K_{k+1}.$
\[EGc\]Let $k\geq2.$ If $G$ is a graph of order $n,$ with no $C_{k+1},$ then $e\left( G\right) \leq k\left( n-1\right) /2,$ with equality holding if and only if $G$ is a union of copies of $K_{k},$ all sharing a single vertex.
For connected graphs Kopylov [@Kop77] has enhanced Theorem \[EGp\] as follows.
\[Kop\] Let $k\geq1,$ and let $G$ be a connected graph of order $n$.
*(i)* If $n\geq2k+2$ and $G$ contains no $P_{2k+2},$ then $$e\left( G\right) \leq\max\left\{ kn-k\left( k+1\right) /2,\binom{2k}{2}+\left( n-2k\right) \right\} ;$$
*(ii)* If $n\geq2k+3$ and $G$ contains no $P_{2k+3},$ then$$e\left( G\right) \leq\max\left\{ kn-k\left( k+1\right) /2+1,\binom
{2k+1}{2}+\left( n-2k-1\right) \right\} .$$
We refer the reader to the more recent paper [@BGLS08], where the conditions for equality in Kopylov’s bounds are determined as well.
We shall use the following sufficient condition for Hamiltonian cycles, proved by Ore [@Ore60].
\[tOB\]If $G$ is a graph of order $n\geq3$ and $$e\left( G\right) >\binom{n-1}{2}+1,$$ then $G$ has a Hamiltonian cycle.
The following structural extension of Theorem \[EGp\] has been established in [@Nik09].
\[Ni\] Let $k\geq1$ and let the vertices of a graph $G$ be partitioned into two sets $A$ and $B$. If $$2e\left( A\right) +e\left( A,B\right) >\left( 2k-1\right) \left\vert
A\right\vert +k\left\vert B\right\vert ,$$ then there exists a path of order $2k+1$ with both endvertices in $A.$
We finish this subsection with two known upper bounds on $q\left( G\right)
.$ The proof of Theorem \[th1\] will be based on a careful analysis of the following bound on $q\left( G\right) ,$ which can be traced back to Merris [@Mer98]. The case of equality was established in [@FeYu09].
\[tM\]For every graph $G,$ $$q\left( G\right) \leq\max\left\{ d_{u}+\frac{1}{d_{u}}\sum_{v\in\Gamma_{u}}d_{v}:u\in V\left( G\right) \right\} .$$ If $G$ is connected, equality holds if and only if $G$ is regular or semiregular bipartite.
Finally, let us mention the following corollary, due to Das [@Das04].
\[Das\]If $G$ is a graph with $n$ vertices and $m$ edges, then $$q\left( G\right) \leq\frac{2m}{n-1}+n-2,$$ with equality holding if and only if $G$ is either complete, or is a star, or is a complete graph with one isolated vertex.
\[pf\]Proofs
============
In the following proofs there are several instances where the bounds can be somewhat improved at the price of more involved arguments and calculations. Such improvements seem not too worthy unless geared towards the complete solution of Conjecture \[con1\]. Instead, we tried to keep the exposition concise, so that the main points are more visible.
\[**Proof of Proposition \[pro1\]**\]It is known that $$q\left( S_{n,k}\right) =\frac{1}{2}\left( n+2k-2+\sqrt{\left(
n+2k-2\right) ^{2}-8\left( k^{2}-k\right) }\right) .$$ Hence, we see that$$q\left( S_{n,k}\right) -\left( n+2k-2\right) =-\frac{4\left(
k^{2}-k\right) }{n+2k-2+\sqrt{\left( n+2k-2\right) ^{2}-8\left(
k^{2}-k\right) }}>-\frac{2\left( k^{2}-k\right) }{n+2k-3},$$ and also $$q\left( S_{n,k}\right) -\left( n+2k-2\right) <-\frac{2\left(
k^{2}-k\right) }{n+2k-2}.$$ To bound $q\left( S_{n,k}^{+}\right) $ let $\mathbf{x}=\left( x_{1},\ldots,x_{n}\right) $ be a unit eigenvector to $q\left( S_{n,k}^{+}\right)
$ and let $x_{1},\ldots,x_{k\text{ }}$ be the entries corresponding to the vertices of degree $n-1$ in $S_{n,k}^{+}.$ Let $k+1$ and $k+2$ be the vertices of the extra edge of $S_{n,k}^{+}.$ By symmetry, $x_{1}=\cdots=x_{k}$ and $x_{k+1}=x_{k+2}.$ Using the eigenequations for $Q\left( G\right) $ and the fact that $$q\left( S_{n,k}^{+}\right) >q\left( S_{n,k}\right) >n+2k-2-\frac{2\left(
k^{2}-k\right) }{n+2k-3}>n+k-1,$$ we see that $$x_{k+1}^{2}=\frac{k^{2}x_{1}^{2}}{\left( q\left( S_{n,k}^{+}\right)
-k-2\right) ^{2}}<\frac{k}{\left( q\left( S_{n,k}\right) -k-2\right)
^{2}}<\frac{k}{\left( n-3\right) ^{2}}.$$ On the other hand, comparing the quadratic forms $\left\langle Q\left(
S_{n,k}^{+}\right) \mathbf{x},\mathbf{x}\right\rangle $ and $\left\langle
Q\left( S_{n,k}\right) \mathbf{x},\mathbf{x}\right\rangle $ of the matrices $Q\left( S_{n,k}^{+}\right) $ and $Q\left( S_{n,k}\right) $, we see that $$q\left( S_{n,k}^{+}\right) -\left( x_{k+1}+x_{k+2}\right) ^{2}=\left\langle Q\left( S_{n,k}^{+}\right) \mathbf{x},\mathbf{x}\right\rangle
-\left( x_{k+1}+x_{k+2}\right) ^{2}=\left\langle Q\left( S_{n,k}\right)
\mathbf{x},\mathbf{x}\right\rangle \leq q\left( S_{n,k}\right) .$$ Thus, after some algebra, we get $$\begin{aligned}
q\left( S_{n,k}^{+}\right) & <q\left( S_{n,k}\right) +\frac{4k}{\left(
n-3\right) ^{2}}<n+2k-2-\frac{2\left( k^{2}-k\right) }{n+2k-2}+\frac
{4k}{\left( n-3\right) ^{2}}\\
& <n+2k-2-\frac{2\left( k^{2}-k\right) }{n+2k+2},\end{aligned}$$ completing the proof of Proposition \[pro1\].
\[**Proof of Lemma \[le1\]**\]Let $H$ be a component of $G.$ Set $m=v\left( H\right) $ and assume that $m\neq2k.$ We shall show that $e\left( H\right) <\left( k-1\right) m.$ If $m\leq2k-1,$ then $$e\left( H\right) \leq\binom{m}{2}=m\left( \frac{m-1}{2}\right) \leq\left(
k-1\right) m,$$ as claimed. If $m\geq2k+1$, then clause *(ii)* of ** Theorem \[Kop\] implies that$$e\left( H\right) \leq\max\left\{ \left( k-1\right) m-\left( \left(
k-1\right) ^{2}+\left( k-1\right) \right) /2+1,\binom{2k-1}{2}+\left(
m-2k-1\right) \right\} . \label{K}$$ This inequality splits into $$e\left( H\right) \leq\left( k-1\right) m-\left( \left( k-1\right)
^{2}+\left( k-1\right) \right) /2+1\leq\left( k-1\right) m,$$ and $$e\left( H\right) \leq\binom{2k-1}{2}+\left( m-2k+1\right) =\left(
2k-1\right) \left( k-2\right) +m\leq\left( k-1\right) m.$$ Thus, in all cases we see that $e\left( H\right) \leq\left( k-1\right) m,$ completing the proof of Lemma \[le1\].
\[**Proof of Lemma \[le2\]**\]Assume for a contradiction that $$2e\left( G\right) -d_{v}\geq\left( 2k-1\right) \left( n-1\right) +1,$$ and that $G$ has no path of order $2k+1$ with both endvertices different from $v.$ Write $H$ for the component containing $v$ and let $F$ be the union of the other components of $G$. Since $P_{2k+1}\nsubseteq F$, Theorem \[EGp\] implies that $$2e\left( F\right) \leq\left( 2k-1\right) v\left( F\right) , \label{in1}$$ and so$$2e\left( H\right) -d_{v}\geq\left( 2k-1\right) \left( v\left( H\right)
-1\right) +1.$$ Noting that $$2e\left( H\right) -d_{v}=\sum_{u\in V\left( H\right) \backslash\left\{
v\right\} }d_{u}\leq\left( v\left( H\right) -1\right) ^{2},$$ we find that $v\left( H\right) \geq2k+1.$
Assume that $v\left( H\right) \geq2k+2.$ Since $$2e\left( H\right) \geq\left( 2k-1\right) \left( v\left( H\right)
-1\right) +1+d_{v}>\left( 2k-1\right) \left( v\left( H\right) -1\right)
,$$ Theorem \[EGc\] implies that $H$ contains a cycle $C$ of order $m\geq2k.$ If $m\geq2k+1,$ then obviously there is a $P_{2k+1}$ with both endvertices different from $v,$ so let $m=2k.$ Choose a vertex $w\in V\left( H\right) $ such that $w\neq v$ and $w\notin C.$ There exists a shortest path $P$ joining $w$ to a vertex $u\in C.$ By symmetry, we can index the vertices of $C$ as $u=u_{1},u_{2},\ldots,u_{2k}.$ Take $u_{0}$ in $P$ at distance $1$ from $C.$ Then the sequences $u_{0},u_{1},u_{2},\ldots,u_{2k}$ and $u_{0},u_{1},u_{2k},\ldots,u_{2}$ induce paths of order $2k+1.$ Since $v$ must be an endvertex to each of them, we see that $u_{0}=v$. But $w\neq v,$ hence $P$ contains a vertex $u_{-1}$ at distance $2$ from $C.$ Now the sequence $u_{-1},u_{0},u_{1},u_{2},\ldots,u_{2k-1}$ induces a path of order $2k+1$ with both endvertices different from $v,$ a contradiction completing the proof whenever $v\left( H\right) \geq2k+2.$
It remains to consider the case $v\left( H\right) =2k+1.$ In this case $H$ is not Hamiltonian, as otherwise there is a path of order $2k+1$ with both endvertices different from $v$; hence, Theorem \[tOB\] implies that $e\left( H\right) \leq k\left( 2k-1\right) +1$ and so$$2k\left( 2k-1\right) +2-d_{v}\geq e\left( H\right) -d_{v}\geq\left(
2k-1\right) 2k+1.$$ This is possible only if $d_{u}=1$ and $e\left( H\right) =k\left(
2k-1\right) +1.$ Since $H-v$ is complete, obviously, $H=K_{2k}+v.$ In addition, in (\[in1\]) we have $2e\left( F\right) =\left( 2k-1\right)
v\left( F\right) ,$ and so the condition for equality in Theorem \[EGp\] implies that $G$ is a union of several copies of $K_{2k}$ and one copy of $K_{2k}+v,$ completing the proof of Lemma \[le2\].
\[**Proof of Lemma \[le3\]**\]Let $q:=q\left( G\right) ;$ assume for a contradiction that $q\geq n+2k-2,$ and let $\mathbf{x}=\left( x_{1},\ldots,x_{n}\right) $ be a unit eigenvector to $q.$ From the eigenequation for $Q\left( G\right) $ and the vertex $w$ we see that$$\left( q-n+1\right) x_{w}\leq\left( q-d_{w}\right) x_{w}\leq\sum_{i\in
V\left( G\right) \backslash\left\{ w\right\} }x_{i}\leq\sqrt{\left(
n-1\right) \left( 1-x_{w}^{2}\right) },$$ and in view of $q\geq n+2k-2,$ it follows that$$x_{w}^{2}\leq\frac{n-1}{\left( q-n+1\right) ^{2}+n-1}<\frac{n-1}{n-1+\left(
2k-1\right) ^{2}}\leq1-\frac{9}{n+8}. \label{in4}$$
On the other hand, let $u\in V\left( F\right) $ be such that $x_{u}=\max\left\{ x_{v}:v\in V\left( F\right) \right\} $. Set $x:=x_{u}$ and note that the eigenequation for $u$ implies that $$qx=d_{u}x+\sum_{i\sim u}x_{i}=d_{u}x+x_{w}+\sum_{\left\{ i,u\right\} \in
E\left( F\right) }x_{i}\leq2kx+x_{w}+\left( 2k-1\right) x=\left(
4k-1\right) x+x_{w}.$$ Hence, the inequality $q\geq n+2k-2$ implies that $$x\leq\frac{x_{w}}{q-4k-1}\leq\frac{x_{w}}{n-2k-1}.$$ Next, expanding the quadratic form $\left\langle Q\left( G\right)
\mathbf{x},\mathbf{x}\right\rangle ,$ we find that $$\begin{aligned}
q & =\sum_{\left\{ i,j\right\} \in E\left( G\right) }\left( x_{i}+x_{j}\right) ^{2}\leq\sum_{\left\{ i,j\right\} \in E\left( G_{0}\right)
}\left( x_{i}+x_{j}\right) ^{2}+2kp\left( x+x_{w}\right) ^{2}+4p\binom
{2k}{2}x^{2}\\
& \leq q\left( G_{0}\right) +2kp\left( x+x_{w}\right) ^{2}+4pk\left(
2k-1\right) x^{2}\\
& =q\left( G_{0}\right) +2pkx_{w}^{2}+4pkxx_{w}+2pk\left( 4k-1\right)
x^{2}\\
& \leq q\left( G_{0}\right) +2pk\left( 1+\frac{2}{n-2k-1}+\frac
{4k-1}{\left( n-2k-1\right) ^{2}}\right) x_{w}^{2}.\end{aligned}$$ Now, plugging here the bound (\[in4\]), we get$$\begin{aligned}
q & \leq q\left( G_{0}\right) +2pk\left( 1+\frac{3}{n-2k-1}\right)
\left( 1-\frac{9}{n+8}\right) \nonumber\\
& \leq q\left( G_{0}\right) +2pk+6pk\left( \frac{1}{n-2k-1}-\frac{3}{n+8}\right) \nonumber\\
& =q\left( G_{0}\right) +2pk-6pk\left( \frac{2n-6k-13}{\left(
n-2k-1\right) \left( n+8\right) }\right) . \label{in6}$$ Note that, in view of $n\geq6k+13$ and $k\geq2,$ we have $$\frac{2n-6k-13}{\left( n-2k-1\right) \left( n+8\right) }\geq\frac
{n}{\left( n-2k-1\right) \left( n+8\right) }\geq\frac{n}{\left(
n-5\right) \left( n+8\right) }>\frac{1}{n+3}.$$ Plugging this inequality back in (\[in6\]) and using (\[co1\]), we obtain $$\begin{aligned}
n+2k-2 & \leq q\leq q\left( G_{0}\right) +2pk-\frac{6pk}{n+3}\leq
m+2k-2+\frac{6pk}{n+3}+2pk-\frac{6pk}{n+3}\\
& =n+2k-2.\end{aligned}$$ Hence $q\leq n+2k-2,$ with equality holding if and only if equality holds in (\[co1\]). The proof of Lemma \[le3\] is completed.
\[**Proof of Corollary \[cor1\]**\]We shall apply Lemma \[le3\] with $H=K_{2k+2}$ and $F=pK_{2k}.$ Clearly $2pk=n-2k-2$ and so$$\begin{aligned}
q\left( H\right) & =q\left( K_{2k+2}\right) =4k+2<v\left( H\right)
+2k-2+\frac{3\left( n-2k-2\right) }{n+3}\\
& =v\left( H\right) +2k-2+\frac{6kp}{n+3}.\end{aligned}$$ In the derivation above we use that the inequality $n\geq6k+13$ implies that $3n-6k-6>2\left( n+3\right) .$ The conditions for Lemma \[le3\] are met and so $q\left( G\right) <n+2k-2,$ completing the proof of Corollary \[cor1\].
\[**Proof of Corollary \[cor2\]**\]Let $F$ be the union of all components of $G-w$ having order exactly $2k,$ and let $p$ be their number, possibly zero. Let $H$ be the graph induced by the vertices in $V\left( G\right)
\backslash V\left( F\right) .$ Note that the hypothesis of Corollary \[cor2\] implies that $e\left( H-w\right) \leq\left( k-1\right) \left(
m-1\right) $ and so $$e\left( H\right) \leq e\left( H-w\right) +m-1\leq\left( k-1\right)
\left( m-1\right) +m-1=k\left( m-1\right) .$$ Now, from Theorem \[Das\] we get $$q\left( H\right) \leq\frac{2e\left( H\right) }{m-1}+m-2\leq\frac{2k\left(
m-1\right) }{m-1}+m-2=v\left( H\right) +2k-2\leq v\left( H\right)
+2k-2+\frac{6kp}{n+3}. \label{i}$$ Since $n\geq6k+13,$ we can apply Lemma \[le3\], obtaining $$q\left( G\right) <n+2k-2,$$ unless equality holds in (\[i\]). Equality in (\[i\]) implies that $p=0,$ that is to say $G=H.$ Also, by the condition for equality in Theorem \[Das\], we see that $G$ is either complete, or is a star, or is a complete graph with one isolated vertex. Since $q\left( G\right) =n+2k-2$, $G$ cannot be a star. If $G$ is complete, then $n+2k-2=2n-2$ and so $n=2k,$ contradicting that $n\geq6k+13.$ For the same reason $n+2k-2<2n-4$ and so $G$ cannot be a complete graph with one isolated vertex either. Corollary \[cor2\] is proved.
\[**Proof of Theorem \[th1\]**\]For short, set $q:=q\left( G\right) $ and $V:=V\left( G\right) .$ Assume for a contradiction that $G$ is a graph of order $n>6k^{2},$ with $q\geq n+2k-2,$ and suppose that $C_{2k+1}\nsubseteq
G$ or $C_{2k+2}\nsubseteq G.$ We may and shall suppose that $G$ is edge maximal, because edge addition does not decrease the $Q$-index. In particular, this assumption implies that $G$ is connected.
Let $w$ be a vertex for which the expression$$d_{w}+\frac{1}{d_{w}}\sum_{i\sim w}d_{i}$$ is maximal. We shall show that $$d_{w}+\frac{1}{d_{w}}\sum_{i\sim w}d_{i}\leq n+2k-2. \label{in3}$$ This is enough to prove Theorem \[th1\], unless $$q=d_{w}+\frac{1}{d_{w}}\sum_{i\sim w}d_{i}.$$ However, $G$ is connected, so if equality holds in (\[in3\]) Theorem \[tM\] implies that $G$ is regular or bipartite semiregular; it is not hard to see that neither of these conditions can hold. Indeed, if $G$ is bipartite, then $q\leq n.$ If $G$ is regular, then $q=2\delta\leq n,$ as otherwise, Bondy’s theorem [@Bol98] implies that $G$ is pancyclic. So to the end of the proof we shall focus on the proof of (\[in3\]).
For short, set $A=\Gamma_{w},$ $B=V\left( G\right) \backslash\left(
\Gamma_{w}\cup\left\{ w\right\} \right) ,$ and $G_{w}=G\left[
V\backslash\left\{ w\right\} \right] .$ Obviously, $\left\vert A\right\vert
=d_{w}$ and $\left\vert A\right\vert +\left\vert B\right\vert =n-1.$
First we shall prove that $C_{2k+1}\subset G.$ Assume thus that $C_{2k+1}\nsubseteq G;$ clearly $P_{2k}\nsubseteq G\left[ A\right] ,$ and so Theorem \[EGp\] implies that $e\left( A\right) \leq\left( k-1\right) \left\vert
A\right\vert $. Now $$\begin{aligned}
d_{w}+\frac{1}{d_{w}}\sum_{i\sim w}d_{i} & =\left\vert A\right\vert
+1+\frac{2e\left( A\right) +e\left( A,B\right) }{\left\vert A\right\vert
}\leq\left\vert A\right\vert +1+\frac{2\left( k-1\right) \left\vert
A\right\vert +\left\vert A\right\vert \left\vert B\right\vert }{\left\vert
A\right\vert }\\
& \leq\left\vert A\right\vert +1+2k-2+\left\vert B\right\vert =n+2k-2.\end{aligned}$$ This completes the proof that $C_{2k+1}\subset G.$
The proof that $C_{2k+2}\subset G$ is somewhat longer. Assume that $C_{2k+2}\nsubseteq G$ and note that if $d_{w}\leq2k-1$, then $$d_{w}+\frac{1}{d_{w}}\sum_{i\sim w}d_{i}=d_{w}+\Delta\leq2k-1+n-1=n+2k-2,$$ so (\[in3\]) holds. Thus, hereafter we shall assume that $d_{w}\geq2k.$
Further, note that the graph $G_{w}$ contains no path with both endvertices in $A,$ as otherwise $C_{2k+2}\subset G$. Hence, Theorem \[Ni\] implies that$$2e\left( A\right) +e\left( A,B\right) \leq\left( 2k-1\right) \left\vert
A\right\vert +k\left\vert B\right\vert =\left( k-1\right) d_{w}+k\left(
n-1\right) ,$$ and therefore $$\begin{aligned}
d_{w}+\frac{1}{d_{w}}\sum_{i\sim w}d_{i} & =d_{w}+1+\frac{2e\left(
A\right) +e\left( A,B\right) }{d_{w}}\leq d_{w}+1+\frac{\left( k-1\right)
d_{w}+k\left( n-1\right) }{d_{w}}\\
& =d_{w}+k+\frac{k\left( n-1\right) }{d_{w}}.\end{aligned}$$ The function $x+k\left( n-1\right) /x$ is convex for $x>0;$ hence, the maximum of the expression $$d_{w}+\frac{k\left( n-1\right) }{d_{w}}$$ is attained for the minimum and maximum admissible values for $d_{w}.$ Since $d_{w}\geq2k,$ in either case we find that $$d_{w}+\frac{1}{d_{w}}\sum_{i\sim w}d_{i}<n+2k-2,$$ unless $d_{w}\geq n-2.$ Therefore, to complete the proof we only need to consider the cases $d_{w}=n-2$ and $d_{w}=n-1.$
First, suppose that $d_{w}=n-2$ and let $v$ be the vertex of $G$ such that $v\neq w$ and $v\notin\Gamma_{w}.$ Note that $G_{w}$ contains no path of order $2k+1$ with both endvertices different from $v,$ as such a path would make a $C_{2k+2}$ with $w.$ Therefore, the hypothesis of Lemma \[le2\] is satisfied, and so either $$2e\left( A\right) +e\left( A,B\right) =2e\left( G_{w}\right) -d_{v}\leq\left( 2k-1\right) \left( n-2\right) \label{in2}$$ or $G_{w}$ is a union of several copies of $K_{2k}$ and one $K_{2k}+v.$ If (\[in2\]) holds, we see that $$\begin{aligned}
d_{w}+\frac{1}{d_{w}}\sum_{i\sim w}d_{i} & \leq n-2+1+\frac{2e\left(
A\right) +e\left( A,B\right) }{n-2}\leq n-1+\frac{\left( 2k-1\right)
\left( n-2\right) }{\left( n-2\right) }\\
& =n+2k-2,\end{aligned}$$ completing the proof of (\[in3\]). On the other hand, if $G_{w}$ is a union of several copies of $K_{2k}$ and one $K_{2k}+v,$ then $G$ is a spanning subgraph of the graph $G^{\prime}=K_{1}\vee\left( \left( pK_{2k}\right)
\cup K_{2k+1}\right) ,$ with $p$ chosen so that $n=2\left( p+1\right) k+2.$ Since $n\geq6k^{2}+1\geq6k+13,$ we can apply Corollary \[cor1\] obtaining that $$q\left( G\right) <q\left( G^{\prime}\right) <n+2k-2,$$ which contradicts the assumption and completes the proof of Theorem \[th1\] if $d_{w}=n-2$.
Finally, let $d_{w}=n-1.$ Since $G_{w}$ contains no $P_{2k+1},$ Lemma \[le1\] implies that for each component $C$ of $G_{w},$ either $v\left(
C\right) =2k$ or $e\left( C\right) \leq\left( k-1\right) v\left(
C\right) .$ Since $n\geq6k^{2}+1\geq6k+13,$ the graph $G$ satisfies the hypothesis of Corollary \[cor2\], and so$$q\left( G\right) <n+2k-2,$$ completing the proof of Theorem \[th1\].
**Acknowledgement**
Thanks are due to the referee for helpful suggestions.
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[^1]: Department of Mathematical Sciences, University of Memphis, Memphis TN 38152, USA; *email: [email protected]*
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abstract: 'We present a model for wave propagation in a monolayer of spheres on an elastic substrate. The model, which considers sagittally polarized waves, includes: horizontal, vertical, and rotational degrees of freedom; normal and shear coupling between the spheres and substrate, as well as between adjacent spheres; and the effects of wave propagation in the elastic substrate. For a monolayer of interacting spheres, we find three contact resonances, whose frequencies are given by simple closed-form expressions. For a monolayer of isolated spheres, only two resonances are present. The contact resonances couple to surface acoustic waves in the substrate, leading to mode hybridization and “avoided crossing” phenomena. We present dispersion curves for a monolayer of silica microspheres on a silica substrate, assuming adhesive, Hertzian interactions, and compare calculations using an effective medium approximation to a discrete model of a monolayer on a rigid substrate. While the effective medium model does not account for discrete lattice effects at short wavelengths, we find that it is well suited for describing the interaction between the monolayer and substrate in the long wavelength limit. We suggest that a complete picture of the dynamics of a discrete monolayer adhered to an elastic substrate can be found using a combination of the results presented for the discrete and effective medium descriptions. This model is potentially scalable for use with both micro- and macroscale systems, and offers the prospect of experimentally extracting contact stiffnesses from measurements of acoustic dispersion.'
author:
- 'S. P. Wallen$^{1}$, A. A. Maznev$^2$, and N. Boechler$^{1}$'
title: Dynamics of a Monolayer of Microspheres on an Elastic Substrate
---
Introduction
============
Granular media are simultaneously one of the most common and complex forms of matter on Earth. This complexity stems, in part, from heterogeneous structure and highly nonlinear particulate interactions [@DuranBook; @GranularPhysicsBook; @NesterenkoBook]. Over the past thirty years, mechanical wave propagation in ordered granular media has become an active field of research as it provides a setting for the broader understanding of granular media dynamics [@NesterenkoBook]. Ordered granular media have also been shown to enable a wide array of novel passive wave tailoring devices that leverage the nonlinear response stemming from the Hertzian relationship between elastic particles in contact [@Hertz; @Johnson], in conjunction with dispersion induced by periodicity [@GranularCrystalReviewChapter] or local resonances [@LocalResonance].
Experimental configurations used to study mechanical wave propagation in ordered granular media typically involve spherical particles confined by elastic media. This type of arrangement is particularly common in one- and two-dimensional configurations, and includes macro- to microscale particles. For example, at the macroscale, elastic rod structures, tracks, and tubes have been used to confine the particles in one-dimensional [@LocalResonance; @Macro1D; @MacroUpshift] and quasi-one-dimensional [@MacroQuasi1D] configurations, and elastic plates have been used in two dimensions [@Macro2D]. More recently, the dynamics of a two-dimensional monolayer of 1 $ \mu\mathrm{m} $ silica particles adhered to an elastic substrate was studied using a laser ultrasonic technique [@Boechler_PRL].
Analytical models used to describe the dynamics of these systems typically only include the interaction between the particles (often just the normal Hertzian contact interaction) and disregard the effect of the substrate. In reality, even for the simple case of a particle monolayer on a substrate, more complex dynamics involving interactions between the particles and elastic waves in the substrate should be expected. Indeed, a recent experiment [@Boechler_PRL] showed that a monolayer of microspheres adhered to a substrate strongly interacts with Rayleigh surface waves in the substrate, leading to the hybridization between Rayleigh waves and a microsphere contact resonance. The results of this experiment were analyzed with a simple model involving only vertical (normal to the substrate surface) vibrations of isolated particles, following the approach adopted in earlier theoretical works on the interaction of surface oscillators with Rayleigh waves [@Baghai1992; @Garova1999]. However, in reality, the particle motion is not confined to the vertical direction, and the Rayleigh wave has a significant horizontal component. Furthermore, the interaction between neighboring particles in contact is expected to significantly influence the dynamics.
A notable theoretical work [@Kosevich1989] provided a model for the dynamics of a monolayer adhered to an elastic substrate which accounted for both normal and horizontal motion and interaction between the particles. However, this study did not take into account particle rotation. A more recent study [@Tournat2011] demonstrated that the rotational degree of freedom has a profound effect on the dynamics of granular monolayers. However, the analysis of monolayers on substrates in Ref. [@Tournat2011] only accounted for normal contact interactions between the particles and the substrate, and the substrate was considered rigid.
The aim of this work is to provide a theoretical model for the contact-based dynamics of a two-dimensional layer of spheres on a substrate, accounting for the elasticity of the substrate, translational and rotational motion of the spheres, and both normal and shear stiffnesses of sphere-to-sphere and sphere-substrate contacts. We focus on a system with microscale particles that interact with each other and with the substrate via van der Waals adhesion forces. Rather than postulate the contact stiffness constants, we derive them from Hertzian contact models. This imposes certain constraints on the values of the constants: for example, the ratio of the normal and shear contact stiffness between the spheres is a constant only weakly dependent on Poisson’s ratio. We consider contact-based modes having frequencies significantly below the intrinsic spheroidal vibrational modes of the spheres, such that they can be described as spring-mass oscillators. Furthermore, we focus on dynamics involving particle and substrate displacements in the sagittal plane, as would be detectable in a laser-based experiment, such as that of Ref. [@Boechler_PRL].
We start with the case of a rigid substrate, where we find three eigenmodes involving vertical, horizontal and rotational motion of the spheres. In the long-wavelength limit these modes yield three contact resonances, for which simple analytical expressions are obtained. One of the resonances only involves motion of the spheres normal to the substrate surface, whereas the other two involve mixed horizontal-rotational motion. We then present our effective medium model, which describes the interaction between the spheres and the substrate. The results show that the contact resonances interact with Rayleigh surface waves, which leads to mode hybridization and avoided crossings. We discuss the behavior for cases involving both isolated (non-touching) and interacting spheres, and demonstrate the important role of rotations in both cases. We also examine the validity of the effective medium approximation, by comparing the calculations using discrete and effective medium models. Finally, we discuss the implications of our findings for past and future studies on granular monolayer systems.
Model
=====
![(a) Side-view schematic of an amplified wave profile for isolated and interacting spheres. (b) Top-down view of the square-packed monolayer, with the arrow indicating the direction of wave propagation. (c) Schematic for the model of a monolayer of spheres coupled to an elastic halfspace.[]{data-label="Model_Schem"}](Improved_Model_Schematic_v12.pdf){width="0.9\columnwidth"}
We consider a monolayer of elastic spheres on a substrate, which can be either close-packed and in contact, or isolated, as shown in Fig. \[Model\_Schem\](a). In either case, the spheres are assumed to form a square lattice, with the wave propagation direction aligned with the lattice vector, as shown in Fig. \[Model\_Schem\](b). We model the layer as an infinite lattice of rigid spheres with diameter $D = 2R$ and mass $m$, coupled to a semi-infinite, isotropic elastic substrate by normal and shear stiffnesses $K_N$ and $K_S$, and to nearest-neighbor spheres by stiffnesses $G_N$ and $G_S$, as schematically shown in Fig. \[Model\_Schem\](c). The subscript $N$ corresponds to forces acting normal to the surface of the sphere, and $S$ to forces acting transverse to the surface of the sphere. The shear springs generate an associated torque about the sphere center, while the normal springs do not. The absolute horizontal, vertical, and angular displacements of sphere $j$ from its equilibrium state are given by $Q_j$, $Z_j$, and $\theta_j$, respectively, and the displacements of the substrate are denoted by $u(x,z)$, corresponding to displacement in the $x$-direction, and $w(x,z)$, corresponding to displacements in the $z$-direction.
Contact Stiffness {#Contact}
-----------------
We derive the stiffnesses $K_N$, $K_S$, $G_N$, and $G_S$ using Derjaguin-Muller-Toporov (DMT) [@DMT1983; @Israelachvili] and Mindlin contact models [@Mindlin]. The DMT theory is typically applicable in weakly-adhesive systems with small, stiff particles [@Bhushan], and assumes that the deformation profile is Hertzian. The Mindlin model describes the shear stiffness of particles in contact, assuming an applied normal force [@Mindlin]. At the microscale, adhesive contact models have been explored experimentally in the quasi-static regime using atomic force microscopy and nanoindentation approaches [@Fuchs].
For contact between two spheres (or a sphere and a halfspace) having elastic moduli $E_1$ and $E_2$, and Poisson’s ratios $\nu_1$ and $\nu_2$, the Hertzian restoring elastic force $F_N$ corresponding to displacement $\delta_N$ of the particle center in the direction normal to the contact surface is given by $$\begin{aligned}
% \begin{split}
F_N &= \frac{4}{3} E^* R_{c}^{1/2} \delta_N^{3/2},\label{contactForceN}
% F_S &= 8 G^* R_{c}^{1/2} \delta_T \delta_N^{1/2},\label{contactForceS}
% F_N &= K R_{c}^{1/2} \delta_N^{3/2}\label{contactForceN}\\
% F_S &= \frac{4GR_{c}^{1/2}}{2-\nu} \delta_T \delta_N^{1/2},\label{contactForceS}
% \end{split}
\end{aligned}$$ where $R_{c}$ is the effective radius of contact (equal to $R$ for sphere-sphere contacts and $R/2$ for sphere-halfspace contacts), and $E^* = [(1-\nu_1^2)/E_1 + (1-\nu_2^2)/E_2]^{-1}$ is the effective modulus. Considering the DMT adhesive force $F_{DMT} = 2 \pi w R_c$ acting normal to the contact surface [@DMT1983; @Israelachvili] (where $w$ is the work of adhesion between two surfaces), the net normal force is given by $$F_{N,net} = F_N - F_{DMT}.
\label{contactForceNnet}$$ To describe the shear contact, we utilize the Mindlin model [@Mindlin], which assumes small relative displacements and no slip at the contact surface. For two elastic bodies with shear moduli $G_1$ and $G_2$, the restoring elastic force $F_S$ to displacement $\delta_S$ of the particle center in the direction transverse to the contact normal is given by [@Mindlin] $$\begin{aligned}
% \begin{split}
% F_N &= \frac{4}{3} E^* R_{c}^{1/2} \delta_N^{3/2},\label{contactForceN}\\
F_S &= 8 G^* R_{c}^{1/2} \delta_S \delta_N^{1/2},\label{contactForceS}
% F_N &= K R_{c}^{1/2} \delta_N^{3/2}\label{contactForceN}\\
% F_S &= \frac{4GR_{c}^{1/2}}{2-\nu} \delta_T \delta_N^{1/2},\label{contactForceS}
% \end{split}
\end{aligned}$$ where $G^* = [(2-\nu_1)/G_1 + (2-\nu_2)/G_2]^{-1}$ is the effective shear modulus. Here, the factor of $\delta_N^{1/2}$ arises from the Hertzian relation between the contact radius and $F_N$.
By substituting the relative displacements $\delta_N = Z - w_0$ and $\delta_S = Q - u_0 + R\theta$ into Eqs. \[contactForceNnet\] and \[contactForceS\], and linearizing about the equilibrium configuration of $\delta_{N,0} = [3 F_{DMT}/(4 E^* R_{c}^{1/2})]^{2/3}$ and $\delta_{S,0} = 0$, we derive linearized normal and shear contact stiffnesses $$\begin{aligned}
\begin{split}
K_N &=\left(6 E^{*^2} R_c F_{DMT}\right)^{1/3}\\
% G_N &= \left[(6 E^{*^2} R_c F_{DMT})^{1/3}\right]_{sph}\\
K_S &= 8 \left(\frac{3}{4} \frac{G^{*^3}}{E^*} R_c F_{DMT}\right)^{1/3},\label{springConst}
\end{split}
\end{aligned}$$ with $ G_N $ and $ G_S $ given by equations of the same form, but with $ E^* $, $ G^* $, $ R_c $, and $F_{DMT}$ adjusted for sphere-sphere contacts. In the special case where the spheres and substrate are composed of the same material, the relative magnitudes of the stiffness constants are determined exclusively by Poisson’s ratio $ \nu $ of the material, $$\begin{aligned}
\begin{split}
% K_N &= (6 E^{*^2} R F_{DMT})^{1/3}\\
G_N &= 2^{-2/3} K_N\\
K_S &= \nu^* K_N\\
G_S &= 2^{-2/3} \nu^* K_N,\label{springConstReduced}
\end{split}
\end{aligned}$$ where $\nu^* = 2(1-\nu)/(2-\nu)$.\
Equations of Motion of the Spheres
----------------------------------
Assuming small displacements (i.e. $Q_j $, $ Z_j $, and $ R \theta_j $ are much less than $ D $), the $j^{th}$ sphere obeys the equations of motion $$\begin{aligned}
\begin{split}
m \ddot{Q}_j = & - K_S(Q_j - u_{0,j} + R \theta_j)\\
& + G_N(Q_{j+1} - 2 Q_j + Q_{j-1})\\
m \ddot{Z}_j = & - K_N(Z_j - w_{0,j})\\
& + G_S[Z_{j+1} - 2 Z_j + Z_{j-1} - R(\theta_{j+1} -\theta_{j-1})]\\
I \ddot{\theta}_j = & - K_S R(Q_j - u_{0,j} + R \theta_j)\\
& - G_S R [R(\theta_{j+1} + 2 \theta_j + \theta_{j-1})\\
& - (Z_{j+1} - Z_{j-1})], \label{EOM_Disc}
\end{split}
\end{aligned}$$ where $u_{0,j}$ and $ w_{0,j} $ are horizontal and vertical displacements of the substrate surface at the point of contact, respectively.
Effective Medium Approximation
------------------------------
Considering wavelengths much longer than the sphere diameter, we treat the monolayer as an effective continuous medium. By substituting the center difference formulas $[(\cdotp)_{j+1} - (\cdotp)_{j-1}]/(2D) \approx \partial (\cdotp) / \partial x$ and $[(\cdotp)_{j+1} - 2(\cdotp)_j + (\cdotp)_{j-1}] / (D^2) \approx \partial^2 (\cdotp) / \partial x^2$ into Eq. (\[EOM\_Disc\]), we arrive at the equations of motion of the monolayer in effective medium form: $$\begin{aligned}
\begin{split}
m \frac{\partial^2 Q}{\partial t^2} = & - K_S(Q - u_0 + R \theta)\\
& + 4 G_N R^2 \frac{\partial^2 Q}{\partial x^2}\\
m \frac{\partial^2 Z}{\partial t^2} = & - K_N(Z - w_0)\\
& + 4 G_S R^2 (\frac{\partial^2 Z}{\partial x^2} - \frac{\partial \theta}{\partial x})\\
I \frac{\partial^2 \theta}{\partial t^2} = & - K_S R(Q - u_0 + R \theta)\\
& - 4 G_S R^2 (R^2 \frac{\partial^2 \theta}{\partial x^2} + \theta + \frac{\partial Z}{\partial x} ). \label{EOM_Cont}
\end{split}
\end{aligned}$$ The coupling between the monolayer and substrate is described by the following boundary conditions at the surface $z = 0$, which describe the average force acting on the surface due to the motion of the spheres: $$\begin{aligned}
\begin{split}
\sigma_{zx} &= \frac{K_S}{A}(Q - u_0 + R \theta)\\
\sigma_{zz} &= \frac{K_N}{A}(Z - w_0),\label{BC}
\end{split}
\end{aligned}$$ where $\sigma_{zx}$ and $\sigma_{zz}$ are components of the elastic stress tensor [@Ewing] and $A$ = $D^2$ is the area of a primitive unit cell in our square-packed monolayer. The combination of Eq. (\[EOM\_Cont\]) and the linear elastic wave equations describing waves in the substrate [@Ewing], coupled by the boundary conditions of Eq. (\[BC\]), comprises the complete effective medium model.
Dispersion Relations {#DispRel}
====================
Rigid Substrate
---------------
### Discrete Model
We first consider the discrete model, which accurately captures the structural periodicity of the monolayer. We assume spatially-discrete traveling wave solutions of the form $\hat{Q} e^{i(\omega t - kDj)}$ (with similar terms for the other displacements) and set the displacements of the substrate surface $u_{0,j}$ and $w_{0,j}$ to zero (rigid substrate). Here, $\hat{(\cdotp)}$ is the amplitude of a plane wave in the displacement variable $(\cdotp)$, $\omega$ is the angular frequency, and $k$ is the wave number. After algebraic manipulation, Eq. (\[EOM\_Disc\]) is reduced to a homogeneous system of three linear algebraic equations in the amplitudes $\hat{(\cdotp)}$. This system possesses non-trivial solutions only for pairs of $ k $ and $ \omega $ that cause the determinant of the system to vanish. Enforcing this condition, we arrive at the following dispersion relation, where the three rows of the determinant correspond to the three equations of Eq. (\[EOM\_Disc\]):
$$\left|\begin{array}{ccc}
\frac{c_N^2}{2 R^2}(1-\cos(kD)) + \phi_S \omega_S^2 & 0 & R\omega_S^2\\
0 & \frac{c_S^2}{2 R^2}(1-\cos(kD)) + \phi_N \omega_N^2 & -\frac{c_S^2}{2 R} i\sin(kD)\\
R\omega_S^2 & \frac{c_S^2}{2 R} i\sin(kD) & \frac{I}{m} [\frac{c_\theta^2}{2 R^2}(1-\cos(kD)) + \phi_\theta \omega_\theta^2]
\end{array}\right| = 0,
\label{DRdet_disc}$$
where $\phi_N = 1 - \omega^2 / \omega_N^2$, $\phi_S = 1 - \omega^2 / \omega_S^2$, $\phi_\theta = 1 - \omega^2 / \omega_\theta^2$, $c_N=\sqrt{G_N / m}(2R)$ and $c_S=\sqrt{G_S / m } (2R)$ are the longitudinal and transverse long-wavelength sound speeds of the discrete monolayer, $c_\theta^2 = -mR^2c_S^2 / I$, $\omega_N^2 =K_N / m$, $\omega_S^2 =K_S / m$, and $\omega_\theta^2 = (K_S + 4G_S)R^2 / I$.
### Effective Medium
In the effective medium model, which approximates the discrete system at long wavelengths, we substitute spatially-continuous traveling wave solutions of the form $\hat{Q} e^{i(\omega t - kx)}$ (with similar terms for the other displacements) into $Q$, $Z$, and $\theta$ in Eq. (\[EOM\_Cont\]) with $u_{0} = w_{0} = 0$. Following the same procedure as in the discrete model, we arrive at the dispersion relation for the effective medium model:
$$\left|\begin{array}{ccc}
c_N^2 k^2 + \phi_S \omega_S^2 & 0 & R \omega_S^2\\
%
0 & c_S^2 k^2 + \phi_N \omega_N^2 & -ik c_S^2\\
%
R\omega_S^2 & ikc_S^2 & \frac{I}{m}(c_\theta^2 k^2 + \phi_\theta \omega_\theta^2)
\end{array}\right| = 0.
\label{DRdet_cont_RB}$$
It is particularly instructive to examine the behavior of the effective medium model in the long wavelength limit $k \rightarrow 0$. In this limit, the displacements vary slowly in space, and the spatial derivative terms of Eq. (\[EOM\_Cont\]) may be neglected. For the case of a rigid base, Eq. (\[EOM\_Cont\]) then reduces to the form
$$\begin{aligned}
\begin{split}
m \frac{\partial^2 Q}{\partial t^2} = & - K_S(Q + R \theta)\\
m \frac{\partial^2 Z}{\partial t^2} = & - K_N Z\\
I \frac{\partial^2 \theta}{\partial t^2} = & - K_S R(Q + R \theta) - 4 G_S R^2 \theta. \label{EOM_Cont_noSD}
\end{split}
\end{aligned}$$
The equation for $Z$ decouples from the other two equations and yields a vertical vibrational mode. The two other equations remain coupled, yielding two modes containing both horizontal and rotational motion. Using the moment of inertia of a solid sphere $I = (2/5) m R^2$, we find three resonance frequencies $$\begin{aligned}
\begin{split}
\omega_N & = \left[\frac{K_N}{m}\right]^{1/2}\\
\omega_{RH} & = \left[\left(\frac{K_S}{4m}\right) \left(20 \gamma + 7 + \sqrt{400 \gamma^2 + 120 \gamma + 49}\right)\right]^{1/2}\\
\omega_{HR} & = \left[\left(\frac{K_S}{4m}\right) \left(20 \gamma + 7 - \sqrt{400 \gamma^2 + 120 \gamma + 49}\right)\right]^{1/2}. \label{Res_longwave}
\end{split}
\end{aligned}$$ where $ \gamma = G_S / K_S $. Here, $\omega_N$ corresponds to a mode with exclusively vertical motion, described by Eq. (\[EOM\_Cont\_noSD\]). The other two modes $ \omega_{RH} $ and $ \omega_{HR} $ exhibit both rotational and horizontal (but not vertical) motion, with relative amplitudes determined by $ \gamma $. The higher of the two horizontal-rotational modes is predominantly rotational and the lower is predominantly horizontal, hence we have used the notations $\omega_{RH}$ and $\omega_{HR}$, where the first letter in the subscript denotes the dominant motion. If the spheres and substrate are made of the same material, then, by using Eq. (\[springConstReduced\]), we can relate the horizontal-rotational frequencies of Eq. (\[Res\_longwave\]) to the vertical resonance frequency, with the expressions $\omega_{RH} = 3.018 \nu^{*^{1/2}} \omega_N$ and $\omega_{HR} = 0.832 \nu^{*^{1/2}} \omega_N$.
In the limiting case of isolated spheres (described by $ \gamma = 0 $), $ \omega_{RH} $ and $ \omega_{HR} $ of Eq. (\[Res\_longwave\]) become $ \omega_{RH,Iso} = \sqrt{7/2} \hspace{1 pt}\omega_S $ and $ \omega_{HR,Iso} = 0 $, respectively. For identical materials, $ \omega_{RH,Iso} = \sqrt{7\nu^*/2} \hspace{1 pt}\omega_N $. The dependence of $ \omega_{RH} $ and $ \omega_{HR} $ on $ \gamma $ is shown in Fig. \[gammaPlot\] (a), where it can be seen that $ \omega_{RH} $ originates at $ \omega_{RH,Iso} $ for $ \gamma = 0 $ and grows unbounded, while $ \omega_{HR} $ originates at $ \omega_{HR,Iso} = 0 $, and approaches $ \omega_S $ asymptotically. In Fig. \[gammaPlot\] (b), we plot the horizontal and rotational displacement amplitudes as functions of $ \gamma $ for these two modes. Different signs of the rotational amplitude indicate that the $\omega_{RH}$ and $\omega_{HR}$ modes have different motion patterns. In the former, a positive displacement is accompanied by a counterclockwise rotation, while in the latter, it is accompanied by a clockwise rotation.
We note that the zero-frequency mode, $\omega_{HR,Iso}$, corresponds to the rolling motion of an isolated sphere. With the inclusion of a bending rigidity, the sphere would not be allowed to freely roll, and instead would undergo rocking motion of a finite frequency. While non-zero bending rigidity is expected to exist in real systems (for instance, a microsphere adhered to a substrate does not freely roll), the frequency of resulting rocking vibrations is predicted to be orders of magnitude lower than the other contact resonances discussed here [@Tielens]. Bending rigidity would thus act as a small perturbation to the predictions of our model, and we do not include it in our analysis.
To illustrate the importance of particle rotations in the model, we note that in the limiting case of $I \rightarrow \infty$, when there is no rotation, Eq. (\[EOM\_Cont\_noSD\]) yields two resonances: a vertical resonance with frequency $\omega_N$, and a horizontal resonance having frequency $\omega_S$. For isolated spheres, the effect of rotations can be thought of as a reduction of the “effective mass" of the sphere to $ (2/7)m $, which increases the horizontal resonance frequency. For interacting spheres, on the other hand, rotations drastically change the dynamics, yielding two horizontal-rotational modes whose frequencies depend on the relative strengths of the sphere-to-sphere and sphere-substrate interactions.
![(a) Resonance frequencies $ \omega_{RH} $ (red line) and $ \omega_{HR} $ (black line) as functions of the stiffness ratio $ \gamma $. (b) Displacement amplitudes of the resonant modes with frequencies $\omega_{RH} $ (red lines) and $ \omega_{HR} $ (black lines), as functions of the stiffness ratio $ \gamma $. Solid and dotted lines correspond to $ Q $ and $ R\theta $, respectively. For each resonance, the amplitudes are normalized such that the sum of squares is unity. The positive sign of $ R \theta $ corresponds to counterclockwise rotation.[]{data-label="gammaPlot"}](DR_ImprovedModel_gammaPlot_v4.pdf){width="\columnwidth"}
Elastic Substrate
-----------------
As in the case of the effective medium approximation for a rigid substrate, we assume traveling wave solutions of the form $\hat{Q} e^{i(\omega t - kx)}$ (with similar terms for the other displacements) into $Q$, $Z$, and $\theta$ in Eq. (\[EOM\_Cont\]). Likewise, we express the variables $u_0$, $w_0$, $\sigma_{zx}$ $\sigma_{zz}$ in terms of surface wave solutions for the elastic potentials [@Ewing] $\phi(x,z,t) = \hat{\phi} e^{k \alpha z + i(\omega t - kx)}$ and $\psi(x,z,t) = \hat{\psi} e^{k \beta z + i(\omega t - kx)}$, and then substitute these expressions into Eq. (\[EOM\_Cont\]) and Eq. (\[BC\]). Here, $\hat{(\cdotp)}$ is the amplitude of a plane wave in the displacement or potential variable $(\cdotp)$, $\alpha = \sqrt{1 - \omega^2/(c_L^2 k^2)}$, $\beta = \sqrt{1 - \omega^2/(c_T^2 k^2)}$, and $ c_L $ and $c_T$ are the longitudinal and transverse sound speeds of the substrate, respectively. After algebraic manipulation, Eq. (\[EOM\_Cont\]) and Eq. (\[BC\]) are reduced to a homogeneous system of five linear algebraic equations in the five plane wave amplitudes $\hat{(\cdotp)}$, with coefficients depending on $k$ and $\omega$. We reach the dispersion relation by seeking nontrivial solutions of this system, which exist only for pairs of $k$ and $\omega$ that cause the determinant of the following coefficient matrix to vanish:
$$\left|\begin{array}{ccccc}
ik\omega_S^2 & k \beta \omega_S^2 & c_N^2 k^2 + \phi_S \omega_S^2 & 0 & R \omega_S^2\\
%
-k \alpha \omega_N^2 & ik \omega_N^2 & 0 & c_S^2 k^2 + \phi_N \omega_N^2 & -ik c_S^2\\
%
ikR\omega_S^2 & kR\beta\omega_S^2 & R\omega_S^2 & ikc_S^2 & \frac{I}{m}(c_\theta^2 k^2 + \phi_\theta \omega_\theta^2)\\
%
1+\beta^2 & -2i\beta & 0 & \frac{m}{\rho A c_T^2 k^2}(c_S^2 k^2 + \phi_N \omega_N^2 - \omega_N^2) & \frac{-m}{\rho A c_T^2 k^2}ikc_S^2\\
%
-2i\alpha & -(1+\beta^2) & \frac{m}{\rho A c_T^2 k^2}(c_N^2 k^2 + \phi_S \omega_S^2 - \omega_S^2) & 0 & 0
\end{array}\right| = 0,
\label{DRdet}$$
where $ \rho $ is the density of the substrate, and $ A = D^2 $ is the area of a primitive unit cell in our square-packed monolayer. We note that the coupling between the spheres and the substrate is represented by elements $(4,4)$, $(4,5)$, and $(5,3)$ of the matrix in Eq. (\[DRdet\]). Thus, the strength of the coupling can be quantified by the ratio $m / (\rho A)$; if this term is made to vanish (e.g. by making the mass of each sphere much less than that of the portion of the substrate below it, extending to the depth of material influenced by Rayleigh waves), then the substrate and monolayer will be effectively decoupled. We note that if rotations are disregarded (e.g. by letting $ I \rightarrow \infty $), Eq. (\[DRdet\]) reduces to the same form as that of the adsorbed monolayer of Ref. [@Kosevich1989].
It is instructive to consider the long-wave limit when the spatial derivatives in Eq. (\[EOM\_Cont\]) can be disregarded. In this case, we find the simplified dispersion relation
$$\left|\begin{array}{ccccc}
ik\omega_S^2 & k \beta \omega_S^2 & \phi_S \omega_S^2 & 0 & R \omega_S^2\\
%
-k \alpha \omega_N^2 & ik \omega_N^2 & 0 & \phi_N \omega_N^2 & 0\\
%
ikR\omega_S^2 & kR\beta\omega_S^2 & R\omega_S^2 & 0 & \frac{I}{m}\phi_\theta \omega_\theta^2\\
%
1+\beta^2 & -2i\beta & 0 & \frac{m}{\rho A c_T^2 k^2}(\phi_N \omega_N^2 - \omega_N^2) & 0\\
%
-2i\alpha & -(1+\beta^2) & \frac{m}{\rho A c_T^2 k^2}(\phi_S \omega_S^2 - \omega_S^2) & 0 & 0
\end{array}\right| = 0.
\label{DRdet_reduced}$$
For isolated spheres, there is no approximation in Eq. (\[DRdet\_reduced\]) with respect to Eq. (\[DRdet\]), because in this case the terms generated by the spatial derivatives in Eq. (\[EOM\_Cont\]) are identically zero. For interacting spheres, the accuracy of dispersion relations calculated with Eq. (\[DRdet\_reduced\]) will be assessed below by a comparison with results obtained with Eq. (\[DRdet\]). We will see that Eq. (\[DRdet\_reduced\]) essentially describes the interaction of contact resonances given by Eq. (\[EOM\_Cont\_noSD\]) with Rayleigh surface waves.
Numerical Results and Discussion
================================
In the following calculations, we consider silica spheres of $1.08$ $\mu$m diameter on a silica substrate, and use the elastic constants (Ref. [@GlassProp]) $E =$ 73 GPa, $\nu =$ 0.17, and work of adhesion (Ref. [@Israelachvili]) $w =$ 0.063 J/m^2^.
Rigid Substrate
---------------
![(a) Dispersion relation of a discrete monolayer adhered to a rigid base. Blue solid and red dotted lines denote, respectively, discrete and effective medium monolayers. Black dashed lines denote the contact resonances. (b)-(d) relative amplitudes of the displacement variables $Q$ (black dotted lines), $Z$ (red dotted lines), and $R\theta$ (blue dotted lines), corresponding to the branches of the same numeral for the dispersion of the discrete monolayer adhered to the rigid base shown in (a). The amplitudes are normalized such that the sum of the squares is unity.[]{data-label="DR_Layer"}](DR_ImprovedModel_Layer_v11.pdf){width="\columnwidth"}
We plot numerical solutions of Eq. (\[DRdet\_disc\]), to obtain the dispersion curves for the discrete model of interacting spheres on a rigid base, as shown in Fig. \[DR\_Layer\](a). In our description of a rigid substrate, we assume that no elastic waves propagate in the substrate, but allow local deformation at the points of contact for the purpose of the contact stiffness calculation; this preserves the same contact stiffnesses as in the elastic substrate analysis. We note that due to the periodicity of the system, all three branches have zero-group velocities at the edge of the first irreducible Brillouin zone [@BrillouinBook] of the monolayer.
By substituting the solutions shown in Fig. \[DR\_Layer\](a) into the coefficient matrix of the corresponding algebraic system, we numerically determine the amplitudes of the sphere displacements, which we plot in Fig. \[DR\_Layer\](b-d). By comparing the calculated displacements of with the dispersion curves, we see that each branch takes on the character of its respective contact resonance in the limit $k \rightarrow 0$. One can see that each of the three modes generally involves both vertical and horizontal, as well as rotational motion (albeit the rotational component of mode II is quite small). The existence of the three modes with mixed displacements is a consequence of the inclusion of the rotational degree of freedom: without rotations, there would be two modes, one vertical and one horizontal.
We note that, in the special case $ K_S = 0 $, the mode originating at $ \omega_{HR} $ becomes purely horizontal and decouples from the other two modes. The remaining modes (characterized by vertical translation and rotation) are generally consistent with the results of Ref. [@Tournat2011], for the case of normal contact with a rigid surface and no bending rigidity. Since Ref. [@Tournat2011] considered hexagonal packing, the behavior is analogous at long wavelengths, but diverges at short wavelengths due to discrete lattice effects.
The dotted lines in Fig. \[DR\_Layer\](a) show dispersion curves calculated with the effective medium model as per Eq. (\[DRdet\_cont\_RB\]). The effective medium approximation yields accurate results at long wavelengths but fails at shorter wavelengths with the unphysical behavior of the first mode, whose frequency goes to zero. At even shorter wavelengths, as shown in Fig. \[DR\_EM\_Compare\], the effective medium dispersion curves of modes II and III asymptotically approach straight lines with slopes given by the longitudinal and transverse sound speeds in the monolayer. This asymptotic behavior has been described by Kosevich and Syrkin [@Kosevich1989]. However, as can be seen from the dispersion curves generated using the discrete model in Fig. \[DR\_EM\_Compare\], this asymptotic behavior does not occur in our system due to the spatial periodicity of the monolayer. As a result, the inclusion of the first- and second-order spatial derivative terms of Eq. (\[EOM\_Cont\]) does not deliver much additional understanding of the dynamics of our system.
. Black dash-dotted lines denote the long-wavelength longitudinal and transverse sounds speeds of the monolayer.[]{data-label="DR_EM_Compare"}](DR_ImprovedModel_Compare_v5.pdf){width="\columnwidth"}
Elastic Substrate
-----------------
### Isolated Spheres {#iso}
![Dispersion relation of SAWs in an elastic half space coupled to a monolayer of isolated elastic spheres, denoted by the blue solid lines. Black dashed lines denote the contact resonances, and black dash-dotted lines denote the transverse and Rayleigh waves speeds of the substrate.[]{data-label="DR_Iso"}](DR_ImprovedModel_noGs_v3.pdf){width="\columnwidth"}
We numerically solve Eq. (\[DRdet\_reduced\]) for the isolated spheres case using $G_S = 0$ and all other parameters derived in Sec. \[Contact\], and plot the resulting dispersion relation for the effective medium model, as shown in Fig. \[DR\_Iso\]. This dispersion relation exhibits classic “avoided crossing" behavior [@Wigner] about the resonance frequencies $\omega_N$ and $\omega_{RH,Iso} = \sqrt{7/2} \hspace{2 pt}\omega_S$. In this model, surface acoustic waves (SAWs) in the substrate behave as classical Rayleigh waves at frequencies far from the contact resonances, and the dispersion curves follow the line corresponding to the substrate Rayleigh wave speed $c_R$ [@Ewing]. Conversely, sphere motions dominate those of the substrate at frequencies close to the contact resonances. For phase velocities greater than $c_T$, which correspond to the region $\omega > c_T k$, the wave numbers that solve Eq. (\[DRdet\]) are complex valued; these solutions are “leaky" modes for which energy leaves the surface of the substrate, and radiates into the bulk. This isolated spheres case is particularly applicable in systems where adhesion between particles is negligible, e.g. for: macroscale particles without lateral compression where the dominant static compression is due to gravity and is between the particles and substrate; or for microscale particles, if the separation distance between particles is larger than the range of adhesion forces.
### Interacting Spheres
![(a) Blue solid lines denote the dispersion relation of SAWs in an elastic halfspace coupled to a monolayer of interacting elastic spheres. Black dashed lines denote the contact resonances, and black dash-dotted lines denote wave speeds in the substrate. (b)-(e) relative amplitudes of the displacement variables $u_0$ (black solid lines), $w_0$ (red solid lines), $Q$ (black dotted lines), $Z$ (red dotted lines), and $R\theta$ (blue dotted lines), corresponding to the branch denoted by the same numeral in (a). The amplitudes are normalized such that the sum of the squares is unity.[]{data-label="DR_Inter"}](DR_ImprovedModel_Reduced_v7.pdf){width="\columnwidth"}
In Fig. \[DR\_Inter\](a), we plot numerical solutions of Eq. (\[DRdet\_reduced\]) for the long wavelength limit of the effective medium model with interacting spheres. The amplitudes of the sphere and substrate displacements are calculated in the same manner as in Fig. \[DR\_Layer\], and are plotted in Fig. \[DR\_Inter\](b-e). In Fig. \[DR\_Inter\](a), we observe features qualitatively similar to the dispersion relation for isolated spheres in Fig. \[DR\_Iso\], with the exception of a third avoided crossing at frequency $\omega_{HR}$. The mode shapes reveal the ways in which each of the branches are influenced by the contact resonances, as well as long and short wavelength asymptotic behavior of our system. In the long wavelength limit, the substrate motions closely resemble Rayleigh SAWs [@Ewing], with a mix of vertical and horizontal motions. Since the frequencies of waves in this regime are well below the contact resonances, the effect of the spheres is negligible, and the monolayer moves in phase with the substrate surface. At short wavelengths, it can be clearly seen that the first, second, and third lowest branches exhibit motions dominated by the displacements $Q$, $Z$, and $\theta$, respectively (each corresponding to a resonant mode of the monolayer), while the highest branch tends toward the Rayleigh SAW. The effects of proximity to the contact resonances are well illustrated, for example, by branch III of Fig. \[DR\_Inter\](a), which exhibits large vertical sphere motions at its starting point near $\omega_N$, resembles the Rayleigh SAW as it approaches and crosses the $c_R$ line, and transitions into large rotational sphere motions after bending around the avoided crossing with $\omega_{RH}$. In order to examine the behavior of our system throughout the entire Brillouin zone, we superimpose the dispersion curves for the effective medium model of interacting spheres on an elastic base including higher order spatial derivative terms (the full Eq. (\[DRdet\])) with the dispersion curves for the discrete monolayer on a rigid substrate (Eq. (\[DRdet\_disc\])), as shown in Fig. \[DR\_RE\_Compare\]. At long wavelengths the discreteness of the monolayer is insignificant, and the dispersion is well described by the effective medium model. Furthermore, we note that at long wavelengths the dispersion curves calculated using the effective medium model including higher order terms shown in Fig. \[DR\_RE\_Compare\], only slightly deviates from the dispersion calculated using the effective medium model with the higher order terms neglected shown in Fig. \[DR\_Inter\](a). The only noticeable effect is a downshift in frequency of the avoided crossing between the Rayleigh wave and the $\omega_{RH}$ resonance; since the latter intersects at the highest wave vector of the three contact resonances, calculations with Eq. (\[DRdet\_reduced\]) in this case are the least accurate. In Fig. \[DR\_RE\_Compare\], at short wavelengths, beyond the avoided crossings with the Rayleigh wave branch, the elasticity of the substrate has little effect on the dynamics, and the dispersion can be described using the discrete model for interacting spheres on a rigid substrate. We suggest that by “stitching together” the effective medium model for spheres on an elastic substrate with the discrete model for spheres on a rigid substrate, we can simultaneously capture the interaction of SAWs with the monolayer at long wavelengths and effects caused by the discreteness of the spheres at short wavelengths. Past the avoided crossings, the two sets of curves in Fig. \[DR\_RE\_Compare\] stitch together smoothly, resulting in a full picture of the monolayer dynamics on the elastic substrate.
.[]{data-label="DR_RE_Compare"}](DR_ImprovedModel_Transition_v2.pdf){width="\columnwidth"}
Experimental Implications
=========================
We expect the presented results to be useful for predicting complex dynamic responses and extracting effective contact stiffnesses from measurements of acoustic dispersion in a manner similar to Boechler [*et al.*]{} [@Boechler_PRL]. The findings described above invite several questions, including whether our model of a square lattice is applicable to results on hexagonally packed monolayers, and why horizontal-rotational resonances were not observed in the experiment [@Boechler_PRL].
We believe that the assumption of the square lattice is not essential. For isolated spheres, Eqs. (\[EOM\_Cont\]) and (\[BC\]) with $ G_N $ and $ G_S $ set to zero can be obtained for any arrangement of the spheres, periodic or random, with the only parameter depending on the arrangement being the surface area per sphere A. For interacting spheres, the results generally do depend on the lattice structure and the propagation direction. However, the contact resonances given by Eq. (\[Res\_longwave\]) correspond to the $ k=0 $ limit and, consequently, do not depend on the propagation direction. The relative positions of the three contact resonances may be different in the long wavelength limit between a hexagonal and square packed lattice, but their presence should still be expected in both cases.
We suggest that the reasons why horizontal-rotational resonances were not observed in Ref. [@Boechler_PRL] may be the following. Since the measurements were not sensitive to horizontal motion, the $ \omega_{RH} $ and $ \omega_{HR} $ resonances could only be detected when they hybridized with SAWs near avoided crossings, and since the avoided crossings with $ \omega_{RH} $ and $ \omega_{HR} $ resonances are more narrow than the one with the $ \omega_N $ resonance, they could have been missed. Furthermore, our model assumes that all spheres are either connected by identical springs or are isolated. If the contact stiffness between spheres were to vary widely (some neighboring spheres being in contact and others not, for example), then distinct resonances may be absent. In addition, the upper ($ \omega_{RH} $) resonance may have been outside the range of the measurements in Ref. [@Boechler_PRL]. Further experimental studies of monolayer dynamics in conjunction with exploration of ways to control sphere-to-sphere contacts should help resolve the discrepancy between the theory and experiment.
While the main focus of this work has been on microgranular monolayers, our theory is equally valid for macroscale systems. In this case, the contact springs would be determined by gravity and, possibly, applied lateral static compression [@MacroUpshift], rather than by adhesion forces. We note that several past experimental works on macroscale granular systems [@MacroUpshift] have observed systematic upshifts in frequency relative to theoretical predictions, and have suggested uncertainties in material parameters and experimental setups, as well as deviations from Hertzian contact behavior as possible causes. As per the results from our model, the presence of additional degrees of freedom and interactions between spheres and substrate may also be possible causes. In the absence of the external lateral compression, highly nonlinear “sonic vacua” [@NesterenkoBook] should also be expected. Generally, as amplitudes are increased, interesting nonlinear dynamics are expected for both micro- and macroscale monolayers due to nonlinearity of Hertzian contacts between the particles [@NesterenkoBook; @GranularCrystalReviewChapter] and between the particles and the substrate [@ISOT2014].
Conclusion
==========
We have developed a model for wave propagation in granular systems composed of a monolayer of spheres on an elastic substrate. Our model expands on those used in previous works by including the elasticity of the substrate, horizontal and rotational sphere motions, shear coupling between the spheres and substrate, and interactions between adjacent spheres. We have shown that a monolayer of interacting spheres on a rigid substrate supports three modes involving vertical, horizontal, and rotational motion. In the long-wavelength limit, these modes yield three contact resonances, one purely vertical and two of mixed horizontal-rotational character. On an elastic substrate, these resonances hybridize with the Rayleigh surface wave yielding three avoided crossings. For isolated spheres, the frequency of the lower horizontal-rotational resonance, in the absence of bending rigidity, tends to zero and only two contact resonances with two respective avoided crossings remain.
By comparing the effective medium (valid for long wavelengths) to the discrete formulation of our model, we have demonstrated that for the presented microsphere monolayer example, the effective medium model can be used to describe the interaction of the contact resonances with the Rayleigh waves in the substrate, but loses accuracy at shorter wavelengths. In that case, the substrate can be considered rigid, and the discrete model is more appropriate. This model is scalable in that it can be adapted for use with both macro- and microscale systems, and provides a means to experimentally extract contact stiffnesses from dynamic measurements. Opportunities for future studies include exploration of analogous models for granular monolayers in the nonlinear regime, as well as analysis of the transverse modes of a monolayer of spheres on an elastic substrate (the latter involves transverse horizontal displacement and rotations of the spheres, as well as shear horizontal acoustic waves in the substrate). Further experiments with macro- and microscale granular monolayers will help guide the modeling effort.
Acknowledgments
===============
The authors greatly appreciate discussions with Vitaly Gusev. S.W. and N.B. gratefully acknowledge support from the US National Science Foundation (grant no. CMMI-1333858) and the US Army Research Office (grant no. W911NF-15-1-0030). The contribution by A.A.M. was supported by the National Science Foundation Grant No. CHE-1111557.
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---
abstract: 'We explain in depth the previously proposed theory of the coherent Van der Waals(cVdW) interaction - the counterpart of Van der Waals (VdW) force - emerging in spatially coherently fluctuating electromagnetic fields. We show that cVdW driven matter is dominated by many body interactions, which are significantly stronger than those found in standard Van der Waals (VdW) systems. Remarkably, the leading 2- and 3-body interactions are of the same order with respect to the distance $(\propto R^{-6})$, in contrast to the usually weak VdW 3-body effects ($\propto R^{-9}$). From a microscopic theory we show that the anisotropic cVdW many body interactions drive the formation of low-dimensional structures such as chains, membranes and vesicles with very unusual, non-local properties. In particular, cVdW chains display a logarithmically growing stiffness with the chain length, while cVdW membranes have a bending modulus growing linearly with their size. We argue that the cVdW anisotropic many body forces cause local cohesion but also a negative effective “surface tension”. We conclude by deriving the equation of state for cVdW materials and propose new experiments to test the theory, in particular the unusual 3-body nature of cVdW.'
author:
- 'Igor M. Kulić$^{1}$'
- 'Miodrag L. Kulić$^{2}$'
title: Theory of Coherent Van der Waals Matter
---
Introduction
============
The major goal of physics is the quest for understanding and controlling the forces of Nature. In recent decades physicists and chemists have begun to invent increasingly creative ways to combine the fundamental forces and to generate new, effective interactions on microscopic and macroscopic scales. Cold atoms could not be trapped and cooled [@ColdAtoms], colloidal suspensions would become unstable and flocculate [@Colloidal; @Suspensions], and magnetic levitation would be impossible [@Levitron] if combined, effective interactions were absent. As we know from condensed matter physics, the interplay of attractive and repulsive forces of different origins can give rise to highly complex structures. They range from gyroid phases in block copolymers [@Block-Copolymers], labyrinthine phases in ferrofluids [@Labyrinthine] to nuclear pasta phases in neutron stars [@Nuclear-Pasta] to name only a few. Not surprisingly, adding more physical interactions naturally increases the structural complexity of the resulting materials. Here we ask the opposite question: how much complexity can emerge from a *single*, simple to generate, effective interaction?
In the recent short paper [@KulicKulic-PRL] we studied, the probably simplest *effective interaction* able to generate surprisingly complex structures. This effective interaction appears, for instance, between dipolar magnetic (or dielectric) particles when a spatially uniform, isotropic but time varying magnetic (or electric) field is externally applied (cf. Fig.1b). The first instance of it was described in a series of important papers by Martin et al. [@Martin1; @Martin2] in a system of superparamagnetic colloids in *balanced* *triaxial magnetic fields* (BTMF)* *- rotating magnetic fields spinning on a cone with the magic opening angle $\theta_{m}\approx54,7^{\circ}$. The emerging effective interaction between two colloids appeared to be, at the first glance, reminiscent of the London-Van der Waals force [@Martin1; @Martin2; @Osterman]. Yet the structures formed, including colloidal membranes and foams, were unexpectedly more intricate and differing strongly from those expected in classical Van der Waals (VdW) systems.
Inspired by the fascinating magnetic colloid superstructures generated experimentally [@Martin1; @Martin2; @NatureCommentMartin] we have begun to systematically investigate the physical ingredients and the consequences of the induced interaction [@KulicKulic-PRL]. By starting out from the analogy with the VdW interaction we have considered a generalization of Martin’s BTMF field structure [@Martin1; @Martin2] and arrived at the concept of the *spatially coherent Van der Waals* (cVdW) interaction, see Fig.1b [@NOTEcVdWName].
In the previous short and rather dense paper [@KulicKulic-PRL], we have answered the important question, why cVdW generates complex structures like chains, membranes and foams while its sister - the Van der Waals-like *incoherent fluctuation interaction* (VdW) – see Fig.1a, merely forms phase-separated lumps or 3D droplets of matter within a two phase system [@Van; @Der; @Waals]. However, many details and numerous subtle questions were omitted in [@KulicKulic-PRL] due to the lack of space. In the following we close the gap and present a fairly complete theory of cVdW.
As a new item going much beyond the previous Letter [@KulicKulic-PRL], we study the bulk and finite size effects in chains, rings, membranes, spherical and cylindrical shells. In particular, we investigate the bending elasticity of chains and membranes and show that there is a qualitative difference with systems with short-range forces. As we will see, most structures formed by cVdW, have properties which are inherently dictated by *long range anisotropic many body forces*. They exhibit collective (i.e. scale and shape dependent) stiffness, surface tension and line tension.
The physical content of the paper is schematically represented in the Diagram in Fig.2, where the free-energy as a function of the *effective non-local susceptibility* $\hat{\chi}_{eff,ij}$ plays the central role in all our studies of cVdW systems. In the microscopic theory the latter contains the 3-body (and higher order) interactions, while in the macroscopic theory it can be expressed via an *effective demagnetization tensor* $\hat{L}$. Both approaches give rise to *the collective, long range, shape-sensitive* nature of the cVdW interaction.
The text is organized as follows. In $Section$ $II$ we briefly introduce the reader into the basic properties of the magnetorheological (MR) colloids and describe the first experimental realization of the cVdW interaction in such systems by Martin et al[@Martin1; @Martin2]. In $Section$ $III$ we develop the basic physical and mathematical machinery to treat the cVdW interaction. We then derive the first central result of this paper: The time-averaged free-energy as the trace of the *effective non-local susceptibility tensor* $\hat{\chi}_{eff}$. The latter tensor describes an effective interaction between two colloids mediated by all the other colloids - thus containing all many body interactions. Based on the microscopic theory for the free-energy in $Section$ $IV$ we discuss cVdW systems and self-assemblies of colloids in various structures.
In $Section$ $V$ we study the formation of chains and membranes within the framework of a macroscopic mean-field theory. The consistency of the latter with the microscopic approach is discussed there as well. There we also show that the cVdW systems can be considered as systems with an effective *negative* surface energy. In $Secton$ $VI$ we generalize the cVdW interaction to anisotropic objects and study the interaction between multiple elementary structures, including the bead-membrane and the two membrane interaction. The interaction turns out to be very rich, anisotropic and changes sign depending on the mutual orientation of the interacting objects. Based on these preparatory results, in $Section$ $VII$ we study the formation of the more complex emergent structures: the colloidal foams. We show that the cVdW theory predicts a positive pressure of the foam and that it swells against the gravitational field to notable heights.
The most notable quantitative results are summarized in two tables at the end of $Section$ $X.$. We conclude by pointing out interesting experimental effects and tests of the theory and by outlining some important open questions concerning cVdW. More detailed derivations of some formulas are contained in several Appendices for the interested reader. The mathematically less interested reader is invited to browse through the figures, each of which explains a new concept, and to run through them towards the Discussion.
Preliminaries - Magnetic Colloids in Balanced Triaxial Fields
=============================================================
The study of responsive “smart materials” with remarkable properties has been intensifying in the last decades. In that respect the magnetorheological (MR) suspensions, made of magnetizable solid microparticles (colloids), dispersed in nonmagnetic fluids and placed in magnetic fields, are of immense interest due to the rapid, large and tunable transformations in their mechanical and rheological properties. Having in mind the numerous applications [@MR-applic], a scientific challenge is to investigate which kind of assembled structures are realized depending on combined static and oscillating magnetic fields.
In a typical MR system consisting of superparamagnetic microbeads the induced magnetic moment of a single bead $\mathbf{m}_{b}=\mathbf{M}_{b}V_{b}$, with $V_{b}=(4\pi/3)d^{3}$ its volume and $\mathbf{M}_{b}$ its magnetization, is proportional to the applied external magnetic field $\mathbf{H}_{0}$ i.e. $\mathbf{m}_{b}=\chi_{b}V_{b}\mathbf{H}_{0}$. Here, $\chi_{b}(>0)$ stands for the shape-dependent *bead susceptibility* with respect to the external field. This one should not be confused with the material susceptibility $\chi_{b,m}(>\chi_{b})$ , which characterizes the physical properties of the material itself (not the shape) out of which the bead is made. In the case when the bead is suspended in a solvent with a material susceptibility $\chi_{s}$ in the magnetostatic limit $\chi_{b}$ is given from $\chi_{b,m}$ by [@Landau; @Jones] $$\chi_{b}=3\frac{\chi_{b,m}-\chi_{s}}{3+\chi_{b,m}+2\chi_{s}}. \label{hi-bead}$$
In the following we will study dipolar magnetic (dielectric) beads, which do not carry permanent moments [@CommentNotation]. The beads, enumerated by an index $i$, are assumed to be all identical, magnetically isotropic and spherically shaped. They are placed in a spatially and temporally fluctuating magnetic field $\mathbf{B}_{0,i}=\mu_{0}\mathbf{H}_{0,i}(t)$ with $\mu_{0}$ the vacuum permeability. We will focus here on the case when the field varies on an intermediate timescale $\tau_{H}=2\pi/\omega$ fulfilling the condition $\tau_{M}\ll\tau_{H}\ll\tau_{visc}$. Here $\tau_{M}$ is the typical magnetic relaxation time of the paramagnetic bead, which is typically in the range of a 10th of second to microseconds. The other relevant characteristic time scale is the aggregation time $\tau_{visc}\propto\eta$ which characterizes the bead’s motion in the surrounding viscous fluid over characteristic distances comparable or larger than the bead diameter $d$. Under these conditions the beads’ magnetization $\mathbf{M}_{i}$ is equilibrated much faster than its positional coordinate $\mathbf{R}_{i}(t)$, the beads move and aggregate slowly and feel a net time-averaged force due to the dipole-dipole interaction. In general , the susceptibility $\chi_{b}(\omega)$ can be a complex frequency dependent function but in the following we will restrict ourselves to the case when $\operatorname{Im}\chi_{b}(\omega)\ll\operatorname{Re}\chi_{b}(\omega)$. In this case, the magnetic dissipation effects are in the first approximation negligible compared to the magnetic free-energy effects.
In general, the dynamics of the $i-th$ bead is determined by the interplay of: (1) the friction force $\mathbf{F}_{v}=-\xi\mathbf{v}_{b}$ with the friction coefficient $\xi=6\pi d_{b}\eta$, (2) the average dipole-dipole force $-\partial\mathcal{\bar{F}(}\mathbf{H}_{0},\mathbf{M}_{i},\{\mathbf{R}_{i}\})/\partial\mathbf{R}_{i}$ (with the dipole-dipole energy $\mathcal{\bar
{F}}(\mathbf{H}_{0},\{\mathbf{R}_{i}\})$) and (3) the fluctuating Brownian force $\mathbf{F}_{B,i}$. However, in the following we shall consider the (quasi-)equilibrium structures first, and postpone the bead dynamics to later works.
The first concrete instance of the cVdW interaction was realized [@Martin1; @Martin2; @Osterman] a MR suspension that was placed in a magnetic field rotating with a frequency $\omega$ on a cone with the opening angle $\theta$, i.e. $\mathbf{H}_{0}(t)=\sqrt{3}H_{0}(\sin\theta\cos\omega
t,\sin\theta\sin\omega t,\cos\theta)$ - see Fig. 3a. More precisely, an ideal, fully isptropic cVdW interaction is only realized when the cone opening angle coincides with the magic angle $\theta=\arccos(1/\sqrt{3})\approx54,7^{\circ}$ . Such a *balanced triaxial magnetic field* (BTMF) has very special correlation properties: Its components have time averaged correlations, denoted by $\overline{\left( ...\right) }$, $$\overline{H_{i,0}^{\alpha}H_{j,0}^{\beta}}=\delta_{\alpha\beta}H_{0}^{2}.
\label{SqIsotropy}$$ that are formally (square) isotropic. Note, that this relation is true even though the BTMF itself has a preferred orientation along the positive $z$ axis, see Fig. 3a. In the most general case we can consider any field realization with such an isotropic correlation property and all particular realizations (cf. Fig. 3(a-b)) will be considered equivalent within our theory. In fact we will abstract away from any concrete representation of the field and take the squared isotropy Eq.(\[SqIsotropy\]) as the defining property of the exciting field.
Note that in the most general case found in literature [@Martin1],[@Martin2] the triaxial field is unbalanced and can have an arbitrary in-plane $H_{\parallel}$ and perpendicular component $H_{\perp}:$ $\mathbf{H}_{0}=(H_{\parallel}\cos\omega t,H_{\parallel}\sin\omega t,H_{\perp
})$ with $2H_{\parallel}^{2}+H_{\perp}^{2}=H_{0}^{2}.$ One can show that in this case the interaction can be linearly decomposed into a balanced triaxial field (BTMF) magic angle interaction and a residual dipole-dipole interaction along the orthogonal direction [@KulicKulic-PRL]. The latter is well understood while the former is new and investigated here.
The cVdW Free-Energy
====================
In this section we develop the mathematical formalism necessary to understand the cVdW interaction. The cVdW interaction is a general phenomenon going beyond the magnetic realm and the theory developed here is equally valid for electrically polarizable colloids, i.e. the electrorheological materials. In order to keep the continuity with Refs. [@KulicKulic-PRL; @Martin1; @Martin2; @Osterman], we arbitrarily follow the magnetic notation. The results for electrically polarizable colloids are obtained by replacing the magnetic quantities with the corresponding electric ones.
Under the condition of quick variations of the external field $\mathbf{H}_{0}$, yet a much quicker equilibration of the magnetization $\mathbf{M}_{i}$ we study equilibrium structures which minimize the effective (time averaged) free-energy $\mathcal{\bar{F}}$, i.e. $$-\frac{\partial\mathcal{\bar{F}(}\mathbf{H}_{i,0},\mathbf{M}_{i},\{\mathbf{R}_{i}\})}{\partial\mathbf{M}_{i}}=0 \label{Min-cond}$$ for fixed particle positions $\mathbf{R}_{i}$.
In the following we will be dealing with purely athermal effects. This is usually well justified: due to the large moments $\mathbf{m}_{i}$ of the beads with diameters $D(=2d)>1$ $\mu m$, the energy per particle will be well in excess of the thermal energies making contributions of a configurational entropy negligible in practice. Therefore, all external field fluctuations will be considered as extrinsically given.
The basic expression for the non-equilibrium free-energy $\mathcal{F}\{\mathbf{M}_{i},\mathbf{H}_{i,0}(t)\}$ of magnetic colloids (beads) with respect to $\mathbf{M}_{i}$ in inhomogeneous external time-dependent field $\mathbf{H}_{i,0}\left( t\right) $ can be written as [@Landau]$$\frac{\mathcal{F}}{\mu_{0}V_{b}}=\sum_{i=1}^{N}\left( \frac{\mathbf{M}_{i}^{2}}{2\chi_{b}}-\mathbf{M}_{i}\mathbf{H}_{i,0}\right) +\frac{1}{2}\sum_{i,j\mathbf{\neq}i}\mathbf{M}_{i}\hat{T}_{ij}\mathbf{M}_{j}, \label{S-1}$$ where $V_{b}$ is the bead-volume of one of the $N$ identical beads. $\mathbf{M}_{i}=\mathbf{m}_{i}/V_{b}$ is the magnetization of the $i$-th bead resulting from its magnetic moment $\mathbf{m}_{i}$ and $\chi_{b}$ is the *bead (sample) susceptibility* in the external field. The first term in Eq.(\[S-1\]) is the “self-energy” of the beads. It ensures that in absence of external fields there is no magnetization. The second term represents the $i-th$ bead’s dipole-dipole interaction with all the other beads. It is mediated by the dipole tensor $\hat{T}_{ij}$$$\begin{aligned}
\hat{T}_{ij} & =\varphi_{ij}\hat{t}(\mathbf{b}_{ij}),\text{ }\varphi
_{ij}=\frac{V_{b}}{4\pi\left\vert \mathbf{R}_{ij}\right\vert ^{3}}\label{T-ensor}\\
\hat{t}(\mathbf{b}_{ij}) & =\hat{1}-3\left\vert \mathbf{b}_{ij}\right\rangle
\left\langle \mathbf{b}_{ij}\right\vert ,\text{ }\mathbf{b}_{ij}=\frac{\mathbf{R}_{ij}}{\left\vert \mathbf{R}_{ij}\right\vert },\nonumber\end{aligned}$$ with $\mathbf{R}_{ij}\equiv\mathbf{R}_{i}-\mathbf{R}_{j}\neq0$ ( $i\neq j$ ) and $\mathbf{b}_{ij}$ the normalized bonding vector, i.e. the unit vector pointing pointing from bead $j$ to $i$. Here, we decompose conveniently the dipole tensor $\hat{T}_{ij}$ into a purely geometric $3\times3$ tensor $\hat{t}(\mathbf{b}_{ij})$ - a linear combination of the unity matrix $\hat
{1}$ and the pure projector on the bonding vector $\left\vert \mathbf{b}_{ij}\right\rangle \left\langle \mathbf{b}_{ij}\right\vert $ (in the “Bra-ket” notation). The second contribution in $\hat{T}_{ij}$ is the purely distance dependent dipolar field-decay factor $\varphi_{ij}\left(
R_{ij}\right) \propto R_{ij}^{-3}.$
Performing the minimization of $\mathcal{F}$ w.r.t. $\mathbf{M}_{i}$ and reintroducing the result into Eq.(\[S-1\]) we arrive at the quasi-equilibrium free-energy for fixed coordinates $\{\mathbf{R}_{i}\}$ [@Landau] $$\mathcal{F}\{\mathbf{H}_{i,0}\}=-\frac{1}{2}\mu_{0}V_{b}\sum_{i}\mathbf{M}_{i}\mathbf{H}_{i,0}, \label{S2}$$ where the quasi-equilibrium magnetization of the i-th bead $\mathbf{M}_{i}\left( t\right) =\chi_{b}\mathbf{H}_{i,loc}\left( t\right) $ is determined by the *total local* fields $\mathbf{H}_{i,loc}$ acting at the position of the bead $i.$ These local fields are given implicitly as a function of the applied external fields $\mathbf{H}_{i,0}(t)$ via $$\sum_{j}\left( \delta_{ij}+\chi_{b}\hat{T}_{ij}\right) \mathbf{H}_{j,loc}(t)=\mathbf{H}_{i,0}(t). \label{Hloc}$$ The last line is strictly valid if we adopt the practical convention that $\hat{T}_{ii}=0$ for two identical bead indices, i.e. excluding a self-interaction of beads. By inverting Eq.(\[Hloc\]), the formal solution for the local fields $\mathbf{H}_{i,loc}$ reads $$\mathbf{H}_{i,loc}=\chi_{b}^{-1}\sum_{j}\hat{\chi}_{eff,ij}\mathbf{H}_{j,0}
\label{Hloc2}$$ Here we encounter a main player in the cVdW interaction - the *effective non-local microscopic susceptibility tensor* defined as: $$\hat{\chi}_{eff}=\chi_{b}(\hat{1}+\chi_{b}\hat{T})^{-1}. \label{chi-eff}$$ This *non-local microscopic susceptibility tensor* $\hat{\chi}_{eff}$ is crucial for understanding all the many body effects that will follow. Lets take a few notes in order to understand some of its features. Mathematically, it is a $3\times N$- dimensional matrix, relating the external fields at all particle $j-th$ locations to the local fields at particle $i-th$ position. For any fixed $i$ and $j,$ the single components $\hat{\chi}_{eff,ij}$ are themselves $3$ dimensional second–rank tensors - i.e. $3\times3$ matrices in $3$-dimensional space. In the following, dealing with $\hat{\chi}_{eff}$ and its components will be conceptually easy with a small caveat and the note of caution: the $i,j$ components $\hat{\chi
}_{eff,ij}$ are to be evaluated *after* performing the operator inversion in Eq. (\[chi-eff\]). This is at the very origin of the many body forces (cf. below).
With all theses issues and precautions about $\hat{\chi}_{eff}$ in mind we can now insert Eq.(\[Hloc2\]) into Eq.(\[S2\] ) and average over the fluctuating fields $\mathbf{H}_{i,0}(t)$ to obtain the averaged free-energy $\mathcal{\bar{F}}$ $$\begin{aligned}
\mathcal{\bar{F}}\left( \mathbf{H}_{0},\{\mathbf{R}_{i}\})\right) &
=-\frac{\mu_{0}}{2}V_{b}\sum_{i,j}\overline{\mathbf{H}_{i,0}\hat{\chi
}_{eff,ij}\mathbf{H}_{j,0}}\label{S4}\\
& =-\frac{\mu_{0}}{2}V_{b}\sum_{i,j,\alpha,\beta}C_{ij}^{\alpha\beta}\chi_{eff,ij}^{\alpha\beta},\nonumber\end{aligned}$$ where $C_{ij}^{\alpha\beta}$ is the *field-field correlation function* defined by $$C_{ij}^{\alpha\beta}=\overline{H_{i,0}^{\alpha}H_{j,0}^{\beta}}. \label{Cij}$$ Here, $\alpha,\beta=x,y,z$ stand for the 3-spatial directions and $i,j=1,2....N$ are the particle indices. The Eq. (\[S4\]) is completely general and forms the backbone for all further analysis. It holds both for the classic VdW interaction (with appropriately chosen $\mathbf{H}_{i,0}$) as well as for cVdW and generally couples the field correlators $C_{ij}^{\alpha\beta}$ with the many-body interaction-encoding susceptibility $\chi_{eff,ij}^{\alpha\beta}$ functions. The former are given by the type of interaction (VdW or cVdW) while the latter depend on the spatial configuration of all particles in an interesting but (for now) very convoluted, little transparent manner that we want to elucidate in the following.
Incoherent vs. Coherent Fields
------------------------------
The decomposition of the free energy in Eq. (\[S4\]) into an “external influence” term (the field correlator $C_{ij}^{\alpha\beta}$) and an “internal response” function (susceptibility tensor components $\chi
_{eff,ij}^{\alpha\beta}$) is conceptually appealing. The remainder of the paper we will spend on exploring the physical features associated with these two terms.
In a first step, let us investigate how different types of driving field correlations change the interactions. There are two important limiting cases for the correlator $C_{ij}^{\alpha\beta}$ :
$(A)$ *strong correlations* with perfect spatial coherence and
$(B)$ *no correlations* in the driving field with perfect decoherence.
These are defined in the following way:
($A$) The *spatially coherent fluctuation interaction* with the correlator $$C_{ij}^{\alpha\beta}=C^{\alpha\beta}=\delta_{\alpha\beta}H_{0}^{2}+h^{\alpha\beta} \label{C-coh}$$ The first term $C_{0}^{\alpha\beta}=\delta_{\alpha\beta}H_{0}^{2}$ describes the *isotropic* and spatially coherent (uniform) excitation, while the second, $h^{\alpha\beta},$ describes a uniform constant field (anisotropic contribution). In the following we restrict ourself only to the *isotropic coherent van der Waals interaction* (the cVdW one) with a vanishing anisotropic component, $h^{\alpha\beta}=0$ i.e. $$C_{ij}^{\alpha\beta}=C_{0}^{\alpha\beta}=\delta_{\alpha\beta}H_{0}^{2}.
\label{C0}$$ It is easy to see that the corresponding *cVdW free-energy*, is then given by $$\mathcal{\bar{F}}_{cVdW}\left( \mathbf{H}_{0},\{\mathbf{R}_{i}\}\right)
=-\frac{\mu_{0}}{2}H_{0}^{2}V_{b}\sum_{i,j}Tr\left( \hat{\chi}_{eff,ij}\right) , \label{S5}$$ where $$Tr\left( \hat{\chi}_{eff,ij}\right) \equiv\chi_{eff,ij}^{xx}+\chi
_{eff,ij}^{yy}+\chi_{eff,ij}^{zz}$$ is the trace of the effective susceptibility. Note, that the coherent isotropic correlation function $C_{\alpha\beta}=\delta_{\alpha\beta}H_{0}^{2}$ comprises also the case of the experimentally realized *balanced triaxial magnetic fields [@Martin1; @Martin2; @Osterman]* (BTMF) - see $Section$ $II$ and Fig.3a. Even though the cone at which the field precesses has a direction (opening) by itself, the square of the field is statistically identical in all directions and mutually uncorrelated in all directions i.e. $\overline{H_{i,0}^{\alpha}H_{j,0}^{\beta}}=\delta_{\alpha\beta}H_{0}^{2}$. The BTMF is therefore only one instance of a general coherent isotropic field. Any other realization, like e.g. one of those in Fig 3b satisfies the relation (\[S4\]) and consequently has the same energy Eq. (\[C0\]).
($B$) *The spatially incoherently excited fields* are realized for $C_{ij}^{\alpha\beta}=\delta_{ij}C^{\alpha\beta}$. The latter may in principle contain isotropic and anisotropic terms, too. Note, that the term proportional to $\delta_{ij}$ means that the correlations of magnetic field fluctuations on different particles vanish. In the completely *isotropic* case one has $$C_{ij}^{\alpha\beta}=\delta_{ij}\delta_{\alpha\beta}H_{0}^{2} \label{C-icFI}$$ and the fully incoherent VdW free-energy - of the *VdW systems*, reads$$\mathcal{\bar{F}}_{VdW}\left( \mathbf{H}_{0},\{\mathbf{R}_{i}\}\right)
=-\frac{\mu_{0}}{2}H_{0}^{2}V_{b}\sum_{i}Tr\left( \hat{\chi}_{eff,ii}\right)
. \label{S6}$$
Note, that there is a *significant difference* between the two excitation cases ($A$) and ($B$), described by Eq.(\[S5\]) and Eq.(\[S6\]), respectively. In the incoherent case ($B$) the free-energy $\mathcal{\bar{F}}_{VdW}\left( \mathbf{H}_{0},\{\mathbf{R}_{i}\}\right) $ *contains a summation over the index* $i$ *only*, i.e. it includes only the diagonal terms $i=j$ - sometimes called the self-energy terms. This is equivalent to the usual Van der Waals (VdW) interaction and in the following this part of the free-energy will be called the VdW one. However in the novel case ($A$) $\mathcal{\bar{F}}_{cVdW}$ contains the more complex, double summation over $i$ and $j$ which gives rise to unusual, non-local and anisotropic many body effects in cVdW system - most of which are absent in a standard VdW system (case $B$). In the coherent case, the terms $Tr(\hat{\chi
}_{eff,ij})$ with $i\neq j$, describe the effective coupling between the $i$-th and $j$-th bead acting directly or indirectly via all other beads, thus giving rise to very specific and anisotropic *many body interactions*. The latter will turn out to be a crucial effect for cVdW matter and will be responsible for the formation of hierarchical assemblies of colloids. This is in strong contrast with the standard VdW systems where 3D bulk structures such as droplets and close-packed 3D crystal structures are favored and realized.
Microscopic cVdW Theory - Many Body Interactions, the Formation of Chains and Membranes
=======================================================================================
In this Section we will explore how a *microscopic cVdW theory*, based on the effective energy Eq.(\[S5\]) and the non-local many body susceptibility Eq.(\[chi-eff\]) works in practice. While the energy Eq.(\[S5\]) appears (deceptively) straightforward to evaluate, the many body susceptibility operator Eq.(\[chi-eff\]) is a sophisticated mathematical object. To grasp physical insights about the latter, except for the simplest case of two spherical beads, seems challenging.
After dealing with the elementary case of two particles, which can be treated exactly, we will resort to the approximation of small bead susceptibility i.e. $\chi_{b}\ll1$ - a limiting case that allows a controllable evaluation of the many body interactions. In this spirit we will be making an energy expansion up to lowest order in $\chi_{b}$. Notably, this lowest order expansion of the energy, as we will see, comprises both the 2-body interaction of beads and the non-local 3-body interactions at the same order.
As it will be shown both these interactions (2 and 3 body) scale identically with distance for cVdW, i.e. $\mathcal{\bar{F}}_{2-body}\propto\mathcal{\bar
{F}}_{3-body}\propto R^{-6}$. The *2+3 body inseparability* is the most peculiar *hallmark signature of the cVdW interaction*, and to our knowledge stands out rather uniquely among other known many-body forces in Nature.
Two-Body Interaction - Dimer Formation
--------------------------------------
Let us start out elementary and consider a very dilute system. In such a case the *pairwise* bead-bead ($2$-body) interaction should dominate in the free-energy $\mathcal{\bar{F}}_{cVdW}$. While this assumption of dominant 2-body forces turns out as too naive (see the next subsection) it is still natural to consider only two interacting particles first. For two beads 1 and 2 the non-local susceptibility operator $\hat{\chi}_{eff}$ is easily calculated by using Eq. (\[chi-eff\]). The detailed derivation is given in $Appendix$ $1A$ while the final result reads $$\begin{aligned}
Tr\hat{\chi}_{eff} & =C\cdot\left(
\begin{array}
[c]{cc}1-3\varphi_{12}^{2}\chi_{b}^{2} & 2\chi_{b}^{3}\varphi_{12}^{3}\\
2\chi_{b}^{3}\varphi_{12}^{3} & 1-3\varphi_{12}^{2}\chi_{b}^{2}\end{array}
\right) \label{TrChieff2Bead}\\
C & =\frac{\allowbreak3\chi_{b}}{\left( 1-4\varphi_{12}^{2}\chi_{b}^{2}\right) \left( 1-\varphi_{12}^{2}\chi_{b}^{2}\right) }$$ Having the trace of $\hat{\chi}_{eff}$ , we can now evaluate the mean free energy for the incoherent and the coherent case.
In the *incoherent (standard) VdW* case the energy $\bar{f}_{VdW}=\mathcal{\bar{F}}_{VdW}/\left( \mu_{0}V_{b}H_{0}^{2}\right) $ is the sum of the $Tr\hat{\chi}_{eff}$ diagonals (note $\left( Tr\hat{\chi}_{eff}\right) _{11}=\left( Tr\hat{\chi}_{eff}\right) _{22}$) i.e. $$\begin{aligned}
\bar{f}_{VdW} & =-\left( Tr\hat{\chi}_{eff}\right) _{11}\\
& =-\frac{3\chi_{b}\left( 1-3\varphi_{12}^{2}\chi_{b}^{2}\right) }{\left(
4\varphi_{12}^{2}\chi_{b}^{2}-1\right) \left( \varphi_{12}^{2}\chi_{b}^{2}-1\right) }.\nonumber\end{aligned}$$ $\bigskip$On the other hand, for the *coherent cVdW* interaction we have to sum all four elements of $Tr\hat{\chi}_{eff}$ (note $\left(
Tr\hat{\chi}_{eff}\right) _{12}=\left( Tr\hat{\chi}_{eff}\right) _{21}$) obtaining $$\begin{aligned}
\bar{f}_{cVdW} & =-\left( Tr\hat{\chi}_{eff}\right) _{11}-\left(
Tr\hat{\chi}_{eff}\right) _{12}\\
& =-\frac{\allowbreak3\chi_{b}\left( 1-\chi_{b}\varphi_{12}\right)
}{\left( 1+\chi_{b}\varphi_{12}\right) \left( 1-2\chi_{b}\varphi
_{12}\right) }.\nonumber\end{aligned}$$
Interestingly, the coherent and the incoherent 2 bead interaction energy look very similar but are *not* identical. After expanding the energies in powers of $\varphi_{12}$ we see that $\bar{f}_{VdW}=\allowbreak-3\chi
_{b}-6\varphi_{12}^{2}\chi_{b}^{3}-18\varphi_{12}^{4}\chi_{b}^{5}+...\acute{}$ and $\bar{f}_{cVdW}\approx-3\chi_{b}-6\varphi_{12}^{2}\chi_{b}^{3}-6\varphi_{12}^{3}\chi_{b}^{4}+...$ or in terms of the bead-bead distance $R_{12}$: $$\begin{aligned}
\bar{f}_{VdW} & \approx-3\chi_{b}-\frac{3V_{b}^{2}\chi_{b}^{3}}{8\pi^{2}}R_{12}^{-6}-\frac{9\chi_{b}^{5}V_{b}^{4}}{128\pi^{4}}R_{12}^{-12}\label{Bead-Bead}\\
\bar{f}_{cVdW} & \approx-3\chi_{b}-\frac{3V_{b}^{2}\chi_{b}^{3}}{8\pi^{2}}R_{12}^{-6}-\frac{3V_{b}^{3}\chi_{b}^{4}}{32\pi^{3}}R_{12}^{-9}\nonumber\end{aligned}$$ We observe that the first interaction terms $\propto R_{12}^{-6}$ exactly coincide. This interesting $2$-body result was first obtained by Martin and coworkers [@Martin1; @Martin2] and confirmed experimentally by Osterman et al. [@Osterman]. However, the higher order terms in $\bar{f}_{VdW}$ and $\bar{f}_{cVdW}$ scale quite differently, and they are $\propto R_{12}^{-12}$ and $\propto R_{12}^{-9}$, respectively. This makes the cVdW interaction slightly stronger (more attractive) than the usual incoherent VdW.
Now, if it was only for this slight difference between the two, investigating the cVdW would hardly be very interesting. But we will see soon that the 3-body forces are a real game changer, giving the cVdW interaction its unique character and flavor.
Many Body Interactions
----------------------
For the standard (incoherent) VdW interaction the $2$-body interaction is $\propto\left\vert \mathbf{R}_{12}\right\vert ^{-6}$ in leading order and since the 3-body (and higher order) interactions are shorter ranged (cf. below) and much smaller in magnitude than the 2–body ones, VdW favors the formation of close packed droplet-like or 3D bulk (crystalline) structures with high symmetries [@Van; @Der; @Waals]. If the 2-body interaction - given by Eq.(\[Bead-Bead\]) - would dominate the behavior of cVdW as well, one would also expect the formation of bulk droplets. However this is in sharp contrast to experimental evidence [@Martin1; @Martin2; @Osterman] which shows a clear *tendency for chain and membrane formation*, i.e. for low-dimensional structures under cVdW. What is the microscopic origin of these complex and low dimensional (anisotropic) structures in cVdW systems?
In the following we will explore how this remarkable difference of the two forces emerges once the three body forces are considered.
To this end we consider the free energy Eq.(\[S5\]) with Eq.(\[chi-eff\]) in the general case of $N\geq3$ particles. We expand the non-local susceptibility tensor $\chi_{b}(\hat{1}+\chi_{b}\hat{T})^{-1}$ for small bead susceptibility and large distances $\chi_{b}\varphi_{ij}\ll1$ (i.e. $\chi
_{b}\hat{T}\ll1$) into a Taylor series and take its trace over spacial directions: $$Tr\hat{\chi}_{eff}=Tr\left( \chi_{b}\hat{1}-\chi_{b}^{2}\hat{T}+\chi_{b}^{3}\hat{T}^{2}-\chi_{b}^{4}\hat{T}^{3}+...\right)
\label{ChiEffTaylorExpansion}$$
The resulting (scaled) free energy in the coherent case is then $\bar
{f}_{cVdW}\equiv\mathcal{\bar{F}}_{cVdW}/(\mu_{0}V_{b}H_{0}^{2})=-\frac{1}{2}\sum_{i,j}Tr\left( \hat{\chi}_{eff,ij}\right) $ can be split in terms of ascending order in $\chi_{b}$ :$$\ \bar{f}_{cVdW}=\bar{f}_{cVdW}^{(1)}+\bar{f}_{cVdW}^{(2)}+\bar{f}_{cVdW}^{(3)}+O\left( \chi_{b}^{4}\hat{T}^{3}\right) .$$ with $\bar{f}_{cVdW}^{(k)}\propto\chi_{b}^{k}\sum_{i,j}Tr\left( \hat{T}^{k-1}\right) _{ij}.$ Similarly we can expand the incoherent VdW energy in terms like $\bar{f}_{VdW}\propto\chi_{b}^{k}\sum_{i}Tr\left( \hat{T}^{k-1}\right) _{ii}$ as $\ $$$\bar{f}_{VdW}=\bar{f}_{VdW}^{(1)}+\bar{f}_{VdW}^{(2)}+\bar{f}_{VdW}^{(3)}+\bar{f}_{VdW}^{(4)}+... \label{f-VdW}$$ Note that for VdW systems, it is necessary to include the term $\bar{f}_{VdW}^{(4)}$ since this term contains the leading 3-body interaction (Axilrod-Teller interaction - see below) in the Van der Waals case.
To shed light on the difference between cVdW and VdW energies, let us have a closer look at the terms of the expansion Eq.(\[ChiEffTaylorExpansion\]). The first term $Tr\left( \chi_{b}\hat{1}\right) =3\chi_{b}$ is particle distance independent and describes noninteracting beads, while the second term of Eq.(\[ChiEffTaylorExpansion\]) trivially vanishes as the tensors $\hat
{T}_{ij}$ are traceless: $Tr\hat{T}_{ij}=\varphi_{ij}Tr\left( \hat{1}-3\hat{N}_{ij}\right) =0.$ Here again $\hat{N}_{ij}=\left\vert
\mathbf{b}_{ij}\right\rangle \left\langle \mathbf{b}_{ij}\right\vert $ is the bond vector projector with $Tr\left( \hat{N}_{ij}\right) =1$. It is only the third term of Eq.(\[ChiEffTaylorExpansion\]) $\propto Tr\left( \hat{T}^{2}\right) _{ij}=\sum_{k=1}^{N}\varphi_{ik}\varphi_{kj}Tr\left[ \left(
\hat{1}-3\hat{N}_{ik}\right) \left( \hat{1}-3\hat{N}_{kj}\right) \right] $ that gives rise to the first non-trivial interaction contribution. Using $Tr\left[ \hat{N}_{ik}\hat{N}_{kj}\right] =$ $Tr\left[ \left\vert
\mathbf{b}_{ik}\right\rangle \left\langle \mathbf{b}_{ik}\right\vert
\left\vert \mathbf{b}_{jk}\right\rangle \left\langle \mathbf{b}_{jk}\right\vert \right] $ $=\left( \left\langle \mathbf{b}_{ik}\right\vert
\left\vert \mathbf{b}_{kj}\right\rangle \right) ^{2}:=\cos^{2}\theta_{k,ij}$ which involves the angle $\theta_{k,ij}$ between the bond vectors $\mathbf{b}_{ik}$ and $\mathbf{b}_{jk}$ (at the particle $k$) we obtain:$$\left[ Tr\left( \hat{T}^{2}\right) \right] _{ij}=3\sum_{k=1}^{N}\varphi_{ik}\varphi_{kj}\left( 3\cos^{2}\theta_{k,ij}-1\right)
\label{TrTSquare}$$ Similarly, using the relation $Tr\left[ \hat{N}_{kl}\hat{N}_{ik}\hat{N}_{lj}\right] =\cos\theta_{kl,ik}\cos\theta_{ik,lj}\cos\theta_{kl,lj}$ with the angles between the bond vectors defined by $\cos\theta_{ik,jl}=\left\langle \mathbf{b}_{lj}\right\vert \left\vert \mathbf{b}_{ik}\right\rangle $ we can expand also the 4-th term $\propto\hat{T}^{3}$ of Eq.(\[ChiEffTaylorExpansion\]) : $$\begin{aligned}
\left[ Tr\left( \hat{T}^{3}\right) \right] _{ij} & =\sum_{k=1}^{N}\sum_{l=1}^{N}\varphi_{ik}\varphi_{kl}\varphi_{lj}C_{iklj}\text{
\ with}\label{TrTQube}\\
C_{iklj} & =9\left( \cos^{2}\theta_{ik,jl}+\cos^{2}\theta_{ik,kl}+\cos
^{2}\theta_{jl,lk}\right) \nonumber\\
& -27\cos\theta_{kl,ik}\cos\theta_{ik,lj}\cos\theta_{kl,lj}-6\nonumber\end{aligned}$$
With these results in our hands we are now well equipped to analyze the 3-body terms of the two interactions, cVdW and VdW, and understand how they differ.
### 3-Body Energy for the Incoherent VdW
From the $Tr\left( \hat{T}^{2}\right) $ term given by Eq.(\[TrTSquare\]) we obtain the incoherent 2-body interaction $\bar{f}_{VdW}^{(3)}=-3\chi
_{b}^{3}\sum_{i,k}\varphi_{ki}^{2}$. To calculate the lowest order VdW 3-body term $\bar{f}_{VdW}^{(4)}$ we define in the triangle $(ikl)$ the angles $\theta_{i},\theta_{k},\theta_{l}$ - cf. Fig.4a, with the properties $\cos\theta_{i,kl}=-\cos\theta_{i}$, etc. By using the geometrical rule $\cos^{2}\theta_{i}+\cos^{2}\theta_{k}+\cos^{2}\theta_{l}=1-2\cos\theta
_{i}\cos\theta_{k}\cos\theta_{l}$ for a triangle we obtain $$\begin{aligned}
\bar{f}_{VdW}^{(4)} & =\frac{1}{2}\sum_{i,k,l}\varphi_{ik}\varphi
_{kl}\varphi_{li}C_{ikl}\label{f-VdW-3}\\
C_{ikl} & =3(3\cos\theta_{i}\cos\theta_{k}\cos\theta_{l}+1).\nonumber\end{aligned}$$ This term coincides exactly with the *Axilrod-Teller* $3$*-body potential* [@Teller] for the Van der Waals interaction. Due to its weaker ($\sim R^{-9}$) scaling than the 2-body force, it is typically *small and overridden* by the $2$-body VdW interaction $\sim R^{-6}$ [@NOTEAxilrodTeller] giving rise to close packed structures and droplets in VdW systems.
### The cVdW 3-Body Energy
In the case of the cVdW interaction the effective free-energy is given by Eq.(\[S5\]) where the double summation over $i,j$ must be performed by including $i=j$, as well . The presence of non-local terms with $i\neq j$ gives rise to qualitatively new many-body effects in cVdW matter with respect to the VdW one. As seen from Eq.(\[TrTSquare\]), the leading 3-body cVdW interaction term arises from $\bar{f}_{cVdW}^{(3)}$ and is thus proportional to $\hat{T}^{2}$, so it scales as $\propto\chi_{b}^{3}R^{-6}$. This means that the leading order 3-body cVdW interaction term is of the same order as the 2-body cVdW one [@Note4Body]. The *3-body free-energy* $\bar{f}_{cVdW}^{(3)}$ is given by $$\bar{f}_{cVdW}^{(3)}=-\bar{\beta}\sum\nolimits_{i,j,k}^{\prime}\frac{3\cos
^{2}\theta_{k,ij}-1}{\left\vert \mathbf{R}_{ik}\right\vert ^{3}\left\vert
\mathbf{R}_{kj}\right\vert ^{3}}, \label{3-body}$$ where $\bar{\beta}=(3/32\pi^{2})\chi_{b}^{3}V_{b}^{2}$ and the sum running over all triplets $\left( i,j,k\right) $ with $k\neq i,j$ (for angles $\theta_{k,ij}$ cf. Fig.4b).
Since the $2$-body interaction in cVdW systems is *contained* in $\bar{f}_{cVdW}^{(3)}$ (for $k\neq i=j$) it is physically* inseparable* *from the* $3$*-body one*. Therefore the $3$-body interaction must be treated on the same footing as the $2$-body one. This fact shows us the pitfall in the dimer formation section which considered the $2$-body interaction alone. Interestingly, $\bar
{f}_{cVdW}^{(3)}$ in Eq.(\[3-body\]) is very anisotropic and has a specific angular dependence. This angular dependence intuitively hints towards the explanation of the tendency of cVdW interaction not to form 3D bulk structures, but to drive the formation of anisotropic 1D and 2D structures. For instance, the $-\cos^{2}\theta_{k,ij}$ term in $\bar{f}_{cVdW}^{(3)}$ favors either $\theta_{k,ij}=0$ or $\pi\ $- i.e. a colloidal chains, membranes are preferred by the many body interactions.
The Principle of “Anisotropic Lumping”: An Emergent 2-Body Interaction from the 3-Body One
------------------------------------------------------------------------------------------
In this section we investigate how the 3-body cVdW interaction works in a simple physical limit. Let us consider only 3 beads and place two of them, say $1$ and $2$, very close to each other at distance $R_{12}=r$, while the bead number $3$ is at large distance from $1$ and $2$, i.e. we assume $R_{13}\approx R_{23}\approx R\gg r$. The free-energy Eq.(\[3-body\]) is calculated in $Appendix$ $2A$ and the expression up to the the lowest order in the distance is $$\frac{\mathcal{\bar{F}}_{cVdW}^{(3)}}{\beta}=-\frac{4}{r^{6}}-\frac{4\left(
3\cos^{2}\theta_{3,12}-1\right) }{r^{3}R^{3}}+O\left( R^{-6}\right) ,
\label{F3-R}$$ where the first term is the 2-body interaction between particles $1$ and $2$. The peculiarity of the second term in Eq.(\[F3-R\]), i.e. the three-body interaction is readily seen, when two particles are very near and the third one is far away. From Eq.(\[F3-R\]) it comes out that for the bead arrangement shown in Fig.5, although the two body interactions $1-3$ and $2-3$ are attractive, the third bead is repelled from the first two as a consequence of the anisotropic 3-body interactions. This is due to the condition $R\gg r$ making $\cos^{2}\theta_{3,12}\ll1$ for $\theta_{3,12}\approx\pi/2$ in the configuration of Fig. 5 .
The second interesting observation that we can make from Eq.(\[F3-R\]) is that for a fixed dimer size $r$, the third particle interacts with the point-like two particle complex via a “long-range” force $\propto R^{-3}$. The $1+2$ dimer, instead of the two single monomers becomes now the emerging elementary unit governed by different laws than single particles.
This example is quite instructive and tells us a lot about the very nature of cVdW. On the one hand particles display many-body effects which override their individual pairwise interaction in most general configurations. However there is another idea emerging from the same example, which simplifies dealing with the cVdW quite a bit. When two beads come together - i.e. “lump” together by attractive forces - and find themselves much closer then to the rest of the beads, they can be considered as a new combined entity- the dimer. Now, the dimer itself interacts with the rest of the world by an anisotropic, angle dependent, longer ranged interaction $\propto R^{-3}$ (instead of $R^{-6}$), and this interaction is now 2-body, pairwise and is attractive for $3\cos
^{2}\theta_{3,12}-1>0$.
The concept of “lumping” works generally, also for more than 3 particles which are forming anisotropic lumps out of closely packed aggregates of particles. The usefulness of the lumping idea will become more clear in sections that will follow. In particular, we will derive how beads lumped together inside of chains or membranes interact with other beads and other lumps in the far field. The consideration of the pairwise interactions of such lumps, instead of all many-body interactions of all particles, will be an enormous simplification.
Anomalous Elasticity of cVdW Chains and Membranes
-------------------------------------------------
Here we study consequences of the many body effects in cVdW chains and membranes with finite number ($N$) of particles, - called finite N-effects. Additionally, we study their unusual elastic properties and compare with more classical systems with short range forces. In order to grasp the physics of the problem in the simplest form, we calculate the free-energy for small bead susceptibility, $\chi_{b}\ll1$ - see Eq.(\[3-body\]), which includes first non-trivial and leading $3$-body effects.
### 3-Body Effects in cVdW Chain - Free-Energy and Elasticity
($i$) *Finite linear chain -* The free-energy per particle of the finite chain $\bar{f}_{cVdW,ch}^{(3),N}(\equiv\mathcal{\bar{F}}_{cVdW,lch}^{(3)}/N)$ is calculated by direct summation of 3-body forces in $Appendix$ $2B$ and the result is $$\frac{\bar{f}_{cVdW,ch}^{(3),N}}{\beta}\approx-\frac{8\zeta^{2}\left(
3\right) }{D^{6}}+\allowbreak\frac{3.4}{ND^{6}}, \label{lin-chain}$$ with $\beta=(3/32\pi^{2})\mu_{0}H_{0}^{2}\chi_{b}^{3}V_{b}^{3}$ and the zeta function $\zeta\left( 3\right) \simeq1.2$. The first $O(1)$ term is the free-energy per particle of an infinite chain, while the leading finite $N$ term is of order $O\left( 1/N\right) $. The latter is the price the last few edge particles (at both free ends) pay for being at the end of the chain. Note that, this edge energy being positive indicates that the chain would like to be closed eventually (i.e. elliminate free ends), provided that the bending energy for doing so is less than the gained edge energy.
($ii$) *Young’s modulus for linear chain* - If one generates a slight increase of the distance $D$ between bead centers, i.e $D\rightarrow
D(1+\varepsilon)$ then the Young’s modulus of the stretched chain can be formally defined by the second derivative of the free energy like $Y(\varepsilon=0)=-(1/DS_{b})(\partial^{2}\mathcal{\bar{F}}_{cVdW,ch}^{(3)}/\partial\varepsilon^{2})$, where the bead cross-section surface is $S_{b}\simeq\pi(D/2)^{2}$. From Eq.(\[lin-chain\]) one obtains $$Y(\varepsilon=0)\approx0.84\cdot\mu_{0}H_{0}^{2}\chi_{b}^{3}\text{,}$$ i.e. the effective Young’s modulus scales quadratically with external field (for small $\chi_{b}$) and is independent of the bead diameter $D$. For $\chi_{b}>0$ (paramagnetic beads) it is of course $Y(\varepsilon=0)>0$, which guarantees stability of the chain. In case of $\chi_{b}<0$ (effective diamagnetic beads - when the medium susceptibility $\chi_{m}>\chi_{b}$) one has $Y(\varepsilon=0)<0$ and the chain is unstable due to the repulsive forces of effectively diamagnetic beads.
($iii$) *Bending energy of ring* - In another situation, instead of stretching the chain, we can bend its center line and close it into a ring. The free-energy per particle in this case is calculated in $Appendix$ $2C$ and reads$$\frac{\mathcal{\bar{F}}_{cVdW,ring}^{(3)}}{N\beta}\approx-\frac{8\zeta
^{2}\left( 3\right) }{D^{6}}+\frac{16\pi^{2}\zeta\left( 3\right) }{D^{6}}\frac{\ln N}{N^{2}}. \label{ring}$$ The first (leading) term is the same as for the infinite linear chain, while the second term in Eq.(\[ring\]) can be related to the bending elasticity energy per particle of the coherent VdW chain. Usually, the bending modulus (stiffness) $K$ for the chain with short range forces is defined as the prefactor in the bending energy$$\mathcal{\bar{F}}_{bend,ring}-\mathcal{\bar{F}}_{ch}=\frac{K}{2}{\displaystyle\int\limits_{0}^{2\pi R}}
ds\left( \frac{\partial\mathbf{t}}{\partial s}\right) ^{2}=K\frac{\pi}{R}.
\label{Fbend}$$ where the latter is true for a ring. Here, $\mathbf{t}$ is the unit tangent vector $\mathbf{t}=(-\cos(s/R),\sin(s/R),0)$ and $R=ND/2\pi$.
By interpreting the cVdW ring in this elasticity framework a first surprise comes out. From Eq.(\[ring\]) and Eq.(\[Fbend\]) we see an anomalous behavior of the effective bending modulus $$K_{cVdW}\simeq\frac{\zeta\left( 3\right) D}{48}(\mu_{0}H_{0}^{2}\chi_{b}^{3}V_{b})\ln N, \label{K-cVdW}$$ i.e. $K_{cVdW}\propto\ln N\propto\ln R/D$ grows logarithmically with the chain size. This behavior is qualitatively very different from the case of the chain with short range interaction, where $K\ $is always a N-independent constant. The logarithmic stiffness of chains is caused by the long-range many body nature of cVdW.
It is interesting to note that even though the bending stiffness is growing, the chain closure cost (per bead) $\propto N^{-2}\ln N$ is becoming quickly smaller with large N. A comparison of the ring energy Eq.(\[ring\]) with the straight chain result Eq.(\[lin-chain\]) tells us that the rings will indeed become more preferable for long enough chains with $N\gtrsim320.$
### cVdW Membranes - Free-Energy and Elastic Properties
As in the case of linear chains we can consider the elastic and finite size properties of flat and curved membranes. Most of the results in this section can be derived from the phenomenological macroscopic (demagnetization tensor) approach for fat cylinders, hollow spheres etc. presented in the forthcoming sections, while others are obtained by discrete summations. Here we only present the main physical results and point out the unusual size dependent scaling of various material properties.
($i$) *The free-energy of the finite flat membrane* - The discrete summations of the free-energy $\mathcal{\bar{F}}_{cVdW,fl-me}^{(3),N}$ ** for large $N$ and the radius $R\propto\sqrt{N}$ are difficult due to absence of a convenient symmetry for all particles (like present for the ring). However, if we combine the results for the tubular membrane in the limit $N\rightarrow\infty$ - see Eq.(\[F-tube\]) and $Appendix$ $2E$, with the finite $N$ corrections in the cylindrical scheme 2 of the macroscopic approach - see Eq.(\[f2-mac-me\]). In this approach we can estimate $\mathcal{\bar{F}}_{cVdW,fl-me}^{(3),N}$ $$\frac{\mathcal{\bar{F}}_{cVdW,fl-me}^{(3),N}}{N\beta}\approx-\frac{8\pi^{4}}{27D^{6}}\allowbreak+\frac{B}{D^{6}}\frac{\ln N}{\sqrt{N}}, \label{fl-mem-N}$$ with $B=192\rho_{pack}^{2}$ . Here, $\rho_{pack}<1$ is the *packing* (*volume*) *fraction* of the beads in membrane. The second term in Eq.(\[fl-mem-N\]) is due to the line tension of the membrane. This line tension, similarly as the bending stiffness of chains, scales logarithmically with the system size.
($ii$) *Bending energy of the spherical membrane (vesicle) -* For a fluid, spherical membrane of radius $R$ with classical elasticity (due to a finite range interaction) the energy density would be proportional to $(1/R^{2})$ (=curvature$^{2}$). In that case classic “Helfrich-like” membrane case the total energy coming solely from the bending is then $\sim\frac
{1}{R^{2}}R^{2}\sim1$, i.e. it is constant for all vesicle sizes. What happens in the case of a cVdW vesicle? To answer that, we can calculate the 3-body part of the free-energy per particle with the diameter $D$ of the spherical shell - see $Appendix$ $2D$, which gives$$\frac{\mathcal{\bar{F}}_{cVdW,sph-me}^{(3),N}}{N\beta}\approx-\frac{50\pi
\rho_{sph}^{2}}{D^{6}}\allowbreak+\frac{50\sqrt{\pi}\rho_{sph}^{9/4}}{D^{6}\sqrt{N}}, \label{sph-mem-N}$$ where $\rho_{sph}\approx0.4$ is the surface packing factor of the spherical beads on the sphere, obtained by comparing Eq.(\[sph-mem-N\]) and Eq.(\[fl-mem-N\]). The relation between $\rho_{sph}$, $N$ and the radius of the shell $R$ is given by $N=4\pi R^{2}/\rho_{sph}^{-1}\pi\left( D/2\right)
^{2}(=\allowbreak16\rho_{sph}R^{2}/D^{2})$. The free-energy expressed in terms of $R(\propto\sqrt{N})$ is given by $$\frac{\mathcal{\bar{F}}_{cVdW,sph-me}^{(3),N}}{\beta}\approx
-const.N\allowbreak+\frac{204\sqrt{\pi}\rho_{sph}^{11/4}}{D^{7}}R,
\label{sph-mem-R}$$ where the first term is the constant energy per particle. The second term is remarkable as the bending energy grows with radius $\propto R(\propto\sqrt
{N})$. This means that the effective bending energy density is $\propto
R^{-1}(\propto N^{-1/2})$ which is much larger than in classical membranes where $\propto R^{-2}(\propto N^{-1})$. To put it differently, the bending stiffness of the membrane $K_{me}$ becomes size dependent with $K_{me}\sim\sqrt{N}$. This behavior is also confirmed in the macroscopic theory (studied below) for the continuous hollow sphere.
After the logarithmic stiffness of the chain (with $K_{ch}\propto\ln N$), the non-locality of the bending stiffness of membrane is another signature of the anisotropic and long range 3-body nature of the coherent interaction in cVdW systems. This is somehow reminiscent, of the physics of classic elastic cross-linked (i.e. non-fluid) membranes where the non-locality is due to the coupling of in-plane strains with the flexural deformation [@Nelson]. These effects of the elastic in plane coupling are neglected in our case, as the curved cVdW membranes can be considered to be a well shaken, i.e. behaving like a fluid and without in-plane stresses. The size dependent stiffness effects emerge in our case entirely from the many-body, long-range nature of the cVdW interaction.
By comparing the free-energies for the flat- and spherical-membrane in Eq.(\[fl-mem-N\]) and Eq.(\[sph-mem-N\]) one sees that for finite, but large, $N$ the spherical membrane has lower energy than the flat one, i.e. $\mathcal{\bar{F}}_{cVdW,sph-me}^{(3),N}<\mathcal{\bar{F}}_{cVdW,fl-me}^{(3),N}$. This result is also confirmed in the macroscopic approach - see below. At the first glance this result is not conform with experiments where only flat membranes were observed [@Martin1; @Martin2; @Osterman]. This can be explained by the fact that, in order to form a spherical membrane large energy barriers, far beyond thermal energies ($\mathcal{\bar{F}}_{cVdW,sp-me}^{(3),N}\gg kT$), have to be overcome. This might prevent the spherical membranes from being observed experimentally, so far.
($iii$) *Bending energy of the tubular membrane* - We consider the case when the thickness of the tube is the bead radius $D$, i.e. $R_{2}-R_{1}=D$. The approximate free-energy is calculated in $Appendix$ $2E$ . The obtained free-energy in the leading order is given by $$\frac{\mathcal{\bar{F}}_{cVdW,tub}^{(3)}}{\beta N}\simeq-\frac{8\pi^{2}}{27}\frac{1}{D^{6}}(\pi^{2}-5\frac{D}{R_{\perp}}), \label{F-tube}$$ where $N=N_{1}N_{2}$, $R_{\perp}\sim N_{1}$ is the external radius of the tube and the limit $(R_{\perp}/D)\gg1$ is assumed - see Fig.11. The first nontrivial term is proportional to $1/R_{\perp}$ which means that the tubular membrane (cylindrical shell) has a similar type of anomalous and non-local elasticity (at least in scaling) as the spherical membrane. This result is also confirmed within the macroscopic approach - see the next Subsection.
Macroscopic Approach to cVdW
============================
By studying the microscopic $3$-body interaction in Eq.(\[3-body\]) we have understood, intuitively and qualitatively, why chains form initially. However, in order to capture quantitatively their transition to membranes for arbitrary values of $\chi_{b}<3$, higher $O\left( \chi_{b}^{4}\varphi_{ij}^{4}\right)
$ terms beyond the 3-body interactions (in Eq.(\[3-body\])) are necessary. This appears as a difficult task at present. In order to study the assembly of magnetic colloids, especially in dense systems, hard and physically much less transparent numerics would be necessary. Therefore, it is conceptually instructive to take a more macroscopic, continuous mean-field approach [@Landau], where *dense chains/membranes* are modelled as a continuum medium. In this approach the dipolar tensor $\hat{T}_{ij}$ is replaced by its macroscopic analogue - the demagnetization tensor $\hat{L}$, while the microscopic effective susceptibility $\hat{\chi}_{eff,ij}$ in Eq.(\[chi-eff\]) is replaced in the continuum limit by the corresponding macroscopic, (shape-dependent) tensor-susceptibility $\hat{\chi}^{(L)}$ given by [@Landau] $$\hat{\chi}^{(L)}=\chi(1+\hat{L}\chi)^{-1}. \label{chi-L}$$ Here, $\chi$ is the *material susceptibility* (with respect to an internal field $\mathbf{H}_{int}$) which is due to local field effects in an aggregate of beads and depends in a nonlinear way on the bead susceptibility $\chi_{b}$. For $\chi_{b}>0$ one has $\chi>\chi_{b}$. The demagnetization tensor $\hat{L}$ depends on the shape of the sample, which is chosen in such a way to mimic the composite structure like e.g. a chain and membrane. The time-averaged cVdW free-energy $\mathcal{\bar{F}}$ in the macroscopic approach is generally given by [@KulicKulic-PRL]$$\mathcal{\bar{F}}^{mac}(\mathbf{H}_{0},\hat{L})=-\frac{1}{2}\mu_{0}H_{0}^{2}VTr\left\{ \hat{\chi}^{(L)}\right\} , \label{F-macro1}$$ together with Eq.(\[chi-L\]), where the demagnetization coefficients $L_{x}$, $L_{y}$,$L_{z}$ - the eigenvalues of $\hat{L}$ - depend on the shape and aspect ratio of the sample.
As already mentioned the main structures that form on the intermediate scales are initially chains and then membranes. In the macroscopic approach we can model them within the framework of two different schemes:
$(1)$ the *spheroid scheme*, where the structure is replaced with a spheroidal shape with the semi-axes $a=b\neq c$ and the volume $V=(4\pi
/3)a^{2}c$, where we have $c\gg a$ for *prolates (chains)* and $c\ll a$ for *oblates (membranes)*, or
($2$) the *cylinder scheme*, where the sample is modelled by a cylinder with height $h$ and radius $R$. For a *long cylinder* we have $h\gg R$ and $h\ll R$ for a *thin* (flat) one.
It will be shown below that for infinite systems both approaches (1) and (2) give the same results, while for a finite number of particles (the $N$-effects) they differ. We study also the energetics of the *spherical membrane (spherical shell - coated sphere)* which in the large $N$ limit mimics a membrane and its bending stiffness. At the end of this chapter, we shall also compare the microscopic and macroscopic approach for various shapes and ask for consistency of the two approaches.
Infinite Chains and Membranes
-----------------------------
Here we consider the case $N\rightarrow\infty$ and obtain asymptotic results for large (infinite) membranes and chains.
($i$) *Chain* - The two perpendicular demagnetization factors of infininte chains are given by $L_{a,\infty}=L_{b,\infty}=1/2$, while for the one along the long chain axis we have $L_{c,\infty}=0$ in both schemes (cylinder and ellipsoid). In that case Eq.(\[F-macro1\]) gives the macroscopic approximation of the free-energy $\mathcal{\bar{F}}_{ch,\infty
}^{mac}$ of chain $$\mathcal{\bar{F}}_{ch,\infty}^{mac}(\mathbf{H}_{0})=-\frac{1}{2}\mu_{0}H_{0}^{2}V\chi(1+\frac{2}{1+\chi/2}) \label{Fchain}$$
($ii$) *Membrane* - In both schemes in the limit $N\rightarrow\infty$ one has $L_{a,\infty}=L_{b,\infty}=0$, $L_{c,\infty}=1$ and the asymptotic *free-energy of the membrane* is $$\mathcal{\bar{F}}_{me,\infty}^{mac}(\mathbf{H}_{0})=-\frac{1}{2}\mu_{0}H_{0}^{2}V\chi(2+\frac{1}{1+\chi}) \label{Fmem}$$
Note, that for both chains and membranes the macroscopic free-energy is dominated by the smallest demagnetization factors $L^{m/ch}\rightarrow0$. For membranes two demagnetization factors vanish, while for chains only one vanishes. Then, for $N\rightarrow\infty$ and by assuming the same $\chi$ for chains and membranes, the free-energy of the membrane is always smaller than that of the chain for any $\chi$, i.e. $\mathcal{\bar{F}}_{me,\infty}^{mac}<\mathcal{\bar{F}}_{ch,\infty}^{mac}$.
The macroscopic approach is well in agreement with the experiments [@Martin1; @Martin2; @Osterman] which show that the formation of 3D spherical droplets is unfavorable. Namely, from Eq.(\[Fmac-exp\]) (further below) it is seen that for a *spherical droplet,* where $L_{x}=L_{y}=L_{z}=1/3$, the free-energy is given by $\mathcal{\bar{F}}_{drop}^{mac}(\mathbf{H}_{0},\hat{L})\approx-(3/2)\mu_{0}H_{0}^{2}V\chi/(1+\chi/3)$. This means that $\mathcal{\bar{F}}_{me,\infty}^{mac}<\mathcal{\bar{F}}_{ch,\infty}^{mac}<\mathcal{\bar{F}}_{drop}^{mac}$, i.e. also in the macroscopic approach the formation of chains and membranes (once they become large) is more favorable than the creation of spherical droplets.
Finite Chains and Membranes
---------------------------
For finite $N$ the demagnetization factors $L_{i}$ depend on the aspect ratio $a/c$, i.e. on $N$. The chain is characterized by the long semi-axis $c\sim
ND$ ($D$ the bead size), and the short semi-axis$\ a\sim D\ll c$ , with the corresponding demagnetization factors $L_{a},L_{b},L_{c}$ ($L_{a}+L_{b}+L_{c}=1$) [@Landau], [@1.Beleggia; @4.Beleggia; @2.Beleggia]. The *chain-membrane transition* in the macroscopic approach is reached on the critical line $N_{c}(\chi)$, where $\mathcal{\bar{F}}_{me}^{mac}(N_{c})=\mathcal{\bar{F}}_{ch}^{mac}(N_{c})$. To demonstrate the existence of such a transition the critical line was calculated for the spheroid scheme 1 by assuming, for simplicity, that the chain and membrane material susceptibilities are the same [@KulicKulic-PRL]. It was found that the critical cluster size $N_{c}$ grows with the material susceptibility $\chi$. For $\chi\approx1-3$, it was estimated $N_{c}\approx10-20$. Such a tendency is also observed in experiments [@Osterman], where for $N_{c}\approx10$ initial signs of the chain-membrane transition are found. In these experiments, chains are formed for small $N\sim10$ in the very dilute limit, while a further addition of colloids results in a branching of chains (via the so called $Y$-junctions), further followed by a network of inter-connections and finally dense membrane patches are formed. The conclusion is that the chain-membrane transition in the macroscopic approach is qualitatively in accord with experiments.
Let us calculate the finite $N$-effects in the macroscopic approach and compare it with the microscopic theory given in Section IV. To remind the reader, in the *microscopic approach* the free-energy in Section IV is calculated for small $\chi_{b}$ by expanding it up to $\chi_{b}^{3}$, i.e. it is given by $\mathcal{\bar{F}}_{cVdW}^{micro}\simeq\mathcal{\bar{F}}_{cVdW}^{(1)}+\mathcal{\bar{F}}_{cVdW}^{(3)}$ (since $\mathcal{\bar{F}}_{cVdW}^{(2)}=0$) where $\mathcal{\bar{F}}_{cVdW}^{(1)}\propto N\chi_{b}$ describes $N$ non-interacting beads, while the first nontrivial term due to the dipole-dipole interaction in Eq.(\[3-body\]) is $\mathcal{\bar{F}}_{cVdW}^{(3)}\propto\chi_{b}^{3}$ . As already discussed, the term $\mathcal{\bar{F}}_{cVdW}^{(3)}$ is due to both, 2-body and 3-body interactions.
In order to compare these two approaches we need to expand the free-energy in the macroscopic approach as a function of $\chi_{b}$ up to $\chi_{b}^{3}$, as well. In the macroscopic approach, described by Eq.(\[chi-L\])-Eq.(\[Fmem\]), the free-energy depends on the material susceptibility $\chi$ which is related to the bead susceptibility $\chi_{b}$ and in systems with small bead susceptibility ($\chi_{b}$) $\chi$ is also small. Therefore, in order to make the expansion of the macroscopic free-energy Eq.(\[F-macro1\]) with respect to $\chi_{b}$ a relation between $\chi$ and $\chi_{b}$ is necessary. In fact $\chi$ is related to the averaged bead susceptibility $\tilde{\chi}_{b}=\rho_{pack}\chi_{b}$, where $\rho_{pack}(<1)$ is the *packing* (*volume*) *fraction* of the beads in a given assembly. (Note that the total volume of the bead $V_{tot}=NV_{b}$ is related to the macroscopic volume $V$ by $V_{tot}=\rho_{pack}V$.) By assuming high local symmetry around each bead within the assembled structures it is easy to show that in that case the Lorenz-Lorenz relation $\chi=\tilde{\chi
}_{b}(1-(\tilde{\chi}_{b}/3))^{-1}$ holds. By making an expansion of $\chi$ up to $\tilde{\chi}_{b}^{3}$ and using that $L_{x}=L_{y}$ and $L_{x}=(1-L_{z})/2$ the *macroscopic free-energy* in Eq.(\[F-macro1\]) reads $$\frac{\mathcal{\bar{F}}^{mac}(\mathbf{H}_{0},\hat{L})}{(\mu_{0}H_{0}^{2}V/2)}\approx-3[\tilde{\chi}_{b}+\frac{\tilde{\chi}_{b}^{3}}{2}(L_{z}-\frac
{1}{3})^{2}]. \label{Fmac-exp}$$ Note, that in the case of the chain one has $L_{z}\rightarrow0$ for $N\rightarrow\infty$, while for the membrane one has $L_{z}\rightarrow1$ for $N\rightarrow\infty$ - see below.
### Linear Chain
As mentioned before in the case of finite $N$ the demagnetization factor $L_{z}(N)$ is different in the spheroid scheme 1 and the cylindrical scheme 2. Let us have a look at the difference [@Landau; @1.Beleggia; @4.Beleggia; @2.Beleggia]:
\(i) the *spheroid scheme 1* - In this case the chain corresponds to a prolate spheroid and the exact demagnetization factor $L_{z}$ of a prolate spheroid is given in $Appendix$ $3A$ Eq.(\[chain-prol\]) with $\tau_{a}=N$ one has $L_{z}\approx\left( \ln N\right) /N^{2}$ and the dimensionless free-energy $f_{ch,N}^{mac,1}$($\equiv\mathcal{\bar{F}}_{ch,N}^{mac,1}(\mathbf{H}_{0},\hat{L})/(\mu_{0}H_{0}^{2}NV_{b}/2)$) given by: $$f_{ch,N}^{mac,1}\approx f_{ch,\infty}^{mac}+\rho_{pack}^{2}\chi_{b}^{3}\frac{\ln N}{N^{2}}, \label{f1-ch-N}$$ where the energy per bead of the infinite chain is $$f_{ch,\infty}^{mac}\approx-(3\chi_{b}+\frac{\rho_{pack}^{2}\chi_{b}^{3}}{6}).
\label{f-ch-inf}$$
\(ii) the *cylinder* *scheme* 2 - In the cylinder scheme 2 a chain corresponds to a long cylinder with the aspect ratio $\tau
\equiv(h/2R)\approx2N/3\gg1$, $h$ is the hight (along the c-axis) and $R$ the radius of cylinder, one has $L_{z}\approx2N/\pi$ - see in $Appendix$ $3A$ Eq.(\[A3-Lz-cyl-l\]) and the corresponding free-energy $f_{ch,N}^{mac,2}$($\equiv\mathcal{\bar{F}}_{ch,N}^{mac,2}(\mathbf{H}_{0},\hat{L})/(\mu
_{0}H_{0}^{2}NV_{b}/2)$) reads$$f_{ch,N}^{mac,2}\approx f_{ch,\infty}^{mac}+\frac{2\rho_{pack}^{2}\chi_{b}^{3}}{\pi N}. \label{f2-ch-N}$$ By comparing Eq.(\[f1-ch-N\]) and Eq.(\[f2-ch-N\]) of the macroscopic approach with the corresponding microscopic free-energy for the finite linear chain in Eq.(\[lin-chain\]), it turns out that the linear chain is slightly better described by the long cylinder in the cylinder scheme 2.
### Flat Membrane
\(i) the *spheroid scheme* 1 - In that case the flat membrane corresponds to extreme oblate spheroids with $L_{z}\approx1-\pi/2\sqrt{N}$ - see Eq.(\[A3-Lz-sp-0\]) in $Appendix$ $3A$ and the dimensionless free-energy $f_{me,N}^{mac,1}$($\equiv\mathcal{\bar{F}}_{me,N}^{mac,1}(\mathbf{H}_{0},\hat{L})/(\mu_{0}H_{0}^{2}NV_{b}/2)$) is given by$$f_{me,N}^{mac,1}\approx f_{me,\infty}^{mac}+\frac{\pi\rho_{pack}^{2}\chi
_{b}^{3}}{\sqrt{N}}, \label{f1-mac-me}$$ where the free-energy per bead of the infinite flat membrane is$$f_{me,\infty}^{mac}\approx-(3\chi_{b}+\frac{2\rho_{pack}^{2}\chi_{b}^{3}}{3}).
\label{f-mac-me}$$
\(ii) *the cylinder scheme* 2 *-* In this case a thin cylinder with aspect ratio $\tau\approx1/\sqrt{N}\ll1$, one has $L_{z}\approx1-\ln
N/\pi\sqrt{N}$ which gives the dimensionless free-energy $f_{me,N}^{mac,2}$($\equiv\mathcal{\bar{F}}_{me,N}^{mac,2}(\mathbf{H}_{0},\hat{L})/(\mu
_{0}H_{0}^{2}NV_{b}/2)$ ) $$f_{me,N}^{mac,2}\approx f_{me,\infty}^{mac}+\rho_{pack}^{2}\chi_{b}^{3}\frac{\ln N}{\sqrt{N}} \label{f2-mac-me}$$ By comparison with the microscopic energy in Eq.(35) one expects that the cylinder scheme 2 mimics the membrane better than the spheroid scheme 1.
Spherical Shell Membrane
------------------------
In the microscopic approach we calculated the energy of a closed, monolayered spherical membrane (spherical shell) under the action of cVdW interaction. It is interesting to study its energetics in the macroscopic approach where the outer and inner radius of the spherical shell are $R_{o}$, $R_{i}$, respectively, while the relative magnetic permeability in the shell is $\mu_{shell}=1+\chi$. Outside this shell we assume $\mu_{out}=1$. In this case the symmetry implies that Eq.(\[F-macro1\]) is simplified to $$\mathcal{\bar{F}}_{cVdW,shell}^{mac}(\mathbf{H}_{0},\hat{L})=-\frac{3}{2}\mu_{0}H_{0}^{2}\alpha_{shell}, \label{Fshell}$$ where $\alpha_{shell}\equiv V\chi_{shell}$ can be calculated in the magneto-static limit by using standard boundary conditions [@Jones]$$\alpha_{shell}=4\pi R_{o}^{3}\frac{\chi(3+2\chi)(1-\phi)}{(\chi+3)(3+2\chi
)-2\phi\chi^{2}}. \label{alpha-shell}$$ with $\phi=(R_{i}/R_{o})^{3}$ where $R_{i}$ is the inner and $R_{o}$ is the outer radius of the spherical shell. For a small shell thickness $D\ll
R_{o},R_{i}$ one has $R_{o}^{3}-R_{i}^{3}\approx N(D/2)^{3}$. For the outer surface with $4\pi R_{o}^{2}\approx\pi N(D/2)^{2}$ one has $\phi
=1-N(D/2R_{o})^{3}=1-8/\sqrt{N}$. For a small bead susceptibility $\chi_{b}<1$ the material susceptibility is given by $\chi\approx\tilde{\chi}_{b}(1+(\tilde{\chi}_{b}^{2}/3)+(\tilde{\chi}_{b}^{3}/9))$. By making an expansion for large $N$ it is straightforward to obtain $\alpha_{shell}$ and $\mathcal{\bar{F}}_{shell}^{mac}$ in Eq.(\[Fshell\])$$f_{shell,\infty}^{mac}=f_{me,\infty}^{mac}+\frac{16}{3}\frac{\varrho
_{pack}^{2}\chi_{b}^{3}}{\sqrt{N}}. \label{Fshell-N}$$ The first term characterizes the infinite flat membrane, while the second one $\propto(16\varrho_{pack}^{2}\chi_{b}^{3}/3)N^{-1/2}$ is asymptotically smaller than that for the flat membrane $\propto(\varrho_{pack}^{2}\chi
_{b}^{3})N^{-1/2}\ln N$ in Eq.(\[Fmac-exp\]. This means that the free-energy of the spherical membrane (with some large but finite $N$) becomes overall smaller than the microscopic free-energy of the finite flat membrane. This confirms our previous analysis that a large spherical membrane is slightly more favorable than the flat one.
Consistency of Microscopic and Macroscopic cVdW Theory
------------------------------------------------------
Let us check if the macro- and microscopic theory agree in concrete cases. For that purpose we compare the corresponding interacting parts of the free-energy, i.e. $\Delta f_{\infty}^{mac}$($=f_{\infty}^{mac}-\frac{3}{2}\chi_{b}$) and $\Delta f_{\infty}^{mic}$, for chains and membranes. In the case of the chain one has $V=\rho_{pack}V_{0}$ and $\Delta f_{ch,\infty}^{mac}\approx-\chi_{b}^{3}\rho_{pack}^{2}/6$, while the first term of the microscopic free-energy in Eq.(\[lin-chain\]) is $\Delta f_{ch,\infty}^{mic}\approx-\varsigma^{2}(3)\chi_{b}^{3}H_{0}^{2}/24$. The equality $\Delta
f_{ch,\infty}^{mac}=\Delta f_{ch,\infty}^{mic}$ gives the consistency condition for the bead packing fraction $\rho_{pack}=0.5\varsigma
(3)\approx0.6$ - indeed a rather plausible and realistic value for the packing density. A similar situation holds for membranes, where Eq.(\[Fmac-exp\]) gives $\Delta f_{me,\infty}^{mic}\approx-2\chi_{b}^{3}\rho_{pack}^{2}/3$. The microscopic free-energy of the flat membrane can be obtained, for instance, from Eq.(\[F-tube\]) for the tubular membrane in the limit $R_{\perp
}\rightarrow\infty$, i.e. $\Delta f_{me,\infty}^{mic}=-(\pi^{4}/162)\chi
_{b}^{3}$. The condition $\Delta f_{me,\infty}^{mac}=\Delta f_{me,\infty
}^{mic}$ gives the packing fraction $\rho_{pack}\approx0.9$ which is a reasonable value.
The good news overall is, that for several types of assemblies we confirm a qualitative and a satisfactory quantitative agreement of the macroscopic approach with the microscopic one in cases considered. This justifies the more coarse grained but simpler macroscopic approach in studying several cVdW structures.
Instability of The Spherical Droplet
------------------------------------
We have seen that the many body interactions in cVdW systems favor the formation of large membrane structures. The latter behavior could be interpreted as coming from an effective negative “surface tension”. It appears that cVdW tends to “flatten out” every aggregate into a thin monolayered sheet down to the smallest cut-off length (the constituent bead size)- as if a negative effective surface tension was at work.
In this $Section$ we show on a concrete example that the cVdW assemblies indeed behave as systems with negative effective surface tension $\sigma_{cVdW}<0$. Although this quasi-surface tension is in reality a rather complex and anisotropic *bulk* term (and depends on the shape of assemblies) it generates effects which are very reminiscent of a real surface tension.
To crystallize out the physics, in the following we will let the cVdW interaction directly compete with an additional real (positive) fluid surface tension. Concretely, we consider a cVdW system made of paramagnetic beads which are embedded in a spherical droplet of the solvent$,$ say oil or water, with the surface tension $\sigma$. Now, the question is : What is the critical surface tension below which the spherical droplet becomes unstable and starts forming an oblate ($\tau_{s}<1$) or prolate ($\tau_{s}>1$) spheroid? Here $\tau_{s}=c/a$ is the aspect ratio of the spheroid.
For shapes very close to the sphere, one can expand the total free-energy in terms of the small $z$-axial stretch $\varepsilon=\tau_{s}-1\ll1$, where $\varepsilon<0$ represents an oblate ellipsoid and $\varepsilon>0$ means a prolate one. By using Eq.(\[A3-Lz-sp\]) given in $Appendix$ $3B$ one obtains the $z$-axis demagnetization factor $L_{z}$. The results for $L_{z}$ holds in both cases the oblate spheroid and prolate one, and we furthermore have $L_{x}=L_{y}=(1-L_{z})/2$. By using the latter property the dimensionless free-energy in Eq.(\[Fmac-exp\]) takes the form $$\frac{\mathcal{\bar{F}}_{cVdW}-\mathcal{\bar{F}}_{0}}{E_{0}}=-\left( \frac
{8}{75}\varepsilon^{2}-\frac{24}{175}\varepsilon^{3}+\frac{1382}{11\,025}\varepsilon^{4}\right) +O\left( \varepsilon^{5}\right) ,\nonumber$$ with $E_{0}=\mu_{0}\tilde{\chi}_{b}^{3}VH_{0}^{2}/2$ the convenient energy scale and $\mathcal{\bar{F}}_{0}$ the non-interacting (self-) free-energy of all beads.
Introducing also the surface energy term $\sigma A$ to the total energy we have then$$\mathcal{\bar{F}}_{tot}=\mathcal{\bar{F}}_{cVdW}+\sigma A$$ where the surface area of spheroid $A$ depends on the aspect ratio $\tau_{s}$ and is given in $Appendix$ $3B$. A short calculation then gives$$\begin{aligned}
\frac{\mathcal{\bar{F}}_{tot}}{E_{0}} & =-\left( \frac{8}{75}\varepsilon^{2}-\frac{24}{175}\varepsilon^{3}+\frac{1382}{11\,025}\varepsilon^{4}\right) \label{Ftot}\\
& +\frac{\sigma A_{0}}{E_{0}}\left( \frac{8}{45}\varepsilon^{2}-\frac
{584}{2835}\varepsilon^{3}+\frac{118}{567}\varepsilon^{4}\right) .\nonumber\end{aligned}$$ We see that the sphere with radius R is only stable when the $\varepsilon^{2}$ term is positive which implies $$\begin{aligned}
\frac{\sigma A_{0}}{E_{0}} & >\allowbreak\frac{3}{5}\equiv\frac{\sigma
_{cr}A_{0}}{E_{0}}\label{sigma-cr}\\
\sigma_{crit} & =\frac{1}{5}\mu_{0}\tilde{\chi}_{b}^{3}H^{2}R,\text{
\ }\nonumber\end{aligned}$$ where $\mu_{0}\tilde{\chi}_{b}^{3}H^{2}R/5$ may be considered as an effective cVdW surface tension $\sigma_{cVdW},$ which has however a negative sign and counteracts (reduces) the actual surface tension of the surrounding liquid.
The phase diagram for the cVdW spherical droplet is shown in Fig.6. Notably for a subcritical fluid surface tension $\sigma<\sigma_{crit}$ the droplet can be either prolate or oblate. Which branch is actually chosen might subtly depend on the dynamics and history of the shape. However, from energetic point of view, the absolute energy minimum in the subcritical regime is reached for the flatter i.e. oblate shape (lower $\varepsilon$ stable branch). The elongated prolate ellipsoid forms only a shallow local minimum and is therefore thermodynamically only metastable.
The effective surface tension $\sigma_{cVdW}\left( R\right) $ is in reality a many-body bulk term and thus size dependent. Let us estimate on which scale it becomes relevant, for instance, in a drop of water, with surface tension $\sigma_{H_{2}O}=\allowbreak0.073\,J/m^{2}$ (at room temperature). For $\chi_{b}\sim1$ and $B=\mu_{0}H_{0}=0.01T$, the cVdW surface tension is $\sigma_{crit}\approx-B^{2}R/5\mu_{0}=\allowbreak-15.\,9(J/m^{3})R$. For this to be of the same order as $\sigma_{H_{2}O}$ we need the radius of the droplet to be $R_{c}=5\sigma_{H_{2}O}/(\mu_{0}\chi_{b}^{3}H^{2})=4.\,\allowbreak
6\times10^{-3}m=4mm$. For larger droplets made of paramagnetic beads with $R>R_{c}$ the spherical shape becomes unstable.
In the dielectric analogue of this phenomenon in cVdW systems we would have $\sigma_{crit}=\varepsilon_{0}\kappa_{el}^{3}E^{2}R/5$. Let us assume that the dielectric droplet has much smaller dielectric susceptibility than of the electric beads ($\kappa_{el,m}\ll\kappa_{el}$) and that the surface tension is of the order as that of water. Then for a feasible electric field $E=10V/mm$, $\ \varepsilon_{0}=$ $8.8\ast10^{-12}J/mV^{2}$ , and for $\kappa_{el}^{3}\sim1$ we have $R_{el}=5\sigma/(\varepsilon_{0}\kappa_{el}^{3}E^{2})\approx\allowbreak500m$ ! This means that this effect is less favorable in the electric case. However, for magnetic colloids (such as ferrofluids) under magnetically induced cVdW, the “negative surface tension” instability effect should be easily observable.
Formation of Superstructures in cVdW Systems
============================================
In magnetically driven cVdW systems chains and membranes are the predominant structures formed on intermediate length and timescales [@Martin1; @Martin2; @Osterman]. However, the experiments also show that in more dense colloid systems which are placed in containers of finite volume, more complex structures like foams are formed on larger scales. The existence of these foam structures is also confirmed in numerical simulations [@Martin1; @Martin2]. The basic motif underlying such foams, is a complex network of interconnected membrane patches, which apart from touching along their edges do not stack and aggregate. Instead the membranes seem, at least by visual inspection, to repel each other and the whole foam structure appears to swell against gravity. What is the origin of such large scale cVdW foam structures?
In this Section we give a plausible physical explanation by combining both the microscopic and macroscopic approaches. First, we will study the cVdW interaction free-energy of two flat membranes. We show below that this interaction switches from an attraction to a repulsion, depending on the mutual orientation.
While in general, the interactions between two membranes can have both signs, it turns out that in the majority of possible configurations the interaction is in fact repulsive on the average. Then, we calculate the free-energy of the foam structure by modelling it by a cubic shelf structure ansatz. We show that such a structure indeed tends to swell and at the end we derive something that reassembles an equation of state of a cVdW foam, i.e. a pressure-concentration-field relation. We show that the magnitude of this pressure is quite notable and can indeed lead to a rise of the foam to measurable heights.
cVdW Interaction Generalization to Anisotropic Objects
------------------------------------------------------
In the previous sections we were concerned with interactions of isotropic spherical particles whose susceptibility tensors were merely diagonal $(\hat{\chi}_{b})_{\alpha\beta}=\chi_{b}\delta_{\alpha\beta}$ i.e. simply a number. Here we generalize the interaction to any two arbitrary shaped bodies. In general this is a complicated problem when the bodies are very close. However, when they are far enough, say much further than their typical body extensions, the field of any object can be replaced with a corresponding effective ellipsoid field. In this sense it is sufficient to consider the interaction of two ellipsoids, with orientation dependent and non-trivial susceptibility tensors $\hat{\chi}_{1},\hat{\chi}_{2}$. The free-energy in this case can be rewritten in the form $$\mathcal{\bar{F}}_{cVdW}\left( \mathbf{H}_{0},\{\mathbf{R}_{i}\}\right)
=-\frac{\mu_{0}H_{0}^{2}}{2}Tr(V_{1}\hat{\chi}_{1,eff}+V_{2}\hat{\chi}_{2,eff}), \label{M1}$$ where $\hat{\chi}_{1,eff}$ and $\hat{\chi}_{2,eff}$ are now effective susceptibility tensors of the two bodies (ellipsoids) with respective volumes $V_{1/2}$. The effective susceptibility tensors are now given by the expression $$\hat{\chi}_{1,eff}=(1-\varphi_{12}^{2}\hat{\chi}_{1}\hat{t}_{12}\hat{\chi}_{2}\hat{t}_{12})^{-1}(\hat{\chi}_{1}-\varphi_{12}\hat{\chi}_{1}\hat{t}_{12}\hat{\chi}_{2}) \label{M2}$$ and same for the second $\hat{\chi}_{2,eff}$ which is obtained by replacing $1\rightarrow2$. The slightly more intricate form of $\hat{\chi}_{1/2,eff},$ which is obviously a generalization of the corresponding isotropic expression Eq.(\[chi-eff\]), comes now from the fact that the operator $\hat{t}_{12}=\hat{1}-3\left\vert \mathbf{b}_{12}\right\rangle \left\langle
\mathbf{b}_{12}\right\vert $ and the susceptibilities $\hat{\chi}_{i}$ are operators with spacial orientations which don’t commute now any more in general.
The expressions in Eq.(\[M1\] -\[M2\]) are general and contain the distance dependence through the scalar factor $\varphi_{12}\propto1/\left\vert
\mathbf{R}_{12}\right\vert ^{3}$ in a slightly scrambled form that hides the leading order scaling. Thus, it is interesting to expand $\hat{\chi}_{1/2,eff}$ to the first order w.r.t. $\varphi_{12}\ $and obtain the leading order interaction part $$\mathcal{\bar{F}}_{cVdW,inter}(1,2)=\frac{\mu_{0}}{2}H_{0}^{2}V\varphi
_{12}Tr\{(\hat{\chi}_{1}\hat{\chi}_{2}+\hat{\chi}_{2}\hat{\chi}_{1})\hat
{t}_{12}\}. \label{M3}$$ Here we omit trivial self-energies, i.e. consider the interaction energy term only, and assume the two bodies to have the same volume $V$. Note that, when the objects are isotropic (e.g. spheres, point like) then the $\hat{\chi
}_{1/2}$ turn simply into numbers. Reminding ourselves that $\hat{t}_{12}$ is traceless we see that the whole term $\propto\varphi_{12}$ vanishes in the isotropic case. This is why the cVdW interaction for two spheres only starts out with a higher order leading $\propto\varphi_{12}^{2}\propto1/\left\vert
\mathbf{R}_{12}\right\vert ^{6}$ interaction term. However in general, for anisotropic objects the trace in Eq.(\[M3\]) is non-zero, giving rise to a strong and long range interaction $\propto$ $1/R^{3}$. That is, anisotropic objects interact much stronger than isotropic ones under cVdW. Once growing aggregates become shape anisotropic and notably they always tend to do so (forming chains and membranes), they interact in a long range manner. This can be seen as another manifestation of the cooperative many-body nature of cVdW.
Interaction of Membrane and Single Spherical Bead
-------------------------------------------------
To understand the content of the anisotropic cVdW interaction from Eq.(\[M3\]), let us have a look at how a flat membrane interacts with a single spherical bead. The effects of beads approaching chains and membranes were previously studied numerically by Osterman et al. [@Osterman]. Interestingly, it turns out, that even in this simplest case the interaction can vary in sign. It is attractive when the membrane is approached by the bead from the edge side. When however approaching from the top, in the direction of the membrane normal, the bead is repelled in the far field. If the bead approaches the membrane even further (against the repulsive force) and comes closer in this normal direction, the interaction switches again to a short range attraction. Obviously there is a barrier to cross in this normal direction. The largest barrier for joining of the bead to the membrane is when the former is above the center of the membrane and we study this case first - see Fig.(7a). By applying Eq.(\[S5\]) on two particles, where the first one is big and anisotropic - it mimics a membrane, while the second one is small and isotropic - it mimics a bead, one obtains the free-energy in the form$$\begin{aligned}
W^{m-b} & =-\frac{1}{2}\mu_{0}H_{0}^{2}V_{0}(q_{m}Tr\hat{\chi}_{m,eff}+q_{b}Tr\hat{\chi}_{b,eff})\label{Wmb}\\
& =W^{m}+W^{b}$$ Here, $V_{0}=V_{m}+V_{b}$, $q_{m,b}=V_{m,b}/V_{0}$ and $V_{m}$, $V_{b}$ is the volume of the membrane and bead, respectively, while $\hat{\chi}_{m,eff}$ and $\hat{\chi}_{b,eff}$ the effective susceptibilities of membrane and bead respectively with $$\hat{\chi}_{m,eff}=\frac{1}{(1-q_{m}q_{b}\varphi_{0}^{2}\hat{\chi}_{m}\hat
{t}\hat{\chi}_{b}\hat{t})}\hat{\chi}_{m}(1-q_{m}\varphi_{0}\hat{t}\hat{\chi
}_{b}). \label{chi-m}$$ and $\hat{\chi}_{b,eff}$ is obtained by replacing $m\leftrightarrow b$ in Eq.(\[chi-m\]) and $\varphi_{0}=V_{0}/4\pi R_{mb}^{3}$. In this approach the membrane is replaced by an oblate spheroid with the susceptibility $\hat{\chi
}_{m}=\chi_{\min}\left\vert \mathbf{n}_{m}\right\rangle \left\langle
\mathbf{n}_{m}\right\vert +\chi_{\max}(1-\left\vert \mathbf{n}_{m}\right\rangle \left\langle \mathbf{n}_{m}\right\vert )$, while for the spherical bead we have $\hat{\chi}_{b}=\chi_{b}\hat{1}$; $\mathbf{n}_{m}$ is the unit vector normal to the membrane. The tensor $\hat{t}$ in Eq.(\[chi-m\]) is $\hat{t}(\equiv\hat{t}_{m-b})=\hat{1}-3\left\vert
\mathbf{b}_{m-b}\right\rangle \left\langle \mathbf{b}_{m-b}\right\vert $. Since we assume that the membrane’s dimensions are much larger than the bead size $D$ and the bead is above the membrane one has $\mathbf{n}_{m}\parallel$ $\mathbf{b}_{m-b}$, i.e. $\hat{t}_{m-b}=$ $1-3\left\vert \mathbf{n}_{m}\right\rangle \left\langle \mathbf{n}_{m}\right\vert $.
The final expression for the dimensionless free-energy $w^{m-b}=W^{m-b}/(\frac{1}{2}\mu_{0}H_{0}^{2}V_{0})=w^{m}+w^{b}$ is given by (cf. $Appendix$ $4A$ ) $$w^{m}=-\frac{q_{m}\chi_{\max}}{1-b}\{2(1-\alpha_{b})+\frac{\chi_{\min}}{\chi_{\max}}\frac{1+2\alpha_{b}}{1+c}\} \label{wm}$$$$w^{b}=-\frac{q_{b}\chi_{\max}}{1-\alpha_{m}\alpha_{b}}\{\frac{3+2c}{1+c}+2\frac{\chi_{\min}}{\chi_{\max}}\frac{\alpha_{m}}{1+c}-2\alpha_{m}\},
\label{wb}$$ where $\alpha_{b}=q_{b}\chi_{b}\psi_{0}$, $\alpha_{m}=q_{m}\chi_{\max}\psi
_{0}$; $c=(\alpha_{m}\alpha_{b}(\chi_{\max}-4\chi_{\min}))/\chi_{\max
}(1-\alpha_{m}\alpha_{b}))$. Let us discuss the energy of the membrane-bead complex as a function of their distance $r=R_{m-b}/D$ by assuming (for simplicity) that $\chi_{\min}\ll\chi_{\max}$ i.e. a very flat membrane. Since $q_{b}\ll q_{m}\simeq1$ one has $\alpha_{m}\alpha_{b}\ll1$ and the expression simplifies to $$\frac{w^{m-b}}{2\chi_{\max}}\simeq-1+\frac{\chi_{b}}{24}(\frac{2}{r^{3}}-\frac{\chi_{\max}}{24q_{b}}\frac{1}{r^{6}}). \label{w-mb}$$ From Eq.(\[w-mb\]) we see that there is an energy barrier for the bead, i.e. for $R_{m-b}>R_{c}=D(\chi_{\max}/24q_{b})^{1/3}$ the membrane repels the bead since $F_{m-b}=-(\partial W^{m-b}/\partial R_{m-b})>0$, while for $R_{m-b}<R_{c}$ the force is attractive ($F_{m-b}<0$). As an example we take $\chi_{\max}\sim10$ and $q_{b}>10^{-3}$ which gives us the barrier at a notable distance $R_{c}>10D$ - larger then the bead size. Based on the same formalism it is straightforward to show that when the bead is placed in the plane containing the membrane, i.e. when $\mathbf{n}_{m}\perp$ $\mathbf{b}_{m-b}$ holds, it is always attracted to the membrane, i.e. $F_{m-b}<0$.
To conclude, the above considerations show that the most favorable and fastest membrane growth pathway is addition of beads along the membrane edges in membrane’s plane. Those beads found above the membrane must move parallel to the membrane and finally descend toward the ends of membranes. This is schematically shown in Fig.7a for a bead interacting with a membrane and in a Fig.7b for a bead interacting with a chain (prolate ellipsoid). In the latter case calculations go along the same lines as for membranes, with the only difference being in flipping the signs of interaction. The beads are repelled laterally and attracted along the symmetry axis of the chains. The easy calculation being very similar as for membranes is omitted here.
Interaction of Two Membranes
----------------------------
Once they emerge, what is the fate of the membranes as they continue growing? At some point the membranes will run out of free beads in the solution and start interacting only with the remaining aggregates which turn into membranes once they are large enough. To understand, how two membranes mutually order, we need the* 2-membrane interaction* for arbitrary membrane orientations $\mathbf{n}_{1,2}$ and anisotropic susceptibilities $\hat{\chi
}_{i}^{(L)}=\chi(1+\hat{L}_{i}\chi)^{-1}$. For identical membranes and at large distances ($\varphi_{12}\ll1$, i.e. for $\left\vert \mathbf{R}_{12}\right\vert \gg V_{m}^{1/3}$) one expands $\hat{\chi}_{eff,12}\approx
\hat{\chi}_{1}^{(L)}(1-\varphi_{12}$ $\hat{t}(\mathbf{b}_{12})\hat{\chi
}_{\mathbf{2}}^{(L)})$ and the long range interaction energy in Eq.(\[chi-eff\]) after a short calculation reads (see details in $Appendix$ $4B$), $$\frac{\mathcal{\bar{F}}_{int}}{\alpha}=\frac{C_{1}^{2}+C_{2}^{2}+\frac{1-\gamma}{3}C_{3}^{2}-(1-\gamma)C_{1}C_{2}C_{3}-\frac{2}{3}}{\left\vert
\mathbf{R}_{12}\right\vert ^{3}} \label{me-me}$$ with $\alpha=3(1-\gamma)\chi_{\max}^{2}\mu_{0}H_{0}^{2}V_{m}^{2}/16\pi,$ and $\gamma=\chi_{\min}/\chi_{\max}$ the ratio of the minimal/maximal eigenvalue of the membrane susceptibility tensor $\hat{\chi}^{(L)}$.
The dimensionless factors $C_{1}=\mathbf{n}_{1}\cdot\mathbf{b}_{12}$, $C_{2}=\mathbf{n}_{2}\cdot\mathbf{b}_{12}$, $C_{3}=\mathbf{n}_{1}\cdot\mathbf{n}_{2}$ reveal all the geometrical beauty of cVdW: the 2-membrane interaction is angle dependent and repulsive in many configurations - see Fig.8. Notably, for a fixed $\left\vert \mathbf{R}_{12}\right\vert $, $\mathcal{\bar{F}}_{int}$ becomes minimal for the orthogonally *twisted* membrane orientation with $\mathbf{n}_{1}\perp\mathbf{n}_{2}$, $\mathbf{n}_{1}\perp\mathbf{b}_{12}$ and $\mathbf{n}_{2}\perp
\mathbf{b}_{12}$ ($C_{1/2/3}=0$). The twisted membranes attract each other since $\mathcal{\bar{F}}_{int}^{\left( tw\right) }<0$ (up to the point of mutual contact), as in the *coplanar* case, yet the *twisted* configuration has lower energy. This interesting result should affect the kinetics of membrane formation: If two distant membranes start growing within a large distance they will rotate to a $90^{\circ}$ position before touching. Therefore, some type of glassy state in their orientation may be kinetically favored. In other relevant configurations, such as the *top*, with two out of plane parallel membranes ($C_{1/2/3}=1$) or the *generic* one (cf. Fig.8), the interaction is repulsive with $0<\mathcal{\bar{F}}_{int}^{(gen)}<\mathcal{\bar{F}}_{int}^{(top)}$.
Emergence of Foams
------------------
Simulations and experiments [@Martin1; @Martin2] provide some empirical evidence for the existence of a hollow foam-like superstructure forming on large scales (cf. Fig 9a). What is the physical mechanism driving such a cVdW foam formation?
We have seen above that large aggregates prefer to form membranes, and that these membranes mutually interact. Specifically, when two distant membranes are stacked over each other they repel each other ($\mathcal{\bar{F}}_{int}^{(top)}>0$). In the opposite limit - in close contact distance- a simple estimate implies their preference to split as well. Namely, when a thick membrane, with the thickness $2D$, radius $R$ and volume $2V_{m}$, is cut into two parallel membranes, with the thickness $D$ and radius $R$ each and separated to infinite distance there is a gain in the energy $\Delta\mathcal{F}=2\mathcal{F}_{1m}-\mathcal{F}_{2m}\approx-V_{2m}L\chi
^{2}(1-(1+\chi)^{-2})<0$ for $L\chi\ll1$, where $L\propto D^{3/2}V_{2m}^{-1/2}$. Physically this means that the second membrane lying above the first one is repelled to increase the local fields with respect to the thicker membrane case.
It is this remarkable reluctance of membranes to mutually stack that in fact sets the microscopic structure of the foam: It is formed out of the thinnest possible membrane patches, whose thickness is collapsed onto the smallest available physical scale - the bead size $D$. The characteristic lateral size $a_{M}$ of these membrane patches, on the other hand, is set by the bead volume fraction in the container $f_{V}=V_{b}^{tot}/V\ll1$ (with $V_{b}^{tot}=NV_{b}$ the total volume of all beads and $V$ the container’s volume). By assuming a cubic shelf structure as an ansatz, cf. Fig 9b, one obtains a patch size $a_{M}\approx3D/f_{V}$.
In order to calculate the pressure in such a foam structure we need the total interaction free-energy of all membranes in the system. It turns out that the interaction part of the free-energy ($\mathcal{\bar{F}}_{int}$) of the assumed cubic shelf structure is positive ($\mathcal{\bar{F}}_{int}>0$) due to global average repulsion of membranes - see $Appendix$ $4C$.
Equation of State of cVdW Foams
===============================
In the previous Section we have argued qualitatively that in the cVdW foam structure the positive interaction free-energy should favor an effective repulsion between constituent membranes forming this structure, i.e. that the foam should exert a pressure on walls of the container and in fact swell. In this Section we calculate this pressure as a function of the volume fraction of magnetic beads, i.e. we derive the equation of state for a cVdW material.
As above we define the volume fraction of all beads in the container $f_{V}=(N_{m}V_{m}/V)\approx V_{b}^{tot}/V$ where $V_{m}(\approx Da_{M}^{2})$ is the volume of the single membrane and $N_{m}$ is the total number of (equal) membranes in the container volume $V\approx Na_{M}^{3}$, and $V_{b}^{tot}$ is the total volume of the beads. Here, $D$ is the bead diameter and $a_{M}$ is the size of the single membrane - see Fig.9b. It follows that $f_{V}\approx3D/a_{M}$.
In the following we fix the total volume of all $N_{m}$ membranes $V_{m}^{tot}=N_{m}V_{m}$ and vary the size of the container $V$. The pressure is then defined by $p=-\partial\mathcal{\bar{F}}^{tot}/\partial V$ where $\mathcal{\bar{F}}^{tot}\mathcal{=\bar{F}}_{self}^{tot}\mathcal{+\bar{F}}_{int}^{tot}$ is the total energy of the membranes. $\mathcal{\bar{F}}_{self}^{tot}$ is the self-energy of (non-interacting) membranes and $\mathcal{\bar{F}}_{int}^{tot}$ is the interaction energy of membranes. From Eq.(\[F-macro1\]) the free-energy of the $N_{m}$ single membranes (the self-energy) with the total volume $V_{m}^{tot}=N_{m}V_{m}$ is given by $$\mathcal{\bar{F}}_{self}^{tot}=-(2\chi_{\max}+\chi_{\min})V_{m}^{tot}\frac{B_{0}^{2}}{2\mu_{0}}, \label{p1}$$$$\chi_{\max}=\frac{\chi}{1+L_{m}\chi},\text{ }\chi_{\min}=\frac{\chi
}{1+(1-2L_{m})\chi}. \label{p2}$$ For simplicity, we study here only the case with large material susceptibility $\chi>1$ (note that, the material susceptibility fulfills $\chi>\chi_{b}\leq
3$, where $\chi_{b}$ is the bead susceptibility with respect to the applied (external) field) and at the same time $L_{m}\chi\ll1$ ($L_{m}\ll1$).
In the following we approximate, for simplicity, the membranes by oblate spheroids. The demagnetization factor of the flat membranes in the plane direction can be related with the membrane aspect ratio, which itself is set by the volume fraction $L_{m}\approx f_{V}/4$ (for $f_{V}\ll1$). After a straightforward expansion with respect to small $L_{m}$ one obtains$$\mathcal{\bar{F}}_{self}^{tot}(V)\approx-(Const-\frac{1}{2}f_{V}\chi^{2})V_{m}^{tot}\frac{B_{0}^{2}}{2\mu_{0}}, \label{p3}$$ where $Const$ is independent of the volume fraction $f_{V}$ .
The *total interaction energy* of membranes $\mathcal{\bar{F}}_{int}^{tot}$ is on the other hand$$\mathcal{\bar{F}}_{int}^{tot}(V)=\frac{1}{2}\sum_{i,j}\mathcal{\bar{F}}_{int}(i,j), \label{p4}$$ where the pair-interaction energy $\mathcal{\bar{F}}_{int}(i,j)$ is given by Eq.(\[me-me\]) where the summation goes over all membranes in the container. Note, that for the nearest neighbor membranes with $\left\vert \mathbf{R}_{12}\right\vert \approx a_{M}$ the far field approximation Eq.(\[me-me\]) holds qualitatively only, while for the next-nearest neighbors it holds already quantitatively. For $\chi\gg1$ and $L_{m}\chi\ll1$ Eq.(\[p4\]) gives $$\mathcal{\bar{F}}_{int}^{tot}(V)\approx(\frac{1}{8\pi}f_{V}\chi^{2}S)V_{m}^{tot}\frac{B_{0}^{2}}{2\mu_{0}}, \label{p5}$$ where the explicit expression for the sum $S(\equiv a_{M}^{3}\mathcal{\bar{F}}_{int}/\alpha)\approx10$ - a numeric dimensionless constant - is calculated by explicitly summing over all the pairwise membrane-membrane interactions (given by Eq.(\[me-me\])) in the cubic shelf lattice , for details cf. $Appendix$ $4C$.
Finally, combining both contributions to the free energy (self-energy and total interactions), the total pressure of the foam in a container with the volume $V$ is given by $$p=-\frac{\partial\mathcal{\bar{F}}^{tot}}{\partial V}\approx(\frac{1}{2}+\frac{S}{8\pi})\chi^{2}f_{V}^{2}(\frac{B_{0}^{2}}{2\mu_{0}}). \label{p7}$$ As a result the foam’s pressure is given by the approximate expression $$p\approx\frac{1}{2}\mu_{0}\chi^{2}f_{V}^{2}H_{0}^{2}. \label{pressure-1}$$ Interestingly, this pressure can assume notable magnitudes in practice. For moderate volume fractions, reasonable fields and susceptibilities ($f_{V}\approx5\cdot10^{-2},$ $\mu_{0}H_{0}\approx20mT,$ and $\chi\approx10$ in densely packed $Ni$-beads membranes) we obtain $p\approx40$ $Pa$. Since $p\propto H_{0}^{2}$, the pressure is very sensitive to the strength of the excitation (field) $H_{0}$ and can lead to strong swelling of the foam against gravity. The latter effect is also observed experimentally [@Jim-swelling] and can be used to practically test the equation of state Eq.(\[pressure-1\]).
Gravitational Pressure of the Foam
----------------------------------
Since a real foam is formed in the gravitation field, the gravity can limit its swelling. As we see from Eq.(\[pressure-1\]) the foam’s pressure $p$ is proportional to $f_{V}^{2}$ and in the gravitational field both are dependent on the vertical height position $h$ along the gravity direction. If one assumes that at $h=0$ the volume fraction takes the value $f_{V,0}$ and the pressure $p_{0}$ then (in case of constant $f_{V}$ and $p$) the foam would grow up to the maximal hight $h_{\max}^{0}=p_{0}/\Delta\rho gf_{V,0}$, where $g\simeq10m/s^{2},$ is the gravitational acceleration and $\Delta\rho
=\rho_{bead}-\rho_{s}$ is the difference in densities of magnetic beads and solvent. For instance, for water immersed $Ni$-beads as in Refs. [@Martin1; @Martin2] one has $\Delta\rho\simeq8\cdot10^{3}kg/m^{3}$. For $f_{V,0}\approx5\cdot10^{-2},$ $\mu_{0}H_{0}\approx20mT,$ and $\chi\approx10$ in densely packed $Ni$-beads membranes one obtains the pressure $p^{(1)}\approx40$ $Pa$ and the equilibrium foam height $h$ is reached once the internal and the gravitational pressure balance, i.e. $p\approx\Delta\rho
gf_{V,0}h_{\max}^{0}$ and the foam will swell strongly up to $h_{\max}^{0}\sim1$ $cm$.
The variation of pressure $p\left( h\right) $ and the volume fraction $f_{V}\left( h\right) $ with the hight in the gravitational field changes this approximative analysis slightly. In the gravitational field one has $$\frac{dp}{dh}=-\Delta\rho gf_{V}. \label{dp-dh}$$ By using the equation of state in Eq.(\[pressure-1\]) - with $f_{V}=C\cdot\sqrt{p}$, the solution of Eq.(\[dp-dh\]) reads $$\begin{aligned}
p\left( h\right) & =p_{0}(1-\frac{h}{2h_{\max}})\label{p1-f-h}\\
f_{V}\left( h\right) & =f_{V,0}(1-\frac{h}{2h_{\max}}).\nonumber\end{aligned}$$ The the maximal height is reached when $p=0$, i.e. when $h_{\max}^{(1)}=2h_{\max}^{0}$. For the above parameters one obtains $h_{\max}^{(1)}\sim2$ $cm$.
Therefore, the strong swelling behavior of magnetic foams can be used as a sensitive test of the theory.
Summary and Discussion
======================
We have studied the formation of hierarchical superstructures in systems driven by the spatially coherent Van der Waals (cVdW) interaction. We have developed a fairly general formalism involving the effective susceptibility tensor which allowed us to walk through all the important aspects of the cVdW interaction. Within this setting, in a bottom up approach we investigated numerous phenomena, from dimer formation, over 3 body forces, then collective elasticity of intermediate structures (chains and membranes) up to the presumably highest scale of pattern formation, i.e. to the cVdW foams.
In the theory we took a bird’s view approach, and we have shown that the cVdW interaction can be equivalently created in many types of excitation fields, generalizing the triaxial balanced fields used in the past. It turned out that the consideration of a general square isotropic uniform field (rather then any particular realization of it), brings the cVdW and its classical incoherent VdW “sister-”interaction onto a common footing. This parallel consideration of cVdW- and VdW-matter allowed us also to crystallize out the common behavior, but more importantly the central differences between the two types of forces behind them.
The most remarkable difference is found in the 3-body interactions. For the standard VdW matter the 3-body forces are recovered in the fully incoherent limit of our formalism and they agree with the classic result of Axilrod and Teller [@Teller]. These VdW 3-body forces are much weaker and shorter ranged than the corresponding 2-body forces, i.e. one could say they are *subdominant* and give only higher order corrections. In sharp contrast, in the cVdW-matter the 3-body forces derived here are as strong and often even stronger than the pairwise 2-body ones. Thus, the 3-body effects under the cVdW interaction can be considered as *essential* and *dominant* forces in the system. To our knowledge, this “many body dominance” makes the cVdW force stand out among other known interactions and gives it a very unique, interesting character. We have studied the physical origin of these dominant cVdW many body forces and we found them originating from the fact that the direct (induced) dipole-dipole interactions between isotropic objects vanish (are averaged out) and only the many- body mutual polarization effects survive the statistical averaging over the external excitation fields.
The pronounced *anisotropy* of the many-body interactions in the cVdW-matter systems also gives rise to a number of phenomena that are qualitatively different from standard VdW-matter, in particular the growth of anisotropic, low-dimensional assemblies - chains, then membranes once a critical size is reached. In a container of finite size, smaller membrane patches are formed, which tend to repel on the average, thus giving rise to dipolar foam structures. The foam exerts a positive pressure onto the walls of the sample container due to the tendency of membranes to increase their surface areas as well as their mutual repulsion. The dipolar foam represents a new and intriguing state of colloidal matter, formed by a delicate interplay of an attractive local interaction and a net repulsive longer range force. Remarkably, both types of forces are born out of a single, conceptually simple cVdW interaction - given by Eq.(\[S5\]).
The interactions driving the hierarchy of the assembly processes, from dimers to foams are summarized in Figs.10 and 11, where the 2-body and anisotropic 3-body interactions are responsible for the formation of chains, membranes and vesicles, while the membrane-membrane interaction is responsible for the formation of foams in a container with finite volume.
We have also argued that the finite size (finite particle number $N$) effects in cVdW-matter are very different from other common interactions like VdW or e.g. for classical magnetic beads with permanent moments. The many-body forces are also found to play a crucial role in the *anomalous elastic properties* of chains and membranes. For, instance the bending stiffness of a cVdW ring and the cVdW spherical membrane’s stiffness are proportional to $\ln
N$ and $\sqrt{N}$, respectively, which is a direct consequence of the specifically induced long-range many body effects in these systems.
The theory suggests a number of interesting and feasible *experiments* that can be performed to test the theoretical predictions about the interactions in the cVdW-matter:
$1$. It would be very interesting to experimentally probe the dynamics of exactly $3$ beads and the behavior of 3-body forces, cf. for instance the surprising attraction/repulsion effects in Fig. 5. The experiment can be performed for microscopic or macroscopic beads (the effect is scale independent). Since 3 beads always span a common plain, the most general dynamics can be observed directly in a single focal plain, e.g. on the microscopy glass-slide on which the beads naturally settle down by gravity.
$2$. The predicted negative effective surface tension and the instability of millimetric ferrofluid droplets, like in Fig. 6, would be a rather simple experimental test of the theory. Also the shape bistability, i.e. the coexistence of prolate and oblate shapes of the droplets, would be an interesting qualitative outcome to be tested.
$3$. Bending cVdW chains and flat membrane patches, either by active forces or passively by their own weight and observing their deflections should experimentally reveal the presence of the predicted anomalous, size dependent stiffness.
$4$. The most telling and fundamental experiment would be to directly probe the equation of state for a foam material. Measuring actively the forces on the container walls or passively observing the rising height of the foam against gravity would be two simple possibilities to test the predicted internal pressure equation for the cVdW foams.
Finally, the central theoretical and experimental question, in our opinion, remains if and how the cVdW can be generally realized in Nature. In particular, we might ask if it can be induced in a truly *equilibrium* system. The previous realizations, in field driven colloidal systems were all non-equilibrium. However there is no principal aspect of the theory that is specific and restricted to a non equilibrium system only. While the driving field amplitudes in our case are externally set, in an equilibrium system they would satisfy a fluctuation dissipation condition which would relate them to the temperature and the susceptibilities of the particles in the system. We might speculate that in some long-range correlated fluctuating media, like those considered in [@Kardar] (see also [@ReviewCasimir]) the cVdW can indeed be realized even in equilibrium. If the fluctuations of the medium are sufficiently longer ranged than the typical sizes of the formed structures, the assembly will be driven by the cVdW interactions instead of the VdW ones on these scales.
It is important to note that a simple tweak in the way how the interaction is induced (by switching from incoherent to coherent excitation), enormously increases the “morphogenic capacity” of the interaction i.e. its ability to form complex structures. If we are interested in the self-assembly of anything more complex than a spherical droplet (for which the standard VdW-matter is good enough), cVdW-matter would be a better candidate than the simple VdW one. The exploration and utilization of novel non-equilibrium (field-driven) or equilibrium realizations of cVdW interactions is an interesting future challenge. It could open the doors to deeper many-body studies of complex self-assembled materials, and more importantly to technological applications of the potentially very versatile and powerful cVdW-matter.
*Acknowledgements*
==================
We thank Jim Martin, A.Johner, H.Mohrbach, for discussions and comments.
Appendix 1
==========
Calculation of The cVdW Dimer Free-Energy
-----------------------------------------
In the two-particle problem the dipole operator $\hat{T}$ in Eq. (\[chi-eff\]) has only one non-vanishing component ($\hat{T}_{12}=\hat
{T}_{21}\neq0$) and can be written as: $$\hat{T}=\varphi_{12}\cdot\left(
\begin{array}
[c]{cc}0 & \hat{1}-3\hat{N}\\
\hat{1}-3\hat{N} & 0
\end{array}
\right)$$ where $\hat{N}=\left\vert \mathbf{b}_{12}\right\rangle \left\langle
\mathbf{b}_{12}\right\vert $ is the projector on the bond vector of the two particles and the scalar factor $\varphi_{12}=V_{b}/4\pi\left\vert
\mathbf{R}_{12}\right\vert ^{3}$ as introduced before. Using the relations $\hat{N}^{2}=\hat{N}$ , $(\hat{1}-a\hat{N})^{-1}=\hat{1}+a(1-a)^{-1}\hat{N}$ and the fact that $\hat{N}$ and $\hat{1}$ (or any scalar function like $\varphi_{12}$) commute the operator inversion in $\hat{\chi}_{eff}=\chi
_{b}(\hat{1}+\chi_{b}\hat{T})^{-1}$ is quickly evaluated $$\hat{\chi}_{eff}=\left(
\begin{array}
[c]{cc}A_{1}\hat{1}+A_{2}\hat{N} & B_{1}\hat{1}+B_{2}\hat{N}\\
B_{1}\hat{1}+B_{2}\hat{N} & A_{1}\hat{1}+A_{2}\hat{N}\end{array}
\right) \label{Chi2Bead}$$ with the bead distance dependent (scalar) coefficients$$\begin{aligned}
A_{1} & =\frac{\chi_{b}}{1-\chi_{b}^{2}\varphi_{12}^{2}}\text{ , }A_{2}=\frac{3\chi_{b}^{3}\varphi_{12}^{2}}{\left( 1-\chi_{b}^{2}\varphi
_{12}^{2}\right) \left( 1-4\chi_{b}^{2}\varphi_{12}^{2}\right) }\nonumber\\
B_{1} & =-\frac{\chi_{b}^{2}\varphi_{12}}{1-\varphi_{12}^{2}\chi_{b}^{2}}\text{ , }B_{2}=\frac{3\chi_{b}^{2}\varphi_{12}\left( 1-2\varphi_{12}^{2}\chi_{b}^{2}\right) }{\left( 1-\varphi_{12}^{2}\chi_{b}^{2}\right)
\left( 1-4\varphi_{12}^{2}\chi_{b}^{2}\right) }\nonumber\end{aligned}$$ The relevant quantity for the coherent and the incoherent VdW - the trace of $\hat{\chi}_{eff}$ over the 3 spacial directions- is directly obtained by taking into account that $Tr\hat{N}=1$ and $Tr\hat{1}=3$, which gives Eq.(\[TrChieff2Bead\]) in the main text.
Appendix 2
==========
The Free-Energy for The Three-Body Problem in cVdW Systems
----------------------------------------------------------
If we consider only $3$ beads the interaction energy is given by $\mathcal{\bar{F}}_{cVdW}^{(3)}=-\beta\sum\nolimits_{k=1,2,3}\sum
\nolimits_{i\neq k,j\neq k}w_{k,ij}$ $\ $with $w_{k,ij}=(3\cos^{2}\theta_{k,ij}-1)/\left\vert \mathbf{R}_{ki}\right\vert ^{3}\left\vert
\mathbf{R}_{kj}\right\vert ^{3}$. We put two beads $1$ and $2$ very close to each other at distance $R_{12}=d$ and the $3$-rd one at distance $R_{13}\approx R_{23}\approx R\gg d$. Altogether we have $3\times2\times2=12$ terms in the sum. The terms with flipped $i\rightarrow j$ indices are identical so we can reorder: $$\begin{aligned}
-\mathcal{\bar{F}}_{cVdW}^{(3)}/\beta & =\left( w_{1,22}+w_{1,33}+2w_{1,23}\right) \label{2b-1}\\
& +\left( w_{2,11}+w_{2,33}+2w_{2,13}\right) \nonumber\\
& +\left( w_{3,11}+w_{3,22}+2w_{3,21}\right) \nonumber\end{aligned}$$ Whenever an index repeats (e.g. as in $w_{3,11},w_{3,22}$ etc) we have a $2$-body force. Then the terms are symmetric and we have $w_{1,22}=w_{2,11}$ , $w_{2,33}=w_{3,22},$ $w_{1,33}=w_{3,11}:$$$\begin{aligned}
-\mathcal{\bar{F}}_{cVdW}^{(3)}/\beta & =\left( 2w_{1,22}+2w_{1,33}+2w_{2,33}\right) \label{2b-2}\\
& +\left( 2w_{2,13}+2w_{1,23}+2w_{3,21}\right) \nonumber\end{aligned}$$ As $R_{13}\approx R_{23}\approx R$ and $R_{12}=d\ll R$ we have $w_{2,13}\approx w_{1,23}$ (as $\theta_{2,13}\approx\pi-\theta_{1,23}$ and so $\cos
^{2}\theta_{2,13}=\allowbreak\cos^{2}\theta_{1,23}$) , one has $w_{1,33}\approx w_{2,33}$ so that$$-\mathcal{\bar{F}}_{cVdW}^{(3)}/\beta\approx\left( 2w_{1,22}+4w_{1,33}\right) +\left( 4w_{1,23}+2w_{3,21}\right) . \label{2b-3}$$ Further we have $w_{1,22}\approx\frac{1}{d^{6}}\left( 3-1\right) ,$ $w_{1,33}\approx w_{3,21}\approx\left( 3-1\right) /R^{6}$ and $w_{1,23}\approx\left( 3\cos^{2}\theta_{1,23}-1\right) /d^{3}R^{3}$. Keeping only the lowest power in $R$ it simplifies to $$-\frac{\mathcal{\bar{F}}_{cVdW}^{(3)}}{\beta}=\frac{4}{d^{6}}+\frac{4\left(
3\cos^{2}\theta_{1,23}-1\right) }{d^{3}R^{3}}+O\left( R^{-6}\right) .
\label{2b-4}$$
3-Body Free-Energy of Finite cVdW Chain
---------------------------------------
For the finite chain we consider the limit of the chain being still long enough that the two ends do not see each other (summations for each particle are infinite in one direction). Then we have $3\cos^{2}\theta_{k,ij}-1=2$ and we can split up the summation $$\mathcal{\bar{F}}_{cVdW,ch}^{(3)}=-\beta\sum\nolimits_{k=1}^{N}f_{k}=-2\beta\sum\nolimits_{k=1}^{N/2}f_{k} \label{2c-1}$$ with$$\begin{aligned}
f_{k} & =\sum\nolimits_{j=1,j\neq k}^{N}\sum\nolimits_{i=1,i\neq k}^{N}\frac{2}{\left\vert \mathbf{R}_{ik}\right\vert ^{3}\left\vert \mathbf{R}_{kj}\right\vert ^{3}}\label{2c-2}\\
& =2\left( \sum\nolimits_{i=1,i\neq k}^{N}\frac{1}{\left\vert \mathbf{R}_{ik}\right\vert ^{3}}\right) ^{2}\nonumber\end{aligned}$$ The upper sum can be subdivided in two parts, one left and one right of the particle $k$ with one of the sums approximated by an infinite boundary $N=\infty$$$\begin{aligned}
\sum\nolimits_{i=1,i\neq k}^{N}\frac{1}{\left\vert \mathbf{R}_{ik}\right\vert
^{3}} & \approx\frac{1}{D^{3}}\left( \sum\nolimits_{i=1}^{k-1}\frac
{1}{i^{3}}+\sum\nolimits_{l=1}^{\infty}\frac{1}{i^{3}}\right) \label{2c-3}\\
& =\frac{2\zeta\left( 3\right) -S_{k}}{D^{3}}\nonumber\end{aligned}$$ with $S_{k}=\sum\nolimits_{i=k}^{\infty}(1/i^{3})$. Therefore $f_{k}=2\frac
{1}{D^{6}}\left( 2\zeta\left( 3\right) -S_{k}\right) ^{2}$ and the free-energy per particle is given by $$\frac{\mathcal{\bar{F}}_{cVdW,ch}^{(3)}}{N\beta}\approx-\frac{2}{D^{6}}\frac{1}{N}\sum\nolimits_{k=1}^{N}\left( 2\zeta\left( 3\right)
-S_{k}\right) ^{2}. \label{2c-7}$$ For large $k$ we approximate $S_{k}$ by integral which gives $S_{k}\approx1/2k^{2}$ and this $$\frac{\mathcal{\bar{F}}_{cVdW,ch}^{(3)}}{N\beta}\approx-\frac{8\zeta
^{2}\left( 3\right) }{D^{6}}+\allowbreak\frac{3.4}{ND^{6}}. \label{A2c-Fch}$$ This first term is for the infinite chain, while the second is the leading order correction as expected $O\left( 1/N\right) $ .
The 3-Body Free-Energy of Ring in cVdW Systems
----------------------------------------------
Here $\mathbf{R}_{i0}=R\left( \cos(2\pi i/N),\sin(2\pi i/N)\right) $ (note $\mathbf{R}_{ij}=(R_{x,ij},R_{y,ij})$ , $\cos\theta_{k,ij}=\mathbf{R}_{ik}\cdot\mathbf{R}_{kj}/\left\vert \mathbf{R}_{ik}\right\vert \left\vert
\mathbf{R}_{kj}\right\vert $. All terms $k=0,1,..N-1$ give the same contribution as the term $k=0$ due to symmetry. We can introduce the angle $\phi_{1}=2\pi i/N$ and $\phi_{2}=2\pi j/N$ with $d\phi\approx2\pi/N$ (for $N\rightarrow\infty)$ . Then $\mathbf{R}_{j0}\approx\mathbf{R}\left(
\phi\right) $ $$\begin{aligned}
\left\vert \mathbf{R}\left( \phi\right) \right\vert ^{3} & =R^{3}\left(
2\left( 1-\cos\phi\right) \right) ^{\frac{3}{2}}\label{2d-1}\\
\text{ }\cos^{2}\theta\left( \phi_{1},\phi_{2}\right) & =\left(
\frac{a_{1}a_{2}+b_{1}b_{2}}{2\sqrt{a_{1}a_{2}}}\right) \nonumber\end{aligned}$$ where $a_{1,2}=(1-\cos\phi_{1,2})$ and $b_{1,2}=\sin\phi_{1,2}$. It also holds $\cos^{2}\theta\left( \phi_{1},\phi_{2}\right) =\cos^{2}\left( \frac
{\phi_{1}-\phi_{2}}{2}\right) $. With a small distance cutoff angle $c=2\pi/N$. Then we can write the free-energy:$$\begin{aligned}
\frac{\mathcal{\bar{F}}_{cVdW,ring}^{(3)}}{\beta N} & =-\sum\nolimits_{j=1}^{N-1}\sum\nolimits_{i=1}^{N-1}\frac{3\cos^{2}\theta_{0,ij}-1}{\left\vert
\mathbf{R}_{i0}\right\vert ^{3}\left\vert \mathbf{R}_{0j}\right\vert ^{3}}\label{2d-2}\\
& =-\frac{1}{2R^{6}}\sum\nolimits_{j=1}^{N-1}\sum\nolimits_{i=1}^{N-1}\frac{1+3\cos\left( \phi_{1}-\phi_{2}\right) }{\left( 2a_{1}\right)
^{\frac{3}{2}}\left( 2a_{2}\right) ^{\frac{3}{2}}}.\nonumber\end{aligned}$$ The latter gives most contribution for $\phi_{1/2}$ small and can be expanded around $\phi_{1/2}=0$. Note that we have $2$ identical contributions around $i,j=1$ and around $N-1$ in each of the terms. Summing these expanded terms (with $R=\frac{ND}{2\pi},$ $\phi=\frac{2\pi}{N}k$ ) we arrive at: $$\frac{\mathcal{\bar{F}}_{cVdW,ring}^{(3)}}{N\beta}\approx-\frac{8\zeta
^{2}\left( 3\right) }{D^{6}}+\frac{16\zeta\left( 3\right) \pi^{2}}{D^{6}}\frac{\ln N}{N^{2}} \label{2d-3}$$
3-Body Free-Energy of The cVdW Spherical Shell
----------------------------------------------
For a spherical membrane (coloidosome) with classical elasticity or radius $R$, we would have an energy density proportional to $1/R^{2}$ (=curvature$^{2}$). The total energy coming from the bending (i.e. without self energy of beads) is then $\sim(1/R^{2})R^{2}\sim1$ constant. What happens in the case of a coherent coloidosome? Starting again from:$$\frac{\mathcal{\bar{F}}_{cVdW,sph}^{(3),N}}{\beta}=-\sum\nolimits_{i,j,k}^{\prime}\frac{3\cos^{2}\theta_{k,ij}-1}{\left\vert \mathbf{R}_{ik}\right\vert
^{3}\left\vert \mathbf{R}_{kj}\right\vert ^{3}} \label{2e-1}$$ with fixed arbitrary $\mathbf{R}_{k}=R\left( 0,0,1\right) $ (north pole of the sphere, $\mathbf{R}=\left( R_{x},R_{y},R_{z}\right) $) and $\mathbf{R}_{kj}=R\left( -\sin\theta_{j}\cos\phi_{j},-\sin\theta_{j}\sin
\phi_{j},1-\cos\theta_{j}\right) $, $|\mathbf{R}_{kj}|=R\sqrt{2\left(
1-\cos\theta_{j}\right) }=2R\sin(\theta/2)$ and the apex angle is given by $\cos\alpha=\mathbf{R}_{ik}\cdot\mathbf{R}_{kj}/\left\vert \mathbf{R}_{ik}\right\vert \left\vert \mathbf{R}_{kj}\right\vert $, where $\alpha
\equiv\alpha\left( \theta_{1},\theta_{2},\phi_{1},\phi_{2}\right) $. In order to pick up the $N$-effects we replace the summation by integration over two spheres where each of the two spheres contains $N=4\pi R^{2}/\rho_{sph}^{-1}\pi\left( D/2\right) ^{2}=\allowbreak16\rho_{sph}R^{2}/D^{2}$ beads giving the sphere radius: $(\sqrt{N}D/4\rho_{sph}^{1/2})=\allowbreak R.$ The summation can be replaced by double integral over the sphere with surface element $dA_{1,2}=R^{2}\sin\theta d\phi d\theta,$ and the energy can be written as: $$\frac{\mathcal{\bar{F}}_{cVdW,sph}^{(3)}}{N\cdot\Gamma}=-\int\left(
\frac{3\cos^{2}\alpha-1}{\left\vert \mathbf{R}\left( \theta_{1}\right)
\right\vert ^{3}\left\vert \mathbf{R}\left( \theta_{2}\right) \right\vert
^{3}}\right) dA_{1}dA_{2} \label{A2e-F-dA}$$ with $\Gamma=\beta N^{2}/\left( 4\pi R^{2}\right) ^{2}$ and the latter is calculated by an integration (in Mathematica) over $\phi_{1}$ and $\phi_{2}$ $$\frac{\mathcal{\bar{F}}_{cVdW,sph}^{(3)}}{N\beta}=-\frac{\rho_{pack}^{2}}{8R^{2}D^{4}}\left( \int_{\theta_{\min}\left( D\right) }^{\pi}\frac
{3\cos\theta-1}{\sin^{3}\left( \frac{\theta}{2}\right) }\sin\theta
d\theta\right) ^{2} \label{A2e-Fthe}$$ with $\theta_{\min}\left( D\right) $ being the angular cut-off resulting from the spherical angle relation (surface area of bead / surface area of whole sphere) : $\Omega\left( D\right) =\int_{0}^{2\pi}\int_{0}^{\theta_{\min}\left( D\right) }\sin\theta d\theta d\phi\approx
2\pi(1/2)\theta_{\min}^{2}$ but on the other hand $\Omega\left( D\right)
=\pi\left( D/2\right) ^{2}/4\pi R^{2}=\rho_{sph}/N$ so that $\theta_{\min
}=\sqrt{\rho_{sph}/\pi N}.$ The integral can be done which gives the energy per particle of the spherical shell: $$\frac{\mathcal{\bar{F}}_{cVdW,sph}^{(3)}}{N\beta}\approx-\frac{50\pi\rho
_{sph}^{2}}{D^{6}}\allowbreak\left( 1-\frac{4\sqrt{\rho_{sph}}}{\sqrt{\pi
}\sqrt{N}}\right)$$
The Microscopic Energy of the cVdW Tubular Membrane
---------------------------------------------------
For an infinite cylinder of radius $R_{\perp}$ again we have a high symmetry (all beads are the same) and we can choose the apex point anywhere, say at $\left( x,y,z\right) =\left( 1,0,0\right) .$ The other points along the cylinder we parameterize cylindrically with coordinates $\left(
\phi,z\right) $ , so that the difference vector becomes $\mathbf{R}_{kj}=\left( R_{\perp}\left( \cos\phi_{j}-1\right) ,R_{\perp}\sin\phi
_{j},z\right) $ and its length $R_{kj}=\sqrt{\allowbreak z_{j}^{2}+2R_{\perp
}^{2}\left( 1-\cos\phi_{j}\right) }.$ The apex angle is $\cos\alpha
_{k,ij}=\mathbf{R}_{ki}\cdot\mathbf{R}_{kj}/R_{1}R_{2}$ with $R_{1/2}=\sqrt{\allowbreak z_{1/2}^{2}+2R_{\perp}^{2}a_{1/2}}$.
The energy density consists of seven terms :$$\frac{\mathcal{\bar{F}}_{cVdW,tub}^{(3)}}{-\beta N}=\sum\nolimits_{i,j}^{\prime}\frac{3\cos^{2}\theta_{k,ij}-1}{\left\vert \mathbf{R}_{ik}\right\vert
^{3}\left\vert \mathbf{R}_{kj}\right\vert ^{3}} \label{2e-Ft}$$$$\begin{aligned}
\frac{\mathcal{\bar{F}}_{cVdW,tub}^{(3)}}{-\beta N} & =\frac{3\left(
C_{1}^{2}+C_{2}^{2}+C_{3}^{2}+2C_{4}^{2}\right) }{R_{\perp}^{6}}\text{
\ }\label{A2f-F-C}\\
& +\frac{3(2C_{5}^{2}+2C_{6}^{2})-C_{7}^{2}}{R_{\perp}^{6}}\nonumber\end{aligned}$$ Some of the sums are trivially zero due to symmetry: $C_{4}=0$ ($R\left(
\phi,z\right) $ even but $\left( \cos\phi-1\right) \sin\phi$ odd in$\phi$) $\ C_{5}=0$ (odd in $z$ ) , $C_{6}=0$ (odd in $z$ and $\phi$ ). Those which are different from zero are $$C_{1}=\sum_{k=\pm1...,\pm\frac{1}{\delta}}\sum_{l=0,\pm1,..,\pm\infty}\frac{\left( \cos\left( \delta k\right) -1\right) ^{2}}{P^{5}(k,l)}
\label{C1}$$$$C_{2}=\sum_{k=\pm1...,\pm\frac{1}{\delta}}\sum_{l=0,\pm1,..,\pm\infty}\frac{\sin^{2}\left( \delta k\right) }{P^{5}(k,l)}, \label{C2}$$$$C_{3}=\sum_{k=\pm1..,\pm\frac{1}{\delta}}\sum_{l=0,\pm1,..,\pm\infty}\frac{\left( \delta l\right) ^{2}}{P^{5}(k,l)} \label{C3-4}$$$$C_{7}=\sum_{k=\pm1..,\pm\frac{1}{\delta}}\sum_{l=0,\pm1,..,\pm\infty}\frac
{1}{P^{3}(k,l)} \label{C5-7}$$ where $P(k,l)=\left( l\allowbreak^{2}\delta^{2}+2\left( 1-\cos
k\delta\right) \right) ^{1/2}$. To obtain the scaling we introduced small scale cutoffs $\delta=D/R_{\perp}$. Parameterizing the angle $\phi=\delta\cdot
k$ with $k=\pm1,\pm2,\pm3,...\pm\frac{1}{\delta}$ and the $z$ displacement as $z=\delta\cdot l$ (with $l=0,\pm1,\pm2,...\pm\infty$) one can approximate the summation over $l$ by integration. As the final result this gives for $C_{1}\approx(2/3\delta^{2})$, $(C_{2}/2)\approx C_{3}$, $\approx
(4\zeta\left( 2\right) /3\delta^{3})\approx C_{7}$. When inserting $\delta=\frac{D}{R_{\perp}}$ we get then the final result for the cylinder free-energy ($N=N_{1}N_{2},R_{\perp}\sim N_{1}$ - see Fig.11): $$\frac{\mathcal{\bar{F}}_{cVdW,tub}^{(3)}}{\beta N}\approx-\frac{8\pi^{4}}{27}\frac{1}{D^{6}}+\frac{40\pi^{2}}{27}\frac{1}{R_{\perp}D^{5}}.
\label{F-tub}$$$$\frac{\mathcal{\bar{F}}_{cVdW,sph}^{(3)}}{N\beta}\approx-\frac{50\pi\rho
_{sph}^{2}}{D^{6}}\allowbreak\left( 1-\frac{4\sqrt{\rho_{sph}}}{\sqrt{\pi
}\sqrt{N}}\right) \label{A2e-Fsph-N}$$
Appendix 3
==========
Demagnetization Tensors of Spheroids and Cylinders
--------------------------------------------------
In calculating magnetic fields of magnetized bodies and the corresponding magnetostatic energy two kind of demagnetization tensors appear [@1.Beleggia; @4.Beleggia; @2.Beleggia]. The first one is related to the demagnetizing field of the uniformly magnetized body $\mathbf{H}_{D}(\mathbf{r})=-\hat{L}(r)\mathbf{M}(\mathbf{r})$ where $\mathbf{M}(\mathbf{r})=\mathbf{M}D(\mathbf{r})$. Here $\mathbf{M}=\mathbf{const}$ and $D(\mathbf{r})$ is the dimensionless shape function which represents the region of the space bounded by the body (sample) surface, i.e. $D(\mathbf{r})=1$ inside the body and $D(\mathbf{r})=0$ outside it. Its Fourier transform $D(\mathbf{k})$ - the shape amplitude, which is related to $\hat{L}(r)$ [@1.Beleggia; @4.Beleggia; @2.Beleggia] $$\begin{aligned}
\hat{L}(\mathbf{r}) & =\frac{1}{(2\pi)^{3}}\int d^{3}k\hat{L}(\mathbf{k})e^{i\mathbf{k\cdot r}}\label{LrLk}\\
\hat{L}(\mathbf{k}) & =\frac{D(\mathbf{k})}{k^{2}}\left\vert \mathbf{k}\right\rangle \left\langle \mathbf{k}\right\vert .\nonumber\end{aligned}$$ Note, that $\left( \hat{L}(\mathbf{k})\right) _{ij}=D(\mathbf{k})k_{i}k_{j}/k^{2}$. The shape amplitude $D(\mathbf{k})$ and $\hat{L}(r)$ are calculated for various bodies. For instance, for sphere of radius $R$ one has $D(\mathbf{k})=(4\pi R^{2}/k)j_{1}(kR)$ where the spherical Bessel function of first order $j_{1}(x)=(\sin x/x^{2})-\cos x/x$. For other body-shapes see more in [@1.Beleggia; @4.Beleggia; @2.Beleggia] and references therein.
The second type of demagnetization tensor(factors) appears in the expression for the magnetostatic (demagnetization) energy with the uniform magnetization $\mathbf{M}(\mathbf{r})=\mathbf{M}$ $$\begin{aligned}
E_{D} & =-\frac{\mu_{0}}{2}\int_{V_{D}}d^{3}r\mathbf{M}(\mathbf{r})\mathbf{H}_{D}(\mathbf{r})\label{A3-Ed}\\
& =\frac{\mu_{0}}{2}V_{D}\mathbf{M}\left\langle \hat{L}(r)\right\rangle
\mathbf{M,}\nonumber\end{aligned}$$ where $V_{D}$ is the volume of the body and $\left\langle \hat{L}(r)\right\rangle $ is the *magnetometric* (volume averaged) *demagnetization tensor*, i.e. $$\begin{aligned}
\left\langle \hat{L}(r)\right\rangle & =\frac{1}{V_{D}}\int_{V_{D}}d^{3}r\hat{L}(\mathbf{r})\label{A3-L-av}\\
& =\frac{1}{(2\pi)^{3}V_{D}}\int d^{3}k\frac{D^{2}(\mathbf{k})}{k^{2}}\left\vert \mathbf{k}\right\rangle \left\langle \mathbf{k}\right\vert
\nonumber\end{aligned}$$ It is easy to show that the trace of $\hat{L}(\mathbf{r})$ inside the body is one, while outside is zero, i.e. $Tr\hat{L}(\mathbf{r})=D(\mathbf{r})$. The magnetometric tensor fulfills $Tr\left\langle \hat{L}(r)\right\rangle =1$. Note, that in Eq.(\[F-macro1\]) enter the diagonal components of the magnetometric demagnetization tensor $\left\langle \hat{L}(r)\right\rangle $. Since we study magnetic bodies where $\hat{L}(r)=const=\hat{L}$ inside the body, then $\left\langle \hat{L}(r)\right\rangle =\hat{L}$. Wi give the exact and asymptotic expressions for the $L_{z}$ demagnetization factor for ellipsoids and cylinders which are important for the studies in the main text.
\(i) *Demagnetization factors for ellipsoids* - If $a$, $b$, $c$ are the semi-axis of ellipsoid with $\tau_{a}=(c/a)$, $\tau_{b}=(c/b)$, $k=\arcsin
\sqrt{1-\tau_{a}^{-2}}$, $m=(1-\tau_{b}^{-2})/(1-\tau_{a}^{-2})$ than one has [@1.Beleggia; @4.Beleggia; @2.Beleggia]$$L_{z}(\tau_{a},\tau_{b})=\frac{1}{\tau_{a}\tau_{b}}\frac{F(k,m)-E(k,m)}{m\sin^{3}k}, \label{A3-Lz-ell}$$ where $E(k,m)$ and $F(k,m)$ are incomplete elliptic integrals [@Abramowitz]. The symmetry implies $L_{x}(\tau_{a},\tau_{b})=L_{z}(\tau_{a}^{-1},\tau_{b}\tau_{a}^{-1})$ and $L_{y}(\tau_{a},\tau_{b})=L_{z}(\tau_{a}\tau_{b}^{-1},\tau_{b}^{-1})$ - see [@1.Beleggia; @4.Beleggia; @2.Beleggia]. For oblate and prolate spheroids, where $\tau
_{a}=\tau_{b}=\tau_{s}$, one has $$L_{z}(\tau_{s})=\frac{1}{1-\tau_{s}^{2}}\left[ 1-\frac{\tau_{s}\arccos
(\tau_{s})}{\sqrt{1-\tau_{s}^{2}}}\right] \label{A3-Lz-sp}$$ For $\tau_{e}\rightarrow0$ (extreme oblate, i.e. membrane-like spheroid) it gives $$L_{z}(\tau_{s})=1-\frac{\pi}{2}\tau_{s}+2\tau_{s}^{2}+O(\tau_{s}^{3}),
\label{A3-Lz-sp-0}$$ and for $\tau_{e}\rightarrow\infty$ (extreme prolate, i.e. chain-like spheroid) one has $$L_{z}(\tau_{s})=\frac{\ln(2\tau_{s}/e)}{\tau_{s}^{2}}+O(\tau_{s}^{-4}).
\label{chain-prol}$$
\(ii) *Demagnetization factors for cylinders* - For cylinders with thickness (height) $t$ and radius $R$ with the aspect ratio $\tau=t/2R$ and $\kappa=1/\sqrt{1+\tau^{2}}$ one has [@1.Beleggia; @4.Beleggia; @2.Beleggia]$$L_{z}^{cyl}(\tau)=1+\frac{4}{3\pi\tau}\left\{ 1-\frac{1}{\kappa}\left[
(1-\tau^{2})E(\kappa^{2})+\tau^{2}K(\kappa^{2})\right] \right\}
\label{A3-Lz-cyl}$$ where $E(\kappa^{2})$ and $K(\kappa^{2})$ are complete elliptic functions [@Abramowitz].
For *very thin cylinder* where $\tau\rightarrow0$ one has$$L_{z}^{cyl}(\tau)=1+\frac{\tau}{\pi}\left( 1+2\ln\frac{\tau}{4}\right)
+O(\tau^{3}). \label{A3-Lz-cyl-t}$$ For *very long cylinder* where $\tau\rightarrow\infty$ one has $$L_{z}^{cyl}(\tau)=\frac{4}{3\pi\tau}-\frac{8}{\tau^{2}}+O(\tau^{-4}).
\label{A3-Lz-cyl-l}$$
Surface and Demagnetization Factors For Deformed Sphere
-------------------------------------------------------
The aspect ratio of the spheroid is $\tau_{s}=c/a$. Close to the sphere one has $\varepsilon=\tau_{s}-1\ll1$, where $\varepsilon<0$ is for oblate ellipsoid while $\varepsilon>0$ is for the prolate one. By using Eq.(\[A3-Lz-sp\]) one has for small $\left\vert \varepsilon\right\vert \ll1$ one has $$\begin{aligned}
L_{z}^{prol/obl} & =\frac{1}{3}-\frac{4}{15}\varepsilon+\frac{6}{35}\varepsilon^{2}\label{Lz}\\
& -\frac{32}{315}\varepsilon^{3}+\frac{40}{693}\varepsilon^{4}+O\left(
\varepsilon^{5}\right) \nonumber\end{aligned}$$ Similarly for the surface of the prolate spheroid one has $$A_{prolate}=2\pi a^{2}\left( 1+\frac{\tau_{s}}{x}\sin^{-1}x\right) \text{,}
\label{Apr}$$ with $\ x^{2}=1-1/\tau_{s}^{2}$ for $\tau_{s}>1$ and for the oblate one $$A_{oblate}=2\pi a^{2}\left( 1+\frac{1-x^{2}}{x}\tanh^{-1}x\right) ,
\label{Aob}$$ for $x^{2}=1-\tau_{s}^{2}$ for $\tau_{s}<1$. In the following analysis the volume is fixed, i.e. $V=4\pi a^{2}c/3=4\pi a^{3}\tau_{s}/3$ , $a^{2}=\left(
3V/4\pi\tau_{s}\right) ^{2/3}$ and in terms of the axial stretching one has$$\frac{A_{prolate}}{A_{0}}=\frac{1+\frac{\left( \varepsilon+1\right) ^{2}}{\sqrt{\varepsilon\left( \varepsilon+2\right) }}\arcsin\sqrt{1-\frac
{1}{\left( \varepsilon+1\right) ^{2}}}}{2\left( \varepsilon+1\right)
^{2/3}} \label{Apr-eps}$$ for $\tau_{s}>1$ and $$\text{\ }\frac{A_{oblate}}{A_{0}}=\frac{1+\frac{\left( \varepsilon+1\right)
^{2}}{\sqrt{-\varepsilon\left( \varepsilon+2\right) }}\operatorname{arctanh}\sqrt{-\varepsilon\left( \varepsilon+2\right) }}{2\left( \varepsilon
+1\right) ^{2/3}} \label{Aobl-eps}$$ for $\tau_{s}<1$,with $A_{0}=4\pi(3V/4\pi)^{2/3}$ the initial area of the sphere. We can expand the surface area $A_{prol/obl}\left( \varepsilon
\right) $ (expansions coincide):$$\begin{aligned}
\frac{A_{prol/obl}\left( \varepsilon\right) \allowbreak}{A_{0}} &
=1+\frac{8}{45}\varepsilon^{2}-\frac{584}{2835}\varepsilon^{3}\label{Apr-ob}\\
& +\frac{118}{567}\varepsilon^{4}+O\left( \varepsilon^{5}\right) \nonumber\end{aligned}$$
Appendix 4
==========
Membrane-Bead Interaction in cVdW Systems
-----------------------------------------
In order to calculate the free-energy in Eq.(\[Wmb\]) we need to know $Tr\hat{\chi}_{m,eff}$ and $Tr\hat{\chi}_{b,eff}$ where $$\hat{\chi}_{m,eff}=(1-q_{m}q_{b}\varphi_{0}^{2}\hat{\chi}_{m}\hat{t}\hat{\chi
}_{b}\hat{t})^{-1}\hat{\chi}_{m}(1-q_{m}\varphi_{0}\hat{t}\hat{\chi}_{b}),
\label{A4-chi-m}$$ with $\hat{t}_{m-b}\equiv\hat{t}$ and $\hat{\chi}_{b,eff}$ is obtained by replacing $m\leftrightarrow b$. For the assumed symmetry and geometry of the problem we have $\left\vert \mathbf{n}_{m}\right\rangle =\left\langle
\mathbf{b}\right\vert $. Here, $q_{m,b}=V_{m,b}/V_{0}$, $V_{0}=V_{m}+V_{b}$, and $\hat{\chi}_{m,b}$ is the membrane and bead susceptibility (with respect to external field), respectively, $\varphi_{0}=V_{0}/4\pi R_{mb}^{3}$. By defining $\hat{N}=\left\vert \mathbf{n}_{m}\right\rangle \left\langle
\mathbf{n}_{m}\right\vert $ and $\hat{P}=1-\hat{N}$ we have $\hat{N}^{2}=\hat{N}$, $\hat{P}^{2}=\hat{P}$, $\hat{N}\hat{P}=0$, $Tr\hat{N}=1$ and $Tr\hat{P}=2$ and $$\hat{t}=-2\hat{N}+\hat{P}. \label{A4-t}$$ Further we parameterize $\hat{\chi}_{m}=\chi_{\max}\hat{\chi}_{m}^{0}$, $\hat{\chi}_{m}^{0}=p_{m}\hat{N}+\hat{P}$, $p_{m}=\chi_{\min}/\chi_{\max}$, $\hat{\chi}_{b}=\chi_{b}\hat{1}$, $b=\alpha_{b}\alpha_{m}$, $\alpha_{m}=q_{m}\chi_{\max}\varphi_{0}$, $\alpha_{b}=q_{b}\chi_{b}\varphi_{0}$. In this parametrization we have $$\begin{aligned}
\hat{\chi}_{m,eff} & =\chi_{\max}\hat{A}^{-1}\hat{\chi}_{m}^{0}(1-\alpha
_{b}\hat{t})\label{A4-chi-mb}\\
\hat{\chi}_{b,eff} & =\chi_{b}\hat{A}^{-1}(1-\alpha_{m}\hat{\chi}_{m}^{0}\hat{t})\nonumber\end{aligned}$$ and $$\hat{A}=1-b\hat{\chi}_{m}^{0}\hat{t}^{2}. \label{A4-A}$$ By using the projecting properties of $\hat{N}$ and $\hat{P}$ one obtains the inverse matrix $\hat{A}^{-1}$ $$\hat{A}^{-1}=\frac{1}{1-b}\left[ 1-\frac{c}{1+c}\hat{N}\right]
\label{A4-A-1}$$ with $c=b(1-4p_{m})/(1-b)$. After some algebra one obtains the effective membrane susceptibility$$\hat{\chi}_{m,eff}=\frac{\chi_{\max}}{1-b}\left\{ a\hat{1}+\left[
\frac{p_{m}(1+2\alpha_{b})}{1+c}-a\right] \hat{N}\right\}
\label{A4-chi-meff}$$ where $a=(1-\alpha_{b})$. Then the trace is$$Tr\hat{\chi}_{m,eff}=\frac{\chi_{\max}}{1-b}\left\{ 2(1-\alpha_{b})+\frac{p_{m}(1+2\alpha_{b})}{1+c}\right\} . \label{A4-Tr-chi-m}$$ Analogously one obtains the effective bead susceptibility $Tr\hat{\chi
}_{b,eff}$$$Tr\hat{\chi}_{b,eff}=\frac{\chi_{b}}{1-b}\left\{ \frac{3+2c}{1+c}+2\alpha
_{m}(\frac{p_{m}}{1+c}-1)\right\} . \label{A4-Tr-chi-b}$$ By inserting Eq.(\[A4-Tr-chi-m\]) and Eq.(\[A4-Tr-chi-b\]) into Eq.(\[Wmb\]) one obtains $w_{m}$ and $w_{b\text{ }}$in Eq.(\[wm\]) and Eq.(\[wb\]), respectively.
Derivation of the 2-Membrane Interaction in cVdW Systems
--------------------------------------------------------
Having in mind two identical membranes (with volume $V_{m}$) we consider susceptibilities $\hat{\chi}_{1}$ and $\hat{\chi}_{2}$ of two oblate spheroids, which are differently oriented in space. In terms of their own local coordinate systems (in Dirac bra-ket notation for tensors) they are given by$$\begin{aligned}
\hat{\chi}_{1} & =\chi_{\min}\left\vert \mathbf{n}_{1}\right\rangle
\left\langle \mathbf{n}_{1}\right\vert +\chi_{\max}(\hat{1}-\left\vert
\mathbf{n}_{1}\right\rangle \left\langle \mathbf{n}_{1}\right\vert
)\label{M4}\\
\hat{\chi}_{2} & =\chi_{\min}\left\vert \mathbf{n}_{2}\right\rangle
\left\langle \mathbf{n}_{2}\right\vert +\chi_{\max}(\hat{1}-\left\vert
\mathbf{n}_{2}\right\rangle \left\langle \mathbf{n}_{2}\right\vert ),\nonumber\end{aligned}$$ where the unit vectors $\left\vert \mathbf{n}_{1}\right\rangle ,$ $\left\vert
\mathbf{n}_{2}\right\rangle $ are the normals of the membranes $1$ and $2$ respectively. By using Eq.(\[M4\]), and noting that $Tr\{\hat{\chi}_{1}\hat{\chi}_{2}\}=$ $Tr\{\hat{\chi}_{2}\hat{\chi}_{1}\}$, $Tr\{\hat{\chi}_{1}\hat{\chi}_{2}(\left\vert \mathbf{b}_{21}\right\rangle \left\langle
\mathbf{b}_{21}\right\vert )\}=Tr\{\hat{\chi}_{2}\hat{\chi}_{1}(\left\vert
\mathbf{b}_{12}\right\rangle \left\langle \mathbf{b}_{12}\right\vert )\}$ and $Tr\{\left\vert \mathbf{n}_{i}\right\rangle \left\langle \mathbf{n}_{j}\right\vert \}=\mathbf{\langle n}_{i}\left\vert \mathbf{n}_{j}\right\rangle $ (with $\mathbf{a}\cdot\mathbf{b\equiv\langle a}\left\vert
\mathbf{b}\right\rangle $ the scalar product) it follows $$Tr\{\hat{\chi}_{1}\hat{\chi}_{2}\}=\chi_{\max}^{2}\left[ 1+2\gamma+c_{3}^{2}(1-\gamma)^{2}\right] \label{M5}$$$$\begin{aligned}
\frac{Tr\{\hat{\chi}_{1}\hat{\chi}_{2}(\left\vert \mathbf{b}_{12}\right\rangle
\left\langle \mathbf{b}_{12}\right\vert )\}}{\chi_{\max}^{2}} &
=[1-(1-\gamma)(c_{1}^{2}+c_{2}^{2})\label{M6}\\
& +(1-\gamma)^{2}c_{1}c_{2}c_{3}],\nonumber\end{aligned}$$ where $\gamma=(\chi_{\min}/\chi_{\max})$ and $c_{1}=\mathbf{n}_{1}\cdot\mathbf{b}_{12}$, $c_{2}=\mathbf{n}_{2}\cdot\mathbf{b}_{12}$, $c_{3}=\mathbf{n}_{1}\cdot\mathbf{n}_{2}$ are factors describing the mutual orientation of membranes. By replacing Eqs.(\[M5\]-\[M6\]) in Eq.(\[M3\]) (where $V_{b}$ in $\varphi_{12}$ is replaced by the membrane volume $V_{m}$) one obtains Eq.(\[me-me\]) in the manuscript.
The Interaction Energy of The Cubic Shelf Structure
---------------------------------------------------
The lattice sum $S(\equiv a_{M}^{3}\mathcal{\bar{F}}_{int}/\alpha)$, $\alpha=9\eta\chi_{\max}^{2}\mu_{0}H_{0}^{2}V_{m}^{2}/16\pi$ (where $\mathbf{r}_{i1}=\mathbf{R}_{i1}/a_{M}$ and $\eta=(1-\chi_{\min}/\chi_{\max
})/3$) for the cubic shelf structure is given by $$S=\sum_{\mathbf{l},\kappa=x,y,z}\frac{\eta C_{3,\kappa}^{2}+C_{1,\kappa}^{2}+C_{2,\kappa}^{2}-3\eta C_{1,\kappa}C_{2,\kappa}C_{3,\kappa}-\frac{2}{3}}{r_{\mathbf{l},\kappa}^{3}}, \label{A4c-S}$$ where for compactness we define $C_{1/2/3,\kappa}=C_{1/2/3,\mathbf{l,}\kappa}$. The coefficients $C_{1}=\mathbf{n}_{1}\cdot\mathbf{b}_{12}$, $C_{2}=\mathbf{n}_{2}\cdot\mathbf{b}_{12}$, $C_{3}=\mathbf{n}_{1}\cdot\mathbf{n}_{2}$ where $\mathbf{n}_{1}$ and $\mathbf{n}_{2}$ are normals to membrane 1 and 2, respectively, while $\mathbf{b}_{12}$ $\mathbf{is}$ the unit bonding vector. The summation over unit cells labeled by **$l$**$=(l_{x},l_{y},l_{z})$ comprises interactions of membrane at the point $\mathbf{r}_{\mathbf{l},\kappa}=(0,0,0)$ and with the normal to the membrane plane parallel to the z-axis, i.e. $\mathbf{n}_{z}^{0}=(0,0,1)$, with all others. The summation over $\kappa=x,y,z$ means the interaction with membranes whose normals $\mathbf{n}_{x}$, $\mathbf{n}_{y}$, $\mathbf{n}_{z}$ are along the $x$, $y$, $z$-axis, respectively. For further calculations we parameterize $\mathbf{n}_{1}=\left( 0,0,1\right) $, $\mathbf{n}_{2}=\cos\phi_{2}\sin\theta_{2},\sin\phi_{2}\sin\theta_{2},\cos\theta_{2}$ and for $\mathbf{b}$ the same as for $\mathbf{n}_{2}$ but $\phi_{2}$, $\theta_{2}$ goes to $\phi_{b}$, $\theta_{b}$. It is straightforward to show that $C_{3,\kappa
=x}=C_{3,\kappa=y}=0$. Similarly, $C_{3,\kappa=z}=1$, $C_{1,\kappa=z}=l_{z}/\sqrt{l_{x}^{2}+l_{y}^{2}+l_{z}^{2}}$ and that $C_{1,\kappa
=z}=C_{2,\kappa=z}$. It turns out that $C_{1,\kappa=x}=(l_{z}-1/2)/\sqrt
{(l_{x}-1/2)^{2}+l_{y}^{2}+(l_{z}-1/2)^{2}}$, and analogously for $C_{2,\kappa=x}$, $C_{1,\kappa=y}$, $C_{2,\kappa=x}$. Based on these results the sum in Eq.(\[A4c-S\]) has the final form $$\begin{aligned}
S & =2\sum_{l_{x},l_{y}}\sum_{l_{z}=1}^{\infty}\frac{A(l_{x},l_{y},l_{z})}{(l_{x}^{2}+l_{y}^{2}+l_{z}^{2})^{3/2}}\label{A4c-S-final}\\
& +2\sum_{l_{x},l_{y}}\sum_{l_{z}=-\infty}^{\infty}\frac{B(l_{x},l_{y},l_{z})}{(l_{x}^{2}+(l_{y}-\frac{1}{2})^{2}+(l_{z}-\frac{1}{2})^{2})^{3/2}}\nonumber\end{aligned}$$ where$$A(l_{x},l_{y},l_{z})=\alpha+\frac{l_{z}^{2}}{l_{x}^{2}+l_{y}^{2}+l_{z}^{2}}-\frac{2}{3}$$$$B(l_{x},l_{y},l_{z})=\frac{(l_{y}-\frac{1}{2})^{2}+(l_{z}-\frac{1}{2})^{2}}{l_{x}^{2}+(l_{y}-\frac{1}{2})^{2}+(l_{z}-\frac{1}{2})^{2}}-\frac{2}{3}$$ $\alpha=(1-\chi_{\min}/\chi_{\max})/3$ and the sum over $l_{x},l_{y}$ goes from $-\infty$ to $\infty$.
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In the previous paper [@KulicKulic-PRL] we called it *spatially coherent fluctuation interaction* but find that the former name (cVdW) is physically more elucidating.
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The $\chi^{2}$ scaling is valid for equilibrium conditions, where a detailed ballance between the moments and (thermal or quantum) bath hold. The scaling for icFI switches to $\propto\chi^{3}$ if the fluctuating field is exogeneous (externally set) and non-equilibrium as in the present cVdW case.
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---
abstract: 'Final states with a vector boson and a hadronic jet allow one to infer the Born-level kinematics of the underlying hard scattering process, thereby probing the partonic structure of the colliding protons. At forward rapidities, the parton collisions are highly asymmetric and resolve the parton distributions at very large or very small momentum fractions, where they are less well constrained by other processes. Using theory predictions accurate to next-to-next-to-leading order (NNLO) in QCD for both and production in association with a jet at large rapidities at the LHC, we perform a detailed phenomenological analysis of recent LHC measurements. The increased theory precision allows us to clearly identify specific kinematical regions where the description of the data is insufficient. By constructing ratios and asymmetries of these cross sections, we aim to identify possible origins of the deviations, and highlight the potential impact of the data on improved determinations of parton distributions.'
author:
- 'A. Gehrmann–De Ridder'
- 'T. Gehrmann'
- 'E. W. N. Glover'
- 'A. Huss'
- 'D. M. Walker'
title: Vector Boson Production in Association with a Jet at Forward Rapidities
---
Introduction
============
The production of a vector boson in association with a hadronic jet is the simplest hadron-collider process that probes both the strong and electroweak interactions at Born level. It has been measured extensively at the Tevatron [@Aaltonen:2018tqp; @Aaltonen:2007ae; @Abazov:2013gpa; @Abazov:2008ez] and the LHC [@Aad:2014qxa; @Aaboud:2017soa; @Aad:2013ysa; @Aad:2014rta; @Aaboud:2017hbk; @Khachatryan:2014uva; @Khachatryan:2016fue; @Sirunyan:2017wgx; @Khachatryan:2014zya; @Khachatryan:2016crw; @Sirunyan:2018cpw; @AbellanBeteta:2016ugk], covering a large range in transverse momentum and rapidity of the final-state particles. When compared to theory predictions, these measurements provide important tests of the dynamics of the Standard Model and help to constrain the momentum distributions of partons in the proton.
The study of the forward-rapidity region for this process is particularly important for our understanding of parton distribution functions (PDFs) at extremal values of Bjorken-$x$, due to the different kinematic regimes that are probed compared to the inclusive case. Owing to the extended rapidity coverage of the LHC experiments, data is now available for both highly boosted leptons and jets, giving direct access to these regions in phenomenological studies.
In order to make this connection more concrete, it is instructive to relate the event kinematics to the Bjorken-$x$ values that are probed. For a given vector-boson-plus-jet event, one can directly infer the valid range in Bjorken-$x$ values from the event kinematics at the hadronic centre-of-mass energy $\sqrt{s}$ through $$\begin{aligned}
\label{eqn:kinematic_constraints_x12}
x_1 &\ge \frac{1}{\sqrt{s}}\left(m_\rT^V \cdot \re^{+y^V} +\ptjo \cdot \re^{+y^{j1}} \right) , \notag\\
x_2 &\ge \frac{1}{\sqrt{s}}\left(m_\rT^V \cdot \re^{-y^V} +\ptjo \cdot \re^{-y^{j1}} \right) ,\end{aligned}$$ with $m_\rT^V = \sqrt{{(\ptv)}^2+{m_{V}^2}}$ denoting the transverse mass. In this equation, $x_1$ and $x_2$ correspond to the momentum fractions of the incoming partons present in the colliding protons, and are the transverse momenta of the vector boson and the leading-jet, $m_{V}$ is the invariant mass of the combined system of the decay products of the vector boson and $y_V$ and $y_{j1}$ are the rapidities of the vector boson and the leading jet. The equality in the above relations holds at Born level.
In general, the smallest $x$ value that can be probed simultaneously ($x_1 \sim x_2$) is $$\begin{aligned}
x_{\min} = \frac{m_{V+j}^{\min}}{\sqrt{s}} \; ,\end{aligned}$$ which is relevant primarily for data where fiducial cuts are symmetric in rapidity. Here $m_{V+j}$ is the invariant mass of the vector-boson-plus-jet final state at LO. In addition, we have the combined kinematic constraint $$\begin{aligned}
x_1 x_2 \ge \frac{1}{s} \left(m_\rT^{V,\min}+\pt^{j1,\min}\right)^{2} ,\end{aligned}$$ where $m_\rT^{V,\min}$ and $\pt^{j1,\min}$ are the minimum values of the vector boson transverse mass and leading jet admitted by the fiducial cuts. This constraint is particularly relevant in phase-space regions that are asymmetric in rapidity, which in turn probes more asymmetric values in $x_1$, $x_2$ and gives rise to a more complex interplay between the kinematics and the event selection cuts.
![\[fig:kinematic\_xq2\] The regions of the ($x$, $Q^2$) plane accessible for the LHCb [@AbellanBeteta:2016ugk] and ATLAS [@Aad:2014rta] selection criteria at LO. Here $Q^2$ is the invariant mass of the full final state including both charged leptons and QCD radiation and $x$ is the Bjorken-x from either of the incoming beams.](kinematic_regions_xq2.pdf){width="0.92\linewidth"}
Precision QCD predictions for the production of a vector boson in association with a jet have advanced considerably in recent years with the completion of fixed-order next-to-next-to-leading (NNLO) QCD calculations [@Boughezal:2015dva; @Ridder:2015dxa; @Boughezal:2015ded; @Boughezal:2016dtm; @Boughezal:2016isb; @Ridder:2016nkl; @Gehrmann-DeRidder:2016jns; @Campbell:2017dqk; @Gauld:2017tww; @Gehrmann-DeRidder:2017mvr], which are now being matched to resummation results [@Bizon:2018foh; @Sun:2018icb] to extend their validity across a wider kinematic range. These are complemented by NLO electroweak corrections [@Denner:2009gj; @Denner:2011vu; @Kallweit:2015dum], which are particularly relevant at large transverse momenta. There is a strong experimental motivation for precise predictions for these processes due to the high statistics and clean decay channels observed at the LHC, and their relevance to determinations of Standard Model parameters and as backgrounds for new physics searches [@Lindert:2017olm]. Fitting procedures for PDFs also benefit greatly from improved predictions, due to the increased sensitivity to the gluon and quark content of the proton [@Ball:2017nwa; @Boughezal:2017nla]. Owing to the large gluon luminosity at the LHC, the dominant initial state for vector-boson-plus-jet production is quark–gluon scattering, with different quark flavour combinations probed by the different bosons.
In this paper, we perform a comparison between NNLO QCD predictions for vector-boson-plus-jet (VJ) production and measurements by the LHCb [@AbellanBeteta:2016ugk] and ATLAS [@Aad:2014rta] experiments. These measurements are highly complementary, allowing one to probe a much larger kinematic region than if either of them were taken alone due to the different rapidity coverages of the two detectors. The region of the ($x$, $Q^2$) plane which is probed at LO in QCD in production is shown in Fig. \[fig:kinematic\_xq2\], where one can see that LHCb covers two distinct sectors corresponding to the $x$ values of the two beams. The corresponding plot for the ($x_1$, $x_2$) plane is shown in Fig. \[fig:kinematic\_x1x2\], where the asymmetry of the LHCb region preferentially probes large $x_1$ and small $x_2$ values in contrast to the symmetric ($x_1$, $x_2$) coverage of the ATLAS fiducial region. The kinematic constraints on the LHCb region are relaxed beyond LO as the presence of radiation permits larger $Q^2$ and $x_2$ values, unlike on the ATLAS region where LO kinematics already fully cover the kinematic region accessible at higher orders. The LO kinematics dominates in the contribution to the total cross section however, and gives a good indication of where the sensitivities of the two experiments lie.
The theoretical predictions are obtained using the framework [@Ridder:2015dxa; @Gehrmann-DeRidder:2017mvr], which implements the relevant NNLO VJ matrix elements [@Garland:2001tf; @Garland:2002ak; @Glover:1996eh; @Bern:1996ka; @Campbell:1997tv; @Bern:1997sc; @Hagiwara:1988pp; @Berends:1988yn] and uses the antenna subtraction method [@GehrmannDeRidder:2005cm; @Daleo:2006xa; @Currie:2013vh] to extract and combine infrared singularities from partonic subprocesses with different multiplicity.
![\[fig:kinematic\_x1x2\] The regions of the ($x_1$, $x_2$) plane accessible for the LHCb [@AbellanBeteta:2016ugk] and ATLAS [@Aad:2014rta] selection criteria at LO. Here $x_1$ and $x_2$ are the Bjorken-x values from beams 1 and 2 respectively.](kinematic_regions_x1x2.pdf){width="0.92\linewidth"}
Throughout this work, the theoretical predictions employ a diagonal CKM matrix. The electroweak parameters are set according to the $G_{\mu}$ scheme with the following input parameters: $$\begin{aligned}
M_\PZ &= 91.1876~\GeV , &
M_\PW &= 80.385~\GeV , \notag\\
\Gamma_\PZ &= 2.4952~\GeV , &
\Gamma_\PW &= 2.085~\GeV , \notag\\
G_\mathrm{F} &= \mathrlap{ 1.1663787 \times 10^{-5}~\GeV^{-2} ,}\end{aligned}$$ and the PDF set used at all perturbative orders is the central replica of `NNPDF31_nnlo` [@Ball:2017nwa] with $\alphas(M_\PZ)=0.118$.
LHC 8 TV Boosted Cuts {#sec:LHCb}
=====================
At the proton–proton centre-of-mass energy of $8~\TeV$, the LHCb experiment has measured both - and -boson production in association with a jet with the vector bosons decaying in the muon channel [@AbellanBeteta:2016ugk]. The acceptance in the forward region of the LHCb experiment allows it to reliably probe PDFs at both much higher and lower momentum fractions $x$ than the general-purpose detectors at the LHC. This sensitivity arises from kinematic configurations that are asymmetric in $x_1$ and $x_2$, which in turn means that the event is boosted into the forward region. PDF uncertainties at large $x$ and $Q^2$ are generally driven by uncertainties in the $d$ content of the proton, which these measurements have the capacity to constrain due to their flavour sensitivity, particularly in the charged-current channels. This provides a strong motivation to use the state-of-the-art NNLO QCD results to test the quantitative agreement of the predictions with the experimental data.
The fiducial cuts applied to the charged leptons and the jets, which we label as the LHCb cuts for both and production are given by $$\begin{aligned}
\pt^{j} &> 20~\GeV , &
2.2 &< \eta^{j} < 4.2 , \notag\\
\pt^{\Pgm} &> 20~\GeV , &
2 &< y^{\Pgm} < 4.5 , \notag\\
\Delta R_{\Pgm,j} &> 0.5 ,\end{aligned}$$ where $\pt^{j}$ and $\pt^{\mu}$ are the transverse momenta of the jets and muons respectively, $\eta^{j}$ is the jet pseudorapidity, $y^{\Pgm}$ is the muon rapidity and $\Delta R_{\mathrm{\mu,j}}$ is the angular separation between the leading jet and the muon. In addition, the requirement $\pt^{\Pgm+j} > 20~\GeV$ is applied for $\PWpmJ$ production, where $\pt^{\Pgm+j}$ is the transverse component of the vector sum of the charged lepton and jet momenta. For $\PZJ$ production, the invariant mass of the dimuon system $m_{\mu\mu}$ is restricted to the window $60~\GeV < m_{\mu\mu} < 120~\GeV$ around the -boson resonance. The anti-$k_\rT$ jet algorithm [@Cacciari:2008gp] is used throughout, with radius parameter $R=0.5$. In the LHCb analysis [@AbellanBeteta:2016ugk], the VJ data were compared to NLO theory predictions, which were observed to overshoot the data throughout, albeit being consistent within the combined theoretical and experimental uncertainties.
For the theoretical predictions presented in this section, we set the central scale as in [@AbellanBeteta:2016ugk], i.e., $$\mur = \muf =
\sqrt{{m^2_{V}+\sum\nolimits_i(p^i_{\rT,j})^2}}
\ \equiv \mu_{0} ,$$ with scale variations performed independently for the factorisation and renormalisation scales $\muf$, $\mur$ by factors of $\frac{1}{2}$ and $2$ subject to the constraint $\frac{1}{2}<\muf/\mur<2$.
[c@l@c]{}\
Process& & Fiducial $\sigma$ \[pb\]\
&&\
$\PWpJ$ & LO & $46.9^{+5.6}_{-2.2}$\
&&\
& NLO & $62.8^{+3.6}_{-3.5}$\
&&\
& NNLO & $63.1^{+0.4}_{-0.5}$\
&&\
& LHCb & $56.9\pm0.2\pm5.1\pm0.7$\
&&\
$\PWmJ$ & LO & $27.2^{+3.2}_{-2.6}$\
&&\
& NLO & $36.7^{+2.2}_{-2.1}$\
&&\
& NNLO & $36.8^{+0.3}_{-0.2}$\
&&\
& LHCb & $33.1\pm0.2\pm3.5\pm0.4$\
&&\
$\PZJ$ & LO & $4.59^{+0.53}_{-0.43}$\
&&\
& NLO & $6.04^{+0.32}_{-3.1}$\
&&\
& NNLO & $6.03^{+0.02}_{-0.04}$\
&&\
& LHCb & $5.71\pm0.06\pm0.27\pm0.07$\
The predictions for the fiducial cross section are shown in Table \[tab:LHCb\_fiducial\] for LO, NLO and NNLO QCD and compared to the results reported by the LHCb experiment for the individual VJ channels. We see large corrections when going from LO to NLO as observed in the NLO/LO K-factor of 1.34 for $\PWm$, $1.35$ for $\PWp$ and 1.32 for $\PZ$. On the other hand, going from NLO to NNLO produces much smaller and more stable corrections, with a NNLO/NLO K-factor of $1.006$ for $\PWm$, $1.003$ for $\PWp$ and 0.998 for $\PZ$. The NNLO corrections lie within the scale bands of the NLO results. We note that the uncertainty bands overlap marginally between theory and data in Table \[tab:LHCb\_fiducial\] for $\PWm$ and $\PZ$ production, but not for $\PWp$ production.
Distributions Differential in Leading Jet
------------------------------------------
![\[fig:LHCb\_Wm\_pt\] Cross section differential in the of the leading jet for production. Predictions at LO (green), NLO (orange), and NNLO (red) are compared to LHCb data from Ref. [@AbellanBeteta:2016ugk], and the ratio to NLO is shown in the lower panel. The bands correspond to scale uncertainties estimated as described in the main text. ](LHCb_gnu.pdf){width=".95\linewidth"}
![\[fig:LHCb\_Wp\_pt\] Cross section differential in the of the leading jet for production. See Fig. \[fig:LHCb\_Wm\_pt\] for details.](LHCb_gnu.pdf){width=".95\linewidth"}
![\[fig:LHCb\_Z\_pt\] Cross section differential in the of the leading jet for production. See Fig. \[fig:LHCb\_Wm\_pt\] for details. ](LHCb_gnu.pdf){width=".95\linewidth"}
Figures \[fig:LHCb\_Wm\_pt\]–\[fig:LHCb\_Z\_pt\] show the distributions for transverse momentum of the leading jet in $\PWm$, $\PWp$ and $\PZ$ production respectively. Similarly to the fiducial cross section, the scale dependence of the differential distributions is considerably reduced when going from NLO to NNLO. The NNLO corrections are stable with respect to NLO, indicating a good convergence of the perturbative series. In addition, these results exhibit a strong similarity in behaviour between the $\PWm$, $\PWp$ and $\PZ$ production channels. We see that the theory overshoots the data by $\sim 5$–$10\%$ over the bulk of the distribution, rising to $30\%$ in the highest bin. This closely mirrors the effects seen at NLO as well as in the total cross section. The considerable decrease in theory uncertainty from NLO to NNLO makes the tension between data and theory more pronounced.
![\[fig:LHCb\_Wm\_etaj1\] Cross section differential in the pseudorapidity $\eta$ of the leading jet for $\PWm$ production. See Fig. \[fig:LHCb\_Wm\_pt\] for details. ](LHCb_gnu.pdf){width=".95\linewidth"}
![\[fig:LHCb\_Wp\_etaj1\] Cross section differential in the pseudorapidity $\eta$ of the leading jet for $\PWp$ production. See Fig. \[fig:LHCb\_Wm\_pt\] for details. ](LHCb_gnu.pdf){width=".95\linewidth"}
![\[fig:LHCb\_Z\_etaj1\] Cross section differential in the pseudorapidity $\eta$ of the leading jet for $\PZ$ production. See Fig. \[fig:LHCb\_Wm\_pt\] for details. ](LHCb_gnu.pdf){width=".95\linewidth"}
For the cuts placed on the final state, we are also able to associate the bins in to lower limits on the Bjorken-$x$ invariants. The lowest bin has the loosest constraint on the forward $x$, with $x_1>0.041$, $x_2>5.4\times 10^{-5}$. However, for the highest bins, between $50$ and $100~\GeV$, the restrictions translate to $x_1>0.075$, $x_2>0.00011$. Due to the invariant mass cuts applied in the case shown in Fig. \[fig:LHCb\_Z\_pt\], the smallest values in Bjorken-$x$ that can be probed only extend down to $x_1 > 0.11$, $x_2 > 0.0002$ in the highest bin. As a result, one probes larger values of $x$ for production than for in general. At large , we see that the same features are present in the neutral and charged current cases. We observe that the NNLO predictions overshoot the data.
Distributions Differential in Pseudorapidity
--------------------------------------------
The leading jet pseudorapidity distributions in Figs. \[fig:LHCb\_Wm\_etaj1\]–\[fig:LHCb\_Z\_etaj1\] show a similar pattern of deviation between NNLO predictions and data to the previous results, with theory predictions exceeding the data at the largest values of $\eta_{j1}$. The behaviour is similar for $\PWp$, $\PWm$ and $\PZ$, which may further indicate that the discrepancy is mainly due to the gluon distribution being overestimated at large $x$. Changes in individual quark or antiquark distributions would instead give a pattern of discrepancy that is more pronounced in one of the channels than in the others. In the pseudorapidity distributions, we probe simultaneously more extreme regions of $x_1$ and $x_2$ than for the distributions as the directional dependence on $y_{j}$ as given in Eq. allows us to more directly discriminate the two Bjorken-$x$ values. This can be seen most explicitly for the case, for which the forward-most bin in pseudorapidity requires implicitly $x_1>0.16$, $x_2>1.1\times 10^{-4}$, meaning that the large $x>\cO(0.1)$ regions are probed efficiently.
![\[fig:LHCb\_Wm\_etalep\] Cross section differential in the pseudorapidity $\eta$ of the lepton for $\PWm$J production. See Fig. \[fig:LHCb\_Wm\_pt\] for details. ](LHCb_gnu.pdf){width=".95\linewidth"}
![\[fig:LHCb\_Wp\_etalep\] Cross section differential in the pseudorapidity $\eta$ of the lepton for $\PWp$J production. See Fig. \[fig:LHCb\_Wm\_pt\] for details. ](LHCb_gnu.pdf){width=".95\linewidth"}
![\[fig:LHCb\_Z\_yz\] Cross section differential in the rapidity of the dilepton system for $\PZ$J production. See Fig. \[fig:LHCb\_Wm\_pt\] for details. ](LHCb_gnu.pdf){width=".95\linewidth"}
The distributions for the rapidity of the charged lepton $\eta_\ell$ are shown in Figs. \[fig:LHCb\_Wm\_etalep\] and \[fig:LHCb\_Wp\_etalep\] for $\PWm$ and $\PWp$ respectively. Here the NNLO predictions lie $\sim 5$–$15\%$ above the data across the entire considered range in $\eta_\ell$. Note that it would be preferable to construct these distributions as a function of the rapidity $y_\PW$, which however can not be unambiguously reconstructed experimentally due to the unknown longitudinal component of the neutrino momentum. For the case of neutral-current production, on the other hand, this is possible and is shown in Fig. \[fig:LHCb\_Z\_yz\] differentially with respect to the rapidity of the reconstructed boson.
From the charged-current data one can further construct the charge asymmetry differentially in the lepton pseudorapidity $A^{\pm}(\eta_\ell)$, $$A^{\pm}(\eta_\ell) =
\frac{\rd\sigma^{\PWp j}/\rd\eta_\ell - \rd\sigma^{\PWm j}/\rd\eta_\ell}
{\rd\sigma^{\PWp j}/\rd\eta_\ell + \rd\sigma^{\PWm j}/\rd\eta_\ell} \; .
\label{eqn:charge_asym_def}$$ The charge asymmetry is a valuable input to PDF fits as many systematic experimental errors cancel due to correlations in luminosity and systematic errors between the measurements of and , giving a higher level of precision than for the total cross sections alone. This is also true for the theory predictions, where many higher-order contributions cancel between and , and the similarity of the two calculations justifies some correlation between scale errors. $A^{\pm}$ directly provides information on the difference between the $u$ and $d$ quark (as well as between the $\bar d$ and $\bar u$ anti-quark) content of the proton.
The advantage of considering the charge asymmetry for events where a jet is produced in association with the boson, which can be regarded as an exclusive asymmetry, as opposed to the inclusive $A^{\pm}$ is that the implicit constraint on Bjorken-$x$ is tightened due to the increase in partonic energy required. Before comparing our predictions with LHCb data for the exclusive charge asymmetry, it is instructive to recall the status of measurements of its inclusive analogue. The LHCb measurement of the inclusive charge asymmetry [@Aaij:2015zlq] probes larger values of $x$ than at ATLAS or CMS. Currently the main constraints on $u$ and $d$ content at $x>0.1$ come primarily from fixed-target DIS experiments and the D0 inclusive lepton charge asymmetry data [@D0:2014kma]. The inclusion of the latest Tevatron results in PDF fits generally results in a harder $u/d$ behaviour in this high-$x$ region [@Dulat:2015mca].
In Fig. \[fig:LHCb\_Asym\_etalep\], we show a comparison between our theoretical predictions for $A^{\pm}$ related to WJ production and the LHCb data. Inside the numerator and the denominator expressions, we fully correlate the scales between the $\PWp$ and $\PWm$ cross sections, which amounts to taking the sum and difference of the cross sections as independent physical quantities $\left[\rd\sigma^{\PWp}\pm\rd\sigma^{\PWm}\right](\muf, \mur)$ instead of the $\PWp$ and $\PWm$ cross sections. The scale uncertainty shown is then obtained by independently varying the factorisation $(\muf)$ and renormalisation $(\mur)$ scales of both the numerator and denominator by factors of $\frac{1}{2}$ and $2$ around the central scale, while imposing the restriction $\frac{1}{2}\leq \mu/\mu'\leq 2$ between all pairs of scales ($\mu,\mu'$) in Eq. .
![\[fig:LHCb\_Asym\_etalep\] $\PWpm$ asymmetry in WJ production differential in the pseudorapidity $\eta$ of the lepton produced from the boson decay. See Fig. \[fig:LHCb\_Wm\_pt\] for details. ](LHCb_gnu.pdf){width=".95\linewidth"}
The shape of $A^{\pm}$ as a function of $\eta_\ell$ is generally determined by two competing effects [@Farry:2015xha]. The first is the (anti-)quark content of the PDF, where the $u/d$ ratio and $q/\bar{q}$ asymmetry increase with momentum fraction $x$, and therefore with $\eta_\ell$. This alone gives an increase in $A^{\pm}$ with $\eta_\ell$ since $u$-initiated production is dominant in $\PWp$ production while $d$-initiated production is dominant for $\PWm$.
![\[fig:LHCb\_pdf\_variation\] $\PWpm$ asymmetry in WJ final states differential in the pseudorapidity $\eta$ of the lepton produced from the boson decay, evaluated with NNPDF3.1 (red), MMHT14 (yellow), CT14 (green) NNLO parton distribution functions. The NNPDF3.1 curve corresponds to a full NNLO calculation with scale uncertainties as described in the main text, and is used to determine a differential NNLO/NLO K-factor. The other two predictions are calculated at NLO and then rescaled by this K-factor.](LHCb_pdf_variation.pdf){width=".95\linewidth"}
The second factor is due to the left-handedness of the couplings in the $\PWpm$ production and decay process, which results in opposite preferential directions of the positive and negative decay leptons relative to the $\PWpm$ spin. As a consequence, for the $\PWp$ case, the lepton is preferentially produced at lower $\eta$ than the $\PWp$, whereas for the $\PWm$ case, the lepton is preferentially produced at higher relative $\eta$. This effect causes the asymmetry to decrease with $\eta_\ell$, and dominates over the quark PDF effects at higher $x$, as can be seen in Fig. \[fig:LHCb\_Asym\_etalep\].
We find that the NNLO predictions for the asymmetry describe the data reasonably well, but in general show a less steep slope with $\eta_\ell$ than the data. This may be indicative of a PDF overestimate in the $u/d$ ratio for $x\gtrsim 0.1$ which would lead to the observed overprediction of the charge asymmetry in this region. It is noted that the large $u/d$ ratio is in particular inferred [@Ball:2017nwa; @Dulat:2015mca] from the Tevatron D0 lepton charge asymmetry data [@D0:2014kma]. It will thus be crucial to combine these data with the LHCb results [@AbellanBeteta:2016ugk] in a global fit to determine whether they are mutually consistent.
The sensitivity of the $\PWpm$ asymmetry in WJ final states on the PDF parametrizations is illustrated in Figure \[fig:LHCb\_pdf\_variation\], which shows this asymmetry at NNLO for NNPDF3.1 [@Ball:2017nwa], MMHT14 [@Harland-Lang:2014zoa] and CT14 [@Dulat:2015mca] parton distributions. The NNPDF3.1 prediction is obtained from a full NNLO calculation of the individual cross sections entering into the ratio, which are also used to extract NNLO K-factors. Predictions for the other two PDF parametrizations are computed at NLO at cross section level, and then rescaled by these K-factors, before computing the ratio. The large spread of the predictions (noting also the different scale in the ratio compared to Figure \[fig:LHCb\_Asym\_etalep\]) in the last bin reflects the different modelling of the quark distributions at large $x$ in the three parametrizations, and demonstrates the discriminating power of the LHCb asymmetry measurement.
ATLAS 7 TV Standard Cuts {#sec:ATLAS}
========================
The second set of experimental data we consider is the $7~\TeV$ (electron and muon) measurement by the ATLAS experiment [@Aad:2014rta], which combines data from the and analyses of [@Aad:2014qxa] and [@Aad:2013ysa] with a small modification to the lepton selection criteria applied in the analysis when taking ratios. This modification is applied in order to better match the selection criteria.
The ATLAS detector has a large rapidity range, capable of measuring pseudorapidities of up to $|\eta|=4.9$ in the endcap region for both hadronic and electromagnetic final states. Unlike the LHCb measurement region, the large pseudorapidity reach of ATLAS also allows to probe large rapidity separations between final state particles, which correspond to configurations in which the Bjorken-$x$ of both incoming protons is relatively large. In the following, we perform a comparison of fixed-order NNLO results to the individual and distributions of [@Aad:2014qxa] and [@Aad:2013ysa], before constructing the ratios of $\PWJ$ ($\equiv\PWpJ+ \PWmJ$) and distributions and comparing those to the results of [@Aad:2014rta]. We consider leading jet distributions in inclusive (at least one jet is required) and exclusive (exactly one jet is required) jet production, as well as inclusive leading jet rapidity distributions. The inclusive distributions have previously been compared to NNLO QCD predictions in [@Boughezal:2016dtm], however exclusive distributions and ratios of distributions were not considered.
The fiducial cuts used in the ATLAS analyses are as follows: $$\begin{aligned}
\pt^{j} &> 30~\GeV , &
|y^{j}| &< 4.4 , \notag\\
\pt^{\ell} &> 25~\GeV , &
|y^{\ell}| &< 2.5 , \notag\\
\Delta R_{\ell,j} &> 0.5 \label{eqn:atlas_cuts}.\end{aligned}$$ For $\PWpmJ$ production, the restrictions $E_{\rT}^\miss>25~\GeV$, and $m_{\rT}^\PW>40~\GeV$ on the missing transverse energy and transverse mass of the boson are imposed. For production the requirements $66~\GeV < m_\rT^{\ell\ell} < 116~\GeV$ and $\Delta R_{\ell\ell} > 0.2$ are applied to the transverse mass of the dilepton system and angular separation of the leptons. In the distributions, we relax the lepton cut from $25$ to $20~\GeV$ in order to compare directly with the results of [@Aad:2013ysa]. However we keep the lepton cut at $25~\GeV$ when constructing ratios of WJ and ZJ distributions.
Jets are reconstructed using the anti-$k_\rT$ algorithm [@Cacciari:2008gp] with radius parameter $R=0.4$, and we choose the central scale of the theory predictions as $$\begin{aligned}
\muf = \mur = \frac{1}{2}H_{\rT} = \frac{1}{2}\sum_{i \,\in\, \mathrm{jets,\,\ell,\,\nu}}\pt^i
\equiv\mu_{0} ,\end{aligned}$$ where $H_\rT$ is the scalar sum of the transverse momenta of all final state jets and leptons/neutrinos as appropriate. We denote the number of jets as $N$, such that in the selection criteria $N=1$ corresponds to the exclusive case and $N\geq1$ corresponds to the inclusive case.
The scale variation uncertainties for the ratios are obtained in a similar manner as for LHCb asymmetries, with fully correlated scales between the and processes in the numerator, but taking the envelope of the scales when taking the ratio to the distributions, imposing $\frac{1}{2}\leq \mu/\mu'\leq 2$ between all pairs of scales.
Exclusive Distributions
-----------------------
![\[fig:ATLAS\_W\_ptj1\_excl\] cross section differential in the transverse momentum of the leading jet for events with exactly one associated jet $(N=1)$ in the ATLAS fiducial region from Eq. \[eqn:atlas\_cuts\]. Predictions at LO (green), NLO (orange), and NNLO (red) are compared to ATLAS data from Ref. [@Aad:2014qxa], and the ratio to NLO is shown in the lower panel. The bands correspond to scale uncertainties estimated as described in the main text. ](ATLAS_gnu.pdf){width=".95\linewidth"}
![\[fig:ATLAS\_Z\_ptj1\_excl\] cross section differential in the transverse momentum of the leading jet for events with exactly one associated jet $(N=1)$. Predictions at LO (green), NLO (orange), and NNLO (red) are compared to ATLAS data from Ref. [@Aad:2013ysa], and the ratio to NLO is shown in the lower panel. The bands correspond to scale uncertainties estimated as described in the main text. ](ATLAS_gnu.pdf){width=".95\linewidth"}
First we consider the exclusive $(N=1)$ distribution of the leading jet for production using the data from [@Aad:2014qxa] as shown in Fig. \[fig:ATLAS\_W\_ptj1\_excl\]. Here we observe agreement of the theory with data within errors up to $\ptjo \sim 80~\GeV$, beyond which the theoretical predictions are systematically below the data. This behaviour is closely replicated in Fig. \[fig:ATLAS\_Z\_ptj1\_excl\], which shows the equivalent distribution. However beyond $\ptjo \sim 80~\GeV$, the agreement with data is noticeably worse than for the distribution. While we neglect electroweak corrections which have a well-known impact on the weak boson distributions [@Denner:2009gj; @Denner:2011vu; @Kallweit:2015dum] from large Sudakov logarithms, these generally give considerable reductive K-factors at large and so would further worsen the agreement with data in both cases (see e.g. [@Kallweit:2015dum]). For these exclusive distributions, it is instructive to note that is equivalent to the transverse momentum of the vector boson due to the absence of extra jet radiation.
Inclusive Distributions
-----------------------
![\[fig:ATLAS\_W\_ptj1\] cross section differential in the transverse momentum of the leading jet for events with one or more associated jets $(N\geq 1)$. See Fig. \[fig:ATLAS\_W\_ptj1\_excl\] for details. ](ATLAS_gnu.pdf){width=".95\linewidth"}
![\[fig:ATLAS\_Z\_ptj1\] cross section differential in the transverse momentum of the leading jet for events with one or more associated jets $(N\geq 1)$. See Fig. \[fig:ATLAS\_Z\_ptj1\_excl\] for details. ](ATLAS_gnu.pdf){width=".95\linewidth"}
For the inclusive $(N\geq1)$ spectrum in production, shown in Fig. \[fig:ATLAS\_W\_ptj1\], we observe marginally improved agreement over a wider range of , with overlapping uncertainty bands between data and theory up to $\ptjo \sim 300~\GeV$. Beyond this point, there are substantial, $\cO(15\%)$, shape corrections when moving from NLO to NNLO which improve the agreement with data with respect to the NLO results. In production, shown in Fig. \[fig:ATLAS\_Z\_ptj1\], the pattern of perturbative corrections is very similar. However we do not observe the same level of improved agreement with data when moving from exclusive to inclusive jet production as for the process and we again see that the theory prediction is systematically below the data from $\ptj\sim 100~\GeV$ onwards.
Allowing extra QCD radiation, as in the inclusive case, entails also allowing for dijet-type configurations where two hard jets are produced alongside a relatively soft vector boson. In the full NNLO calculation, these $\cO(\alphas)$ contributions are first described at NLO, and give rise to a large QCD K-factors at high [@Rubin:2010xp]. This is the dominant cause of the distinct structure of the perturbative corrections between exclusive and inclusive production; for $N=1$ we see a decrease in the high- cross-sections with the inclusion of higher orders as opposed to an increase in $N\geq1$ production. The difference in theory-to-data agreement between the and distributions persists however, and may be a related to the different quark flavour combinations probed by the different processes. Whilst not as constraining as the / ratio, the / ratio still retains some sensitivity to the $u/d$ ratio due to different coupling strengths, and some dependence on the strange quark distributions, albeit suppressed compared to the inclusive Drell-Yan cross sections due to the Born-level gluon contribution. The inclusion of higher-order EW terms are unlikely to describe the difference with respect to data at high , as the EW corrections to the leading distribution in vector-boson-plus-dijet events behave in a very similar manner for and production as demonstrated in [@Kallweit:2015dum].
Exclusive/Inclusive Ratios
--------------------------
![\[fig:ATLAS\_W\_ptj1\_exc\_inc\] Ratio of exclusive/inclusive $(N=1/N\geq 1)$ production differential in the transverse momentum of the leading jet. Errors on the ATLAS data are approximated using uncertainties from the $N=1$ distribution normalised to the $N\geq 1$ results. See Fig. \[fig:ATLAS\_W\_ptj1\_excl\] for details. ](ratios.pdf){width=".95\linewidth"}
![\[fig:ATLAS\_Z\_ptj1\_exc\_inc\] Ratio of exclusive/inclusive $(N=1/N\geq 1)$ production differential in the transverse momentum of the leading jet. Errors on the ATLAS data are approximated using uncertainties from the $N=1$ distribution normalised to the $N\geq 1$ results. See Fig. \[fig:ATLAS\_Z\_ptj1\_excl\] for details. ](ratios.pdf){width=".95\linewidth"}
In order to better understand the description of real emission by the fixed order predictions, one can construct the ratio between the exclusive and inclusive leading jet distributions for both the and the case, shown in Figures \[fig:ATLAS\_W\_ptj1\_exc\_inc\] and \[fig:ATLAS\_Z\_ptj1\_exc\_inc\]. The experimental measurements [@Aad:2013ysa; @Aad:2014qxa] do not explicitly quote the data in terms of exclusive/inclusive ratios. We have therefore reconstructed it here using the central values of the relevant distributions with the errors approximated using uncertainties from the $N=1$ distribution normalised to the $N\geq 1$ results. For both distributions we observe similar behaviour, with good description of the data across the range of , albeit with the general trend that the predictions systematically undershoot the central values of the data below $\ptjo\sim200~\GeV$, from which we can conclude that the extra jet radiation is well-described by the fixed order predictions.
W/Z Ratios Differential in Leading Jet
---------------------------------------
![\[fig:ATLAS\_ptj1\_exc\] $\PWJ/\PZJ$ ratio differential in the exclusive of the leading jet $(N=1)$. Predictions at LO (green), NLO (orange), and NNLO (red) are compared to ATLAS data from Ref. [@Aad:2014rta], and the ratio to NLO is shown in the lower panel. The bands correspond to scale uncertainties estimated as described in the main text. ](ATLAS_gnu.pdf){width=".95\linewidth"}
![\[fig:ATLAS\_ptj1\_inc\] $\PWJ/\PZJ$ ratio differential in the inclusive of the leading jet $(N\geq 1)$. See Fig. \[fig:ATLAS\_ptj1\_exc\] for details. ](ATLAS_gnu.pdf){width=".95\linewidth"}
Figure \[fig:ATLAS\_ptj1\_exc\] shows the /ratio as a function of , for the exclusive $(N=1)$ case. The large scale variation bands visible at NLO are a result of large NLO corrections at high that increase the scale uncertainties when propagated through ratios. In particular, as shown in Fig. \[fig:ATLAS\_W\_ptj1\_excl\] and Fig. \[fig:ATLAS\_Z\_ptj1\_excl\], we observe large reductive NLO/LO K-factors at high for the individual and distributions, reaching $K=0.3$ in the highest bin, whereas the absolute size of the scale variation bands does not reduce significantly when going from LO to NLO. This has the effect of making the exclusive $\PWJ/\PZJ$ ratio much more sensitive to scale variation in the constituent distributions at NLO than LO, artificially inflating the scale uncertainties at this order. The inclusive ($N\geq 1$) ratio, shown in Fig. \[fig:ATLAS\_ptj1\_inc\], has very similar central values at LO, NLO and NNLO, but does not display the inflated NLO scale uncertainty.
When taking the ratio, the impact of the extra jet activity is strongly suppressed, while the PDF sensitivity is enhanced. As mentioned in the case of the individual distributions, the / ratio can be used to provide constraints on the ratio of up and down valence quark distributions inside the PDFs, as well as on the strange distribution, due to the different couplings of the vector bosons. Taking only the dominant incoming $qg$ partonic configurations, we can see that naïvely the ratio behaves as $$\frac{\sigma^\PWJ}{\sigma^\PZJ}\sim\frac{ug+dg}{0.29ug+0.37dg},$$ where the numerical factors are the appropriate sums of the squares of the vector and axial vector quark- couplings. Discarding the common factor of the gluon PDF, this can be used to interpret a theory-to-data excess in the / ratio as an overestimate of the $u/d$ ratio. If we look back to the individual distributions, we see that for each of the and cases, the theory falls below the data. From this, it can be deduced that the most probable cause is an underestimate in the $d$ quark content of the PDF.
Inclusive Leading Jet Rapidity Distributions
--------------------------------------------
The leading jet rapidity distribution $|y_{j1}|$ for events is shown in Fig. \[fig:ATLAS\_W\_yj1\], and for events in Fig. \[fig:ATLAS\_Z\_yj1\]. Here we observe that the higher-order QCD predictions are relatively stable for all orders up to $|y_{j1}| \sim 3$. Beyond this point, we see a change in shape when transitioning from LO to NLO. The shape is kept unmodified under the inclusion of the NNLO corrections. There is an increase in scale uncertainty at higher rapidities $|y_{j1}|\gtrsim3.5$ due to large subleading jet contributions in this region, which are only described at lower orders for inclusive observables in the NNLO VJ calculation. In both cases, we see good agreement for all rapidities, with overlapping scale errors and experimental error bars for the entire distribution. However, the shape corrections induced at NNLO for $|y_{j1}|\gtrsim3.5$ modify the central values of the theory predictions such that the tension with data increases compared to NLO.
![\[fig:ATLAS\_W\_yj1\] cross section differential in the absolute rapidity $|y_j|$ of the leading jet. See Fig. \[fig:ATLAS\_W\_ptj1\_excl\] for details. ](ATLAS_gnu.pdf){width=".95\linewidth"}
![\[fig:ATLAS\_Z\_yj1\] cross section differential in the absolute rapidity $|y_j|$ of the leading jet. See Fig. \[fig:ATLAS\_Z\_ptj1\_excl\] for details. ](ATLAS_gnu.pdf){width=".95\linewidth"}
![\[fig:ATLAS\_yj1\] $\PWJ/\PZJ$ ratio differential in the absolute rapidity $|y_j|$ of the leading jet. See Fig. \[fig:ATLAS\_ptj1\_exc\] for details. ](ATLAS_gnu.pdf){width=".95\linewidth"}
If one associates the higher-energy incoming parton with $x_1$ and the lower-energy incoming parton with $x_2$, such that the sum of all final state momenta lies in the same direction as parton 1, the forward-most bin $(3.8<y_{j1}<4.4)$ in rapidity here corresponds to $x_1>0.19$, $x_2>0.00012$ for production and $x_1>0.19$, $x_2>0.00019$ in production. One can then analyse the distributions here in a similar manner to the LHCb predictions in Figs. \[fig:LHCb\_Wm\_etaj1\]–\[fig:LHCb\_Z\_etaj1\]. As is the case for the LHCb data, we see a theory excess in the jet rapidity bins corresponding to $x\gtrsim 0.1$. This is again indicative of an overestimate of the gluon contributions to the PDF in this region since this excess is present in both and distributions. The central rapidity bins allow us to quantify better the PDF description at intermediate Bjorken-$x$, with the central-most bin in $y_{j1}$ requiring $x_1>0.0044$ and $x_2>0.0036$ for both neutral- and charged-current production. Here we see good agreement with the data, indicating that the behaviour in this region is well under control. The ratio of to differential in the absolute rapidity $|y_{j1}|$ of the leading jet is shown in Fig. \[fig:ATLAS\_yj1\]. Due to the cross-cancellation in the ratios, we see that these predictions display a considerably better perturbative stability than the individual distributions at high rapidities. We observe excellent agreement with the ATLAS data across the entire rapidity range. In the ratio, the PDF dependence of the predictions is in general lowered, particularly for gluonic contributions due to their similarity between the and cases. The agreement on the ratio demonstrates that the NNLO QCD description of the underlying parton-level process is reliable. It indicates that the discrepancies observed in the individual distributions are of parametric origin and can be remedied by an improved determination of the gluon distribution.
Conclusions
===========
The recent calculations [@Boughezal:2015dva; @Ridder:2015dxa; @Boughezal:2015ded; @Boughezal:2016dtm; @Boughezal:2016isb; @Ridder:2016nkl; @Gehrmann-DeRidder:2016jns; @Campbell:2017dqk; @Gauld:2017tww; @Gehrmann-DeRidder:2017mvr] of NNLO QCD corrections to all observables related to the production of a massive vector boson in association with a jet open up a new level of precision in the phenomenological interpretation of these observables. In this context, final states at forward rapidity are particularly interesting, since they correspond to initial states with very asymmetric momentum fractions of the incoming partons, thereby probing the parton distributions in regions where they are insufficiently constrained by other data sets.
In this paper, we performed an in-depth comparison of forward vector-boson-plus-jet data from LHCb [@AbellanBeteta:2016ugk] and ATLAS [@Aad:2014rta] with precise NNLO QCD predictions, obtained using the code [@Ridder:2016nkl; @Gehrmann-DeRidder:2016jns; @Gehrmann-DeRidder:2017mvr]. Inclusion of NNLO QCD corrections leads to a substantial reduction of the theory uncertainty on the predictions, thereby matching the accuracy of the LHC precision data. Deviations between data and theory are observed in various distributions, which are further investigated by constructing ratios between different vector bosons, and between inclusive and exclusive vector-boson-plus-jet cross sections. The pattern of vector boson ratios and related asymmetries points to an overestimate of the PDF parametrisation in the gluon distribution for Bjorken-$x\gtrsim 0.1$ and equally to an overestimate in the $u/d$ quark ratio in the same region.
Our results highlight the unique sensitivity of forward vector-boson-plus-jet production to the PDF content of the proton. We expect that the results presented here will enable improved determinations of the gluon distribution and of the quark flavour decomposition at large Bjorken-$x\gtrsim 0.1$, thereby enhancing the accuracy of theory predictions for signal and background processes at the highest invariant masses.
Acknowledgements
================
The authors thank Rhorry Gauld for assistance with the uncertainty propagation in the LHCb data, and Xuan Chen, Juan Cruz-Martinez, James Currie, Marius Höfer, Imre Majer, Tom Morgan, Jan Niehues, Joao Pires and James Whitehead for useful discussions and their many contributions to the code. This research was supported in part by the UK Science and Technology Facilities Council, by the Swiss National Science Foundation (SNF) under contracts 200020-175595 and 200021-172478, by the ERC Consolidator Grant HICCUP (No. 614577) and by the Research Executive Agency (REA) of the European Union through the ERC Advanced Grant MC@NNLO (340983).
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---
abstract: 'A proposal for maintaining CP conservation in $K_{L}^{0}$ decays is resumed, by adding a heterodox hypothesis. One recovers a consistent picture with a $2\pi$ $K_{L}^{0}$ decay branching ratio proportional to the vertical $K_{L}^{0}$ displacement. This can be clearly tested at the KLOE experiment, where important vertical $K_{L}^{0}$ displacements occur.'
author:
- |
Giuseppe Mambriani and Luca Trentadue [^1]\
Dipartimento di Fisica dell’ Università\
43100 - Parma - Italy
title: Testing CP Conservation at KLOE
---
Introduction
============
Bernstein, Cabibbo, and Lee [@Bern], after the discovery [@Christ] of 2$\pi$ decay of $K_{L}^{0}$ and the explanatory hypothesis of CP violation, considered the possibility that 2$\pi$ decay of $K_{L}^{0}$ could be due to the effect of an external field, thus saving CP symmetry [@Bell]. In this paper we try to resume the external-field hypothesis and the related CP conservation, by introducing the heterodox assumption that the gravitational field has a scalar component acting in opposite ways on the $K^{0}$ and the $\overline{K^{0}}$ mixed in the$\ |K_{L}^{0}\rangle$ state. When suitably developed, this assumption gives a picture which allows some simple predictions for the KLOE experiment at the DA$\mathit{\Phi}$NE collider in Frascati. For the first time, KLOE allows great vertical $K_{L}^{0}$ displacements, thus permitting the Earth’s gravitational field to manifest its possible effects clearly. The chief attraction of the present approach is on one hand the easiness of testing it at KLOE, and on the other hand the simplicity introduced by CP conservation. Some of the challenging questions opened by the repulsive gravitational coupling are briefly considered.
$K_{L}^{0}$ decays and a scalar component of the gravitational field
====================================================================
As is well known, the $K^{0}-\overline{K^{0}}$ system has two $\it{eigenstates}$ of the CP operator. If one assumes that CP is conserved in all decays, the physical $K_{S}^{0}$ coincides with the state $|K_{S}^{0}\rangle=$ $(|K^{0}\rangle+|\overline{K^{0}%
}\rangle)/\sqrt{2}$ having CP $\it{eigenvalue}$ $+1$. It quickly decays with mean life $\tau_{S}$, practically always into two pions (CP $\it{eigenvalue}$ $+1$). The other state $|K_{L}^{0}%
\rangle=(|K^{0}\rangle-|\overline{K^{0}}\rangle)/\sqrt{2}$ with CP $\it{eigenvalue}$ $-1$, coincides with the physical $K_{L}^{0}$, which decays slowly (mean life $\tau_{L}\approx600$ $\tau_{S}$) along a lot of channels. Some of these channels have final states with CP eigenvalue $+1$. This long-lived state can be thought of as a fast mixing of $|K^{0}\rangle$ and $|\overline{K^{0}}\rangle$ performing the virtual oscillation: $K^{0}\rightarrow2\pi\rightarrow$ $\overline{K^{0}}\rightarrow2\pi\rightarrow K^{0}$, with a frequency of the order of $1/\tau_{S}$, and a mixing energy $\hbar/\tau_{S}\approx7\times10^{-6}$ eV.
Although it’s been almost forty years since the experimental discovery of $2\pi$ decay of $K_{L}^{0}$ [@Christ], the origin of the CP violation, usually assumed to be at the root of $2\pi$ $K_{L}^{0}$ decay, is still judged as not fully understood. Indeed as is well known, within the Standard Model, one can accommodate CP violation through a parametrization of Cabibbo-Kobayashi-Maskawa’s matrix (see, for instance, [@DafneH] [@Jul] [@Fry]).
Bernstein, Cabibbo, and Lee [@Bern] a year after the discovery of $2\pi$ $K_{L}^{0}$ decay, attempted to reconcile CP conservation with it. They analyzed the possibility that $2\pi$ decay could be explained in terms of the effect of an external field (they considered the three values of the intermediate-boson spin $J$: $0$, $1$, or $2$), producing a potential-energy difference $+V/2$ for $K^{0}$ but $-V/2$ for $\overline
{K^{0}}$. By means of a relativistic time evolution equation, they found for the complex $2\pi$ $K_{L}^{0}$ branching ratio $\epsilon$, when $V \ll \hbar/\tau_{S}$: $$\epsilon\approx\frac{\gamma V(\gamma)}{2}\left[
\left( m_{L}-m_{S}\right) c^{2}-\frac{i\hbar}{2} \left( \frac{1}{\tau_{S}}-\frac{1}{\tau_{L}%
}\right) \right] ^{-1/2},\label{EpsC}%$$ where $\gamma$ is the kaon Lorentz factor in the laboratory frame, and $m_{L}$ and $m_{S}$ are the masses of $K_{L}^{0}$ and $K_{S}^{0}$, respectively. Starting from general assumptions for the classical Lagrangian, the intrinsic $V$ dependence on $\gamma$ must be expected to be: $V(\gamma
)=V_{0}$ $\gamma^{J-1}$ [@Thirr], where $V_{0}$ is the potential energy seen by the particle when at rest, relative to the field source. In Ref.[@Bern], the assumption that $V_{0}$ is the fourth component of an unknown vector field ($J=1$), has been made, then, from (\[EpsC\]), it follows that $|\epsilon|$ must be proportional to $\gamma$. After a few years, the experiments [@K-66][@Fitch][@K-72] showed that $|\epsilon|$ was independent of $\gamma$. Therefore the external-field CP-conserving interpretation of $2\pi$ $K_{L}^{0}$ decay was dismissed.
Actually, if in (\[EpsC\]) one considers a scalar field ($J=0$), one has $\left| \epsilon\right| $ independent of $\gamma$, as the experiments require. The questions arise whether this field can be identified with a scalar component of the Earth’s gravitational field, as well as through this scalar component, the Earth can exert an antigravitational force on $\overline{K^{0}}$. These questions raise a web of problems, which will be only briefly considered below.
By assuming that the Earth’s gravitational field has a scalar component, and by identifying $V_{0}/2$ with $m_{K}g_{E}\Delta\zeta$ [@Good], whith $m_{K}$ the kaon mass, $g_{E}$ the Earth fall acceleration, and $\Delta\zeta$ the vertical $K_{L}^{0}$ displacement, one finds:
$$|\epsilon|\approx m_{K}g_{E}\Delta\zeta \left [
\left( m_{L}-m_{S} \right ) ^{2} c^{4}+\frac{{\hbar}^{2}}{4} \left (
\frac{1}{\tau_{S}}-\frac{1}{\tau_{L}}\right )^{2} \right ]
^{-1/2}=m_{K} g_{E}\;\Delta\zeta \Lambda
,\label{Meps}%$$
where $\Lambda=(1.231\pm0.002)\times10^{24}$ J$^{-1}$, as follows from using standard values [@PDG] for kaon properties.
For KLOE, $\Delta\zeta$ is the vertical displacement between the small intersection region of DA$\mathit{\Phi}$NE $e^{-}$ and $e^{+}$ beams (within which $\mathit{\Phi}$ decays into kaons) and the $K_{L}^{0}$ decay vertex. The maximum effective $\Delta\zeta$ is roughly 1.5 m, for which (\[Meps\]) gives $|\epsilon|=|\eta_{+-}|\approx15\times10^{-3}$, that is, roughly seven times the standard value ($|\eta_{+-}|=(2.27\pm0.02)\times
10^{-3}$ [@PDG]. According to this approach, the first effect that KLOE should have to observe, is an average $|\eta_{+-}|$ value three to four times greater than the standard one. This large average $|\eta_{+-}|$ could be observed with a number of $K_{L}^{0}$ much smaller than the number forecast for KLOE design targets.
For instance, when collecting a sample of 10$^{4}$ $K_{L}^{0}$ with a $\Delta\zeta$ within $0.95$ m and $1.05$ m (inside the horizontal slabs below and above the production point), one finds one hundred of $K_{L}^{0}$ decaying into two charged pions, that is, a branching ratio $|\eta_{+-}|=(10\pm1)\times10^{-3}$. Such a value would practically exclude the standard picture. Better information can of course be obtained by collecting *e.g.* $10^{6}$ $K_{L}^{0}$ inside the whole KLOE volume. By subdividing them, for instance, in six bins of $\Delta\zeta$ between 0.3 m and 1.5 m, one may look for the possible proportionality relationship between $|\eta_{+-}|$ and $\Delta\zeta$, and should it be found, by comparing the experimental slope with $\Lambda
m_{K}g_{E}=(1070\pm2)\times10^{-5}$ m$^{-1}$, where $m_{K}=$ $497.672\pm0.031$ MeV [@PDG] and $g_{E}=9.8$ ms$^{-2}$.
Moreover, one must find small values of $|\eta_{+-}|$ at low $\Delta\zeta$. For instance, once $10^{6}$ $K_{L}^{0}$ are collected, roughly $7.5\times10^{4}$ $K_{L}^{0}$ decays along all channels, will be observed inside a horizontal slab $20$ cm thick placed from $10$ cm below to $10$ cm above the production point. With the standard $|\eta_{+-}|$ value one must expect $170\pm13$ decays into two charged pions, while following the present approach one must detect only $40\pm7$ decays, with a difference of nearly $9$ standard deviations.
Most information on the $K^{0}-\overline{K^{0}}$ system, has been obtained from horiz- ontal-beam experiments, to which of course the present gravitational interpretation must apply as well. When considering a nearly horizontal $K_{L}^{0}$ beam, $V_{0}$ can be due to the gravitational fields of the Earth and of the Sun. At the Earth’s surface, the latter is much smaller than the former. For the Earth’s field the vertical effective displacements are rather small since they are mainly linked to the $K_{L}^{0}$ beam divergency. For the Sun’s field the displacements projected along the field direction, can also be a few thousand times greater when considering long $K_{L}^{0}$ beams.
The papers reporting experimental $\left| \epsilon\right| $ values, give in general little information, if any, concerning beam geometry. However, the first relevant information on decay amplitudes, comes from the experiments [@Christ][@K-66][@Fitch][@K-72][@K--79], where the $K_{L}^0$ beam lenght was markedly shorter than $100$ m. Thus, the unique possibility of having $V_{0}$ different from zero, is linked to a possible effect of the Earth’s field, when a vertical displacement due to a small vertical component of $K_{L}^{0}$ velocity, occurs. A first set of experiments up to 1972 [@Christ][@K-66][@Fitch][@K-72], reported $\left| \epsilon\right| $ values grouped around $1.95\times10^{-3}$, and all have roughly $\Delta\zeta\approx0.2$ m, as far as is possible to judge from the beam size in the decay region. By taking $\Delta\zeta$=$(0.20\pm0.04)$ m (a somewhat arbitrary $20\%$ error has been assumed), and $\left| \epsilon\right| =(1.95\pm0.2)\times10^{-3}$ as correlated values, eq.(\[Meps\]) gives:
$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;g_{E}=\left| \epsilon \right| /(\Lambda m_{K} \Delta \varsigma)= 8.9\pm 2.7 $ m [s]{}$^{-2}$,
0.3cm which is consistent with the standard* *$g_{E}%
$* *value. Since 1973 higher values of $\left| \epsilon\right| $ have been reported [@K--79] grouped finally around $2.27\times10^{-3}$ (the presently accepted value), perhaps with slightly larger $\Delta\zeta$ values.
If the present gravitational interpretation applies, the effect of the Earth’s field can introduce important biases in an experiment such as CPLEAR at CERN, where the effective maximum vertical displacement is roughly 0.4 m. This would make it necessary to reanalyze all CPLEAR results, such as those concerning CP violation parameters and the so-called direct T-reversal violation.
The four experiments NA31 and NA48 at CERN, and E731 and KTeV at Fermilab, have utilized and utilize long $K_{L}^{0}$ horizontal beams. Thus, their results could be biased by the effects of both the Earth’s and the Sun’s field. The latter effect depends on the beam length projected along the direction of the solar field, and this projected length continuously varies owing to Earth rotation and revolution. Thus, these long-beam experiments could be affected by non-trivial systematic errors, and their results, such as those concerning $Re(\epsilon^{\prime}/\epsilon)$ and time reversal $T$ violation, would all have to be reanalyzed.
Among the various topics concerning the $K^{0}$-$\overline{K^{0}}$ system, let us consider an argument due to Sakurai and Wattemberg [@Sakur]. They elegantly argued that CP violation in $2\pi$ $K_{L}^{0}$ decay is conclusively demonstrated by so-called soft regeneration, first observed by Fitch *et al*. [@Fitch]. This argument does not apply if one assumes that $K^{0}$ and $\overline{K^{0}}$ have opposite gravitational behavior, with the possibility that CP is an exact symmetry [@AntiS].
Coherently with the starting hypothesis of antigravity, one must expect that, for instance, antiprotons, antineutrons, and antiatoms are gravitationally repelled from the Earth, thus *antifalling* with the acceleration modulus $g_{E}$. One must take into account that antigravity, besides introducing a lot of problems in the standard physical picture, cannot coexist with the equivalence principle [@Eq-Prin]. Actually, it seems possible to maintain the agreement with a very large part of the known phenomena, by embedding the antigravity assumption in a set of suitable hypotheses [@Mamb]. In this way, it seems possible to avoid all paradoxes and difficulties (such as violation of energy conservation, causality violation, possible CPT violations, etc.), which antigravity implies when directly inserted into the standard physical picture, It also seems possible to circumvent any conflict with the so-well tested proportionality between the inertial and the gravitational mass [@MiMg].
Concluding remarks
==================
For the first time the KLOE experiment at DA$\mathit{\Phi}$NE offers the possibility of having important vertical $K_{L}^{0}$ displacements, thus allowing the Earth’s gravitational field to manifest its possible effects clearly. The proposal advanced in 1964 by Bernstein, Cabibbo, and Lee has been resumed, by adding to it the heterodox hypothesis that the gravitational field has a scalar component acting in opposite ways on the $K^{0}$ and $\overline{K^{0}}$ mixed in $|K_{L}^{0}\rangle$. We have shown that the CP-conserving mechanism seems to fit the known $K_{L}^{0}$ properties well. The main new consequence is the proportionality between the modulus of $2\pi$ $K_{L}^{0}$ decay branching ratio and the vertical $K_{L}^{0}$ displacement. At KLOE, this proportionality should be observed with a relatively small number of $2\pi$ $K_{L}^{0}$ decays. If KLOE data should validate this proportionality, it would become necessary to consider the above sketched questions, and many other related topics and problems as well.
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Good (Phys. Rev. **121** (1961) 311) assumed $V_{0}/2=m_{K}g_{E}\Delta
\zeta$ and argued by means of a *Gedanken* experiment, that $K_{L}^{0}$ would not exist if $K^{0}$ and $\overline{K^{0}%
}$ have opposite gravitational interactions with the Earth (the final part of Good’s argument is not consistently correct: see, for instance, T.Goldman and M.M.Nieto: Phys. Rep. **205** (1991) 221). Indeed, the *Gedanken* experiment can be seen as nothing else than an actual experiment with a horizontal $K_{L}^{0}$ beam, if one identifies Good’s device which can raise and lower $K_{L}^{0}$, with a small vertical component of the velocity of $K_{L}^{0}$ itself.
*Particle Data Group ’98*: Europ. Phys. J. C **3** (1998) 1.
R.Messner, *et al.*: Phys. Rev. Lett. **30** (1973) 876; C.Geweniger, *et al.*: Phys. Lett. B **48** (1974) 483; *idem*: *ibidem* **48** (1974) 487; G.Gjesdal, *et al.*: Phys. Lett. B **52** (1974) 119; R.Devoe, *et al.*: Phys. Rev. **16** (1977) 565; J.H.Christenson, *et al.*: Phys. Rev. Lett. **43** (1979) 1209; *idem*: *ibidem* **43** (1979) 1212.
J.J.Sakurai and A.Wattemberg: Phys. Rev. **161** (1967) 1449.
In the so-called soft regeneration experiments [@Fitch], low-density regenerators are used, so that the modulus of the complex regeneration parameter $\rho$ is roughly equal to $\left| \epsilon\right|
$. The experiments show that there is interference between $\rho$ and $\epsilon$ and that, therefore, the final two-pion states of $K_{S}^{0}$ and $K_{L}^{0}$ decays, are identical. This identity provides an absolute criterion for distinguishing matter from antimatter, because in an experiment performed on an Antiearth the relative $\rho$ and $\epsilon$ sign would change, thus exhibiting a matter-antimatter asymmetry, that is, a clear signature of CP violation. The Sakurai-Wattemberg argument does not apply if one assumes that $K^{0}$ and $\overline{K^{0}}$ have opposite gravitational behavior. In fact, when observing $2\pi\ K_{L}^{0}$ decay on an Antiearth, not only $\rho$ would reverse its sign as in the standard physical picture, but also $\epsilon$ would do the same, owing to the reversed gravitational interactions of both $K^{0}$ and $\overline{K^{0}}$. Then, the relative sign of $\rho$ and $\epsilon$ would not change, showing complete symmetry between matter and antimatter, thus reopening the possibility that CP is an exact symmetry.
In fact, it would be easy in principle for an experimenter to distinguish whether he belongs to an accelerated frame, or if he is inside a homogeneous gravitational field. It is enough to free a hydrogen atom $H$ and an antihydrogen atom $\overline{H}$ and observe their motion, because relative to an accelerated frame both atoms would have the same acceleration, while inside a gravitational field they would move with opposite accelerations. The use of a $H,$ $\overline{H}$ pair, instead of a charged particle-antiparticle pair (*e.g.*, a $p,$ $\overline{p}$ pair), can eliminate the many troubles linked to the weakness of gravitational coupling compared to electromagnetic coupling: see, for instance, the analysis made by Holzscheiter and Charlton (Rep. Progr. Phys.** 62** (1999) 1), and by Gabrielse *et al.* (Phys. Lett. B **455** (1999) 311); the latter authors have also shown that the possibility of producing “cold” $\overline{H}$ atoms, is open.
G.Mambriani: Riv. Mat. Univ. Parma (5) **2** (1993) 129.
P.G.Roll, R.Krotov, and R.H. Dicke: Ann. Phys. **26** (1964) 442; V.B. Braginskiĭ and V.Panov: Sov. Phys. JEPT **34** (1972) 463; Y.Su, *et al.*: Phys. Rev. D **50** (1994) 3614; J.O.Dickey, *et al.*: Science **265** (1994) 482; K.Nordtvedt: Icarus **114** (1995) 51.
[^1]: INFN Gruppo Collegato di Parma.
|
---
author:
- 'Yusuke Sugita[^1] and Yukitoshi Motome'
title: Topological Insulators from Electronic Superstructures
---
Quantum phenomena originating from the geometrical properties of electronic wave functions have been a central issue in modern condensed matter physics. The study of the anomalous Hall effect in ferromagnetic metals has revealed that the relativistic spin-orbit coupling (SOC) plays a crucial role in such phenomena [@RevModPhys.82.1539]. In particular, the strong SOC may bring about intriguing topological states of matter, such as the topological insulator (TI) [@RevModPhys.82.3045; @RevModPhys.83.1057; @doi:10.7566/JPSJ.82.102001]. The TI is a nontrivial band insulator, which is distinguished from conventional ones by a $Z_{2}$ topological invariant under the time-reversal symmetry. It exhibits a peculiar metallic edge (or surface) state, which gives rise to the quantized spin Hall effect in two-dimensional TIs. Such an unusual edge or surface state has been observed experimentally in several systems, e.g., a two-dimensional quantum well of CdTe/HgTe/CdTe [@Konig766] and three-dimensional bulk crystals of Bi$_{x}$Sb$_{1-x}$ [@hsieh2008topological].
Recently, electron correlations in the systems with strong SOC have attracted much interest. In weakly correlated systems, the band topology survives and the spontaneous symmetry breaking by electron correlations may lead to new types of topological phases, such as Weyl semimetals by spatial-inversion or time-reversal symmetry breaking [@doi:10.1146/annurev-conmatphys-020911-125138; @doi:10.1146/annurev-conmatphys-031113-133841; @doi:10.7566/JPSJ.83.061017]. On the other hand, strong electron correlations in the presence of strong SOC give rise to highly anisotropic exchange interactions in the Mott insulator, which may lead to unconventional quantum phases, e.g., quantum spin liquids [@doi:10.1146/annurev-conmatphys-020911-125138; @doi:10.1146/annurev-conmatphys-031115-011319]. Thus, the interplay between the SOC and electron correlations provides a key for new quantum phenomena, but the survey has only been initiated and many aspects remain unexplored.
In this Letter, we propose a new route to realize topological states of matter through the interplay between the SOC and electron correlations. In our scenario, spontaneous symmetry breaking takes place to form an electronic superstructure, such as a charge density wave, which brings about a topological nature in the band structure. We examine this scenario in a minimal model on a triangular lattice, which mimics some delafossite-type oxides [@ong2004electronic; @PhysRevLett.99.157204] and transition metal dichalcogenides [@0953-8984-23-21-213001; @chhowalla2013chemistry]. Using the mean-field approximation, we clarify the ground-state phase diagram at commensurate electron fillings while changing the SOC and electron correlations. We find that the system becomes TIs, in some specific charge-ordered states, where the charge disproportionation comprises a honeycomb or kagome superstructure. We show that such charge-ordered TIs are stabilized by the cooperation of the SOC and electronic correlations. Our results indicate the new possibility of realizing and controlling TIs through electronic superstructures.
![ (Color online) (a) Schematic picture of edge-sharing octahedra. The large (red) spheres inside the octahedra denote the transition metal cations and the small (gray) ones on the vertices indicate the ligand ions. (b) Atomic $d$ orbital levels of the transition metal cations under the octahedral and trigonal crystalline electric fields corresponding to (a). The $e'_{g}$ orbitals are further split by the SOC; see the text for details. (c) Schematic picture of the triangular lattice of transition metal cations. $\bm{\eta}_n$ ($n=1,2,3$) are the primitive translational vectors. (d) Energy levels and hopping processes in the two-orbital model in Eq. (\[ModOne\]). []{data-label="Setup"}](Fig1.eps){width="48.00000%"}
To investigate the spontaneous formation of electronic superstructures and resultant topological nature, we consider a minimal model on a triangular lattice. We begin with the edge-sharing octahedra composed of transition metal cations and ligands, as shown in Fig. \[Setup\](a). Note that a similar situation is realized in delafossite compounds [@ong2004electronic; @PhysRevLett.99.157204] and 1T-type transition metal dichalcogenides [@0953-8984-23-21-213001; @chhowalla2013chemistry]. When the octahedral and trigonal crystalline electric fields are sufficiently large, the $d$ orbitals in the transition metal cations are split into three groups, $e_g$, $e'_g$, and $a_{1g}$, as shown in Fig. \[Setup\](b). Assuming that the Fermi level is at the $e'_g$ manifold (otherwise, the SOC is rather irrelevant), we take into account only the $e'_g$ orbitals and omit the others. The $e'_g$ states are denoted by $|m=\pm1,\sigma\rangle = (|xy,\sigma\rangle + e^{\pm i\omega}|yz,\sigma\rangle + e^{\mp i\omega}|zx,\sigma\rangle)/\sqrt{3}$, where $\omega = 2\pi/3$ and $xyz$-axes are taken as shown in Fig. \[Setup\](a); $\sigma=\pm1$ denotes the spin, whose quantization axis is taken along the (111) direction. Under these assumptions, we construct a tight-binding model for the triangular lattice composed of transition metal cations with $e'_g$ orbitals, whose one-body Hamiltonian is given by $$\begin{aligned}
H_{0}
=
&-t_{0}\sum_{\bm{k}}\sum_{m,\sigma=\pm1}
\gamma_{0\bm{k}}c^{\dagger}_{\bm{k}m\sigma}c_{\bm{k}m\sigma}
-t_{1}\sum_{\bm{k}}\sum_{m,\sigma=\pm1}
\gamma_{m\bm{k}}c^{\dagger}_{\bm{k}m\sigma}c_{\bm{k}-m\sigma} \nonumber \\
&+\frac{\lambda}{2}\sum_{\bm{k}}\sum_{m,\sigma=\pm1}
(m\sigma)c^{\dagger}_{\bm{k}m\sigma}c_{\bm{k}m\sigma},
\label{ModOne}\end{aligned}$$ where $c^{\dagger}_{\bm{k} m \sigma}$($c_{\bm{k} m \sigma}$) is the creation (annihilation) operator of an electron for the wave vector $\bm{k}$, orbital $m = \pm1$, and spin $\sigma =\pm1$. $t_{0}$ and $t_{1}$ are the intra- and interorbital hopping elements between nearest-neighbor sites, respectively. The factors $\gamma_{\alpha\bm{k}}$ ($\alpha=0,\pm1$) in the hopping terms in Eq. (\[ModOne\]) are given by $$\begin{aligned}
\gamma_{\alpha\bm{k}} = \sum_{n=1,2,3} 2 e^{2(n-1)i \alpha\omega}\cos{(\bm{k}\cdot \bm{\eta}_{n})},
\label{ModGam}\end{aligned}$$ which originate from the directional dependences of the overlaps between $|xy \rangle$, $|yz \rangle$, and $|zx \rangle$ orbitals. Hereafter, we take the triangular plane as the $x'y'$ plane, and set the primitive translational vectors for the triangular lattice as $\bm{\eta}_{1} = (1, 0)$, $\bm{\eta}_{2} = (1/2, \sqrt{3}/2)$, and $\bm{\eta}_{3} = (-1/2, \sqrt{3}/2)$ \[see Fig. \[Setup\](c)\]. $\lambda$ is the SOC constant, which splits the energy levels of $|m,\sigma\rangle$ into two Kramers doublets, $\{|+1,+1\rangle,|-1,-1\rangle\}$ and $\{|+1,-1\rangle,|-1,+1\rangle\}$, when $t_0=t_1=0$, as shown in Fig. \[Setup\](b). The energy levels and hopping processes are schematically shown in Fig. \[Setup\](d). Note that a similar model was studied on a honeycomb lattice [@PhysRevB.90.081115; @1742-6596-592-1-012131]. Although the honeycomb lattice model exhibits topologically nontrivial states even in the noninteracting case [@1742-6596-592-1-012131], our triangular lattice model in Eq. (\[ModOne\]) does not for any values of the parameters $t_{0}$, $t_{1}$, and $\lambda$. Hereafter, we set $t_{0}=0.5$ and $t_{1}=0.25$, which are reasonable when considering the $d$-$d$ direct and $d$-$p$-$d$ indirect hoppings in the Slater–Koster scheme [@PhysRev.94.1498].
In addition to the one-body part, we take into account both the onsite and intersite Coulomb interactions. The onsite one is given by $$H^{\rm onsite}_{1} =
\frac{1}{2} \sum_{mnm'n'} U_{mnm'n'}
\sum_{i}
\sum_{\sigma \sigma'}
c^{\dagger}_{i m\sigma} c^{\dagger}_{i n\sigma'} c_{i n'\sigma'} c_{im'\sigma}.
\label{ModOns}$$ Assuming the rotational symmetry of the Coulomb interaction, we set $U_{mmmm} = U$, $U_{mnmn} = U-2J$, and $U_{mnnm}=U_{mmnn}=J$ ($m\neq n$), where $U$ is the intraorbital Coulomb interaction and $J$ is the Hund’s coupling, respectively. We set $U=1.0$ and $J/U=0.1$ in the following calculations. For the intersite interaction, we consider the density-density repulsions given by $$H^{\rm intersite}_{1}
=
V_1 \sum_{\langle i,j \rangle} n_{i}n_{j} + V_2\sum_{\langle\langle i,j \rangle\rangle} n_{i}n_{j},
\label{ModInt}$$ where $n_{i} = \sum_{m\sigma}c^{\dagger}_{i m\sigma} c_{i m\sigma}$. The sum of $\langle i,j \rangle$ ($\langle\langle i,j \rangle\rangle$) is taken for the nearest- (next-nearest-) neighbor sites.
To clarify the ground states of the two-orbital model given by Eqs. (\[ModOne\]), (\[ModOns\]), and (\[ModInt\]), we use the mean-field approximation. In the mean-field calculation, we employ 12 sublattices \[see Figs. \[BandHon\](c) and \[BandKag\](c)\] and approximate the integration in the folded Brillouin zone by the summation over $64\times64$ $\bm{k}$ points. We apply the Hartree-Fock approximation to the onsite interaction in Eq. (\[ModOns\]) and the Hartree approximation to the intersite interaction in Eq. (\[ModInt\]) to focus on charge order[@note1]. The mean fields are determined self-consistently, until they converge within a precision of less than $10^{-6}$. We investigate the ground state and find several interesting charge orders at nearly 1/3 electron filling, $\langle \sum_i n_i \rangle/(4N) \sim 1/3$, where $N$ is the number of lattice sites.
In addition, we compute the spin Hall conductivity, which signals the nontrivial topological nature of the system; it can be quantized at a nonzero integer multiple value of $e/2\pi$ for two-dimensional TIs [@RevModPhys.82.3045] ($e$ is the elementary charge). Using the standard Kubo formula in the linear response theory, we calculate the spin Hall conductivity as $$\sigma^{s}_{xy}
=
\frac{e}{2}\frac{1}{i\Omega}
\sum_{\bm{k} \alpha\beta}
\frac{f(\varepsilon_{\alpha\bm{k}}) - f(\varepsilon_{\beta\bm{k}})}{\varepsilon_{\alpha\bm{k}} - \varepsilon_{\beta\bm{k}}}
\frac{\langle \alpha\bm{k}| j^{s}_{x} |\beta\bm{k} \rangle \langle \beta\bm{k}| j_{y} |\alpha\bm{k} \rangle}{\varepsilon_{\alpha\bm{k}}-\varepsilon_{\beta\bm{k}}+i\delta},
\label{SHC}$$ where $\Omega$ is the system volume, $f$ is the Fermi distribution function, $\varepsilon_{\alpha\bm{k}}$ and $|\alpha\bm{k} \rangle$ are the eigenvalues and eigenvectors of the $\alpha$th electronic band with the wave vector $\bm{k}$ in the mean-field solution, respectively, and $\delta$ is the infinitesimal positive parameter. In the calculation of Eq. (\[SHC\]), we take the summation over $512\times512$ $\bm{k}$ points and set $T=10^{-3}$ and $\delta=10^{-3}$. The current operator is defined by $j_{y} \equiv \partial H_{\rm MF}/\partial k_{y}$, where $H_{\rm MF}$ is the mean-field Hamiltonian. We define the spin current operator as $j^{s}_{x} \equiv \{ j_{x}, \sigma _{z} \}/2$, where $\sigma_z$ is the $z$-component of the Pauli matrices for spin; note that $H_{\rm MF}$ for our mean-field solutions commutes with $\sigma_z$. Hereafter, we denote $\tilde{\sigma}^{s}_{xy} \equiv \sigma^{s}_{xy}/(e/2\pi)$ as the normalized spin Hall conductivity. We note that, for two-dimensional systems whose Hamiltonian commutes with $\sigma_z$, $\tilde{\sigma}^{s}_{xy}$ is directly related to the $Z_{2}$ topological invariant [@RevModPhys.82.3045].
![ (Color online) (a) Ground-state phase diagram for the model given by Eqs. (\[ModOne\]), (\[ModOns\]), and (\[ModInt\]) at 1/3 filling obtained by the mean-field approximation. We set $U=1.0$, $J/U=0.1$, and $V_2=0$. A schematic picture of the charge ordering pattern is shown in each phase. The size of the circle represents the magnitude of the local charge density at each sublattice. In the type-B COI phase, the band gap closes on the red dashed line, which corresponds to the phase boundary between the TI and trivial band insulator. (b) $V_1$ dependence of the local charge density at each sublattice at $\lambda=2.0$. (c) $\lambda$ dependences of the band gap and the normalized spin Hall conductivity at $V_{1}=0.6$. []{data-label="MF1/3"}](Fig2.eps){width="48.00000%"}
First, we consider the situation where $V_{1}$ is dominant rather than $V_{2}$, and thus, set $V_{2}=0$. In this case, the system shows an interesting behavior at 1/3 filling. Figure \[MF1/3\](a) shows the ground-state phase diagram obtained by the mean-field approximation while changing $\lambda$ and $V_{1}$. We find three different phases in this parameter region: paramagnetic metal (PM) for small $V_1$ and two charge-ordered insulators (COIs) for large $V_1$. In both COIs, the local charge density is disproportionated to form a honeycomb superstructure; in the COI in the larger $V_1$ region, the local charge density is lower at the sites belonging to the honeycomb network than at the isolated sites, while they are opposite in the COI in the intermediate $V_1$ and large $\lambda$ region \[see the schematic pictures in Fig. \[MF1/3\](a)\]. The local charge densities are plotted in Fig. \[MF1/3\](b). We call the former (latter) the honeycomb type-A(B) COI.
In the small $\lambda$ region, there is a transition from the paramagnetic metal to the honeycomb type-A COI with increasing $V_1$. This is easily understood by considering that the electrons tend to avoid each other under large $V_1$ and the lowest energy configuration at 1/3 filling is given by the type-A charge ordering. On the other hand, in the large $\lambda$ region, the type-B COI appears between the type-A COI and PM phases. The intervening type-B COI is stabilized by the synergy between the strong SOC and intersite Coulomb repulsion. This is understood by considering the large $\lambda$ limit as follows. As the bands for the Kramers pair are largely split from each other, the two-orbital model at 1/3 filling reduces to a single-band model at 2/3 filling for the lower-energy band. In the single-band model at 2/3 filling, the lowest energy state under $V_1$ is given by the type-B charge ordering, which explains why the type-B COI is stabilized in the large $\lambda$ and $V_1$ region in Fig. \[MF1/3\](a). We note that all the phase boundaries in Fig. \[MF1/3\](a) are of first order with discontinuous changes in local charge densities.
Figure \[MF1/3\](c) shows $\lambda$ dependences of the energy gap and normalized spin Hall conductivity at $V_1=0.6$. The results clearly indicate that both CO states are gapped insulators, while the type-A COI has a larger gap than the type-B COI in this parameter region. Remarkably, in the type-B CO state, the band gap once closes around $\lambda \sim 3.2$. The gapless line inside the type-B CO phase is shown by the dashed line in Fig. \[MF1/3\](a). The result indicates that the type-B CO phase may include two different insulating states separated by the gapless boundary. Indeed, as shown in Fig. \[MF1/3\](c), the normalized spin Hall conductivity in the type-B COI is quantized at $-1$ for $\lambda \lesssim 3.2$, while it changes discontinuously to zero when crossing the gapless point. Therefore, the gapless boundary in the type-B COI corresponds to a topological transition between a TI for smaller $\lambda$ and a trivial band insulator for larger $\lambda$.
![ (a) Electronic band structure for the type-B COI at $V_{1}=0.6$ and $\lambda=3.242$. (b) Enlarged figure of the band structures near the Fermi level around the $\Gamma$ point along the K’-K line, for $V_{1}=0.6$ and $\lambda=3.220$, $3.242$, and $3.260$. (c) Schematic picture of the three-site unit cell (gray triangle) used for drawing the electronic band structures. $\bm{a}_{n}$ ($n=1,2$) are the primitive translational vectors. (d) Schematic picture of the folded Brillouin zone for the unit cell in (c). $\bm{b}_{n}$ ($n=1,2$) are the reciprocal lattice vectors. []{data-label="BandHon"}](Fig3.eps){width="48.00000%"}
To clarify the electronic states in the type-B COI further, we show the electronic band structure of the mean-field solution for the type-B COI near the gapless boundary in Figs. \[BandHon\](a) and \[BandHon\](b) \[the unit cell and Brillouin zone are shown in Figs. \[BandHon\](c) and \[BandHon\](d), respectively\]. As shown in Fig. \[BandHon\](a), the Kramers doublets are split by the strong SOC into two ‘copies’ of three bands; the highest band in the lower three bands hybridizes with the lowest one in the higher three bands, resulting in a small gap at $\varepsilon\sim1.1$. The three bands in each copy are composed of two subsets, reflecting the honeycomb CO superstructure; the lower two bands comprise the dispersive bands similar to those of the single-band model on the honeycomb lattice, and the remaining higher band is less dispersive as it comes from the isolated sites in the honeycomb hexagons. The lower honeycomb-like bands are occupied (the Fermi level is set at zero). This result supports the above discussion for the origin of the type-B COI.
Figure \[BandHon\](b) shows more details of the band structures near the Fermi level at 1/3 filling around the $\Gamma$ point at $V_{1}=0.6$. With increasing $\lambda$, the band gap at 1/3 filling decreases and closes at $\lambda\sim3.242$. In the gapless state, the low-energy dispersions are well approximated by the massless Dirac cone. The Dirac cone is gapped out again by further increasing $\lambda$. We note that, although this topological transition appears to share the fundamental mechanism with that found for the similar two-orbital model on a honeycomb lattice [@1742-6596-592-1-012131], the critical value of $\lambda$ is largely reduced by the mean-field contribution from electron correlations. In other words, electron correlations enhance the effective SOC for realizing the topological state of matter.
![ (Color online) (a) Ground-state phase diagram for the model given by Eqs. (\[ModOne\]), (\[ModOns\]), and (\[ModInt\]) at 3/8 filling obtained by the mean-field approximation. We set $U=1.0$, $J/U=0.1$, and $V_{1}=0.6$. Schematic picture of the charge ordering pattern is shown in each phase. The size of the circle represents the magnitude of the local charge density at each sublattice. In the kagome CO phase, the orange dashed line separates the metallic and insulating regions. (b) $V_2$ dependences of the charge density at each sublattice, the band gap, and the normalized spin Hall conductivity at $\lambda=0.6$. []{data-label="MF3/8"}](Fig4.eps){width="48.00000%"}
Next, we take into account the next-nearest-neighbor repulsion $V_{2}$. We find that $V_2$ leads to different types of electronic superstructures around 1/3 filling. In particular, here, we discuss interesting CO states appearing at 3/8 filling. Figure \[MF3/8\](a) shows the ground-state phase diagram at $V_{1}=0.6$ while changing $\lambda$ and $V_{2}$. In the small $V_{2}$ region, the system exhibits a honeycomb type-A CO metal (COM) as well as PM, whose charge patterns are also seen in the 1/3 filling case above [@note2]. When increasing $V_{2}$, we find two new CO phases: kagome and stripy CO phases \[see the schematic picture in the phase diagram in Fig. \[MF3/8\](a)\]. In the kagome CO state, the charge density is disproportionated so that the charge-poor sites comprise a kagome superstructure, as plotted in Fig. \[MF3/8\](b). (The charge-poor sites have a very small charge disproportionation among them, which does not affect the following topological nature of this phase.) On the other hand, the stripy CO state has a four-sublattice order, where the charge density is disproportionated into three groups: charge-rich, charge-poor, and intermediate at one, two, and one sublattices, respectively \[see Fig. \[MF3/8\](b)\].
![ (a) and (b) Electronic band structure for the kagome CO phase at $V_{1}=0.6$, $V_{2}=0.15$, and (a) $\lambda=0.0$ and (b) $\lambda=0.6$. (c) Schematic picture of the unit cell used for drawing the electronic band structures of the kagome CO phase. The gray region indicates the unit cell composed of four sites. $\bm{a}_{n}$ ($n=1,2$) are the primitive translational vectors. (d) Schematic picture of the folded Brillouin zone for the unit cell in (c). $\bm{b}_{n}$ ($n=1,2$) are the reciprocal lattice vectors. []{data-label="BandKag"}](Fig5.eps){width="48.00000%"}
The kagome CO state is intriguing from the topological viewpoint, as discussed below. In Fig. \[MF3/8\](b), we plot the $V_2$ dependence of the energy gap at $\lambda=0.6$. The result shows that the kagome CO phase is metallic in the small $V_2$ region but becomes insulating with increasing $V_2$. The band gap is opened by the cooperation between the SOC and $V_{2}$ \[see also Fig. \[MF3/8\](a)\]. This is explicitly shown in the band structures for $V_2=0.15$ in Figs. \[BandKag\](a) and \[BandKag\](b) at $\lambda=0.0$ and $\lambda=0.6$, respectively \[the unit cell and Brillouin zone are shown in Figs. \[BandKag\](c) and \[BandKag\](d), respectively\]. Although the bands near the Fermi level do not have a gap at $\lambda=0.0$, they are separated by a gap for $\lambda=0.6$. We find that the kagome COI is a TI by calculating the normalized spin Hall conductivity, as shown in Fig. \[MF3/8\](b). Although the normalized spin Hall conductivity is already nonzero in the honeycomb type-A and kagome COM phases for smaller $V_2$, it is quantized at a nonzero integer number, $\tilde{\sigma}^{s}_{xy}=-1.0$ in the kagome COI.
Finally, let us discuss our results. We found two different types of COIs which are topologically nontrivial: the honeycomb type-B and kagome COIs. The important physics here is the role of the SOC under the electronic superstructures. As remarked above, the noninteracting model including the SOC \[Eq. (\[ModOne\])\] does not exhibit any topological nature owing to the high symmetry of the triangular lattice. The formation of the honeycomb and kagome superstructures activates the hidden SOC effect and changes the system into TIs. This is, for instance, understood from the relationship between the electronic states of the honeycomb type-B COI in the present model and the PM in the honeycomb-lattice model studied in a previous work [@1742-6596-592-1-012131], as discussed above. The situation is distinct from other interaction-driven TIs, the so-called topological Mott insulators [@doi:10.7566/JPSJ.83.061017; @PhysRevLett.100.156401; @PhysRevB.82.045102; @PhysRevB.82.075125], where the atomic SOC does not play an important role [@note1].
Similar mechanisms activating the SOC effect by superstructure formation were discussed for spatial inversion symmetry breaking, which induces the antisymmetric SOC [@PhysRevB.90.081115; @doi:10.7566/JPSJ.83.014703; @doi:10.7566/JPSJ.83.114704]. Thus, our results point to a much broader route to activate the nontrivial SOC physics and realize topological states of matter, with the aid of the change of spatial symmetry by electronic correlations. This has richer implications, since, in addition to charge ordering, the superstructure formation can be caused by other degrees of freedom, e.g., magnetic ordering in spin-charge coupled systems [@PhysRevLett.109.237207; @PhysRevB.88.100402; @PhysRevB.91.155132] and bond ordering in electron-phonon coupled systems. Interestingly, there are many candidate materials exhibiting various superstructures, e.g., delafossite-type oxides [@ong2004electronic; @PhysRevLett.99.157204] and transition metal dichalcogenides [@0953-8984-23-21-213001; @chhowalla2013chemistry]. In particular, the latter compounds are intriguing, as they show a variety of charge density waves with longer periodicities accompanied by lattice distortions. The topological nature in these interesting states with electronic superstructures is left for a future study.
The authors thank Satoru Hayami, Hiroaki Kusunose, Takahiro Misawa, and Youhei Yamaji for constructive suggestions. Y.S. is supported by the Japan Society for the Promotion of Science through the Program for Leading Graduate Schools (MERIT). This work was supported by Grants-in-Aid for Scientific Research (Nos. 24340076 and 15K05176), the Strategic Programs for Innovative Research (SPIRE), MEXT, and the Computational Materials Science Initiative (CMSI), Japan.
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[^1]: Email:[email protected]
|
---
abstract: '*Time-reversal symmetry*, which requires that the dynamics of a system should not change with the reversal of time axis, is a fundamental property that frequently holds in classical and quantum mechanics. In this paper, we propose a novel loss function that measures how well our ordinary differential equation (ODE) networks comply with this time-reversal symmetry; it is formally defined by the discrepancy in the time evolution of ODE networks between forward and backward dynamics. Then, we design a new framework, which we name as *Time-Reversal Symmetric ODE Networks (TRS-ODENs)*, that can learn the dynamics of physical systems more sample-efficiently by learning with the proposed loss function. We evaluate TRS-ODENs on several classical dynamics, and find they can learn the desired time evolution from observed noisy and complex trajectories. We also show that, even for systems that do not possess the full time-reversal symmetry, TRS-ODENs can achieve better predictive errors over baselines.'
author:
- |
In Huh$^{1}$, Eunho Yang$^{2,3}$, Sung Ju Hwang$^{2,3}$, Jinwoo Shin$^{2}$\
$^{1}$Samsung Electronics; $^{2}$Korea Advanced Institute of Science and Technology (KAIST); $^{3}$AItrics\
`[email protected], [email protected], {sjhwang82, jinwoos}@kaist.ac.kr`\
bibliography:
- 'ms.bib'
title: 'Time-Reversal Symmetric ODE Network'
---
Introduction {#sec:1}
============
Recent advances in artificial intelligence allow researchers to recover laws of physics and predict dynamics of physical systems from observed data by utilizing machine learning techniques, e.g., evolutionary algorithms [@schmidt09; @lonie11], sparse optimizations [@schaeffer17; @brunton16], Gaussian process regressions [@seikel12; @cui2015], and neural networks [@iten20; @battaglia16; @greydanus19; @zhong19; @sanchez19]. Among various models, the neural networks are considered as one of the most powerful tools to model complicated physical phenomena, owing to their remarkable ability to approximate arbitrary functions [@hornik89]. One notable aspect of the observations in physical systems is that they manifest some fundamental properties including conservation or invariance [@goldstein02; @bluman13]. However, it is not straightforward for neural networks to learn and model the embedded physical properties from observed data only. Consequently, they often overfit to short-term training trajectories and fail to predict the long-term behaviors of complex dynamical systems [@greydanus19; @zhong19].
To overcome these issues, it is important to introduce appropriate inductive biases based on knowledge of physics, dynamics and their properties [@zhong19; @sanchez19]. Common approaches to incorporate physics-based inductive bias include modifying neural network architectures [@schutt17; @schutt19] or introducing regularization terms based on specialized knowledge of physics and natural sciences [@nabian18; @long18]. These methods demonstrate impressive performance on their target problems, but such a problem-specific model cannot generalize across domains. As for more general approaches, the authors in [@chen18; @chang19] propose the *ordinary differential equation (ODE) networks*, which view the neural networks as parameterized ODE functions. They are shown to be able to represent the vast majority of dynamical systems with higher precision over vanilla recurrent neural networks and their variants [@chen18; @chang19], but are still unable to learn underlying physics such as the law of conservation [@greydanus19]. Recent works [@greydanus19; @zhong19; @sanchez19; @chen19; @toth19] apply the Hamiltonian mechanics to ODE networks, and succeed in enforcing the energy conservation as well as the accurate time evolution of classical conservative systems. However, these Hamiltonian ODE networks have inherent limitations that they cannot be applied to non-conservative systems, since the Hamiltonian structures require to strictly conserve the total energy [@greydanus19].
To address such limitations of existing works on modeling classical dynamics, we introduce a physics-inspired, general, and flexible inductive bias, *symmetries*. It is at the heart of the physics: the laws of physics are invariant under certain transformations in space and time coordinates, thus show the universality [@elliott79; @livio12]. For example, the classical dynamics possess the *time-reversal symmetry*, which means the classical equations of motion should not change under the transformation of time reversal: $t \mapsto -t$ [@lamb98; @roberts92] (see Figure \[fig:1\]). Therefore, if the target underlying physics being approximated has some symmetries, it is natural that the approximated physics using neural networks should also comply with these properties. Motivated by this, we feed the symmetry as an additional information to help neural networks learn the physical systems more efficiently.
![(a) **Time-reversal symmetry of dynamical systems.** The gray ellipse is a phase space trajectory, which do not change under $t \mapsto -t$. The reversing of forward time evolution (blue arrows) of an arbitrary state should yield an equal state to what is estimated by the the backward time evolution of the reversed state (orange arrows). For more mathematical details, see Section \[sec:3.2\]. Examples of (b) non-linear and (c) non-ideal dynamical systems modeled by various ODE networks including TRS-ODENs. TRS-ODENs can learn appropriate long-term dynamics from noisy and short-term training samples.[]{data-label="fig:1"}](figure1.png){width="1.0\linewidth"}
Specifically, we focus on the *time-reversal symmetry* of classical dynamics described above, due to its simplicity and popularity. We propose a new ODE learning framework, which we refer to as *Time-Reversal Symmetry ODE Network (TRS-ODEN)*, that utilize the time-reversal symmetry as a regularizer in training ODE networks, by unifying recent studies of ODE networks [@chen19] and classical symmetry theory for ODE systems [@lamb98]. Our scheme can be easily implemented with a small modification of codes for conventional ODE networks, and is also compatible with extensions of ODE networks, such as Hamiltonian ODE networks [@zhong19; @sanchez19; @chen19]. It can be used to predict many branches of physical systems, because the isolated classical and quantum dynamics exhibit the perfect time-reversal symmetry [@lamb98; @sachs87]. Moreover, even for the case when the full time-reversal symmetry are broken [@lamb98], e.g., in the presence of interaction with environments through friction or energy transfer, we also show that TRS-ODENs are beneficial to learn such system by annealing the strength of the proposed regularizer appropriately. This flexibility with regard to the target problem is the main advantage of the proposed framework, in contrast to prior methods, e.g., only for suitable explicitly conservative systems [@greydanus19]. In summary, our contribution is threefold:
- We propose a novel loss function that measures the discrepancy in the time evolution of ODE networks between forward and backward dynamics, thus estimate whether the ODE networks are time-reversal symmetric or not.
- We show ODE networks with the proposed loss, coined TRS-ODENs, achieve better predictive error than baselines, e.g., from 50.81 to 10.85 for non-linear oscillators.
- We validate even for time-irreversible systems, the proposed framework still works well compared to baselines, e.g., from 3.68 to 0.12 in terms of error for damped oscillators.
Background and Setup {#sec:2}
====================
Predicting dynamical systems {#sec2:1}
----------------------------
In a dynamical system, its states evolve over time according to the governing time-dependent differential equations. The state is a vector in the phase space, which consists of all possible positions and momenta of all particles in the system. If one knows the governing differential equation and initial state of the system, the future state is predictable by solving the equation analytically or numerically. On the other hand, if one does not know the exact governing equation, but has some state trajectories of the system, one can try to model the dynamical system, e.g., by using neural networks. More specifically, one can build a neural network whose input is current state (or trajectory) and the output is the next state, from the perspective of the sequence prediction. However, such a method may overfit to short-term training trajectories and fail to predict the long-term behaviors [@zhong19]. It is also not straightforward to predict the continuous-time dynamics, because neural network models typically assume the discrete time-step between states [@greydanus19].
Neural ODE and its applications [@chen18; @chang19; @greydanus19; @zhong19; @sanchez19; @chen19; @toth19], alias ODE networks (ODENs), tackle these issues by learning the governing equations, rather than the state transitions directly. Moreover, some of them use special ODE functions such as Hamilton’s equations to incorporate physical properties to neural network structurally [@greydanus19; @zhong19; @sanchez19; @chen19; @toth19]. In the rest of this section, we briefly review ODENs and Hamiltonian ODE networks (HODENs), which are closely related to our work.
ODE networks (ODENs) for learning and predicting dynamics {#sec:2.2}
---------------------------------------------------------
We consider dynamics of state $\mathbf{x}$ in phase space $\Omega$ ($=\mathbb{R}^{2n}$, in classical dynamics[^1]) given by: $$\label{eq:1}
\frac{d\mathbf{x}}{dt} = f(\mathbf{x}) \quad \text{for } t \in \mathbb{R},\, \mathbf{x} \in \Omega,\, f: \Omega \mapsto T\Omega.$$ The continuous time evolution between arbitrary two time points $t_i$ and $t_{i+1}$ by (\[eq:1\]) is equal to: $$\label{eq:2}
\mathbf{x}({t_{i+1}}) = \mathbf{x}({t_i}) + \int_{t_i}^{t_{i+1}}{f(\mathbf{x})dt}.$$ The recent works [@zhong19; @sanchez19; @chen18; @chen19] propose the ODENs, which represent the ODE functions $f$ in (\[eq:1\]) by neural networks and learn the unknown dynamics from data. For ODENs, fully-differentiable numerical ODE solvers are required to train the black-box ODE functions, e.g., Runge-Kutta method [@dormand80] or symplectic integrators such as leapfrog method [@leimkuhler04]. With an ODE solver, say $\mathtt{Solve}$, one can obtain the estimate time evolution by ODENs: $$\label{eq:3}
\tilde{\mathbf{x}}({t_{i+1}})=\mathtt{Solve}\{\tilde{\mathbf{x}}({t_i}),f_{\theta},{\Delta}t_i\},\, \tilde{\mathbf{x}}({t_0}) = \mathbf{x}({t_0}),$$ where $f_{\theta}$ is a $\theta$-parameterized neural network, $\tilde{\mathbf{x}}(t_i)$ is a prediction of ${\mathbf{x}}(t_i)$ using ODENs, ${\Delta}t_i = t_{i+1} - t_i$ is a time-step, and $\mathbf{x}(t_0)$ is a given initial value. Given observed trajectory ${\mathbf{x}}(t_1),...,{\mathbf{x}}(t_T)$, ODENs can learn the dynamics by minimizing the mean-squared lose function $\mathcal{L}_\text{ODE} \equiv \sum\nolimits_{i=0}^{T-1} \| {\mathtt{Solve}\{\tilde{\mathbf{x}}(t_i), f_\theta, \Delta{t_i}\} - \mathbf{x}}(t_{i+1}) \|_2^2$.
Hamiltonian ODE networks (HODENs) {#sec:2.3}
---------------------------------
The Hamiltonian mechanics describes the phase space equations of motion for conservative systems by following two first-order ODEs called Hamilton’s equations [@goldstein02]: $$\label{eq:4}
\frac{d\mathbf{q}}{dt}= {\nabla_{\mathbf{p}}}\mathcal{H}(\mathbf{q}, \mathbf{p}),\, \frac{d\mathbf{p}}{dt}= -{\nabla_{\mathbf{q}}}\mathcal{H}(\mathbf{q}, \mathbf{p}),$$ where $\mathbf{q} \in \mathbb{R}^n$, $\mathbf{p} \in \mathbb{R}^n$, and $\mathcal{H}: \mathbb{R}^{2n} \mapsto \mathbb{R}$ are positions, momenta, and Hamiltonian of the system, respectively. Recent works [@zhong19; @sanchez19; @chen19] apply the Hamilton’s equations to ODENs, by parameterizing the Hamiltonian as $\mathcal{H}_\theta$, and replacing $f_\theta(\mathbf{q}, \mathbf{p})$ to the gradients of $\mathcal{H}_\theta$ with respect to inputs $(\mathbf{p}, \mathbf{q})$ according to (\[eq:4\]). Thus, the time evolution of HODENs is equal to: $$\label{eq:5}
(\tilde{\mathbf{q}}({t_{i+1}}), \tilde{\mathbf{p}}({t_{i+1}})) = \mathtt{Solve}\{(\tilde{\mathbf{q}}({t_i}), \tilde{\mathbf{p}}({t_i})), (\nabla_{p}\mathcal{H}_{\theta}, -\nabla_{q}\mathcal{H}_{\theta}), {\Delta}t_i\}.$$ HODENs shows better predictive performance for conservation systems. Furthermore, they can lean the underlying law of conservation of energy automatically, because they fully exploit the nature of the Hamiltonian mechanics [@greydanus19]. However, a fundamental limitation of HODENs is that they do not work properly for the non-conservative systems [@greydanus19], because they always conserve the energy.
Time-Reversal Symmetry Inductive Bias for ODENs {#sec:3}
===============================================
Target problems {#sec:3.1}
---------------
Before introducing the time-reversal symmetry, we briefly explain two perspectives of the classical dynamical systems: *conservative* and *reversible*. The former is the system that its Hamiltonian does not depend on time explicitly, i.e., $\partial{\mathcal{H}}/\partial{t} = 0$. The latter is the system that possesses the time-reversal symmetry, whose mathematical details will be discussed in the following section.
[**Conservative and reversible systems.**]{} All conservative systems that their Hamiltonians satisfy $\mathcal{H}(\mathbf{q}, \mathbf{p})$ = $\mathcal{H}(\mathbf{q}, -\mathbf{p})$ are also reversible [@lamb98]. It means that many kinds of classical dynamics are both conservative and reversible[^2]. For these systems, both Hamiltonian and time-reversal symmetry inductive biases are appropriate. Furthermore, combining two inductive biases can improve the sample efficiency of a learning scheme.
[**Non-conservative and reversible systems.**]{} It is noteworthy that reversible systems are not necessarily conservative systems. Some examples about non-conservative but reversible systems can be found in [@lamb98; @roberts92]. Clearly, baselines such as HODENs that enforce conservative property would break down in this environment. On the other hand, our scheme, named TRS-ODEN, presented in Section \[sec:3.3\] would accurately model the dynamics of given data by exploiting time-reversal symmetry.
[**Non-conservative and irreversible systems.**]{} Under interactions with environments, the dynamical systems become non-conservative and often irreversible[^3]. Depending on the intensity of such interactions, the Hamiltonian or time-reversal symmetry inductive bias can be beneficial or harmful. HODENs strictly enforce the conservation, thus they are not suitable for this [@greydanus19]. On the other hand, TRS-ODENs are more flexible, since they use the inductive bias as a form of regularizer, which is easily controlled via hyper-parameter tuning [@schmidhuber15].
Time-reversal symmetry in dynamics {#sec:3.2}
----------------------------------
First-order ODE systems (\[eq:1\]) are said to be *time-reversal symmetric* if there is an invertible transformation $R: \Omega \mapsto \Omega$, that reverses the direction of time: $$\label{eq:6}
\frac{dR(\mathbf{x})}{dt} = -f(R(\mathbf{x})),$$ where $R$ is called *reversing operator* [@lamb98]. Comparing (\[eq:1\]) and (\[eq:6\]), one can find that the equation is invariant under the transformations of phase space $R$ and time-reversal $t \mapsto -t$. For notational simplicity, let’s introduce a time evolution operator $U_{\tau}: \Omega \mapsto \Omega$ for (\[eq:1\]) as follows [@lamb98]: $$\label{eq:7}
U_{\tau}: \mathbf{x}(t) \mapsto U_{\tau}(\mathbf{x}(t)) = \mathbf{x}(t + \tau),$$ for arbitrary $t, \tau \in \mathbb{R}$. Then, in terms of the time evolution operator (\[eq:7\]), (\[eq:6\]) imply: $$\label{eq:8}
R \circ U_{\tau} = U_{-\tau} \circ R,$$ which means that *the reversing of the forward time evolution of an arbitrary state should be equal to the backward time evolution of the reversed state* (see Figure \[fig:1\]).
In classical dynamics, generally, even-order and odd-order derivatives with respect to $t$ are respectively preserved and reversed under $R$ [@lamb98; @roberts92]. For example, consider a conservative and reversible Hamiltonian $\mathcal{H}(\mathbf{q}, \mathbf{p})$ = $\mathcal{H}(\mathbf{q}, -\mathbf{p})$, as mentioned in Section \[sec:3.1\]. Because $\mathbf{q}$ and $\mathbf{p}$ are respectively zeroth and first order derivatives with respect to $t$, $R$ is simply given by $R(\mathbf{q}, \mathbf{p}) = (\mathbf{q}, -\mathbf{p})$. In this case, one can easily check the Hamilton’s equations (\[eq:4\]) are invariant under $R$ and $t \mapsto -t$.
Time-reversal symmetry ODE networks (TRS-ODENs) {#sec:3.3}
-----------------------------------------------
Inspired from ODENs (\[eq:3\]) and time-reversal symmetry (\[eq:8\]), here we propose a novel *time-reversal symmetry loss function*. First, the backward time evolution of the reversed state for ODENs can be obtained as follows: $$\label{eq:9}
\tilde{\mathbf{x}}_R({t_{i+1}})=\mathtt{Solve}\{\tilde{\mathbf{x}}_R({t_i}),f_{\theta},-{\Delta}t_i\},\, \tilde{\mathbf{x}}_R({t_0}) = R(\tilde{\mathbf{x}}({t_0})).$$
Then, using (\[eq:3\]) and (\[eq:9\]), we define the time-reversal symmetry loss $\mathcal{L}_\text{TRS}$ as an ODEN version of (\[eq:8\]): $$\label{eq:10}
\mathcal{L}_{\text{TRS}} \equiv \sum^{T-1}_{i=0} \left \| R(\mathtt{Solve}\{\tilde{\mathbf{x}}(t_i), f_\theta, {\Delta}t_i\}) - \mathtt{Solve}\{\tilde{\mathbf{x}}_R(t_i), f_\theta, -{\Delta}t_i\} \right \|^2_2 \\.$$ Finally, we define the TRS-ODEN as a class of ODENs whose loss function $\mathcal{L}_\text{TRS-ODEN}$ is given by the sum of standard ODEN error $\mathcal{L}_\text{ODE}$ and time-reversal symmetry regularizer $\mathcal{L}_\text{TRS}$ as follows: $$\label{eq:11}
\mathcal{L}_{\text{TRS-ODEN}}(\mathbf{x}(t), \tilde{\mathbf{x}}(t), \tilde{\mathbf{x}}_R(t), R, \theta) \equiv \mathcal{L}_{\text{ODE}}(\mathbf{x}(t), \tilde{\mathbf{x}}(t), \theta) + \lambda\cdot \mathcal{L}_{\text{TRS}}(\tilde{\mathbf{x}}(t), \tilde{\mathbf{x}}_R(t), R, \theta),$$ where $\lambda \geq 0$ is a hyper-paremeter. It is noteworthy that $\lambda$ can be also a function of time $t$. This is owing to the heuristic that although the target dynamics do not possess the full time-reversal symmetry over time, they can be partially reversible when the symmetry breaking terms become negligible at certain time points.
Experiments {#sec:4}
===========
Setups
------
[**Default model setting.**]{} We compare three models: vanilla ODENs, HODENs, and TRS-ODENs. A single neural network $f_\theta(\mathbf{q}, \mathbf{p})$ is used for ODENs and TRS-ODENs, while HODENs consist of two neural networks $K_{\theta_1}(\mathbf{p})$ and $V_{\theta_2}(\mathbf{q})$, i.e., separable $\mathcal{H}_\theta(\mathbf{q}, \mathbf{p}) = K_{\theta_1}(\mathbf{p}) + V_{\theta_1}(\mathbf{q})$. We use the leapfrog integrator for $\mathtt{Solve}$, following the recent work [@chen19]. The maximum allowed value of trajectory length at training phase is set to 10. If training trajectories are longer that 10, we divide them properly. We train models by using the Adam [@kingma14] with initial learning rate of $2 \times 10^{-4}$ during 5,000 epochs. We use the full-batch training because the training sample sizes are quite small.
[**Performance metric.**]{} As primary performance metrics, we use the mean-squared error (MSE) between test ground truths and models’ predictive phase space trajectories as well as total energies[^4] (see Table \[table:1\] for summary). The predictive trajectories are obtained by recursively solving (\[eq:3\]) or (\[eq:5\]), thus errors accumulate and diverge over time if the models do not learn the accurate time evolution. [**Default data generation method.**]{} In this paper, we focus on the Duffing oscillators [@kovacic11], which are generalized equations of motion for oscillating systems and given by[^5]: $$\label{eq:12}
\frac{d\mathbf{q}}{dt} = \mathbf{p},\; \frac{d\mathbf{p}}{dt} = -\alpha{\mathbf{q}} -\beta{\mathbf{q}^3} - \gamma{\mathbf{p}} + \delta{\cos({t})},$$ where $\alpha$, $\beta$, $\gamma$, and $\delta$ are scalar parameters that determine the linear stiffness, non-linear stiffness, damping, and driving force terms, respectively. For non-zero parameters, Duffing oscillators are neither conservative nor reversible. Furthermore, they often exhibit chaotic behaviors [@kovacic11]. However, the characteristics of Duffing oscillator can be changed greatly by adjusting parameters. Thus, by this single coupled equations, we can simulate several dynamical systems mentioned in Section \[sec:3.1\].
We generate 50 trajectories each for training and test sets. For each trajectory, The initial state $(\mathbf{q}(t_0), \mathbf{p}(t_0))$ are uniformly sampled from $[0.2, 1]$. The length of training and test trajectories are 30 and 200, respectively, while the time-step is fixed at 0.1, i.e., $\Delta{t_i} = 0.1$ for all $i$. Thus, we can evaluate whether the models can mimic the untrained long-term dynamics. We add Gaussian noise $0.1{n}, n \sim \mathcal{N}(0, 1)$ to training set. We use fourth order Runge-Kutta method to get trajectories.
Conservative and reversible systems {#sec:4.1}
-----------------------------------
First, we evaluate our proposed method for conservative and reversible systems, where we demonstrate that TRS-ODENs are comparable with or even outperform HODENs. Moreover, we confirm combining HODENs and time-reversal symmetry loss can lead further improvement for these systems.
[**Experiment I: Simple oscillator.**]{} For a toy example, we choose simple oscillators, i.e., $\alpha = 1$ and $\beta = \gamma = \delta = 0$. We use single hidden layer neural networks consists of 1,000 hidden units and $\texttt{tanh}$ activations for all models. Figure \[fig:2\] (a-b) show that TRS-ODENs with $\lambda = 10$ outperform both ODENs and HODENs. For qualitative analysis, we plot a test trajectory and its total energy (see Figure \[fig:2\] (c-h)). It shows the TRS-ODENs lean the energy conservation as well as accurate dynamics.
![**Summary of Experiment I.** (a-b) Test (a) trajectory MSE and (b) energy MSE across the models. (c-h) Sampled trajectory and its total energy for (c-d) ODENs, (e-f) HODENs, and (g-h) TRS-ODENs.[]{data-label="fig:2"}](figure2.png){width="1.0\linewidth"}
[**Experiment II: Non-linear oscillator.**]{} As a more interesting problem, we choose the undamped and unforced non-linear oscillators, i.e., $\alpha = -1$, $\beta = 1$, and $\gamma = \delta = 0$. We use neural networks consist of two hidden layers with 100 units and $\texttt{tanh}$ activations.
In this experiment, TRS-ODENs outperform HODENs in terms of the trajectory MSE, and vice-versa for total energy MSE (see Figure \[fig:3\] (a-b)). For qualitative analysis, we sample five trajectories and their energy values (see Figure \[fig:3\] (c-h)). It shows HODENs fail to lean time evolution especially near the origin point, while TRS-ODENs shows undesirable peaks in energy. This room for improvement leads us to combining the HODENs and TRS-ODENs, the *Time-Reversal Symmetric Hamiltonian ODE Networks (TRS-HODENs)*[^6]. After estimation, We find that TRS-HODENs can achieve almost same performance as HODEN in terms of energy MSE, and clearly outperform baselines for trajectory MSE (see Figure \[fig:3\] (a-b) and \[fig:4\] (a-b)). Furthermore, we evaluate the sample efficiency and find that the combination of two inductive bias improves the learning process more reliable (see Figure \[fig:4\] (c)).
![**Summary of Experiment II.** (a-b) Test (a) trajectory MSE (b) and energy MSE across the models. (c-h) Sampled five trajectories and their total energies for (c-d) ODENs, (e-f) HODENs, and (g-h) TRS-ODENs.[]{data-label="fig:3"}](figure3.png){width="1.0\linewidth"}
![(a-b) Sampled five (a) trajectories and (b) their total energies in the case of TRS-HODENs. (c) Test trajectory MSE *vs.* the number of training samples across the models. The means and error bars of MSE are calculated from results of five different test sets, each consist of 50 trajectories.[]{data-label="fig:4"}](figure4.png){width="1.0\linewidth"}
Non-conservative and reversible systems
---------------------------------------
Second, we evaluate proposed framework for non-conservative and reversible systems. To avoid getting some trivial results, we try to model the chaotic systems using TRS-ODENs.
[**Experiment III: Forced non-linear oscillator.**]{} We set $\alpha = -0.2$, $\beta = 0.2$, $\gamma = 0$, and $\delta = 0.15$ for system parameters. Due to the periodic driving force $\delta\cos{t}$, the systems are non-autonomous. Therefore, we use a tuple $(\mathbf{q}, \mathbf{p}, t)$ as an input of the neural networks for this experiment[^7]. Hyper-parameters of neural networks are same as them for Experiment II, except for $\lambda$: $\lambda \in \{0.5, 1, 5\}$ is estimated in here. We generate 200 and 50 trajectories whose lengths are 50 and 100, respectively, for train and test sets in this experiment, considering the complexity of the target system.
We find that TRS-ODENs clearly outperform their baselines with significant margin in both trajectory and energy MSE metrics (see Figure \[fig:5\] (a-b)). From Figure \[fig:5\] (c-h), one can check the dynamics predicted by ODENs or HODENs diverge as times passes, while TRS-ODENs shows reliable long-term behaviors. As a result, the total energy of TRS-ODENs follow the ground truth reasonably, while that estimated by baselines soar explosively in $t > 8$.
![**Summary of Experiment III.** (a-b) Test (a) trajectory MSE (b) and energy MSE across the models. (c-h) Sampled five trajectories and their total energies for (c-d) ODENs, (e-f) HODENs, and (g-h) TRS-ODENs.[]{data-label="fig:5"}](figure5.png){width="1.0\linewidth"}
Non-conservative and irreversible systems
-----------------------------------------
Finally, we validate our proposed framework for non-conservative and irreversible damped systems. HODENs cannot learn this system because of their strong tendency to conserve the energy, as previously reported in [@greydanus19]. We demonstrate TRS-ODENs can learn this system flexibly.
[**Experiment IV: Damped oscillator.**]{} We simulate damped oscillators by setting the system parameters as follows: $\alpha = 1,\, \beta = 0,\, \gamma = 0.1, \delta = 0$. In this experiment, we assume the time-reversal symmetry tends to hold as $t\to \infty$, thus evaluate the time-dependent $\lambda$ approach. This assumption is quite reasonable for various disspative irreversible systems, because their irreversibility is typically originated from the (odd powers of) $\mathbf{p}$[^8] in their governing ODEs, e.g., $\gamma \mathbf{p} $ in (\[eq:12\]). Since dissipative systems lose their kinetic energy as time passes, i.e., $\mathbf{p} \to 0$ as $t \to \infty$, we can design $\lambda$ as a linear increasing function of min-max normalized $t$. In this experiment, we evaluate four cases of $\lambda$: $\lambda \in \{0.5, 0.5t, 1, t\}$. Other hyper-parameters are same with them of Experiment I.
It is shown that the TRS-ODENs can outperform ODENs and HODENs, except for $\lambda = 1$ case (see Figure \[fig:6\] (a-b)). Especially, $\lambda = 0.5t$ case shows great predictability in both time evolution and total energy of the damped system, while ODENs lose their energy too excessively and HODENs conserve their energy too strictly (see Figure \[fig:6\] (c-h)). We believe it is owing to the balance between physics-based inductive bias and data-driven learning process.
![**Summary of Experiment IV.** (a-b) Test (a) trajectory MSE and (b) energy MSE across the models. (c-h) Sampled trajectory and its total energy for (c-d) ODENs, (e-f) HODENs, and (g-h) TRS-ODENs.[]{data-label="fig:6"}](figure6.png){width="1.0\linewidth"}
Metric Model Experiment I Experiment II Experiment III Experiment IV
-------- ----------- --------------------- ----------------------- --------------------- ---------------------
ODEN 4.05 $\pm$ 2.66 50.81 $\pm$ 26.80 39.21 $\pm$ 21.19 1.28 $\pm$ 0.82
HODEN 0.84 $\pm$ 0.37 17.40 $\pm$ 17.74 24.09 $\pm$ 14.29 3.68 $\pm$ 2.19
TRS-ODEN **0.31 $\pm$ 0.19** 13.78 $\pm$ 14.86 **6.50 $\pm$ 5.59** **0.12 $\pm$ 0.06**
TRS-HODEN N/A **10.85 $\pm$ 12.62** N/A N/A
ODEN 9.04 $\pm$ 10.14 6.14 $\pm$ 9.13 242.06 $\pm$ 204.59 1.04 $\pm$ 1.17
HODEN 0.08 $\pm$ 0.09 **0.22 $\pm$ 0.17** 80.50 $\pm$ 128.94 8.26 $\pm$ 9.60
TRS-ODEN **0.07 $\pm$ 0.09** 0.53 $\pm$ 0.75 **1.52 $\pm$ 3.20** **0.03 $\pm$ 0.03**
TRS-HODEN N/A 0.29 $\pm$ 0.18 N/A N/A
: Summary of test MSEs across all experiments. All MSE values are multiplied by $10^2$.[]{data-label="table:1"}
Conclusion {#sec:5}
==========
Introducing physics-based inductive bias for neural networks is actively studied. e.g., ODE [@chen18], Hamiltonian [@greydanus19; @sanchez19; @toth19; @zhong19; @chen19], and other domain knowledge [@schutt17; @schutt19; @long18; @nabian18]. We have proposed a simple yet effective approach to incorporate the time-reversal symmetry into ODEN, coined TRS-ODEN, which is not shown in previous works. The proposed method can learn the dynamical system accurately and efficiently. We have validated our proposed framework with various experiments including non-conservative and irreversible systems.
There are some papers discuss the use of symmetry for neural networks. For example, the rotational or reflection symmetries are frequently used in computer vision tasks [@funk17; @dieleman16; @worrall17]. Some researchers have focused on finding symmetries using neural networks, especially in theoretical physics [@decelle19; @li2020; @bondesan19]. Among them, [@bondesan19; @li2020] are closely related to our work because they discuss the method of searching a canonical transformation that satisfies the symplectic symmetry of Hamiltonian systems. Combining these approaches, i.e., finding symmetry, with our proposed framework, i.e., exploiting symmetry, would be an interesting direction for future work.
Broader Impact {#broader-impact .unnumbered}
==============
In this paper, we introduce a neural network model that regularized by a physics-originated inductive bias, the symmetry. Our proposed model can be used to identify and predict unknown dynamics of physical systems. In what follows, we summarize the expected broader impacts of our research from two perspectives.
[**Use for current real world applications.**]{} Predicting dynamics plays a important role in various practical applications, e.g., robotic manipulation [@hersch08], autonomous driving [@levinson11], and other trajectory planning tasks. For these tasks, the predictive models should be highly reliable to prevent human and material losses due to accidents. Our propose model have a potential to satisfy this high standard on reliability, considering its robustness and efficiency (see Figure \[fig:4\] (c) as an example).
[**First step for fundamental inductive bias.**]{} According to the CPT theorem in quantum field theory, the CPT symmetry, which means the invariance under the combined transformation of charge conjugate (C), parity transformation (P), and time reversal (T), exactly holds for all phenomena of physics [@kostelecky98]. Thus, the CPT symmetry is a fundamental rule of nature: that means, it is a fundamental inductive bias of deep learning models for natural science. However, this symmetry-based bias has been unnoticed previously. We study one of the fundamental symmetry, the time-reversal symmetry in classical mechanics, as a proof-of-concept in this paper. We expect our finding can encourage researchers to focus on the fundamental bias of nature and extend the research from classical to quantum, and from time-reversal symmetry to CPT symmetry. Our work would also contribute to bring together experts in physics and deep learning in order to stimulate interaction and to begin exploring how deep learning can shed light on physics.
**Supplementary Material:**
Time-Reversal Symmetric ODE Network
Time-reversal symmetry loss for non-autonomous systems {#sec:A}
======================================================
Here, we consider the time-reversal symmetry of non-autonomous ODE systems, i.e., systems that depend on time $t$ explicitly as follows:
$$\label{eq:s1}
\frac{d\mathbf{x}}{dt} = f(\mathbf{x}, t).$$
This non-autonomous systems are said to be time-reversal symmetric if there is a reversing operator $R_a: (\mathbf{x}, t) \mapsto (R(\mathbf{x}), -t + a)$ which satisfies [@lamb98]:
$$\label{eq:s2}
\frac{dR(\mathbf{x})}{dt} = -f(R(\mathbf{x}), -t + a),$$
for some $a \in \mathbb{R}$. It means that we should consider the time $t$ itself carefully, as well as the direction of time, unlike the autonomous case (6-10) in the main paper. For example, consider forced non-linear oscillators estimated in Experiment III (Section 4.3 in the main paper):
$$\label{eq:s3}
\frac{d\mathbf{q}}{dt} = \mathbf{p},\; \frac{d\mathbf{p}}{dt} = -\alpha{\mathbf{q}} -\beta{\mathbf{q}^3} + \delta{\cos({\omega{t} + \phi})}.$$
(\[eq:s3\]) is time-reversal symmetric under $R_{-2\phi/\omega}: (\mathbf{q}, \mathbf{p}, t) \mapsto (\mathbf{q}, -\mathbf{p}, -t - 2\phi / \omega)$.
The forward time evolution of non-autonomous ODENs is given by:
$$\label{eq:s4}
\tilde{\mathbf{x}}({t_{i+1}})=\mathtt{Solve}\{\tilde{\mathbf{x}}({t_i}), t_i, f_{\theta},{\Delta}t_i\},\, \tilde{\mathbf{x}}({t_0}) = \mathbf{x}({t_0}).$$
On the other hand, the backward time evolution is equal to:
$$\label{eq:s5}
\tilde{\mathbf{x}}_R({\tau_{i+1}})=\mathtt{Solve}\{\tilde{\mathbf{x}}_R({\tau_i}),\tau_i,f_{\theta},-{\Delta}t_i\},\, \tilde{\mathbf{x}}_R(\tau_0) = R_a(\tilde{\mathbf{x}}({t_0})),$$
where $\tau_i = -t_i + a$. As a result, the time-reversal symmetry loss of autonomous ODE systems is given by:
$$\label{eq:s6}
\mathcal{L}_{\text{TRS}} \equiv \sum^{T-1}_{i=0} \left \| R(\mathtt{Solve}\{\tilde{\mathbf{x}}(t_i), t_i, f_\theta, {\Delta}t_i\}) - \mathtt{Solve}\{\tilde{\mathbf{x}}_R(\tau_i), \tau_i, f_\theta, \tau_i), -{\Delta}t_i\} \right \|^2_2.$$
Reasoning on the improvement made by TRS-HODENs {#sec:B}
===============================================
As mentioned in Section 3.1 in the main paper, the Hamiltonian $\mathcal{H}$ of conservative and reversible systems satisfies $\mathcal{H}(\mathbf{q}, \mathbf{p}) = \mathcal{H}(\mathbf{q}, -\mathbf{p})$. With this symmetry property, we analyze the reason of improvement made by TRS-HODENs over HODENs in Experiment II (Section 4.2 in the main paper). Note that the ground truth Hamiltonian of non-linear oscillator tested in Experiment II is described as:
$$\label{eq:s7}
\mathcal{H}(\mathbf{q}, \mathbf{p}) = \frac{\mathbf{p}^2}{2} + \frac{\alpha \mathbf{q}^2}{2} + \frac{\beta \mathbf{q}^4}{4},$$
which clearly possesses $\mathcal{H}(\mathbf{q}, \mathbf{p}) = \mathcal{H}(\mathbf{q}, -\mathbf{p})$.
We find that the time-reversal symmetry loss helps the learned $\theta$-parameterized Hamiltonian $\mathcal{H}_\theta(\mathbf{q}, \mathbf{p})$ possess the above property thanks to the symmetry under the momentum-reversing operator $R(\mathbf{q}, \mathbf{p}) = (\mathbf{q}, -\mathbf{p})$. To show this, we calculate $\mathcal{H}_\theta(\mathbf{q}, \mathbf{p}) - \mathcal{H}_\theta(\mathbf{q}, -\mathbf{p})$ for HODEN and TRS-HODEN ($\lambda = 10$) tested in Experiment II, with varying $\mathbf{p}$ from 0 to 1.5 and fixing $\mathbf{q}$ to 0 (see Figure \[fig:s1\] (a)). It shows that the Hamiltonian of HODEN does not follow $\mathcal{H}(\mathbf{q}, \mathbf{p}) = \mathcal{H}(\mathbf{q}, -\mathbf{p})$ precisely, while that of TRS-HODEN is almost even function of $\mathbf{p}$. As a result, TRS-HODENs can learn the ground truth Hamiltonian from noisy data more structurally and efficiently.
To confirm the above discussion, we compare their kinetic energies $K_{\theta_1}(\mathbf{p}) = \mathcal{H}_\theta(\mathbf{0}, \mathbf{p}) + \text{const.}$ (see Figure \[fig:s1\] (b)). It shows the kinetic energy of TRS-HODEN is almost indistinguishable from that of the ground truth. On the other hand, the kinetic energy of HODEN does not match well with that of the ground truth. In Figure \[fig:s2\], We plot the Hamiltonian (total energy) surfaces across the models. One can check the Hamiltonian surface of HODEN shows highly asymmetric double well shape, unlike that of the ground truth and TRS-HODEN.
![(a) Calculated $\mathcal{H}(\mathbf{q}, \mathbf{p}) - \mathcal{H}(\mathbf{q}, -\mathbf{p})$ of ground truth, HODEN, and TRS-HODEN. (b) Comparison of kinetic energy profiles obtained from ground truth, HODEN, and TRS-HODEN. Note that we calibrate the ground energy level to make $\mathcal{H}(\mathbf{0}, \mathbf{0})$ = 0 for all models.[]{data-label="fig:s1"}](figureS1.png){width="1.0\linewidth"}
![Hamiltonian surfaces obtained from (a) ground truth, (b) HODEN, and (c) TRS-HODEN. The ground truth Hamiltonian shows symmetric double well shape.[]{data-label="fig:s2"}](figureS2.png){width="1.0\linewidth"}
Predicting stable centers and homoclinic orbits of non-linear oscillators {#sec:C}
=========================================================================
The non-linear oscillator systems in Experiment II have two stable centers at $(1, 0)$, $(-1, 0)$, and saddle point at $(0, 0)$ (see Figure \[fig:s2\] (a)). Clearly, at the stable centers, states do not evolve with time at all, i.e., the equilibrium states. At the saddle point, there are two interesting trajectories, that appear to start and end at the same saddle point. These trajectories are called homoclinic orbits [@strogatz01]. Note that the homoclinit orbits lie on $\mathbf{q} > 0$ and $\mathbf{q} < 0$ respectively start from $(\epsilon, \epsilon)$ and $(-\epsilon, -\epsilon)$, for some small positive constants $\epsilon$.
Here, we estimate whether the learned dynamics can represent the special trajectories originated from these critical points well. To do this, we generate trajectories, whose initial states are given by the centers or saddle point[^9], by using the models trained in Experiment II: ODENs, HODENs, TRS-ODENs ($\lambda = 10$), and TRS-HODENs ($\lambda = 10$). Figure \[fig:s3\] demonstrates the generated phase space trajectories. For ODENs, they cannot achieve the accurate time evolution at all. HODENs show relatively reasonable behaviors, but they predict the same direction of homoclinic orbits for $(\epsilon, \epsilon)$ and $(-\epsilon, -\epsilon)$. Also, periodic motions near the stable centers are observed for HODENs. TRS-ODENs and TRS-HODENs show two separated homoclinic orbits, clearly. Moreover, TRS-HODENs show stable equilibrium behaviors at the center points. In summary, TRS-HODENs can predict physically-consistent behaviors even for critical points. We summarize the phase space trajectory and total energy MSE metrics in Table \[table:s1\].
![The critical phase space trajectories obtained from (a) ODENs, (b) HODENs, (c) TRS-ODENs, and (d) TRS-HODENs.[]{data-label="fig:s3"}](figureS3.png){width="1.0\linewidth"}
Model ODEN HODEN TRS-ODEN TRS-HODEN
-------------- ------------------- ------------------- ----------------- ---------------------
MSE (Traj.) 14.28 $\pm$ 10.47 15.26 $\pm$ 25.15 3.88 $\pm$ 5.92 **2.03 $\pm$ 2.17**
MSE (Energy) 9.31 $\pm$ 16.11 0.32 $\pm$ 0.53 0.52 $\pm$ 0.78 **0.21 $\pm$ 0.21**
: Summary of phase space trajectory and total energy MSEs evaluated in Section \[sec:C\]. All MSE values are multiplied by $10^2$.[]{data-label="table:s1"}
[^1]: For Hamiltonian as an example, $\mathbf{x} = (\mathbf{q}, \mathbf{p})$, where $\mathbf{q} \in \mathbb{R}^n$ and $\mathbf{p} \in \mathbb{R}^n$ are positions and momenta.
[^2]: Note that the most basic definition of the Hamiltonian is the sum of kinetic and potential energy, i.e., $\mathcal{H}(\mathbf{q}, \mathbf{p}) = \mathbf{p}^2/2 + V(\mathbf{q})$ (if we omit the mass) [@goldstein02], which possess $\mathcal{H}(\mathbf{q}, \mathbf{p})$ = $\mathcal{H}(\mathbf{q}, -\mathbf{p})$ naturally.
[^3]: Let’s consider a damped pendulum. They are irreversible since one can distinguish the motion of the pendulum in forward (amplitude increases) and that in backward directions (amplitude decreases).
[^4]: They can be calculated from trajectories. For example, a total energy of simple oscillator is $\mathbf{q}^2 + \mathbf{p}^2$.
[^5]: Typically, Duffing oscillator is given by a second order ODE $\ddot{\mathbf{x}} + \alpha{\mathbf{x}} + \beta{\mathbf{x}^3} + \gamma\dot{\mathbf{x}} = \delta{\cos({t})}$. We separate this equation from the perspective of the pseudo-phase space, although they are not in canonical coordinates.
[^6]: It can be obtained straightforwardly by combining (\[eq:5\]) and (\[eq:8\]), similar to (\[eq:9\]-\[eq:10\]).
[^7]: In [@greydanus19; @chen19], the authors say for HODENs, time dependency should be modeled separately from them. However, we use time-dependent HODENs in here to prevent large modifications of HODENs for fair comparison.
[^8]: It is because of the definition of the classical reversing operator $R(\mathbf{q}, \mathbf{p}) = (\mathbf{q}, -\mathbf{p})$.
[^9]: We use $10^{-8}$ and $10^{-2}$ instead of 0 and $\epsilon$, respectively, considering numerical stability.
|
---
author:
- |
Peng Zhao, Engui Fan[^1],Yu Hou\
Institue of Mathematics, Fudan University, Handan Road 220,\
[Shanghai 200433, P R China]{}
title: 'The Algebro-Geometric Initial Value Problem for the Relativistic Lotka-Volterra Hierarchy and Quasi-Periodic Solutions'
---
We provide a detailed treatment of relativistic Lotka-Volterra hierarchy and a kind of initial value problem with special emphasis on its the theta function representation of all algebro-geometric solutions. The basic tools involve hyperelliptic curve $\mathcal{K}_n$ associated with the Burchnall-Chaundy polynomial, Dubrovin-type equations for auxiliary divisors and associated trace formulas. With the help of a foundamental meromorphic function $\tilde{\phi}$ on $\mathcal{K}_p$ and trace formulas, the complex-valued algebro-geometric solutions of of RLV hierarchy are derived.\
Introduction
============
Nonlinear integrable lattice systems have been studied extensively in relation with various aspects and they usually possess rich mathematical structure such as Lax pairs, Hamilton structure, conservation law, etc. The Toda lattice (TL), $$\label{1.1}
\text{TL}:
\begin{cases}
a_{t}=a(b^+-b),\cr
b_t=a-a^-,\cr
\end{cases}$$ is one of the most important integrable systems. It is well-known soliton equations such as the KdV, modified KdV, and nonlinear Schrödinger equations are closely related to or derived from the Toda equation by suitable limiting procedures [@119; @120]. Another celebrated integrable lattice system is Lotka-Volterra (LV) lattice, $$\label{1.2}
\text{LV}:
\begin{cases}
u_t=u(v-v^-),\cr
v_t=v(u^+-u),\cr
\end{cases}$$ or $$\label{1.3}
a_t=a(a^+-a^-),$$ by setting $a(2n-1)=u(n), a(2n)=v(n).$ Ruijsenaars found a relativistic integrable generalization of non-relativistic Toda lattice through solving a relativistic version of the Calogero-Moser system [@RT].The Lax representation, inverse scattering problem of the Ruijsenaars-Toda lattice and its connection with soliton dynamics were investigated. A general approach to constructing relativistic generalizations of integrable lattice systems, applicable to the whole lattice KP hierarchy, was proposed by Gibbons and Kupershmidt [@i]. However, nobody have known what the relativistic Lotka-Volterra lattice is until Y. B. Suris and O. Ragnisco found it [@RLV].
There are very close connnection among the Toda lattice, the Lotka-Volterra lattice, the relativistic Toda lattice and the relativistic Lotka-Volterra lattice. It is well known that (\[1.1\]) and (\[1.2\]) are both discrete version of the KdV equation in the sense of different limiting process and in the non-relativistic limit $h\rightarrow 0$, the relativistic Toda system (RT), $$\label{1.3}
\text{RT}:
\begin{cases}
a_t=a(b^+-b+ha^+-ha^-),\cr
b_t=(1+hb)(a-a^-),\cr
\end{cases}$$ and the “relativistic splitting” relativistic Lotka-Volterra systems (RLV), $$\label{1.4}
\text{RLV}:
\begin{cases}
u_t=u(v-v^-+huv-hu^-v^-),\cr
v_t=v(u^+-u+hu^+v^+-huv),\cr
\end{cases}$$ reduced to the well-known Toda lattice equation (\[1.1\]) and Lotka-Volterra lattice equation (\[1.2\]), respectively. Moreover, the Miura relation can be summarized in the following diagram: $$\xymatrix{\text{RLV} \ar[r]^{h\rightarrow 0} \ar[d]_{\tau_{1}} & \text{LV} \ar[d]^{\tau_{2}} \\ \text{RT} \ar[r]_{h\rightarrow 0} & \text{TL}}$$ $$\text{RLV}_1\xrightarrow{\tau_1}\text{RT}, \quad\text{RLV}_2\xrightarrow{\tau_2}\text{RT}$$ Here $$\tau_1:
\begin{cases}
a=uv,\cr b=u+v^-,\cr
\end{cases}$$ $$\tau_2:
\begin{cases}
a=u^+v,\cr b=u+v.\cr
\end{cases}$$
Mathematical structure related to relativistic volterra lattice (\[1.4\]) such as Lax integrablity [@300], $2\times 2$ Lax representation [@Laxpresentation], conservation laws [@zwxz], bilinear structure and determinant solution [@KMMO] have been closely studied and hence the purpose of this paper is to uniformly construct algebro-geometric solutions of the relativistic Lotka-Volterra hierarchy which invariably is connected with geometry and Riemann theta functions, parameterized by some Riemann surface. Algebro-geometric solutions (finite-gap solutions or quasi-period solutions), as an important character of integrable system, is a kind of explicit solutions closely related to the inverse spectral theory [@99; @t]. Around 1975, several independent groups in UUSR and USA, namely, Novikov, Dubrovin and Krichever in Moscow, Matveev and Its in Leningrad, Lax, McKean, van Moerbeke and M. Kac in New York, and Marchenko, Kotlyarov and Kozel in Kharkov, developed the so-called finite finite-gap theory of nonlinear KdV equation based on the works of Drach, Burchnall and Chaunchy, and Baker [@21; @5]. The algebro-geometric method they established allowed us to find an important class of exact solutions to the soliton equations. As a degenerated case of this solutions, the multisoliton solutions and elliptic functions may be obtained [@t; @1]. Its and Matveev first derived explicit expression of the quasi-period solution of KdV equation in 1975 [@4], which is closely related to the finite-gap spectrum of the associated differential operator. Further exciting results appeared later, including the finite-gap solutions of Toda lattice, the Kadomtsev-Petviashvili equation and others [@d; @5; @1], which could be found in the wonderful work of Belokolos, et al [@t]. In recent years, a systematic approach based on the nonlinearization technique of Lax pairs or the restricted flow technique to derive the algebro-geometric solutions of (1+1)- and (2+1)-dimensional soliton equations has been obtained [@7; @11]. An alternate systematic approach proposed by Gesztesy and Holden can be used to construct algebro-geometric solutions has been extended to the whole (1+1) dimensional continuous and discrete hierarchy models [@A1; @A2; @u; @19; @13].
The outline of our present paper is as follow. In section 2, the relativistic Lotka-Volterra equation (\[1.4\]) is extended to a whole hierarchy through the polynomial recursive relation. The hyperelliptic curve associated with RLV hierarchy is given in terms of the polynomial. In section 3, We focus on the stationary RLV hierarchy. based on the polynomial recursion formalism introduced in Section 2 and a fundamental meromorphic function $\tilde{\phi}$ on the hyperelliptic curve $\mathcal{K}_n$, we study the Baker-Akhiezer function $\Psi,$ trace formulas, from which the algebro-geometric solutions for stationary RLV hierarchy are constructed in terms of Riemann theta functions. In section 4, we extend the algebro-geometric analysis of Section 3 to the time-dependent RLV hierarchy based on a kind of special initial value problem. Finally, in section 5 we give Lagrange interpolation representation that will be used in this paper.
Relativistic Lotka-Volterra hierarchy and associated hyperelliptic curve
=========================================================================
In this section, we investigate the relativistic Lotka-Volterra hierarchy and then derive the hyperelliptic algebraic curves associated with the algebro-geometric solutions of the newly constructed hierarchy. Throughout this paper, we have the following definition.
We denote by $\ell$($\mathbb{Z}$) the set of all the complex-valued sequences $\{f(n)\}_{n=-\infty}^{+\infty}$. This is a vector space with respect to the naturally defined operation. A subspace $\ell^{2}(\mathbb{Z})\subset\ell(\mathbb{Z})$ is defined by the set of {$f\in\ell(\mathbb{Z})|\sum^{+\infty}_{n=-\infty}|f(n)|^{2}<+\infty,n\in\mathbb{Z}$}.
We denote by $S^{\pm}$ the shift operators acting on $\psi=\{\psi(n)\}_{n=-\infty}^{+\infty}\in\ell(\mathbb{Z})$ according to $(S^{\pm}\psi)(n)=\psi(n\pm1)$. The identity operator I acting on $\psi=\{\psi(n)\}_{n=-\infty}^{+\infty}\in\ell(\mathbb{Z})$ according to $(I\psi)(n)=\psi(n)$. We also define $\psi^{\pm}=S^{\pm}\psi,\psi\in\ell(\mathbb{Z})$.
We introduce the following $2\times2$ matrix problem [@Laxpresentation] $$\label{2.1}
\begin{split}
&S^{+}\Psi=U(\lambda)\Psi,\\
&U(\lambda)=\left(
\begin{array}{cc}
\lambda \tilde{p}-\lambda^{-1} & \tilde{q} \\
\tilde{r} & \lambda \\
\end{array}
\right), \quad \lambda\in\mathbb{C},
\end{split}$$ where $\Psi=(\Psi_1,\Psi_2)^{T}\in\ell(\mathbb{Z})\times\ell(\mathbb{Z}), \tilde{r}=(\tilde{p}-1)\tilde{q}^{-1}, \tilde{p}=p, \tilde{q}=e^{(S^+-I)^{-1}\ln q}$ and $\tilde{p}, \tilde{q}, p, q$ are potential functions and $\lambda$ is the spectral parameter. Here $p=p(n,t),q=q(n,t)\in\ell(\mathbb{Z}), (n,t)\in\mathbb{Z}\times\mathbb{R}$ in time-dependent case and $p=p(n),q=q(n)\in\mathit{l}(\mathbb{Z}), n\in\mathbb{Z}$ in stationary case.
Define sequences $\{a_\ell(n)\}_{\ell\in\mathbb{N}_0}, \{b_\ell(n)\}_{\ell\in\mathbb{N}_0}$ and $\{c_\ell(n)\}_{\ell\in\mathbb{N}_0}$ in $\ell(\mathbb{Z})$ recursively by $$\label{2.2}
\tilde{p}a_\ell^{-}-\tilde{p}a_\ell=a_{\ell+1}^--a_{\ell+1}+\tilde{r}b_{\ell}-\tilde{q}c_{\ell}^{-},\quad\ell\in\mathbb{N}_{0},$$ $$\label{2.3}
\tilde{p}b_\ell^{-}-b_{\ell+1}^{-}-\tilde{q}a_{\ell+1}^{-}=\tilde{q}a_{\ell+1}+b_{\ell},\quad\ell\in\mathbb{N}_{0},$$ $$\label{2.4}
\tilde{r}a_{\ell+1}^{-}+c_{\ell}^{-}=\tilde{p}c_{\ell}-c_{\ell+1}-\tilde{r}a_{\ell+1},\quad\ell\in\mathbb{N}_{0},$$ $$\label{2.5}
a_0=1/2, b_{0}=-\tilde{q}^{+}, c_0=-\tilde{r}.$$ Explicitly, one obtains $$\label{2.6}
\begin{split}
&a_1=-\tilde{q}^{+}\tilde{r}+\delta_1/2,\\
&b_1=-\tilde{p}^{+}\tilde{q}^{+}+(\tilde{q}^{+})^{2}\tilde{r}
+\tilde{q}^{+}\tilde{q}^{++}\tilde{r}^{+}+\tilde{q}^{++}-\tilde{q}^{+}\delta_1,\\
&c_1=-\tilde{p}\tilde{r}+\tilde{q}^{+}\tilde{r}^{2}+\tilde{r}^{-}+\tilde{q}\tilde{r}\tilde{r}^{-}-\tilde{r}\delta_1,\\
&a_2=-\tilde{p}^+\tilde{q}^+\tilde{r}+(\tilde{q}^+)^2\tilde{r}^2-\tilde{p}\tilde{q}^+\tilde{r}+\tilde{q}^{++}\tilde{r}
+\tilde{q}^+\tilde{r}^{-}+\tilde{q}^{++}\tilde{q}^+\tilde{r}^+\tilde{r}+\tilde{q}^+\tilde{q}\tilde{r}\tilde{r}^-\\
&-\tilde{q}^+\tilde{r}\delta_1+\delta_2/2,\\
&\dotsi \dotsi.
\end{split}$$ Here $\{\delta_{\ell}\}_{\ell\in\mathbb{N}}$ denote summation constants which naturally arise when solving (\[2.2\])-(\[2.5\]).
If we denote by $\bar{a}_{\ell}=a_{\ell}|_{\delta_{j}=0,j=1,\dotsi,\ell}, \bar{b}_{\ell}=b_{\ell}|_{\delta_{j}=0,j=1,\dotsi,\ell}, \bar{c}_{\ell}=c_{\ell}|_{\delta_{j}=0,j=1,\dotsi,\ell}$, $\ell\in\mathbb{N}$ the homogeneous coefficients of $a_\ell, b_\ell, c_\ell$, that is, $$\label{2.7}
\begin{split}
&\bar{a}_1=-\tilde{q}^{+}\tilde{r},\\
&\bar{b}_1=-\tilde{p}^{+}\tilde{q}^{+}+(\tilde{q}^{+})^{2}\tilde{r}
+\tilde{q}^{+}\tilde{q}^{++}\tilde{r}^{+}+\tilde{q}^{++},\\
&\bar{c}_1=-\tilde{p}\tilde{r}+\tilde{q}^{+}\tilde{r}^{2}+\tilde{r}^{-}+\tilde{q}\tilde{r}\tilde{r}^{-},\\
&\bar{a}_2=-\tilde{p}^+\tilde{q}^+\tilde{r}+(\tilde{q}^+)^2\tilde{r}^2-\tilde{p}\tilde{q}^+\tilde{r}+\tilde{q}^{++}\tilde{r}
+\tilde{q}^+\tilde{r}^{-}+\tilde{q}^{++}\tilde{q}^+\tilde{r}^+\tilde{r}+\tilde{q}^+\tilde{q}\tilde{r}\tilde{r}^-\\
&\dotsi\dotsi.
\end{split}$$ By induction one infers that $$\label{2.8}
a_{\ell}=\sum_{k=0}^{\ell}\delta_{\ell-k}\bar{a}_{k},\quad \ell\in\mathbb{N}_0,$$ $$\label{2.9}
b_{\ell}=\sum_{k=0}^{\ell}\delta_{\ell-k}\bar{b}_{k},\quad \ell\in\mathbb{N}_0,$$ $$\label{2.10}
c_{\ell}=\sum_{k=0}^{\ell}\delta_{\ell-k}\bar{c}_{k},\quad \ell\in\mathbb{N}_0,$$ introducing $\delta_0=1$.
To construct the relativistic Lotka-Volterra hierarchy we will consider the following ansatz $$\label{2.11}
V_n(\lambda)=\left(
\begin{array}{cc}
A_{2n+2}^-(\lambda) & B_{2n+1}^-(\lambda) \\
C_{2n+1}^-(\lambda) & -D_{2n+2}^-(\lambda)
\end{array}\right),\quad n\in\mathbb{N}_0,$$ where $A_{2n+2}, B_{2n+1}, C_{2n+1}$ and $D_{2n+2}$ are chosen as polynomials, namely $$\label{2.12}
A_{2n+2}(\lambda)=\sum_{\ell=0}^{n+1}a_\ell\lambda^{-\left(2n+2-2\ell\right)}=
a_0\lambda^{-(2n+2)}+a_1\lambda^{-2n}+\dotsi+a_{n+1},$$ $$\label{2.13}
B_{2n+1}(\lambda)=\sum_{\ell=0}^{n}b_\ell\lambda^{-\left(2n+1-2\ell\right)}=
b_0\lambda^{-(2n+1)}+b_1\lambda^{-(2n-1)}+\dotsi+b_{n}\lambda^{-1},$$ $$\label{2.14}
C_{2n+1}(\lambda)=\sum_{\ell=0}^{n}c_\ell\lambda^{-\left(2n+1-2\ell\right)}=
c_0\lambda^{-(2n+1)}+c_1\lambda^{-(2n-1)}+\dotsi+c_{n}\lambda^{-1},$$ $$\label{2.15}
D_{2n+2}(\lambda)=\sum_{\ell=0}^{n+1}d_\ell\lambda^{-\left(2n+2-2\ell\right)}=
d_0\lambda^{-(2n+2)}+d_1\lambda^{-2n}+\dotsi+d_{n+1}$$ and the coefficient $\{a_\ell\}_{\ell=0}^{n+1}, \{b_\ell\}_{\ell=0}^{n}, \{c_\ell\}_{\ell=0}^{n}, \{d_\ell\}_{\ell=0}^{n+1}$ are defined in $\ell(\mathbb{Z})$.
The stationary zero-curvature equation $$\label{2.16}
0=U(\lambda)V_n(\lambda)-V_n^+(\lambda)U(\lambda)$$ is equivalent to the following equalities $$\label{2.17}
(\tilde{p}\lambda-\lambda^{-1})A_{2n+2}^-+\tilde{q}C^{-}_{2n+1}=
(\tilde{p}\lambda-\lambda^{-1})A_{2n+2}+\tilde{r}B_{2n+1},$$ $$\label{2.18}
(\tilde{p}\lambda-\lambda^{-1})B_{2n+1}^--\tilde{q}D_{2n+2}^-=\tilde{q}A_{2n+2}+B_{2n+1}\lambda,$$ $$\label{2.19}
\tilde{r}A_{2n+2}^-+C_{2n+1}^-\lambda=(\tilde{p}\lambda-\lambda^{-1})C_{2n+1}-\tilde{r}D_{2n+2},$$ $$\label{2.20}
\tilde{r}B_{2n+1}^--D_{2n+2}^-\lambda=\tilde{q}C_{2n+1}-D_{2n+2}\lambda.$$ From (\[2.16\]), one finds the matrix $V_n(\lambda)$ is similar to $V_{n}^{+}(\lambda)$ and then we have $$\text{trace}\left(V_n(\lambda)\right)=\text{trace}\left(V_{n}^+(\lambda)\right),$$ namely, $$A^{-}_{2n+2}-D^{-}_{2n+2}=A_{2n+2}-D_{2n+2}.$$ Hence $A_{2n+2}-D_{2n+2}$ is $n$-independence. Without loss of generality we can choose $$\label{2.21}
d_{\ell}=a_{\ell},\quad \ell=0,1,2,\dotsi,n, \quad d_{n+1}=\triangle,$$ where $\triangle=1=(\dotsi,1,1,1,\dotsi)\in\ell(\mathbb{Z})$ is a constant value sequence. In stationary case we have $$\label{2021}A_{2n+2}=D_{2n+2},$$ since this can be always be achieved by adding a constant coefficient polynomial times the identity matrix to $V_n$, which not affect the stationary zero-curvature equation (\[2.16\]).
Plugging the ansatz (\[2.12\])-(\[2.15\]) (\[2.21\]) into (\[2.17\])-(\[2.20\]) and comparing the coefficients yields the following relations for $\{a_{\ell}\}_{\ell=0}^{n+1},\{b_{\ell}\}_{\ell=0}^{n},\{c_{\ell}\}_{\ell=0}^{n}.$ That is, $$\label{2.22}
\tilde{p}a_\ell^{-}-\tilde{p}a_\ell=a_{\ell+1}^--a_{\ell+1}+\tilde{r}b_{\ell}-\tilde{q}c_{\ell}^{-},\quad\ell=0,1,\dotsi,n-1,$$ $$\label{2.23}
\tilde{p}b_\ell^{-}-b_{\ell+1}^{-}-\tilde{q}a_{\ell+1}^{-}=\tilde{q}a_{\ell+1}+b_{\ell},\quad\ell=0,1,\dotsi,n-1,$$ $$\label{2.24}
\tilde{r}a_{\ell+1}^{-}+c_{\ell}^{-}=\tilde{p}c_{\ell}-c_{\ell+1}-\tilde{r}a_{\ell+1},\quad\ell=0,1,\dotsi,n-1,$$ $$\label{2.25}
\tilde{r}b_{\ell+1}^--a_{\ell+1}^-=\tilde{q}c_{\ell+1}-a_{\ell+1},\quad\ell=0,\dotsi,n-1,$$ $$\label{2.26}
\tilde{p}a_{n+1}^--\tilde{p}a_{n+1}=0,$$ $$\label{2.27}
\tilde{p}b_{n}^--\tilde{q}\triangle=\tilde{q}a_{n+1}+b_{n},$$ $$\label{2.28}
\tilde{r}a_{n+1}^-+c_n^-=\tilde{p}c_n-\tilde{r}\triangle,$$ $$\label{2.29}
-\triangle^-=-\triangle.$$ Hence if $\{a_{\ell}\}_{\ell=0}^{n+1},\{b_{\ell}\}_{\ell=0}^{n},\{c_{\ell}\}_{\ell=0}^{n}$ are defined by (\[2.2\])-(\[2.5\]), then one finds (\[2.22\])-(\[2.24\]) (\[2.29\]) hold naturally. From (\[2.22\])-(\[2.24\]), one calculate $$\label{2.30}
\begin{split}
&\tilde{r}b_{\ell+1}-\tilde{q}a_{\ell+1}\\
&=\tilde{r}\left(\tilde{p}b_{\ell}^--\tilde{q}a_{\ell+1}^--\tilde{q}a_{\ell+1}-b_{\ell}\right)-\tilde{q}
\left(\tilde{p}c_{\ell}-\tilde{r}a_{\ell+1}-\tilde{r}a_{\ell+1}-c_{\ell}^-\right)\\
&=\tilde{p}(\tilde{r}b_{\ell}^--\tilde{q}c_{\ell})-\tilde{r}b_{\ell}+\tilde{q}c_{\ell}^-\\
&=\tilde{p}(a_{\ell}^--a_{\ell})-\tilde{r}b_{\ell}+\tilde{q}c_{\ell}^-\\
&=a_{\ell+1}^--a_{\ell+1},\\
\end{split}$$ where we used the induction $$\tilde{r}b_{k}^--a_{k}^-=\tilde{q}c_{k}-a_{k},\quad k=0,1,\dotsi,\ell$$ in the third equality of (\[2.30\]). Therefore (\[2.25\]) is the direct result of (\[2.22\])-(\[2.24\]). Using (\[2.2\])-(\[2.5\]), (\[2.26\])-(\[2.29\]) are equivalent to $$\label{2.31}
\tilde{p}a_{n+1}^--\tilde{p}a_{n+1}=0,$$ $$\label{2.32}
b_{n+1}^{-}+\tilde{q}a_{n+1}^--\tilde{q}\triangle=0,$$ $$\label{2.33}
c_{n+1}+\tilde{r}a_{n+1}+\tilde{r}\triangle=0.$$ Noticing the condition (\[2.25\]) and above analysis, (\[2.31\]) and (\[2.32\]) give rise to the stationary relativistic Lotka-Volterra hierarchy, which we introduce as follows $$\label{2.34}
\text{s-RLV}_n(p,q)=\text{s-}\widetilde{\text{RLV}_n}(\tilde{p},\tilde{q},\tilde{r})
=\left(
\begin{array}{c}
a_{n+1}^--a_{n+1} \\
b_{n+1}^{-}+\tilde{q}a_{n+1}^--\tilde{q}\triangle \\
\end{array}
\right)=0,\quad p\in\mathbb{N}_0.$$ Explicitly, $$\label{2.35}
\begin{split}
&\text{s-RLV}_0(p,q)=\text{s-}\widetilde{\text{RLV}_0}(\tilde{p},\tilde{q},\tilde{r})\\
&=\left(\begin{array}{c}
-\tilde{q}\tilde{r}^-+\tilde{q}^+\tilde{r} \\
-\tilde{p}\tilde{q}+\tilde{q}\tilde{q}^+\tilde{r}+\tilde{q}^+-\tilde{q}(\triangle+\delta_1/2)
\end{array}\right)\\
&=\left(
\begin{array}{c}
pq-p^-q^-+q^--q \\
\tilde{q}\left(pq-p-(\delta_1/2+\triangle)\right) \\
\end{array}
\right)=0,
\end{split}$$ where we use the relation $$\label{2.36}
\begin{split}
&\tilde{q}^+/\tilde{q}=e^{(S^+-I)^{-1}q^+}/e^{(S^+-I)^{-1}q}=q,\\
&\tilde{q}\tilde{r}=\tilde{p}-1=p-1, \\
&\tilde{q}\tilde{r}^+=(p-1)q.
\end{split}$$ Taking $\delta_1=-2\triangle=-2$, (\[2.35\]) gives rise to the stationary relativistic Lotka-Volterra equation $$\label{2.37}
\left(
\begin{array}{c}
pq-p^-q^-+q^--q\\
pq-p \\
\end{array}
\right)=0,$$ which means $$\label{2.38}
\begin{split}
&\left(
\begin{array}{c}
pq-p^-q^-+q^--q \\
p^+q^+-pq-p^++p \\
\end{array}
\right)\\
&=\left(
\begin{array}{cc}
I & 0 \\
0 & S^+-I \\
\end{array}
\right)
\left(
\begin{array}{c}
pq-p^-q^-+q^--q \\
pq-p \\
\end{array}
\right)\\
&=0.
\end{split}$$ Next we turn to the time-dependent relativistic Lotka-Volterra hierarchy. The zero-curvature equation is $$\label{2.39}
U_{t_n}(\lambda)+U(\lambda)V_n(\lambda)-V_n^+(\lambda)U(\lambda)=0.$$ $U(\lambda), V_n(\lambda),\{a_\ell(\cdot,t)\}_{\ell=0}^{n+1},\{b_\ell(\cdot,t)\}_{\ell=0}^{n},
\{c_\ell(\cdot,t)\}_{\ell=0}^{n},\{d_\ell(\cdot,t)\}_{\ell=0}^{n+1}$ are defined by (\[2.1\]), (\[2.2\])-(\[2.5\]), (\[2.11\])-(\[2.15\]) and (\[2.21\]). Then (\[2.39\]) implies $$\label{2.40}
\left(
\begin{array}{cc}
-\tilde{p}_{t_n}\lambda & -\tilde{q}_{t_n} \\
-\tilde{r}_{t_n} & 0 \\
\end{array}
\right)+
\left(
\begin{array}{cc}
\Delta_{11}&\Delta_{12}\\
\Delta_{21}&\Delta_{22} \\
\end{array}
\right)=0,$$ where $$\label{2.41}
\Delta_{11}=(\tilde{p}\lambda-\lambda^{-1})A_{2n+1}^-+\tilde{q}C_{2n+1}^--(\tilde{p}\lambda-\lambda^{-1})A_{2n+2}+\tilde{r}B_{2n+1},$$ $$\label{2.42}
\Delta_{12}=(\tilde{p}\lambda-\lambda^{-1})B_{2n+1}^--\tilde{q}D_{2n+2}^--\tilde{q}A_{2n+2}-B_{2n+1}\lambda,$$ $$\label{2.43}
\Delta_{21}=\tilde{r}A_{2n+2}^-+C_{2n+1}^-\lambda-(\tilde{p}\lambda-\lambda^{-1})C_{2n+1}+\tilde{r}D_{2n+2}$$ $$\label{2.44}
\Delta_{22}=\tilde{r}B_{2n+2}^--D_{2n+2}^-\lambda-\tilde{q}C_{2n+1}+D_{2n+2}\lambda.$$ Inserting (\[2.2\])-(\[2.5\]) into (\[2.40\]) then yields $$\label{2.45}
-\tilde{p}_{t_n}+\tilde{p}(a_{n+1}^--a_{n+1})=0,$$ $$\label{2.46}
-\tilde{q}_{t_n}+b_{n+1}^{-}+\tilde{q}a_{n+1}^--\tilde{q}\triangle=0$$ $$\label{2.47}
-\tilde{r}_{t_n}+c_{n+1}+\tilde{r}a_{n+1}+\tilde{r}\triangle=0.$$ Using (\[2.23\]) (\[2.25\]) (\[2.45\]) (\[2.46\]), we directly calculate $$\label{2.48}
\begin{split}
&\tilde{r}_{t_n}=\partial_{t_n}((\tilde{p-1})/\tilde{q})=\tilde{p}_{t_n}/q-(\tilde{p}-1)\tilde{q}_{t_n}/\tilde{q}^2\\
&=\tilde{p}(a_{n+1}^--a_{n+1})/\tilde{q}-(\tilde{p}-1)(b_{n+1}^{-}+\tilde{q}a_{n+1}^--\tilde{q}\triangle)/\tilde{q}^2\\
&=\tilde{p}(a_{n+1}^--a_{n+1})/\tilde{q}-\tilde{r}(\tilde{p}b_{n}^--b_n-\tilde{q}a_{n+1}-\tilde{q}\triangle)/\tilde{q}\\
&=\tilde{p}(a_{n+1}^--a_{n+1})/\tilde{q}-\tilde{r}\tilde{p}b_{n}^-/\tilde{q}+\tilde{r}b_n/\tilde{q}+(\tilde{r}a_{n+1}+\tilde{r}\triangle)\\
&=[\tilde{p}(a_{n+1}^--a_{n+1})+\tilde{r}(b_n-\tilde{p}b_n)]/\tilde{q}+(\tilde{r}a_{n+1}+\tilde{r}\triangle)\\
&=[\tilde{p}(a_{n+1}^--a_{n+1})+\tilde{r}(-b_{n+1}^--\tilde{q}a_{n+1}^--\tilde{q}a_{n+1}]/\tilde{q}+(\tilde{r}a_{n+1}+\tilde{r}\triangle)\\
&=[\tilde{p}(a_{n+1}^--a_{n+1})+(1-\tilde{p})(a_{n+1}^--a_{n+1})-\tilde{r}b_{n+1}^-]/\tilde{q}+(\tilde{r}a_{n+1}+\tilde{r}\triangle)\\
&=[a_{n+1}^--a_{n+1}-\tilde{r}b_{n+1}^-]/\tilde{q}+(\tilde{r}a_{n+1}+\tilde{r}\triangle)\\
&=c_{n+1}+\tilde{r}a_{n+1}+\tilde{r}\triangle.\\
\end{split}$$ Thus, (\[2.45\])-(\[2.47\]) are essentially two independent equations (\[2.45\]) and (\[2.46\]). Varying $n\in\mathbb{N}_0$, (\[2.45\]) and (\[2.46\]) give rise to time-dependent relativistic Lotka-Volterra hierarchy, which we introduce as follows $$\label{2.49}
\text{RLV}_n(p,q)=\widetilde{\text{RLV}_n}(\tilde{p},\tilde{q},\tilde{r})
=\left(
\begin{array}{c}
-\tilde{p}_{t_n}+\tilde{p}(a_{n+1}^--a_{n+1})\\
-\tilde{q}_{t_n}+b_{n+1}^{-}+\tilde{q}a_{n+1}^--\tilde{q}\triangle \\
\end{array}
\right)=0,\quad p\in\mathbb{N}_0.$$ Explicitly, $$\label{2.50}
\begin{split}
&\text{RLV}_0(p,q)=\widetilde{\text{RLV}_0}(\tilde{p},\tilde{q},\tilde{r})\\
&=\left(\begin{array}{c}
-\tilde{p}_{t_0}+\tilde{p}(-\tilde{q}\tilde{r}^-+\tilde{q}^+\tilde{r}) \\
-\tilde{q}_{t_0}-\tilde{p}\tilde{q}+\tilde{q}\tilde{q}^+\tilde{r}+\tilde{q}^+-\tilde{q}(\triangle+\delta_1/2)
\end{array}\right)\\
&=\left(
\begin{array}{c}
-p_{t_0}+p(pq-p^-q^-+q^--q) \\
\partial_{t_0}(e^{(S^{+}-I)^{-1}\ln q})+e^{(S^+-I)^{-1}\ln q}\left(pq-p-(\delta_1/2+\triangle)\right) \\
\end{array}
\right)\\
&=\left(
\begin{array}{c}
-p_{t_0}+p(pq-p^-q^-+q^--q) \\
e^{(S^+-I)^{-1}\ln q}(S^{-1}-I)^{-1}(q_{t_n}/q)+e^{(S^+-I)^{-1}\ln q}\left(pq-p-(\delta_1/2+\triangle)\right)\\
\end{array}\right)\\
&=0
\end{split}$$ where we used $$\label{2.51}
\partial_{t_n}(e^{(S^{+}-I)^{-1}\ln q})=e^{(S^+-I)^{-1}}\ln q\partial_{t_n}((S^+-I)^{-1}\ln q)=e^{(S^+-I)^{-1}\ln q}(S^{-1}-I)^{-1}(q_{t_n}/q),$$ in the third equality of (\[2.50\]). Taking $\delta_1=-2\Delta=-2$, (\[2.50\]) is just the relativistic Lotka-Volterra equation, which is equivalent to the form of $$\label{2.52}
\begin{split}
&\text{RLV}_0(p,q)=\widetilde{\text{RLV}_0}(\tilde{p},\tilde{q},\tilde{r})\\
&=\left(\begin{array}{c}
-p_{t_0}+p(pq-p^-q^-+q^--q) \\
-q_{t_0}+q(p^+q^+-p^+-pq+p)
\end{array}\right)\\
&=0.\\
\end{split}$$ In order to derive the algebraic curve associated with the relativistic Lotka-Volterra hierarchy, we need the some change about $U(\lambda)$, which is $$\label{2.53}
U(\lambda)\rightarrow \tilde{U}(\xi)=\xi U(\xi^{-1})=
\left(
\begin{array}{cc}
\tilde{p}-z & \tilde{q}\xi \\
\tilde{r}\xi & 1 \\
\end{array}
\right),\quad z=\xi^2.$$ Let $$\label{2.54}
\tilde{V}_{n}(\xi)=V_{n}(\xi^{-1}),$$ then $\tilde{U}(\xi), \tilde{V}_{n}(\xi)$ remain satisfy the zero-curvature equation (\[2.16\]) and (\[2.39\]), that is $$\label{2.55}
\tilde{U}(\xi)\tilde{V}_n(\xi)-\tilde{V}_n^+(\xi)\tilde{U}(\xi)=0$$ $$\label{2.56}
\tilde{U}_{t_n}(\xi)+\tilde{U}(\xi)\tilde{V}_n(\xi)-\tilde{V}_n^+(\xi)\tilde{U}(\xi)=0$$ and hence the final form of (\[2.34\]) (\[2.49\]) is invariant. Assume $$\label{2.57}
\tilde{V}_{n}(\xi)=V_{n}(\xi^{-1})
=\left(
\begin{array}{cc}
\tilde{A}_{n+1}(z) & \xi\tilde{B}_{n}(z) \\
\xi\tilde{C}_{n}(z) & -\tilde{D}_{n+1}(z) \\
\end{array}
\right)$$ and then one derives $$\label{2.58}
\tilde{A}_{n+1}(z)=A_{2n+2}(\xi^{-1}),$$ $$\label{2.59}
\tilde{B}_{n}(z)=\xi^{-1}B_{2n+1}(\xi^{-1}),$$ $$\label{2.60}
\tilde{C}_{n}(z)=\xi^{-1}C_{2n+1}(\xi^{-1}),$$ $$\label{2.61}
\tilde{D}_{n+1}(z)=D_{2n+2}(\xi^{-1})$$ by comparing with (\[2.11\]). Then (\[2.17\])-(\[2.10\]) change into $$\label{2.17a}
(\tilde{p}-z)\tilde{A}_{n+1}^-+z\tilde{q}\tilde{C}_n^{-}=
(\tilde{p}-z)\tilde{A}_{n+1}+z\tilde{r}\tilde{B}_{n+1},$$ $$\label{2.18a}
(\tilde{p}-z)\tilde{B}_{n}^--\tilde{q}\tilde{D}_{n+1}^-=\tilde{q}\tilde{A}_{n+1}+\tilde{B}_{n},$$ $$\label{2.19a}
\tilde{r}\tilde{A}_{n+1}^-+\tilde{C}_{n}^-=(\tilde{p}-z)\tilde{C}_{n}-\tilde{r}\tilde{D}_{n+1},$$ $$\label{2.20a}
z\tilde{r}\tilde{B}_{n}^--D_{n+1}^-=z\tilde{q}\tilde{C}_{n}-\tilde{D}_{n+1},$$ respectively.
Taking into account (\[2.55\]) (\[2021\]), one infers that the expression $R_{2n+2}(z)$, defined as $$\label{2.62}
R_{2n+2}(z)=-(\tilde{A}_{n+1})^2-z\tilde{B}_n\tilde{C}_n,$$ is a lattice constant, that is, $R_{2n+2}=R_{2n+2}^-$, since taking determinants in the stationary zero-curvature equation (\[2.55\]) immediately yields $$\label{2.63}
\tilde{p}(1-z)[-(\tilde{A}_{n+1})^2-\tilde{B}_n^-\tilde{C}_n^-+(\tilde{A}_{n+1})^2+\tilde{B}_{n}\tilde{C}_n]=0.$$ Hence, $R_{2n+2}$ only depends on $z$, and one may write $R_{2n+2}$ as $$\label{2.64}
R_{2n+2}(z)=-(1/4)\prod_{m=0}^{2n+1}(z-E_m),\quad n\in\mathbb{N}_0.$$
\[remark3\] (i) Taking the transformation $p\rightarrow hp, q\rightarrow hq,$ one concludes that the equation (\[2.34\]) and (\[2.49\]) change into the normal form of relativistic Lotka-Volterra system [@RLV; @Laxpresentation].
\(ii) Taking the transformation $\tilde{p}=p=\tilde{v}^+/\tilde{v},\tilde{v}\in\ell(\mathbb{Z})$, (\[2.34\]) and (\[2.49\]) change into $$\label{2.65}
\text{s-}\overline{\widetilde{\text{RLV}}}_n(\tilde{v},\tilde{q},\tilde{r})=\text{s-}\widetilde{\text{RLV}_n}(\tilde{v}^+/\tilde{v},\tilde{q},\tilde{r})=0,$$ $$\label{2.66}
\overline{\widetilde{\text{RLV}}}_n(\tilde{v},\tilde{q},\tilde{r})=\widetilde{\text{RLV}_n}(\tilde{v}^+/\tilde{v},\tilde{q},\tilde{r})=0.$$
Algebro-geometric Solutions of Stationary Relativistic Lotka-Volterra Hierarchy
===============================================================================
In this section, we present a detailed study of the stationary Toda hierarchy. Our principle tools are derived from the polynomial recursion formalism introduced in section 2 and a fundamental meromorphic function $\phi$ on $\mathcal{K}_n$. With the help of $\phi$ we study the Baker-Akhiezer vector $\Psi$, the common eigenfunctionof $\tilde{U}(\xi)$ and $\tilde{V}_n(\xi)$, trace formulas, and theta function representations of $\phi$, $\Psi$, $p$ and $q$.
Throughout this paper we have the following hypothesis
$$\label{3.1}
\tilde{p}=p\neq 0,1,\quad\tilde{p}(n),\tilde{q}(n),\tilde{r}(n),p(n),q(n)\in\ell(\mathbb{Z}),\quad n\in\mathbb{Z},$$ and $n\in\mathbb{N}_0$ in (\[2.34\]) fixed.
The affine part of the algebraic curve associated with the $n$-th equation in the stationary relativistic Lotka-Volterra hierarchy (\[2.34\]), which takes the form of $$\label{3.2}
\mathcal{K}_n:\mathcal{F}_n(z,y)= y^2+4R_{2n+2}(z)=y^2-\prod_{m=0}^{2n+1}(z-E_m)=0,\quad\{E_m\}_{m=0}^{2n+1}\subseteq\mathbb{C}\backslash\{0\}$$ is nonsingular. That is, $$\label{3.3}
\begin{split}
&E_m\neq E_{m^{'}}\quad for\quad m\neq m^{'}, m, m^{'}=0,1,2,\dotsi,2n+1.\\
\end{split}$$
$\mathcal{K}_{n}$ defined in (\[3.2\]) is compactified by joining two points $P_{\infty\pm}, P_{\infty+}\neq P_{\infty-}$ at infinity, but for notational simplicity the compactification is also denoted by $\mathcal{K}_n$.
One can introduce the complex structure on $\mathcal{K}_{n}$ to yield a compact Riemann surface of genus $n$ [@r; @s; @16]. This is a hyperelliptic Riemann surface. It can be regarded as a double covering of sphere surface $\mathbb{C}_{\infty}=\mathbb{C}\bigcup\{\infty\}$. Points $P$ on $\mathcal{K}_p\backslash\{P_{\infty\pm}\}$ are represented as pairs $P=(z,y)$, where $y(\cdot)$ is the meromorphic function on $\mathcal{K}_n$ satisfying $\mathcal{F}_{n}(z,y)=0$.
We write $$\label{3.4}
\tilde{B}_{n}(z)=(-\tilde{q}^{+})\prod_{j=1}^{n}(z-\mu_j),\quad \tilde{C}_{n}(z)=(-\tilde{r})\prod_{j=1}^{n}(z-\nu_j),\quad\mu_j,\nu_j\in\ell(\mathbb{Z}).$$ Next we ¡¯lift¡¯ the point $\mu_j,$ $\nu_j$ from the $\mathbb{C}_{\infty}$ to the compact Riemann surface $\mathcal{K}_n,$ $$\label{3.5}
\hat{\mu}_j=(\mu_j,-2\tilde{A}_{n+1}(\mu_j,n)),\quad \hat{\nu}_j=(\nu_j,2\tilde{A}_{n+1}(\nu_j,n)),
\quad j=1,2,\dotsi,n$$ and introduce the points $P_{0,\pm}$ on $\mathcal{K}_n$ $$\label{3.6}
P_{0,\pm}=(0,\pm2\tilde{A}_{n+1}(0,n)),\quad \tilde{A}_{n+1}^2(0,n)=\frac{1}{4}\prod_{m=0}^{2n+1}E_m.$$ We introduce the holomorphic sheet exchange map on $\mathcal{K}_n$ $$*:\quad \mathcal{K}_n\rightarrow\mathcal{K}_n\quad P=(z,y)\mapsto P^{*}=(z,-y)\quad P_{\infty\pm}\mapsto P_{\infty\pm}^{*}=P_{\infty\mp}.$$
Next one can define a fundamental meromorphic function $\tilde{\phi}$ on $\mathcal{K}_n$ $$\label{3.7}
\phi(P,n)=\frac{\frac{1}{2}y-\tilde{A}_{n+1}(z,n)}{B_{n}(z,n)}
=\frac{z\tilde{C}_{n}(z,n)}{\frac{1}{2}y+\tilde{A}_{n+1}(z,n)},
\quad P=(z,y)\in\mathcal{K}_n.$$ The divisor $(\tilde{\phi}(\cdot))$ of $\tilde{\phi}$ is $$\label{3.8}
(\phi(\cdot,n))=\mathcal{D}_{P_{0,+}\underline{\hat{\nu}}(n)}-\mathcal{D}_{P_{\infty+}\underline{\hat{\mu}}(n)},$$ where we abbreviated $$\label{3.9}
\underline{\hat{\mu}}(n)=\left(\mu_1(n),\mu_2(n),\dotsi,\mu_n(n)\right),\quad \underline{\hat{\nu}}(n)=\left(\nu_1(n),\nu_2(n),\dotsi,\nu_n(n)\right).$$ Given $\phi(\cdot,n)$, the stationary Baker-Akhiezer vector $\Psi(\cdot,n,n_0)$ on $\mathcal{K}_n$ is then defined by $$\label{3.10}
\Psi(P,\xi,n,n_0)=\left(
\begin{array}{c}
\Psi_1(P,\xi,n,n_0) \\
\Psi_2(P,\xi,n,n_0)
\end{array}
\right),$$
$$\label{3.11}
\Psi_1(P,\xi,n,n_0)=
\begin{cases}
\prod_{n^{'}=n_0+1}^{n}\left(\tilde{p}(n^{'})-z+\tilde{q}(n^{'})\tilde{\phi}^-(P,n^{'})\right),&n\geq n_0+1,\cr
1,&n=n_0,\cr
\prod_{n^{'}=n+1}^{n_0}\left(\tilde{p}(n^{'})-z+\tilde{q}(n^{'})\tilde{\phi}^-(P,n^{'})\right)^{-1},&n\leq n_0-1,
\end{cases}$$
$$\label{3.12}
\Psi_2(P,\xi,n,n_0)=\xi^{-1}\tilde{\phi}(P,n_0)
\begin{cases}
\prod_{n^{'}=n_0+1}^{n}\left(\frac{\tilde{r}(n^{'})z}{\tilde{\phi}^-(P,n^{'})}+1\right),&n\geq n_0+1,\cr
1,&n=n_0,\cr
\prod_{n^{'}=n+1}^{n_0}\left(\frac{\tilde{r}(n^{'})z}{\tilde{\phi}^-(P,n^{'})}+1\right)^{-1},&n\leq n_0-1.
\end{cases}$$
Some properties of $\tilde{\phi}, \Psi_1, \Psi_2$ are discussed in the following lemma.
\[lemma1\] Suppose $p,q$ satisfy the $n$-th relativistic Lotka-Volterra hierarchy (\[2.34\]) and (\[3.1\])-(\[3.12\]) holds. Let $P=(z,y)\in\mathcal{K}_n\backslash\{P_{\infty\pm},P_{0,\pm}\}, (n,n_0)\in\mathbb{Z}^2.$ Then $\tilde{\phi}$ satisfies the Riccati-type equation $$\label{3.13}
(\tilde{p}-z)\tilde{\phi}+\tilde{q}\tilde{\phi}\tilde{\phi}^-=\tilde{r}z+\tilde{\phi}^{-},$$ as well as $$\label{3.14}
\tilde{\phi}(P)\tilde{\phi}(P^{*})=\frac{-z\tilde{C}_{n}(z)}{\tilde{B}_{n}(z)},$$ $$\label{3.15}
\tilde{\phi}(P)+\tilde{\phi}(P^{*})=\frac{-2\tilde{A}_{n+1}}{\tilde{B}_n},$$ $$\label{3.16}
\tilde{\phi}(P)-\tilde{\phi}(P^{*})=\frac{y}{\tilde{B}_n(z)}.$$ Moreover, the vector $\Psi$ satisfy $$\label{3.17}
\tilde{U}(\xi)\Psi^-(P)=\Psi(P),$$ $$\label{3.18}
\tilde{V}_n(\xi)\Psi(P)=(1/2)y\Psi(P),$$ $$\label{3.19}
\Psi_2(P,\xi,n,n_0)=\xi^{-1}\tilde{\phi}(P,n)\Psi_1(P,\xi,n,n_0).$$
Using (\[2021\]) (\[2.18a\]) (\[2.20a\]) (\[2.62\]) (\[2.64\]) (\[3.7\]), we directly calculate
$$\label{3.20}
\begin{split}
&(\tilde{p}-z)\tilde{\phi}+\tilde{q}\tilde{\phi}\tilde{\phi}^--\tilde{r}z-\tilde{\phi}^{-}\\
&=1/(\tilde{B}_n\tilde{B}_n^-)[\tilde{B}_n^-(\tilde{p}-z)(\frac{1}{2}y-\tilde{A}_{n+1})+\tilde{q}(\frac{1}{2}y-\tilde{A}_{n+1})(\frac{1}{2}-\tilde{A}_{n+1}^-)\\
&-\tilde{r}\tilde{B}_n\tilde{B}_n^--(\frac{1}{2}y-\tilde{A}_{n+1})\tilde{B}_n]\\
&=0.
\end{split}$$
Equalities (\[3.14\])-(\[3.16\]) hold from (\[3.7\]). Next we use induction to prove (\[3.19\]). Obviously, for $n=n_0$, we have $$\label{3.21}
\Psi_2(P,\xi,n_0,n_0)=\xi^{-1}\tilde{\phi}(P,n_0)\Psi_1(P,\xi,n_0,n_0)$$ from (\[3.11\]) and (\[3.12\]). In the case $k>n_0$, assume (\[3.19\]) holds for $n=n_0,n_0+1,\dotsi,k-1$. From the definition of $\Psi_1$, $\Psi_2$ in (\[3.11\]) (\[3.12\]), one finds $$\label{3.22}
\begin{split}
&\xi\Psi_2(P,\xi,k,n_0)/\Psi_1(P,\xi,k,n_0)=\xi\times[\Psi_2(P,\xi,k-1,n_0)/\Psi_1(P,\xi,k-1,n_0)]\\
&\times[\left(\frac{\tilde{r}z+1}{\tilde{\phi}^-(P,k)}\right)/\left(\tilde{p}-z+\tilde{q}\tilde{\phi}^-(P,k)\right)]\\
&=\xi\times\xi^{-1}\tilde{\phi}(P,k-1)\times[\left(\frac{\tilde{r}z+1}{\tilde{\phi}^-(P,k)}\right)/\left(\tilde{p}-z+\tilde{q}\tilde{\phi}^-(P,k)\right)]\\
&=\left(\tilde{r}z+1\right)/\left(\tilde{p}-z+\tilde{q}\tilde{\phi}^-(P,k)\right).
\end{split}$$ The Riccati-type equation (\[3.13\]) shows that $\tilde{\phi}(P,k)$ satisfies $$\label{3.23}
\tilde{\phi}(P,k)=\left(\tilde{r}z+1\right)/\left(\tilde{p}-z+\tilde{q}\tilde{\phi}^-(P,k)\right).$$ Noted that $\xi\Psi_2(P,\xi,k,n_0)/\Psi_1(P,\xi,k,n_0)$ and $\tilde{\phi}(P,k)$ take the same value at $n=n_0$ and (\[3.22\]) (\[3.23\]), we have (\[3.19\]) for $n=k$. The proof of the case $n<n_0$ is similar with $n>n_0$. From (\[3.11\]) (\[3.12\]), one finds $$\label{3.24}
\begin{split}
&\Psi_1=\left(\tilde{p}-z+\tilde{q}\tilde{\phi}^-\right)\Psi_1^-\\
&=(\tilde{p}-z)\Psi_1^-+\xi\tilde{q}\Psi_2^-\\
\end{split}$$ and $$\label{3.25}
\begin{split}
&\Psi_2=\left(\frac{\tilde{r}z}{\tilde{\phi}^-}+1\right)\Psi_2^-\\
&=\xi\tilde{r}\Psi_1^-+\Psi_2^-,\\
\end{split}$$ where we used (\[3.19\]) in the last equalities of (\[3.24\]) and (\[3.25\]). (\[3.18\]) comes from (\[3.7\]) and (\[3.19\]) by direct calculation.
Combining the polynomial recursive relation defined in section 1 with (\[3.4\]) then yields the following trace formula for $a_\ell$, and $ b_\ell$ in terms of symmetric functions of $\mu_j$ and $\nu_j$, respectively.
\[lemma2\] Suppose $p, q$ satisfy the $n$-th stationary relativistic Lotka-Volterra system (\[2.34\]) and (\[3.1\]). Then we have the following trace formula $$\label{3.26}
\tilde{p}^{+}-\tilde{q}^{+}\tilde{r}-\tilde{q}^{++}\tilde{r}^+-\tilde{q}^{++}/\tilde{q}^{+}+\delta_1=-\sum_{j=1}^{n}\mu_j,$$ $$\label{3.27}
\tilde{p}-\tilde{q}^{+}\tilde{r}-\tilde{r}^{-}/\tilde{r}-\tilde{q}\tilde{r}^{-}+\delta_1=-\sum_{j=1}^{n}\nu_j.$$
Comparing the coefficients of $z^{n-1}$ of $\tilde{B}_n(z), \tilde{C}_n(z)$ in (\[2.59\]) (\[2.60\]) and (\[3.4\]) the yield the above formulas (\[3.26\]) (\[3.27\]).
Next we turn to study the asymptotic behavior of $\tilde{\phi}, \Psi_1, \Psi_2$ in the neigborhood of $P_{\infty\pm}$ and $P_{0,\pm}$. This is a crucial step to construct the algebro-geometric solutions of relativistic Lotka-Volterra hierarchy.
\[lemma3\] Suppose $p, q$ satisfy the $n$-th stationary relativistic Lotka-Volterra system (\[2.34\]) and (\[3.1\]). Moreover, let $P=(z,y)\in\mathcal{K}_n\backslash\{P_{\infty\pm}, P_{0,\pm}\}, (n,n_0)\in\mathbb{Z}\times\mathbb{Z}.$ Then $\tilde{\phi}$ defined in (\[3.7\]) has the following asymptotic property $$\label{3.28}
\tilde{\phi}(P)=
\begin{cases}
(\tilde{q}^{+})^{-1}\zeta^{-1}+\left(((\tilde{q}^{+}/\tilde{q}-1)\tilde{p})/\tilde{q}\right)^{+}+O(\zeta)
&\text{as}\quad P\rightarrow P_{\infty+}, \cr
-\tilde{r}+(\tilde{p}\tilde{r}^--\tilde{p}\tilde{r})\zeta+O(\zeta^2)
&\text{as}\quad P\rightarrow P_{\infty-}, \cr
\end{cases}$$ where we use the local coordinate $z=\zeta^{-1}$ near the points $P_{\infty\pm}$. $$\label{3.29}
\tilde{\phi}(P)=
\begin{cases}
c_{n}/(\prod_{m=0}^{2n+1}E_{m})\zeta+O(\zeta^2)
&\text{as}\quad P\rightarrow P_{0,+}, \cr
-\left(\prod_{m=0}^{2n+1}E_m\right)/b_n+O(\zeta)
&\text{as}\quad P\rightarrow P_{0,-}, \cr
\end{cases}$$ where we use the local coordinate $z=\zeta$ near the points $P_{0,\pm}.$
The components $\Psi_1, \Psi_2$ of the Baker-Akhiezer $\Psi$ have the following asymptotic properties $$\label{3.30}
\Psi_1(P,\xi,n,n_0)=
\begin{cases}
\left(\tilde{q}^{+}(n)\tilde{v}^{+}(n)\right)/\left(\tilde{q}^{+}(n_0)\tilde{v}^{+}(n_0)\right)+O(\zeta)
&\text{as}\quad P\rightarrow P_{\infty+}, \cr
(-1)^{n-n_0}\times\zeta^{n_0-n}\left(1+O(\zeta)\right)
&\text{as}\quad P\rightarrow P_{\infty-},\cr
\end{cases}$$
$$\label{3.31}
\Psi_1(P,\xi,n,n_0)=
\begin{cases}
\tilde{v}^{+}(n)/\tilde{v}^{+}(n_0)+O(\zeta)
&\text{as}\quad P\rightarrow P_{0,+}, \cr
\Gamma(\tilde{p}-\tilde{q}\frac{\prod_{m=0}^{2n+1}E_m}{b_n^-})(n,n_0)+O(\zeta)
&\text{as}\quad P\rightarrow P_{0,-}, \cr
\end{cases}$$
and $$\label{3.32}
\Psi_2(P,\xi,n,n_0)=\xi^{-1}\times
\begin{cases}
\tilde{v}^{+}(n)/\left(\tilde{q}^{+}(n_0)\tilde{v}^{+}(n_0)\right)\zeta^{-1}\left(1+O(\zeta)\right)
&\text{as}\quad P\rightarrow P_{\infty+}, \cr
(-1)^{n+1-n_0}\times\tilde{r}(n)\zeta^{n_0-n}\left(1+O(\zeta)\right)
&\text{as}\quad P\rightarrow P_{\infty-}, \cr
\end{cases}$$ $$\label{3.33}
\Psi_2(P,\xi,n,n_0)=\xi^{-1}\times
\begin{cases}
[\left(\tilde{v}^{+}(n)c_n\right)/\left(\prod_{m=0}^{2n+1}E_m\tilde{v}^{+}(n_0)\right)]\zeta+O(\zeta^2)\cr
\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\text{as}\quad P\rightarrow P_{0,+},\cr
-\left(\prod_{m=0}^{2n+1}E_m\left(\Gamma(\tilde{p}-\tilde{q}\frac{\prod_{m=0}^{2n+1}E_m}{b_n^-})(n,n_0)\right)\right)/b_n +O(\zeta)\\
\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\text{as}\quad P\rightarrow P_{0,-}, \cr
\end{cases}$$ where $\tilde{v}$ are defined in Remark \[remark3\] and $$\label{3.34}
\Gamma(f)(n,n_0)=
\begin{cases}
\prod_{n^{'}=n_0+1}^{n}f(n^{'})& n>n_0,\cr
1&n=n_0,\cr
\prod_{n^{'}=n+1}^{n_0}f(n^{'})^{-1}& n<n_0,\cr
\end{cases}\quad \forall f\in\ell(\mathbb{Z}).$$
The existence of the asymptotic expansion of $\tilde{\phi}$ in terms of the local coordinate $z=\zeta^{-1}$ near $P_{\infty\pm}$, respectively, $\zeta=z$ near $P_{0,\pm}$ is clear from the explicit form of $\tilde{\phi}$ in (\[3.7\]) (\[3.8\]). Assume $\tilde{\phi}$ has the following asymptotic expansions $$\label{3.35}
\tilde{\phi}=
\begin{cases}
\phi_{-1}\zeta^{-1}+\phi_{0}+\phi_{1}\zeta+O(\zeta^2),&\text{as}\quad P\rightarrow P_{\infty+},\cr
\phi_{0}+\phi_{1}\zeta+\phi_2\zeta^2+O(\zeta^3), &\text{as}\quad P\rightarrow P_{\infty-}.\cr
\end{cases}$$ Inserting the asymptotic expansions (\[3.35\]) into the Riccati-type equation (\[3.13\]) and comparing coefficients of powers of $\zeta$, which determines the coefficients $\phi_k$ in (\[3.25\]), one concludes (\[3.28\]). Insertion of the polynomials $\tilde{B}_{n}, \tilde{C}_{n}$ defined in (\[2.59\]) (\[2.60\]) into (\[3.7\]) then yields the explicit coefficients in (\[3.29\]). Next we compute the asymptotic expansion of $\Psi_1$. Noted the definition of $\Psi_1$ in (\[3.11\]), we first investigate the expression $\tilde{p}-z+\tilde{q}\tilde{\phi}^-$. With the help of (\[3.28\]) (\[3.29\]), one finds $$\label{3.36}
\tilde{p}-z+\tilde{q}\tilde{\phi}^-=
\begin{cases}
\left(\tilde{q}^{+}\tilde{v}^{+}\right)/\left(\tilde{q}\tilde{v}\right)+O(\zeta),&\text{as}\quad P\rightarrow P_{\infty+},\cr
-\zeta^{-1}+O(1),&\text{as}\quad P\rightarrow P_{\infty-},\cr
\tilde{v}^{+}/\tilde{v}+O(\zeta),&\text{as}\quad P\rightarrow P_{0,+},\cr
\tilde{p}-\tilde{q}\frac{\prod_{m=0}^{2n+1}E_m}{b_n^-}+O(\zeta),&\text{as}\quad P\rightarrow P_{0,-},\cr
\end{cases}$$ which give rise to (\[3.30\]) (\[3.31\]). Obviously, the exact $n$ poles of $\tilde{p}-z+\tilde{q}\tilde{\phi}^-$ in $\mathcal{K}_n\backslash\{P_{\infty\pm},P_{0,\pm}\}$ coincide with the ones of $\tilde{\phi}^-$ and there are $n+1$ poles of $\tilde{p}-z+\tilde{q}\tilde{\phi}^-$ in $\mathcal{K}_n$. However, it is easy to know that $\Psi_1$ is a meromorphic function on $\mathcal{K}_n$ from the definition (\[3.11\]) and the meromorphic property of $\tilde{\phi}.$ Therefore the meromorphic function $\tilde{p}-z+\tilde{q}\tilde{\phi}^-$ possess exact $n+1$ zero points on $\mathcal{K}_n.$ So we need some deformation of function $\tilde{p}-z+\tilde{q}\tilde{\phi}^-$. Using (\[2.18a\]) (\[2.62\]) (\[3.2\]) and (\[3.7\]), one may calculate $$\label{3.37}
\begin{split}
&\tilde{p}-z+\tilde{q}\tilde{\phi}^-\\
&=\tilde{p}-z+\tilde{q}\frac{\frac{1}{2}y-\tilde{A}_{n+1}^{-}}{\tilde{B}_{n}^{-}}\\
&=\frac{(\tilde{p}-z)\tilde{B}_{n}^{-}+\tilde{q}(\frac{1}{2}y-\tilde{A}_{n+1}^-)}{\tilde{B}_{n}^{-}}\\
&=\frac{\tilde{q}\tilde{A}_{n+1}+\tilde{B}_n+\frac{1}{2}\tilde{q}y}{\tilde{B}_n^{-}}\\
&=\frac{\tilde{B}_n}{\tilde{B}_{n}^{-}}+\tilde{q}\frac{\frac{1}{2}y+\tilde{A}_{n+1}}{\tilde{B}_{n}^{-}}\\
&=\frac{\tilde{B}_n}{\tilde{B}_{n}^{-}}+
\tilde{q}\frac{\frac{1}{4}y^2-\tilde{A}_{n+1}^2}{\tilde{B}_{n}^{-}\left(\frac{1}{2}y-\tilde{A}_{n+1}\right)}\\
&=\frac{\tilde{B}_n}{\tilde{B}_{n}^{-}}\left(1+\tilde{q}\frac{z\tilde{C}_{n}}{\frac{1}{2}y-\tilde{A}_{n+1}}\right).\\
\end{split}$$ Thus, $$\label{3.38}
\tilde{p}-z+\tilde{q}\tilde{\phi}^-(P)\equfill{P\rightarrow \hat{\mu}_j}{}\frac{\tilde{B}_{n}(P)}{\tilde{B}_{n}^-(P)}O(1),$$ which shows $\hat{\mu}_j, \quad j=1,\dotsi,n$ are $n$ zero points of function $\tilde{p}-z+\tilde{q}\tilde{\phi}^-(P).$ The remaining one zero point is denoted by $P_{\sharp}$. Finally, (\[3.32\]) and (\[3.33\]) follows from (\[3.19\]) and (\[3.28\])-(\[3.31\]).
We choose a fixed base point $Q_{0}$ on $\mathcal{K}_{p}\backslash\{P_{0,+,\pm},P_{\infty\pm}\}$. Let $\omega_{P_{0}P_{\infty+}}^{(3)}$ be a normal differential of the third kind holomorphic on $\mathcal{K}_{p}\backslash\{P_{\infty+},P_{0,+}\}$ with simple poles at $P_{\infty+}$ and $P_{0}$ and residues -1 and 1, respectively, that is, $$\begin{aligned}
\omega_{P_{0,+}P_{\infty+}}^{(3)}\equfill{\zeta\rightarrow 0}{}
\begin{cases}
(-\zeta^{-1}+O(1))d\zeta&P\rightarrow P_{\infty+}\cr
(\zeta^{-1}+O(1))d\zeta&P\rightarrow P_{0,+}
\end{cases}\end{aligned}$$ and $$\label{3.321}
\omega_{P_{\infty+}P_{\infty-}}^{(3)}=\frac{1}{y}\prod_{j=1}^p\left(z-\lambda_j^{'}\right)dz$$ be a normal differential of the third kind holomorphic on $\mathcal{K}_{p}\backslash\{P_{\infty+},P_{\infty-}\}$ with simple poles at $P_{\infty+}$ and $P_{\infty-}$ and residues 1 and -1, where the local coordinates $z=\zeta^{-1}$ for $P$ near $P_{\infty\pm}$ , $z=\zeta$ for $P$ near $P_{0}$, and $\{\lambda_{j}\}_{j=1,\dotsi,p}$, $\{\lambda_{j}^{'}\}_{j=1,\dotsi,p}$ are constants uniquely determined by normalized process.
Moreover, $$\int_{a_{j}}\omega_{P_{0,+}P_{\infty+}}^{(3)}=0,\quad j=1,\dotsi,p,$$ $$\int_{Q_{0}}^{P}\omega_{P_{0,+}P_{\infty+}}^{(3)}\equfill{\zeta\rightarrow 0}{}\left(\begin{array}{cccc}0\\\ln\zeta\end{array}\right)+\left(\begin{array}{cccc}e_{0.-}\\e_{0,+}\end{array}\right)+O(\zeta)\quad\begin{array}{cccc}P\rightarrow P_{\infty-}\\P\rightarrow P_{\infty+}\end{array},$$ $$\int_{Q_{0}}^{P}\omega_{P_{0,+}P_{\infty+}}^{(3)}\equfill{\zeta\rightarrow 0}{}-\ln\zeta+d_{0}+O(\zeta)\quad P\rightarrow P_{0},$$ where we choose a homology basis $\{a_{j},b_{j}\}_{j=1}^{n}$ on $\mathcal{K}_{n}$ in such a way that the intersection matrix of the cycles satisfies $$a_{j}\circ b_{k}=\delta_{j,k}, \quad a_{j}\circ a_{k}=0,\quad b_{j}\circ b_{k}=0,\quad j,k=1,\dotsi,n$$ and $e_{0,\pm},d_{0}\in\mathbb{C}.$ One easily verifies that $dz/y$ is a differential on $\mathcal{K}_{n}$ with zeros of order $p-1$ at $P_{\infty\pm}$ and hence $$\eta_{j}=\frac{z^{j-1}}{y}dz,\quad j=1,\dotsi,n$$ form a basis for the space of holomorphic differentials on $\mathcal{K}_{n}$. Introducing the following invertible matrix $C_{j,k}\in\mathbb{C}$ $$C=(C_{j,k})_{j,k=1,\dotsi,p},\quad C_{j,k}=\int_{a_{k}}\eta_{j},$$ $$\underline{c}(k)=(c_{1}(k),\dotsi,c_{p}(k)),\quad c_{j}(k)=(C^{-1})_{j,k} \quad k=1,\dotsi,n.$$ It’s easy to show that the normalized holomorphic differentials $\{\omega_{j}\}_{j=1,\dotsi,p}$ can be written into $$\omega_{j}=\sum_{l=1}^{p}c_{j}(l)\eta_{l},\quad \int_{a_{k}}\eta_{j}=\delta_{j,k},\quad j,k=1,\dotsi,n,$$ $$\underline{\omega}=\left(\omega_1,\dotsi,\omega_n\right).$$ Assume $\eta\in\mathbb{C}$ and $|\eta|<$min$\{|E_0|^{-1},|E_{1}|^{-1},|E_{2}|^{-1},\dotsi,|E_{2n+1}|^{-1}\}$ and abbreviate $$\underline{E}=(E_{0},E_{1},\dotsi,E_{2n+1}).$$ Then $$\left(\prod_{m=0}^{2n+1}(1-E_{m}\eta)\right)^{-1/2}=\sum_{k=0}^{+\infty}\hat{c}_{k}(\underline{E})\eta^{k},$$ where $$\hat{c}_{0}(\underline{E})=1,\quad \hat{c}_{1}(\underline{E})=\frac{1}{2}\sum_{m=0}^{2n+1}E_{m}, \quad etc.$$ Similarly, $$\left(\prod_{m=0}^{2n+1}(1-E_{m}\eta)\right)^{1/2}=\sum_{k=0}^{+\infty}{c}_{k}(\underline{E})\eta^{k},$$ where $${c}_{0}(\underline{E})=1,\quad {c}_{1}(\underline{E})=-\frac{1}{2}\sum_{m=0}^{2n+1}E_{m}, \quad etc.$$ Obviously, $$\begin{split}
&y(P)=\mp\zeta^{-n-1}\sum_{k=0}^{+\infty}c_{k}(\underline{E})\zeta^{k}\\
&=\mp\left(1-\frac{1}{2}\left(\sum_{m=0}^{2n+1}E_m\zeta+O(\zeta^{2})\right)\right)
\quad\text{as}\quad P\rightarrow P_{\infty\pm},\quad z=\zeta^{-1}.\\
\end{split}$$ In the following it will be convenient to introduce the abbreviations $$\underline{z}(P,\underline{Q})=\underline{\Xi}_{Q_{0}}-\underline{A}_{Q_{0}}(P)+\underline{\alpha}_{Q_{0}}(\mathcal{D}_{\underline{Q}}),\quad P\in\mathcal{K}_{p},\quad\underline{Q}=\{Q_{1},\dotsi,Q_{p}\}\in\text{Sym}^{n}(\mathcal{K}_{p}),$$ where $\underline{\Xi}_{Q_{0}}$ is the vector of Riemann constants and the Abel maps $\underline{A}_{Q_0}(\cdot), \underline{\alpha}_{Q_0}(\cdot)$ are defined by (period lattice $L_p=\{\underline{z}\in\mathbb{Z}^g|\underline{z}=\underline{n}+\underline{m}\tau, \underline{n},\underline{m}\in\mathbb{Z}^g\}$) $$\begin{split}
&\underline{A}_{Q_0}: \mathcal{K}_p\rightarrow \mathcal{J}(\mathcal{K}_p)=\mathbb{Z}^p/L_p\\
&P\mapsto\underline{A}_{Q_0}(P)=\left(A_{Q_0,1}(P),\dotsi,A_{Q_0,p}(P)\right)=\left(\int_{Q_0}^p\omega_1,\dotsi,\int_{Q_0}^{P}\omega_p\right)\\
\end{split}$$ and $$\underline{\alpha}_{Q_0}: Div(\mathcal{K}_p)\rightarrow\mathcal{J}(\mathcal{K}_p),\mathcal{D}\mapsto\underline{\alpha}_{Q_0}(\mathcal{D})=\sum_{P\in\mathcal{K}_g}\mathcal{D}(P)\underline{A}_{Q_0}(P).$$
\[TH4\] Suppose that $p, q$ satisfy the $n$-th stationary RLV hierarchy,let $P\in\mathcal{K}_{n}$\\$\{P_{\infty\pm},P_{0,\pm}\}$ and $(n,n_{0})\in\mathbb{Z}^{2}.$ Then $\mathcal{D}_{\underline{\hat{\mu}}(n)}$ is non-special. Moreover,\
$$\label{3.51}
\phi(P,n)=C(n)\frac{\theta(\underline{z}(P,\underline{\hat{\nu}}(n)))}{\theta(\underline{z}(P,\underline{\hat{\mu}}(n)))}\exp\left(\int_{Q_{0}}^{P}\omega_{P_{0,+}P_{\infty+}}^{(3)}\right),$$ $$\label{3.a0}
\Psi_1(P,n,n_0)=C(n,n_0)\frac{\theta(\underline{z}(P,\underline{\hat{\mu}}(n)))}{\theta(\underline{z}(P,\underline{\hat{\mu}}(n_0)))}\\
\times\exp\left(\left(n-n_0\right)\int_{Q_0}^{P}\omega_{P_{\sharp}(n) P_{\infty-}}\right),$$ $$\label{3.a1}
\begin{split}
&\Psi_2(P,n,n_0)=\xi^{-1}\times
C(n)C(n,n_0)\frac{\theta(\underline{z}(P,\underline{\hat{\nu}}(n)))}{\theta(\underline{z}(P,\underline{\hat{\mu}}(n_0)))}\\
&\times\exp\left(\int_{Q_{0}}^{P}\omega_{P_{0,+}P_{\infty+}}^{(3)}+\left(n-n_0\right)\int_{Q_0}^{P}\omega_{P_{\sharp} (n)P_{\infty-}}\right),
\end{split}$$ where $$\begin{aligned}
\label{3.a2}
C(n,n_0)=\frac{\theta(\underline{z}(P_{\infty-},\underline{\hat{\mu}}(n_0)))}{\theta(\underline{z}(P_{\infty-},\underline{\hat{\mu}}(n)))}\end{aligned}$$ and finally $p, q$ are the form of $$\label{3.100}
p^{+}=\frac{1}{2}\left(-\Delta_3-\Delta_3^{+}-\delta_1+1-\Delta_1\pm\left((\Delta_3+\Delta_3^++\delta_1-1+\Delta_1)^{2}+4\Delta_2\right)^{\frac{1}{2}}\right)$$ $$\label{3.101}
\begin{split}
&q^{+}=\Delta_2/p^{+}+1\\
&=2\Delta_2\left(-\Delta_3-\Delta_3^{+}-\delta_1+1-\Delta_1\pm\left((\Delta_3+\Delta_3^++\delta_1-1+\Delta_1)^{2}+4\Delta_2\right)^{-\frac{1}{2}}\right)^{-1}\\
&+1.\\
\end{split}$$ Here $$\label{3.81}
\Delta_1=\sum_{j=1}^n\lambda_j^{'}-\sum_{j=1}^n c_j(k)\partial_{\omega_j}\ln\left(\frac{\theta(\underline{z}(P_{\infty+},\underline{\hat{\mu}}(n))+\underline{\omega})}{\theta(\underline{z}(P_{\infty-},\underline{\hat{\mu}}(n))+\underline{\omega})}\right)|_{\underline{\omega}=0},$$ $$\label{3.82}
\Delta_2=\kappa_{\infty+}-\sum_{j=1}^{n}c_{j}(n)\partial_{\omega_{j}} \ln\left(\frac{\theta(\underline{z}(P_{\infty+},\underline{\hat{\nu}}(n))+\underline{\omega})}
{\theta(\underline{z}(P_{\infty+},\underline{\hat{\mu}}(n))+\underline{\omega})}\right)
|_{\underline{\omega}=0},$$ $$\label{3.83}
\Delta_3=\frac{\theta(\underline{z}(P_{\infty-},\underline{\hat{\nu}}(n)))}{\theta(\underline{z}(P_{\infty-},\underline{\hat{\mu}}(n)))}\frac{\theta(\underline{z}(P_{\infty+},\underline{\hat{\mu}}(n)))}{\theta(\underline{z}(P_{\infty+},\underline{\hat{\nu}}(n)))}\frac{\tilde{a}_2}{\tilde{a}_1}$$ and $\tilde{a}_1, \tilde{a}_2$, $ \{\lambda_j^{'}\}_{j=1,\dotsi,n}\in\mathbb{C}$ in (\[3.32\]).
The proof that the divisor $\mathcal{D}_{\underline{\hat{\mu}}(n)}$ is non-special see Lemma \[lemma8\], where $t_r$ is regarded as a parameter. Hence the theta functions defined in this lemma are meaningful and not identical to zero. Obviously, $$\tilde{\phi}(P,n)\frac{\theta(\underline{z}(P,\underline{\hat{\mu}}(n)))}{\theta(\underline{z}(P,\underline{\hat{\nu}}
(n)))}\exp\left(-\int_{Q_{0}}^{P}\omega_{P_{0,+}P_{\infty+}}^{(3)}\right)$$ is holomorphic function on compact Riemann surface $\mathcal{K}_{n}$(Riemann-Roch Theorem \[14\]). So it is a constant $C(n)$ related to $n$ and $\phi(P,n)$ has the form (\[3.51\]). One have the following expansion as $P\rightarrow P_{\infty+},\quad( z=\zeta^{-1})$
$$\begin{split}
&\frac{\theta(\underline{z}(P,\underline{\hat{\nu}}(n)))}{\theta(\underline{z}(P,\underline{\hat{\mu}}(n)))}=\frac{\theta(\underline{z}(P_{\infty+},\underline{\hat{\nu}}(n)))}{\theta(\underline{z}(P_{\infty+},\underline{\hat{\mu}}(n)))}\\
&\times\left(1-\sum_{j=1}^{n}c_{j}(n)\frac{\partial}{\partial\omega_{j}}\ln\left(
\frac{\theta(\underline{z}(P_{\infty+},\underline{\hat{\nu}}(n))+\underline{\omega})}{\theta(\underline{z}(P_{\infty+},\underline{\hat{\mu}}(n))+\underline{\omega})}\right)|_{\underline{\omega}=0}\zeta+O(\zeta^{2})\right).
\end{split}$$
Then as $P\rightarrow P_{\infty+}$, $$\label{3.90}
\begin{split}
&\tilde{\phi}(P,n)\\
&=\tilde{a}_1C(n)\frac{\theta(\underline{z}(P_{\infty+},\underline{\hat{\nu}}(n)))}{\theta(\underline{z}(P_{\infty+},\underline{\hat{\mu}}(n)))}\\&\times\left(1-\sum_{j=1}^{n}c_{j}(n)\frac{\partial}{\partial\omega_{j}}\ln\left(\frac{\theta(\underline{z}(P_{\infty+},\underline{\hat{\nu}}(n))+\underline{\omega})}{\theta(\underline{z}(P_{\infty+},\underline{\hat{\mu}}(n))+\underline{\omega})}\right)|_{\underline{\omega}=0}\zeta+O(\zeta^{2})\right)\\
&\times\zeta^{-1}\left(1+\kappa_{\infty+}\zeta+O(\zeta^2)\right),
\end{split}$$ where $\tilde{a}_1, \kappa_{\infty+}\in\mathbb{C}$ are constants generated in the limit procedure. In another way, meromorphic function $\tilde{\phi}$ has the asymptotic expansion (\[3.28\]). Comparing the coefficients of (\[3.28\]) with the ones of (\[3.90\]) then yields $$\label{3.91}
(\tilde{q}^{+})^{-1}=\tilde{a}_1C(n)\frac{\theta(\underline{z}(P_{\infty+},\underline{\hat{\nu}}(n)))}{\theta(\underline{z}(P_{\infty+},\underline{\hat{\mu}}(n)))}.$$ and $$\label{3.93}
\begin{split}
&\left(\tilde{q}^{++}/\tilde{q}^{+}-1\right)\tilde{p}^{+}\\&=\kappa_{\infty+}-\sum_{j=1}^{n}c_{j}(n)\frac{\partial}{\partial\omega_{j}}
\ln\left(\frac{\theta(\underline{z}(P_{\infty+},\underline{\hat{\nu}}(n))+\underline{\omega})}
{\theta(\underline{z}(P_{\infty+},\underline{\hat{\mu}}(n))+\underline{\omega})}\right)
|_{\underline{\omega}=0},\\
\end{split}$$ where $\underline{\omega}=(\omega_1,\omega_2,\dotsi,\omega_n).$ Similarly, as $P\rightarrow P_{\infty-}$, one finally derives $$\label{3.92}
-\tilde{r}=\tilde{a}_2C(n)\frac{\theta(\underline{z}(P_{\infty-},\underline{\hat{\nu}}(n)))}{\theta(\underline{z}(P_{\infty-},\underline{\hat{\mu}}(n)))},$$ where $\tilde{a}_2\in\mathbb{C}.$ Noted the assumption (\[2.1\]) (\[3.1\]) that $\tilde{q}(n)\neq 0, \tilde{r}(n)\neq 0$ for all $n\in\mathbb{Z}$, we conclude $$\label{3.94}
-\tilde{q}^{+}\tilde{r}=\frac{\tilde{a}_2}{\tilde{a}_1}\frac{\theta(\underline{z}(P_{\infty-},\underline{\hat{\nu}}(n)))\theta(\underline{z}(P_{\infty+},\underline{\hat{\mu}}(n)))}{\theta(\underline{z}(P_{\infty-},\underline{\hat{\mu}}(n)))\theta(\underline{z}(P_{\infty+},\underline{\hat{\nu}}(n)))}$$ from (\[3.91\]) and (\[3.92\]). Let us consider the trace formula (\[3.26\]). After a standard residue calculation at $P_{\infty\pm}$ [@A1; @A2], $\sum_{j=1}^{n}\mu_j$ has the following theta function representation $$\label{3.103}
\sum_{j=1}^{n}\mu_j=\sum_{j=1}^n\lambda_j^{'}-\sum_{j=1}^n c_j(k)\partial_{\omega_j}\ln\left(\frac{\theta(\underline{z}(P_{\infty+},\underline{\hat{\mu}}(n))+\underline{\omega})}{\theta(\underline{z}(P_{\infty-},\underline{\hat{\mu}}(n))+\underline{\omega})}\right)|_{\underline{\omega}=0}$$ Then $\tilde{p},\tilde{q} $ satisfy the following equations $$\label{3.104}
\left(\tilde{q}^{++}/\tilde{q}^{+}-1\right)\tilde{p}^{+}=\Delta_2,$$ $$\label{3.79}
-\tilde{q}^{+}\tilde{r}=\Delta_3,$$ Plugging (\[3.104\]) (\[3.79\]) into (\[3.26\]) then yields $$\label{3.80}
(\tilde{p}^{+})^2+\left(\Delta_3+\Delta_3^++\Delta_1-\delta_1+1\right)\tilde{p}^{+}-\Delta_2=0.$$ The two solutions of (\[3.80\]) are (\[3.100\]). (\[3.101\]) follows from (\[3.104\]) and (\[3.100\]).
\[remark5\] The theta function representations of $\Psi_1, \Psi_2$ in (\[3.a0\]) (\[3.a1\]) is merely a formal expression and they are of little use in the process of solving the stationary relativistic Lotka-Volterra system (\[2.34\]). In fact the expression (\[3.a0\]) itself is indeterminate as $P_{\sharp}$ is unknown and may relate to the lattice variant $n$.
Algebro-geometric Solutions of Time-dependent Relativistic Lotka-Volterra Hierarchy
===================================================================================
In this section, we mainly derive the algebro-geometric solutions of time-dependent relativistic Lotka-Volterra hierarchy defined in section 2 by extending the method employed in the section 3 to time-dependent cases.
$$\label{4.1}
\begin{split}
&p=\tilde{p}\neq 0,1\quad p(\cdot,t),q(\cdot,t),\tilde{p}(\cdot,t),\tilde{q}(\cdot,t)\tilde{r}(\cdot,t)\in\ell(\mathbb{Z}),\\
&p(n,\cdot),q(n,\cdot),\tilde{p}(n,\cdot),\tilde{q}(n,\cdot),\tilde{r}(n,\cdot)\in\textbf{C}^{1}(\mathbb{R})\\
&\text{and}\quad (n,t)\in\mathbb{Z}\times\mathbb{R}.\\
\end{split}$$
Throughout this section we suppose Hypothesis 2 and Hypothesis 3 holds. The basic algebro-geometric initial value problem is that if we consider a solution $p^{1}(n), q^{1}(n)$ of the $n$-th stationary relativistic Lotka-Volterra system $s$-$RLV_p(p^{1},q^{1})$\
=0, associated with the hyperelliptic curve $\mathcal{K}_n$ and a corresponding of the summation $\{\delta_\ell\}_{\ell=0}^{n}\subseteq\mathbb{C}$, then we construct a solution $p,q$ of the $r$-th time-dependent relativistic Lotka-Volterra flow $RLV_r(p,q)=0$ satisfying $p(n,t_{0,r})=p^{1}(n), q(n,t_{0,r})$\
$=q^{1}(n)$ for some $t_{0,r}\in\mathbb{R}$ and any $n\in\mathbb{Z}.$ We shall use the notation $\bar{\tilde{V}}_r,\bar{\tilde{A}}_{r+1},\bar{\tilde{B}}_{r},$ $\bar{\tilde{C}}_{r},\bar{\tilde{D}}_{r+1},\bar{\tilde{a}}_{\ell},\bar{\tilde{b}}_{\ell},
\bar{\tilde{c}}_{\ell},\bar{\tilde{d}}_{\ell},\bar{\tilde{\delta}}_{\ell}$ in the $r$-th time-dependent flow to distinguish $\tilde{A}_{n+1},\tilde{B}_{n},$\
$\tilde{C}_{n},\tilde{D}_{n+1},
\tilde{a}_{\ell},\tilde{b}_{\ell},\tilde{c}_{\ell},\tilde{d}_{\ell},\tilde{\delta}_{\ell}$ in the $n$-th stationary relativistic Lotka-Volterra system. The algebro-geometric initial value problem discussed above can be summed up in the form of zero-curvature equation $$\label{4.2}
\tilde{U}_{t_{r}}(\xi,t_{r})+\tilde{U}(\xi,t_{r})\bar{\tilde{V}}_{r}(\xi,t_{r})-\bar{\tilde{V}}^{+}_{r}(\xi,t_{r})U(\xi,t_{r})=0,$$ $$\label{4.3}
\tilde{U}(\xi,t_{0,r})\tilde{V}_{n}(\xi,t_{0,r})-\tilde{V}^{+}_{n}(\xi,t_{0,r})\tilde{U}(\xi,t_{0,r})=0,$$ Considering the isospectral property of the Lax operator $L$ corresponding to $U$, we may impose more strong condition on equation (\[4.3\]), which is $$\label{4.4}
\tilde{U}(\xi,t_{r})\tilde{V}_{n}(\xi,t_{r})-\tilde{V}^{+}_{n}(\xi,t_{r})\tilde{U}(\xi,t_{r})=0, \quad t_r\in\mathbb{R}.$$ For further reference, we recall the relevant quantities here: $$\label{4.5}
\tilde{U}(\xi)=\left(
\begin{array}{cc}
\tilde{p}-z & \tilde{q}\xi\\
\tilde{r}\xi & 1
\end{array}
\right),$$ $$\label{4.6}
\tilde{V}_{n}(\xi)=\left(
\begin{array}{cc}
\tilde{A}_{n+1}^-(z)& \xi\tilde{B}_{n}^-(z)\\
\xi\tilde{C}_{n}^-(z) & -\tilde{D}_{n+1}^-(z)
\end{array}
\right),$$ $$\label{4.7}
\tilde{V}_{r}(\xi)=\left(
\begin{array}{cc}
\bar{\tilde{A}}_{r+1}^-(z)& \xi\bar{\tilde{B}}_{r}^-(z)\\
\xi\bar{\tilde{C}}_{r}^-(z) &-\bar{\tilde{D}}_{r+1}^-(z)
\end{array}
\right),$$ where $$\label{4.8}
\tilde{A}_{n+1}(z)=\sum_{\ell=0}^{n+1}\tilde{a}_{n+1-\ell}z^{\ell},$$ $$\label{4.9}
\tilde{B}_{n}(z)=\sum_{\ell=0}^{n}\tilde{b}_{n-\ell}z^{\ell},$$ $$\label{4.10}
\tilde{C}_{n}(z)=\sum_{\ell=0}^{n}\tilde{c}_{n-\ell}z^{\ell},$$ $$\label{4.11}
\tilde{D}_{n+1}(z)=\sum_{\ell=0}^{n+1}\tilde{a}_{n+1-\ell}z^{\ell},$$ $$\label{4.12}
\bar{\tilde{A}}_{r+1}(z)=\sum_{\ell=0}^{r+1}\bar{\tilde{a}}_{r+1-\ell}z^{\ell},$$ $$\label{4.13}
\bar{\tilde{B}}_{r}(z)=\sum_{\ell=0}^{r}\bar{\tilde{b}}_{r-\ell}z^{\ell},$$ $$\label{4.14}
\bar{\tilde{C}}_{r}(z)=\sum_{\ell=0}^{r}\bar{\tilde{c}}_{r-\ell}z^{\ell},$$ $$\label{4.15}
\bar{\tilde{D}}_{r+1}(z)=\sum_{\ell=0}^{r+1}\bar{\tilde{a}}_{r+1-\ell}z^{\ell},$$ Here $\{\tilde{a}_\ell\}_{\ell=0}^{n+1}, \{\tilde{b}_\ell\}_{\ell=0}^{n}, \{\tilde{c}_{\ell}\}_{\ell=0}^{n}, \{\tilde{d}_{\ell}\}_{\ell=0}^{n+1}$ and $\{\bar{\tilde{a}}_\ell\}_{\ell=0}^{n+1}, \{\bar{\tilde{b}}_\ell\}_{\ell=0}^{n}, \{\bar{\tilde{c}}_{\ell}\}_{\ell=0}^{n}, \{\bar{\tilde{d}}_{\ell}\}_{\ell=0}^{n+1}$ are defined by (\[2.2\])-(\[2.6\]) corresponding to different constants $\tilde{\delta}_\ell$ and $\bar{\tilde{\delta}}_\ell$, respectively. Explicitly, equation (\[4.2\]) and (\[4.4\]) are equivalent to $$\label{4.16}
-\tilde{p}_{t_r}=(\tilde{p}-z)\bar{\tilde{A}}_{r+1}^-+z\tilde{q}\bar{\tilde{C}}_r^{-}-
(\tilde{p}-z)\bar{\tilde{A}}_{r+1}-z\tilde{r}\bar{\tilde{B}}_{r},$$ $$\label{4.17}
-\tilde{q}_{t_r}=(\tilde{p}-z)\bar{\tilde{B}}_{r}^--\tilde{q}\bar{\tilde{D}}_{r+1}^--\tilde{q}\bar{\tilde{A}}_{r+1}-\bar{\tilde{B}}_{r},$$ $$\label{4.18}
-\tilde{r}_{t_r}=\tilde{r}\bar{\tilde{A}}_{r+1}^-+\bar{\tilde{C}}_{r}^--(\tilde{p}-z)\bar{\tilde{C}}_{r}+\tilde{r}\bar{\tilde{D}}_{r+1},$$ $$\label{4.19}
0= z\tilde{r}\bar{\tilde{B}}_{r}^--\bar{\tilde{D}}_{r+1}^--z\tilde{q}\bar{\tilde{C}}_{r}+\bar{\tilde{D}}_{r+1},$$ $$\label{4.20}
0=(\tilde{p}-z)\tilde{A}_{n+1}^-+z\tilde{q}\tilde{C}_n^{-}-
(\tilde{p}-z)\tilde{A}_{n+1}-z\tilde{r}\tilde{B}_{n+1},$$ $$\label{4.21}
0=(\tilde{p}-z)\tilde{B}_{n}^--\tilde{q}\tilde{D}_{n+1}^--\tilde{q}\tilde{A}_{n+1}-\tilde{B}_{n},$$ $$\label{4.22}
0=\tilde{r}\tilde{A}_{n+1}^-+\tilde{C}_{n}^--(\tilde{p}-z)\tilde{C}_{n}+\tilde{r}\tilde{D}_{n+1},$$ $$\label{4.23}
0=z\tilde{r}\tilde{B}_{n}^--\tilde{D}_{n+1}^--z\tilde{q}\tilde{C}_{n}+\tilde{D}_{n+1},$$ respectively. In particular, (\[2.62\]) holds in the present $t_r$-dependence setting, that is, $$\label{4.24}
R_{2n+2}(z,t_r)=-(\tilde{A}_{n+1}(z,t_r))^2-z\tilde{B}_n(z,t_r)\tilde{C}_n(z,t_r).$$ Obviously the algebraic curve defined in (\[4.24\]) is $n$-independent and may depend on the parameter $t_{r}.$ In fact we can prove $$\partial_{t_r}R_{2n+2}(z,t_r)=0$$ under the initial value condition (\[4.4\]), which means $R_{2n+2}(z,t_r)$ is $t_r$-independent (see lemma \[lemma5\]). We write $$\label{4.24a}
\begin{split}
&\tilde{B}_{n}(z,n,t_r)=\left(-\tilde{q}^{+}(n,t_r)\right)\prod_{j=1}^{n}\left(z-\mu_j(n,t_r)\right),\\ &\tilde{C}_{n}(z,n,t_r)=\left(-\tilde{r}(n,t_r)\right)\prod_{j=1}^{n}\left(z-\nu_j(n,t_r)\right),\\
&\text{and}\quad\mu_j(n,t_r),\nu_j(n,t_r)\in\ell(\mathbb{Z}), (n,t_r)\in\mathbb{Z}\times\mathbb{R}.
\end{split}$$ As in the stationary context (\[3.5\]) (\[3.7\]) we introduce $$\label{4.25}
\begin{split}
&\hat{\mu}_j(n,t_r)=(\mu_j(n,t_r),-2\tilde{A}_{n+1}(\mu_j(n,t_r),n,t_r)),\\ &\hat{\nu}_j(n,t_r)=(\nu_j(n,t_r),2\tilde{A}_{n+1}(\nu_j(n,t_r),n,t_r)),\\
&j=1,2,\dotsi,n, (n,t_r)\in\mathbb{Z}\times\mathbb{R}.
\end{split}$$ on $\mathcal{K}_n$ and define the following meromorphic function $\tilde{\phi}(\cdot,n,t_r)$ on $\mathcal{K}_n$, $$\label{4.26}
\tilde{\phi}(P,n,t_r)=\frac{\frac{1}{2}y-\tilde{A}_{n+1}(z,n,t_r)}{B_{n}(z,n,t_r)}
=\frac{z\tilde{C}_{n}(z,n,t_r)}{\frac{1}{2}y+\tilde{A}_{n+1}(z,n,t_r)},
\quad P=(z,y)\in\mathcal{K}_n.$$ The divisor $(\tilde{\phi}(\cdot))$ of $\tilde{\phi}$ is $$\label{4.26a}
(\tilde{\phi}(\cdot,n))=\mathcal{D}_{P_{0,+}\underline{\hat{\nu}}(n,t_r)}-\mathcal{D}_{P_{\infty+}\underline{\hat{\mu}}(n,t_r)},$$ and the time-dependent Baker-Akhiezer vector is then defined in term of $\tilde{\phi}$ by $$\label{4.27}
\Psi(P,\xi,n,n_0,t_r,t_{0,r})=\left(
\begin{array}{c}
\Psi_1(P,\xi,n,n_0,t_r,t_{0,r}) \\
\Psi_2(P,\xi,n,n_0,t_r,t_{0,r}) \\
\end{array}
\right),$$ $$\label{4.28}
\begin{split}
&\Psi_1(P,\xi,n,n_0,t_r,t_{0,r})
=\exp\left(\int_{t_{0,r}}^{t_r}\left(\bar{\tilde{A}}_{r+1}(z,n_0,s)+\bar{\tilde{B}}_{r}(z,n_0,s)\tilde{\phi}(P,n_0,s)\right)ds
\right)\\
&\times\begin{cases}
\prod_{n^{'}=n_0+1}^{n}\left(\tilde{p}(n^{'},t_r)-z+\tilde{q}(n^{'})\tilde{\phi}^-(P,n^{'},t_r)\right),&n\geq n_0+1,\cr
1,&n=n_0,\cr
\prod_{n^{'}=n+1}^{n_0}\left(\tilde{p}(n^{'},t_r)-z+\tilde{q}(n^{'},t_r)\tilde{\phi}^-(P,n^{'},t_r)\right)^{-1},&n\leq n_0-1,
\end{cases}\\
\end{split}$$ $$\begin{aligned}
\label{4.29}
\begin{split}
&\Psi_2(P,\xi,n,n_0,t_r,t_{0,r})=\exp\left(\int_{t_{0,r}}^{t_r}\left(\bar{\tilde{A}}_{r+1}(z,n_0,s)+\bar{\tilde{B}}_{r}(z,n_0,s)
\tilde{\phi}(P,n_0,s)\right)ds\right)\\
&\times\xi^{-1}\times\tilde{\phi}(P,n_0,t_r)
\begin{cases}
\prod_{n^{'}=n_0+1}^{n}\left(\frac{\tilde{r}(n^{'},t_r)z}{\tilde{\phi}^-(P,n^{'},t_r)}+1\right),&n\geq n_0+1,\cr
1,&n=n_0,\cr
\prod_{n^{'}=n+1}^{n_0}\left(\frac{\tilde{r}(n^{'},t_r)z}{\tilde{\phi}^-(P,n^{'},t_r)}+1\right)^{-1},&n\leq n_0-1.
\end{cases}\\
\end{split}
\end{aligned}$$ $$\label{4.30}
P=(z,y)\in\mathcal{K}_n\backslash\{P_{0,\pm},P_{\infty\pm}\},\quad (n,t_r)\in\mathbb{Z}\times\mathbb{R}.$$ One observes that $$\label{4.31}
\begin{split}
&\Psi_1(P,n,n_0,t_r,t_{0,r})=\Psi_1(P,n_0,n_0,t_r,t_{0,r})\Psi_1(P,n,n_0,t_r,t_{0,r}),\\
&P=(z,y)\in\mathcal{K}_n\backslash\{P_{0,\pm},P_{\infty\pm}\},\quad (n,n_0,t_r,t_{0,r})\in\mathbb{Z}^2\times\mathbb{R}^2.\\
\end{split}$$ The following lemma shows the properties of $\tilde{\phi}, \Psi_1$ and $\Psi_2$ as discussed in the stationary case.
Assume Hypothesis 2 and Hypothesis 3 hold and suppose $p(n,t_r),q(n,t_r)$ satisfy (\[4.2\]) (\[4.4\]). In addition, let $P=(z,y)\in\mathcal{K}_n\backslash\{P_{\infty\pm},P_{0,\pm}\}, (n,n_0,t_r,t_{0,r})\in\mathbb{Z}^2\times\mathbb{R}^2.$ Then $\tilde{\phi}(P,t_r)$ satisfies the following equations $$\label{4.32}
(\tilde{p}-z)\tilde{\phi}(P)+\tilde{q}\tilde{\phi}(P)\tilde{\phi}^-(P)=\tilde{r}z+\tilde{\phi}^{-}(P),$$ $$\label{4.33}
\tilde{\phi}_{t_r}(P)=z\bar{\tilde{C}}_r-\bar{\tilde{D}}_{r+1}\tilde{\phi}(P)-\bar{\tilde{A}}_{r+1}\tilde{\phi}(P)-\bar{\tilde{B}}_r\tilde{\phi}^2(P),$$ as well as $$\label{4.34}
\tilde{\phi}(P)\tilde{\phi}(P^{*})=\frac{-z\tilde{C}_{n}(z)}{\tilde{B}_{n}(z)},$$ $$\label{4.35}
\tilde{\phi}(P)+\tilde{\phi}(P^{*})=\frac{-2\tilde{A}_{n+1}}{\tilde{B}_n},$$ $$\label{4.36}
\tilde{\phi}(P)-\tilde{\phi}(P^{*})=\frac{y}{\tilde{B}_n(z)}.$$ The vector $\Psi$ satisfy $$\label{4.37}
\tilde{U}(\xi)\Psi^-(P)=\Psi(P),$$ $$\label{4.38}
\tilde{V}_n(\xi)\Psi(P)=(1/2)y\Psi(P),$$ $$\label{4.39}
\Psi_2(P,\xi,n,n_0)=\xi^{-1}\tilde{\phi}(P,n)\Psi_1(P,\xi,n,n_0),$$ $$\label{4.40}
\Psi_{t_r}(P)=\bar{\tilde{V}}_r(\xi)\Psi(P).$$ Moreover, Moreover, as long as the zeros of $\mu_j(n_0,s)$ of $B_n(\cdot,n_0,s)$ are all simple and nonzero for all $s\in\Omega, \Omega\subseteq\mathbb{R}$ is an open interval, $\Psi_1$ is meromorphic on $\mathcal{K}_n\backslash\{P_{0,\pm}, P_{\infty\pm}\}$ for $(n,t_r,t_{0,r})\in\mathbb{Z}\times\Omega^2.$
The proof of (\[4.32\]), (\[4.34\])-(\[4.39\]) is the same with lemma \[lemma1\], where $t_r$ is regarded as a parameter. From (\[4.32\]) we have $$\label{4.41}
\tilde{p}_{t_r}\tilde{\phi}+(\tilde{p}-z)\tilde{\phi}+\tilde{q}_{t_r}\tilde{\phi}\tilde{\phi}^-
+\tilde{q}\tilde{\phi}_{t_r}\tilde{\phi}^{-}+\tilde{q}\tilde{\phi}\tilde{\phi}^{-}_{t_r}=\tilde{r}_{t_r}z
+\tilde{\phi}^{-}_{t_r},$$ that is, $$\label{4.42}
[\tilde{p}-z+\tilde{q}\tilde{\phi}^{-}+(\tilde{q}\tilde{\phi}-1)S^{-}]\tilde{\phi}_{t_r}=\tilde{r}_{t_r}z-
\tilde{p}_{t_r}\tilde{\phi}-
\tilde{q}_{t_r}\tilde{\phi}\tilde{\phi}^{-}.$$ Using (\[4.16\])-(\[4.18\]), one finds $$\label{4.43}
\begin{split}
&[\tilde{p}-z+\tilde{q}\tilde{\phi}^-+(\tilde{q}\tilde{\phi}-1)S^{-}]\left(z\bar{\tilde{C}}_r-\bar{\tilde{D}}_{r+1}
\tilde{\phi}-\bar{\tilde{A}}_{r+1}\tilde{\phi}-\bar{\tilde{B}}_r\tilde{\phi}^2\right)\\
&=[\frac{\tilde{r}z+\tilde{\phi}^{-}}{\tilde{\phi}}+(\tilde{q}\tilde{\phi}-1)S^{-}]
\left(z\bar{\tilde{C}}_r-\bar{\tilde{D}}_{r+1}\tilde{\phi}-
\bar{\tilde{A}}_{r+1}\tilde{\phi}-\bar{\tilde{B}}_r\tilde{\phi}^2\right)\\
&=(\tilde{p}-z+\tilde{q}\tilde{\phi}^{-})z\bar{\tilde{C}}_{r}-(\tilde{r}z+\tilde{\phi}^{-})\bar{\tilde{D}}_{r+1}
-(\tilde{r}z+\tilde{\phi}^{-})\bar{\tilde{A}}_{r+1}-(\tilde{r}z+\tilde{\phi}^{-})\bar{\tilde{B}}_r\tilde{\phi}^-\\
&+(\tilde{q}\tilde{\phi}-1)z\bar{\tilde{C}}_{r}^--(\tilde{q}\tilde{\phi}-1)\bar{\tilde{D}}_{r+1}^-\tilde{\phi}^{-}
-(\tilde{q}\tilde{\phi}-1)\bar{\tilde{A}}_{r+1}^-\tilde{\phi}^{-}-(\tilde{q}\tilde{\phi}-1)\bar{\tilde{B}}_r
\tilde{\phi}^2\\
&=(\tilde{p}-z+\tilde{q}\tilde{\phi}^{-})z\bar{\tilde{C}}_{r}-(\tilde{r}z+\tilde{\phi}^{-})\bar{\tilde{D}}_{r+1}
-(\tilde{r}z+\tilde{\phi}^{-})\bar{\tilde{A}}_{r+1}-(\tilde{r}z+\tilde{\phi}^{-})\bar{\tilde{B}}_r\tilde{\phi}^-\\
&+(\tilde{q}\tilde{\phi}-1)z\bar{\tilde{C}}_{r}^--(\tilde{q}\tilde{\phi}-1)\bar{\tilde{D}}_{r+1}^-\tilde{\phi}^{-}
-(\tilde{q}\tilde{\phi}-1)\bar{\tilde{A}}_{r+1}^-\tilde{\phi}^{-}-\tilde{r}z\bar{\tilde{B}}_r^-\tilde{\phi}^-
+\bar{\tilde{B}}_r(\tilde{p}-z)\tilde{\phi}\tilde{\phi}^-\\
&=[\tilde{r}\bar{\tilde{A}}_{r+1}^--\bar{\tilde{C}}_{r}^-+(\tilde{p}-z)\bar{\tilde{C}}_{r}-\tilde{r}\bar{\tilde{D}}_{r+1}]z+
[(\tilde{p}-z)\bar{\tilde{A}}_{r+1}^-+z\tilde{q}\bar{\tilde{C}}_r^{-}
-(\tilde{p}-z)\bar{\tilde{A}}_{r+1}\\&-z\tilde{r}\bar{\tilde{B}}_{r+1}]\tilde{\phi}+[(\tilde{p}-z)\bar{\tilde{B}}_{r}^-
-\tilde{q}\bar{\tilde{D}}_{r+1}^--\tilde{q}\bar{\tilde{A}}_{r+1}-\bar{\tilde{B}}_{r}]\tilde{\phi}\tilde{\phi}^-\\
&+[(\bar{\tilde{A}}_{r+1}^--\bar{\tilde{A}}_{r+1})(\tilde{\phi}^--(\tilde{p}-z)\tilde{\phi}-\tilde{q}\tilde{\phi}\tilde{\phi}^-+\tilde{r}z)]
+[z\tilde{r}\bar{\tilde{B}}_{r}^--\bar{\tilde{D}}_{r+1}^--z\tilde{q}\bar{\tilde{C}}_{r}+\bar{\tilde{D}}_{r+1}]\tilde{\phi}^-\\
&=\tilde{r}_{t_r}z-\tilde{p}_{t_r}\tilde{\phi}-\tilde{q}_{t_r}\tilde{\phi}\tilde{\phi}^-.\\
\end{split}$$ Considering (\[4.42\]), one may conclude $$\label{4.44}\begin{split}
&[\tilde{p}-z+\tilde{q}\tilde{\phi}^{-}+(\tilde{q}\tilde{\phi}-1)S^{-}]\left(\tilde{\phi}_{t_r}-(z\bar{\tilde{C}}_r-\bar{\tilde{D}}_{r+1}\tilde{\phi}-\bar{\tilde{A}}_{r+1}\tilde{\phi}-\bar{\tilde{B}}_r\tilde{\phi}^2)\right) \\&=0.\\\end{split}$$ Thus, the expression $$\label{4.45}
\tilde{\phi}_{t_r}-(z\bar{\tilde{C}}_r-\bar{\tilde{D}}_{r+1}\tilde{\phi}-\bar{\tilde{A}}_{r+1}\tilde{\phi}
-\bar{\tilde{B}}_r\tilde{\phi}^2)\\
=C\begin{cases}\prod_{n^{'}=n_0+1}^{n}B(z,n^{'},t_r),&n\geq n_0+1,\cr
1,&n=n_0,\cr
\prod_{n^{'}=n_0}^{n}B(z,n^{'},t_r)^{-1},&n\leq n_0-1,
\end{cases}$$ where $$\label{4.46}
B(z,n,t_r)=\frac{1-\tilde{q}(n,t_r)\tilde{\phi}(z,n,t_r)}{\tilde{p}(n,t_r)-z+\tilde{q}(n,t_r)
\tilde{\phi}^{-}(z,n,t_r)},\quad (n,t_r)\in\mathbb{Z}\times\mathbb{R}.$$ Obviously, the expression $\tilde{\phi}_{t_r}-(z\bar{\tilde{C}}_r-\bar{\tilde{D}}_{r+1}\tilde{\phi}-\bar{\tilde{A}}_{r+1}\tilde{\phi}
-\bar{\tilde{B}}_r\tilde{\phi}^2)$ in the left-hand of (\[4.45\]) is a meromorphic function on Riemann Surface $\mathcal{K}_n$ and its order is finite at $P_{\infty-}$. However, the order of the right-hand in (\[4.45\]) is $O(z^{n-n_0})$ as $P$ near the point $P_{\infty-}.$ Hence, taking $n$ sufficiently large, then yields a contradiction at the both bides of (\[4.45\]) unless $C=0.$ Therefore we have (\[4.33\]). From the definition of $\Psi_1(P,\xi,n,n_0,t_r,t_{0,r})$ in (\[4.28\]) it is easy to check $$\label{4.47}
\begin{split}
&(\tilde{p}-z+\tilde{q}\tilde{\phi}^-)_{t_r}=\tilde{p}_{t_r}+\tilde{q}_{t_r}\tilde{\phi}^-
+\tilde{q}\tilde{\phi}^-_{t_r}\\
&=-(\tilde{p}-z)\bar{\tilde{A}}_{r+1}^--z\tilde{q}\bar{\tilde{C}}_r^{-}+
(\tilde{p}-z)\bar{\tilde{A}}_{r+1}+z\tilde{r}\bar{\tilde{B}}_{r}\\
&\times\left(-(\tilde{p}-z)\bar{\tilde{B}}_{r}^-+\tilde{q}\bar{\tilde{D}}_{r+1}^-+\tilde{q}\bar{\tilde{A}}_{r+1}+
\bar{\tilde{B}}_{r}\right)\tilde{\phi}^-\\
&+\tilde{q}\left(z\bar{\tilde{C}}_r^--\bar{\tilde{D}}_{r+1}^-
\tilde{\phi}^--\bar{\tilde{A}}_{r+1}^-\tilde{\phi}^--\bar{\tilde{B}}_r^-(\tilde{\phi}^-)^2\right)\\
&=\left(\tilde{p}-z+\tilde{q}\tilde{\phi}^-\right)\left(\bar{\tilde{A}}_{r+1}+\bar{\tilde{B}}_r\tilde{\phi}-
\bar{\tilde{A}}_{r+1}^--\bar{\tilde{B}}_r^-\tilde{\phi}^-\right).\\
\end{split}$$ If we note $$\label{4.48}
\Upsilon(P,n_0,t_r,t_{0,r})=\exp\left(\int_{t_{0,r}}^{t_r}\left(\bar{\tilde{A}}_{r+1}(z,n_0,s)+\bar{\tilde{B}}_{r}(z,n_0,s)\tilde{\phi}(P,n_0,s)\right)ds
\right),$$ then $$\label{4.49}
\begin{split}
&\Psi_{1,t_r}=\left(\Upsilon\prod_{n^{'}=n_0+1}^{n}(\tilde{p}-z+\tilde{q}\tilde{\phi}^-)(n^{'})\right)_{t_r}\\
&=\Upsilon_{t_r}\prod_{n^{'}=n_0+1}^{n}(\tilde{p}-z+\tilde{q}\tilde{\phi}^-)(n^{'})+\Upsilon\sum_{n^{'}=n_0+1}^{n}
(\tilde{p}-z+\tilde{q}\tilde{\phi}^-)_{t_r}(n^{'})\prod_{n^{''}\neq n^{'}}(\tilde{p}-z+\tilde{q}\tilde{\phi}^-)(n^{''})\\
&=\Upsilon\prod_{n^{'}=n_0+1}^{n}(\tilde{p}-z+\tilde{q}\tilde{\phi}^-)(n^{'})[\bar{\tilde{A}}_{r+1}(n_0,t_r)+
\sum_{n^{'}=n_0+1}^{n}(\bar{\tilde{A}}_{r+1}(n^{'})+\bar{\tilde{B}}_r(n^{'})-\\
&\bar{\tilde{A}}_{r+1}^-(n^{'})-\bar{\tilde{B}}_r^-(n^{'}))]\\
&=(\bar{\tilde{A}}_{r+1}+\bar{\tilde{B}}_{r})\Psi_1=\bar{\tilde{A}}_{r+1}\Psi_1+\xi\bar{\tilde{B}}_{r}\Psi_2,\\
&\text{as} \quad n\geq n_0+1.\\
\end{split}$$ The proof for the case $n\leq n_0-1$ is similarly with (\[4.49\]). Using (\[4.32\]) and (\[4.49\]) we have $$\label{4.50}
\begin{split}
&\Psi_{2,t_r}=\xi^{-1}(\tilde{\phi}_{t_r}\Psi_1+\tilde{\phi}\Psi_{1,t_r})\\
&=\xi^{-1}\left(z\bar{\tilde{C}}_r-\bar{\tilde{D}}_{r+1}\tilde{\phi}-
\bar{\tilde{A}}_{r+1}\tilde{\phi}-\bar{\tilde{B}}_r\tilde{\phi}^2+(\bar{\tilde{A}}_{r+1}+\bar{\tilde{B}}_{r+1}
\tilde{\phi})\tilde{\phi}\right)\Psi_1\\
&=\xi^{-1}(z\bar{\tilde{C}}_r-\bar{\tilde{D}}_{r+1})\Psi_1=\xi\bar{\tilde{C}}_r\Psi_1-\bar{\tilde{D}}_{r+1}\Psi_2.\\
\end{split}$$ To illustrate that $\Psi_1(\cdot,\xi,n,n_0,t_r,t_{0,r})$ is a meromorphic function on $\mathcal{K}_n\backslash\{P_{0,\pm},P_{\infty\pm}\}$ we only need to prove $\Upsilon$ is a meromorphic function on $\mathcal{K}_n\backslash\{P_{0,\pm},P_{\infty\pm}\}$. Taking into account (\[4.25\]) (\[4.26\]) (\[4.52\]), one may derive $$\label{4.51}
\bar{\tilde{B}}_r(z,n_0,s)\tilde{\phi}(P,n_0,s)\thicksim \partial_s\ln\left(B_r(z,n_0,s)\right)+O(1)$$ as $P\rightarrow\mu_j(n_0,s).$ Hence we conclude $\Upsilon$ is a meromorphic function on $\mathcal{K}_n\backslash\{P_{0,\pm},P_{\infty\pm}\}$ with the help of analysis the asymptotic behavior of the possible poles $\{\mu_j(n_0,s)\}$ of the function $\bar{\tilde{A}}_{r+1}(z,n_0,s)+\bar{\tilde{B}}_{r}(z,n_0,s)\tilde{\phi}(P,n_0,s).$
Next we consider the $t_r$-dependence of $\tilde{A}_{n+1}, \tilde{B}_n, \tilde{C}_n.$
\[lemma5\] Assume Hypothesis 2 and Hypothesis 3 hold and suppose $p(n,t_r),q(n,t_r)$ satisfy (\[4.2\]) (\[4.4\]). In addition, let $(z,n,t_r)\in\mathbb{C}\times\mathbb{Z}\times\mathbb{R}.$ Then, $$\label{4.52}
\tilde{\bar{B}}_{n,t_r}=-2\tilde{A}_{n+1}\bar{\tilde{B}}_r+(\bar{\tilde{A}}_{r+1}+\bar{\tilde{D}}_{r+1})\tilde{B}_n,$$ $$\label{4.53}
\tilde{A}_{n+1,t_r}=z(\bar{\tilde{B}}_r\tilde{C}_n-\tilde{B}_n\bar{\tilde{C}}_r),$$ $$\label{4.54}
\tilde{C}_{n,t_r}=2z^{-1}\tilde{A}_{n+1}\bar{\tilde{C}}_r-\tilde{C}_n(\bar{\tilde{A}}_{r+1}+\bar{\tilde{D}}_{r+1}).$$ In particular, (\[4.52\])-(\[4.54\]) is equivalent to $$\label{4.55}
\tilde{V}_{n,t_r}=[\bar{\tilde{V}}_r,\tilde{V}_n].$$ and the algebraic curve defined in (\[4.24\]) is $t_r$-independent.
To prove (\[4.52\]) one first differentiates equation (\[4.36\]) $$\label{4.56}
\tilde{\phi}_{t_r}(P)-\tilde{\phi}_{t_r}(P^{*})=\frac{-y\tilde{B}_{n,t_r}}{\tilde{B}_n^2}.$$ The time derivative of $\tilde{\phi}$ given in (\[4.32\]) and (\[4.35\]) (\[4.36\]) yields $$\label{4.57}
\begin{split}
&\tilde{\phi}_{t_r}(P)-\tilde{\phi}_{t_r}(P^{*})\\
&=\bar{\tilde{B}}_r(\tilde{\phi}(P)+\tilde{\phi}(P^{*}))(\tilde{\phi}(P)-\tilde{\phi}(P^{*}))+(\bar{\tilde{A}}_{r+1}
+\bar{\tilde{D}}_{r+1})(\tilde{\phi}(P)-\tilde{\phi}(P^{*}))\\
&=\frac{2\tilde{A}_{n+1}\bar{\tilde{B}}_ry}{\tilde{B}_n^2}-(\bar{\tilde{A}}_{r+1}+\bar{\tilde{D}}_{r+1})
\frac{y}{\tilde{B}_n}\\
\end{split}$$ and hence $$\label{4.58}
\tilde{\bar{B}}_{n,t_r}=-2\tilde{A}_{n+1}\bar{\tilde{B}}_r+(\bar{\tilde{A}}_{r+1}+\bar{\tilde{B}}_{r+1})\tilde{B}_n.$$ Similarly, starting from (\[4.35\]) $$\label{4.59}
\tilde{\phi}_{t_r}(P)+\tilde{\phi}_{t_r}(P^{*})=2z\tilde{C}_r+(\bar{\tilde{A}}+\bar{\tilde{D}}_{r+1})
\frac{2\tilde{A}_{n+1}}{\tilde{B}_{n}}-\bar{\tilde{B}}_r(\frac{4\tilde{A}_{n+1}^2}{\tilde{B}_n^2}
+\frac{2z\tilde{C}_n}{\tilde{B}_n})$$ yields (\[4.53\]). Differentiating the equation (\[4.23\]) then yields $$\label{4.60}
z\tilde{r}_{t_r}\tilde{B}_{n}^--z\tilde{r}\tilde{B}_{n,t_r}^--\tilde{D}^-_{n+1,t_r}-z\tilde{q}_{t_r}\tilde{C}_n-
z\tilde{q}\tilde{C}_{n,t_r}+\tilde{D}_{n+1,t_r}=0.$$ Using (\[2021\]) (\[4.17\]) (\[4.18\]) (\[4.19\]) (\[4.21\]) (\[4.22\]) (\[4.23\]) (\[4.52\]) and (\[4.53\]) we have $$\label{4.61}
\begin{split}
&z\tilde{r}_{t_r}\tilde{B}_{n}^--z\tilde{r}\tilde{B}_{n,t_r}^--\tilde{D}^-_{n+1,t_r}-z\tilde{q}_{t_r}\tilde{C}_n-
z\tilde{q}\tilde{C}_{n,t_r}+\tilde{D}_{n+1,t_r}\\
&=-z\tilde{B}_{n}^-[\tilde{r}\bar{\tilde{A}}_{r+1}^-+\bar{\tilde{C}}_{r}^--(\tilde{p}-z)\bar{\tilde{C}}_{r}+
\tilde{r}\bar{\tilde{D}}_{r+1}]-z\tilde{r}[-2\tilde{A}_{n+1}^-\bar{\tilde{B}}_r^-\\
&+(\bar{\tilde{A}}_{r+1}^-+\bar{\tilde{D}}_{r+1}^-)\tilde{B}_n^-]+z[\tilde{C}_n\bar{\tilde{B}}_r-\tilde{B}_n\bar{\tilde{C}}_r
-\tilde{C}_n^-\bar{\tilde{B}}_r^-+\tilde{B}_n^-\bar{\tilde{C}}_r^-]\\
&+z\tilde{C}_n[(\tilde{p}-z)\bar{\tilde{B}}_{r}^-
-\tilde{q}\bar{\tilde{D}}_{r+1}^--\tilde{q}\bar{\tilde{A}}_{r+1}-\bar{\tilde{B}}_{r}]-z\tilde{q}\tilde{C}_{n,t_r}\\
&=-z\tilde{B}_{n}^-[-(\tilde{p}-z)\bar{\tilde{C}}_r+\tilde{r}\bar{\tilde{D}}_{r+1}]+z\tilde{r}[-2\tilde{A}_{n+1}^-
\bar{\tilde{B}}_r^-+\bar{\tilde{D}}_{r+1}^-\tilde{B}_n^-]\\
&+z[-\tilde{B}_n\bar{\tilde{C}}_r
-\tilde{C}_n^-\bar{\tilde{B}}_r^-]+z\tilde{C}_n[(\tilde{p}-z)\bar{\tilde{B}}_{r}^-
-\tilde{q}\bar{\tilde{D}}_{r+1}^--\tilde{q}\bar{\tilde{A}}_{r+1}]-z\tilde{q}\tilde{C}_{n,t_r}\\
&=z[(\tilde{p}-z)\tilde{B}_n^--\tilde{B}_n]\bar{\tilde{C}}_r+z[-2\tilde{r}\tilde{A}_{n+1}^--\tilde{C}_{n}^-+
(\tilde{p}-z)\tilde{C}_n]\bar{\tilde{B}}_r^-\\
&+z[\tilde{r}\tilde{B}_n^-(\bar{\tilde{D}}_{r+1}^--\bar{\tilde{D}}_{r+1})-\tilde{q}\tilde{C}_n
\bar{\tilde{D}}_{r+1}^-]-z\tilde{q}\tilde{C}_n\bar{\tilde{A}}_{r+1}-z\tilde{q}\tilde{C}_{n,t_r}\\
&=z\tilde{q}\bar{\tilde{C}}_r[2\tilde{A}_{n+1}+(-\tilde{A}_{n+1}+\tilde{A}_{n+1}^-)]+z\tilde{r}\bar{\tilde{B}}_r^-
(\tilde{A}_{n+1}-\tilde{A}_{n+1}^-)\\
&+[(-\tilde{D}_{n+1}+\tilde{D}_{n+1}^-+z\tilde{q}\tilde{C}_n)(\bar{\tilde{D}}_{r+1}^--\bar{\tilde{D}}_{r+1})
-\tilde{q}\tilde{C}_n\bar{\tilde{D}}_{r+1}^-]\\
&-z\tilde{q}\tilde{C}_n\bar{\tilde{A}}_{r+1}-z\tilde{q}\tilde{C}_{n,t_r}\\
&=2z\tilde{q}\tilde{A}_{n+1}\bar{\tilde{C}}_r+(\tilde{A}_{n+1}-\tilde{A}_{n+1}^-)(z\tilde{r}\bar{\tilde{B}}_r^-
-z\tilde{q}\bar{\tilde{C}}_r)+[(-\tilde{D}_{r+1}+\\
&\tilde{D}_{r+1}^-+z\tilde{q}\tilde{C}_n)(\bar{\tilde{D}}_{n+1}^--\bar{\tilde{D}}_{n+1})-\tilde{q}\tilde{C}_n
\bar{\tilde{D}}_{r+1}^-]-z\tilde{q}\tilde{C}_n\bar{\tilde{A}}_{r+1}-z\tilde{q}\tilde{C}_{n,t_r}\\
&=2z\tilde{q}\tilde{A}_{n+1}\bar{\tilde{C}}_r+(\tilde{A}_{n+1}-\tilde{A}_{n+1}^-)(\bar{\tilde{D}}_{r+1}^-
-\bar{\tilde{D}}_{r+1})+[(-\tilde{D}_{n+1}+\\
&\tilde{D}_{n+1}^-+z\tilde{q}\tilde{C}_n)(\bar{\tilde{D}}_{r+1}^--\bar{\tilde{D}}_{r+1})-\tilde{q}\tilde{C}_n
\bar{\tilde{D}}_{r+1}^-]-z\tilde{q}\tilde{C}_n\bar{\tilde{A}}_{r+1}-\tilde{q}\tilde{C}_{n,t_r}\\
&=2z\tilde{q}\tilde{A}_{n+1}\bar{\tilde{C}}_r-z\tilde{q}\tilde{C}_n\bar{\tilde{D}}_{r+1}-z\tilde{q}\tilde{C}_n
\bar{\tilde{A}}_{r+1}-z\tilde{q}\tilde{C}_{n,t_r}=0,\\
\end{split}$$ which is equivalent to (\[4.54\]). Finally one can directly differentiate (\[4.24\]) $$\label{4.60}
R_{t_r}=-2\tilde{A}_{n+1}\tilde{A}_{n+1,t_{r}}-z\tilde{B}_{n,t_r}\tilde{C}_{n}-z\tilde{B}_{n}\tilde{C}_{n,t_r}$$ and insert (\[4.52\])-(\[4.54\]) into (\[4.60\]) to yield $R_{t_r}=0.$
Next we derive Dubrovin-type equations, that is, first-order ordinary differential equations, which govern the dynamics of $\mu_j$ and $\nu_j$ with respect to variations of $t_r.$ We recall that the affine part of $\mathcal{K}_n$ is nonsingular if $$\label{4.61}
\{E_m\}_{m=0,\ldots,2n+1}\subset\mathbb{C},\quad E_m\neq E_{m^{'}}\quad\text{for}\quad m\neq m^{'},m,m^{'}=0,\ldots,2n+1.$$
Assume Hypothesis 2 and Hypothesis 3 and suppose (\[4.2\]) (\[4.4\]) hold on $\mathbb{Z}\times\Omega_\mu$ with $\Omega_\mu\subseteq\mathbb{R}$ an open interval. In addition, assume that the zeros $\mu_j$ of $\tilde{B}_n(\cdot)$ remain distinct and nonzero on $\mathbb{Z}\times\Omega_{\mu}$. Then $\{\hat{\mu}_j\}_{j=1,\dotsi,n}$ defined in (\[4.25\]) satisfy the following first-order system of differential system on $\mathbb{Z}\times\Omega_{\mu}$, $$\label{4.62}
\mu_{j,t_r}=\frac{(\tilde{q^+})^{-1}y(\hat{\mu_j})\bar{\tilde{B}}_r(\mu_j)}{\prod_{k=1,k\neq j}^{n}(\mu_k-\mu_j)},\quad j=1,\dotsi,n,$$ with $$\hat{\mu}_j(n,\cdot)\in\textit{C}^{\infty}(\Omega_{\mu},\mathcal{K}_n),\quad j=1,\ldots,n, n\in\mathbb{Z}.$$ For the zeros $\nu_j$ of $\tilde{C}_n,$ identical statements hold with $\mu_j$ and $\Omega_\mu$ replaced by $\nu_j$ and $\Omega_\nu,$ etc. In particular, $\{\hat{\nu}_j\}_{j=1,\dotsi,n},$ defined in (\[4.25\]), satisfies the following first-order system on $\mathbb{Z}\times\Omega_\nu,$ $$\label{4.63}
\nu_{j,t_r}=\frac{2\tilde{r}^{-1}y(\hat{\nu}_j)\bar{\tilde{C}}_r(\nu_j)}{\nu_j\prod_{k=1,k\neq j}^{n}(\mu_k-\mu_j)},\quad j=1,\dotsi,n,$$ with $$\hat{\nu}_j(n,\cdot)\in\textit{C}^{\infty}(\Omega_{\nu},\mathcal{K}_n),\quad j=1,\ldots,n, n\in\mathbb{Z}.$$
It suffices to consider (\[4.62\]) for $\mu_{j,t_r}.$ Using the product representation for $\tilde{B}_{n}$ in (\[4.24a\]) $$\tilde{B}_{n}=(-\tilde{q}^{+})\prod_{j=1}^{n}(\lambda-\mu_j)$$ and employing the (\[4.25\]) and (\[4.52\]), one computes $$\begin{aligned}
\begin{split}
&\tilde{B}_{n,t_r}(\mu_j)=-2\tilde{A}_{n+1}(\mu_j)\bar{\tilde{B}}_{r}(\mu_j)\\
&=y(\hat{\mu}_j)\bar{\tilde{B}}_{r}(\mu_j)
=\tilde{q}^+\mu_{j,t_r}\prod_{k\neq j,j=1}^n\left(\mu_j-\mu_k\right),\quad j=1,\dotsi,n,\\
\end{split}\end{aligned}$$ proving (\[4.62\]). The case of (\[4.63\]) for $\nu_j$ is analogous using the product presentation for $\tilde{C}_n$ in (\[4.24a\]) and employing (\[4.25\]) and (\[4.54\]).
Since the stationary trace formulas for $\tilde{a}_\ell, \tilde{b}_\ell$ in terms of symmetric functions of $\mu_j$ and $\nu_j$ in Lemma \[lemma2\] extend line by line to the corresponding time-dependent setting, we next record their $t_r$-dependent analogs without proof. For simplicity we confine ourselves to the simplest ones only.
Assume Hypothesis 2 and Hypothesis 3 hold and suppose $p(n,t_r),q(n,t_r)$ satisfy (\[4.2\]) (\[4.4\]). Then, we have the following trace formula $$\label{4.64}
\tilde{p}^{+}-\tilde{q}^{+}\tilde{r}-\tilde{q}^{++}\tilde{r}^+-\tilde{q}^{++}/\tilde{q}^{+}+\delta_1=-\sum_{j=1}^{n}\mu_j,$$ $$\label{4.65}
\tilde{p}-\tilde{q}^{+}\tilde{r}-\tilde{r}^{-}/\tilde{r}-\tilde{q}\tilde{r}^{-}+\delta_1=-\sum_{j=1}^{n}\nu_j.$$
\[lemma8\] Suppose $p,q$ satisfy the (\[4.1\]) (\[4.2\]) and the $n$-th stationary RLV system (\[2.34\]). Moreover, let $(n,t_r)\in\mathbb{Z}\times\mathbb{R}$ and $\mathcal{D}_{\underline{\hat{\mu}}}, \underline{\hat{\mu}}=\left(\hat{\mu}_1,\dotsi,\hat{\mu}_{n}\right)\in \text{Sym}^n(\mathcal{K}_n)$, $\mathcal{D}_{\underline{\hat{\nu}}},\underline{\hat{\nu}}=\left(\hat{\mu}_1,\dotsi,\hat{\nu}_{n}\right)\in \text{Sym}^n(\mathcal{K}_n)$ be the pole and zero divisors of degree $n$, respectively, associated with $p, q$ and $\tilde{\phi}$ defined according to (\[4.26\]), that is, $$\hat{\mu}_j(n,t_{r})=\left(\mu_j(n,t_r),-2\tilde{A}_{n+1}(\mu_j(n,t_r),n,t_r)\right)\in\mathcal{K}_n,\quad j=1,\dotsi,n,$$ $$\hat{\nu}_j(n,t_{r})=\left(\nu_j(n,t_r),2\tilde{A}_{n+1}(\nu_j(n,t_r),n,t_r)\right)\in\mathcal{K}_n,\quad j=1,\dotsi,n.$$ Then $\mathcal{D}_{\underline{\hat{\mu}}(n,t_r)}$ and $\mathcal{D}_{\underline{\hat{\nu}}(n,t_r)}$ are non-special for all $(n,t_{r})\in\mathbb{Z}\times\mathbb{R}$.
We are only to prove the conclusion for $\mathcal{D}_{\underline{\hat{\mu}}(n,t_r)}$. $\mathcal{D}_{\underline{\hat{\mu}}(n)}$ is non-special if and only if $\{\hat{\mu}_1(n),\dotsi,\hat{\mu}_n(n)\}$ contains one pair of $\{\hat{\mu}_j,\hat{\mu}^*_j\}$. Hence, $\mathcal{D}_{\underline{\hat{\mu}}(n)}$ is non-special as long as the projection $\mu_j$ of $\hat{\mu}_j$ are mutually distinct, $\mu_j(n)\neq\mu_k(n)$ for $j\neq k$. If two or more projection coincide for some $n_0\in\mathbb{Z}$, for instance,$$\mu_{j_1}(n_0)=\dotsi=\mu_{j_k}(n_0)=\mu_0,\quad k>1.$$ There are two cases in the following associated with $\mu_0$. If $\mu_0\in\mathbb{C}\backslash\{E_0,\dotsi,E_{2n+1}\}$, then $\tilde{A}_{n+1}(\mu_0,n_0,t_r)\neq 0$. It is obvious that $\hat{\mu}_{j_1}(n_0,t_r),\dotsi,\hat{\mu}_{j_k}(n_0,t_r)$ all meet in the same sheet and hence no special divisor can arise in this manner. If $\mu_0$ equals to some $E_{m_0}$ and $k>1$, one concludes $$\tilde{B}_{n}(z,n_0,t_r)\equfill{z\rightarrow E_{m_0}}{}O\left((z-E_{m_0})^2\right)$$ and $$\tilde{A}_{n+1}(E_{m_0},n_0,t_r)=0.$$ But one observes $R_{2p+2}(z,n_0,t_r)=-\tilde{A}_{n+1}^2-z\tilde{B}_n\tilde{C}_n=O\left((z-E_{m_0})^2\right).$ This conclusion contradict with the hypothesis that the curve is nonsingular. We have $k=1 $. Therefore no special divisor can arise in this manner. Then we have completed the proof.
Next, Now we turn to asymptotic properties of $\tilde{\phi}, \Psi_1, \Psi_2$ defined in (\[4.26\]) (\[4.28\]) and (\[4.29\]) in a neighborhood of $P_{0,\pm}$ and $P_{\infty\pm}.$
Assume Hypothesis 2 and Hypothesis 3 hold and suppose $p(n,t_r),q(n,t_r)$ satisfy (\[4.2\]) (\[4.4\]). Moreover, let $P=(z,y)\in\mathcal{K}_{n}\backslash\{P_{\infty\pm},P_{0}\}$, $(n,n_{0},t_r)\in\mathbb{Z}\times\mathbb{Z}\times\mathbb{R}$. Then, the meromorphic function $\tilde{\phi}$ on $\mathcal{K}_n$ has the following asymptotic behavior $$\label{4.66}
\tilde{\phi}(P)=
\begin{cases}
(\tilde{q}^{+})^{-1}\zeta^{-1}+\left(((\tilde{q}^{+}/\tilde{q}-1)\tilde{p})/\tilde{q}\right)^{+}+O(\zeta)
&\text{as}\quad P\rightarrow P_{\infty+}, \cr
-\tilde{r}+(\tilde{p}\tilde{r}^--\tilde{p}\tilde{r})\zeta+O(\zeta^2)
&\text{as}\quad P\rightarrow P_{\infty-}, \cr
\end{cases}$$ where we use the local coordinate $z=\zeta^{-1}$ near the points $P_{\infty\pm}$. $$\label{4.67}
\tilde{\phi}(P)=
\begin{cases}
c_{n}/(\prod_{m=0}^{2n+1}E_{m})\zeta+O(\zeta^2)
&\text{as}\quad P\rightarrow P_{0,+}, \cr
-\left(\prod_{m=0}^{2n+1}E_m\right)/b_n+O(\zeta)
&\text{as}\quad P\rightarrow P_{0,-}, \cr
\end{cases}$$ where we use the local coordinate $z=\zeta$ near the points $P_{0,\pm}.$
By the definition of $\tilde{\phi}$ in (\[4.26\]) the time parameter $t_r$ can be viewed as an additional but fixed parameter, the asymptotic behavior of $\tilde{\phi}$ remains the same as in Lemma \[lemma3\].
Assume Hypothesis 2 and Hypothesis 3 hold and suppose $p(n,t_r),q(n,t_r)$ satisfy (\[4.2\]) (\[4.4\]). Moreover, let $P\in\mathcal{K}_{n}$\\$\{P_{\infty\pm},P_{0,\pm}\}$ and $(n,n_{0})\in\mathbb{Z}^{2}.$ Then $\mathcal{D}_{\underline{\hat{\mu}}(n,t_r)}$ is non-special. Moreover,\
$$\label{4.69}
\phi(P,n,t_r)=C(n,t_r)\frac{\theta(\underline{z}(P,\underline{\hat{\nu}}(n,t_r)))}{\theta(\underline{z}(P,\underline{\hat{\mu}}(n,t_r)))}\exp\left(\int_{Q_{0}}^{P}\omega_{P_{0,+}P_{\infty+}}^{(3)}\right),$$ and $p, q$ are the form of $$\label{4.70}
p^{+}(n,t_r)=\frac{1}{2}\left(-\Delta_3-\Delta_3^{+}-\delta_1+1-\Delta_1\pm\left((\Delta_3+\Delta_3^++\delta_1-1+\Delta_1)^{2}+4\Delta_2\right)^{\frac{1}{2}}\right)$$ $$\label{4.71}
\begin{split}
&q^{+}(n,t_r)=\Delta_2/p^{+}+1\\
&=2\Delta_2\left(-\Delta_3-\Delta_3^{+}-\delta_1+1-\Delta_1\pm\left((\Delta_3+\Delta_3^++\delta_1-1+\Delta_1)^{2}+4\Delta_2\right)^{-\frac{1}{2}}\right)^{-1}\\
&+1.\\
\end{split}$$ Here $$\label{4.72}
\Delta_1=\sum_{j=1}^n\lambda_j^{'}-\sum_{j=1}^n c_j(k)\partial_{\omega_j}\ln\left(\frac{\theta(\underline{z}(P_{\infty+},\underline{\hat{\mu}}(n,t_r))
+\underline{\omega})}{\theta(\underline{z}(P_{\infty-},\underline{\hat{\mu}}(n,t_r))+\underline{\omega})}\right)|_{\underline{\omega}=0},$$ $$\label{4.73}
\Delta_2=\kappa_{\infty+}-\sum_{j=1}^{n}c_{j}(n)\partial_{\omega_{j}} \ln\left(\frac{\theta(\underline{z}(P_{\infty+},\underline{\hat{\nu}}(n,t_r))+\underline{\omega})}
{\theta(\underline{z}(P_{\infty+},\underline{\hat{\mu}}(n,t_r))+\underline{\omega})}\right)
|_{\underline{\omega}=0},$$ $$\label{4.74}
\Delta_3=\frac{\theta(\underline{z}(P_{\infty-},\underline{\hat{\nu}}(n,t_r)))}
{\theta(\underline{z}(P_{\infty-},\underline{\hat{\mu}}(n,t_r)))}\frac{\theta(\underline{z}(P_{\infty+},\underline{\hat{\mu}}(n,t_r)))}
{\theta(\underline{z}(P_{\infty+},\underline{\hat{\nu}}(n,t_r)))}\frac{\tilde{a}_2}{\tilde{a}_1}$$ and $\tilde{a}_1, \tilde{a}_2$, $ \{\lambda_j^{'}\}_{j=1,\dotsi,n}\in\mathbb{C}$ in (\[3.32\]). The Abel map evolving with respect to $t_r$ are $$\label{4.74a}
\frac{\partial}{\partial_{t_r}}\underline{\alpha}_{Q_{0},\ell}(\mathcal{D}_{\underline{\hat{\mu}}(n,t_r)})
=\frac{\partial}{\partial_{t_r}}\underline{\alpha}_{Q_{0},\ell}(\mathcal{D}_{\underline{\hat{\nu}}(n,t_r)})
=-\sum_{k=1}^{p}\sum_{s=0}^{r}c_\ell(k)\bar{\tilde{\delta}}_{r-s}\hat{c}_{k+s-p}(\underline{E}).$$
The proof of (\[4.69\])-(\[4.74\]) is analogous with theorem \[TH4\]. Here $t_r$ can be regarded as a parameter. Next we prove (\[4.74a\]). $$\begin{split}
&\frac{\partial}{\partial_{t_r}}\underline{\alpha}_{Q_{0},\ell}(\mathcal{D}_{\underline{\hat{\mu}}(n,t_r)})
=\frac{\partial}{\partial_{t_r}}\sum_{j=1}^n\int_{Q_0}^{\hat{\mu}_j(n,t_r)}\omega_\ell\\
&=\sum_{j=1}^n \omega_\ell(\hat{\mu}_j)\mu_{j,t_r}\\
&=\sum_{j=1}^n\left(\sum_{k=1}^n c_\ell(k)\frac{\mu_j^{k-1}}{y(\hat{\mu}_j(n,t_r))}\right)\left((\tilde{q}^{+})^{-1}\bar{\tilde{B}}_{r}(\mu_{j}(n,t_r))
y(\hat{\mu}_j(n,t_r))\prod_{k=1,k\neq j}^{n}\left(\mu_{j}-\mu_{k}\right)^{-1}\right)\\
&=\sum_{j=1}^n\left(\sum_{k=1}^n c_\ell(k)\frac{\mu_j^{k-1}}{\prod_{k=1,k\neq j}^{n}\left(\mu_{j}-\mu_{k}\right)}\right)\left((\tilde{q})^{-1}\bar{\tilde{B}}_{r}(\mu_{j}(n,t_r))\right)\\
&=-\sum_{k=1}^{n}c_\ell(k)\sum_{j=1}^{n}\frac{\mu_j^{k-1}}{\prod_{k=1,k\neq j}^{n}\left(\mu_{j}-\mu_{k}\right)}\left(\sum_{s=0}^{r}\bar{\tilde{\delta}}_{r+1-s}\left(\sum_{t=\text{max}\{0,s-p\}}^s\hat{c}_t(\underline{E})\Psi_{s-t}^{(j)}(\underline{\mu})\right)\right)\\
&=-\sum_{k=1}^{p}\sum_{s=0}^{r}c_\ell(k)\bar{\tilde{\delta}}_{r-s}\hat{c}_{k+s-p}(\underline{E}).\\
\end{split}$$
Appendix: The Lagrange Interpolation Representation of $-(\tilde{q}^{+})^{-1}\tilde{B}_{r+1}(\mu_j(n,t_r))$
===========================================================================================================
We search for the interpolation representation of $\tilde{B}_r(\mu_j(n,t_r))$ as in the KdV, AKNS, Toda cases. Introducing the notation in [@A1; @A2] , $$\begin{aligned}
\begin{split}
&\Psi_k(\underline{\mu})=(-1)^k\sum_{\underline{\ell}\in\mathcal{S}_k}\mu_{\ell_1}\dotsi,\mu_{\ell_k},\\ &\mathcal{S}_k=\{\underline{l}=(\ell_1,\dotsi,\ell_k)\in\mathbb{N}^k|\ell_1<\dotsi<\ell_k\leq n\},\quad k=1,\dotsi n,\\
&\Phi_k^{(j)}(\underline{\mu})=(-1)^k\sum_{\underline{\ell}\in\mathcal{\tau}_k^{(j)}}\mu_{\ell_1}\dotsi,\mu_{\ell_k},\\ &\mathcal{\tau}_k^{(j)}=\{\underline{l}=(\ell_1,\dotsi,\ell_k)\in\mathbb{N}^k|\ell_1<\dotsi<\ell_k\leq n,\quad \ell_m\neq j,\quad m=1,\dotsi,k\}, \\
&k=1,\dotsi n-1,\quad j=1,\dotsi,n\\
\end{split}\end{aligned}$$ and the formula $$\label{6.3}
\sum_{\ell=0}^{k}\Psi_{k-\ell}(\underline{\mu})\mu_j^\ell=\Phi_{k}^{(j)}(\underline{\mu}),\quad k=0,\dotsi,n,\quad j=1,\dotsi,n.$$ Let $B_n=(\tilde{q}^{+})^{-1}\tilde{B}_n, b_s=(\tilde{q}^{+})^{-1}\tilde{b}_s(s=0,1,\dotsi,n),$ then one finds $$B_{n}(z)=\sum_{\ell=0}^{n}b_{n-s}z^s=\prod_{j=1}^n\left(z-\mu_j\right)=\sum_{l=0}^{n}\Psi_{p-l}(\underline{\mu})
z^\ell$$ and $$b_\ell=\Psi_{\ell}\left(\underline{\mu}\right),\quad \ell=0,\dotsi,n.$$ In the case $r<n$, $$\begin{aligned}
\label{6.4}
\begin{split}&\bar{B}_{r}=\sum_{s=0}^{r}\bar{b}_{r-s}z^s=\sum_{s=0}^{r}\left(\sum_{k=0}^{\text{min}\{r-s,n\}}\hat{c}_{r-s-k}
(\underline{E})b_k\right)z^s\\
&=\sum_{s=0}^{r}\left(\sum_{k=0}^{r-s}\hat{c}_{r-s-k}(\underline{E})b_k\right)z^s\\
&=\sum_{s=0}^{r}\left(\sum_{k=0}^{r-s}\hat{c}_{r-s-k}(\underline{E})\Psi_{k-1}(\underline{\mu})\right)z^s\\
&=\sum_{s=0}^{r}\hat{c}_{s}(\underline{E})\sum_{t=0}^{r-s}\Psi_{r-s-t}(\underline{\mu})z^t.\\
\end{split}\end{aligned}$$ Using (\[6.3\]), we have $$\label{6.5}
\begin{split}
&\bar{B}_{r}(\mu_j)=\sum_{s=0}^{r}\hat{c}_{s}(\underline{E})\sum_{t=0}^{r-s}\Psi_{r-s-t}(\underline{\mu})\mu_j^t\\
&=\sum_{s=0}^{r}\hat{c}_{s}(\underline{E})\Psi_{r-s}^{(j)}(\underline{\mu}).\\
\end{split}$$ In the case $r>n$, $$\begin{aligned}
\begin{split}
&\bar{B}_{r}(z)=\sum_{s=0}^{r}\bar{b}_{r-s}z^s=\sum_{s=0}^{r}\left(\sum_{k=0}^{\text{min}\{r-s,n\}}\hat{c}_{r-s-k}
(\underline{E})b_k\right)z^s\\
&=\sum_{s=0}^{r-n}\sum_{k=0}^{n}\hat{c}_{r-s-k}(\underline{E})\Psi_{k}(\underline{\mu})z^s+\sum_{s=r-p+1}^{r}
\sum_{k=0}^{r-s}\hat{c}_{r+1-s-k}(\underline{E})\Psi_{k}(\underline{\mu})z^s\\
&=\sum_{s=0}^{r-n}\sum_{k=0}^{n}\hat{c}_{r-s-k}(\underline{E})\Psi_{k}(\underline{\mu})z^s+
\sum_{s=r-n+1}^{r}\sum_{k=0}^{n}\hat{c}_{r-s-k}(\underline{E})\Psi_{k}(\underline{\mu})z^s\\
&=\sum_{k=0}^{n}\sum_{s=0}^{r}\hat{c}_{r-s-k}(\underline{E})\Psi_{k}(\underline{\mu})z^s\\
&=\sum_{s=0}^{r}\sum_{k=0}^{n}\hat{c}_{r-s-k}(\underline{E})\Psi_{k}(\underline{\mu})z^s\\
&=\sum_{s=0}^{r}\sum_{k=0}^{n}\hat{c}_{s}(\underline{E})\Psi_{k}(\underline{\mu})z^{r-s-k}\\
&=\sum_{s=0}^{r-n}\hat{c}_{s}(\underline{E})\left(\sum_{k=0}^{n}\Psi_{k}(\underline{\mu})z^{p-k}\right)z^{r-n-s}
+\sum_{s=r-p+1}^{r}\hat{c}_{s}(\underline{E})\left(\sum_{k=0}^{n}\Psi_{k}(\underline{\mu})z^{r-s-k}\right)\\
&=\sum_{s=0}^{r-n}\hat{c}_{s}(\underline{E})\left(B_{n}(\lambda)\right)z^{r-n-s}+\sum_{s=r-n+1}^{r}\hat{c}_{s}
(\underline{E})\left(\sum_{k=0}^{r-s}\Psi_{k}(\underline{\mu})z^{r-s-k}\right).\\
\end{split}\end{aligned}$$ Then we have $$\label{6.7}
\bar{B}_{r}(\mu_j)=\sum_{s=r-n+1}^{r}\hat{c}_{s}(\underline{E})\left(\sum_{k=0}^{r-s}\Psi_{k}(\underline{\mu})
\mu_j^{r-s-k}\right)=\sum_{s=r-n+1}^{r}\hat{c}_s(\underline{E})\Psi_{r-s}^{(j)}(\underline{\mu}).$$ Combining (\[6.5\]) with (\[6.7\]), one finds $$\bar{B}_{r}(z)=\sum_{s=\text{max}\{0,r-n\}}^{r}\hat{c}_{s}(\underline{E})\Psi_{r-s}^{(j)}(\underline{\mu}).$$ Hence $$\begin{split}
&\bar{\tilde{B}}_{r}(\mu_j)=\sum_{s=0}^{r}\bar{\tilde{\delta}}_{r-s}\bar{B}_s(\mu_j)=\sum_{s=0}^{r}
\tilde{\delta}_{r-s}\left(\sum_{t=\text{max}\{0,s-n\}}^{s}\hat{c}_{t}(\underline{E})\Psi_{s-t}^{(j)}
(\underline{\mu})\right).\\
\end{split}$$
[99]{} S. N. M. Ruijsennaars, *Relaticistic Toda Systems*, Commun. Math. Phys. **133** 217 (1990). Y. B. Suris and O. Ragnisco, *What is the Relativistic Volterra Lattice*, Commun. Math. Phys. **200** 445 (1999). Y. B. Suris, *The Problem of Integrable Discretization: Hamiltonian Approach*, Birkh¡§auser, Boston, (2003). Y. B. Suris, *Miura transformations for Toda-type integrable systems, with applications to the problem of integrable discretizations*, arXiv:solv-int/9902003v1. M. Toda, *Theory of Nonlinear Lattices*, Springer, Berlin, (1989). H. Flaschka, *The Toda lattice. II. Existence of integrals*, Phys. Rev. B **9** 1924 (1974). S. P. Novikov, S. V. Manakov, L. P. Pitaevskii, V.E. Zakharov, *Theory of Solitons the Inverse Scattering Methods*, Consultants Bureau, New York, (1984). A. Bloch, *Hamiltonian and Gradient Flows, Algorithms, and Control*, Amer. Math. Soc, 155 (1994). J. Gibbons, B. A. Kupershmidt: in: B. A. Kupershmidt (Ed.), *Integrable and Superintegrable Systems*, World Scientific, Singapore, 207 (1990). Z. N. Zhu, H. C. Huang, W. M. Xue, *New Lax Representation and Integrable Discretization of the Relativistic Volterra Lattice*, J. Phys. Soc. Jpn. **69** (1999). Z. N. Zhu, W. M. Xue, X. N. Wu and Z. M. Zhu, *Infinitely many conservation laws and integrable discretizations for some lattice soliton equations*, J. Phys. A: Math. Gen. **35** 5079 (2002). K. Maruno, M. Oikawa, *Bilinear structure and determinant solution for the relativistic Lotka-Volterra equation*, Phys. Lett. A **270** 122 (2000). P. D. Lax, *Periodic solutions of the KdV equation*, Commun. Pure Appl. Math. **28** 141 (1975). A. R. Its and V. B. Matveev, *The Schödinger operator in a finite-zone spectrum and N-soliton solutions of the Korteweg-de Vries equation*, Theor. Mat. Fiz. **9** 51 (1975). V. Batchenko and F. Gesztesy, *On the spectrum of Schrödinger operators with quasi-periodic algebro-geometric KDV potentials*, Spinger, (2007). M. Kac and P. van Moerbeke, *A complete solution of the periodic Toda problem*, Proc. Natl. Acad. Sci. USA. **72** 2879 (1975). E. D. Belokolos, A. I. Bobenko, V. Z. Enolskii, A. R. Its, V. B. Matveev, *Algebro-Geometric Approach to Nonlinear Integrable Equations*, Springer, (1994). F. Gesztesy, H. Holden, *Soliton Equations and Their Algebro-Geometric Solutions, Volume I: (1+1)-Dimensional Continuous Models*, Cambridge University Press, (2003). F. Gesztesy, H. Holden, J. Michor, G.Teschl, *Soliton Equations and Their Algebro-Geometric Solutions, Volume II: (1+1)-Dimensional Discrete Models*, Cambridge University Press, (2008). G. Teschl, *Jacobi Operators and Completely Integrable Nonlinear Lattices*, Amer. Math. Soc, **72**. H. M. Farkas, and I. Kra, *Riemann Surfaces. Second ed.* Springer, New York, (1992). P. A. Griffiths, *Introduction to Algebraic Curves, Providence, RI. Amer. Math. Soc*, (1989). P. A. Griffiths, J. Harris, *Principles of Algebraic Geometry*, Wiley, New York, (1994). D. Mumford, *Tata Lectures on Theta II*, Birkhäser, Boston, (1984). X. G. Geng, Y. T. Wu, and C. W. Cao, *Quasi-periodic solutions of the modified Kadomtsev-Petviashvili equation* J. Phys. A: Math. Gen. **32** 3733 (1999). C. W. Cao, X. G. Geng, H. Y. Wang, *Algebro-geometric solutions of the 2+1 dimensional Burgers equation with a discrete variable*, J. Math. Phys. **43** 621 (2002). W. Bulla, F. Gesztesy, H. Holden, G. Teschl, *Algebro-Geometric Quasi-Periodic Finite-Gap Solutions of the Toda and Kac-van Moerbeke Hierarchies*, Amer. Math. Soc, **135** 79 (1998). J. S. Geronimo, F. Gesztesy, H. Holden, *Algebro-geometric solutions of the Baxter-Szegö difference equation*, Comm. Math. Phys. **258** 149 (2005). R. Dickson , F. Gesztesy , K. Unterkofler *Algebro-geometric solutions of the Boussinesq hierarchy*, Rev. Math. Phys. **11** 823 (1999).
[^1]: Corresponding author and E-mail: [email protected]
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abstract: 'We investigate whether inclusion of dimension six terms in the Standard Model lagrangean may cause the unification of the coupling constants at a scale comprised between $10^{14}$ and $10^{17}$ GeV. Particular choice of the dimension 6 couplings is motivated by the spectral action. Given the theoretical and phenomenological constraints, as well as recent data on the Higgs mass, we find that the unification is indeed possible, with a lower unification scale slightly favoured.'
---
ICCUB-14-062
**Unification of Coupling Constants, Dimension six Operators and the Spectral Action**[ ]{}
[Agostino Devastato$^{1,2}$, Fedele Lizzi$^{1,2,3}$, Carlos Valcárcel Flores$^{4}$ and Dmitri Vassilevich$^{4}$]{}\
$^{1}$*Dipartimento di Fisica, Università di Napoli* *Federico II*\
$^{2}$*INFN, Sezione di Napoli*\
*Monte S. Angelo, Via Cintia, 80126 Napoli, Italy*\
$^{3}$ *Departament de Estructura i Constituents de la Matèria,*\
*Institut de Ciències del Cosmos, Universitat de Barcelona,*\
*Barcelona, Catalonia, Spain*\
$^{4}$ *CMCC-Universidade Federal do ABC, Santo Andrè, S.P., Brazil*\
`[email protected], [email protected], [email protected], [email protected]`[ ]{}
2 cm
Introduction
============
The coupling constants of the three gauge interactions run with energy [@particledata]. The ones relating to the nonabelian symmetries are relatively strong at low energy, but decrease, while the abelian interaction increases. At an energy comprised between $10^{13}-10^{17}$ GeV their values are very similar, around $0.52$, but, in view of present data, and *in absence of new physics*, they fail to meet at a single scale. Here by absence of new physics we mean extra terms in the Lagrangian of the model. The extra terms may be due for example to the presence of new particles, or new interaction. A possibility could be supersymmetric models which can alter the running and cause the presence of the unification point [@Susy].
The standard model of particle interaction coupled with gravity may be explained to some extent as a particular for of Noncommutative, or spectral geometry, see for example [@WalterBook] for a recent introduction. The principles of noncommutative geometry are rigid enough to restrict gauge groups and their fermionic representations, as well as to produce a lot of relations between bosonic couplings when applied on (almost) commutative spaces. All these restrictions and relations are surprisingly well compatible with the Standard Model, except that the Higgs field comes out too heavy, and that the unification point of gauge couplings is not exactly found. We have nothing new to say about the first problem, which has been solved in [@Stephan; @coldplay; @resilience; @CCvS; @BoyleFarnsworthsigma] with the introduction of a new scalar field $\sigma$ suitably coupled to the Higgs field, but we shall address the second one.
Some years ago the data were compatible with the presence of a single unification point $\Lambda$. This was one of the motivations behind the building of grand unified theories. Such a feature is however desirable even without the presence of a larger gauge symmetry group which breaks to the standard model with the usual mechanisms. In particular, the approach to field theory, based on noncommutative geometry and spectral physics [@spectralaction], needs a scale to regularize the theory. In this respect, the finite mode regularization [@fujikawa; @AndrianovBonora1; @AndrianovBonora2] is ideally suited. In this case $\Lambda$ is also the field theory cutoff. In fact using this regularization it is possible to generate the bosonic action starting form the fermionic one [@AndrianovLizzi; @AndrianovKurkovLizzi; @KurkovLizzi], or describe induced gravity on an equal footing with the anomaly-induced effective action [@KurkovSakellariadou]
The aim of this paper is to investigate whether the presence of higher dimensional terms in the standard model action $-$ dimension six in particular $-$ may cause the unification of the coupling constants. The paper may be read in two contexts: as an application of the spectral action, or independently on it, from a purely phenomenologically point of view.
From the spectral point of view, the spectral action [@spectralaction] is solved as a heath kernel expansion in powers in the inverse of an energy scale. The terms up to dimension four reproduce the standard model qualitatively, but the theory is valid at a scale in which the couplings are equal. The expansion gives, however, also higher dimensional terms, suppressed by the power of the scale, and depending on the details of the cutoff. This fixes relations among the coefficients of the new terms. The analysis of this paper gives the conditions under which the spectral action can predict the unification of the three gauge coupling constants.
On the other side, it is also possible to read the paper at a purely phenomenological level, using the spectral action as input only for the choice of the subset of all possible higher dimensions terms in the action, and as a guide for the setting of the low energy values for the couplings of the coefficients of the extra terms. We show that the presence of these terms enables the possibility of a unification.
In both cases the scale of unification $\Lambda$ is considered the cutoff, and we run the theory below it. We assume, therefore, that perturbation theory is valid. There appears a hierarchy problem. From the point of view of the spectral action this implies a rather strange (though admissible) cutoff function. From a phenomenological point of view this entails either unnaturally large dimensionless quantities, or the presence of a new intermediate scale, $\Gamma$. The latter option is, of course, more desirable and we will discuss it below.
The paper is organized as follows: in section \[SMRCC\], we present the action of Standard Model (SM) of particle physics, and the standard running coupling constants. We show how the spectral action approach $-$ whose principles are summarized in the appendix \[appspectralaction\] $-$ fits the SM. In section \[RGEs\] we give the new renormalization group equations at one loop, due to the dimensions-6 operators; then, we show how these new operators affect the SM phenomenology. In section \[RGF\] we run the renormalization group equations to study the new coupling constants behavior, checking the possibility to improve the gauge unification point. A final section contains conclusions and some comments and open questions.
Standard Model Running Coupling Constans\[SMRCC\]
=================================================
The standard model action (including right handed neutrinos) is: $$\begin{aligned}
\mathcal{L}_{SM} & = & -\frac{1}{4}V_{\mu\nu}^{A}V^{A\mu\nu}-\frac{1}{4}W_{\mu\nu}^{I}W^{I\mu\nu}-\frac{1}{4}B_{\mu\nu}B^{\mu\nu}+\left(D_{\mu}H\right)^{\dagger}D^{\mu}H-\frac{m^{2}}{2}H^{\dagger}H-\lambda \left(H^{\dagger}H\right)^{2}+\nonumber \\
& & -\left(\overline{e}\mathbf{F}_{L}H^{\dagger}l+\overline{d}\mathbf{F}_{D}H^{\dagger}q+\overline{u}\mathbf{F}_{U}H^{c\dagger}q+h.c.\right)+\mbox{Majorana mass terms} \label{eq: SM Lagrangian}\end{aligned}$$ where $B_{\mu\nu}$, $W_{\mu\nu}^{I}$ and $V_{\mu\nu}^{A}$ are respectively the field strengths associated with the gauge groups $U(1),\, SU(2)$ and $SU(3)$; the gauge covariant derivative is $D_{\mu}=\partial_{\mu}+ig_{3}T^{A}A_{\mu}^{A}+ig_{2}t^{I}W_{\mu}^{I}+ig_{1}\mbox{y}B_{\mu}$, where $T^{A}$ are the $SU(3)$ generators, $t^{I}=\tau^{I}/2$ are the $SU(2)$ generators, and y is the $U(1)$ hypercharge generator. $H$ is the Higgs field, a $SU(2)$ scalar doublet with hypercharge $1/2$ and its charged conjugated field defined as $H^{c}\equiv i\tau_{2}H^{*}$. The three families of fermions are grouped together so that $\mathbf{F}_{L}$, $\mathbf{F}_{U}$, $\mathbf{F}_{D}$ are the $3\times3$ complex Yukawa matrices acting on the hidden flavor index of every fermion field. Since these matrices are dominated by the Yukawa coupling $y_t$ of the top quark, in the following we will consider this parameter only. Likewise we will consider a single mass term for the Majorana masses: $y_\nu$.
This Lagrangian can be obtained from first principles using the spectral action [@AC2M2; @NCPart1], which is a regularized trace, with $\Lambda$ appearing as the cutoff. We give the details of the spectral action calculations in the appendix. For the economy of this paper the relevant part is the fact that the spectral action requires the coupling constants of the three gauge groups to be equal at a scale $\Lambda$, which is also the cutoff of the theory. There is no need for a unified gauge group at the scale $\Lambda$, which in fact may signify a phase transition to a pre geometric phase [@Kuliva], although larger symmetries are also possible [@coldplay; @CCvS]. As explained in the appendix, the spectral action is an expansion in inverse powers of $\Lambda^{2}$, and it enables the presence of a set of new dimension six operators. Dimension five operators, which violate lepton number, and do not change the properties of the Higgs boson are not present in the expansion. The spectral action also gives relations among the coefficients of the required dimension six operators, which are described in detail in the appendix \[appspectralaction\]. The reader interested only in the phenomenological aspect of this paper may skip the appendix, and accept our choice of operator as a convenient one.
A complete classification of the dimension-six operators in the standard model is given in [@Dim6Classification]. There it is shown that there are 59 independent operators, preserving baryon number, after eliminating redundant operators using the equations of motion. Here we consider only the following dimension-six operators, mixing the gauge field strength and the Higgs field. They are the ones coming from the spectral action expansion: $$\begin{aligned}
\mathcal{L}^{(6)} & = & C_{HB}H\overline{H\,}B_{\mu\nu}B^{\mu\nu}+C_{HW}H\overline{H}\,\mathbf{W}_{\mu\nu}\mathbf{W}^{\mu\nu}+C_{HV}H\overline{H}\,\mathbf{V}_{\mu\nu}\mathbf{V}^{\mu\nu}+\nonumber \\
& & +C_{W}\mathbf{W}_{\mu\nu}\mathbf{W}^{\nu\alpha}\mathbf{W}_{\alpha}^{\mu}+C_{V}\mathbf{V}_{\mu\nu}\mathbf{V}^{\nu\alpha}\mathbf{V}_{\alpha}^{\mu}+C_{H}\left(\overline{H}H\right)^{3}\label{dimsixlagrangian}\end{aligned}$$ The coefficients $C$ have the dimension of an inverse energy square. The spectral action fixes their value at the cutoff $\Lambda$. To these terms we have to add a coupling between the Higgs, the $W$ and the $B$ which is absent in the spectral action at scale $\Lambda$, but is dynamically created. With the couplings considered here no other term is induced.
The SM running coupling constants at one loop, associated to (\[eq: SM Lagrangian\]), are ruled by the following equations, where we defined the dot derivations as $16\pi^{2}\mu\frac{d}{d\mu}$: $$\begin{aligned}
\dot{g}_{i} & = & \left(b_{i}g_{i}^{3}\right)\,\,\mbox{with}\,\left(b_{1},b_{2},b_{3}=\frac{41}{6},-\frac{19}{6},-7\right)\nonumber \\
\dot{\lambda} & = & \left(24\lambda^{2}-\left(3g_{1}^{2}+9g_{2}^{2}\right)\lambda+\frac{3}{8}\left(g_{1}^{4}+2g_{1}^{2}g_{2}^{2}+3g_{2}^{4}\right)+\left(12y_t^{2}+4y_\nu^{2}\right)\lambda-6y_t^{4}-2y_\nu^{4}\right)\nonumber \\
\dot{y}_{t} & = & \left(\frac{9}{2}y_t^{2}+y_\nu^{2}-\frac{17}{12}g_{1}^{2}-\frac{9}{4}g_{2}^{2}-8g_{3}^{2}\right)\nonumber \\
\dot{y}_{\nu} & = & \left(\frac{5}{2}y_\nu^{2}+3y_t^{2}-\frac{3}{4}g_{1}^{2}-\frac{9}{4}g_{2}^{2}\right)\label{eq: SM RGE}\end{aligned}$$ For the purposes of this paper one loop is sufficient, the running up to three loops can be found in [@Machacek1; @Machacek2; @Machacek3; @3loops] and references therein. In the present case, one separately solves the equations for the gauge coupling constants and the other couplings; for the former, the boundary conditions are given at the electro-weak scale by the experimental values [@particledata], $$\begin{aligned}
g_{1}(m_{Z}) = 0.358,\,\, g_{2}(m_{Z})=0.651,\,\, g_{3}(m_{Z})=1.221\,\label{eq:InitialCondition}\end{aligned}$$ while for the other coupling constants $\lambda,y_t$ and $y_\nu$ the boundary conditions are taken at the cut-off scale $\Lambda$ that is the scale at which the spectral action lives. These boundary conditions use the parameters of the fermions that are the inputs in the Dirac operator [(\[Diracoperator\])]{}, as shown in the appendix \[appspectralaction\], $$\begin{aligned}
\lambda(\Lambda) & =\frac{4\left(\rho^2+3\right)}{\left(\rho+3\right)^{2}}g^{2}\nonumber \\
y_t(\Lambda) & =\sqrt{\frac{4}{\rho^2+3}}g\nonumber \\
y_\nu(\Lambda) & =\sqrt{\frac{4\rho^2}{\rho^2+3}}g\label{eq:Boundary conditions}\end{aligned}$$ $\rho$ is a free parameter such that $y_\nu(\Lambda)=\rho\,y_t(\Lambda)$ and $g\equiv g_{3}(\Lambda)=g_{2}(\Lambda)=\frac{5}{3}g_{1}(\Lambda)$. Since the coupling constants $g_{i}$ do not meet exactly, forming a triangle, one takes for $\Lambda$ a range of values beetwen the extremal points of the triangle. The results, for a particular set of values of the parameters *$(g,\rho,\Lambda)=(0.530,\,1.25,10^{16}GeV)$,* are plotted in Fig. \[fig:Standard gi\].
![*The standard model running for the gauge coupling constant (left) and the Yukawa coupling (right) in the spectral action approach for $(g,\rho,\Lambda)=(0.530,\,1.25,\,10^{16}GeV)$. The red dot indicates the starting value of the parameters (Log$_{10} (\frac{\Lambda}{GeV}),g$) and (Log$_{10} (\frac{\Lambda}{GeV}),\lambda(\Lambda)$). \[fig:Standard gi\]*](Standardgi.pdf "fig:"){width="9cm" height="7cm"}![*The standard model running for the gauge coupling constant (left) and the Yukawa coupling (right) in the spectral action approach for $(g,\rho,\Lambda)=(0.530,\,1.25,\,10^{16}GeV)$. The red dot indicates the starting value of the parameters (Log$_{10} (\frac{\Lambda}{GeV}),g$) and (Log$_{10} (\frac{\Lambda}{GeV}),\lambda(\Lambda)$). \[fig:Standard gi\]*](StandardYt.pdf "fig:"){width="9cm" height="7cm"}
After running these couplings from unification energy $\Lambda$ to low energy $M_Z$, we compare the values of $y_t(M_Z)$ and $\lambda(M_Z)$ with their experimental values $$\begin{aligned}
y^{exp}_t(M_Z)=0.997,\,\,\ & \lambda^{exp}(M_Z)=0.130 \label{expvalues}\end{aligned}$$ In Fig. \[fig:Standard gi\] we can see the good agreement between $y_t(M_Z)$ predicted by the spectral action and its experimental value. Very different is the case for the Higgs self-coupling $\lambda$, Fig. \[lambdabehaviours\], whose predicted value, in the spectral action approach, is around 0.240 with a resulting Higgs mass $M_H=\sqrt{2 \lambda v^2 }\simeq 170$ GeV. On the other hand, the experimental value for the Higgs mass ($\simeq$125 GeV) leads to the instability problem for the self-interaction parameter $\lambda$, which becomes negative at a scale of the order of $10^{8}GeV$; two loop calculations make the situation slightly worse as one can see on the left side of Fig. \[lambdabehaviours\].
![*On the left, the standard model running for the coupling constant $\lambda$ starting from $\lambda(m_{Z})=0.130$ corresponding to $M_{H}=125GeV$. The dashed and solid lines represent the one and two loop respectively. On the right, the $\lambda$ behaviour starting from the red point of the spectral action and culminating in the prediction $\lambda(m_{Z})=0.240$.* \[lambdabehaviours\]]("LambdaOneTwoloop".png "fig:"){width="9cm" height="7cm"}![*On the left, the standard model running for the coupling constant $\lambda$ starting from $\lambda(m_{Z})=0.130$ corresponding to $M_{H}=125GeV$. The dashed and solid lines represent the one and two loop respectively. On the right, the $\lambda$ behaviour starting from the red point of the spectral action and culminating in the prediction $\lambda(m_{Z})=0.240$.* \[lambdabehaviours\]](StandardlambdaYaxe.pdf "fig:"){width="9cm" height="7cm"}
A negative $\lambda$ means an instability and renders the model inconsistent, although it may just mean the presence of a long lived metastable state [@metastable1; @metastable2]. The spectral action model can be fixed [@Stephan; @coldplay; @resilience; @CCvS; @BoyleFarnsworthsigma] with the introduction of a scalar field, $\sigma$, possibly coming from a larger symmetry, connected with the fluctuations of a Majorana neutrino mass term in the action. Since the running of the Higgs parameters do not affect strongly the running of the coupling constants (which are the true aim of this paper), nor does this field $\sigma$, we will not consider it in what follows. However, a more complete and accurate analysis will necessitate also this element, and it is in progress. Also the presence of gravitational couplings in the spectral action could alter significativly the running at high energy leading to an asymptotically free theory at the Planck scale [@Devastato].
Coupling Constants RGEs\[RGEs\]
===============================
In this section we give the new renormalization group equations (RGEs) at one loop due to the dimensions-6 operators in the Lagrangian [(\[dimsixlagrangian\])]{}. Although the choice of the dimension six operators and some of characteristics of the Lagrangian are coming from the spectral action, this section can be read independently of it.
The full one-loop contributions to the SM running for dimension six operators have been calculated in [@Grojean; @Manohar1; @Manohar2; @JenkinsIII]. The modifications to the standard model RGEs are given by the following new terms to be added to the rhs of [(\[eq: SM RGE\])]{}: $$\begin{aligned}
\delta\dot{g_{3}} & = & -4m_{H}^{2}g_{3}C_{HV}\nonumber \\
\delta\dot{g_{2}} & = & -4m_{H}^{2}g_{2}C_{HW}\nonumber \\
\delta\dot{g_{1}} & = & -4m_{H}^{2}g_{1}C_{HB}\nonumber \\
\delta\dot{\lambda}\ & = & m_{H}^{2}\left(9g_{2}^{2}C_{HW}+3g_{1}^{2}C_{HB}+12C_{H}+3g_{1}g_{2}C_{HWB}\right)\nonumber \\
\delta\dot{y}_{t,\nu} & = & 0\label{eq:modification beta}\end{aligned}$$ and the RGEs for the dim-6 coupling constants are given by $$\begin{aligned}
\dot{C}_{HB} & =C_{HB}\left(12\lambda+2\left(3y_t^{2}+y_\nu^{2}\right)+\frac{85}{6}g_{1}^{2}-\frac{9}{2}g_{2}^{2}\right)+3C_{HWB}g_{1}g_{2}\nonumber \\
\dot{C}_{HW} & =C_{HW}\left(12\lambda+2\left(3y_t^{2}+y_\nu^{2}\right)-\frac{47}{6}g_{1}^{2}-\frac{5}{2}g_{2}^{2}\right)+C_{HWB}g_{1}g_{2}-15C_{W}g_{2}^{3}\nonumber \\
\dot{C}_{HV} & =C_{HV}\left(12\lambda+2\left(3y_t^{2}+y_\nu^{2}\right)-\frac{3}{2}g_{1}^{2}-\frac{9}{2}g_{2}^{2}-14g_{3}^{2}\right)\nonumber \\
\dot{C}_{HWB} & =C_{HWB}\left(4\lambda+2\left(3y_t^{2}+y_\nu^{2}\right)+\frac{19}{3}g_{1}^{2}+\frac{4}{3}g_{2}^{2}\right)+2g_{1}g_{2}\left(C_{HB}+C_{HW}\right)+3C_{W}g_{1}g_{2}^{2}\nonumber \\
\dot{C}_{W} & =\frac{29}{2}C_{W}g_{2}^{2}\nonumber \\
\dot{C}_{V} & =15C_{V}g_{3}^{2}\nonumber \\
\dot{C}_{H} & =C_{H}\left(108\lambda+6\left(3y_t^{2}+y_\nu^{2}\right)-\frac{9}{2}g_{1}^{2}-\frac{27}{2}g_{2}^{2}\right)-3C_{B}g_{1}^{2}\left(g_{1}^{2}+g_{2}^{2}-4\lambda\right)+\nonumber \\
& \,\,\,\,+3C_{W}g_{2}^{2}\left(12\lambda-3g_{2}^{2}-g_{1}^{2}\right)+C_{HWB}\left(12\lambda g_{1}g_{2}-3g_{1}^{3}g_{2}-3g_{1}g_{2}^{3}\right)\label{eq: RGE Correction-1}\end{aligned}$$ Although the spectral action does not contain explicitly the term $C_{HWB}H^{2}W_{\mu\nu}B^{\mu\nu}$, due to the unimodular condition, the coupling constant $C_{HWB}$ is however induced by the running of $C_{HB},\, C_{HW}$ and $C_{W}$.
In the framework of the spectral action these equations are solved with boundary conditions at the cut-off scale $\Lambda$ given by the coefficients appearing in (\[eq:A6 Expression\]): $$\begin{aligned}
C_{HB}(\Lambda) & =-\frac{f_{6}}{16\pi^{2}\Lambda^{2}}\frac{4\left(3\rho^2+17\right)}{9\left(\rho^2+3\right)}g^{4}\,\,,\, C_{HW}(\Lambda)=-\frac{f_{6}}{16\pi^{2}\Lambda^{2}}\frac{4}{3}g^{4}\,\,,\, C_{HV}(\Lambda)=-\frac{f_{6}}{16\pi^{2}\Lambda^{2}}\frac{16}{3\left(\rho^2+3\right)}g^{4}\,\,,\nonumber \\
C_{H}(\Lambda) & =-\frac{f_{6}}{16\pi^{2}\Lambda^{2}}\frac{512(\rho^{6}+3)}{3\left(\rho^2+3\right)^{3}}g^{6}\,\,,\,\,\,\,\, C_{W}(\Lambda)=-\frac{f_{6}}{16\pi^{2}\Lambda^{2}}\frac{26}{15}g^{3}\,\,,\,\,\,\,\, C_{V}(\Lambda)=-\frac{f_{6}}{16\pi^{2}\Lambda^{2}}\frac{26}{15}g^{3}\,.\,\,\,\label{eq: constrains new}\end{aligned}$$ The coupling $C_{HWB}$ is set to zero at the cut-off scale $C_{HWB}(\Lambda)=0$ since it does not appear in the spectral action.
In (\[eq: constrains new\]) $g\equiv g_{3}(\Lambda)=g_{2}(\Lambda)=\frac{5}{3}g_{1}(\Lambda)$ is the value of the gauge coupling constants at the cut-off scale which, therefore, is identified with the unification scale. These two constants, $g$ and $\Lambda$, together with the ratio $\rho$ and the parameter $f_{6}$ appearing in the spectral action, will be the four free parameters of this model.
There are also constraints at low energy to satisfy. The values of the $g_{i}$’s are known at the scale of the top mass with very high precision, and the parameters $\lambda$ and the $y_t$ are related to the Higgs and top mass. As we said earlier, the spectral action requires a positive value of $\lambda$ at the cutoff scale $\Lambda$, (\[eq:Boundary conditions\]), and without the field $\sigma$, it predicts a mass of the Higgs at 170 GeV. However, the presence of higher-order operators in the action alters the form of the usual coupling constants, leading to a new phenomenology which we outline in the following section.
New phenomenology
-----------------
In this section, following [@JenkinsIII sect.5] we give the main modifications to the SM phenomenology due to the dim-6 Lagrangian, i.e. the new form of the observables measured at the electroweak scale. The new operators, in fact, alter the definition of the SM parameters at tree level in several ways.
First of all, we focus on the effects of the dimension-six Lagrangian on the Higgs mass $m_{H}$ and the self-coupling $\lambda$. The dim-6 operator $C_{H}\bar{H}H$ changes the shape of the scalar doublet potential at order $C_{H}v^{2}$ to $$V(H)=-\frac{m^{2}}{2}H^{\dagger}H+\lambda\left(H^{\dagger}H\right)^{2}-C_{H}\left(H^{\dagger}H\right)^{3}\label{eq:new potential}$$ generating the new minimum $$\begin{aligned}
\left\langle H^{\dagger}H\right\rangle & = & \frac{1}{3C_{H}}\left(\lambda-\sqrt{\lambda^{2}-3C_{H}\lambda v^{2}}\right)\nonumber \\
& \simeq & \frac{v^{2}}{2}\left(1+\frac{3C_{H}v^{2}}{4\lambda}\right)\equiv\frac{v_{T}^{2}}{2}\end{aligned}$$ in the second line we have expanded the exact solution to first order in $C_{H}.$ Therefore the shift in the vacuum expectation value is proportional to $C_{H}v^{2},$ which is of order $f_{6}\frac{v^{2}}{\Lambda^{2}}.$ On expanding the potential (\[eq:new potential\]) around the minimum and neglecting kinetics corrections, $$H=\frac{1}{\sqrt{2}}\left(\begin{array}{c}
0\\
h+v_{T}
\end{array}\right),$$ we find for the Higgs boson mass $$m_{H}^{2}=2\lambda v_{T}^{2}\left(1-\frac{3C_{H}v^{2}}{2\lambda}\right)\label{Higgsmass}$$ At the same time the gauge fields and the gauge couplings are also affected by the dim-6 couplings.
In the broken theory the $X^{2}H^{2}$ operators (with $X$ being any field strength) contribute to the gauge kinetic energies, through the Lagrangian terms $$\begin{aligned}
\left(\mathcal{L}_{SM}+\mathcal{L}_{6}\right)_{kin} & = & -\frac{1}{4}B_{\mu\nu}B^{\mu\nu}-\frac{1}{4}G_{\mu\nu}G^{\mu\nu}-\frac{1}{4}W_{\mu\nu}^{3}W_{3}^{\mu\nu}-\frac{1}{2}W_{\mu\nu}^{+}W_{-}^{\mu\nu}+ \\
& & +\frac{1}{2}v_{T}^{2}\left(C_{HB}B_{\mu\nu}B^{\mu\nu}+C_{HW}W_{\mu\nu}W^{\mu\nu}+C_{HG}G_{\mu\nu}G^{\mu\nu}-C_{HWB}W_{\mu\nu}^{3}B^{\mu\nu}\right) \nonumber\end{aligned}$$ while for the mass terms of the gauge bosons, arising from $(D_{\mu}H)^{\dagger}(D^{\mu}H$), we have $$\left(\mathcal{L}_{SM}+\mathcal{L}_{6}\right)_{mass}=\frac{1}{4}g_{2}^{2}v_{T}^{2}W_{\mu\nu}^{+}W_{-}^{\mu\nu}+\frac{1}{8}v_{T}^{2}\left(g_{2}W_{\mu}^{3}-g_{1}B_{\mu}\right)^{2}+\frac{1}{16}v_{T}^{4}C_{HD}\left(g_{2}W_{\mu}^{3}-g_{1}B_{\mu}\right)^{2}$$ The gauge fields have to be redefined, so that the kinetic terms are properly normalized and diagonal, $$G_{\mu}=\mathcal{G}_{\mu}\left(1+C_{HG}v_{T}^{2}\right),\,\, W_{\mu}=\mathcal{W}_{\mu}\left(1+C_{HW}v_{T}^{2}\right),\,\, B_{\mu}=\mathcal{B}_{\mu}\left(1+C_{HB}v_{T}^{2}\right),$$ so that the modified coupling constants become $$\bar{g}_{3}=g_{3}\left(1+C_{HG}v_{T}^{2}\right),\,\,\bar{g}_{2}=g_{2}\left(1+C_{HW}v_{T}^{2}\right),\,\,\bar{g}_{1}=g_{1}\left(1+C_{HB}v_{T}^{2}\right),$$ and the products $g_{1}B_{\mu}=\bar{g}_{1}\mathcal{B}_{\mu}$ etc. are unchanged. Therefore, the electroweak Lagrangian is $$\begin{aligned}
\mathcal{L} & = & -\frac{1}{4}\mathcal{B}_{\mu\nu}\mathcal{B}^{\mu\nu}-\frac{1}{4}\mathcal{W}_{\mu\nu}^{3}\mathcal{W}_{3}^{\mu\nu}-\frac{1}{2}\mathcal{W}_{\mu\nu}^{+}\mathcal{W}_{-}^{\mu\nu}-\frac{1}{2}\left(v_{T}^{2}C_{HWB}\right)\mathcal{W}_{\mu\nu}^{3}\mathcal{B}^{\mu\nu}\nonumber \\
& & +\frac{1}{4}\bar{g}_{2}^{2}v_{T}^{2}\mathcal{W}_{\mu\nu}^{+}\mathcal{W}_{-}^{\mu\nu}+\frac{1}{8}v_{T}^{2}\left(\bar{g}_{2}\mathcal{W}_{\mu}^{3}-\bar{g}_{1}\mathcal{B}_{\mu}\right)^{2}+\frac{1}{16}v_{T}^{4}C_{HD}\left(\bar{g}_{2}\mathcal{W}_{\mu}^{3}-\bar{g}_{1}\mathcal{B}_{\mu}\right)^{2}.\end{aligned}$$ The mass eigenstate basis is given by, [@JenkinsIII eq.5.21], $$\left[\begin{array}{c}
\mathcal{W}_{\mu}^{3}\\
\mathcal{B}_{\mu}
\end{array}\right]=\left[\begin{array}{cc}
1 & -\frac{1}{2}v_{T}^{2}C_{HWB}\\
-\frac{1}{2}v_{T}^{2}C_{HWB} & 1
\end{array}\right]\left[\begin{array}{cc}
\mbox{cos}\bar{\theta} & \mbox{sin}\bar{\theta}\\
\mbox{-sin}\bar{\theta} & \mbox{cos}\bar{\theta}
\end{array}\right]\left[\begin{array}{c}
\mathcal{Z}_{\mu}^{3}\\
\mathcal{A}_{\mu}
\end{array}\right],$$ with $\bar{\theta}$, rotation angle, given by $$\mbox{tan}\bar{\theta}=\frac{\bar{g}_{1}}{\bar{g}_{2}}+\frac{v_{T}^{2}}{2}C_{HWB}\left[1-\frac{\bar{g}_{1}^{2}}{\bar{g}_{2}^{2}}\right]\,.$$ The photon remains massless and the $W$ and $Z$ masses are $$\begin{aligned}
M_{W}^{2} & = & \frac{\bar{g}_{2}^{2}v_{T}^{2}}{4},\nonumber \\
M_{Z}^{2} & = & \frac{(\bar{g}_{1}^{2}+\bar{g}_{2}^{2})v_{T}^{2}}{4}+\frac{1}{8}v_{T}^{4}C_{HD}\left(\bar{g}_{1}^{2}+\bar{g}_{2}^{2}\right)+\frac{1}{2}v_{T}^{4}\bar{g}_{1}\bar{g}_{2}C_{HWB}\label{eq: new gauge masses}\end{aligned}$$ The covariant derivative has the form $$D_{\mu}=\partial_{\mu}+i\frac{\bar{g}_{2}}{\sqrt{2}}\left[\mathcal{W}_{\mu}^{+}T^{+}+\mathcal{W}_{\mu}^{-}T^{-}\right]+ig_{Z}\left[T_{3}-\mbox{sin}\bar{\theta}^{2}Q\right]\mathcal{Z}_{\mu}+i\bar{e}Q\mathcal{A}_{\mu},$$ where $Q=T_{3}+Y$ and the effective couplings become, $$\begin{aligned}
\bar{e} & = & \frac{\bar{g}_{1}\bar{g}_{2}}{\sqrt{\bar{g}_{1}^{2}+\bar{g}_{2}^{2}}}\left[1-\frac{\bar{g}_{1}\bar{g}_{2}}{\bar{g}_{1}^{2}+\bar{g}_{2}^{2}}v_{T}^{2}C_{HWB}\right]=\bar{g}_{2}\mbox{sin}\bar{\theta}-\frac{1}{2}\mbox{cos}\bar{\theta}\bar{g}_{2}v_{T}^{2}C_{HWB}\,,\nonumber \\
\bar{g}_{Z} & = & \sqrt{\bar{g}_{1}^{2}+\bar{g}_{2}^{2}}+\frac{\bar{g}_{1}\bar{g}_{2}}{\sqrt{\bar{g}_{1}^{2}+\bar{g}_{2}^{2}}}v_{T}^{2}C_{HWB}=\frac{\bar{e}}{\mbox{sin}\bar{\theta}\mbox{cos}\bar{\theta}}\left[1+\frac{\bar{g}_{1}^{2}+\bar{g}_{2}^{2}}{2\bar{g}_{1}\bar{g}_{2}}v_{T}^{2}C_{HWB}\right]\,,\nonumber \\
\mbox{sin}\bar{\theta}^{2} & = & \frac{\bar{g}_{1}^{2}}{\bar{g}_{1}^{2}+\bar{g}_{2}^{2}}+\frac{\bar{g}_{1}\bar{g}_{2}\left(\bar{g}_{2}^{2}-\bar{g}_{1}^{2}\right)}{\bar{g}_{1}^{2}+\bar{g}_{2}^{2}}v_{T}^{2}C_{HWB}\,.\label{eq: new effective couplings}\end{aligned}$$ Considering (\[eq: new effective couplings\]) and (\[eq: new gauge masses\]), the experimental values for the $W$ and $Z$ masses and couplings fix $\bar{g}_{1},$$\bar{g}_{2},$$v_{T}$, $C_{HWB}$ and $C_{HD}.$ This procedure consists of solving 5 equations in 5 variables: the unique solution of this system is given by the classical values for $\bar{g}_{1},$$\bar{g}_{2}$ and $v_{T}$, i.e. $$\bar{g}_{1}=0.358,\,\,\bar{g}_{2}=0.651,\,\, v_{T}=246\, GeV$$ while the dim-6 parameters $C_{HWB}$ and $C_{HD}$ have to give negligble corrections to the standard results. This means the products $v_{T}^{2}C_{HWB}$ and $v_{T}^{2}C_{HD}$ have to be, at least, of the order $10^{-3}$, i.e. $C_{HWB,D}\lesssim10^{-7}GeV^{-2}$.
Running of the constants\[RGF\]
===============================
In the following section we run the renormalization group equations, presented in sect. \[RGEs\], to study the modification of the coupling constants behavior, due to the dim-6 operators. We check the possibility, for these new terms, to give a gauge unification point and to return values for the coupling constants compatible with the spectral action predictions.
Renormalization group flow
--------------------------
One can run the equations of the renormalization group in two directions. A “bottom-up” running assumes boundary values for the various constants at low energy (usually the $Z$ or top mass) and runs toward higher energies. This is the way Fig. \[fig:Standard gi\] has been obtained. On the contrary the spectral action is defined at the high energy scale $\Lambda$, and its strength lies in the fact that it specifies the boundary conditions of all constants there. Therefore a “top-down” approach is more natural. In this paper we follow a combined approach.
We start at the scale $\Lambda$ in the range $10^{13-17}$ GeV. At this energy we give the boundary values given by the spectral action. In particular we use for the dimension six terms the values we have calculated and presented in [(\[eq: constrains new\])]{}. The top-down running depends on four other parameters (described below) and gives a set of values for all of physical parameters at low energy. The parameters we find are not too distinct from the experimentally known ones, but there are discrepancies. As it should be: the heat kernel expansion is akin to a one loop calculation and, apart form any other incomplete aspect of the theory, it would be unreasonable to find the correct values for all parameters. The values one finds are however close to the experimental ones for the three $g_i$ and $y_t$, while as remarked earlier $\lambda$, which is the parameter appearing in the Higgs mass, is off by nearly a factor two. The top-down running gives a set of values of the dimension six couplings $C_i$ at $M_Z$ .
We then performed a bottom-up running to see if the presence of the new terms could give a unification point, and we found that in several cases it does. As boundary conditions we used the experimental values for the $g_i$’s and $y_t$ and the low energy values of the $C_i$’s obtained in the top-down running. The case of $\lambda$ deserves a little discussion. Since the experimental and spectral action values are quite different, the qualitative behaviour in the two cases are different. On the other side, it is known that the problem is fixed by the presence of another field ($\sigma$), which we do not discuss in this paper. We have therefore performed our analysis in the two cases, i.e. the value of $\lambda$ obtained by the spectral action, and the experimental one. The strategy we followed is synthesized in Table \[scheme\].
$$\left[\begin{array}{ccc}
\Lambda\ \mbox{scale}\overset{\mbox{\small RGEs}}{\longrightarrow}M_{Z}\,\mbox{scale} & & M_{Z}\,\mbox{scale}\overset{\mbox{\small RGEs}}{\longrightarrow}\Lambda\mbox{scale}\\
\mbox{\small In:}\{\mbox{\small eq.(\ref{eq: constrains new})}\} & & \mbox{\small In:}\{C_{i}(M_{Z}),\, g_{i}^{exp}\}\\
\mbox{\small Out:}\{\mbox{\small}C_{i}(M_{Z})\} & & \mbox{\small Out:}\{C_{i}(\Lambda),g_{i}(\Lambda)\}\\
\underbrace{\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,} & & \underbrace{\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\\
{\color{blue}\mbox{top-down\,\ running}} & & {\color{blue}\mbox{bottom-up\,\ running}}
\end{array}\right]$$
The second case, in which we used the experimental values as initial conditions, can be considered on a purely phenomenological basis, to show that higher dimension operators may cause unifications of the constants at one loop.
Top-Down running
----------------
In the spectral action model we have four free parameters: the value of the gauge coupling constants at the unification, $g$. The value of the cut-off and unification scale, $\Lambda$. The ratio between top and neutrino Yukawa couplings, $\rho$. The momentum $f_{6}$ which will fix the new physics scale $\Gamma$. This last parameter appears as coefficient to the dimension six operators with the combination $f_6/\Lambda^2$, and therefore effectively defines a new energy scale.
All parameters have a particular range in which we expect they could be chosen. From the SM running of the gauge coupling constants we know $g$ is expected around $0.55\pm0.03$, while $\Lambda$ has a more significant range between $10^{13}$ GeV and $10^{17}$ GeV. The ratio $\rho$ between the top and neutrino Yukawa couplings should be expected of $O(1)$. The value of the parameter $f_{6}$ requires a separate discussion. From the internal logic of the spectral action its “natural” value would be of order unity, or not much larger. Such a value would however make the corrections to the running totally irrelevant. The parameter appears with a denominator in $\Lambda^{2}$, and the corrections are often quadratic in this ratio. On the other side, from the phenomenology of electroweak processes it can be expected the effects of these new physics terms on the measured signal strength for $H\rightarrow\mathrm{\gamma\gamma}$ decay, whose measured value is given by ATLAS and CMS [@Atlas; @CMS]. To obtain comparable data the new physics scale has to be fixed around $\Gamma\thicksim1-10$ TeV. This leads to expected values for the dim-6 coefficients $C_{i}$ around $10^{-6}-10^{-8}\mbox{GeV}^{-2}$ . The range for $f_{6}$ will be $\thicksim\Lambda^{2}/\Gamma^{2}$, i.e. $10^{20-28}$. Given the fact that the cutoff function is undetermined in the scheme, such numbers are allowed, although a more physical explanation of their size would be preferable. The spectral action, given by an expansion valid below the unification scale, gives a framework to use a perturbative expansion valid beyond the scale of new physics, although it does not explain it. From the spectral point of view this is a weak point, the presence of such a high value for $f_6$ is very strange and creates an unnatural hierarchy with the other coefficients.
Since the point of the calculation was to verify the possibility of unification, the top-down calculation has been performed with the aim of obtaining values which would be a good starting point for the bottom up calculation. We did search for the best solutions for the range of parameters above. We performed first a coarse search to restrict the range, and then optimized the input parameters to find a good unification point. For the scope of this paper, i.e. to show that dimension six operators could give unification, this is sufficient.
The boundary conditions at $M_Z$ for the subsequent bottom-up run approach are the experimental values for the $g_i$ and $y_t$, and the values obtained from the top-down for the $C_i$’s. In the case of $\lambda$ we have the two choices: either the values obtained from the top down, or the one from experiment. Since these two are different, in the following we present both cases.
### Spectral action value for $\lambda$
In the following table we describe the values of the free parameters we used which will enable the best unification.
Table \[topdownvariation\] shows, for various values of $\Lambda$, the parameters used for the top-down running, and the value of the couplings at low energy, shown as ratio with respect to the experimental value, corrected as described in the previous section: $\gamma_i=\frac{g_i(M_Z)}{\bar g^{exp}_i}$ and $\gamma_t=\frac{y_t(M_Z)}{y_t^{exp}}$. The values for $\lambda$ are not shown since, for the reasons described above, they are not significant.
$\Lambda$ GeV $g(\Lambda)$ $\rho(\Lambda)$ $\frac{f_6}{16\pi^2\Lambda^2}\mbox{Gev}^{-2}$ $ \gamma_1$ $ \gamma_2$ $ \gamma_3$ $ \gamma_t$
--------------- -------------- ----------------- ----------------------------------------------- ------------- ------------- ------------- -------------
$10^{14}$ 0.580 1.6 $4.8~10^{-6}$ $ 1.0 $ $ 1.0 $ $1.0 $ $1.0 $
$10^{15}$ 0.570 1.9 $7.3~10^{-6}$ $0.98 $ $1.0 $ $1.0 $ $1.0$
$10^{16}$ 0.550 1.9 $6.9~10^{-6}$ $0.95 $ $0.99 $ $1.0$ $1.0$
$10^{17}$ 0.540 2.0 $8.3~10^{-6}$ $0.93$ $0.97 $ $1.1 $ $1.0$
: *The values of the coupling constants at $M_Z$ compared with the experimental values for the top-down running. The values of the free parameters are optimized for the subsequent bottom-up run. \[topdownvariation\]*
Note that the choice of parameters has been made to optimize the subsequent bottom-up running. The amount of variations with respect to the experimental values for the couplings could be made smaller with a different choice of $g, f_6$ and $\rho$. This top-down running gives values for the $C_i$’s, which are shown in Table \[topdownCi\].
$\Lambda$ $C_{HWB}$ $C_{W}$ $C_{V}$ $C_{HV}$ $C_{H}$ $C_{HB}$ $C_{HW}$
----------- ---------------- ----------------- ----------------- ----------------- ---------------- ----------------- -----------------
$10^{14}$ $1.1\,10^{-7}$ $-5.8\,10^{-7}$ $-2.7\,10^{-7}$ $-1.1\,10^{-6}$ $3.8\,10^{-8}$ $-1.7\,10^{-7}$ $-7.5\,10^{-7}$
$10^{15}$ $1.4\,10^{-7}$ $-8.1\,10^{-7}$ $-3.3\,10^{-7}$ $-1.4\,10^{-6}$ $5.6\,10^{-8}$ $-2.1\,10^{-7}$ $-9.9\,10^{-7}$
$10^{16}$ $1.2\,10^{-7}$ $-6.7\,10^{-7}$ $-2.6\,10^{-7}$ $-1.3\,10^{-6}$ $4.2\,10^{-8}$ $-1.7\,10^{-7}$ $-8.2\,10^{-7}$
$10^{17}$ $1.3\,10^{-7}$ $-7.4\,10^{-7}$ $-2.5\,10^{-7}$ $-1.4\,10^{-6}$ $4.6\,10^{-8}$ $-1.7\,10^{-7}$ $-8.8\,10^{-7}$
: *The values of the coefficients of the dimension six operators at $M_Z$ . The values of the free parameters are the ones in Table \[topdownvariation\]. All $C_i$’s are in GeV${}^{-2}$.*[]{data-label="topdownCi"}
One can see that with the choice of parameters, mainly $f_6$, the $C_i$’s are in the range expected by a new physics scale of the order of 1 TeV.
### Experimental value for $\lambda$
The values described above are made with parameters which are natural in the framework of the spectral action, but from the phenomenological point of view, since we now have the mass of the Higgs, and therefore the value of $\lambda(M_Z)$, we can also perform the analysis using as boundary condition the experimental value. As in the previous subsection the parameters are chosen in such a way to optimize the subsequent bottom-up run. Tables \[topdownvariationexp\] and \[topdownCiexp\] are the counterparts of \[topdownvariation\] and \[topdownCi\] for the case optimized for unification using as input the experimental value of $\lambda$ at $M_Z$. Of course some principle like the spectral action must be operating in the background, to make sense of the fact that we are running the theory above the scale $\Gamma$ all the way to the unification point.
$\Lambda$ GeV $g(\Lambda)$ $\rho(\Lambda)$ $\frac{f_6}{16\pi^2\Lambda^2}\mbox{Gev}^{-2}$ $ \gamma_1$ $ \gamma_2$ $ \gamma_3$ $ \gamma_t$
--------------- -------------- ----------------- ----------------------------------------------- ------------- ------------- ------------- -------------
$10^{14}$ 0.580 1.1 $1.1~10^{-5}$ $ 0.98 $ $ 0.95 $ $0.80 $ $1.0 $
$10^{15}$ 0.560 0.7 $8.3~10^{-6}$ $0.98 $ $0.96 $ $0.85 $ $1.1$
$10^{16}$ 0.550 1.0 $9.6~10^{-6}$ $0.98 $ $0.96 $ $0.89$ $1.1$
$10^{17}$ 0.540 0.9 $8.3~10^{-6}$ $0.99$ $0.98 $ $0.95 $ $1.2$
: *The values of the coupling constants at $M_Z$ compared with the experimental values for the top-down running. The values of the free parameters are optimized for the subsequent bottom-up run. The initial value of $\lambda(M_Z)$ is the experimental one.*[]{data-label="topdownvariationexp"}
One can see that with respect to the previous case the values of the $\gamma$’s are slightly worse, showing that in this case the result of the top-down running spectral action “predictions” are off. This is not surprising because for the subsequent running (for which these values are optimized) the connections with the spectral action are weaker.
$\Lambda$ $C_{HWB}$ $C_{W}$ $C_{V}$ $C_{HV}$ $C_{H}$ $C_{HB}$ $C_{HW}$
----------- ---------------- ----------------- ----------------- ----------------- ---------------- ----------------- -----------------
$10^{14}$ $2.3\,10^{-7}$ $-1.4\,10^{-6}$ $-7.3\,10^{-7}$ $-2.9\,10^{-6}$ $9.4\,10^{-8}$ $-4.4\,10^{-7}$ $-1.6\,10^{-6}$
$10^{15}$ $1.5\,10^{-7}$ $-9.0\,10^{-7}$ $-4.5\,10^{-7}$ $-2.4\,10^{-6}$ $6.1\,10^{-8}$ $-5.7\,10^{-8}$ $-1.3\,10^{-6}$
$10^{16}$ $1.6\,10^{-7}$ $-9.3\,10^{-7}$ $-4.0\,10^{-7}$ $-2.6\,10^{-6}$ $6.1\,10^{-8}$ $3.7\,10^{-7}$ $-1.1\,10^{-6}$
$10^{17}$ $1.3\,10^{-7}$ $-7.3\,10^{-7}$ $-2.6\,10^{-7}$ $-3.1\,10^{-6}$ $4.9\,10^{-8}$ $8.6\,10^{-7}$ $-8.6\,10^{-7}$
: *The values of the coefficients of the dimension six operators at $M_Z$. The values of the free parameters are the ones in Table \[topdownvariation\]. All $C_i$’s are in GeV${}^{-2}$.*[]{data-label="topdownCiexp"}
One can notice that the values for the couplings in the two cases are not drastically different.
Bottom-up running
-----------------
In this section we present the result of the running from low to high energy, with the parameters chosen to have the three coupling constants meet near a common value in the range $10^{14}-10^{17}$ GeV. As in the previous subsection we first discuss the case in which the boundary condition for $\lambda$ is the one obtained from the running of the spectral action.
### Spectral action value for $\lambda$
A good solution is one for which the common intersection is the starting point for the top-down running, and the $C_i$ come back to the original values given by the spectral action. We optimized our search for the unification, therefore the fact that the values of the $C_i$ “come back” to the same order within a factor of two or so, and are not off by an order a magnitude, is a check. The coefficient $C_{HWB}$ is not present at the $\Lambda$ scale in the spectral action, in this case one should expect it to be smaller than the other. A further check is the value of the top Yukawa at $\Lambda$ which should be close to the value determined by the spectral action. The results for the coupling constants are in Table \[unificationspectral\]. The quantities $\delta g_i(\%)$ indicate (in percent) how different is the value of the runned constants ($g_i^{run})$ with respect to the original spectral action value $g(\Lambda)$ we started with, as shown in Table \[topdownvariation\]. g\_i %= 100 with an analogous definition for $\delta y_t$.
$\Lambda$ $\delta g_1 \%$ $\delta g_2 \%$ $\delta g_3 \%$ $\delta y_t \%$ $\delta\lambda \%$
----------- ----------------- ----------------- ----------------- ----------------- --------------------
$10^{14}$ 1.4 2.1 0.17 0.30 4.1
$10^{15}$ 3.3 0.02 0.54 3.6 4.7
$10^{16}$ 7.8 0.078 0.97 6.4 2.3
$10^{17}$ 13 1.7 1.1 6.6 3.9
: *The percent variation of the values of the three coupling constants and the top Yukawa coupling compared with the initial values of the top-down run.*[]{data-label="unificationspectral"}
One can see that for the smaller values of $\Lambda\simeq 10^{14}-10^{15}$, one finds a good unification point, while for higher values the unification is worse. This can also be seen in Figs. \[figunif14\] and \[figunif17\] for the two extreme cases of $10^{14}$ and $10^{17}$ respectively, compared with the standard model running.
![*Running of the self-interaction parameter $\lambda$(on the rigth) and gauge coupling constants (on the left) in the presence of dimension six operators (thick lines) and their standard behaviour (dashed lines) for $\Lambda=10^{14}$GeV. The values of the parameters are discussed in the text. The red dot indicates the starting value of the parameter. The dashed lines are the values of the $g_i$’s in the standard model.*[]{data-label="figunif14"}](gaugecoupling14.pdf "fig:")![*Running of the self-interaction parameter $\lambda$(on the rigth) and gauge coupling constants (on the left) in the presence of dimension six operators (thick lines) and their standard behaviour (dashed lines) for $\Lambda=10^{14}$GeV. The values of the parameters are discussed in the text. The red dot indicates the starting value of the parameter. The dashed lines are the values of the $g_i$’s in the standard model.*[]{data-label="figunif14"}]("yt14".pdf "fig:")
In the first case there is a good unification, while in the second case the point at which the constants meet is some way off the initial energy.
![*Same as in Fig. \[figunif14\] for $\Lambda=10^{17}$GeV.*[]{data-label="figunif17"}](gaugecoupling17.pdf "fig:")![*Same as in Fig. \[figunif14\] for $\Lambda=10^{17}$GeV.*[]{data-label="figunif17"}]("yt17".pdf "fig:")
The values of the $C_i$’s at the scale $\Lambda$ are usually close to the one we started with in the top-down running, checking the consistency of the model. In particular $C_{HWB}$, which was zero, is constantly about one order of magnitude smaller than the other. We show this in Table \[Ccomparison\] for the two extreme values of $\Lambda$.
$\Lambda=10^{14}$ $C_{HWB}$ $C_{W}$ $C_{V}$ $C_{HV}$ $C_{H}$ $C_{HB}$ $C_{HW}$
------------------- ---------------- ----------------- ----------------- ----------------- ----------------- ----------------- -----------------
[Spec. Act.]{} $ 0 $ $-3.0\,10^{-6}$ $-3.0\,10^{-6}$ $-5.2\,10^{-7}$ $-3.7\,10^{-6}$ $-8.1\,10^{-7}$ $-7.3\,10^{-7}$
[Run]{} $1.3\,10^{-8}$ $-1.5\,10^{-6}$ $-1.6\,10^{-6}$ $-5.5\,10^{-7}$ $-6.8\,10^{-6}$ $-6.7\,10^{-7}$ $-9.2\,10^{-7}$
$\Lambda=10^{17}$ $C_{HWB}$ $C_{W}$ $C_{V}$ $C_{HV}$ $C_{H}$ $C_{HB}$ $C_{HW}$
[Spec. Act.]{} $ 0 $ $-4.2\,10^{-6}$ $-4.2\,10^{-6}$ $-5.4\,10^{-7}$ $6.9\,10^{-6}$ $-1.0\,10^{-6}$ $-9.4\,10^{-7}$
[Run]{} $5.0\,10^{-8}$ $-2.4\,10^{-6}$ $-1.9\,10^{-6}$ $-6.5\,10^{-7}$ $7.5\,10^{-6}$ $-8.7\,10^{-7}$ $-7.2\,10^{-7}$
: *Comparison of the values of the coefficients of the dimension six operators at $\Lambda$. The second and fifth line are the initial values of the top-down running, as predicted by the spectral action for $\Lambda=10^{14}$. The third and the last lines refer to the $10^{17}$ case. All $C_i$’s are in GeV${}^{-2}$.*[]{data-label="Ccomparison"}
Also in this case, the lower value for $\Lambda$ fares slightly better.
### Experimental value for $\lambda$
If one ignores the spectral action, and trusts it only in that it gives some boundary values for the dimension six operator coefficients, then the bottom-up running can be performed independently. In this subsection we present, therefore, the running of the coupling constants using as boundary conditions at $M_Z$ the experimental values for the $g_i$, $y_t$, $\lambda$, (eq.\[eq:InitialCondition\], \[expvalues\]), and the values of Table \[topdownCiexp\] for $C_i$’s and we check if the unification is possible. As we can see from Fig. \[figunif14and17exp\],
![*Gauge couplings unification for two different unification scale $\Lambda=10^{14}$GeV (left) and $\Lambda=10^{17}$GeV (right) if one relaxes the spectral action boundaries.*[]{data-label="figunif14and17exp"}](gaugecoupling14exp.pdf "fig:")![*Gauge couplings unification for two different unification scale $\Lambda=10^{14}$GeV (left) and $\Lambda=10^{17}$GeV (right) if one relaxes the spectral action boundaries.*[]{data-label="figunif14and17exp"}](gaugecoupling17exp.pdf "fig:")
for two different unification scales, the answer is positive if one relaxes the values of the dim-6 coefficients with respect to that suggested by the spectral action. In fact, in this case, the value of the $\gamma$’s are slightly different from 1, as shown in table \[topdownvariationexp\], but these allow to correct the unification point within an error of 1%, as summarized in Table \[unificationexp\].
$\Lambda(GeV)$ $\delta g_1 \%$ $\delta g_2 \%$ $\delta g_3 \%$
---------------- ----------------- ----------------- ----------------- -- --
$10^{14}$ 0.62 0.74 1.0
$10^{15}$ 1.4 0.38 0.56
$10^{16}$ 1.2 0.50 0.50
$10^{17}$ 0.14 0.98 1.1
: *The percent variation of the values of the there coupling constants compared with the initial value of the unification point.*[]{data-label="unificationexp"}
Conclusions and Outlook
=======================
In this paper we have calculated the sixth order terms appearing in the spectral action Lagrangian. We have then verified that the presence of these terms, with a proper choice of the free parameters, could cause the unification of the three constants at a high energy scale. Although the motivation for this investigation lies in the spectral noncommutative geometry approach to the standard model, the result can be read independently on it, showing that if the current Lagrangian describes an effective theory valid below the unification point, then the dimension six operator would play the proper role of facilitating the unification. In order for the new terms to have an effect it is however necessary to introduce a scale of the order of the TeV, which for the spectral action results in a very large second momentum of the cutoff function.
We note that we did not require a modification of the standard model spectral triple, although such a modification, and in particular the presence of the scale field $\sigma$, could actually improve the analysis. From the spectral action point of view the next challenge is to include the ideas currently come form the extensions of the standard model currently being investigated. From the purely phenomenological side instead a further analysis of the effects of the dimension six operators for phenomenaology at large, using the parameters suggested by this paper, can be a useful pointer to new physics.
[**Acknowledgements**]{} We thank Giancarlo D’Ambrosio and Giulia Ricciardi for discussions. DV was supported in parts by FAPESP, CNPq and by the INFN through the Fondi FAI Guppo IV [*Mirella Russo*]{} 2013. F.L. is partially supported by CUR Generalitat de Catalunya under projects FPA2013-46570 and 2014 SGR 104. A.D. and F.L. were partially supported by UniNA and Compagnia di San Paolo under the grant Programma STAR 2013. C.V.F. is supported by FAPESP.
Spectral geometry \[appspectralaction\]
=======================================
Noncommutative geometry is a way to describe noncommutative as well as commutative space on equal footing. Being quite general, this approach happens to be sufficiently rigid to make predictions about the standard model. Noncommutative geometry uses many tools of spectral geometry.
Generally, the geometry of a noncommutative space is defined through a *Spectral Triple* $\left(\mathcal{A},\mathcal{H},D\right)$ consisting of an algebra $\mathcal{A}$, a Hilbert space $\mathcal{H}$ and a Dirac operator $D$.
The algebra $\mathcal{A}$ should be thought of as a generalization of the algebra of functions to the case when the underlying space is possibly noncommutative. The noncommutative algebra $\mathcal{A}$ relevant for the standard model is the product of the ordinary algebra of functions on $\mathbb{R}^{4}$ times a finite dimensional matrix algebra $\mathcal{A}_{sm}=\mathbb{C}\oplus{\mathbb{H}}\oplus M_{3}(\mathbb{C})$. Here $\mathbb{H}$ is the algebra of quaternions, $M_{3}(\mathbb{C})$ is the algebra of complex $3\times 3$ matrices. $\mathcal{A}_{sm}$ may be interpreted as an algebra of functions on a finite “internal” space. Since just the internal part $\mathcal{A}_{sm}$ is noncommutative, the geometry corresponding to Standard Model is called *almost commutative*. The gauge group is the group of automorphisms of $\mathcal{A}_{sm}$.
The algebra $\mathcal{A}$ acts on the Hilbert space $\mathcal{H}$. We take $\mathcal{H}=L^{2}(\mathrm{sp}(\mathbb{R}^{4}))\otimes\mathcal{H}_{F}$ being a tensor product of the space of square-integrable spinors and a finite-dimensional space $\mathcal{H}_F$. The algebra $\mathcal{A}_{sm}$ acts on $\mathcal{H}_F$, and this imposes severe restrictions on possible representations of the gauge group. Remarkably, these restrictions are satisfied by the Standard Model fermions. To incorporate all of these fermions one takes $\mathcal{H}_F=\mathcal{H}_R\oplus\mathcal{H}_L\oplus\mathcal{H}^c_R\oplus\mathcal{H}^c_L\oplus$, being $\mathcal{H}_R=\mathbb{C}^{24}$ ($8$ fermions with $3$ generations) the space of the right fermion, $\mathcal{H}_L=\mathbb{C}^{24}$ the space of the left fermions and the super index $c$ denotes their respectives antifermions. This give us the total of the $96$ SM degrees of freedom.
The Dirac operator $D$ also has to satisfy some consistency requirements, that all are respected in the Standard Model. These conditions =0,\[a,JbJ\^[-1]{}\]=0,\[\[D,a\],JbJ\^[-1]{}\]=0 include the chirality operator $\gamma$ and the real structure $J$. Specifically, for the Standard Model the Dirac operator has the form $$\begin{aligned}
D = D_M \otimes 1_{96} + \gamma^5 \otimes D_F,
\label{Diracoperator}\end{aligned}$$ where $D_M=\gamma^\mu (\partial_\mu + \omega_\mu)$ is the canonical Dirac operator and the chirality and real structure are $\gamma=\gamma^5\otimes\gamma_F$, $J=\mathcal{J}\otimes J_F$, with $\mathcal{J}$ being the charge conjugation. The finite dimensional Dirac operator $D_F$ is a matrix including the Yukawa couplings of leptons, Dirac and Majorana neutrinos. To introduce he gauge fields and the Higgs we replace $D$ by D\_A = D + + JJ\^[-1]{}, where $\mathbb{A}=\sum a[D,b]$, with $a,b\in \mathcal{A}$. In contrast to the usual Standard Model, the Noncommutative Standard Model includes a singlet scalar field $\sigma$. Roughly speaking, this field is a result of “fluctuating” Majorana mass term of the Dirac operator and is responsible for adjusting the Higgs’ mass of noncommutative spectral action to the experimental values. Some other approaches to the Higgs mass problem in spectral action can be found in [@Stephan; @coldplay; @CCvS].
Following Chamseddine and Connes tensorial notation[^1] [@NCPart1], an element $\Psi_M$ of the Hilbert space $\mathcal{H}$ is denoted as $$\begin{aligned}
\Psi_M = \left(
\begin{array}{c}
\psi_{\alpha I} \\
\psi_{\alpha' I'} \end{array}
\right).
\label{notation}\end{aligned}$$ First, the primed indices denote the conjugate spinor, this is $\psi_{\alpha' I'} = \psi^c_{\alpha I}$. The index $\alpha$ acts on $\mathbb{C}\oplus\mathbb{H}$ and is decomposed as $\alpha=\dot{a},a$ where $\dot{a}=\dot{1},\dot{2}$ acts on $\mathbb{C}$ and $a=1,2$ acts on $\mathbb{H}$. The index $I$ acts on $\mathbb{C}\oplus M_3(\mathbb{C})$ and is decomposed as $I=1,i\,$: the index $1$ acts on $\mathbb{C}$, that is another copy of the algebra of complex numbers, and $i$ acts on $M_3(\mathbb{C})$.
Next, let us consider the action principle. One cannot construct too many invariants by using the spectral triple data. One obvious choice is the ordinary fermionic action $$S_{F}=\langle \psi, D_{A}\psi\rangle \,.
\label{SF}$$ As well, one can use the operator trace ${\rm Tr}$ in $\mathcal{H}$ to construct invariants from the Dirac operator alone. In this way one obtains the *Spectral action* $$S_{\Lambda}\left(D\right)={\rm Tr}\, \left[ f \left(\frac{D^{2}}{\Lambda^{2}}\right)\right],\label{SLambda}$$ where $f$ is a function restricted only by the requirement that trace in (\[SLambda\]) exists. $f$ is usually called the cutoff function since it has to regularize (\[SLambda\]) at large eigenvalues of $D$. $\Lambda$ is a cutoff scale.
One can use the heat kernel expansion[^2] $${\rm Tr}\, \left[ e^{-tD^2}\right] \simeq \sum_{p=0}^\infty
t^{-2+p} a_{2p}\bigl( D^2\bigr)\,,\qquad t\to+0,\label{heatex}$$ to find a large $\Lambda$ expansion of the Spectral Action. Suppose that $f$ is a Laplace transform, $$f(z)=\int_0^\infty dt\,e^{-tz} \tilde f(t)\,.\label{Lap}$$ Then $$S_{\Lambda}\left(D\right)\sim\sum_{p=0}\Lambda^{4-2p}f_{2p}a_{2p}\left(D^{2}\right), \label{asymp}$$ where $$f_{2p}=\int_{0}^{\infty}dt\, t^{-2+p}\tilde f\left(t\right).$$ Note, that we have restricted ourselves to four dimensions where the first four terms of the asymptotic expansion are $$S_{\Lambda}\left(D\right)\sim\Lambda^{4}f_{0}a_{0}\left(D^{2}\right)+\Lambda^{2}f_{2}a_{2}\left(D^{2}\right)+f_{4}a_{4}\left(D^{2}\right)+\frac{1}{\Lambda^{2}}f_{6}a_{6}\left(D^{2}\right)+\dots
\label{Sexp}$$ $D^2$ is an operator of Laplace type. It can be represented as $$D^2=-(\nabla^2+E)\,,\label{Laptype}$$ where $\nabla$ is a covariant derivative, $\nabla_\mu=\partial_\mu +\omega_\mu$, $E$ is a zeroth order term. Denoting by $\rm tr$ the usual matrix trace one may write $$\begin{aligned}
a_{0} & = & \frac{1}{16\pi^{2}}\int d^{4}x\sqrt{g}{\rm tr}\, \left(1\right),\nonumber\\
a_{2} & = & \frac{1}{16\pi^{2}}\frac{1}{6}\int d^{4}x\sqrt{g}{\rm tr}\,\left[6E+R\cdot1\right],\nonumber\\
a_{4} & = & \frac{1}{16\pi^{2}}\frac{1}{360}\int d^{4}x\sqrt{g}
{\rm tr}\,
\left[\left(12R;_{\mu}^{\mu}+5R^{2}-2R_{\mu\nu}R^{\mu\nu}+2R_{\mu\nu\varrho\sigma}R^{\mu\nu\varrho\sigma}\right)\cdot1\right]\\
& + & \frac{1}{16\pi^{2}}\frac{1}{360}\int d^{4}x\sqrt{g}{\rm tr}\,
\left[60E;_{\mu}^{\mu}+60ER+180E^{2}+30\Omega_{\mu\nu}\Omega^{\mu\nu}\right].\end{aligned}$$ Here $R_{\mu\nu\rho\sigma}$, $R_{\mu\nu}$ and $R$ are the Riemann tensor, the Ricci tensor and the curvature scalar, respectively. The semicolon denotes covariant derivatives, and $\Omega_{\mu\nu}=\partial_\mu \omega_\nu - \partial_\nu \omega_\mu +[\omega_\mu,\omega_\nu]$.
The expression for $a_{6}$ is rather long (see [@Vassilevich]), but it simplifies if one considers a flat-space time $$a_{6}^{flat}=\frac{1}{16\pi^{2}}\int d^{4}x\left(\Sigma_{\Omega}+\Sigma_{E}+\Sigma_{E\Omega}\right),\label{S01}$$ where $$\begin{aligned}
\Sigma_{\Omega} & = & {\rm tr}\,\left[-\frac{1}{90}\Omega_{\mu\nu;\tau}\Omega^{\mu\nu;\tau}+\frac{1}{180}\Omega_{\;\;\;\;;\nu}^{\mu\nu}\Omega_{\mu\rho}^{\;\;\;\;;\rho}
-\frac{1}{30}\Omega_{\mu\nu}\Omega^{\nu\tau}\Omega_{\tau}^{\;\;\mu}\right],\label{S02}\\
\Sigma_{E\Omega} & = & {\rm tr}\,\left[\frac{1}{12}E\Omega_{\mu\nu}\Omega^{\mu\nu}\right],\label{S03}\\
\Sigma_{E} & = & {\rm tr}\,\left[-\frac{1}{12}E^{;\mu}E_{;\mu}+\frac{1}{6}E^{3}\right],\label{S04}\end{aligned}$$
Using the notation introduced in (\[notation\]), the matrix elements of the connection $\omega_\mu$ are given by $$\begin{aligned}
\left(\omega_{\mu}\right)_{\dot{1}1}^{\dot{1}1} & = & 0,\nonumber\\
\left(\omega_{\mu}\right)_{\dot{2}1}^{\dot{2}1} & = & ig_{1}B_{\mu}\otimes1_{4}\otimes1_{3},\nonumber\\
\left(\omega_{\mu}\right)_{b1}^{a1} & = & i \left(\frac{g_{1}}{2}B_{\mu}\delta_{b}^{a}-\frac{g_{2}}{2}W_{\mu}^{\tau}\left(\sigma^{\tau}\right)_{b}^{a}\right)\otimes1_{4}\otimes1_{3},\nonumber\\
\left(\omega_{\mu}\right)_{\dot{1}j}^{\dot{1}i} & = & i \left(-\frac{2g_{1}}{3}B_{\mu}\delta_{j}^{i}-\frac{g_{3}}{2}V_{\mu}^{m}\left(\lambda^{m}\right)_{j}^{i}\right)\otimes1_{4}\otimes1_{3},\nonumber\\
\left(\omega_{\mu}\right)_{\dot{2}j}^{\dot{2}i} & = & i \left(\frac{g_{1}}{3}B_{\mu}\delta_{j}^{i}-\frac{g_{3}}{2}V_{\mu}^{m}\left(\lambda^{m}\right)_{j}^{i}\right)\otimes1_{4}\otimes1_{3},\nonumber\\
\left(\omega_{\mu}\right)_{bj}^{ai} & = & i \left(-\frac{g_{1}}{6}B_{\mu}\delta_{b}^{a}\delta_{j}^{i}-\frac{g_{2}}{2}W_{\mu}^{\tau}\left(\sigma^{\tau}\right)_{b}^{a}
\delta_{j}^{i}-\frac{g_{3}}{2}V_{\mu}^{m}\left(\lambda^{m}\right)_{j}^{i}\delta_{b}^{a}\right)\otimes1_{4}\otimes1_{3}.\end{aligned}$$ Therefore, the components of the curvature $\Omega_{\mu\nu}$ are $$\begin{aligned}
\left(\Omega_{\mu\nu}\right)_{\dot{1}1}^{\dot{1}1} & = & 0,\nonumber\\
\left(\Omega_{\mu\nu}\right)_{\dot{2}1}^{\dot{2}1} & = & i g_{1}B_{\mu\nu}\otimes1_{4}\otimes1_{3},\nonumber\\
\left(\Omega_{\mu\nu}\right)_{b1}^{a1} & = & i \left(\frac{g_{1}}{2}B_{\mu\nu}\delta_{b}^{a}-\frac{g_{2}}{2}W_{\mu\nu}^{\tau}\left(\sigma^{\tau}\right)_{b}^{a}\right)\otimes1_{4}\otimes1_{3},\nonumber\\
\left(\Omega_{\mu\nu}\right)_{\dot{1}j}^{\dot{1}i} & = & i \left(-\frac{2g_{1}}{3}B_{\mu\nu}\delta_{j}^{i}-\frac{g_{3}}{2}V_{\mu\nu}^{m}\left(\lambda^{m}\right)_{j}^{i}\right)\otimes1_{4}\otimes1_{3},\nonumber\\
\left(\Omega_{\mu\nu}\right)_{\dot{2}j}^{\dot{2}i} & = & i \left(\frac{g_{1}}{3}B_{\mu\nu}\delta_{j}^{i}-\frac{g_{3}}{2}V_{\mu\nu}^{m}\left(\lambda^{m}\right)_{j}^{i}\right)\otimes1_{4}\otimes1_{3},\nonumber\\
\left(\Omega_{\mu\nu}\right)_{bj}^{ai} & = & i \left(-\frac{g_{1}}{6}B_{\mu\nu}\delta_{b}^{a}\delta_{j}^{i}-\frac{g_{2}}{2}W_{\mu\nu}^{\tau}\left(\sigma^{\tau}\right)_{b}^{a}\delta_{j}^{i}
-\frac{g_{3}}{2}V_{\mu\nu}^{m}\left(\lambda^{m}\right)_{j}^{i}\delta_{b}^{a}\right)\otimes1_{4}\otimes1_{3}.\end{aligned}$$ The quantity $E$, which is defined through the operator $D^2$, has diagonal and non-diagonal components. The diagonal terms, i.e., $E_{\dot{1}1}^{\dot{1}1},E_{\dot{2}1}^{\dot{2}1},E_{b1}^{a1}$ and $E_{\dot{1}j}^{\dot{1}i},E_{\dot{2}j}^{\dot{2}i},E_{bj}^{ai}$ have the form $E^{\rm diag}=-\frac{1}{2}\gamma^{\mu\nu}\Omega_{\mu\nu}-U\otimes1_{4}$, where $$\begin{aligned}
U_{\dot{1}1}^{\dot{1}1} & = & \left(\left|y_{\nu}\right|^{2}\overline{H}H+\left|y_{\nu_{R}}\right|^{2}\sigma^{2}\right),\nonumber\\
U_{\dot{2}1}^{\dot{2}1} & = & \left(\left|y_{e}\right|^{2}\overline{H}H\right),\\
U_{b1}^{a1} & = & \left(\left|y_{e}\right|^{2}H_{a}\overline{H}^{b}+\left|y_{\nu}\right|^{2}\epsilon_{bc}\epsilon^{ad}\overline{H}^{c}H_{d}\right),\nonumber\\
U_{\dot{1}j}^{\dot{1}i} & = & \left(\left|y_{t}\right|^{2}\overline{H}H\right)\delta_{j}^{i},\nonumber\\
U_{\dot{2j}}^{\dot{2}i} & = & \left(\left|y_{d}\right|^{2}\overline{H}H\right)\delta_{j}^{i},\nonumber\\
U_{bj}^{ai} & = & \left(\left|y_{t}\right|^{2}H_{a}\overline{H}^{b}+\left|y_{d}\right|^{2}\epsilon_{bc}\epsilon^{ad}\overline{H}^{c}H_{d}\right)\delta_{j}^{i}.\end{aligned}$$ and the non-diagonal components are $$\begin{aligned}
E^{a1}_{\dot{1}1} & = & -\gamma^\mu \gamma_5 \otimes y^*_{\nu}\otimes \epsilon^{ab}\nabla_\mu H_b,\nonumber\\
E^{a1}_{\dot{2}1} & = & -\gamma^\mu \gamma_5 \otimes y^*_{e}\otimes \nabla_\mu \overline{H}^a,\nonumber\\
E^{\dot{1}1}_{a1} & = & -\gamma^\mu \gamma_5 \otimes y_{\nu}\otimes \epsilon_{ab}\nabla_\mu \overline{H}^b,\nonumber\\
E^{\dot{2}1}_{a1} & = & -\gamma^\mu \gamma_5 \otimes y_{e}\otimes \nabla_\mu H_a,\nonumber\\
E^{aj}_{\dot{1}i} & = & -\gamma^\mu \gamma_5 \otimes y^*_{t}\otimes \epsilon^{ab}\nabla_\mu H_b \delta^j_i,\nonumber\\
E^{aj}_{\dot{2}i} & = & -\gamma^\mu \gamma_5 \otimes y^*_{d}\otimes \nabla_\mu \overline{H}^a \delta^j_i,\nonumber\\
E^{\dot{1}j}_{ai} & = & -\gamma^\mu \gamma_5 \otimes y_{t}\otimes \epsilon_{ab}\nabla_\mu \overline{H}^b \delta^j_i,\nonumber\\
E^{\dot{2}j}_{ai} & = & -\gamma^\mu \gamma_5 \otimes y_{d}\otimes \nabla_\mu H_a \delta^j_i,\end{aligned}$$ and the non-diagonal primed components $$\begin{aligned}
E^{a'1'}_{\dot{1}1} & = & - y^*_{\nu_R} \overline{y^*_{\nu_R}}\otimes \epsilon^{ab} \overline{H}_b \sigma,\nonumber\\
E^{\dot{1}1}_{a'1'} & = & - y_{\nu_R} \overline{y^*_{\nu}}\otimes \epsilon_{ab} H^b \sigma,\nonumber\\
E^{\dot{1}'1'}_{a1} & = & - y^*_{\nu_R} y_{\nu} \otimes \epsilon_{ab} \overline{H}^b \sigma,\nonumber\\
E^{a1}_{\dot{1}'1'} & = & - y_{\nu_R} y^*_{\nu_R}\otimes \epsilon^{ab} H_b \sigma,\nonumber\\
E^{\dot{1}'1'}_{\dot{1}1} & = & -\gamma^\mu \gamma_5 \otimes y^*_{\nu_R}\otimes \partial_\mu \sigma,\nonumber\\
E^{\dot{1}1}_{\dot{1}'1'} & = & -\gamma^\mu \gamma_5 \otimes y_{\nu_R}\otimes \partial_\mu \sigma.\end{aligned}$$
Taking just the contributions from $a_0$, $a_2$ and $a_4$ to the expansion (\[Sexp\]) one reproduces quite well bosonic part of the Standard Model action, modulo the problem with the Higgs mass and with the unification point that we have already mentioned above $$\begin{aligned}
S &=& \int d^4x \left[- \frac{2}{\pi^2} f_2 \Lambda^2\left(\frac{1}{2}a\overline{H}H+\frac{1}{4}\left|y_{\nu_R}\right|^{2}\sigma^2\right)
+\frac{1}{2\pi^2}f_4 \left(\frac{5}{3}g^2_1B^2_{\mu\nu}+g^2_2W^2_{\mu\nu}+g^2_3V^2_{\mu\nu}+a(\nabla_\nu H)^2\right)\right]\nonumber\\
&+&\int d^4x \frac{1}{2\pi^2}f_4 \left[ b(\overline{H}H)^2 +2\left|y_{\nu}\right|^{2}\left|y_{\nu_R}\right|^{2}(\overline{H}H)\sigma^2+\frac{1}{2}\left|y_{\nu_R}\right|^{4}\sigma^4
+ \frac{1}{2}\left|y_{\nu_R}\right|^{2}(\partial_\mu \sigma)^2\right],\label{C01}\end{aligned}$$ where $$\begin{aligned}
a &=& \left|y_{\nu}\right|^{2}+\left|y_{e}\right|^{2}+3(\left|y_{t}\right|^{2}+\left|y_{d}\right|^{2}),\nonumber\\
b &=& \left|y_{\nu}\right|^{4}+\left|y_{e}\right|^{4}+3(\left|y_{t}\right|^{2}+\left|y_{d}\right|^{2})^2.\end{aligned}$$
Higher order terms have been given considerably less attention. The papers [@WvS1; @WvS2] studied the influence of higher order terms on renormalizablity of Yang-Mills spectral actions, while the works [@ILV1; @ILV2; @Kuliva] studied the spectral action beyond the asymptotic expansion (\[asymp\]).
The term $\Sigma_{\Omega}$ contains contributions from the gauge fields only $$\begin{aligned}
\Sigma_{\Omega} & = & 8\left[\frac{1}{9}g_{1}^{2}B_{\mu\nu;\tau}^{2}+\frac{1}{15}g_{2}^{2}W_{\mu\nu;\tau}^{2}+\frac{1}{15}g_{3}^{2}V_{\mu\nu;\tau}^{2}\right]\nonumber \\
& + & 8\left[-\frac{1}{18}g_{1}^{2}B_{\mu\nu;\nu}^{2}-\frac{1}{30}g_{2}^{2}W_{\mu\nu;\nu}^{2}-\frac{1}{30}g_{3}^{2}V_{\mu\nu;\nu}^{2}\right]\nonumber \\
& + & 8\left[\frac{1}{10}g_{2}^{3}\varepsilon^{\delta\eta\kappa}W_{\mu\nu}^{\delta}W_{\nu\tau}^{\eta}W_{\tau\mu}^{\kappa}
+\frac{1}{10}g_{3}^{3}f^{mnr}V_{\mu\nu}^{m}V_{\nu\tau}^{n}V_{\tau\mu}^{r}\right].\label{C02}\end{aligned}$$ where: F\_[;]{} = \_F\_+ ig, is the usual covariant derivative for gauge fields.
Note the presence of dimension six operators $X^{3}$ : $g_{2}^{3}\varepsilon^{\delta\eta\kappa}W_{\mu\nu}^{\delta}W_{\nu\tau}^{\eta}W_{\tau\mu}^{\kappa}$ and $g_{3}^{3}f^{mnr}V_{\mu\nu}^{m}V_{\nu\tau}^{n}V_{\tau\mu}^{r}$. Also note that the kinetic gauge terms, i.e, $F_{\mu\nu;\tau}^{2}$ and $F_{\mu\nu;\nu}^{2}$, are dimension six operators $X^2D^2$. (We remind that $X$ is any field strength and by $D$ we indicate the fact that there are two derivatives). Since under the trace and integral F\_[;]{}\^[2]{}=2F\_[;]{}\^[2]{}-4igF\_F\_F\_\[C02a\], these operators are not independent.
The term $\Sigma_{E\Omega}$ contains the operators $X^{2}H^{2}$: $$\begin{aligned}
\Sigma_{E\Omega} & = & 8\left[\frac{g_{1}^{2}}{144}\left(15\left|y_{e}\right|^{2}+3\left|y_{\nu}\right|^{2}+17\left|y_{t}\right|^{2}+5\left|y_{d}\right|^{2}\right)
B_{\mu\nu}B_{\mu\nu}\left(\overline{H}H\right)\right]\nonumber \\
& + & 8\left[\frac{g_{2}^{2}}{48}\left(3\left|y_{t}\right|^{2}+3\left|y_{d}\right|^{2}+\left|y_{e}\right|^{2}+\left|y_{\nu}\right|^{2}\right)
W_{\mu\nu}^{\eta}W_{\mu\nu}^{\eta}\left(H\overline{H}\right)\right]\nonumber \\
& + & 8\left[\frac{g_{3}^{2}}{12}\left(\left|y_{t}\right|^{2}+\left|y_{d}\right|^{2}\right)V_{\mu\nu}^{m}V_{\mu\nu}^{m}\left(H\overline{H}\right)\right].\label{C03}\end{aligned}$$
For the term $\Sigma_{E}$, let us write $\Sigma_{E}=\Sigma^0_{E}+\Sigma^{kin}_{E}+\Sigma^\sigma_{E}$, where $$\begin{aligned}
\Sigma^0_{E}
&=&-8\left[\frac{1}{48}g_{1}^{2}\left(15\left|y_{e}\right|^{2}+3\left|y_{\nu}\right|^{2}+5\left|y_{d}\right|^{2}+17\left|y_{t}\right|^{2}\right)
H\overline{H}B_{\mu\nu}B^{\mu\nu}\right]\nonumber\\
&&-8\left[\frac{3}{48}g_{2}^{2}\left(\left|y_{e}\right|^{2}+\left|y_{\nu}\right|^{2}+3\left|y_{t}\right|^{2}+3\left|y_{d}\right|^{2}\right)
H\overline{H}W_{\mu\nu}^{\delta}W^{\delta\mu\nu}\right]\nonumber\\
&&-8\left[\frac{3}{12}g_{3}^{2}\left(\left|y_{t}\right|^{2}+\left|y_{d}\right|^{2}\right)H\overline{H}V_{\mu\nu}^{m}V^{m\mu\nu}\right]\nonumber \\
&&-8\left[\frac{1}{3}\left(\left|y_{e}\right|^{6}+\left|y_{\nu}\right|^{6}+3\left|y_{t}\right|^{6}+3\left|y_{d}\right|^{6}\right)
\left(\overline{H}H\right)^{3}\right] \nonumber\\
&&-8\left[\frac{1}{2}g_{2}^{3}\varepsilon^{\delta\eta\lambda}W_{\mu\nu}^{\delta}W_{\nu\tau}^{\eta}W_{\tau\mu}^{\lambda}
+\frac{1}{2}g_{3}^{3}f^{mnr}V_{\mu\nu}^{m}V_{\nu\tau}^{r}V_{\tau\mu}^{r}\right]\nonumber\\
&&-8\left[\frac{5}{12}g_{1}^{2}B_{\mu\nu;\tau}^{2}+\frac{3}{12}g_{2}^{2}W_{\mu\nu;\tau}^{2}+\frac{3}{12}g_{3}^{2}V_{\mu\nu;\tau}^{2}\right].
\label{C04}\end{aligned}$$ is a contribution containing the operators: $X^2H^2$, $X^3$, $H^6$ and $X^2D^2$. The expression $\Sigma^{kin}_{E}$ contains higher order kinetic terms for the Higgs field, these are $H^4D^2$ and $H^2D^4$ operators: $$\begin{aligned}
\Sigma^{kin}_E &=&-8\left[\frac{1}{2}\left(\left(\left|y_{\nu}\right|^{2}+\left|y_{e}\right|^{2}\right)^{2}+6\left|y_{t}\right|^{4}+6\left|y_{d}\right|^{4}\right)
\overline{H}H\left|\nabla_{\tau}H_{a}\right|^{2}\right]\nonumber\\
&&-8\left[\frac{1}{2}\left(\left(\left|y_{\nu}\right|^{2}-\left|y_{e}\right|^{2}\right)^{2}-3\left(\left|y_{t}\right|^{2}-\left|y_{d}\right|^{2}\right)^{2}\right)
\left|\overline{H}^{a}\nabla_{\tau}H_{a}\right|^{2}\right]\nonumber\\
&&-8\left[\frac{1}{12}\left(\left(\left|y_{e}\right|^{2}+\left|y_{\nu}\right|^{2}\right)^{2}+3\left(\left|y_{t}\right|^{2}+\left|y_{d}\right|^{2}\right)^{2}\right)
\left|\partial_{\tau}\left(\overline{H}H\right)\right|^{2}\right]\nonumber\\
&&-8\left[\frac{1}{6}\left(\left|y_{\nu}\right|^{2}+\left|y_{e}\right|^{2}+3\left|y_{t}\right|^{2}
+3\left|y_{d}\right|^{2}\right)\left(\partial_{\tau}\nabla_{\mu}\overline{H}^{a}\right)\left(\partial_{\tau}\nabla_{\nu}H_{a}\right)\right]\nonumber\\
&&-8\left[\frac{1}{12}\left(\left(\left|y_{e}\right|^{2}-\left|y_{\nu}\right|^{2}\right)^{2}
+3\left(\left|y_{t}\right|^{2}-\left|y_{d}\right|^{2}\right)^{2}\right)\left|\partial_{\tau}\left(H_{a}\overline{H}^{b}\right)\right|^{2}\right],
\label{C05}\end{aligned}$$ and $$\begin{aligned}
\Sigma^\sigma_E & = & -8\left[\frac{1}{2}\left|y_{\nu}\right|^{4}\left|y_{\nu_{R}}\right|^{2}\left(\overline{H}H\right)^{2}\sigma^{2}+
\frac{1}{2}\left|y_{\nu}\right|^{2}\left|y_{\nu_{R}}\right|^{4}\left(\overline{H}H\right)\sigma^{4}+\left|y_{\nu_{R}}\right|^{6}\sigma^{6}\right]\nonumber\\ &&-8\left[\frac{1}{6}\left|y_{\nu}\right|^{2}\left|y_{\nu_{R}}\right|^{2}\partial_{\tau}\left(\overline{H}^{a}\sigma\right)\partial_{\tau}
\left(H_{a}\sigma\right)+\frac{1}{12}\left|y_{\nu_{R}}\right|^{2}\left(\partial_{\mu}\partial_{\nu}\sigma\right)^{2}\right]\nonumber\\
&&-8\left[\frac{1}{2}\left|y_{\nu_{R}}\right|^{2}\left|y_{\nu}\right|^{2}\sigma^{2}\left|\nabla_{\tau}H_{a}\right|^{2}
+\frac{1}{2}\left|y_{\nu}\right|^{2}\left|y_{\nu_{R}}\right|^{2}\left(\sigma\partial^{\tau}\sigma\right)\nabla_{\tau}\left(\overline{H}H\right)\right]\nonumber\\ &&-8\left[\frac{1}{2}\left|y_{\nu}\right|^{2}\left|y_{\nu_{R}}\right|^{2}\left(\overline{H}H\right)\left(\partial_{\tau}\sigma\right)^{2}+
\frac{1}{2}\left|y_{\nu_{R}}\right|^{4}\left(\sigma\partial_{\tau}\sigma\right)^{2}\right]\nonumber\\
&&-8\left[\left|y_{\nu}\right|^{4}\left|y_{\nu_{R}}\right|^{2}\sigma^{2}\left(\overline{H}H\right)^{2}
+\frac{1}{2}\left|y_{\nu_{R}}\right|^{4} \left|y_{\nu}\right|^{2}\overline{H}H\sigma^{4}\right]\nonumber\\
&&-8\left[\frac{1}{12}\left|y_{\nu_{R}}\right|^{4}\left(\partial_{\tau}\sigma^{2}\right)^{2}+\frac{1}{6}\left|y_{\nu}\right|^{2}
\left|y_{\nu_{R}}\right|^{2}\partial_{\tau}\left(\overline{H}H\right)\left(\partial_{\tau}\sigma^{2}\right)\right],\label{C06}\end{aligned}$$ is the contribution of the $\sigma$ singlet scalar field. As shown in (\[C01\]), this field already appears in the $a_ 4$ coefficient. As has been explained in the main text, we do not consider this field in the present work and, therefore, discard corresponding contributions to the action.
We consider that $y_{t}$, $y_\nu$ and $y_{\nu_R}$ are dominant and also define the variable $\rho$ as the ratio of the Dirac Yukawa couplings $y_{\nu}= \rho y_{t}$. Under this approximation and replacing (\[C02\],\[C03\],\[C04\],\[C05\]) in (\[S01\]) we have $$\begin{aligned}
a_{6}^{flat} &=& \frac{1}{2\pi^{2}}\int d^{4}x\left(\mathcal{O}_{X^2H^2}+\mathcal{O}_{H^6}+\mathcal{O}_{X^3}
+\mathcal{O}_{X^2D^2}+\mathcal{O}_{Kin}\right),\label{C07}\end{aligned}$$ where $$\begin{aligned}
\mathcal{O}_{X^{2}H^{2}} & = & -\frac{g_{1}^{2}}{72}\left(3\rho^{2}+17\right)\left|y_{t}\right|^{2}
B_{\mu\nu}B_{\mu\nu}\left(\overline{H}H\right)-\frac{g_{2}^{2}}{24}\left(3+\rho^{2}\right)\left|y_{t}\right|^{2}
W_{\mu\nu}^{\eta}W_{\mu\nu}^{\eta}\left(H\overline{H}\right)\nonumber \\
&&-\frac{g_{3}^{2}}{6}\left|y_{t}\right|^{2}V_{\mu\nu}^{m}V_{\mu\nu}^{m}\left(H\overline{H}\right).\label{C08}\\
\mathcal{O}_{H^6} & = & -\frac{1}{3}\left(\rho^{6}+3\right)\left|y_{t}\right|^{6}\left(\overline{H}H\right)^{3},\label{C09}\\
\mathcal{O}_{X^{3}} & = & -\frac{2}{5}g_{2}^{3}\varepsilon^{\delta\eta\lambda}W_{\mu\nu}^{\delta}W_{\nu\tau}^{\eta}W_{\tau\mu}^{\lambda}
-\frac{2}{5}g_{3}^{3}f^{mnr}V_{\mu\nu}^{m}V_{\nu\tau}^{r}V_{\tau\mu}^{r}\label{C10}.\end{aligned}$$ are independent operators. $\mathcal{O}_{X^{2}D^{2}}$ is given by $$\begin{aligned}
\mathcal{O}_{X^{2}D^{2}} & = & -\frac{11}{36}g_{1}^{2}B_{\mu\nu;\tau}^{2}-\frac{11}{60}g_{2}^{2}W_{\mu\nu;\tau}^{2}-\frac{11}{60}g_{3}^{2}V_{\mu\nu;\tau}^{2}\nonumber\\
&&+\frac{1}{18}g_{1}^{2}B_{\nu\mu;\nu}^{2}+\frac{1}{30}g_{2}^{2}W_{\nu\mu;\nu}^{2}+\frac{1}{30}g_{3}^{2}V_{\nu\mu;\nu}^{2},\label{C11}\end{aligned}$$ and, as we have mentioned, they are dependent operators. The term $\mathcal{O}_{Kin}$ contains the $H^2D^4$, $H^4D^2$ operators.
As stated in [@Dim6Classification], there are eight possible classes of dimension six operators: $XD^4$, $XH^2D^2$, $X^3$, $H^3$, $H^2X^2$, $H^4D^2$, $H^2D^4$ and $X^2D^2$. The first two classes $XD^4$, $XH^2D^2$ do not appear in the Spectral Action, since the unimodular condition excludes the terms proportional to $F_{\mu\nu}$. This condition also excludes any mixing between the gauge fields. The set of $X^3$, $H^3$, $H^2X^2$, $H^4D^2$ operators are independents. The class of $H^2D^4$ operators can be “reduced” to the independent $H^4D^2$. The operator $X^2D^2$ can be rewritten as a combination of $X^3$ and $H^4D^2$ operators. This can be done with the use of (\[C02a\]): we write $F^2_{\mu\nu;\tau}$ in terms of $F_{\mu\nu}F_{\nu\tau}F_{\tau\mu}$ and $F^2_{\mu\nu;\mu}$, then we use the equation of motion for the gauge fields to obtain kinetic terms $H^4D^2$.
The operators $X^{3}$ and $X^{2}H^{2}$ affect the gauge coupling constants and the triple unification point. On the other hand, the operators $H^{6}$ and $\mathcal{O}_{kin}$ modify the Higgs mass. However, while $H^6$ produce a shift of order $C_H v^2 / \lambda$ (see eq. (\[Higgsmass\])), the operators $\mathcal{O}_{kin}$ produce a shift of order $C_{kin}v^2$ (see [@JenkinsIII]) which will be negligible if $C_H$ and $C_{kin}$ are of the same order. Therefore, as a first approximation we will only focus on the RG contribution of the operators $X^{3}$, $X^{2}H^{2}$ and $H^6$ $$\begin{aligned}
a_{6}^{flat} = \frac{1}{2\pi^{2}}\int d^{4}x\left(\mathcal{O}_{X^2H^2}+\mathcal{O}_{H^6}+\mathcal{O}'_{X^3}\right),\label{C12}\end{aligned}$$ where $$\begin{aligned}
\mathcal{O}'_{X^{3}} = -\frac{13}{60}g_{2}^{3}\varepsilon^{\delta\eta\lambda}W_{\mu\nu}^{\delta}W_{\nu\tau}^{\eta}W_{\tau\mu}^{\lambda}
-\frac{13}{60}g_{3}^{3}f^{mnr}V_{\mu\nu}^{m}V_{\nu\tau}^{r}V_{\tau\mu}^{r}\label{C13}.\end{aligned}$$
Finally, in order to normalize the kinetic term $\frac{1}{2}a\left|\nabla_\mu H_a\right|^2$ in (\[C01\]), we perform a rescaling H H= g , where $g$ is the gauge coupling to the unification scale. In the end, we have $$\begin{aligned}
\frac{f_6}{\Lambda^{2}}a_{6}^{flat} &=& \frac{1}{16\pi^{2}}\int d^{4}x\left(C_{HB}B^2_{\mu\nu}H\overline{H}+C_{HW}W^2_{\mu\nu}H\overline{H}+C_{HV}V^2_{\mu\nu}H\overline{H}\right)\nonumber\\
&+&\frac{1}{16\pi^{2}}\int d^{4}x\left(C_{H}(H\overline{H})^3+C_{W}\varepsilon^{\delta\eta\lambda}W_{\mu\nu}^{\delta}W_{\nu\tau}^{\eta}W_{\tau\mu}^{\lambda}
+C_{V}f^{mnr}V_{\mu\nu}^{m}V_{\nu\tau}^{r}V_{\tau\mu}^{r}\right)\label{eq:A6 Expression}\end{aligned}$$ where $$\begin{aligned}
C_{HB}(\Lambda) & =-\frac{f_{6}}{16\pi^{2}\Lambda^{2}}\frac{4\left(3\rho^2+17\right)}{9\left(\rho^2+3\right)}g^{4}\,\,,\, C_{HW}(\Lambda)=-\frac{f_{6}}{16\pi^{2}\Lambda^{2}}\frac{4}{3}g^{4}\,\,,\, C_{HV}(\Lambda)=-\frac{f_{6}}{16\pi^{2}\Lambda^{2}}\frac{16}{3\left(\rho^2+3\right)}g^{4}\,\,,\nonumber \\
C_{H}(\Lambda) & =-\frac{f_{6}}{16\pi^{2}\Lambda^{2}}\frac{512(\rho^{6}+3)}{3\left(\rho^2+3\right)^{3}}g^{6}\,\,,\,\,\,\,\, C_{W}(\Lambda)=-\frac{f_{6}}{16\pi^{2}\Lambda^{2}}\frac{26}{15}g^{3}\,\,,\,\,\,\,\, C_{V}(\Lambda)=-\frac{f_{6}}{16\pi^{2}\Lambda^{2}}\frac{26}{15}g^{3}\end{aligned}$$
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[^1]: Not to be confused with the notation used in [@coldplay], where the meaning of dotted and undotted indices is different.
[^2]: See [@Gilkey02; @Vassilevich] for a detailed overview of the heat trace asymptotics.
|
---
abstract: 'The characteristics of the particle emitting source are deduced from low $p_T$ identified hadron spectra ($(m_T-m_0) < 1$ GeV) and HBT radii using a hydrodynamic interpretation. From the most peripheral to the most central data, the single particle spectra are fit simultaneously for all $\pi^{\pm}$, $K^{\pm}$, and $\overline{p}/p$ using the parameterization in [@ref1] and assuming a linear transverse flow profile. Within the systematic uncertainties, the expansion parameters $T_{fo}$ and $\beta_{T}$, respectively decrease and increase with the number of participants, saturating for both at mid-centrality. The expansion using analytic calculations of the $k_T$ dependence of HBT radii in [@ref2] is fit to the data but no $\chi^{2}$ minimum is found.'
address: 'Physics and Applied Technology Directorate, Lawrence Livermore National Laboratory, 7000 East Avenue L-305, Livermore, California 94550'
author:
- 'J. M. Burward-Hoy, for the PHENIX Collaboration[^1]'
title: 'Source Parameters from Identified Hadron Spectra and HBT Radii for Au-Au Collisions at $\sqrt{s_{NN}} = $200 GeV in PHENIX'
---
INTRODUCTION
============
Identified charged hadrons in 11 different centrality selections [@ref3] and the transverse momentum ($p_T$) dependence of HBT radii in 9 $k_T$ bins [@ref4] are measured in Au-Au collisions at 200 GeV by the PHENIX Experiment [@ref5]. In both the 200 GeV and previously measured 130 GeV data [@ref6], the $\langle p_{T} \rangle$ of all particles increases from the most peripheral to the most central events and with heavier particle mass ($m_0$). The dependence of the $\langle p_{T} \rangle$ on $m_0$ suggests a radial expansion, and its dependence on the number of participant nucleons ($N_{part}$) may be due to an increasing radial expansion from peripheral to central events. The $k_T$ dependence of the HBT radii was also observed and interpreted as a radial expansion.
Both the spectra and the $k_T$ dependence of the HBT radii are fit using parameterizations based on a simple model for the source, where fluid elements are each in local thermal equilibrium and move in space-time with a hydrodynamic expansion [@ref1; @ref2]. The assumptions are: $(1)$ no temperature gradients, $(2)$ longitudinal boost invariance along the collision axis z, $(3)$ infinite extent in space-time rapidity $\eta$, and $(4)$ cylindrical symmetry with radius r. The particles are emitted along a hyperbola of constant proper time $\tau_{0} = \sqrt{t^2 - z^2}$ and short emission duration, $\Delta t < $ 1 fm/c.
The $m_T$ dependence of the yield $\frac{d^2N}{m_t dm_t dy}|_{y=0}$ is calculated after integrating the source over space-time (azimuthal and rapidity coordinates) [@ref1]. It is assumed that all particles decouple kinematically on the freeze-out hypersurface at the same freeze-out temperature $T_{fo}$, and that the particles collectively expand with a velocity profile $\beta_T \left( r \right) = \beta_T r/R$ where $R$ is the geometric radius, $r$ is the transverse coordinate, and $\beta_T$ is the surface velocity. (For a box profile, the average velocity is $\langle \beta_T \rangle$ $= 2\beta_T/3$ [@ref7]). The particle density distribution is assumed to be independent of the radial position in the fits to the single particle spectra. In the previous 130 GeV analysis [@ref8], a Gaussian density profile increases $\beta_T$ by $\approx$ 2% with a negligible difference in $T_{fo}$, while a parabolic velocity profile increases $\beta_T$ by 13% and $T_{fo}$ by 5%.
We use analytic expressions to calculate the HBT radii [@ref2]. A linear flow rapidity profile in the transverse plane is assumed and a Gaussian distribution is used for the particle density dependence on r. The parameters are the geometric radius R, the freeze-out temperature T, the flow rapidity at the surface $\eta_T$ ($\beta_T$ $= \rm{tanh} \left( \eta_T \right)$) and the freeze-out proper time $\tau_0$.
RESULTS
=======
Fitting the single particle spectra
-----------------------------------
In order to minimize contributions from hard processes, all $m_T$ dependent particle yields are fit in the range $(m_T-m_0) < 1$ GeV. As resonance decays are known to produce pions at low $p_T$ [@ref9], we place a lower $p_T$ threshold of 500 MeV/c on $\pi$ in the fit. A similar approach was followed by NA44, E814, and other experiments at lower energies. In Fig. \[fig1\], $\pi^{-}$ and $\overline{p}$ yields are shown as a function of $p_T$ for each event centrality [@ref4]. The top 5 centralities are scaled for visual clarity. The solid lines are the simultaneous fits in the limited $p_T$ range. Similar results are obtained for $K^{\pm}$, $\pi^{+}$, and p yields.
![The expansion in each centrality. The top panel is $T_{fo}$ and the bottom is $\beta_T$, both plotted as a function of $N_{part}$. []{data-label="fig2"}](fig1)
![The expansion in each centrality. The top panel is $T_{fo}$ and the bottom is $\beta_T$, both plotted as a function of $N_{part}$. []{data-label="fig2"}](fig2)
The systematic uncertainty in $T_{fo}$ is determined by adding in quadrature the change in inverse slope due to the $p_T$ dependent uncertainties in each particle yield at low $p_T$. For $\pi^{\pm}$, $K^{\pm}$, and $\overline{p}/p$, the uncertainty is $\pm10$, $\pm13$, and $16$ MeV respectively. Added in quadrature, the total systematic uncertainty in the inverse slope is $\pm23$ MeV. The systematic uncertainty in $\beta_T$ is dominated by the uncertainty in the $\overline{p}/p$ spectral shape at low $p_T$ and is determined by measuring the change in $\beta_T$ after fitting for $p_T>0.85$ GeV/c. The systematic uncertainty in $\beta_T$ is $17.5\%$.
For the 5% most central events, particles are coupled to an expanding system with a surface velocity of $\beta_T = 0.7\pm0.2 (syst.)$ and decouple at a common temperature of $T_{fo} = 110\pm23 (syst.)$ MeV with negligible statistical errors. For the most peripheral events, $T_{fo} = 135\pm3 (stat.) \pm23 (syst.)$ and $\beta_T =
0.46\pm0.02 (stat.)\pm0.2 (syst.)$. The statistical error only is included in the fit, resulting in $\chi^{2}/dof =
260.9/52$ for the most central and $321.5/52$ for the most peripheral events. At 130 GeV, similar results were obtained, with $\beta_T =
0.70\pm0.01$, $T_{fo} = 121\pm4$, and $\chi^{2}/dof = 34.0/40.0$ for the most central events (statistical and systematic errors are added in quadrature before the fit) [@ref10].
The fit results of all particles within each event centrality are shown in Fig. \[fig2\]. The top panel is $T_{fo}$ and the bottom panel is $\beta_T$, both plotted as a function of $N_{part}$. Within the systematic uncertainties, the expansion parameters respectively decrease and increase with the number of participants, saturating at mid-centrality.
Fitting the $k_T$ dependence of the HBT radii
---------------------------------------------
The HBT radii are measured from identical charged $\pi$ pairs in 9 $k_T$ selections for 10% central events [@ref5]. The systematic uncertainty in the data is 8.2%, 16.1%, and 8.3% for $R_{\rm{s}}$, $R_{\rm{o}}$, and $R_{\rm{L}}$, respectively. A simultaneous fit to the HBT data could not be found over a broad range of parameter space. As an example, if the parameters $\beta_T$ and $T_{fo}$ are set to the values from the spectra analysis, then the fit to the HBT results constrains R and $\tau_{0}$ from the $R_{\rm{s}}$ and $R_{\rm{L}}$ data respectively, yet the model overpredicts $R_{o}$ by more than 3$\sigma$ for all but the first $m_T$ data point (Fig. \[fig3\]). The systematic uncertainties in $\beta_T$ and $T_{fo}$ are represented by the shaded region. Within these boundaries, R ranges between $6.9-16.8$ fm and $\tau_{0}$ ranges between $11.2-16.7$ fm/c.
The $\chi^{2}$ contour levels of the expansion parameters $T_{\rm{fo}}$ (vertical) and $\eta_{\rm{T}}$ (horizontal) are shown for simultaneous fits to the spectra and separate fits to each HBT radius in Fig. \[fig4\]. We note that no $\chi^{2}$ minima are found, hence the contours are not closed. For the spectra, the contours are closed and show an anticorrelation, however there is no overlap with the HBT contours. The HBT radius $R_{\rm{s}}$ prefers large flow rapidity $\eta_{\rm{T}}>1.0$ and low temperatures $T_{\rm{fo}}<50$ MeV. The parameterization has the most difficulty reproducing $R_{\rm{o}}$.
CONCLUSION
==========
The single particle spectra are qualitatively described by a hydrodynamic parameterization that assumes boost invariance and a linear transverse flow profile. The transverse expansion in 11 different centrality classes is extracted from the single particle spectra. Within the systematic uncertainties, the expansion parameters $T_{\rm{fo}}$ and $\beta_{\rm{T}}$, respectively decrease and increase with the number of participants, saturating at mid-centrality. Expressions for the HBT radii based on similar hydrodynamic assumptions and Gaussian density profiles do not describe the identical $\pi$ pair data. Such fits worked well at CERN SPS energies [@ref11], but fail at RHIC energies.
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[^1]: For the full PHENIX Collaboration author list and acknowledgements, see Appendix “Collaborations” of this volume.
|
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address: |
Low Temperature Laboratory, Aalto University, School of Science and Technology, P.O. Box 15100, FI-00076 AALTO, Finland\
Landau Institute for Theoretical Physics RAS, Kosygina 2, 119334 Moscow, Russia
author:
- 'G.E.Volovik [^1]'
title: 'Flat band in the core of topological defects: bulk-vortex correspondence in topological superfluids with Fermi points'
---
Introduction
============
When the fermion zero modes localized on the surface or on the topological defects are studied in topological media, the investigation is mainly concentrated on the fully gapped topological media, such as topological insulators and superfluids/superconductors of the $^3$He-B type [@TeoKane2010; @SilaevVolovik2010; @FukuiFujiwara2010]. However, the gapless topological media may also have fermion zero modes with interesting properties, in particular they may have the dispersionless branch of spectrum with zero energy – the flat band [@SchnyderRyu2010; @HeikkilaVolovik2010b].
The dispersionless bands, where the energy vanishes in a finite region of the momentum space, have been discussed in different systems. Originally the flat band has been discussed in the fermionic condensate – the Khodel state [@Khodel1990; @NewClass; @Volovik2007; @Shaginyan2010], and for fermion zero modes localized in the core of vortices in superfluid $^3$He-A [@KopninSalomaa1991; @Volovik1994; @MisirpashaevVolovik1995]. The flat band has also been discussed on the surface of the multi-layered graphene (see [@Guinea2006; @CastroNeto2009] and references therein). In particle physics, the Fermi band (called the Fermi ball) appears in a 2+1 dimensional nonrelativistic quantum field theory which is dual to a gravitational theory in the anti-de Sitter background with a charged black hole [@Sung-SikLee2009].
Recently it was realized that the flat band can be topologically protected in gapless topological matter. It appears in the 3D systems which contain the nodal lines in the form of closed loops [@SchnyderRyu2010] or in the form of spirals [@HeikkilaVolovik2010b]. In these systems the surface flat band emerges on the surface of topological matter. The boundary of the surface flat band is bounded by the projection of the nodal loop or nodal spiral onto the corresponding surface. Here we extend this bulk-surface correspondence to the bulk-vortex correspondence, which relates the flat band of fermion zero modes in the vortex core to the topology of the point nodes (Dirac or Fermi points) in the bulk 3D topological superfluids.
Vortex-disgyration
==================
As generic example we consider topological defect in 3D spinless chiral superfluid/superconductor of the $^3$He-A type, which contains two Fermi points (Dirac points). Fermions in this chiral superfluid are described by Hamiltonian $$H=\tau_3\epsilon(p) +c\left( \tau_1 {\bf p}\cdot {\bf e}_1
+\tau_2 {\bf p}\cdot {\bf e}_2\right)~~,~~\epsilon(p)=\frac{p^2-p_F^2}{2m}
\,,
\label{Hamiltonian}$$ where $\tau_{1,2,3}$ are Pauli matrices in the Bogoliubov-Nambu space, and in bulk liquid the vectors ${\bf e}_1$ and ${\bf e}_2$ are unit orthogonal vectors. There is only one topologically stable defect in such superfluid/superconductor, since the homotopy group $\pi_1(G/H)=\pi_1(SO_3)=Z_2$. We choose the following order parameter in the topologically non-trivial configuration (in cylindrical coordinates ${\bf r}=(\rho,\phi,z)$): $${\bf e}_1({\bf r})=f_1(\rho)\hat{\boldsymbol{\phi}} ~~,~~ {\bf e}_2({\bf r})=\hat{\bf z}\sin \lambda - f_2(\rho)\hat{\boldsymbol{\rho}}\cos\lambda\,,
\label{OrderParameter}$$ with $f_{1,2}(0)=0$, $f_{1,2}(\infty)=1$. The unit vector $\hat{\bf l}$, which shows the direction of the Dirac points in momentum space, ${\bf p}_\pm= \pm p_F \hat{\bf l}$, is $$~~ \hat{\bf l}({\bf r})=\frac{{\bf e}_1\times {\bf e}_2}{|{\bf e}_1\times {\bf e}_2|}=\frac{f_2(\rho)\hat{\bf z}\cos \lambda + \hat{\boldsymbol{\rho}}\sin\lambda}
{\sqrt{f_2^2(\rho)\cos^2 \lambda + \sin^2\lambda}}
\,.
\label{l}$$
Asymptotically at large distance from the vortex core one has $$\begin{split}
& {\bf e}_1(\rho=\infty)= \hat{\boldsymbol{\phi}} ~~,~~ {\bf e}_2(\rho=\infty)=\hat{\bf z}\sin \lambda - \hat{\boldsymbol{\rho}}\cos\lambda \,,
\\
& \hat{\bf l}(\rho=\infty)= \hat{\bf z}\cos \lambda + \hat{\boldsymbol{\rho}}\sin\lambda\,,
\label{OrderParameterAsymptote}
\end{split}$$ which means that changing the parameter $\lambda$ one makes the continuous deformation of the pure phase vortex at $\lambda=0$ to the disgyration in the $\hat{\bf l}$ vector without vorticity at $\lambda=\pi/2$, and then to the pure vortex with opposite circulation at $\lambda=\pi$ (circulation of the superfluid velocity around the vortex core is $\oint d{\bf s}\cdot{\bf v}_{\rm s}=\kappa \cos\lambda$, where $\kappa =\pi\hbar/m$). We consider how the flat band in the core of the defect evolves when this parameter $\lambda$ changes. In bulk, i.e. far from the vortex core, the Dirac points are at $${\bf p}_\pm= \pm p_F \hat{\bf l}(\rho=\infty) =\pm p_F\left(\hat{\bf z}\cos \lambda + \hat{\boldsymbol{\rho}}\sin\lambda\right)
\,.
\label{DiracPointsAsymptote}$$ Due to the bulk-vortex correspondence, which we shall discuss in the next section, the projection of these two points on the vortex axis gives the boundary of the flat band in the core of the topological defect: $$E(p_z)=0~~,~~ p_z^2<p_F^2\cos^2 \lambda
\,.
\label{FlatBand}$$ This is the central result of the paper: in general the boundaries of the flat band in the core of the linear topological defect (a vortex) are determined by the projections on the vortex axis of the topologically protected point nodes in bulk. In the next section we consider the topological origin of the flat band and geometrical derivation of its boundaries. In Sec. \[QC\] , the boundaries of the flat band are obtained analytically.
Bulk-vortex correspondence {#Bulk-vortexCorrespondence}
==========================
Let us first give the topological arguments, which support the existence of the flat band inside the vortex-disgyration line. Let us consider the Hamiltonian in bulk (i.e. far from the vortex core) treating the projection $p_z$ as parameter of the 2D system. At each $p_z$ except for two values $p_z=\pm p_F\cos \lambda$ corresponding to two Fermi points (see Fig. 1), the Hamiltonian has fully gapped spectrum and thus describes the effective 2D insulator. One can check that this 2D insulator is topological for $|p_z| < p_F|\cos \lambda|$ and is topologically trivial for $|p_z| > p_F|\cos \lambda|$. For that one considers the following invariant describing the 2D topological insulators or fully gapped 2D supefluids [@Volovik2003]: $$\begin{split}
& \tilde N_3(p_z)
\\
& =\frac{1}{4\pi^2}~
{\bf tr}\left[ \int dp_xdp_yd\omega
~G\partial_{p_x} G^{-1}
G\partial_{p_y} G^{-1}G\partial_{\omega} G^{-1}\right]\,,
\end{split}
\label{N2+1}$$ where $G$ is the Green’s function matrix, which for noninteracting case has the form $G^{-1}=\i \omega - H$. This invariant, which is applicable both to interacting and non-interacting systems, gives $$\begin{aligned}
\tilde N_3(p_z)=1~~,~~|p_z| < p_F|\cos \lambda|
\,,
\label{TopInsulator}
\\
\tilde N_3(p_z)=0~~,~~|p_z| > p_F|\cos \lambda|
\,.
\label{Non-TopInsulator}\end{aligned}$$
![ Fig. 1. Projections of Dirac (Fermi) points on the direction of the vortex axis (the $z$-axis) determine the boundaries of the flat band in the vortex core. Fermi point in 3D systems represents the hedgehog (monopole) in momentum space [@Volovik2003]. For each plane $p_z={\rm const}$ one has the effective 2D system with the fully gapped energy spectrum $E_{p_z}(p_x,p_y)$, except for the planes with $p_{z\pm}=\pm p_F \cos\lambda$, where the energy $E_{p_z}(p_x,p_y)$ has a node due to the presence of the hedgehogs in these planes. Topological invariant $\tilde N_3(p_z)$ in is non-zero for $|p_z| < p_F |\cos\lambda|$, which means that for any value of the parameter $p_z$ in this interval the system behaves as a 2D topological insulator or 2D fully gapped topological superfluid. Point vortex in such 2D superfluids has fermionic state with exactly zero energy. For the vortex line in the original 3D system with Fermi points this corresponds to the dispersionless spectrum of fermion zero modes in the whole interval $|p_z| < p_F |\cos\lambda|$ (thick line). []{data-label="fig:vortex"}](Fermion_Vortex){width="8cm"}
At $p_z =\pm p_F|\cos \lambda|$, there is the topological quantum phase transition between the topological 2D “insulator” and the non-topological one. The difference of 2D topological charges on two sides of the transition, $\tilde N_3(p_z=p_F\cos\lambda+0)-\tilde N_3(p_z=p_F\cos\lambda -0)=N_3$, represents the topological charge of the Dirac point in the 3D system – hedgehog in momentum space [@Volovik2003]. As we know, the topological quantum phase transitions are accompanied by reconstruction of the spectrum of fermions bound to the topological defect: fermion zero modes appear or disappear after topological transition in bulk [@SilaevVolovik2010; @Nishida2010; @Nishida2010b; @Lutchyn2010]. For the pure vortex, i.e. at $\lambda=0$ or $\lambda=\pi$, we know from [@KopninSalomaa1991] that the vortex contains the fermionic level with exactly zero energy for any $p_z$ in the region $|p_z| < p_F$, i.e. in the region of parameters where the 2D medium has non-trivial topological charge, $ \tilde N_3=1$. On the other hand no such levels are present after the topological transition to the state of matter with $\tilde N_3=0$.
The similar reconstruction of the spectrum at the topological quantum phase transition takes place for any parameter $\lambda \neq \pi/2$ of the considered defect. This can be understood using the topology in the mixed real and momentum space [@GrinevichVolovik1988; @SalomaaVolovik1988]. To study fermions with zero energy (Majorana fermions) in the core of a point vortex in a 2D topological superconductor, the Pontryagin invariant for mixed space has been exploited in Ref. [@TeoKane2010]. The Pontryagin invariant describes classes of mappings $S^2\times S^1\rightarrow S^2$. Here the mixed space $S^2\times S^1$ is the space $(p_x,p_y,\phi)$, where $\phi$ is the coordinate around the vortex-disgyration far from the vortex core. This space is mapped to the sphere $S^2$ of unit vector $\hat{\bf g}(p_x,p_y,\phi)={\bf g}(p_x,p_y,\phi)/|{\bf g}(p_x,p_y,\phi)|$ describing the 2D Hamiltonian. In our case it is the Hamiltonian outside the vortex core: $$\begin{aligned}
H_{p_z,\lambda}(p_x,p_y,\phi)=\tau_i g^i(p_x,p_y,\phi;p_z,\lambda)\,,
\label{Hamiltonian_g}
\\
g^3=\frac{p_x^2+p_y^2}{2m}-\mu(p_z)~~,~~\mu(p_z)=\frac{p_F^2-p_z^2}{2m}\,,
\nonumber
\\
g^1=c(p_y\cos\phi -p_x\sin\phi) \,,
\nonumber
\\
g^2=c(p_z\sin\lambda -\cos\lambda(p_x\cos\phi +p_y\sin\phi))
\,,
\label{g-vector}\end{aligned}$$ with $p_z$ and $\lambda$ being the parameters of this effective 2D Hamiltonian. The Pontryagin $Z_2$ invariant is non-trivial and thus the zero energy state exists in the core of the defect the effective 2D superconductor, if the parameters $p_z$ and $\lambda$ of the 2D Hamiltonian satisfy condition $|p_z| < p_F|\cos \lambda|$.
For the considered linear topological defect (vortex-disgyration) in the 3D system this implies that the core of this defect contains the dispersionless band in the interval of momentum $|p_z| < p_F|\cos \lambda|$, i.e. one obtains equation .
Flat band from quasi-classical approach {#QC}
=======================================
Let us now support the above topological arguments by explicit calculation of the fermionic flat band in the vortex-disgyration, which is described by the order parameter . The Bogoliubov-de Gennes Hamiltonian for fermions localized on the defect line is obtained from by substitution of the classical transverse momentum by the quantum-mechanical operator, $${\bf p}_\perp \rightarrow (-i\nabla_x, -i\nabla_y) \,,
\label{MomentumOperator}$$ while $p_z$ remains the good quantum umber which serves as parameter of the effective 2D system. The zero energy states in this 2D system can be studied using the quasiclassical approximation, see details in Chapter 23 of the book [@Volovik2003]. For our purposes it is sufficient to consider the Hamiltonian on the trajectory $s$ which crosses the center of the vortex. The modification of quasiclassical Hamiltonian in Eq.(23.16) in [@Volovik2003] for the considered vortex-disgyration is $$\begin{split}
& H_{\rm qcl}(p_z)=-i\frac{q}{m}\tau_3\partial_s +U(s) \tau_2 \,,
\\
& U(s)= cp_z\sin\lambda - cq f_2(|s|) {\rm sign} (s) \cos\lambda
\,,
\\
& q=\sqrt{p_F^2 -p_z^2} \,.
\end{split}
\label{QuasiclassicalH}$$ The Hamiltonian $H_{\rm qcl}(p_z)$ is super-symmetric if the asymptotes of the potential $U(s)$ have different sign for $s=- \infty$ and $s=+ \infty$. The latter takes place if $$|p_z|\sin\lambda < q|\cos\lambda| \,.
\label{Constraint}$$ The super-symmetric Hamiltonian $H_{\rm qcl}(p_z)$ has the state with zero energy, $E_{\rm qcl}(p_z)=0$, for any $p_z$ in the interval . For vortices in chiral superfluids it is known [@Volovik2003] that the zero energy state of the quasiclassical Hamiltonian, $E_{\rm qcl}=0$, automatically results in the true zero energy state, $E=0$, obtained in the exact quantum-mechanical problem using the Bogoliubov-de Gennes Hamiltonian. This proves the existence of the flat band in the range of momentum , which coincides with equation and is in agreement with the topological analysis in previous section.
Discussion
==========
We discussed the 3D matter with topologically protected Fermi points. Topological defects (vortices and vortex disgyrations) in such matter contain the dispersionless fermionic band with zero energy – the flat band. The boundaries of the flat band are determined by projections of the Fermi points on the axis of the topological defect. This bulk-vortex correspondence for flat band is similar to the bulk-surface correspondence discussed in the media with topologically protected lines of zeroes [@HeikkilaVolovik2010b; @SchnyderRyu2010]. In the latter case the flat band is formed on the surface of the system, and its boundary is determined by projection of the nodal line (closed loop [@SchnyderRyu2010] or spiral [@HeikkilaVolovik2010b]) on the corresponding surface.
This work is supported in part by the Academy of Finland, Centers of excellence program 2006–2011. It is my pleasure to thank V.B. Eltsov, T.T. Heikkilä and N.B. Kopnin for helpful discussions.
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[^1]: e-mail: [email protected]
|
---
author:
- |
G. Fasano , B. Poggianti , D. Bettoni , E. Pignatelli , C. Marmo ,\
M. Moles , P. Kj[æ]{}rgaard , J. Varela , W. Couch , A. Dressler
title: 'The WINGS Survey: a progress report'
---
Introduction
============
Clusters of Galaxies are the largest, yet well defined, known entities in the Universe. The identification of their properties and content could led to use them as tracers of cosmic evolution since they can be detected at large distances.
Over the past five years, the cluster environment has been discovered to be the site of morphological transformations at a relatively recent cosmological epoch. Thanks to the high spatial resolution achieved with the Hubble Space Telescope (HST), it has been ascertained that the morphological properties of galaxies in the cores of rich distant clusters largely differ from those in nearby clusters: at $z=0.4-0.5$, spirals are a factor of 2-3 more abundant and S0 galaxies are proportionally less abundant, while the fraction of ellipticals is already as large or larger [@D97; @smail].
On the basis of excellent seeing images taken at the NOT and La Silla-Danish telescopes [@F02], we have filled in the existing gap of information between local (z$\le$0.1) and distant (z$\sim$0.4-0.5) clusters, confirming that, as the redshift decreases, the S0 population tends to grow at the expense of the spiral population [@FP00]. This work has also highlighted the role played by the cluster type in determining the relative occurrence of S0 and elliptical galaxies at a given redshift: clusters at z$\sim$0.1-0.25 have a low/(high) S0/E ratio if they display/(lack) a strong concentration of elliptical galaxies towards the cluster centre.
Concerning the evolution of the stellar populations in cluster galaxies, ground-based spectroscopic surveys of the intermediate-redshift clusters observed by HST have offered a detailed comparison of the spectral and morphological properties, elucidating the link between the evolution of the stellar populations and the changes in galaxy structure [@D99; @PA99; @CA94; @CA98; @FA98; @LA98]. The spiral population includes most of the star-forming galaxies, a large number of post-starburst galaxies and a sizeable fraction of the red, passive galaxies; in contrast, the stellar populations of the ellipticals and of (the few) bright S0 galaxies appear to be old and passively evolving. These observations are consistent with the post-starburst and star-forming galaxies being recently infallen field spirals whose star formation is truncated upon entering the cluster and that will evolve into S0’s at a later time.
A crucial ingredient for all these evolutionary studies should be the comprehensive knowledge of the physical properties of galaxies in nearby clusters, in order to control their local variance, prior to draw any conclusion on cosmic evolution. When confronted with such kind of requirement one realizes that, while a large amount of high quality data for distant clusters is continuously gathering from HST imaging and ground based spectroscopy, our present knowledge of the systematic properties of galaxies in nearby clusters remains surprisingly poor. Actually, the only reference sample available in the nearby universe for photometry and morphology is the ’historical’ one of Dressler (1980a,b), who lists positions, visual magnitudes and morphological classifications of galaxies, relying on photographic plates taken at Las Campanas 2.5m, Kitt Peak 4m and Palomar 1.5m telescopes. Instead, no reference sample is available for spectroscopy. It is obvious that an adequate photometric and spectroscopic information on nearby clusters is missing and is crucial for studying the morphology and the stellar populations of galaxies in a systematic way, as well as for setting the zero-point for evolutionary studies.
The WINGS photometric survey
============================
-3truecm
The above mentioned lack of systematic information from CCD material on nearby galaxy clusters is mainly due to their huge angular size, which prevented astronomers from gathering large datasets until wide field CCD cameras (WFC) became available. In the last years excellent WFCs became operative in imaging mode (WFI), such as those at the INT-2.5m and ESO-2.2m telescopes, as well as wide-field multifiber spectrographs, e.g. at the WHT-4.2m and AAT-3.9m telescopes. We exploited the new opportunity starting with a program named Wide–field Imaging Nearby Galaxy–cluster Survey (WINGS). WFI proposals were presented for both the northern and the southern hemispheres, resulting in seven obeserving runs, for a total of 18 nights (9 at the INT-2.5m telescope and 9 at the ESO-2.2m telescope).
The sample of nearby clusters
-----------------------------
![distribution of some cluster properties in our sample[]{data-label="FigCP"}](01_fasanog2.eps){width="6.5cm"}
The sample we observed has been selected from essentially complete X-ray (0.1-2.4 keV), flux-limited samples of clusters \[XBACs: Abell clusters with $F_X\ge$5$\times 10^{-12}$erg cm$^{-2}$ s$^{-1}$ [@Eb96]; BCS+eBCS: northern clusters with $F_X\ge$2.8$\times 10^{-12}$erg cm$^{-2}$ s$^{-1}$ [@Eb98; @Eb00]\] compiled from ROSAT All-Sky Survey data. This sample is uncontaminated from AGNs and foreground stars and is not affected by the risk of projection effects as optically selected catalogs are.
Only clusters with galactic latitude $|b|>$20$^\circ$ in the redshift range $0.04<z<0.07$ have been included in the sample. The redshift upper limit ensures sufficient spatial resolution (1$^{\prime\prime} =$ 1.3 kpc at $z=0.07$, $H_0$=70), while the lower limit allows us to efficiently survey a sufficiently large area of the cluster (the central 1.5 $\rm Mpc^2$ at $z=0.04$), comparable to that observed at higher redshift with HST. Our aim is to cover a well defined area in physical terms, such as the $r_{200}$ radius (the radius within which the average cluster density is 200 times the critical density) or the virial radius.
The final sample includes 78 clusters (42 in the southern hemisphere and 36 in the northern one, see Figure \[FigAitoff\]), of which about a third are in common with the @D80b sample. This partial overlap will be useful for comparing the two data sets and cross-check the respective morphological classifications.
Figure \[FigCP\] presents the distribution of some “popular” cluster properties in our sample, showing that it spans a broad range in both optical richness and X-Ray luminosity.
Observational strategy and data reduction
-----------------------------------------
0.5truecm
We decided to take images in the V and B bands. The former one allows us to compare our results with previous studies of nearby clusters, as well as with HST-WFPC2 ($I_{814}$) studies of clusters at z$\sim$0.5. The second filter is valuable in order to get colors of galaxies and to compare with future HST(ACS)/NGST studies of clusters at z$\sim$1.
For each cluster, three slightly offseted frames per filter have been obtained in order to improve the cosmetics of the final mosaics. The exposures taken with insufficient transparency and/or seeing were repeated in different nights and/or runs, until matching the required conditions (FWHM$\le$1.2 arcseconds). A total exposure time of 1200s (in each band) was usually enough to reach a S/N per pixel of $\sim$2.1(1.7) in the V(B) band.
During each night several photometric standard fields were taken at different positions and zenith distances in order to secure a careful determination of the calibration coefficients as a function of airmass, color and chip number. Also a number of astrometric and empty fields have been observed. The latter ones will allow us to estimate the field contamination to galaxy counts for different bins of magnitude. The former ones have been used to determine, for each filter, the astrometric solutions relative to each observing run. This is well known to be an important ingredient in the reduction procedure of WFI data, even as far as the photometric accuracy is concerned.
The standard IRAF-MSCRED package has been used for the basic reduction procedures (debiasing, flat-fielding), while the package WFPRED [@RH] was used to perform astrometry, co-adding, mosaicing and alignment of images in different bands. A number of additional IRAF/shell scripts have been produced in order to make the whole reduction procedure fully automatic. As an example of the final result, Figure \[FigClu\] shows the INT-mosaic of the cluster Abell 2457 in the B band.
The Catalogs
------------
-1cm ![Automatic S/G classification compared with the expected S/G partition line (see text) for the cluster Abell 85[]{data-label="FigPsf"}](01_fasanog4.eps "fig:"){width="7.5cm"}
-2cm ![Histogram of magnitudes (upper panel) and color–magnitude diagram (lower panel) of the cluster Abell 119[]{data-label="FigHCM"}](01_fasanog5.eps "fig:"){width="10cm"} -0.5cm
We used SExtractor [@BA96](Bertin and Arnouts 1996) to produce a number of source catalogs for each cluster. Relying on extensive simulations of cluster fields, the S/N of our final V band images turned out to be sufficient to make SExtractor able to provide a robust star/galaxy(S/G) separation down to a threshold (1.5$\times\sigma$) magnitude (area) of $\sim$23.5–24.3 (10–35 pixels), depending on both the sky surface brightness and seeing. Thus, for each cluster mosaic in the V band, the proper extraction limits of magnitude and area have been derived and four deep catalogs have been produced for: galaxies (Dcat; S/G$\le$0.2), stars (Scat; S/G$\ge$0.8), other objects (Ocat; 0.2$<$S/G$<$0.8) and saturated sources (Satcat). Almost all objects with V$>$23 turned out to belong to the Ocat catalog. An additional catalog for surface photometry and morphology of galaxies (Mcat; see next subsection) has been extracted from Dcat including only galaxies with threshold area A$\ge$200 pixels. Besides the different kinds of magnitudes (aperture, isophotal, total), all derived neglecting the color term, these catalogs contain some global information about size, ellipticity and position angle of objects. The V band catalogs have been then used as reference lists to extract the corresponding catalogs in the B band, allowing us to measure the 5-kpc ($H_0$=70) aperture colors and to evaluate the color corrected magnitudes in both the V and B bands.
The automatic S/G classifier can be sometime worsened by space–varying PSFs and blending. It has been improved interactively looking at the position in a plane like that of Figure \[FigPsf\] (threshold area–V$_{mag}$), where the Star/Galaxy partition line, analitically derived from the proper PSF, is reported for the cluster Abell 85. This interactive cleaning task moves some automatically classified galaxies in the star catalog and viceversa.
In the upper panel of Figure \[FigHCM\] we report the distributions of the total SExtractor magnitudes (V$_{best}$) from the galaxy catalogs Dcat and Mcat, relative to the cluster Abell 119. In the same panel also the number of galaxies in the two catalogs are indicated. The lower panel of the same figure illustrates the corresponding color–magnitude plot. Figure \[FigHCM\] shows that the completeness of the bright and deep galaxy catalogs (Mcat and Dcat) is typically achieved down to V$\sim$20 and V$\sim$22, respectively, while the corresponding cutoff magnitudes turn out to be tipically $\sim$1 mag fainter. Also typical are the sizes of the catalogs Mcat and Dcat indicated in the figure.
It is worth to note that, due to the extrapolation of the luminosity profiles (assumed gaussian), the total galactic magnitudes given by SExtractor could be over–estimated up to 0.4-0.5 mag, the early–types being more biased than late–types [@F98]. This bias disappears if we consider the magnitudes from our surface photometry tool (see next subsection). However, the magnitudes of galaxies not belonging to the catalog Mcat need to be statistically corrected for the bias using concentration indices.
Surface photometry and morphology
---------------------------------
![FWHM along the X and Y axes for the cluster Z1261[]{data-label="FigFwhm"}](01_fasanog6.eps){width="6.5cm"}
The catalog Mcat contains those galaxies from the deep list (Dcat) that are large enough (A$\ge$200 pixels) to be suitable for surface photometry and morphological analysis. This defines the reference list of galaxies to be processed by the automatic surface photometry tool GASPHOT.
The need for such a software has become more and more evident in the last years, as deep and/or wide imaging became more and more common. The usual ’one at a time’ surface photometry tools (IRAF-ELLIPSE, Jedrzejewski 1987; AIAP, Fasano 1990) are clearly inadequate to process CCD frames containing several hundreds (or even thousands) of galaxies. For this reason we have developed a Galaxy Automatic Surface PHOtometry Tool (GASPHOT, Pignatelli & Fasano 1999) which is able to process ’all at once’ a galaxy list like Mcat. It consists of four main tools:
- STARPROF produces a space–varying PSF profile. This is achieved by using an analytical (multi–gaussian) representation of the PSF and turns out to be crucial in order to obtain reliable results in the morphological analysis (next steps), particularly when the variation of the FWHM along the axes is not negligible (see Figure \[FigFwhm\]);
- SEXISOPH exploits SExtractor capabilities to produce luminosity, ellipticity and position angle profiles of the whole galaxy sample;
-6.5truecm
- GALPROF simultaneously fits the major and minor axis luminosity profiles of each galaxy by using a Sersic $r^{1/n}$ law and/or a two component ($r^{1/4}$ + exponential) profile, convolved in any case with the proper PSF. At variance with SExtractor, GALPROF produces unbiased estimates of the total magnitudes, independently on the morphological type. Figures \[FigSers1\] and \[FigSers2\] illustrate the performances of GALPROF in recovering the Sersic’s index of toy galaxies with r$^{1/4}$ and exponential luminosity profiles.
- MORPHOT exploits some characteristic features of the luminosity and geometrical profiles, together with the Sersic index, to estimate the morphological type of individual galaxies.
For each cluster GASPHOT produces two more catalogs. The first one includes the whole set of luminosity and geometrical profiles obtained by SEXISOPH. The second one contains the photometric and morphological information extracted from the profiles by GALPROF and MORPHOT (effective radius and average surface brightness, total magnitude, Sersic index, B/D ratio, morphological type, etc..).
Luminosity profiles of faint and/or small galaxies are usually well enough represented by a simple Sersic law, even in case of multi–component objects (see Figure \[FigSers\]). On the contrary, for bright and/or big galaxies two components are often necessary to model the luminosity profiles.
-2cm -2cm ![Histograms of the Sersic index obtained by running GALPROF on a crowded artificial field containing 200 toy galaxies with different magnitudes and radii. Galaxies with exponential profiles (100 objects) are binned with the solid line histogram, whereas the dotted line histogram refers to galaxies with r$^{1/4}$ profiles.[]{data-label="FigSers2"}](01_fasanog8.eps "fig:"){width="10cm"} -2cm
![Correlation between the Sersic index $n$ resulting from GALPROF and the ’true’ Bulge/Disk ratio for a sample of faint toy galaxies[]{data-label="FigSers"}](01_fasanog9.eps){width="6.5cm"}
Status and perspectives
-----------------------
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The data collection for the photometric survey has been completed in August 2002. The vast majority of the clusters in the sample ($>$90%) have been observed in good (sometime very good) weather conditions, making us confident that reliable photometric and morphological results will be obtained.
The INT–WFI frames have been fully reduced and most of the corresponding catalogs have been already extracted. They will be included in the first paper of the WINGS series (Fasano et al. 2003, in preparation). The morphological analysis of these clusters, which is presently being carried out with GASPHOT, will be the subject of a forthcoming paper. Concerning the ESO-WFI frames, the data reduction is currently in progress.
As stated in Section 1, The photometric WINGS survey has been conceived to fill in the lack of a systematic investigation of nearby clusters and their galaxy content. This is schematically illustrated in Figure \[FigWings\], where the limiting absolute magnitude and space resolution (in kpc) are reported as a function of redshift for most of the available and ongoing galaxy surveys. It can be seen that WINGS is the deepest (M$_V\sim$-14), best resolution (FWHM$\sim$1 kpc) survey of a complete sample of galaxies in nearby clusters to date. For instance, even if the nominal resolution (FWHM in kpc) of WINGS is only slightly better than that of the survey of Dressler (1980), its data quality (CCD) is definitively better and its deepness is incomparably better ($\sim$6 mag) with respect to the Dressler’s survey. This survey will produce detailed surface photometry and morphology for about 5$\times$10$^4$ galaxies, while integrated photometry and rough structural parameters will be obtained for about 2$\times$10$^5$ galaxies. This will represent the first CCD-based systematic investigation of the properties of cluster galaxies in the nearby universe and will provide a local reference sample for distant cluster studies.
What could we do with this huge dataset ? The only limit we have is our imagination !
In the next few years deep imaging surveys of distant clusters will become an even stronger scientific drive for HST with the new Advanced Camera and will be paralleled by extensive spectroscopic surveys with 8-m class telescopes. While the high-z studies occupy the headlines, the knowledge of the local universe is crucial to allow a full exploitation of the high-z data. Given the large cluster-to-cluster variations in properties and morphological content, a large and well-defined sample is needed to investigate in a systematic way what cluster property/ies are driving the variations in galactic properties.
We can do that ! In fact, apart from a robust analysis of the global cluster properties (characteristic radius, total luminosity, ellipticity and shape, subclustering, etc..), this dataset will allow us to study with high statistical significance the global properties of cluster galaxies: luminosity function of different morphological types, morphological fractions E:S0:Sp:Irr, integrated colors, colour-magnitude diagrams, scaling relations ($<\mu_e>-r_e$–Sersic index). Besides, we will be able to look at all the above mentioned galactic properties as a function of the cluster properties (structure and concentration, X-ray and total optical luminosity), local density and clustercentric distance.
We plan to release the WINGS catalogs to the astronomical community in a couple of years.
The WINGS spectroscopic survey
==============================
The spectroscopic follow-up of WINGS (WINGS-SPE) is obtaining optical spectra of 300 to 500 galaxies per cluster in a statistically significant subsample of the WINGS clusters. This subsample consists of about 55 clusters with a range of X-ray and optical properties sufficiently large to explore the dependence on cluster properties. Clusters are known to lie on a “fundamental plane” in a three-dimensional parameter space identified by optical luminosity, half-light radius and X-ray luminosity (e.g. Miller et al. 1999, Fritsch & Buchert 1999), and the deviations from such a plane are strongly correlated with substructure in the cluster. The cluster selection for WINGS-SPE ensures a coverage of this fundamental-plane over a factor of 30 in X-ray luminosity.
The only criterion for spectroscopic target selection is the galaxy total magnitude limit $V<20$, corresponding to $M_B \sim -17$ (on average) over our redshift range. In addition, galaxies lying above the color-magnitude sequence in the B-V color magnitude diagram are sampled at a lower completeness rate than those on and below the sequence. This selection criteria provide an unbiased magnitude-limit sample of galaxies representative of the whole cluster populations, while enhancing the probability to reject non-cluster members.
The magnitude limit for spectroscopy reaches more than 5 magnitudes down the galaxy luminosity function, thus 1.5 magnitude deeper than large area spectroscopic surveys such as the Sloan or the 2dF Galaxy Redshift Survey. The depth of the spectroscopy is important for two reasons. First of all, because it allows an unprecedented view of both massive and intermediate-mass galaxies in clusters, allowing an investigation of the stellar content and morphology as a function of the galaxy luminosity. Second, a wide magnitude coverage is needed for a useful comparison with distant cluster studies. In fact, a large fraction of the luminous star-forming galaxies at high z are expected to fade significantly as a consequence of the decline in their star formation, thus populating the intermediate-to-faint luminosity regime in the nearby clusters.
WINGS-SPE is a long term spectroscopic campaign that has begun in semester 2002B and will stretch over (at least) two semesters. Fifteen nights of spectroscopy have been allocated so far. The spectra are obtained with a multifiber technique with the WYFFOS spectrograph at the William Herschel Telescope and the 2dF spectrographs at the Anglo-Australian Telescope. The use of both facilities is vital for the project, because the Northern and the Southern emispheres contain the X-ray faintest and X-ray brightest subsets of the clusters, respectively.
Spectra cover the range 3600-8000 Å$\,$ (with 2dF) and 3800-7000 Å$\,$ (with WYFFOS), at a resolution of 9 and 6 Å, respectively. The wide magnitude range of the galaxies requires two different fibre configurations per cluster with different exposure times. From the spectra we are measuring redshifts, line indices and equivalent widths of the main absorption and emission features, which provide cluster membership, star formation rates and histories, and metallicity estimates.
Obviously, this dataset is suitable for addressing numerous scientific issues. Our primary goal is to study the issues described below.
The link between star formation and galaxy morphology in dense environments.
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The WINGS sample is unique in terms of the detailed, quantitative morphological information that is available for its galaxy populations. This virtue is particularly important when combined with similarly detailed information about the star formation activity and metal content of these galaxies.
Extensive work on galaxy clusters at high redshift has demonstrated that the combination of morphological and spectroscopic information is a powerful tool in the study of galaxy evolution (Dressler et al. 1999; Poggianti et al. 1999; Couch et al. 1998). High-$z$ observations suggest that stellar populations in cluster ellipticals are old, while star formation in disk galaxies had a steeper evolution in clusters than in the field.
A similar study is missing in the nearby Universe. Studies of a few low-$z$ clusters (Kuntschner & Davies 1998, Smail et al. 2001, Poggianti et al. 2001) have begun to uncover a difference in the age distributions of S0 and E galaxies, which is consistent with the hypothesis that star-forming spirals are transformed into passive S0s.
Based on WINGS spectra, we are currently exploring the stellar ages and metallicity of galaxies as a function of their Hubble type and luminosity. This will clarify whether the differences between S0s vs Es are a widespread phenomenon in clusters, will elucidate the spectroscopic properties of the cluster spirals and what is the incidence of passive, red spirals in high- to low-density regions.
The dependence of star formation on clustercentric position, local density and cluster properties.
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The effects that different physical mechanisms (ram pressure stripping, tidal encounters, loss of halo gas etc.) have on galaxy evolution are expected to vary with the local density and/or the global cluster properties and accretion history. Having a wide-area dataset of a large and variegate sample of clusters, we hope to isolate the different effects to understand what processes have a noticeable influence on galaxy evolution, and how they modify galaxies’ star formation history. In fact, we are carrying out a simultaneous analysis of the galactic properties [**and**]{} their environment, with the aim of understanding how galaxy evolution is related with the cluster/group mass, with the amount of substructure and ongoing merging of groups and clusters, with the intracluster medium local density and metallicity.
The comparison with distant clusters.
-------------------------------------
Finally, WINGS-SPE represents the first local spectroscopic database of its kind that can be used as a baseline for comparison with clusters at higher redshift. For this work, aperture effects need to be taken into account, since slit spectra at high z typically cover much larger galactic areas than the fibre spectra at low z. However, this problem is mitigated by the fact that aperture effects can be estimated from radial color gradients within each galaxy, based on the precision photometry of WINGS, using the fact that color and spectral type are largely correlated.
Part of this work was supported by the Italian Scientific Research Ministery (MIUR).
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abstract: 'As spin glass materials have extremely slow dynamics, devious numerical methods are needed to study low-temperature states. A simple and fast optimization version of the classical Kasteleyn treatment of the Ising model is described and applied to two-dimensional Ising spin glasses. The algorithm combines the Pfaffian and matching approaches to directly strip droplet excitations from an excited state. Extended ground states in Ising spin glasses on a torus, which are optimized over all boundary conditions, are used to compute precise values for ground state energy densities.'
author:
- 'Creighton K. Thomas, A. Alan Middleton'
title: Matching Kasteleyn Cities for Spin Glass Ground States
---
The Ising spin glass is a model for disordered magnetic alloys which captures the complexity of materials with frozen randomness and competing interactions, including frustration, extremely slow dynamics, and intricate memory effects.[@YoungBook] The spins in the model are coupled by random choices of ferromagnetic or antiferromagnetic bonds, leading to a complex free energy landscape. There are at least two theoretical approaches,[@pictures] including the droplet and replica-symmetry-breaking pictures, used to describe the non-equilibrium dynamics and low-free-energy structure of the spin glass phase space. As these theoretical approaches differ significantly and exact results for spin glasses are rare,[@Exactish] computational work has been essential for computing scaling exponents and as a qualitative check of theoretical pictures.[@OptBook]
The history of the relationship between the physical analysis and the mathematics of numerical approaches to spin glasses is long and rich. In general, characterizing the complex free energy landscape of disordered materials is challenging. Direct Monte Carlo simulations are hindered by the same high free-energy barriers that inhibit equilibration in the physical system. It is expected[@YoungBook; @pictures] that times $t$ satisfying $\ln(t)\sim L^{\psi}$ are required to equilibrate systems of size $L$, where $\psi\ge\theta$ and $\theta$ determines the energy scale $\Delta E(\ell)$ of excitations or domain walls on length scale $\ell$, $\Delta E\sim\ell^{\theta}$. To replicate the many decades of experimental time scales and to develop a better understanding of disordered systems for $t\rightarrow\infty$, algorithms for either accelerating the approach to equilibrium or finding ground states in spin glasses have been developed. Many of these techniques (which are often generally applicable to disordered materials) are inspired by, or have inspired, approaches for combinatorial optimization problems. Parallel tempering, genetic algorithms, and extremal optimization are examples of heuristic algorithms to find close approximations to equilibrated and ground state configurations.[@Heuristics] Exact general algorithms such as transfer matrix methods[@Transfer] and branch-and-cut methods[@BranchCut] require times that are exponential in powers of the system size, though extensive development has led to computing ground states in three-dimensional Ising spin glasses with up to $12^{3}$ spins.[@BranchCut]
We have found a simple algorithm for studying two-dimensional (2D) Ising spin glasses that combines use of the classical Kasteleyn city[@Kasteleyn] and application of a standard combinatorial optimization algorithm. Besides solving the problem on planar graphs and linking together these methods, we use this algorithm to study “extended” ground states, which optimize the energy over choices of periodic or anti-periodic boundary conditions, as well as the spin configuration. This approach dramatically improves the treatment of boundary-free samples, so that the finite-size effects are greatly reduced. We have used this algorithm to determine very precisely the energy of the Ising spin glass in the large volume limit.
The Edwards-Anderson Hamiltonian that is used for Ising spin glasses is ${\cal H}\left(\left\{ s_{i}\right\} \right)=-\sum_{\langle ij\rangle}J_{ij}s_{i}s_{j}$, where the couplings $J_{ij}$ between nearest neighbor pairs of spins $\langle ij\rangle$ are independent identically distributed variables, fixed in a given sample, and the $s_{i}$ are Ising spin variables, $s_{i}=\pm1$, with $L^{d}$ sites $i$ on a $d$-dimensional lattice. The distribution for $J_{ij}$ is generally taken either to be Gaussian or bimodal (the “$\pm J$” case). Barahona[@Barahona] has shown that computing the ground state energy of a 3D spin glass (or even two coupled 2D layers) is an NP-hard problem.[@TSP] This implies that if the ground state of the 3D spin glass could be efficiently computed, i.e., found in a time polynomial in $L$, many outstanding computational problems that are believed to require worst-case exponential time to solve, such as the Traveling Salesperson Problem, could also be solved in time polynomial in the size of the problem. Improvements in 3D spin glass calculations therefore focus on reducing the numerical constants in the exponent for the expected solution time.
The two-dimensional Ising spin glass (2DISG) is a case where exact algorithms have allowed for study of the ground state and density of states. These approaches have used two methods: the dimer-Pfaffian method (Pfaffian method) and matching to minimize frustration.
The partition function for Ising models with arbitrary couplings can be solved for either open or toroidal boundary conditions by using techniques developed for the pure Ising model,[@Kasteleyn] i.e., computing and summing Pfaffians, antisymmetric combinations of ordered statistical weights, from $L^{2}\times L^{2}$ sparse matrices. The ground state energy can be computed[@PfaffianSpinGlass] in $O(L^{5})$ time for discrete-valued disorder; the spectrum is discrete and bounded by a power of $L$. Note that the Pfaffian method uses perfect matchings (dimer coverings) on a decorated lattice and requires a sum over four combinations of odd and even constraints on these matchings on a torus.
The fastest ground state algorithms for the 2DISG map the Ising spin glass problem to the weighted perfect matching problem, a common problem in combinatorial optimization. Given a graph $G=(V,E)$, with vertices $V$ and edges $E$, with a weight function $w:E\rightarrow\mathbb{R}$, the problem is to select a perfect matching, a subset of edges $M\subset E$ where every vertex in $V$ belongs to a single edge $e\in E$, such that the total weight $w(M)=\sum_{e\in M}w_{e}$ is minimal. The solution can be found in time polynomial in the number of edges.[@TSP] Matching is the core routine in two mappings for finding 2DISG ground states. The mapping by Bieche *et al*.[@Bieche] uses a graph where the vertex set $V$ contains the frustrated plaquettes (primitive polygons $p$ with $\Pi_{\langle ij\rangle\in p}J_{ij}<0$). The edges connect points in $V$ within some distance $r_{\mathrm{max}}$. This algorithm is exact as $r_{\mathrm{max}}\rightarrow\infty$, but it works for a large fraction of cases even with small values of $r_{\mathrm{max}}$, especially for $\pm J$ disorder. Barahona’s mapping[@Barahona] replaces each plaquette with a subgraph that is connected to neighboring subgraphs by dual bonds, with each dual bond crossing one edge in G \[see Fig. \[cap:GadgetGraph\](a)\]. The subgraph edges have zero weight and the dual edges that cross bonds of strength $J_{ij}$ have weight $w_{ij}=\left|J_{ij}\right|$; the subgraph comes in two types, assigned according to the frustration of the plaquette. These algorithms have been extremely useful, e.g., in studying domain walls and the nature of the ground state as $L\rightarrow\infty$.[@CHK2004; @SLEDW] Note that the graphs used are derived from the (sample-dependent) plaquette frustrations; this is not the case with our algorithm, where the graph is independent of the $J_{ij}$, so its implementation is simpler.
Matching algorithms have been used for planar graphs. The case of the torus, with periodic boundary conditions in both directions, has not been addressed in very large systems, as Pfaffian methods are much slower (and in practice, mean-time exponential run-time algorithms are still commonly used). Studies of smaller toroidal systems with Gaussian disorder have used the branch-and-cut algorithm[@CHK2004] or the transfer matrix; such studies confirm that the finite-size corrections vanish much more quickly in toroidal geometries rather than planar geometries. It would therefore be useful to have a fast algorithm for finding information about the ground states for the 2DISG on the torus.
We have developed an approach which is not limited to planar graphs; it also provides significant information about the ground state on the torus. One component of this approach is a ground-state algorithm that combines a representation from the Pfaffian method with matching. The other component is applying this algorithm on the torus to find an extended ground state: the minimum energy state over all spin configurations and over the set of four boundary conditions (BCs). That is, we find the extended state $\left(\left\{
s_{i}^{0}\right\} ,\sigma_{h}^{0},\sigma_{v}^{0}\right)$ which minimizes ${\cal H}^*=-\sum_{\langle
ij\rangle}J_{ij}s_{i}s_{j}\sigma_{ij}$, with $\sigma_{ij}=1$ except on one vertical column of horizontal bonds, where $\sigma_{ij}=\sigma_{v}$, and on one horizontal row of vertical bonds, where $\sigma_{ij}=\sigma_{h}$ and $\sigma_{h}$ and $\sigma_{v}$ take values $\sigma_{h,v}=\pm1$. The extended ground state on the torus is the minimum energy state over the four possible combinations of BCs given by choosing (anti-)periodic BCs for each direction. The standard ground state for given BCs is therefore exactly found for $\frac{1}{4}$ of the samples. Note that, in general, when all $\sigma_{ij} = 1$, ${\mathcal
H}^* = {\mathcal H}$, so the extended ground state is equal to the standard ground state (this is always the case for planar graphs, so the algorithm finds ground states of planar graphs without modification). The extended ground state on the torus is also of interest in its own right. For example, it can be used as an edge-free background for studying equilibration and droplets[@CMWprep] and to rapidly compute the energy density for large samples.
We first give an overview of our algorithm. A spin and bond configuration is used to define a weighted dual lattice $D$ which in turn is mapped to a weighted graph $G$. A minimum weight perfect matching for $G$ is computed and used to identify a set of negative weight loops in $D$ with the most negative total weight. These loops are exactly the excitations of the current configuration relative to an extended ground state. The configuration is thus set to the ground state by flipping the spins “within” each loop. This method can be applied to any planar graph by supplying the appropriate boundary conditions (i.e. in the same way as Bieche [*et al*]{}. and Barahona algorithms).
A more detailed description of the method for the $L\times L$ toroidal square lattice starts with a list of the inputs: an initial configuration $c=(\{s_{i}\},\sigma_h,\sigma_v)$ and couplings $J_{ij}$. The dual lattice $D=(V,E)$ has edges $e_{ij}\in E$ ($e_{ij}$ crosses the bond $\langle ij \rangle$ in the original lattice) connecting neighboring plaquettes (these make up $V$) on the original spin lattice; it also is an $L\times L$ torus. Given $c$, weights $w_{c}$ for edges in $E$ are set by $w_c(e_{ij})=J_{ij}s_{i}s_{j}\sigma_{ij}$; see Fig. \[cap:GadgetGraph\](a). The value of the extended Hamiltonian is then ${\cal
H}^*(c)=-w_{c}(E)\equiv-\sum_{e_{ij}\in E}w_{c}(e_{ij})$.
To minimize ${\cal H}^*$, we find the extremal (i.e., minimum total weight) set of negative weight loops in the dual graph $D$ by computing a minimal weight perfect matching $M$ on a related graph $G$. In the case of a square lattice, we form $G$ by replacing each vertex in $D$ by a “Kasteleyn city” subgraph, a complete graph with $4$ nodes \[see Fig. \[cap:GadgetGraph\](b)\]; such mappings exist for any lattice. Weights for edges in $G$ are zero on city edges and are given by $w_{c}(e_{ij})$ on edges $e_{ij}$ kept from $D$ (cf. the Barahona algorithm, which instead uses $|w(e_{ij})|$, which is independent of $c$; frustration is incorporated via the use of two distinct graph decorations). Matchings in $G$ can be mapped to sets of loops in $D$: one simply contracts out the Kasteleyn cities from $M$ to arrive at a set of loops made of edges $S\subseteq E$ \[see Fig. \[cap:GadgetGraph\](c,d)\]. The Kasteleyn cities enforce the constraint that an even number of edges in $S$ meet at each dual vertex (i.e. $S$ is a collection of Eulerian subgraphs of $D$).
To prove the correctness of the algorithm, we first show that the weight of the loops that relate two configurations is proportional to the energy difference between the configurations. For an extended spin configuration $c$, let $b_{ij}(c) = s_i s_j \sigma_{ij}$. When comparing two extended configurations $c$ and $c'$, call $S$ the set of bonds in which $b_{ij}(c) = -b_{ij}(c')$ (note that $b_{ij}(c) = \pm b_{ij}(c')$ always). Since in $S$, $b_{ij}(c) = -b_{ij}(c')$ and in $E\backslash S$, $b_{ij}(c)
= b_{ij}(c')$, we have that $$\begin{aligned}
{\mathcal H}^*(c)-{\mathcal H}^*(c') & = &
-\sum_{\langle ij \rangle} J_{ij}b_{ij}(c)
+\sum_{\langle ij \rangle}
J_{ij}b_{ij}(c') \nonumber \\
&=& -\sum_{e_{ij}\in S} J_{ij}b_{ij}(c)
-\sum_{e_{ij}\in E\backslash S}
J_{ij}b_{ij}(c) \nonumber \\
& &
+\sum_{e_{ij}\in S} J_{ij}b_{ij}(c')
+\sum_{e_{ij}\in E\backslash S}
J_{ij}b_{ij}(c') \nonumber\\
&=& -2\sum_{e_{ij}\in S} J_{ij}b_{ij}(c)
\nonumber \\
&=& -2\sum_{e_{ij}\in S} w(e_{ij}(c)),\end{aligned}$$ so that the energy difference between configurations is given by twice the weight of $S$.
The proof that the minimum weight even-degree subgraph always finds the ground state, then, is as follows. Assume, for the sake of contradiction, that there exists some extended spin configuration $c^{0}$ with a lower total energy than the $c'$ returned by our algorithm from initial configuration $c$. Call $S$ the set of bonds for which $b_{ij}(c) = -b_{ij}(c')$, and $S^{0}$ the set of bonds for which $b_{ij}(c) = -b_{ij}(c^{0})$. Since ${\mathcal
H}^*(c^{0}) < {\mathcal H}^*(c')$, the energy difference ${\mathcal H}^*(c)- {\mathcal H}^*(c^{0}) > {\mathcal
H}^*(c) - {\mathcal H}^*(c')$ gives $2\sum_{e_{ij}\in
S^{0}}w_{c}(e_{ij}) < 2\sum_{e_{ij}\in S}w_c(e_{ij})$, which means $S^{0}$ is an even-degree subgraph of D with a more negative weight than $S$, in contradiction with the assumption that $S$ is the extremal weight even degree subgraph of $D$.
Note that Kasteleyn cities are often described on the original lattice, where loops represent a high-temperature expansion, but here on the dual lattice these loops contain clusters in a low-temperature expansion. The terms that contribute to the Pfaffian[@Kasteleyn] are products of statistical weights $\pm e^{-\beta J_{ij}}$ over edges in loops in $D$ and statistical weights of unit norm from the Kasteleyn cities. The dominant term in the Pfaffian that maximizes the norm of such a product minimizes the sums of the $w_{ij}$ consistent with a perfect matching in the graph $G$. We note that there has been at least one mention of using matching on the torus,[@LandryCoppersmith] where one of the four ground states was found using the Bieche [*et al.*]{} algorithm, but the utility of the extended ground state has been made apparent and proven by this algorithmic framework.
This algorithm is simple to implement (given a standard matching algorithm) and fast. On a 3.2 GHz Pentium IV processor, the extended ground state for a $100^{2}$ square lattice on a torus is computed in $0.8\,\mathrm{s}$ for Gaussian disorder, where we use Blossom IV [@Blossom] for the matching routine. The mean solution time scales approximately as $L^{3.5}$ through toroidal lattices of size $400^{2}$. On toroidal graphs with $L\le 128$, our algorithm, which finds exact ground states, is at least three times as fast as our implementation of the Bieche [*et al*]{}. algorithm. Note that the Bieche [*et al*]{}. algorithm does not find the exact optimal state in in all cases – in this case 1.5% of the samples (when $r_{\mathrm{max}}=8$). Because the structure of the graph used in the Barahona algorithm is similar in structure to that of our algorithm, the two algorithms have similar performance, with the Barahona algorithm using slightly less time (about 20%) and more memory (about 20%).
 An outline of the steps that convert a spin and bond configuration first to the dual weighted lattice $D$ and then to the weighted graph $G$, in order to compute the extended ground state. (a) The original spin system, here with initial spins $s_{i}=1$ indicated by white circles, and bond configuration $J_{ij}$ (dashed lines) determine edge weights $w_{ij}=J_{ij}s_{i}s_{j}$ (taking the initial BCs to be periodic) in the dual graph $D$ (solid vertices and edges), with periodic boundary conditions. (b) The vertices in $D$ are replaced by Kasteleyn cities (light lines have zero weight in $G$). (c) An example set $S$ of negative weight loops in $D$ is shown, with two simple loops and one winding loop. (d) Heavy lines indicate the minimum weight perfect matching $M$ for $G$ (light lines are free edges and solid circles are vertices in $G$). The negative weight loops $S$ (heavy dashed lines) are found by clipping out the Kasteleyn cities and keeping the remaining edges. Finally, spins are assigned by scanning across the sample: each time an odd number of loops is crossed, the spins are flipped (gray circles indicate $s_{i}=-1$). In this case, the inconsistency at the right side is corrected by changing horizontal boundary conditions from periodic to antiperiodic.](FIG1){width="1\columnwidth"}
On a torus, we use this algorithm to exactly solve for the extended ground state, which is closely related to, but different than, finding the ground state of the spin glass for given BCs. The ground state energies for the four possible BCs differ by $O(L^{\theta})$, which is the energy of a system-spanning domain wall. The extended ground state, the minimum of the four, therefore has at most an energy difference of $O(L^{\theta})$ from that for specified BCs.
This $O(L^{\theta})$ difference is the same order as the expected finite-size correction to the ground state energy in a periodic system, so the extended ground state is useful for studying energy densities. We computed the sample average of the extended ground state energy ${\cal H}^*_{0}$, using at least $5\times10^{6}$ samples for $L\le64$ and at least $10^{6}$ samples for sizes $128\ge L>64$, both for Gaussian disorder ($\overline{J_{ij}^{2}}$=1, $\overline{J_{ij}}=0$) and for the $\pm J$-model, $J_{ij}=\pm1$ with equal probability. We then plotted the sample average of the ground-state energy density, $e_{0}(L)=\overline{{\cal H}^*_{0}L^{-2}}$, vs. $L^{\theta-2}$, which will give a straight line where the leading finite-size correction dominates. For Gaussian disorder, we find a linear fit to be very good for $L\ge32$, as shown in Fig. \[cap:Results\](a,c), for a wide range of $\theta\approx-0.28(4)$ ($\theta$ is not precisely determined by this method; see a summary of results in Ref. ) and a highly precise estimate $e_{0}=-1.314788(4)$ (cf., e.g., $e_{0}=-1.31479(2)$ from Ref. ). Taking $\theta=0$ for the $\pm J$ data also gives a good fit, with $e_{0}^{\pm}=-1.401925(3)$ (cf. $e_{0}^{\pm}=-1.40193(2)$ from Ref. ; finite-size effects in our $L=48$ samples are less than those for $L=1800$ samples with open BCs). The extra precision results from the rapid convergence to the thermodynamic limit in boundary-free samples, which can be solved much faster than standard periodic samples solved using branch-and-cut.
 The extended ground state energy density $e_{0}(L)$ for the 2D Ising spin glass on a torus is plotted vs. scaled system size $L^{\theta-2}$. Two scales for each disorder type are used, to show the linear fit at large $L$ and the higher-order corrections at small $L$. (a,b) Assuming $\theta\approx-0.28$ for Gaussian disorder gives $e_{0}(\infty)=-1.314788(4)$. (c,d) A similar plot using $\theta=0$ for discrete values of $J_{ij}=\pm1$ gives $e_{0}^{\pm}(\infty)=-1.401925(3)$.](FIG2){width="1\columnwidth"}
In conclusion, we have linked together Pfaffian and matching methods to develop a fast algorithm for finding extended ground states in the two-dimensional Ising spin glass on a torus or standard ground states on planar graphs. For many purposes, the extended ground states on a torus are as useful as ground states that are computed for a fixed choice of periodic and antiperiodic boundary conditions, as we show by precisely computing ground state energy densities. In the Pfaffian method for computing the partition function $Z$ using the dual lattice (i.e., low temperature expansion), the dominant term in any of the four Pfaffians used to compute $Z$ is due to this extended ground state; the partition function for a specified BC combination is found by carefully cancelling out configurations with other boundary conditions in the sum. Our method therefore is a combinatorial method, based on matching, for finding the term that dominates the contributing Pfaffians at low temperature.
This work was supported in part by NSF grant DMR 0606424; we thank the KITP (NSF grant PHY0551164) for its hospitality. The valuable spin glass server at the University of Köln was used to test our algorithm. We thank M. Jünger and S. Coppersmith each for a discussion of toroidal boundary conditions.
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|
---
abstract: |
The motion and variability of the radio components in the low mass X-ray binary system Sco X-1 have been monitored with extensive VLBI imaging at 1.7 and 5.0 GHz over four years, including a 56-hour continuous VLBI observation in 1999 June. We detect one strong and one weak compact radio component, moving in opposite directions from the radio core. Their relative motion and flux densities are consistent with relativistic effects, from which we derive an average component speed of v/c=$0.45\pm 0.03$ at an angle of $44^\circ\pm
6^\circ$ to the line of sight. This inclination of the binary orbit suggests a mass of the secondary star that is $<0.9~M_\odot$, assuming a neutron star mass of $1.4 M_\odot$. We suggest that the two moving radio components consist of ultra-relativistic plasma that is produced at a working surface where the energy in dual-opposing beams disrupt. The radio lobe advance velocity is constant over many hours, but differs among lobe-pairs: 0.32c, 0.46c, 0.48c, and 0.57c. A lobe-pair lifetime is less than two days, with a new pair formed near the core within a day. The lobe flux has flux density that is variable over a time-scale of one hour, a measured minimum size of 1 mas ($4\times
10^{8}$ km), and is extended perpendicular to its motion. This time-scale and size are consistent with an electron radiative lifetime of $<1$ hr. Such a short lifetime can be caused by synchrotron losses if the lobe magnetic field is 300 G or by adiabatic expansion of the electrons as soon as they are produced at the working surface. The lobes also show periods of slow expansion and a steepening radio spectrum. Two of the core flares are correlated with the lobe flares under the assumption that the flares are produced by a coherent energy burst traveling down the beams with a speed $>0.95$c.
The radio morphology for Sco X-1 differs from most other Galactic jet sources. Possible reasons for the morphology difference are: Sco X-1 is associated with a neutron star, it is a persistent X-ray source, the source is viewed significantly away from the angle of motion. However, the lobes in Sco X-1 are similar to the hot-spots found in many extragalactic radio double sources. Scaling the phenomena observed in Sco X-1 to extragalactic sources implies radio source hot-spot variability time-scales of $10^4$ yr and hot-spot lifetimes of $10^5$ yr. [\
]{}
author:
- 'E. B. Fomalont'
- 'B. J. Geldzahler & C. F. Bradshaw'
title: |
Sco X-1: The Evolution and Nature of the\
Twin Compact Radio Lobes
---
Introduction
============
Since Sco X-1 was first detected at radio frequencies in 1969 [@abl69], many observations at radio, optical and X-ray frequencies have been made in order to understand the physical processes associated this object. It is identified with a 13-mag binary system with an orbital period of 0.787d [@got75]. The degenerate object is probably a neutron star, and the companion is an unknown spectral type with about a solar mass [@cow75]. Sco X-1 is one of the most intense persistent X-ray sources and one of about 30 presently known low-mass X-ray binary (LMXB) systems. It is a Z-type LMXB, named for a characteristic shape of its X-ray color-color diagram. It also exhibits quasi-periodic X-ray oscillations (QPO) from a few Hertz to kilo-Herz [@has89; @vdk96].
Radio observations in 1970 detected two symmetrical radio sources about one arcmin from Sco X-1 [@hje71]. They were thought to be radio lobes of Sco X-1, but were later shown to be stationary extragalactic background objects having no relationship to Sco X-1 which has a significant measured proper motion [@fom91]. The results from a multiwavelength campaign [@hje90] also suggested that there was only weak correlation of the radio emission with the optical and X-ray emission.
High-resolution radio observations at 5.0 GHz with the Very Long Baseline Array (VLBA) were initiated in 1995 in order to determine the trigonometric parallax of Sco X-1. From eight VLBA observations through 1998 August, a distance of $2.8\pm 0.3$ kpc was determined [@bra99]. This distance (previous estimates ranged from 200 pc to 2000 pc [@wes68]) established that the X-ray luminosity was near the Eddington limit as inferred from X-ray models of Sco X-1 and other z-type LMXB’s [@lam89; @pen89; @vrt91]. The radio emission during these VLBA observations showed variability in intensity and structure over periods less than one hour, with discrete components moving away from the core at relativistic velocities.
With such variable radio emission in Sco X-1, this source was established as another example of a Galactic-jet variable radio source [@mir99] with strong similarities to the jet and lobe phenomena associated with luminous radio galaxies and quasars [@bla74]. The evolutionary time scales associated with accretion disks and jets scale roughly with the luminosity and mass of the degenerate object. Thus, significant changes should occur over hours and days in the evolution of Galactic sources that would take millions of years in quasars.
Experience gained from the eight VLBA observations used to determine the parallax of Sco X-1 showed that the four to six hour observations made every six months were not sufficient to determine a coherent story for the radio emission from Sco X-1. Each observation was too short to follow much of a component’s evolution, and the individual observations were too separated in time to follow the evolution. For this reason, we observed Sco X-1 with VLBI resolution for a continuous 56-hour period in 1999 June. The results from this series of observations were crucial to understanding the radio evolution of Sco X-1 and formed a framework for understanding the previous VLBA observations.
We have already reported results associated with some of these data. The parallax was given by [@bra99], and Paper I [@fom01] discussed the kinematic properties of the radio source; the space motion of the lobes and the speed of the energy flux in the beam. This paper is organized as follows. The nine VLBI observations, and the data reduction methods are described in $\S$2. Because we display VLBI snapshots (images made from observations less than about two hours), the difficulties and ambiguities with such imaging are described. The images in $\S$3 from the 1999 June 11-13, 56-hour VLBI observation and the 1998 February 27-28, observations are presented and discussed. Then, images from the other less extensive observations are shown and compared with the more extensive observations. In $\S$4 we review the kinematic properties of Sco X-1 (Paper I) and estimate the stellar component masses of Sco X-1 from the orbital inclination. The detailed physical properties and suggested production mechanisms associated with the lobes are discussed in $\S$5. The optical, X-ray and radio properties and evolution of the core will be discussed elsewhere [@gel02]. The comparison with extragalactic radio sources is given in $\S$6, and our conclusions are summarized in $\S$7.
Observations and Data Processing
================================
Table 1 contains the log of all VLBI observations of Sco X-1. The first eight observations with the VLBA, between 1995 August and 1998 August, were planned to optimize the determination of the trigonometric parallax. The 56-hour monitoring of Sco X-1 in 1999 June was composed of a series of seven consecutive eight-hour observations amongst three different VLBI arrays: VLBA+VLA (the Very Large Array), the APT (Asia-Pacific Telescope), and the EVN (European VLBI Network + additional telescopes). Extensive optical and X-ray observations were also made during this period [@tit01; @bra02; @gel02].
Observations and Data Correlation
---------------------------------
The VLBA observations alternated between Sco X-1 for 9 minutes and the calibrator (1504-166) for 2 minutes. Three pairs of observations were made at 5.0 GHz followed by one pair at 1.7 GHz. The APT and EVN observations were made solely at 5.0 GHz and the observing schedule alternated between Sco X-1 for 19 minutes and the calibrator source for 3 minutes. The VLBA and EVN observations were observed with a total bandwidth of 128 MHz with one-bit sampling, whereas the APT observations used a total bandwidth of 32 MHz, but with two-bit sampling. The VLA was used in two ways: as one element of the VLBA and as a stand-alone array from which arc-second resolution images were obtained for Sco X-1 [@gel83; @bra97].
A background radio source, about $70''$ NE of Sco X-1, has a correlated flux density of 8 mJy at 5.0 GHz and 14 mJy at 1.7 GHz. This object is the north-eastern of the two sources near Sco X-1 which were previously thought to be related to the binary system [@fom91]. This source is sufficiently strong and compact to be used as the primary phase calibrator for Sco X-1 [@bra99] at both frequencies. We will designate this radio source as the North-East in-beam calibrator (NEIBC). Because the effective primary beam of the phased arrays (VLA, Westerbork and ATCA) were smaller than the $70''$ separation between Sco X-1 and the NEIBC, these arrays alternated observations between Sco X-1 and the NEIBC with three-minute cycles. All other telescopes pointed midway between the two sources and observed them simultaneously.
The VLBA and EVN data were processed by the VLBA correlator in Socorro, NM, USA. The data from the APT observations were processed with the S2-correlator in Penticton, BC, Canada. All data associated with the observations of Sco X-1 were correlated at two positions: (1) Sco X-1 at $\alpha=16^{\hbox{h}}19^{\hbox{m}}55.085^{\hbox{s}}$, $\delta= -15^\circ38'24.90''$; (2) NEIBC at $\alpha=16^{\hbox{h}}19^{\hbox{m}}57.439^{\hbox{s}}$, $\delta=
-15^\circ37'24.0''$ (epoch J2000).
Calibration
-----------
The observations of 1504-166 were used to determine the delay offset and delay rate for each antenna, predominantly caused by the differences in the independent clocks and oscillators, and to check the general quality of the data. The delay and delay rates were determined by using routine FRING in the AIPS software package [@aips]. The total flux density of the calibrator was measured to 2% accuracy using the VLA observations of 3C286 to determine the absolute flux density scale of the observations. The milliarcsecond structure of 1504-166 was determined from the VLBA observations, and this model was used to determine the gain calibration for all of the data, with an accuracy of about 3%.
The data for the VLBA and EVN observations consisted of 8 independent streams, at 4 contiguous frequencies each with 2 polarizations, and the APT observations contained 2 data streams. After the 1504-166 calibrations, all data streams were combined to increase the signal-to-noise ratio (SNR) for the next calibration step using the relatively weak NEIBC. With the AIPS routine CALIB, we determined the residual phase error for each antenna every few minutes from the observations correlated at the NEIBC position. At 1.7 GHz, its correlated flux density was $>10$ mJy on the longest VLBA baseline; hence, an accurate antenna phase error could be determined from two minutes of data. This time is shorter than the time-scale of phase changes except during rare periods of ionospheric turbulence. When the phase could not be connected unambiguously between consecutive phase solutions, the data between these two times were omitted from the analysis.
At 5.0 GHz the calibration strategy was more complicated. The NEIBC correlated flux density decreased from 8 mJy at the shorter baselines to 3 mJy for the longer baselines. At least two minutes (five minutes for the APT baselines) integration was needed to determine phase errors for these baselines. For low elevation observations or during periods of poor phase stability, phase solutions were ambiguous. Hence, at 5.0 GHz the NEIBC could reliably calibrate baselines shorter than 3000 km. After this initial calibration of the shorter baselines, further self-calibration of Sco X-1 successfully reinstated the longer baselines when Sco X-1 was stronger than about 5 mJy.
With the simultaneous observations of the small-diameter NEIBC, the calibration of Sco X-1 allowed us to register all images over the four-year period on the same astrometric grid with an estimated accuracy of $<0.1$ mas (Bradshaw et al. 1999). The image registration of Sco X-1 was not altered when the data were further self-calibrated in order to add the longer spacings. This registration assumed that the NEIBC was stationary with the same centroid at the two frequencies. This assumption is reasonable because: (1) The source had not varied at either frequency over the last 15 years of VLA monitoring, (2) its structure is symmetric, with an angular size of 2.5 mas at 5.0 GHz and 3.5 mas at 1.7 GHz, and (3) its spectral index is $-0.4~ (S\propto \nu^\alpha$), suggesting the lack of a significant small-diameter opaque component, which might be variable. The source was not identified to a magnitude limit of 25$^m$ (Malin, priv comm, cited in [@gel81]).
Imaging, Self-Consistency and Parameterization
----------------------------------------------
Deconvolved images of Sco X-1 were made using the routine IMAGR in AIPS and with the Caltech software package Difmap [@she97]. Images of the NEIBC were also made as a check on the quality of the calibration. Due to significant variability of Sco X-1 during the observations, image artifacts associated with the aperture synthesis of variable sources often affected an image made from more than one hour of data. For the VLBA observations, when the SNR was sufficient (the flux density was more than about 5 mJy), images could be made every hour at 1.7 GHz from one 9-minute observation and at 5.0 GHz from three 9-minute observations. If Sco X-1 was weaker than 5 mJy, longer periods of data were combined in order to obtain images with a useful SNR.
The u-v coverage in the hour-to-hour VLBA snapshots was sparse, especially during the first and last hour of an observation, when the coverage contained large gaps. An rms noise of 0.2 mJy/beam for each snapshot, a peak intensity in the range 1.5 and 20 mJy/beam, and a somewhat noisy phase calibration using the relatively weak NEIBC, when combined with the relatively poor u-v coverage and the source variability, conspired to make the determination of reliable radio images ambiguous. For some snapshots, an unrestrained deconvolution of the images produced physically unacceptable image characteristics which often had properties of the dirty beam which varied through the observation day. However, the relatively simple structure of Sco X-1 (a few bright small-diameter components along a well-defined position angle) made the deconvolution less ambiguous than with a more complicated source. The justification of this assumption is that the use of this simple model (which means that only a small region of the image was searched for emission using the deconvolution algorithms) led to consistent results for the entire body of data.
Two methods were used to judge the quality of the deconvolution of the images of Sco X-1. First, reasonable continuity of the images between consecutive epochs was expected. Sudden changes in the source structure or image features which resembled the dirty-beam suggested a poor deconvolution or bad data quality. The data were then scrutinized, the NEIBC calibration stability was checked, the source reimaged, and deconvolved. Secondly, the hourly data sets were also processed by the Difmap processing package. By fitting simple Gaussian component models directly to the visibility data, some of the ambiguities of image-plane deconvolution were avoided. If the Difmap model did not agree well with the image, further checking was necessary. Finally, the independent images made at 5.0 GHz and 1.7 GHz had to be consistent; that is, the components at the two frequencies were nearly coincident, although the 1.7 GHz lower resolution observations did occasionally show more extended emission, as expected on physical grounds. The difference in angular scale between the 1.7 and 5.0 GHz beam properties by a factor three also aided in determining real structure from beam-induced artifacts. Simulations were also made in order to quantify the reliability and proscribe the proper reduction methods. Data errors associated only with the expected level of stochastic receiver noise should cause little ambiguity in the images. Other potential data errors (unknown data drop-outs, strong source variations, rapid atmospheric phase variations over minute time-scales) were modeled, showing that relatively large errors could lead to significant image errors. For this reason we removed marginal data, especially for the longer baselines where these non-random errors occurred, with the concomitant loss of resolution for some snap-shots.
For the 5.0 GHz EVN and APT observations, a similar imaging and reduction strategy was used. The NEIBC was only useful in calibrating the European baselines in the EVN and the Australian baselines for the APT. If Sco X-1 was sufficiently strong ($>5$ mJy for the EVN, $>10$ mJy for the APT), then the other antennas could be self-calibrated using Sco X-1. However, the u-v coverage for these arrays was significantly worse than that for the VLBA, so the data had to be averaged over several hours before reliable snapshot images could be made. For the APT observations, two images separated by about three hours were made. For the EVN observation, three images, separated by about two hours, were made.
The resolution of an image depended on the SNR and the elevation of the source. For VLBA observations at 1.7 GHz, all images were convolved to a full-width half-power resolution of $10\times 5$ mas at position angle $0^\circ$, although the resolution of the observations varied somewhat during the day. At 5.0 GHz, imaging using only the shorter spacings, produced the same resolution as that at 1.7 GHz. These image pairs were used to determine the spectral index distribution across Sco X-1. At 5.0 GHz, when the source was frequently stronger than 5 mJy and self-calibration of the longer baselines was possible, the resolution was about $4.5\times 1.5$ mas during the middle part of the day. The resolution degraded to $6\times 2$ mas at low elevation angles. For the occasional periods when Sco X-1 was stronger than about 15 mJy, a resolution of $3\times 1$ mas was obtained.
The flux density, position and angular size for the discrete radio components in Sco X-1 were determined in two ways: (1) fitting Gaussian-components directly to the image, and (2) fitting Gaussian-components to the visibility data. Although errors were estimated from each of the two methods, the difference in the radio source parameters from the two analysis were used as a better accurate estimate of the real uncertainties.
The Radio Images of Sco X-1
===========================
1999 June 11-13; MJD 51340-51342
--------------------------------
### The Images and Component Parameters
Figure 1 shows the changing structure of Sco X-1 with $10\times 5$ mas resolution at 1.7 GHz and 5.0 GHz from the VLBA observations. These images, although only covering 40% of the observation, form a consistent set from which general evolution properties are evident. The vertical dashed line shows the position of the binary system, as determined from its radio parallax [@bra99]. The skew dashed-line to the left shows the location of the component northeast of the core as a function of time. On MJD 51342, the vertical line to the right shows the approximate position of the component southwest of the core. The locations of the dashed lines on the 1.7 GHz and 5.0 GHz columns are identical.
More details of the changing structure of Sco X-1 are shown in Figure 2. These 5.0-GHz images are at the highest resolution obtained during the 56-hour experiment. The image snapshots are separated by about 50 minutes for the VLBA observations and about 2.5 hours for the APT and EVN observations. The APT and EVN images have a resolution of $10\times 5$ mas, and most of the VLBA images have a resolution of $4.5\times 1.5$ in position angle $0^\circ$. The image field of view is $35\times 25$ mas for MJD 51340 and 51341 and $90\times 25$ mas on MJD 51342. The vertical lines follow the same tracks as in Figure 1.
Two estimates of the total flux density for Sco X-1 are shown in the top two plots of Figure 3. The flux density, shown in Figure 3a, was determined from the VLA observations with determination of the flux density made every 10 minutes. With a resolution of $3''$ at 5.0 GHz and $10''$ at 1.7 GHz, these measurements contain the entire emission from Sco X-1. Figure 3b shows the total flux density obtained from the VLBA images. The good agreement between the VLA and the VLBA measurements indicates that no significant large-scale structure associated with Sco X-1 was resolved out by the VLBA observations. Figures 3c, 3d and 3e show the flux density for each of the three components. In Figure 4, the spectral index $\alpha$ of each of the three components during the dual-frequency VLBA observations is shown.
### A Description of the Changes in Sco X-1
The 1999 June observations form a basis for the interpretation of the radio emission from Sco X-1. Thus, we will describe the evolution seen in the images in some detail.
The basic morphology of Sco X-1 is simple. The source is usually composed of three relatively compact components: a radio core that is nearly coincident with the binary system, a compact NE component moving away from the core; a weak SW component moving away on the opposite side of the core. The three components lie along an axis of position angle $54^\circ$ to an accuracy of a few degrees.
Changes associated with the emission from Sco X-1 are as follows:
- [**MJD 51340.1 to 51340.4.**]{} The NE component reached a maximum intensity, with $\alpha=-0.5$, at MJD 51340.2. Subsequently, it faded, became more extended with a steepening spectral index, and fell below the VLBA detection level at MJD 51340.4. It did not reappear over the next ten hours in the images from the APT and EVN. During this period, the radio core flux density steadily increased with $\alpha>0.0$.
- [**MJD 51340.4 to 51341.3.**]{} The radio core decreased in flux density from 20 mJy to 8 mJy in a few hours (the rms error for a flux density measurement is about 0.5 mJy.) No emission to the north-east was present. At MJD 51340.7 the core flux density began to increase, and by MJD 51341.0, the flux density had reached 20 mJy. The images clearly show that this rise in flux density was associated with a component emerging from the core to the northeast. Between MJD 51341.0 to 51341.3, this new NE lobe moved outward and decreased in flux density. Emission between the lobe and core was also present. The core flux density generally decreased in flux density during this period, but a small flare occurred at MJD 51341.3.
- [**MJD 51341.3 to 51341.8.**]{} After moving about 20 mas from the core, the NE component brightened, slowly at first, and then abruptly flared from 2.8 mJy to 20 mJy in less than one hour. The spectrum quickly flattened to $\alpha=-0.5$, then NE flux density slowly decayed to 1.5 mJy over the next 15 hours and its spectral index dropping to $\alpha\approx -1.2$. The radio core, during this period, remained relatively constant at 4 mJy, except for a flare to 8 mJy at MJD 51342.3
- [**MJD 51341.8 to 51342.2.**]{} The radio core brightened from 4 mJy to 20 mJy, reaching the peak at MJD 51342.0. The radio core then decreased in flux density and by MJD 51342.1 extended emission towards the northeast was clearly present. However, unlike the core flare and subsequent expulsion of emission to the NE about 1.3 days earlier, this extended emission decreased below the detection level by MJD 51342.2 and with no apparent motion. The SW component, seen better in the lower resolution 1.7 GHz images, is visible during this period. The NE component decreased slowly in flux density.
- [**MJD 51342.2 to 51342.4.**]{} For the last four hours of the observation, the NE component brightened from 1.5 to 4 mJy and the spectral index flattened again to $-0.5$. This NE component flare is less spectacular than the one about 0.8 days earlier, but has similar properties. The core remained steady at about 4 mJy with no resolvable extended structure.
1998 February 27-28; MJD 50871-2
--------------------------------
The only previous Sco X-1 observation on consecutive days occurred on 1998 February 27-28, from two six-hour VLBA observations separated by 18 hours. The upper part of Figure 5 shows three sets of snapshots, each 52 minutes apart on MJD 50871. The lower two panes show the structure on the following day when the source was weaker. Figure 6 shows the flux density of the three components on both days. Although there was some indication of extended structure between the core and the NE component, the high-resolution 5.0 GHz images show that most of the emission was contained in compact components.
On February 27, the core spectrum was inverted with flux density about 10 mJy at 5.0 GHz. The NE component had a flux density $>15$ mJy and was moving from the core in the same position angle as that in the 1999 June observations. Near the end of the observation period the flux density decreased, its spectral index became steeper and the angular size increased. This behavior will be discussed in more detail in $\S$5.7 and with Figure 14.
Sco X-1 was much fainter on the following day and individual snapshots were too noisy to show rapid changes. At the bottom of Figure 5, the 1.7 GHz and 5.0 GHz images, both with $10\times 5$ mas resolution, were made from the entire 5-hour observation. The core was unresolved with a peak flux density of 3.0 mJy, and decreased during the day. The SW component, clearly detected at about 2 mJy, was also relatively compact. The radio emission northeast of the core was complex but confined along the major axis of the radio structure, between 30 mas to 60 mas from the core at 1.7 GHz. In the 5.0 GHz image, only two features in this northeast emission region were detected above the noise level. A linear extrapolation of the NE component motion on MJD 50871 intercepts the most distant component of emission in the north-east direction on MJD 50872. Motion of the SW component was detected on both two days, and is discussed in the following section.
The Component Velocities and Orientations
-----------------------------------------
The 1998 February and 1999 June images show the major dynamical properties of Sco X-1. A compact component was always present near the binary system. It varied significantly in flux density and occasionally faint emission extended several mas towards the north-east. A compact component was usually detected northeast of the radio core and its motion was clearly directed away from the radio core. Its flux density also varied significantly over hour time scales. A compact component southwest of the core was detected about 50% of the time and was on average about 10 times weaker than the NE component.
The relative simplicity of these images of Sco X-1 is contrasted by the more complicated emission from other well-studied Galactic-jet radio sources [@mir99]. Although their structures tend to lie on a well-defined axis, they are often composed of many components on both sides of the radio core. For Sco X-1, we find that [*only one compact component was detected on the NE and SW sides of the radio core*]{} (See Figure 11 for a possible exception).
The changing separation of the NE and SW components from the core for the 1999 June (top) and 1998 February (bottom) observations is shown in Figure 7. For the 1999 June observations, the velocity in the plane of the sky for the NE component \#1, detected for 6 hours on MJD 51340, was $v=0.73\pm 0.07$ mas hr$^{-1}$ = 0.28c. The NE component \#2 was not resolved from the core until about MJD 51341.0. Then, from MJD 51341.0 to 51341.35, the NE component speed was $v=1.74\pm 0.16$ mas hr$^{-1}$ =0.68c. After MJD 51341.4, when the NE component flared, the component speed decreased to $v=1.25\pm 0.05$ mas hr$^{-1}$ = 0.49c. Although the NE component varied considerably in flux density during the last 24 hours of the observations, its speed remained nearly constant.
For the 1998 February observations, the velocity of the NE component on MJD 50871 was $v=1.11\pm 0.06$ mas hr$^{-1}$ = 0.43c over the five hours of observations, with no significant departure from linearity except for the last frame. The single point for the NE component on MJD 50872 was taken as the position of the most distance part of the NE component seen in Figure 5.
Motion of the SW components was more difficult to detect. On MJD 50872 the component was both strong and long lasting and moving away from the core with $v=0.6\pm 0.2$ mas hr$^{-1}$. On MJD 51342 the velocity was $v=0.5\pm 0.3$ mas hr$^{-1}$. In Figure 7, the dashed-lines passing through the positions of the SW components are not linear fits to its position, but to [**50% of the fitted angular separation to the NE component**]{} (Figure 12 shows a plot of this ratio from all Sco X-1 observations). Thus, we surmise that the SW component was also moving away from the core, but at a projected velocity of about 50% of that of the NE component. These motions are described in more detail in $\S$4.1.
The orientation of the NE and SW components with respect to the core for the 1999 June and 1998 February observations are shown in Figures 8a and 8b. The radio structure of Sco X-1 remained near position angle $54^\circ$ during these these two observations, and the component orientations from the previous observations of Sco X-1 also lie within a few degrees of $54^\circ$. Hence, any long-term variation in the orientation of the radio emission from Sco X-1 is less than about three degrees. It is possible to fit the position angle distribution in 1999 June observations with a mean value of $54^\circ$ plus a sinusoidal term of $\approx 3^\circ$ with a period of between two to four days. Modeling of the binary system X-ray QPO’s suggest that the neutron star/accretion disk interaction could produce a $5^\circ$ precession over several days [@tit00].
The 1995-1996 VLBA Observations at 5.0 GHz
------------------------------------------
The first four VLBA observations of Sco X-1 were made on 1995 August 19 (MJD 49948), 1996 March 16 (MJD 50158), 1996 September 16 (MJD 50340) and 1996 August 3 (MJD 50663), and contour plots of the radio emission are shown in Figure 9. These observations had lower sensitivity than later observations, were only at 5 GHz, and were about four hours in duration. The radio emission from Sco X-1 during these observations was relatively weak so that images made from one-hour slices of the data were too noisy in order to determine variability and motion. Nevertheless, all four observations showed similar properties in the emission of Sco X-1.
- [**MJD 49948.**]{} The radio core had a peak flux density of 0.32 mJy. There was a trace of both the NE component and the SW component 19 mas and 12.2 mas from the core respectively, each with a peak flux density of $<0.20$ mJy.
- [**MJD 50158.**]{} The radio core had a peak flux density of 0.50 mJy. There was a slight indication of the NE component 14 mas from the core, with a peak flux density $<0.25$ mJy.
- [**MJD 50340.**]{} The source was stronger during this observation and all three components were present well above the noise. The SW component had an integrated flux density of 2.5 mJy, located 5 mas from the core. The core flux density increased from 0.30 mJy to 0.60 mJy during the four-hour observations. The NE component was about 8 mas from the core with a peak flux density of about 0.6 mJy. The distance ratio of the NE and SW components from the core was about 2:1.
- [**MJD 50663.**]{} The core had a peak flux density of 2.0 mJy and the NE component was detected, about 26 mas away, with a peak flux density of 0.35 mJy. Any SW component was $<0.15$ mJy.
The relative weakness of Sco X-1 in 1995-1996, compared with later observations, is consistent with the general characteristics of the long-term flux density behavior of Sco X-1, as shown from the Green Bank Interferometer monitoring between MJD 50600 and 51700 (see [www.gb.nrao.edu/fgdocs/gbi/pubgbi/ScoX-1](www.gb.nrao.edu/fgdocs/gbi/pubgbi/ScoX-1)). At the monitoring frequency of 8 GHz, the quiescent level of Sco X-1 remained at 5 mJy for a period of 100 to 150 days, but there were periods of enhanced emission at 10 to 15 mJy which persisted for 50 to 100 days. About 5% of the time there were large outbursts ($>50$ mJy), which lasted over a few days. These outburst occurred randomly and were not associated with the periods of enhanced emission from Sco X-1.
1998 August 29-30, MJD 51054-5
------------------------------
Figure 10 displays the structure of Sco X-1 for a five-hour observation on August 29-30, 1998. Because of the weakness of the source, each image contains data from the entire five-hour observation. All three components were present, although the core was barely detectable at either frequency. The NE component was clearly extended and contained several emission peaks along the source major axis. The SW component was compact and relatively strong. The appearance of Sco X-1 on this day was similar to that on MJD 50872, shown at the bottom of Figure 5. However, the linear size of the source on MJD 51054 was a factor of three smaller. Again, the ratio of the separation from the core of the NE and SW components was about 2:1.
1997 August 22, MJD 50682
-------------------------
During this five hour VLBA observation, Sco X-1 was extremely bright and nearly all of the emission was contained within 5 mas of the core, as shown in Figure 11. The high-resolution image showed emission moving outward from the binary system, particularly on the north-east side of the core. The relatively bright spot of emission at the extreme edge of the extended emission was advancing at a velocity of $1.4\pm 0.3$ mas h$^{-1}$. Some emission was also detected SW of the core. This morphology and evolution was similar to that seen on MJD 51341.0 when the second NE component was ejected from the core. The spectral index of the emission within 5 mas of the core rose from $\alpha=0$ to $\alpha=0.3$ during the five hour observation.
A faint component, about 20 mas NE of the core, was also detected. This is the only example of two radio components on one side of the core. However, the distant component is extremely weak with no measured motion. The radio structure on this day will be discussed in more detail elsewhere [@gel02].
Orientation of the Binary system
================================
Doppler-Beaming of NE and SW Components
---------------------------------------
The kinematic results discussed in the section, have already been reported in Paper I. We present more detail in this section.
The images from the 1998 February and 1999 June observations showed that there was often a compact NE component moving away from the core. Although this component varied significantly in flux density, its velocity remained relatively constant over periods of many hours. When the NE component was well-separated from the core, its measured speeds [*in the plane of the sky*]{} were 0.28c, 0.43c, 0.49c and 0.68c. On MJD 51341 and MJD 50682 a NE component emerged from the core. Because of a lack of sufficient resolution, the emergent velocity was not well-defined, although it was about 0.5c.
Motion of the SW component away from the core was observed during the 1998 February and 1999 June observations, with $v=0.6\pm 0.2$ mas hr$^{-1}$, and $v=0.5\pm 0.3$ mas hr$^{-1}$, respectively. Further detection of the motion was not possible because of the weakness and intermittent detection of this component. A comparison of the NE-to-core and the SW-to-core component distance ratio of all the observations is shown in Figure 12. There is no systematic variation with distance and the average ratio is $0.51\pm 0.02$. Since the NE component was moving away from the core, we infer that the SW component was also moving away from the core in the opposite direction with 51% of the velocity of the NE component.
Another comparison of the radiative properties of the NE and SW components is given in Figure 13, which displays the spectral index versus flux density for the two components in Sco X-1 from all of the observations whenever it was determined. For comparison, the spectral properties for the core are shown [@gel02]. When the flux density of the NE component was stronger than 4 mJy its spectral index was $\alpha\approx -0.6$ but decreases to $<-0.9$ at fainter flux densities. The SW component was detected less than 50% of the time and often only at 1.7 GHz. It greatest flux density was 5 mJy at 1.7 GHz and the spectral index was about $-0.6$; hence, the flux density at 5.0 GHz is about half of that at 1.7 GHz. We conclude that on average the apparent brightness of the SW component is about ten times fainter than that of the NE component. Furthermore, the spectral index of the SW component at its brightest was about equal to that of the NE component at its brightest. We will discuss the flux density ratio between the two lobes in more detail in $\S$5.9
The relative positions and velocities of the NE and SW components, their relative flux density (in a statistical sense since both components vary considerably), and their spectral properties suggest strongly that the NE and SW components have similar properties, with the observed emission differences resulting from relativistic aberration and Doppler beaming. The observed apparent motion in the plane of the sky of a component moving with a velocity v (usually given in term of c as $\beta=$v/c), at an angle of $\theta$ to the line of sight is [@bla77]:
$$\beta_{NE} = \frac {\beta~\hbox{sin}(\theta)} {1-\beta~\hbox{cos}(\theta)}$$
$$\beta_{SW} = \frac {\beta~\hbox{sin}(\theta)} {1+\beta~\hbox{cos}(\theta)}$$
where $\beta_{NE}$ is the observed velocity in the plane of the sky of the component approaching the observer and $\beta_{SW}$ is the velocity of the receding components.
We have measured a range of speeds for the NE component. However, the nearly constant position angle of the axis of Sco X-1 in the plane of the sky (Figure 8) strongly suggests that the direction of the space velocity of the lobes with respect to the line of sight is also constant. To determine an average speed of the two Sco X-1 components, we considered the time ranges MJD 51341.0 to 51342.4 and MJD 50871.3 to 50872.4. These periods are instances when the NE component speed was determined over a long period of time and the SW component was relatively strong. These two periods yield a measured proper motion of $v=1.25\pm 0.05$ and $v=1.11\pm 0.06$ mas hr$^{-1}$. We will thus assume that the average speed of the NE component of Sco X-1 of $1.18\pm 0.08$ mas hr$^{-1}$. At a distance of $2.8\pm 0.3$ kpc for Sco X-1, the average speed of the NE component in the plane of the sky is then $\beta_{NE}=0.46\pm 0.08$, with the distance uncertainty incorporated in the error. Next, we adopt the ratio of the SW speed to the NE speed as $0.51\pm 0.02$. This ratio is a function of the speed, but should be the appropriate value that we have adopted. Equations (1) and (2) become:
$$\frac{\beta_{SW}}{\beta_{NE}} = 0.51\pm 0.02 = \frac{1-\beta~\hbox{cos}(\theta)}{1+\beta~\hbox{cos}(\theta)},$$
$$\beta_{NE} = 0.46\pm 0.08 = \frac{\beta~\hbox{sin}(\theta)}
{1-\beta~\hbox{cos}(\theta)},$$
from which we obtain $$\beta = 0.45\pm 0.03;~~~\theta = 44^\circ \pm 6^\circ.$$ The quoted errors are the one-sigma uncertainties and include the measurement errors.
The determination of the Doppler boosting associated with the flux densities of the NE and SW components requires a specific model of their internal structure, particle motion and magnetic field geometry as discussed in $\S$5.5. If we assume that the moving components are transparent, with all particles moving at the bulk velocity and radiating isotropically, then the flux density ratio, $R$ is given by [@bla79; @caw91]: $$R= \left[\frac{\beta_{SW}} {\beta_{NE}}\right]^{k-\alpha}$$ where k is a geometric factor which equals 2 for an optically thin source, and $\alpha$ is the spectral index of the source emission. From the measured $\alpha=-0.6$ we obtain a predicted flux density ratio of R=0.17, which is somewhat larger than that observed (see $\S$5.9 for more details). Thus, the space motion derived solely from the relative motions of the two components is in reasonable agreement, considering many uncertainties about the radiation properties and variability of the lobes.
Sco X-1 Binary Masses
---------------------
Assuming that the radio beam lies along the rotation axis of the accretion disk which in turn lies in or close to the orbital plane of the binary system, we can better determine some of the properties of the binary. The relative constancy of the radio axis over five years are supports the above association.
The mass function for the Sco X-1 binary has been derived from the measured radial velocity of the HeII $\lambda$-4686 line, presumably from the accretion disk, and is given by $F(M)$ = $(M_2~\hbox{sin}~i)^3 / (M_1+M_2)^2$ = $0.016\pm 0.004$ (our estimated error), where $M_1$ and $M_2$ are the masses of the neutron and secondary star, respectively, and $i$ is the inclination of the orbit [@cow75]. The binary is composed of a degenerate star, almost certainly a neutron star [@has89]. Measurements of many neutron star systems produced estimated masses between 1.2 to 1.6 $M_\odot$ [@lew95]. The intrinsic X-ray luminosity of Sco X-1, for a distance of 2.8 kpc, also suggests a neutron star mass of about $1.4~M_\odot$. The calculated secondary mass $M_2$, from the best value of the mass function, inclination and neutron star mass, is ($0.63\pm 0.26)~M_\odot$. Such a low companion mass is a problem for most models of the energetics in Sco X-1. First, the star may not fill its Roche lobe resulting in an accretion rate that is too low; perhaps the companion is an evolved star. Secondly, infra-red spectroscopic observations of the binary system suggest that the secondary star is earlier than G5 [@ban99]. However, the X-rays from Sco X-1 in all likelihood heat up the atmosphere of the secondary, puffing out its atmosphere until it overflows the Roche lobe reulsting in mass accretion onto the compact star.
The secondary star mass estimates can be increased in the several ways. If the degenerate star mass is $>3.5~M_\odot$, then the secondary star would have a mass $>1$ M$_\odot$. But, then what is the nature of the degenerate star? A mass function $>0.025$ would also increase the secondary star mass associated with a $1.4~M_\odot$ neutron star. This implies that the optical lines are coming from a much larger region than the accretion disk near the neutron star. Thus, the masses of the neutron star and the secondary member of the Sco X-1 binary are still uncertain even with knowledge of the orbit inclination.
The Nature of the NE Component
==============================
Angular and Linear Size
-----------------------
The angular size of the NE component could be measured with an accuracy of 0.5 mas when it had a flux density $>5$ mJy. Figure 14 shows a plot of the angular size and motion of the NE and core components for the seven snapshots on MJD 50871 (see Figure 5 also) when both the core and NE components were relatively strong. The angular sizes were determined by fitting the visibility data directly to an elliptical Gaussian model and by fitting the component brightness on the image and deconvolving the effects of the beam shape. Both methods gave consistent results.
The comparison of the positions and sizes of the core and the NE components provides a good indication of the robustness of these measurements. The core was about 10 mJy, relatively stationary and extended in the direction along the source major axis, as was commonly observed at other epochs. Alternatively, the NE component was extended orthogonal to the direction of motion and was clearly moving away from the core. Since the effects of noise and poor phase calibration will blur all of the emission in the same way, the different size and orientation of the core and NE components strongly suggests that these measured angular sizes and orientations are valid.
For snapshots 1 through 3 in Figure 14, the measured average angular size and orientation of the NE component itself was ($1.5\pm
0.15) \times (0.9\pm 0.2$) mas in position angle (pa) $140^\circ\pm
15^\circ$. The angular size in snapshot 4 is marginally bigger and the size for snapshot 5 is $2.4\times 1.3$ mas in pa $120^\circ$. In the 50-minute separation between snapshot 5 and 6, the component increased in area by a factor of 4.5, to $5.0\times 2.9$ mas in pa $150^\circ$, an expansion at a velocity close to c. In snapshot 7 the size of the component did not increase substantially, but the bulk motion of the emission may have decreased. The angular size of the NE component was also accurately determined on MJD 51341.4 and MJD 51342.3. The angular sizes were ($1.6\pm 0.1) \times (0.7\pm 0.3$) mas in pa $135^\circ\pm 10$ and ($2.4\pm 0.3) \times (0.8\pm 0.4)$ ma in pa $155^\circ\pm 30^\circ$, respectively, in good agreement with the minimum size of the NE component on MJD 50871.
The NE component is clearly extended in the direction perpendicular to its motion. In addition, the measured size in this direction of motion is slightly increased by the component motion during each snapshot over a 33-minute span. During this period, the component moves 0.6 mas; hence, the measured size of 0.9 mas corresponds to a true size of about 0.7 mas after removing the blurring from the motion. Thus, we derive a minimum component size (full-width at half power) of $1.5\times 0.7$ mas, corresponding to $6.3\times 10^{8}$ km by $2.9\times 10^{8}$ km with the major axis perpendicular to the orientation of the source. More detailed measurement of the NE component structure, other than its angular size, cannot be obtained with the present resolution and SNR.
The flux densities and spectral indices for the NE components are also listed in the Figure 14. The relationship between the angular size, flux density and spectrum of the NE component are discussed in $\S$5.7.
Component Energetics
--------------------
The calculation of the energetics of the NE component used the following parameters: a source distance of 2.8 kpc ($8.6\times
10^{16}$ km), a flux density at 1.7 GHz of 30 mJy, $\alpha=-0.6$, with the radio cutoff frequencies of $10^7$ to $10^{11}$ Hz. The volume of the component (ellipsoid of diameters $1.5\times 1.5\times 0.7$ mas with the narrow dimension in the direction of the component advance) is $6.1\times 10^{40}$ cm$^3$. The integrated flux density is $6.5\times 10^{-21}$ erg s$^{-1}$cm$^{-2}$, the total observed radio luminosity is $6.1\times 10^{30}$ erg s$^{-1}$. We assume a factor of 3 to correct for the Doppler boosting of the NE component (and Doppler attenuation of 3 for the SW component) as implied by the average flux density ratio of the two component . The radio luminosity in the frame of the source is $2.0\times 10^{30}$ erg s$^{-1}$, corresponding to a brightness temperature of about $1\times 10^9$ K. The total energy in relativistic particles $E_{e}$, and total energy in the magnetic field $E_{H}$, assuming the above radiating volume, are $$E_{e} (\hbox{ergs}) = 4.0\times 10^{37}~H^{-1.5}$$ $$E_{H}(\hbox{ergs}) = 2.4\times 10^{39}~H^2,$$ where $H$ is the magnetic field in Gauss.
The radiative lifetime of the emitting particles is about one year if magnetic field energy and kinetic energies are about equal. For example, if we assume that the protons contain 100 times the energy as the electrons, the equipartition magnetic field is about 1.0 G with a minimum total energy of $6\times 10^{39}$ erg in the component. If the electron energies dominate, then the minimum total energy is about $5\times 10^{38}$ erg and has a field strength of about 0.3 G. Such a long radiative lifetime is not possible for the NE component since it is variable on an hour time-scale and a size of $5\times 10^8$ km (30-min light travel time). This problem will be discussed in $\S$5.6.
What is the NE component?
-------------------------
Emission confined to relatively discrete components, and which show relativistic motion have been observed in many Galactic X-ray binary systems [@mir99]. Three explanations are generally given for the energetics of these components: (1) Ejected clouds, where the components are clouds of radiating particles, expelled from the binary in a preferred direction, but containing their own energy source; (2) Interaction within beams; where the energy flow within twin-beams is collimated by an accretion disk of the massive object and interacts with embedded matter or entrained material to produce radiating electrons; (3) a working surface where the energy flow in the efficient twin-beams is relatively invisible until the flow impinges on external material and forms a bright, small region of highly relativistic particles. The beam also penetrates through the external medium at relativistic velocities. We suggest that the moving components on opposite sides of Sco X-1 are regions of intense radio emission, generally denoted as [*lobes*]{}, formed from the interaction of a twin-beam with the ambient medium in a confined region called a [*working surface*]{}.
There are two properties of the components in Sco X-1 that that are best explained by this lobe model. First, the rapid variability and lack of systematic decay of these components implies that the radiating electrons have a lifetime of less than one hour and are constantly resupplied with energy. In fact, the NE component variability is similar to that of the core. In contrast, the emission from the relativistic moving components associated with the other well-studied Galactic-jet radio sources are not as variable and tend to decrease fairly uniformly as they move away from the radio core [@fen99; @hje95b]. Secondly, the observations of Sco X-1 suggest that only one pair of components (NE and SW) occur at any given time (see Figure 11 for a possible exception). The core does show activity and extended emission towards the northeast direction, but more than one compact component is not observed on either side of the core. For the other Galactic-jet sources, several components are usually visible on one side of the core and they are often associated with X-ray and radio flaring events in the core region.
However, the later stages of the lobe evolution in Sco X-1 did show a more complicated structure. For example, the radio emission from Sco X-1 on MJD 50872 and MJD 51054 had an extended NE component containing several bright regions. On MJD 50872 this extended NE component is clearly associated with the moving compact component observed on the previous day and consists of an extended lobe towards the core with a few small brighter regions. Perhaps this extended emission is back flow from the lobe region seen in many extragalactic objects [@bla92].
The Advance Velocity of the Lobe
--------------------------------
The speed of advance for three different lobe-pairs were measured over the range 0.32$<\beta<$0.57. The NE lobe on MJD 51340 moved at 0.32c just before evaporating. The lobe on MJD 50871 moved at 0.43c. The lobe on MJD 51341-2 moved initially at 0.57c, but abruptly decreased to 0.46c just after a lobe flare. The advance speed during each epoch was remarkably constant over periods of many hours, up to one day.
A prediction of the advance speed of the lobe requires a detailed model of the interaction of the beam with the ambient medium, A more general approach [@bla79] has derived the velocity of the advance of a lobe under the assumption that the relativistic shock produces an ultra-relativistic equation of state on [*both*]{} sides of the shock. This region is generally called the [*working surface*]{} and this region produces a radio hot-spot which may be a small, but intense, part of the lobe. If $p_{1}$ and $p_{2}$ are the pressures and $v_{1}$ and $v_{2}$ are the velocities in the unshocked (beam) and shocked (working-surface) regions, then $$\frac{v_1}{c}= \sqrt{\frac{p_1+3p_2}{3(3p_1+p_2)}}~~~~\frac{v_2}
{c}= \sqrt{\frac{3p_1+p_2}{3(p_1+3p_2)}}.$$ When the beam pressure is much greater than the shock pressure, the advance speed is 0.33c. For more dominant shock pressure, the advance speed eventually reaches c. This model gives results which are consistent with the range of observed speeds using reasonable pressure ratios across the hot-spot. However, the small variation of the observed speed with time is still puzzling.
Lobe Emission Geometry
----------------------
The ultra-relativistic fluid produced at the working surface fills the larger volume of the radiating lobe. Figure 15 shows a schematic of the general appearance of the lobe based on some very simple assumptions. There are two velocities associated with the lobe: the average bulk motion velocity of $\approx 0.45$c, and a diffusion velocity of 0.57c associated with this relativistic gas. We assume further that the radiative lifetime of the emitting particles is about 30-minutes as required by the variability time-scale. The wagon-wheel diameter (in the frame of the lobe) is then $6.2\times 10^{8}$ km in diameter, and the thickness is the distance traveled by the lobe in 30 minutes, $2.4\times 10^{8}$ km. The lobe, however, viewed at an angle of $44^\circ$ to the line of sight, would make the component appear somewhat more circular than the above model.
A deeper analysis of the physical processes in the lobes requires detailed hydrodynamic jet modeling and the radiative transfer of the electrons. The modeling of [@mio97], for example, shows striking similarities between the structure of Sco X-1 with some of their simulated models at a viewing angle of $45^\circ$ to the line of sight (see their Figures 1d, 5c, 16a). The predicted peak brightness ratio between receding and advancing lobes is R=0.05. However, this model assumes an isotropic magnetic field distribution. In any case, there seems to be a small family of models whose parameters can be chosen to match the morphology as well as the physical properties and characteristics of the lobes.
Lobe Emission Lifetime
----------------------
The lifetime of the radiating plasma in the lobe cannot be much longer than 30 minutes, the variability time-scale. The two most likely sources of energy depletion are from synchrotron losses or from adiabatic expansion of the plasma. First, if equipartition holds between the magnetic energy and particle energies as discussed in $\S$5.2, then the time scale for synchrotron losses is about one year for the equipartition field strength of about 0.5 G. The radiative lifetime decreases with as H$^{-1.5}$; hence a one hour decay time requires a magnetic field strength of H=300 G. The derived lobe energies are then $$E_H\hbox{(ergs)} = 2\times 10^{44} \hbox{ for H= }300~G$$ $$E_{e}\hbox{(ergs)} = 7\times 10^{33} \hbox{ for H= }300~G.$$ The magnetic energy is then a factor of $10^8$ larger than the equipartition value, even assuming a density of protons about 100 times that of the electrons. Magnetic field amplification just in advance of the shock has been suggested [@dey80; @hug81], although not to this extreme. With a lobe brightness temperature $<10^9$K, the rate from inverse-Compton losses is much less than for synchrotron losses even with this large magnetic field. Such a large magnetic field requires a lobe energy density which is about $10^5$ times larger than the minimum energy value. The outer part of the radio lobe would also have a steeper radio spectrum than near the working surface and the 1.7 GHz angular size would be larger than that at 5.0 GHz. Unfortunately, the present observations just barely resolve the lobe at 5.0 GHz and such a spectral gradient is not observable.
The radiating electrons generated within the working surface can also rapidly lose energy if they expand adiabatically to fill the lobe region. Assuming that the radio emission is optically thin (the radio spectral index $\alpha\approx -0.6$ is consistent with this) and neglecting the motion of the lobe for the moment, the steady-state emission profile of an expanding spherical cloud which contains a steady source of accelerated electrons within a radius $r_0$, the working surface region, is [@bal93; @hje95a] $$S(r) = S_0(r/r_0)^{4\alpha-1}$$ where $S_0$ is related to the inserted energy per unit time in the working surface volume, $r$ is the distance from the center of the working surface. For $\alpha=-0.6$, the intensity at $r/r_0=1.22$, is already less than half of the intensity at the working surface. Hence the adiabatically expanding electrons need not diffuse far from the working surface in order to lose most of their energy. From the measured lobe size of about 1 mas and a diffusion speed of 0.57c, the electrons will lose most of their energy in about 40 minutes. The bulk motion of the lobe is somewhat less than the diffusion speed and does not substantially change the above calculation. Adiabatic expansion does not modify the spectral index of the emission across the lobe, in contrast to the spectral steeping with synchrotron ageing. In the case of pure adiabatic expansion, the working surface is an appreciable part of the observed radio lobe.
A combination of both loss mechanisms may be operating in the lobes of Sco X-1. For example, the volume close to the working surface may have a large magnetic field which completely dominates the plasma dynamics. Further away from the working surface, adiabatic expansion may be the dominent loss mechanism. This would reduce the total energy content needed to supply the lobe emission compared with that needed if only synchrotron losses were operating.
Lobe Expansion
--------------
A slow expansion of the lobe angular size over an hour or two time-scale was observed several times during the observations of Sco X-1. The best example of the expansion of a lobe was on MJD 50871 after frame 3 in Figure 14. Between frames 5 and 6 the angular size increased by a linear factor of 2.2 in 50 minutes. As discussed in the above section, for pure adiabatic expansion with a uniform input energy source, the flux density in frame 6 should have decreased $(2.2)^{4\alpha-1}$ [@hje88] or about 4% of that in frame 5. However, the flux density at snapshot 6 was 60% of that in snapshot 5.
Without higher resolution observations of the lobe we can only speculate on this slow lobe expansion phenomenon. Since adiabatic expansion alone would produce a much more rapid decrease of flux density than that observed, we suggest that the increasing size and decreasing intensity of the lobe reflects the energy flow and size of the working surface. The somewhat decreasing spectral index could be related to the acceleration process in the working surface, or the effect of synchrotron ageing. Note that this lobe, although decreasing in flux density at the end of MJD 50871, is present on MJD 50872 as an extended lobe along the axis of the source. Hence, this lobe was reinvigorated sometime after the end of the observations on MJD 50871. Much more data are needed in order to explain the general appearance and evolution of these components.
A similar expansion and spectral steepening of a lobe occurred during the first four hours on MJD 51340. From the images in Figure 2, the component clearly expanded, the flux density decreased and the spectral index steepened to $-1.5$ until the component fell below the detection level of the observations.
Energy Flow Velocity in the Beam
--------------------------------
Paper I we showed that the the flux density variations of the radio components have good temporal correlation when assuming a model of energy flow from the core to the lobe where: (1) a core flare is associated with an event near the binary system (2) a burst of energy associated with this event travels down the twin-beam at a velocity of $\beta_{j}$, (3) this increased energy flux intercepts the NE and SW lobes and produces a flare.
The appropriate time delays between the NE and SW lobes with respect to the observer/core reference frame can be determined using the known lobe velocity $\beta$ and orientation $\theta$ to the line of sight. This model predicts the delay of the lobe flares $\tau_{NE}$ and $\tau_{SW}$, with respect to the driving core flare as viewed by the observer, to be: $$\tau_{NE} = (t_1-t_0)\beta
\frac{(1-\beta_j\hbox{cos}\theta)}{(\beta_j-\beta)},$$ $$\tau_{SW} = (t_1-t_0)\beta
\frac{(1+\beta_j\hbox{cos}\theta)}{(\beta_j-\beta)}.$$ Here $t_0$ is the time of ejection of the lobes from the core, and $t_1$ is the time of a core flare. With this model, the NE lobe should flare at time $t_1 + \tau_{NE}$ and the SW lobe at time $t_1 +
\tau_{SW}$ later than the core, as viewed by the observer. The term $(t_1-t_0)\beta$ is the true distance of the lobes from the core at time $t_1$.
Figure 16a shows the observed flux density versus time for the three components. Figure 16b shows the flux density correlations after correcting for the model delay, assuming $\beta_j=1$. Figure 16c shows the same data assuming $\beta_j=0.90$ and the correlations between the core with the NE and SW lobe flux density flares are significantly worse. Additional discussions of the correlations and its relationship to various properties of Sco X-1 were presented in Paper I.
A lower limit to the flow speed in the beam, as indicated by the difference in correlation between Figures 16b and 16c, can be determined more quantitatively. It seems unlikely that the apparent correlations in Figure 16b are pure chance. There are three pairs of flares (C3-L3, C4-L4, C3-S3) which are clearly correlated; all are associated with a significantly different delay, using a model with only one free parameter, derived solely from the positional information of the components. The correlation of the flares for $\beta_j=1.0$ is clearly better than that for $\beta_j=0.9$ (and by symmetry for $\beta_j=1.1$), but how much better?
An estimate of the uncertainty can be obtained by determining the speed necessary to align each of the three flare correlations (C3-N3), (C4-N4) and (C3-S3). These results yield the three independent estimates of $\beta_j$ of $1.10\pm 0.10$, $1.00\pm 0.04$, and $1.02\pm
0.04$ respectively, which gives a weighted average $\beta_j=1.02\pm
0.04$. Changing the space velocity of the lobes within the estimated uncertainty affects $\beta_j$ only at the few percent level. The relatively large beam speed of 1.10c associated with flare (C3-N3) may be caused by an additional delay in the flare of the core. We have suggested that the core may be at the base of the jet, at a distance of one or two mas from the binary. This would cause a delay in the core flare reaching the observer by $\approx 0.1$ hours, compared the event in the binary system.
Thus, we derive a lower limit of $\beta_j>0.95$ based on the above analysis. Although the analysis indicates that this limit is significant at the two-sigma level, the model may be too simplistic. We believe that the limit has a confidence level of 70% (one-sigma).
Flux Density Variations and Doppler-Boosting
--------------------------------------------
Based on the space motion of the lobes, we calculate that Doppler boosting will produce a flux density ratio of the SW lobe to the NE lobe of R=0.17 assuming a uniformly filled lobe with an isotropic magnetic field. The observed flux density ratio is difficult to determine because of the variability of the lobes and the time delay in their reference frames. However, an estimate can be obtained by two methods. First, Figure 13 shows the distribution of the flux densities of the NE and SW components over all of the observations. At 5.0 GHz the highest flux densities observed for the NE component are about 18 mJy. For the SW component, the largest flux densities are 4.5 mJy at 1.7 GHz, which corresponds to about 2.2 mJy at 5.0 GHz assuming $\alpha=-0.6$. This ratio is R=0.12. A discussion of the core and its relationship to the x-ray and optical emission is given elsewhere [@gel02]. Second, another estimate of the Doppler-boosting can be made from the flare peak \#3 for the NE lobe and the SW lobe, shown in Figure 16. The increased flux densities at 5 GHz in the flare, compared with the baseline derived from a few hours before and after the flare, are: Core flare = 3.5 mJy, NE lobe = 17.0 mJy, and SW lobe = 1.0 mJy. The ratio between the SW and NE lobe flare height is R=0.06. This ratio assumes that the energy flow in the twin-beams are the same. Finally, hydrodynamic modeling of a source with similar properties as Sco X-1 [@mio97] gives suggests a ratio of R=0.07 for the peaks of the simulated lobes–which is resolution dependent. All of these estimates are lower than the R=0.17 expected purely from the space motion; however, any complexity in the emission properties of the lobes (eg. internal structure, non-isotropic magnetic field) tends to decrease this expected ratio [@caw91]
Comparison with other Galactic-Jet Sources
------------------------------------------
Why does Sco X-1 show a lobe phenomenon when the observations of other well studied Galactic-jet sources generally show multiple components? The general consensus from the other well-studied Galactic-jet objects like GRS1915+105 (Dhawan et al. 2000, Fender et al. 1999), GRSJ1655 (Tingay et al. 1995; Hjellming & Rupen 1995), CygX-3 (Geldzahler et al. 1984; Marti et al. 1992. Schalinski et al. 1993; Mioduszewski et al. 2001), SS433 (Margon 1984; Spencer 1984), and Cir X-1 (Fender et al. 1998), is the radiating material is probably associated with discrete clouds or shocks being energized within a collimated flow of energy. Most of these objects are transient X-ray sources, preferentially observed just during or after a strong flare. These flares may be associated with the ejection of a large amount of material that is entrained in a collimated flow of energy, producing strong radio emission. However, there are several periods when the core of Sco X-1 showed changing radio emission usually within 5 mas towards the northeast. Two of these events are associated with the emergence of the new NE component on MJD 51341.0 (Figure 2) and MJD 50682.0 (Figure 11), and have already been discussed. After the core flare at MJD 51341.95 (Figure 2), extended emission to the NE persisted for about four hours. A compact NE component more than 15 mas from the core was also present. Figure 14 shows that the core on MJD 50871 was also slightly extended by about 3 mas toward the north-east direction. The extended emission near the core may be associated with material in the beam flow.
The degenerate object associated with Sco X-1 is probably a neutron star, although the improved orbital parameters, based on the orientation of the source discussed in $\S$4.2, do not confidently constrain the degenerate star mass to less than about 2 M$_\odot$. For most other Galactic-jet sources, the degenerate star is probably too massive to be a neutron star. Sco X-1 is also a persistent X-ray source, unlike many of the other well-studied Galactic transient X-ray sources. The relationship between the X-ray properties with the nature of the degenerate star and with the properties of the radio emission is still unknown.
Finally, the major difference between Sco X-1 and the other sources may be the viewing angle of the source as seen by the observer. The simulated images [@mio97], which showed a radio structure structure surprisingly similar to that seen in Sco X-1, take on the appearance of the other Galactic-jet sources when the viewing angle is decreased. At small viewing angles, the shocked material moving near c are greatly Doppler boosted and dominate over the emission from the more slowly-moving lobes. Thus, like the extra-galactic sources, the radio appearance of Galactic-jet sources may also be a strong function of viewing angle.
Comparison with Extragalactic Sources
=====================================
One of the major reasons for studying Galactic-jet sources is to understand better the jet phenomenon in extragalactic radio sources. The appropriate scaling laws come from considerations of the accretion disks and infalling mass [@beg84]. The radio emission is a second order product of the accretion process, but should also follow the scaling. The luminosity ratio of Sco X-1 to Cygnus A, for example, is $10^{-7}$. The mass ratio of the neutron star in Sco X-1 and the massive object in Cygnus A is $10^{-9}$. We will adopt as a typical scaling parameter $x$, from a radio galaxy to Sco X-1 $x\approx 10^{-8}$. The luminosity, the time scale and the size will scale as $x$, the density as $x^{-1}$, the magnetic field as $x^{-0.5}$, and velocity and temperature are invariant to first order. The following are some examples of simple scaling of the properties of Sco X-1’s radio lobes to those in extragalactic radio sources.
Two of the best studied extragalactic radio sources with strong lobes are Cygnus A [@car91] and Pictor A (Perley et al. 1997). Comparisons of these and other sources with Sco X-1 indicates:
- [**Lobe Variability.**]{} The lobes in Sco X-1 vary on an hour time scale. This scales to $1\times 10^4$ yr for radio galaxies. Variability in extragalactic lobes have not yet been observed over about a 30-year span.
- [**Lobe Lifetime.**]{} The lifetime for a pair of lobes in Sco X-1 is about one day. This scales to $5\times 10^5$ yr for a typical extragalactic source, and is about the estimated age of the Cygnus A hot spots, based on the curvature of their radio spectrum [@car91; @cox91].
- [**Lobe Regeneration.**]{} In Sco X-1 there appears to be replenishment of the radio lobes. That is, after a pair of lobes fades away, another pair emerges from the core, probably in less than one day. The one well-documented disappearance of the lobes (MJD 51840) occurs after an X-ray flare and the emergence of an optically-thick core component. The suggestion that the the beam flow has been disrupted and later reforms is discussed elsewhere [@gel02]. Although a short lifetime of $10^5$ yr for hot-spots in radio galaxies has been suggested from lobe modeling [@cox91] and observations of radio galaxy lobes [@bla92], new hot-spots are believed to appear in the established radio-lobe region rather than being reformed near the core. Hence, unlike Sco X-1, the beam flow in extragalactic sources may not be completely disrupted.
- [**Lobe Size.**]{} The linear size of the lobe in Sco X-1 of $2\times 10^{8}$ km scales to 500 pc for radio galaxies, similar to that observed in radio galaxies [@bla92]. The western lobe in Pictor A [@per97] is well-resolved and is somewhat more elongated perpendicular to the presumed input energy flow, as found for Sco X-1.
- [**Lobe Distance from Core.**]{} The lobe separation of Sco X-1 is about 40 mas or $1.6\times 10^{10}$km. This scales to a linear size of $1.6\times 10^{18}$km or 50 kpc for the typical size of a radio galaxy.
- [**Spectral Index.**]{} The spectral indices associated with the lobes of Sco X-1 are similar to those seen in extragalactic sources, where the most confined lobes have $\alpha\approx -0.5$ and the less confined have steeper spectral index.
- [**Opening angle.**]{} The ratio of the lobe size to the distance from the core is probably related to the angle opening in the beam. For Sco X-1 this angle is typically about $1^\circ$. A similar ratio is observed for extragalactic double sources that have prominent hot-spots in their radio lobes.
- [**Magnetic Field Strength.**]{} The equipartition magnetic field in Sco X-1 was $\sim 1$ G. If adiabatic expansion of the plasma in the lobes is not occurring, the field needed to decrease the electron emitting lifetimes from synchrotron losses to one hour was $\sim 300$ G. These two magnetic fields strengths scale to $1\times 10^{-4}$ G (which is similar to the equipartition field in Cygnus A) and $3\times
10^{-2}$ G respectively. The configuration of the magnetic field in the lobes of Pictor A appears to be circumferential [@per97]. The degree of polarization in Pictor A is also high, indicating an ordered field. No lobe polarization information is available for Sco X-1. However, if the lobes are dynamically controlled by a large magnetic field, a radial magnetic field would be more likely with Sco X-1.
A major difference between the radio lobes of Sco X-1 and those associated with radio galaxies is the apparent simplicity of the Sco X-1 lobes, usually consisting of two hot-spots and no extended emission. In contrast, radio galaxies often show an extended lobe, with a cocoon of emission, and a hot-spot. Emission from the beam between the core and the lobe is also observed in some cases. Often a secondary hot-spot is observed in a radio lobe, not far from the more compact one. These secondary hot-spots are thought to be caused by either the splatter of material from the primary hot-spot, or the remnants of older hot-spots which are still radiating [@cox91].
The difference in appearance between Sco X-1 and a typical radio galaxy is [*not*]{} the result of the high dynamic range images available for many extragalactic sources. Even at the modest dynamic range of the Sco X-1 images (peak flux density to rms noise level of about 50) nearly all double radio sources have more complicated lobes than that observed in Sco X-1. Perhaps the suggested strong magnetic field in Sco X-1 confines the relativistic fluid to a small region. In contrast, on MJD 50872 and MJD 51054 the NE lobe of Sco X-1 resembled the morphology of that in FRII-type radio galaxies. The lobe was extended along the source axis and the hot-spot was not dominant. For the MJD 50872 observations, we know that the lobe was formed more than 1.5 days earlier, suggesting that the confinement of relativistic particles within a small region is less likely as Sco X-1 evolves.
Explanations for the differences seen between the radio lobes in Sco X-1 and those of extragalactic sources require a better understanding of the formation of the working-surface and the relativistic particles and fields and their transport into the hot-spot. One significant difference between Sco X-1 and radio galaxies may be the advance speed of the hot-spot. For Sco X-1, the advance velocity is 0.5c. It is believed that the advance velocity in radio galaxies is only mildly relativistic $\approx 0.1$c. Another difference may be that the lobes in extragalactic sources are probably confined by ram pressure. In Sco X-1, the lobes may be confined by a strong magnetic field, or, not confined at all with adiabatic expansion occurring outside of the working surface region.
Conclusion
==========
These extensive VLBI observations of Sco X-1 are among the most detailed for a Galactic-jet radio source. The high-resolution images over the four-year study show relativistic motion and variability of all components and a recurrent phenomenon of lobe production. The resolution and sensitivity are sufficient to determine accurate parameters of the discrete components as a function of time. Since we have observed Sco X-1 without regard to its X-ray and radio flux density, we may be sampling the more stationary properties of a Galactic-jet source rather than those associated with the evolution during the more explosive periods.
The relatively non-speculative conclusions from these observations are the following.
- Sco X-1 is composed of three components: a radio core near the compact component, a NE component which moves away from the core, and a weaker SW component which is detected about half of the time.
- Occasionally weak extended emission emanates from the core in the north-eastern direction, but generally only a NE and SW component are detected.
- All components are variable on time scales of about one hour. The rapid increase and somewhat less rapid decrease in the flux density for all of the components has a characteristic time scale of about three hours. This time-scale of the radio variations in Sco X-1 has already been noted [@hje90; @bra97].
- A pair of moving components persists up to about two days. Recurrent generation of pairs occurs often enough that a naked core component is not commonly observed.
- The comparison of the radio properties between the NE and SW components strongly suggest that they are intrinsically similar, but differ from the effects of relativistic aberration and Doppler beaming.
- The average speed of the NE and SW components is $0.45c\pm
0.03c$, radially away from the core at an angle $44^\circ\pm 7^\circ$ to the line of sight.
- The component speeds remain unchanged for many hours although the speeds for different pairs of components at different times range between 0.31c and 0.57c.
- The components of Sco X-1 lie on a axis which has not varied by more than about $4^\circ$ over five years. A variation of axis direction of $\approx 3^\circ$, with a period of a few days, may be present.
- The minimum angular size of the NE (and presumably the SW) component is $1.5\times 0.7$ mas ($6\times 3 10^13$ cm) with an orientation [*perpendicular*]{} to the motion of the lobe. This corresponds to a volume of $6.1\times 10^{40}$ cm$^3$. The equipartition magnetic field is between 0.3 to 1.0 G depending on the ratio of electrons to protons.
- The lobes of Sco X-1 are dominated by a compact hot-spot. The lobes of most extragalactic sources often contain one or more hot-spots and much more extended structure.
More speculative conclusions are:
- The NE and SW components are lobes generated from a working surface where the energy flow within a twin-beam from the binary impinges on the ISM. This conclusion is based on the variability of the components, the lack of a systematic decrease in the lobe flux density with time, and the simplicity of the components.
- If the above suggestion is correct then the temporal correlation of the core flare times with those of the NE lobe and SW lobe suggest a beam velocity $>0.95$c.
- In Sco X-1 the lobe emission dominates. In most other Galactic-jet sources, emission from material or shocks within the jets dominate. This may be an intrinsic difference between the sources, perhaps related to either the nature of the degenerate star or the persistence of the X-ray emission. Alternatively, the difference may be associated with the angle between the observer and the direction of source motion.
- The variability of the lobes and their linear size suggest a radiative lifetime of less than one hour. For synchrotron losses in the emitting plasma, a magnetic field of $\sim 300$ G is needed, which implies a lobe dominated by magnetic energy. Alternatively, an adiabatic expansion of the ultra-relativistic plasma, after generation at the working surface, is also consistent with the lobe emission characteristics.
- The simple pressure balance in the ultra-relativistic plasma on both sides of the shock at the working surface predicts an advance speed of the lobe in the range of that observed. The relatively constant speed of the lobes over many hours is surprising and no explanations are offered.
- The lobe looks like a wagon wheel. The working surface is at the hub and the electrons diffuse with 0.57c away rom the working-surface shock. The advance of the lobe over the lifetime of the radiating electrons (30 minutes) also contributes to its size inthe direction of motion.
- The slow expansion of the lobes, with a decrease in flux density and steepening spectrum, may be associated with a decreasing beam flow and exlarging of the working surface, as well as from adiabatic expansion of material in the lobes.
- The scaling of the Sco X-1 phenomena to that of radio galaxies suggests that hot-spot variability and life-times may be only $10^5$ yr in radio galaxy lobes.
These observations show that VLBI resolutions are sufficient to determine the internal properties of Galactic-jet radio sources. The time scales of the evolution vary from less than one hour to a few days or months in some cases. Hence, detailed, long-term, multi-frequency and reasonably continuous observations are required to follow and understand the emission properties of the sources. Because of the possible dominant effect of the magnetic field, linear polarization imaging would be very useful.
The National Radio Astronomy Observatory is a facility of the National Science Foundation, operated under cooperative agreement by Associated Universities, Inc. We thank the European VLBI Network (EVN) and the Asia-Pacific Telescope (APT) for their support and observation time. The data were correlated with the VLBA correlator in Socorro and the Penticton Correlator that is supported by the Canadian Space Agency. It is a pleasure to thank Dr. Jean Swank for granting us RXTE time, to Dr. Tasso Tzioumis for help with the APT scheduling, and to Dr.Sean Dougherty for processing the APT data. We thank Dr. Michael McCollough and Dr. Vivek Dhawan for comments on the draft.
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[rllccrr]{} 19-Aug-1995 & 49948 & 23.7 to 26.7 & VLBA & 4.98 & 64 & 0.4\
16-Mar-1996 & 50158 & 10.1 to 13.0 & VLBA & 4.98 & 64 & 0.6\
14-Sep-1996 & 50340 & 22.1 to 25.0 & VLBA & 4.98 & 64 & 4.5\
03-Aug-1997 & 50663 & 23.9 to 29.3 & VLBA & 4.98 & 64 & 3.5\
21-Aug-1997 & 50681 & 22.8 to 27.8 & VLBA+Y1 & 4.98,1.67 & 128 & 15.0\
27-Feb-1998 & 50871 & 10.3 to 16.0 & VLBA+Y1 & 4.98,1.67 & 128 & 22.0\
28-Feb-1998 & 50872 & 10.2 to 16.0 & VLBA+Y1 & 4.98,1.67 & 128 & 5.0\
29-Aug-1998 & 51054 & 23.8 to 27.0 & VLBA+Y1 & 4.98,1.67 & 128 & 9.0\
\
11-Jun-1999 & 51340 & 02.2 to 10.2 &VLBA+Y27 & 4.98,1.67 & 128 & 20.0\
11-Jun-1999 & 51340 & 10.2 to 18.2 &APT$^1$ & 4.98 & 64 & 8.0\
11-Jun-1999 & 51340 & 18.2 to 26.2 &EVN+$^2$ & 4.98 & 128 & 18.0\
12-Jun-1999 & 51341 & 02.2 to 10.2 &VLBA+Y27 & 4.98,1.67 & 128 & 14.0\
12-Jun-1999 & 51341 & 10.2 to 18.2 &APT$^1$ & 4.98 & 64 & 12.0\
12-Jun-1999 & 51341 & 18.2 to 26.2 &EVN+$^2$ & 4.98 & 128 & 14.0\
13-Jun-1999 & 51342 & 02.2 to 10.2 &VLBA+Y27 & 4.98,1.67 & 128 & 8.0\
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---
abstract: |
We define extender sequences, generalizing measure sequences of Radin forcing.
Using the extender sequences, we show how to combine the Gitik-Magidor forcing for adding many Prikry sequenes with Radin forcing.
We show that this forcing satisfies a Prikry like condition destroys no cardinals, and has a kind of properness.
Depending on the large cardinals we start with this forcing can blow the power of a cardinal together with changing its’ cofinality to a prescirbe value. It can even blow the power of a cardinal while keeping it regular or measurable.
address: |
School of Mathematical Sciences\
Tel-Aviv University\
Tel-Aviv 69978\
ISRAEL
author:
- Carmi Merimovich
date: 'October 10, 1998'
title: Extender Based Radin Forcing
---
Introduction
============
We give some background on previous work relating directly to the present work. The first forcing which changed the cofinality of a cardinal without changing the cardinal structure was Prikry forcing [@Prikry]. In this forcing a measurable cardinal, ${\kappa}$, was ‘invested’ in order to get $\operatorname{cf}({\kappa})={\omega}$ without collapsing any cardinal. Developing that idea, Magidor [@Menachem:cf] used a coherent sequence of measures of length ${\lambda}< {\kappa}$ in order to get $\operatorname{cf}({\kappa})={\lambda}$ without collapsing any cardinals. In [@Radin] Radin, introducing the notion of measure sequence, showed that it is useful to continue the coherent sequence to ${\lambda}> {\kappa}$. For example, ${\kappa}$ remains regular when ${\lambda}= {\kappa}^+$. In general the longer the measure sequence the more resemblance there is between ${\kappa}$ in the generic extension and the ground model.
As is well known, and unlike regular cardinals, blowing the power of a singular cardinal is not an easy task. A natural approach to try was to blow the power of a cardinal while it was regular and after that make it singular by one of the above methods. A crucial idea of Gitik and Magidor [@Moti] was to combine the power set blowing and the cofinality change in one forcing. They introduced a forcing notion which added many Prikry sequences at once and still collapsed no cardinals. The ‘investment’ they needed for this was an extender of length which is the size of power they wanted. Building on the idea of Gitik and Magidor, Segal [@Miri] implemented the idea of adding many sequences to Magidor forcing. So by investing a coherent sequence of extenders of length ${\lambda}< {\kappa}$ she was able to get a singular cardinal of cofinality ${\lambda}$ together with power as large as the length of the extenders in question. Our work also builds on the idea of Gitik and Magidor. However, we implement the idea of adding many sequences to Radin forcing. So we introduce the notion of extender sequence and show that it makes sense to deal with quite long extender sequences. As in Radin forcing, for long enough sequences we are left with ${\kappa}$ which is regular and even measurable. The power size will be the length of the extenders we start with.
The structure of this work is as follows. In section \[ExtenderSequences\] we define extender sequences. In section \[RadinForcing\] we define Radin forcing. The definition is not the usual one and is used to introduce the idea used in section \[PEForcing\]. In section \[PEForcing\] we define ${{\ensuremath{P_{{{\ensuremath{\bar{E}}\/}}}\/}}}$, the forcing notion which is the purpose of this work. In section \[BasicProperties\] we show the chain-conditions satisfied by ${{\ensuremath{P_{{{\ensuremath{\bar{E}}\/}}}\/}}}$ and how ‘locally’ it resembles Radin forcing. We also show here that there are many new subsets in the generic extension. In section \[HomogenDense\] we investigate the structure of dense open subsets of ${{\ensuremath{P_{{{\ensuremath{\bar{E}}\/}}}\/}}}$. We show that they satisfy a strong homogeneity property. In section \[Prikry’sCondition\] we prove Prikry’s condition for ${{\ensuremath{P_{{{\ensuremath{\bar{E}}\/}}}\/}}}$. The proof is a simple corollary of the strong homogeneity of dense open subsets. In section \[Properness\] we show that ${{\ensuremath{P_{{{\ensuremath{\bar{E}}\/}}}\/}}}$ satisfies a kind of properness. In section \[Cardinals\] we combine the machinery developed so far in order to show that no cardinals are collapsed. In section \[PropertiesK\] we show how the length of the extender sequences affect the properties of ${\kappa}$. Section \[TheTheorem\] summarizes what the forcing ${{\ensuremath{P_{{{\ensuremath{\bar{E}}\/}}}\/}}}$ does. In section \[ByIteration\] we have a result concerning ${{\ensuremath{P_{{{\ensuremath{\bar{E}}\/}}}\/}}}$ when $\operatorname{l}({{\ensuremath{\bar{E}}\/}})=1$. We show that there is, in $V$, a generic filter over an elementary submodel in an ${\omega}$-iterate of $V$. We were not able to prove something equivalent (or weaker) for the general case. Section \[ConcludingRemarks\] contains a list of missing or unknown points to check. The last point in this list is in preparation.
Our notation is standard. We assume fluency with forcing and extenders. Some basic properties of Radin forcing are taken for granted.
Extender sequences {#ExtenderSequences}
==================
Constructing from elementary embedding
--------------------------------------
Suppose we have an elementary embedding $j{\mathord{:}}V \to M \supset V_{\lambda}$, $\operatorname{crit}(j)={\kappa}$. The value of ${\lambda}$ is determined later, according to the different applications we have.
Construct from $j$ a nice extender like in [@Moti]: $$\begin{aligned}
E(0) = {\ensuremath{\langle {\ensuremath{{\ensuremath{\langle E_{\alpha}(0) \mid {\alpha}\in {{\mathcal{A}}}\rangle}}}},
{\ensuremath{{\ensuremath{\langle {\pi}_{{\beta},{\alpha}} \mid {\beta}\geq {\alpha}\ {\alpha},{\beta}\in {{\mathcal{A}}}\rangle}}}} \rangle}}.\end{aligned}$$ We recall the properties of this extender:
1. ${{\mathcal{A}}}\subseteq {\lvertV_{\lambda}\rvert}\setminus{\kappa}$,
2. ${\lvert{{\mathcal{A}}}\rvert} = {\lvertV_{\lambda}\rvert}$,
3. ${{\mathcal{A}}}$ is ${\kappa}^+$-directed,
4. ${\kappa}$ is minimal in ${{\mathcal{A}}}$ and we write ${\pi}_{{\alpha},0}$ instead of ${\pi}_{{\alpha},{\kappa}}$,
5. $\forall {\alpha},{\beta}\in {{\mathcal{A}}}$ ${\nu}^0 = {\pi}_{{\alpha},0}({\nu}) = {\pi}_{{\beta},0}({\nu})$,
6. $\forall {\alpha},{\beta}\in {{\mathcal{A}}}$ ${\pi}_{{\beta},0}({\nu}) = {\pi}_{{\alpha},0}(
{\pi}_{{\beta},{\alpha}}({\nu}))$,
7. $\forall {\alpha},{\beta},{\gamma}\in {{\mathcal{A}}}$ $\exists A \in E_{\gamma}(0)$ $\forall {\nu}\in A$ ${\pi}_{{\gamma},{\alpha}}({\nu}) = {\pi}_{{\beta},{\alpha}}(
{\pi}_{{\gamma},{\beta}}({\nu}))$.
If, for example, we need ${\lvertE(0)\rvert} = {\kappa}^{+3}$ then, under $\text{GCH}$, we require ${\lambda}= {\kappa}+3$. A typical large set in this extender concentrates on singletons.
If $j$ is not sufficiently closed , then $E(0) \notin M$ and the construction stops. We set $$\begin{aligned}
\forall {\alpha}\in {{\mathcal{A}}}\ {{\ensuremath{\bar{E}}\/}}_{\alpha}= {\ensuremath{\langle {\alpha},E(0) \rangle}}.\end{aligned}$$ We say that ${{\ensuremath{\bar{E}}\/}}_{\alpha}$ is an extender sequence of length $1$. ($\operatorname{l}({{\ensuremath{\bar{E}}\/}}_{\alpha})=1$)
If, on the other hand, $E(0) \in M$ we can construct for each ${\alpha}\in \operatorname{dom}E(0)$ the following ultrafilter $$\begin{aligned}
A \in E_{{\ensuremath{\langle {\alpha},E(0) \rangle}}}(1) \iff {\ensuremath{\langle {\alpha}, E(0) \rangle}} \in j(A).\end{aligned}$$ Such an $A$ concentrates on elements of the form ${\ensuremath{\langle {\xi}, e(0) \rangle}}$ where $e(0)$ is an extender on ${\xi}^0$ and ${\xi}\in \operatorname{dom}e(0)$. Note that $e(0)$ concentrates on singletons below ${\xi}^0$. If, for example, ${\lvertE(0)\rvert} = {\kappa}^{+3}$ then on a large set we have ${\lverte(0)\rvert} = ({\xi}^{0})^{+3}$.
We define ${\pi}_{{\ensuremath{\langle {\beta},E(0) \rangle}},{\ensuremath{\langle {\alpha},E(0) \rangle}}}$ as $$\begin{aligned}
{\pi}_{{\ensuremath{\langle {\beta},E(0) \rangle}},{\ensuremath{\langle {\alpha},E(0) \rangle}}}({\ensuremath{\langle {\xi}, e(0) \rangle}}) =
{\ensuremath{\langle {\pi}_{{\beta},{\alpha}}({\xi}), e(0) \rangle}}.\end{aligned}$$ From this definition we get $$\begin{aligned}
j({\pi}_{{\ensuremath{\langle {\beta},E(0) \rangle}},{\ensuremath{\langle {\alpha},E(0) \rangle}}})({\ensuremath{\langle {\beta}, E(0) \rangle}}) =
{\ensuremath{\langle {\alpha}, E(0) \rangle}}.\end{aligned}$$ Hence we have here an extender $$\begin{aligned}
E(1) = {\ensuremath{\langle
{\ensuremath{{\ensuremath{\langle E_{{\ensuremath{\langle {\alpha},E(0) \rangle}}}(1) \mid {\alpha}\in {{{\mathcal{A}}}} \rangle}}}},
{\ensuremath{{\ensuremath{\langle {\pi}_{{\ensuremath{\langle {\beta},E(0) \rangle}},{\ensuremath{\langle {\alpha},E(0) \rangle}}} \mid {\beta}\geq {\alpha}\ {\alpha},{\beta}\in {{\mathcal{A}}}\rangle}}}}
\rangle}}.\end{aligned}$$ Note that the difference between ${\pi}_{{\beta},{\alpha}}$ and ${\pi}_{{\ensuremath{\langle {\beta},E(0) \rangle}},{\ensuremath{\langle {\alpha},E(0) \rangle}}}$ is quite superficial. We can define ${\pi}_{{\ensuremath{\langle {\beta},E(0) \rangle}},{\ensuremath{\langle {\alpha},E(0) \rangle}}}$ in a uniform way for both extenders. Just project the first element of the argument using ${\pi}_{{\beta},{\alpha}}$.
If ${\ensuremath{\langle E(0),E(1) \rangle}} \notin M$ then the construction stops. In this case we set $$\begin{aligned}
\forall {\alpha}\in {{\mathcal{A}}}\ {{\ensuremath{\bar{E}}\/}}_{\alpha}= {\ensuremath{\langle {\alpha}, E(0), E(1) \rangle}}.\end{aligned}$$ We say that ${{\ensuremath{\bar{E}}\/}}_{\alpha}$ is an extender sequence of length $2$. ($\operatorname{l}({{\ensuremath{\bar{E}}\/}}_{\alpha})=2$)
If ${\ensuremath{\langle E(0),E(1) \rangle}} \in M$ then we construct the extender $E(2)$ in the same way as we constructed $E(1)$ from $E(0)$.
The above private case being worked out we continue with the general case. Assume we have constructed $$\begin{aligned}
{\ensuremath{{\ensuremath{\langle E({\tau}') \mid {\tau}' < {\tau}\rangle}}}}.\end{aligned}$$ If ${\ensuremath{{\ensuremath{\langle E({\tau}') \mid {\tau}' < {\tau}\rangle}}}} \notin M$ then the construction stops here. We set $$\begin{aligned}
\forall {\alpha}\in {{\mathcal{A}}}\ {{\ensuremath{\bar{E}}\/}}_{\alpha}= {\ensuremath{{\ensuremath{\langle {\alpha}, E({\tau}') \mid {\tau}' < {\tau}\rangle}}}},\end{aligned}$$ and we say that ${{\ensuremath{\bar{E}}\/}}_{\alpha}$ is an extender sequence of length ${\tau}$. ($\operatorname{l}({{\ensuremath{\bar{E}}\/}}_{\alpha})= {\tau}$)
If, on the other hand, ${\ensuremath{{\ensuremath{\langle E({\tau}') \mid {\tau}' < {\tau}\rangle}}}} \in M$ then we construct $$\begin{gathered}
A \in E_{{\ensuremath{{\ensuremath{\langle {\alpha}, E(0),\dotsc, E({\tau}'),\dotsc \mid {\tau}' < {\tau}\rangle}}}}}({\tau}) \iff
\\
{\ensuremath{{\ensuremath{\langle {\alpha}, E(0), \dotsc, E({\tau}'), \dotsc \mid {\tau}' < {\tau}\rangle}}}}
\in j(A).\end{gathered}$$ Defining ${\pi}_{{\ensuremath{{\ensuremath{\langle {\beta}, E(0),\dotsc, E({\tau}'),\dotsc \mid {\tau}' < {\tau}\rangle}}}},
{\ensuremath{{\ensuremath{\langle {\alpha}, E(0),\dotsc, E({\tau}'),\dotsc, \mid {\tau}' < {\tau}\rangle}}}}}$ using the first coordinate as before gives the needed projection.
We are quite casual in writing the indices of the projections and ultrafilters. By this we mean that we sometimes write $\pi_{{\beta},{\alpha}}$ when we should have written ${\pi}_{{\ensuremath{{\ensuremath{\langle {\beta}, E(0),\dotsc, E({\tau}'),\dotsc \mid {\tau}' < {\tau}\rangle}}}},
{\ensuremath{{\ensuremath{\langle {\alpha}, E(0),\dotsc, E({\tau}'),\dotsc, \mid {\tau}' < {\tau}\rangle}}}}}$ and $E_{\alpha}({\tau})$ when we should have written $E_{{\ensuremath{{\ensuremath{\langle {\alpha}, E(0),\dotsc, E({\tau}'),\dotsc, \mid {\tau}' < {\tau}\rangle}}}}}({\tau})$.
With this abuse of notation the projection we just defined satisfies $$\begin{gathered}
j({\pi}_{{\beta},{\alpha}})
({\ensuremath{{\ensuremath{\langle {\beta}, E(0), \dotsc, E({\tau}'), \dotsc \mid {\tau}' < {\tau}\rangle}}}})=
\\
{\ensuremath{{\ensuremath{\langle {\alpha}, E(0), \dotsc, E({\tau}'), \dotsc \mid {\tau}' < {\tau}\rangle}}}},\end{gathered}$$ and we have the extender $$\begin{aligned}
E({\tau}) = {\ensuremath{\langle {\ensuremath{{\ensuremath{\langle E_{\alpha}({\tau}) \mid {\alpha}\in {{{\mathcal{A}}}} \rangle}}}},
{\ensuremath{{\ensuremath{\langle {\pi}_{{\beta},{\alpha}} \mid {\beta}\geq {\alpha}\ {\alpha},{\beta}\in {{\mathcal{A}}}\rangle}}}} \rangle}}.\end{aligned}$$
We let the construction run until it stops due to the extender sequence not being in $M$.
We call ${{\ensuremath{\Bar{{\mu}}}\/}}$ an extender sequence if there is an elementary embedding $j{\mathord{:}}V \to M$ such that ${{\ensuremath{\Bar{{\nu}}}\/}}$ is an extender sequence generated as above and ${{\ensuremath{\Bar{{\mu}}}\/}}= {{\ensuremath{\Bar{{\nu}}}\/}}{\mathord{\restriction}}{\tau}$ for ${\tau}\leq \operatorname{l}({{\ensuremath{\Bar{{\nu}}}\/}})$. ${\kappa}({{\ensuremath{\Bar{{\mu}}}\/}})$ is the ordinal at the beginning of the sequence. (i.e. ${\kappa}(\bar{E}_{\alpha})={\alpha}$). ${\kappa}^0({{\ensuremath{\Bar{{\mu}}}\/}})$ is $({\kappa}({{\ensuremath{\Bar{{\mu}}}\/}}))^0$. (i.e. ${\kappa}^0(\bar{E}_{\alpha})={\kappa}$).
That is, we do not have to construct the extender sequence until it is not in $M$. We can stop anywhere on the way.
A sequence of extender sequences ${\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_1, \dotsc, {{\ensuremath{\Bar{{\mu}}}\/}}_n \rangle}}$ is called $^0$-increasing if ${\kappa}^0({{\ensuremath{\Bar{{\mu}}}\/}}_1) < \dotsb <{\kappa}^0({{\ensuremath{\Bar{{\mu}}}\/}}_n)$.
Let ${\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_1, \dotsc, {{\ensuremath{\Bar{{\mu}}}\/}}_n \rangle}}$ be $^0$-increasing. An extender sequences ${{\ensuremath{\Bar{{\mu}}}\/}}$ is called permitted to ${\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_1, \dotsc, {{\ensuremath{\Bar{{\mu}}}\/}}_n \rangle}}$ if ${\kappa}^0({{\ensuremath{\Bar{{\mu}}}\/}}_n) < {\kappa}^0({{\ensuremath{\Bar{{\mu}}}\/}})$.
We say $A \in \bar{E}_{\alpha}$ if $\forall {\xi}< \operatorname{l}(\bar{E}_{\alpha}) \ A\in E_{\alpha}({\xi})$.
${{\ensuremath{\bar{E}}\/}}={\ensuremath{{\ensuremath{\langle {{\ensuremath{\bar{E}}\/}}_{\alpha}\mid {\alpha}\in {{\mathcal{A}}}\rangle}}}}$ is an extender sequence system if there is an elementary embedding $j{\mathord{:}}V \to M$ such that all ${{\ensuremath{\bar{E}}\/}}_{\alpha}$ are extender sequences generated from $j$ as prescribed above and $\forall {\alpha},{\beta}\in{{{\mathcal{A}}}} \ \operatorname{l}({{\ensuremath{\bar{E}}\/}}_{\alpha})=\operatorname{l}({{\ensuremath{\bar{E}}\/}}_{\beta})$. This common length is called the length of the system, $\operatorname{l}({{\ensuremath{\bar{E}}\/}})$. We write ${{\ensuremath{\bar{E}}\/}}({{\ensuremath{\Bar{{\mu}}}\/}})$ for the extender sequence system to which ${{\ensuremath{\Bar{{\mu}}}\/}}$ belongs (i.e. ${{\ensuremath{\bar{E}}\/}}({{\ensuremath{\bar{E}}\/}}_{\alpha})={{\ensuremath{\bar{E}}\/}}$).
The generalization of the measure on the ${\alpha}$ coordinate in Gitik-Magidor forcing [@Moti] is ${{\ensuremath{\bar{E}}\/}}_{\alpha}$.
$\bar{E}_{\alpha}$-tree
-----------------------
A tree $T$ is an $\bar{E}_{\alpha}$-tree if its’ elements are of the form $${\ensuremath{\langle {\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_1,\linebreak[0] \dotsc, {{\ensuremath{\Bar{{\mu}}}\/}}_n \rangle}},S \rangle}}$$ where
1. Set $\operatorname{dom}T = {\ensuremath{\left\{ {\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_1,\dotsc,{{\ensuremath{\Bar{{\mu}}}\/}}_n \rangle}} \mid {\ensuremath{\langle {\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_1,\dotsc,{{\ensuremath{\Bar{{\mu}}}\/}}_n \rangle}},S \rangle}} \in T \right\}}}$. Then the function $$\begin{aligned}
{\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_1,\dotsc,{{\ensuremath{\Bar{{\mu}}}\/}}_n \rangle}} \mapsto {\ensuremath{\langle {\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_1,\dotsc,{{\ensuremath{\Bar{{\mu}}}\/}}_n \rangle}},S \rangle}}
\end{aligned}$$ from $\operatorname{dom}T$ to $T$ is $1-1$ and onto,
2. $t \in \operatorname{Lev}_n(\operatorname{dom}T) \implies {\lvertt\rvert}=n+1$,
3. ${\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_1,\dotsc,{{\ensuremath{\Bar{{\mu}}}\/}}_n \rangle}}$ are $^0$-increasing extender sequences,
4. $\operatorname{Lev}_0(\operatorname{dom}T)\in \bar{E}_{\alpha}$ and for each $t\in \operatorname{dom}T$ $\operatorname{Suc}_{\operatorname{dom}T}(t)\in \bar{E}_{\alpha}$,
5. $S$ is a ${{\ensuremath{\Bar{{\mu}}}\/}}_n$-tree. When $\operatorname{l}({{\ensuremath{\Bar{{\mu}}}\/}}_n)=0$ we set $S=\emptyset$.
Note that this clause is recursive.
Later on, we abuse notation and use $T$ instead of $\operatorname{dom}T$. i.e. $\operatorname{Suc}_T(t)$ instead of $\operatorname{Suc}_{\operatorname{dom}T}(t)$.
Assume $T$ is a $\bar{E}_{\alpha}$-tree and $t\in T$, then:
1. $T_t={\ensuremath{\left\{ {\ensuremath{\langle s, S \rangle}} \mid {\ensuremath{\langle t{\mathop{{}^\frown}}s, S \rangle}} \in T \right\}}}$.
2. $T_t({{\ensuremath{\Bar{{\mu}}}\/}})$ is the tree, $S$, satisfying ${\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}, S \rangle}}\in \operatorname{Suc}_T(t)$.
Let $T$, $S$ be $\bar{E}_{\alpha}$-trees, where $\operatorname{l}(\bar{E})=1$. We say that $T \leq S$ if
1. $\operatorname{Lev}_0(T) \subseteq \operatorname{Lev}_0(S)$,
2. $\forall t \in T \operatorname{Suc}_T(t)\subseteq \operatorname{Suc}_S(t)$.
Let $T$, $S$ be $\bar{E}_{\alpha}$-trees. We say that $T \leq S$ if
1. $\operatorname{Lev}_0(T) \subseteq \operatorname{Lev}_0(S)$,
2. $\forall t \in T \operatorname{Suc}_T(t)\subseteq \operatorname{Suc}_S(t)$,
3. $\forall {\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}\rangle}}\in\operatorname{Lev}_0(T) \ T({{\ensuremath{\Bar{{\mu}}}\/}})\leq S({{\ensuremath{\Bar{{\mu}}}\/}})$,
4. $\forall t \in T \ \forall {\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}\rangle}} \in T_t \
T_t({{\ensuremath{\Bar{{\mu}}}\/}})\leq S_t({{\ensuremath{\Bar{{\mu}}}\/}})$.
Note that the last 2 conditions are recursive.
Let $S$ be $\bar{E}_{\alpha}$-tree and ${\beta}>{\alpha}$. Define $T = \pi^{-1}_{{\beta},{\alpha}}(S)$ by
1. $\operatorname{dom}T = {\pi}^{-1}_{{\beta},{\alpha}} (\operatorname{dom}S)$,
2. $T_{{\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_1,\dotsc,{{\ensuremath{\Bar{{\mu}}}\/}}_{n-1} \rangle}}}({{\ensuremath{\Bar{{\mu}}}\/}}_n) =
{\pi}^{-1}_{{{\ensuremath{\Bar{{\mu}}}\/}}_n, {\pi}_{{\beta},{\alpha}}({{\ensuremath{\Bar{{\mu}}}\/}}_n)}
S_{{\ensuremath{\langle {\pi}_{{\beta},{\alpha}}({{\ensuremath{\Bar{{\mu}}}\/}}_1),\dotsc,
{\pi}_{{\beta},{\alpha}}({{\ensuremath{\Bar{{\mu}}}\/}}_{n-1}) \rangle}}}({\pi}_{{\beta},{\alpha}}({{\ensuremath{\Bar{{\mu}}}\/}}_{n}))$.
Let $T$, $S$ be $\bar{E}_{\beta}$, $\bar{E}_{\alpha}$-trees respectively, where ${\beta}\geq {\alpha}$. We say that $T \leq S$ if
1. $T \leq \pi^{-1}_{{\beta},{\alpha}} (S)$.
Assume we have $A_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}, R \rangle}}}$ where ${{\ensuremath{\Bar{{\nu}}}\/}}$ is an extender sequence such that each element in $A_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}},R \rangle}}}$ is of the form ${\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}, S \rangle}}$ where ${{\ensuremath{\Bar{{\mu}}}\/}}$ is an extender sequence and $S$ is a tree. (in this work we always have $S$ is ${{\ensuremath{\Bar{{\mu}}}\/}}$-tree). We define $\operatorname*{\triangle}^0_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}},R \rangle}}} A_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}},R \rangle}}}$ as $$\begin{aligned}
{\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}},S \rangle}} \in \sideset{}{^0}\operatorname*{\triangle}_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}},R \rangle}}} A_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}},R \rangle}}}
\iff
\forall {\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}},R \rangle}} \, {\kappa}({{\ensuremath{\Bar{{\nu}}}\/}}) < {\kappa}^0({{\ensuremath{\Bar{{\mu}}}\/}})
\rightarrow
{\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}},S_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}, R \rangle}}} \rangle}} \in A_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}},R \rangle}}}.\end{aligned}$$
Radin Forcing {#RadinForcing}
=============
The main aims of this section are \[fill-missing\], \[skeleton\]. As the simplest way we found to formulate them was with Radin forcing [@Radin; @Mitchell; @Woodin] we took the opportunity to depart from usual formulation in order to introduce ideas that we use in the extender based forcing later.
The main point is that possible extensions of a condition are stored in ${{\ensuremath{\bar{E}}\/}}_{\alpha}$-tree and not in a set. The ${\alpha}$ is be fixed, so practically we deal here with a measure sequence and not an extender sequence.
A condition in $R_{\alpha}$ is of the form $$\begin{aligned}
{\ensuremath{\langle {\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_n,s^n \rangle}},S^n, \dotsc, {\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_{1},s^1 \rangle}},S^{1},
{\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_0,s^0 \rangle}}, S^0 \rangle}}\end{aligned}$$ where
1. ${{\ensuremath{\Bar{{\mu}}}\/}}_0 = {{\ensuremath{\bar{E}}\/}}_{\alpha}$,
2. $\forall i\leq n$ $S^i$ is ${{\ensuremath{\Bar{{\mu}}}\/}}_i$-tree,
3. $\forall i\leq n$ $s^i \in V_{{\kappa}^0({{\ensuremath{\Bar{{\mu}}}\/}}_i)}$ is an extender sequence.
Let $p,q \in R_{\alpha}$. We say that $p$ is Prikry extension of $q$ ($p \leq^* q$ or $p \leq^0 q$) if $p,q$ are of the form $$\begin{aligned}
& p = {\ensuremath{\langle {\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_n,s^n \rangle}},S^n, \dotsc, {\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_{1},s^1 \rangle}},S^{1},
{\ensuremath{\langle {{\ensuremath{\bar{E}}\/}}_{\alpha},s^0 \rangle}}, S^0 \rangle}},
\\
& q = {\ensuremath{\langle {\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_n,t^n \rangle}},T^n, \dotsc, {\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_{1},t^1 \rangle}},T^{1},
{\ensuremath{\langle {{\ensuremath{\bar{E}}\/}}_{\alpha},t^0 \rangle}},T^0 \rangle}},\end{aligned}$$ and
- $\forall i \leq n$ $S^i \leq T^i$,
- $\forall i \leq n$ $s^i = t^i$.
Let $p = {\ensuremath{\langle {\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_n,s^n \rangle}},S^n, \dotsc, {\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_{1},s^1 \rangle}},S^{1},
{\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_0,s^0 \rangle}}, S^0 \rangle}}$ where ${{\ensuremath{\Bar{{\mu}}}\/}}_0={{\ensuremath{\bar{E}}\/}}_{\alpha}$. Let ${\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}\rangle}}\in S^i$. We define $(p)_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}\rangle}}}$ to be $$\begin{aligned}
(p)_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}\rangle}}} =
{\ensuremath{\langle
{\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_n,s^n \rangle}},S^n, \dotsc,
{\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_{i+1},s^{i+1} \rangle}},S^{i+1},
\\
{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}, s^i \rangle}}, S^i({{\ensuremath{\Bar{{\nu}}}\/}}),
{\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_i, {{\ensuremath{\Bar{{\nu}}}\/}}\rangle}}, S^i_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}\rangle}}},
\\
{\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_{i-1},s^{i-1} \rangle}},S^{i-1},
\dotsc,
{\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_0,s^0 \rangle}}, S^0
\rangle}}.
$$
Note the degenerate case in this definition when $\operatorname{l}({{\ensuremath{\Bar{{\nu}}}\/}})=0$. In this case $S^i({{\ensuremath{\Bar{{\nu}}}\/}})=\emptyset$.
Let $p,q \in R_{\alpha}$ where $$\begin{aligned}
q = {\ensuremath{\langle {\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_n,s^n \rangle}},S^n, \dotsc, {\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_{1},s^1 \rangle}},S^{1},
{\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_0,s^0 \rangle}}, S^0 \rangle}}.\end{aligned}$$ We say that $p$ is 1-point extension of $q$ ($p \leq^1 q$) if there is ${\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}\rangle}} \in S^i$ such that $p \leq^* (q)_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}\rangle}}}$.
Let $p,q \in R_{\alpha}$. We say that $p$ is $n$-point extension of $q$ ($p \leq^n q$) if there are $p^n, \dotsc, p^0$ such that $$\begin{aligned}
p = p^n \leq^1 \dotsc \leq^1 p^0 = q.\end{aligned}$$
Let $p,q \in R_{\alpha}$. We say that $p$ is an extension of $q$ ($p \leq q$) if there is $n$ such that $p \leq^n q$.
\[fill-missing\] is needed in the proof of \[PrikryCondition\]. Very loosly speaking \[fill-missing\] means that if “something” happens on a measure one set for one of the measures, that “something” is happening on a measure one set for all the measures.
\[fill-missing\] is proved by induction and \[fill-missing2\] is the first case of the induction.
\[fill-missing2\] Suppose $\operatorname{l}({{\ensuremath{\bar{E}}\/}}_{\alpha})=2$ and $T$ is a tree such that $\operatorname{Lev}_0(T)\in E_{\alpha}(i)$ for $i<2$, and $\forall {\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}\rangle}}\in T$ $T_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}\rangle}}}$ is an ${{\ensuremath{\bar{E}}\/}}_{{\alpha}}$-tree. Then there is an ${{\ensuremath{\bar{E}}\/}}_{{\alpha}}$-tree, $T^*$, satisfying
1. $\forall {\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}\rangle}}\in T {\cap}T^*$ $T^*_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}\rangle}}} \leq T_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}\rangle}}}$, $T^*({{\ensuremath{\Bar{{\nu}}}\/}}) \leq T({{\ensuremath{\Bar{{\nu}}}\/}})$,
2. If $p
\leq
{\ensuremath{\big\langle {\ensuremath{\langle {{\ensuremath{\bar{E}}\/}}_{\alpha},{\ensuremath{\langle \rangle}} \rangle}}, T^* \big\rangle}}$ then there is ${\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}\rangle}}\in T^* {\cap}T$ such that $$\begin{aligned}
p
{\parallel}\big(
{\ensuremath{\big\langle {\ensuremath{\langle {{\ensuremath{\bar{E}}\/}}_{\alpha},{\ensuremath{\langle \rangle}} \rangle}}, T^* \big\rangle}}
\big)_{{\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}\rangle}}}.
\end{aligned}$$
There are 2 cases which to deal with:
- $A_0 = \operatorname{Lev}_0(T) \in E_{\alpha}(0)$: If $A_0 \in E_{\alpha}( 1)$ we set $T^*=T$ and the proof is finished. So suppose $A_0 \notin E_{\alpha}( 1)$. We would like to build $A_1 \in E_{\alpha}( 1)$. Set $$\begin{aligned}
& \forall {\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_0 \rangle}} \in T \ A_{{\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_0 \rangle}}, 1}=
{\ensuremath{\left\{ {\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_1,S \rangle}} \in T_{{\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_0 \rangle}}} \mid A_0 {\cap}{\kappa}^0({{\ensuremath{\Bar{{\mu}}}\/}}_1) \in
{{\ensuremath{\Bar{{\mu}}}\/}}_1(0)_{{\kappa}({{\ensuremath{\Bar{{\mu}}}\/}}_1)} \right\}}}.\end{aligned}$$ As $A_0 \in E_{\alpha}(0)$ and $\operatorname{Suc}_T({\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_0 \rangle}}) \in E_{\alpha}(1)$ we get that $A_{{\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_0 \rangle}},1} \in E_{\alpha}(1)$. Let $$\begin{aligned}
& A_1 = {\sideset{}{^0}\operatorname*{\triangle}}_{{\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_0 \rangle}}\in T} A_{{\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_0 \rangle}},1}.\end{aligned}$$ We can construct now $T^*$: $$\begin{aligned}
&\operatorname{Lev}_0(T^*)= A_0 {\cup}A_1,
\\
&\forall{{\ensuremath{\Bar{{\mu}}}\/}}_0 \in A_0 \ T^*_{{\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_0 \rangle}}}= T_{{\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_0 \rangle}}},
\\
&\forall{{\ensuremath{\Bar{{\mu}}}\/}}_1 \in A_1\ T^*_{{\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_1 \rangle}}}=
{\bigcap}_{{\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_0,{{\ensuremath{\Bar{{\mu}}}\/}}_1 \rangle}}\in T}T_{{\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_0,{{\ensuremath{\Bar{{\mu}}}\/}}_1 \rangle}}}.\end{aligned}$$
- $A_1 = \operatorname{Lev}_0(T)\in E_{\alpha}( 1)$: If $A_1 \in
E_{\alpha}( 0)$ we set $T^*=T$ and finish the proof. So assume $A_1 \notin E_{\alpha}( 0)$. We would like to build $A_0 \in E_{\alpha}( 0)$. Set $$\begin{aligned}
&S=j(T) \big( {\ensuremath{\langle {\alpha}, E( 0) \rangle}} \big),
\\
&A_0 = \operatorname{Lev}_0(S) \setminus A_1.\end{aligned}$$ We construct $T^*$: $$\begin{aligned}
&\operatorname{Lev}_0(T^*)=A_0 {\cup}A_1,
\\
&\forall{{\ensuremath{\Bar{{\mu}}}\/}}_1\in A_1 \ T^*_{{\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_1 \rangle}}} = T_{{\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_1 \rangle}}}.\end{aligned}$$ We are left with the construction of $T^*_{{\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_0 \rangle}}}$ for ${{\ensuremath{\Bar{{\mu}}}\/}}_0 \in A_0$. For all ${{\ensuremath{\Bar{{\mu}}}\/}}\in A_0$ set $$\begin{aligned}
&A_{{\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_0 \rangle}},0}=\operatorname{Suc}_S \big({\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_0 \rangle}} \big),
\\
&A_{{\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_0 \rangle}},1}=
{\ensuremath{\left\{ {\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_1, T({{\ensuremath{\Bar{{\mu}}}\/}}_1)_{{\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_0 \rangle}}} \rangle}} \mid {\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_1 \rangle}}\in T, \ {\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_0 \rangle}}\in T({{\ensuremath{\Bar{{\mu}}}\/}}_1) \right\}}},
\\
&\operatorname{Suc}_{T^*} \big( {\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_0 \rangle}} \big)=A_{{\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_0 \rangle}},0} {\cup}A_{{\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_0 \rangle}},1},
\\
&\forall{{\ensuremath{\Bar{{\mu}}}\/}}_1\in A_{{\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_0 \rangle}},1} \ T^*_{{\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_0,{{\ensuremath{\Bar{{\mu}}}\/}}_1 \rangle}}}=
T_{{\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_1 \rangle}}}.\end{aligned}$$ We continue one more level and hope this will convince the reader we indeed can complete $T^*$. We are left with the construction of $T^*_{{\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_0,{{\ensuremath{\Bar{{\mu}}}\/}}_1 \rangle}}}$ for ${\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_0,{{\ensuremath{\Bar{{\mu}}}\/}}_1 \rangle}}\in A_0 \times A_{{\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_0 \rangle}},0}$. For all ${\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_0,{{\ensuremath{\Bar{{\mu}}}\/}}_1 \rangle}}\in A_0 \times A_{{\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_0 \rangle}},0}$ set $$\begin{aligned}
&A_{{\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_0,{{\ensuremath{\Bar{{\mu}}}\/}}_1 \rangle}}, 0}=\operatorname{Suc}_S \big( {\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_0,{{\ensuremath{\Bar{{\mu}}}\/}}_1 \rangle}} \big),
\\
&A_{{\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_0,{{\ensuremath{\Bar{{\mu}}}\/}}_1 \rangle}}, 1}={\ensuremath{\left\{ {\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_2,
T({{\ensuremath{\Bar{{\mu}}}\/}}_2)_{{\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_0,{{\ensuremath{\Bar{{\mu}}}\/}}_1 \rangle}}} \rangle}} \mid {\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_2 \rangle}} \in T,{\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_0,{{\ensuremath{\Bar{{\mu}}}\/}}_1 \rangle}}\in T({{\ensuremath{\Bar{{\mu}}}\/}}_2) \right\}}},
\\
&\operatorname{Suc}_{T^*} \big( {\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_0,{{\ensuremath{\Bar{{\mu}}}\/}}_1 \rangle}} \big)=A_{{\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_0,{{\ensuremath{\Bar{{\mu}}}\/}}_1 \rangle}},0}
{\cup}A_{{\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_0,{{\ensuremath{\Bar{{\mu}}}\/}}_1 \rangle}},1},
\\
&\forall{{\ensuremath{\Bar{{\mu}}}\/}}_2\in A_{{\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_0,{{\ensuremath{\Bar{{\mu}}}\/}}_1 \rangle}},1}\
T^*_{{\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_0,{{\ensuremath{\Bar{{\mu}}}\/}}_1,{{\ensuremath{\Bar{{\mu}}}\/}}_2 \rangle}}} = T_{{\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_2 \rangle}}}.\end{aligned}$$ We are left with the construction of $T^*_{{\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_0,{{\ensuremath{\Bar{{\mu}}}\/}}_1,{{\ensuremath{\Bar{{\mu}}}\/}}_2 \rangle}}}$ for ${\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_0,{{\ensuremath{\Bar{{\mu}}}\/}}_1,{{\ensuremath{\Bar{{\mu}}}\/}}_2 \rangle}}\in A_0 \times
A_{{\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_0 \rangle}},0} \times A_{{\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_0,{{\ensuremath{\Bar{{\mu}}}\/}}_1 \rangle}},0}$ and we hope that by now the continuation is clear.
\[fill-missing\] Let $\xi_0 < \operatorname{l}({{\ensuremath{\bar{E}}\/}}_{\alpha})$, $T$ a tree such that $\operatorname{Lev}_0(T) \in
E_{\alpha}( \xi_0)$ and $\forall{\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}\rangle}}\in T$ $T_{{\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}\rangle}}}$ is an ${{\ensuremath{\bar{E}}\/}}_{\alpha}$-tree. Then there is an ${{\ensuremath{\bar{E}}\/}}_{\alpha}$-tree, $T^*$, satisfying
1. $\forall {\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}\rangle}}\in T {\cap}T^*$ $T^*_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}\rangle}}} \leq T_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}\rangle}}}$, $T^*({{\ensuremath{\Bar{{\nu}}}\/}}) \leq T({{\ensuremath{\Bar{{\nu}}}\/}})$,
2. If $p
\leq
{\ensuremath{\big\langle {\ensuremath{\langle {{\ensuremath{\bar{E}}\/}}_{\alpha},{\ensuremath{\langle \rangle}} \rangle}}, T^* \big\rangle}}$ then there is ${\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}\rangle}}\in T^* {\cap}T$ such that $$\begin{aligned}
p
{\parallel}\big(
{\ensuremath{\big\langle {\ensuremath{\langle {{\ensuremath{\bar{E}}\/}}_{\alpha},{\ensuremath{\langle \rangle}} \rangle}}, T^* \big\rangle}}
\big)_{{\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}\rangle}}}.
\end{aligned}$$
Our induction hypothesis is that this lemma is true for ${{\ensuremath{\Bar{{\mu}}}\/}}$’s with $\operatorname{l}({{\ensuremath{\Bar{{\mu}}}\/}}) < \operatorname{l}({{\ensuremath{\bar{E}}\/}}_{\alpha})$. The previous lemma is the case for $\operatorname{l}({{\ensuremath{\Bar{{\mu}}}\/}})=2$.
Let $S=j(T)({{\ensuremath{\bar{E}}\/}}_{\alpha}{\mathord{\restriction}}{\xi}_0)$. The tree $S$ is an ${{\ensuremath{\bar{E}}\/}}_{\alpha}{\mathord{\restriction}}\xi_0$-tree. We extend it step by step to a full ${{\ensuremath{\bar{E}}\/}}_{\alpha}$-tree as requested.
Let $$\begin{aligned}
& A_{\xi_0} = \operatorname{Lev}_0 (T),
\\
& A_{{\mathord{<}}\xi_0} = \operatorname{Lev}_0 (S) \setminus A_{{\xi}_0}.\end{aligned}$$ For $\xi_0 < \xi < \operatorname{l}({{\ensuremath{\bar{E}}\/}}_{\alpha})$ do the following: $$\begin{aligned}
& N_{\xi}= \operatorname{Ult}(V, E_{\alpha}({\xi})),
\\
& k_{\xi}([h]_{E_{\alpha}({\xi})}) = j(h)({{\ensuremath{\bar{E}}\/}}_{\alpha}{\mathord{\restriction}}{\xi}).\end{aligned}$$ $$\begin{diagram}
\node{V} \arrow[1]{s} \arrow[1]{e,t}{j} \node[1]{M}
\\
\node{N_{\xi}=\operatorname{Ult}(V,E_{\alpha}({\xi}))} \arrow[1]{ne,b}{{k_{\xi}}}
\end{diagram}$$ and recall that $\operatorname{crit}(k_{\xi}) = ({\kappa}^{++})^{N_{\xi}}$. As $$\begin{aligned}
& {\ensuremath{{\ensuremath{\langle {\alpha},E(0),\dotsc,E({\tau}),\dotsc \mid {\tau}< {\xi}\rangle}}}} =
k_{\xi}([id]_{E_{\alpha}({\xi})}) \in \operatorname{ran}(k_{\xi})\end{aligned}$$ there is in $N_{\xi}$ a preimage for it: $$\begin{aligned}
& {\ensuremath{{\ensuremath{\langle {\alpha}',E'(0),\dotsc,E'({\tau}') \dotsc \mid {\tau}' < {\xi}' \rangle}}}}.\end{aligned}$$ As $A_{{\xi}_0} \in E_{\alpha}({\xi}_0)$, $\xi_0 < {\xi}$ we have ${\tau}' < {\xi}'$ such that $A_{{\xi}_0} \in E'_{{\alpha}'}({\tau}')$, (where ${\alpha}'=[{\kappa}(id)]_{E_{\alpha}({\xi})}$. Taking a function $h_{\xi}$ such that $[h_{\xi}]_{E_{\alpha}({\xi})} = {\tau}'$ we get $$\begin{aligned}
{\ensuremath{\left\{ {{\ensuremath{\Bar{{\mu}}}\/}}\mid A_{{\xi}_0} {\cap}{\kappa}^0({{\ensuremath{\Bar{{\mu}}}\/}}) \in
{{\ensuremath{\Bar{{\mu}}}\/}}(h_{\xi}({{\ensuremath{\Bar{{\mu}}}\/}}))_{{\kappa}({{\ensuremath{\Bar{{\mu}}}\/}})} \right\}}}
\in E_{\alpha}({\xi}).\end{aligned}$$ For each ${{\ensuremath{\Bar{{\mu}}}\/}}_0 \in A_{{\xi}_0}$ we set $$\begin{aligned}
& A_{{\xi}, {\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_0, T({{\ensuremath{\Bar{{\mu}}}\/}}_0) \rangle}}} = {\ensuremath{\left\{ {\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_1,R \rangle}} \in
T_{{\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_0 \rangle}}} \mid A_{{\xi}_0} {\cap}{\kappa}^0({{\ensuremath{\Bar{{\mu}}}\/}}_1) \in
{{\ensuremath{\Bar{{\mu}}}\/}}_1(h_{\xi}({{\ensuremath{\Bar{{\mu}}}\/}}_1))_{{\kappa}({{\ensuremath{\Bar{{\mu}}}\/}})} \right\}}},
\\
& A'_{\xi}= \sideset{}{^0}\operatorname*{\triangle}_{{\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_0, T({{\ensuremath{\Bar{{\mu}}}\/}}_0) \rangle}}}
A_{{\xi}, {\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_0, T({{\ensuremath{\Bar{{\mu}}}\/}}_0) \rangle}}}.\end{aligned}$$ For any tree $R$ which appears in a pair ${\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_1, R \rangle}} \in A'_{\xi}$ we can invoke by induction our lemma and generate $R^*$ which is a ${{\ensuremath{\Bar{{\mu}}}\/}}_1$-tree. Define now $A_{\xi}$ as: $$\begin{aligned}
{\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_1, R^* \rangle}} \in A_{\xi}\iff {\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_1, R \rangle}} \in A'_{\xi}.\end{aligned}$$ When we have ${\ensuremath{\left\{ A_{\xi}\mid {\xi}_0 < {\xi}< \operatorname{l}({{\ensuremath{\bar{E}}\/}}_{\alpha}) \right\}}}$ we set $$\begin{aligned}
& A_{{\mathord{>}}{\xi}_0} = {\bigcup}_{{\xi}_0 < {\xi}< \operatorname{l}({{\ensuremath{\bar{E}}\/}}_{\alpha})} A_{\xi}\setminus (A_{{\mathord{<}}{\xi}_0} {\cup}A_{{\xi}_0}),
\\
& \operatorname{Lev}_0(T^*) = A_{{\mathord{<}}{\xi}_0} {\cup}A_{{\xi}_0} {\cup}A_{{\mathord{>}}{\xi}_0},
\\
& \forall {\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_0 \rangle}} \in A_{{\xi}_0} \
T^*_{{\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_0 \rangle}}} = T_{{\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_0 \rangle}}},
\\
& \forall {\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_1 \rangle}} \in A_{{\mathord{>}}{\xi}_0} \
T^*_{{\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_1 \rangle}}} = {\bigcap}T_{{\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_0, {{\ensuremath{\Bar{{\mu}}}\/}}_1 \rangle}}}.\end{aligned}$$ We are left to define $T_{{\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_0 \rangle}}}$ for ${\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_0 \rangle}} \in
A_{{\mathord{<}}{\xi}_0}$. For each ${{\ensuremath{\Bar{{\mu}}}\/}}_0 \in A_{{\mathord{<}}{\xi}_0}$ set: $$\begin{aligned}
& A_{{\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_0 \rangle}}, {\xi}_0} =
{\ensuremath{\left\{ {\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_1,R_{{\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_0 \rangle}}} \rangle}} \mid {\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_1,R \rangle}} \in A_{{\xi}_0},\ {\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_0 \rangle}} \in R \right\}}},
\\
& A_{{\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_0 \rangle}},{\mathord{<}}\xi_0} = \operatorname{Suc}_S ({\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_0 \rangle}}) \setminus
A_{{\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_0 \rangle}},{\xi}_0}.\end{aligned}$$ For each ${{\ensuremath{\Bar{{\mu}}}\/}}_1 \in A_{{\xi}_0}$ we set $$\begin{aligned}
& A_{{\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_0 \rangle}}, {\xi}, {\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_1,T({{\ensuremath{\Bar{{\mu}}}\/}}_1)_{{\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_0 \rangle}}} \rangle}}} =
{\ensuremath{\left\{ {\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_2, R \rangle}} \in T_{{\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_1 \rangle}}} \mid \begin{aligned}
A_{{\xi}_0} {\cap}{\kappa}^0({{\ensuremath{\Bar{{\mu}}}\/}}_2) \in
{{\ensuremath{\Bar{{\mu}}}\/}}_2(h_{\xi}({{\ensuremath{\Bar{{\mu}}}\/}}_2))_{{\kappa}({{\ensuremath{\Bar{{\mu}}}\/}}_2)}
\end{aligned} \right\}}},
\\
& A'_{{\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_0 \rangle}},{\xi}} =
\sideset{}{^0}\operatorname*{\triangle}_{{\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_1,T({{\ensuremath{\Bar{{\mu}}}\/}}_1)_{{\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_0 \rangle}}} \rangle}}}
A_{{\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_0 \rangle}}, {\xi}, {\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_1, T({{\ensuremath{\Bar{{\mu}}}\/}}_1)_{{\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_0 \rangle}}} \rangle}}}.\end{aligned}$$ We define $$\begin{aligned}
{\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_0, R^* \rangle}} \in A_{{\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_0 \rangle}}, {\xi}} \iff
{\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_0, R \rangle}} \in A'_{{\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_0 \rangle}}, {\xi}}\end{aligned}$$ where $R^*$ is generated from R using the current lemma by induction. Now we set $$\begin{aligned}
& A_{{\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_0 \rangle}},{\mathord{>}}{\xi}_0} =
{\bigcup}_{{\xi}_0 < {\xi}< \operatorname{l}({{\ensuremath{\bar{E}}\/}}_{\alpha})} A_{{\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_0 \rangle}}, {\xi}}
\setminus
(A_{{\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_0 \rangle}}, {\mathord{<}}{\xi}_0} {\cup}A_{{\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_0 \rangle}}, {\xi}_0}),
\\
& \operatorname{Suc}_{T^*} ({\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_0 \rangle}}) = A_{{\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_0 \rangle}}, {\mathord{<}}{\xi}_0} {\cup}A_{{\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_0 \rangle}}, {\xi}_0} {\cup}A_{{\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_0 \rangle}}, {\mathord{>}}{\xi}_0},
\\
& \forall{\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_1 \rangle}} \in A_{{{\ensuremath{\Bar{{\mu}}}\/}}_0, {\xi}_0}\
T^*_{{\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_0, {{\ensuremath{\Bar{{\mu}}}\/}}_1 \rangle}}} = T_{{\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_1 \rangle}}},
\\
& \forall{\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_2 \rangle}} \in A_{{{\ensuremath{\Bar{{\mu}}}\/}}_0, {\mathord{>}}{\xi}_0}\
T^*_{{\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_0, {{\ensuremath{\Bar{{\mu}}}\/}}_2 \rangle}}} =
{\bigcap}T_{{\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_1, {{\ensuremath{\Bar{{\mu}}}\/}}_2 \rangle}}}.\end{aligned}$$ This leaves us with the definition of $T_{{\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_0, {{\ensuremath{\Bar{{\mu}}}\/}}_1 \rangle}}}$ for ${\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_0, {{\ensuremath{\Bar{{\mu}}}\/}}_1 \rangle}} \in A_{{\mathord{<}}{\xi}_0} \times
A_{{\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_0 \rangle}},{\mathord{<}}{\xi}_0}$ which is done exactly as in this step.
\[skeleton\] is needed in the proof of \[DenseHomogen\]. Loosly speaking it says that if “something” happens on all extensions which are taken from $\operatorname{dom}T$, then that “something” happens on all extensions from $T$.
\[skeleton\] is proved by induction where \[skeleton:2\] is the first case.
\[skeleton:2\] Assume $\operatorname{l}({{\ensuremath{\bar{E}}\/}}_{\alpha})=2$ and let $T$ be ${{\ensuremath{\bar{E}}\/}}_{\alpha}$-tree. Then there is $T^* \leq T$ such that if $$\begin{aligned}
p \leq
{\ensuremath{\big\langle {\ensuremath{\langle {{\ensuremath{\bar{E}}\/}}_{\alpha},{\ensuremath{\langle \rangle}} \rangle}}, T^* \big\rangle}}\end{aligned}$$ then there is ${\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_n \rangle}} \in T$ such that $$\begin{aligned}
p \leq^*
\big({\ensuremath{\big\langle {\ensuremath{\langle {{\ensuremath{\bar{E}}\/}}_{\alpha},{\ensuremath{\langle \rangle}} \rangle}}, T \big\rangle}}
\big)_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_n \rangle}}}.\end{aligned}$$
As is usual in this section the proof is done level by level. Let us set $T^1 = T$ and it is trivially true that if $$\begin{aligned}
p \leq^1
{\ensuremath{\big\langle {\ensuremath{\langle {{\ensuremath{\bar{E}}\/}}_{\alpha},{\ensuremath{\langle \rangle}} \rangle}}, T^1 \big\rangle}}\end{aligned}$$ then there is ${\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1 \rangle}} \in T^1=T$ such that $$\begin{aligned}
p \leq^*
\big({\ensuremath{\big\langle {\ensuremath{\langle {{\ensuremath{\bar{E}}\/}}_{\alpha},{\ensuremath{\langle \rangle}} \rangle}}, T^1 \big\rangle}}
\big)_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1 \rangle}}}.\end{aligned}$$ We continue to the second level. Let us set $$\begin{aligned}
& A_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,S^1 \rangle}}} = \operatorname{Suc}_{T^1}({\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1 \rangle}}),
\\
& A_{{\ensuremath{\langle \rangle}}} = {\sideset{}{^0}\operatorname*{\triangle}}_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,S^1 \rangle}}}
A_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,S^1 \rangle}}}
\\
& B_{{\ensuremath{\langle \rangle}}} ={\ensuremath{\left\{ {\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_2 \rangle}} \mid \operatorname{l}({{\ensuremath{\Bar{{\nu}}}\/}}_2)=0\text{ or }\operatorname{Lev}_0(T^1){\cap}{\kappa}^0({{\ensuremath{\Bar{{\nu}}}\/}}_2)
\in {{\ensuremath{\Bar{{\nu}}}\/}}_2 \right\}}},
\\
& \operatorname{Lev}_0(T^{(0)})=A_{{\ensuremath{\langle \rangle}}} {\cap}B_{{\ensuremath{\langle \rangle}}},
\\
& T^{(0)}_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_2 \rangle}}} = {\bigcap}_
{\substack{
{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1 \rangle}} \in T^{(0)}({{\ensuremath{\Bar{{\nu}}}\/}}_2)
}}
T^1_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,{{\ensuremath{\Bar{{\nu}}}\/}}_2 \rangle}}},
\\
& T^2 = T^1 {\cap}T^{(0)}.\end{aligned}$$ Let us assume that $$\begin{aligned}
p \leq^2
{\ensuremath{\big\langle {\ensuremath{\langle {{\ensuremath{\bar{E}}\/}}_{\alpha},{\ensuremath{\langle \rangle}} \rangle}}, T^2 \big\rangle}}.\end{aligned}$$ There are 2 cases to consider here:
1. $p \leq^2
\big({\ensuremath{\big\langle {\ensuremath{\langle {{\ensuremath{\bar{E}}\/}}_{\alpha},{\ensuremath{\langle \rangle}} \rangle}}, T^2 \big\rangle}}
\big)_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,{{\ensuremath{\Bar{{\nu}}}\/}}_2 \rangle}}}$ where ${\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1, {{\ensuremath{\Bar{{\nu}}}\/}}_2 \rangle}} \in T^2$: At once we have ${\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,{{\ensuremath{\Bar{{\nu}}}\/}}_2 \rangle}}\in T^1 \leq T$.
2. $p \leq^2
\big({\ensuremath{\big\langle {\ensuremath{\langle {{\ensuremath{\bar{E}}\/}}_{\alpha},{\ensuremath{\langle \rangle}} \rangle}}, T^2 \big\rangle}}
\big)_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_2,{{\ensuremath{\Bar{{\nu}}}\/}}_1 \rangle}}}$ where ${\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_2 \rangle}} \in T^2$, ${\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1 \rangle}}\in T^2({{\ensuremath{\Bar{{\nu}}}\/}}_2)$: By construction $\forall {\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}\rangle}}\in T^2({{\ensuremath{\Bar{{\nu}}}\/}}_2)\
{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_2 \rangle}}\in
T^1_{{\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}\rangle}}}$. As ${\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1 \rangle}} \in
T^2({{\ensuremath{\Bar{{\nu}}}\/}}_2)$ we get ${\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,{{\ensuremath{\Bar{{\nu}}}\/}}_2 \rangle}} \in T^1 \leq T$.
We show how to continue to the third level. Let us set $$\begin{aligned}
& A_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,S^1,{{\ensuremath{\Bar{{\nu}}}\/}}_2,S^2 \rangle}}} = \operatorname{Suc}_{T^2}({\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,{{\ensuremath{\Bar{{\nu}}}\/}}_2 \rangle}}),
\\
& A_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,S^1 \rangle}}} = {\sideset{}{^0}\operatorname*{\triangle}}_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_2,S^2 \rangle}}}
A_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,S^1,{{\ensuremath{\Bar{{\nu}}}\/}}_2,S^2 \rangle}}},
\\
& B_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,S^1 \rangle}}} = {\ensuremath{\left\{ {\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_3 \rangle}} \mid \operatorname{l}({{\ensuremath{\Bar{{\nu}}}\/}}_3)=0\text{ or }
\operatorname{Suc}_{T^2}({\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,S^1 \rangle}}){\cap}{\kappa}^0({{\ensuremath{\Bar{{\nu}}}\/}}_3) \in {{\ensuremath{\Bar{{\nu}}}\/}}_3 \right\}}},
\\
& \operatorname{Lev}_0(T^{(0)}) = \operatorname{Lev}_0(T^2),
\\
& \operatorname{Suc}_{T^{(0)}}({\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1 \rangle}})= A_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,S^1 \rangle}}} {\cap}B_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,S^1 \rangle}}},
\\
& T^{(0)}_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,{{\ensuremath{\Bar{{\nu}}}\/}}_3 \rangle}}} = {\bigcap}_
{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_2 \rangle}} \in
T^{(0)}_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1 \rangle}}}({{\ensuremath{\Bar{{\nu}}}\/}}_3)}
T^2_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,{{\ensuremath{\Bar{{\nu}}}\/}}_2,{{\ensuremath{\Bar{{\nu}}}\/}}_3 \rangle}}},
\\
& A_{{\ensuremath{\langle \rangle}}}={\sideset{}{^0}\operatorname*{\triangle}}_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,S^1 \rangle}}}
A_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,S^1 \rangle}}},
\\
& B_{{\ensuremath{\langle \rangle}}}={\ensuremath{\left\{ {\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_3 \rangle}} \mid \operatorname{l}({{\ensuremath{\Bar{{\nu}}}\/}}_3)=0\text{ or }
\operatorname{Lev}_0(T^2){\cap}{\kappa}^0({{\ensuremath{\Bar{{\nu}}}\/}}_3) \in {{\ensuremath{\Bar{{\nu}}}\/}}_3 \right\}}},
\\
& \operatorname{Lev}_0(T^{(1)}) = A_{{\ensuremath{\langle \rangle}}} {\cap}B_{{\ensuremath{\langle \rangle}}},
\\
& \operatorname{Suc}_{T^{(1)}}({\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_3 \rangle}})= {\bigcap}_
{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1, {{\ensuremath{\Bar{{\nu}}}\/}}_2 \rangle}} \in
T^{(0)}({{\ensuremath{\Bar{{\nu}}}\/}}_3)}
T^2_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,{{\ensuremath{\Bar{{\nu}}}\/}}_2,{{\ensuremath{\Bar{{\nu}}}\/}}_3 \rangle}}},
\\
& T^3 = T^2 {\cap}T^{(0)} {\cap}T^{(1)}.\end{aligned}$$ Let us assume that $$\begin{aligned}
p \leq^3
{\ensuremath{\big\langle {\ensuremath{\langle {{\ensuremath{\bar{E}}\/}}_{\alpha},{\ensuremath{\langle \rangle}} \rangle}}, T^3 \big\rangle}}.\end{aligned}$$ There are 3 cases to consider here:
1. $p \leq^*
\big({\ensuremath{\big\langle {\ensuremath{\langle {{\ensuremath{\bar{E}}\/}}_{\alpha},{\ensuremath{\langle \rangle}} \rangle}}, T^3 \big\rangle}}
\big)_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,{{\ensuremath{\Bar{{\nu}}}\/}}_2,{{\ensuremath{\Bar{{\nu}}}\/}}_3 \rangle}}}$ where ${\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,{{\ensuremath{\Bar{{\nu}}}\/}}_2,{{\ensuremath{\Bar{{\nu}}}\/}}_3 \rangle}} \in T^3$: At once we get ${\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,{{\ensuremath{\Bar{{\nu}}}\/}}_2,{{\ensuremath{\Bar{{\nu}}}\/}}_3 \rangle}}\in \linebreak[0] T$.
2. $p \leq^*
\big({\ensuremath{\big\langle {\ensuremath{\langle {{\ensuremath{\bar{E}}\/}}_{\alpha},{\ensuremath{\langle \rangle}} \rangle}}, T^3 \big\rangle}}
\big)_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,{{\ensuremath{\Bar{{\nu}}}\/}}_3,{{\ensuremath{\Bar{{\nu}}}\/}}_2 \rangle}}}$ where ${\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,{{\ensuremath{\Bar{{\nu}}}\/}}_3 \rangle}} \in T^3$, ${\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_2 \rangle}}\in T^3_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1 \rangle}}}({{\ensuremath{\Bar{{\nu}}}\/}}_3)$: In this case $\forall {\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}\rangle}}\in T^3_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1 \rangle}}}({{\ensuremath{\Bar{{\nu}}}\/}}_3)\
{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_3 \rangle}}\in
T^3_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,{{\ensuremath{\Bar{{\mu}}}\/}}\rangle}}}$. As ${\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_2 \rangle}} \in
T^3_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1 \rangle}}}({{\ensuremath{\Bar{{\nu}}}\/}}_3)$ we get ${\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_3 \rangle}} \in T_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,{{\ensuremath{\Bar{{\nu}}}\/}}_2 \rangle}}}$.
3. $p \leq^*
\big({\ensuremath{\big\langle {\ensuremath{\langle {{\ensuremath{\bar{E}}\/}}_{\alpha},{\ensuremath{\langle \rangle}} \rangle}}, T^3 \big\rangle}}
\big)_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_3,{{\ensuremath{\Bar{{\nu}}}\/}}_1,{{\ensuremath{\Bar{{\nu}}}\/}}_2 \rangle}}}$ where ${\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_3 \rangle}} \in T^3$, ${\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1, {{\ensuremath{\Bar{{\nu}}}\/}}_2 \rangle}} \in T^3({{\ensuremath{\Bar{{\nu}}}\/}}_3)$: $\forall {\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_1,{{\ensuremath{\Bar{{\mu}}}\/}}_2 \rangle}}\in T^3({{\ensuremath{\Bar{{\nu}}}\/}}_3)\
{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_3 \rangle}}\in
T^3_{{\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_1,{{\ensuremath{\Bar{{\mu}}}\/}}_2 \rangle}}}$ hence ${\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_3 \rangle}}\in
T_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,{{\ensuremath{\Bar{{\nu}}}\/}}_2 \rangle}}}$.
In this way we continue to all levels.
\[skeleton\] Let $T$ be ${{\ensuremath{\bar{E}}\/}}_{\alpha}$ tree. Then there is $T^* \leq T$ such that if $$\begin{aligned}
p \leq
{\ensuremath{\big\langle {\ensuremath{\langle {{\ensuremath{\bar{E}}\/}}_{\alpha},{\ensuremath{\langle \rangle}} \rangle}}, T^* \big\rangle}}\end{aligned}$$ then there is ${\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_n \rangle}} \in T$ such that $$\begin{aligned}
p \leq^*
\big({\ensuremath{\big\langle {\ensuremath{\langle {{\ensuremath{\bar{E}}\/}}_{\alpha},{\ensuremath{\langle \rangle}} \rangle}}, T \big\rangle}}
\big)_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_n \rangle}}}.\end{aligned}$$
The proof is by induction on $\operatorname{l}({{\ensuremath{\bar{E}}\/}}_{\alpha})$. The first case was done in \[skeleton:2\]. The proof is almost the same. We just make sure to invoke the induction hypothesis while repeating the construction.
Construction of $T^1$ and $T^2$ is exactly like in \[skeleton:2\]. We show the construction at the 3rd level.
Let us set $$\begin{aligned}
& A_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,S^1,{{\ensuremath{\Bar{{\nu}}}\/}}_2,S^2 \rangle}}} = \operatorname{Suc}_{T^2}({\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,{{\ensuremath{\Bar{{\nu}}}\/}}_2 \rangle}}),
\\
& A_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,S^1 \rangle}}} = {\sideset{}{^0}\operatorname*{\triangle}}_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_2,S^2 \rangle}}}
A_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,S^1,{{\ensuremath{\Bar{{\nu}}}\/}}_2,S^2 \rangle}}},
\\
& B_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,S^1 \rangle}}} = {\ensuremath{\left\{ {\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_3 \rangle}} \mid \operatorname{l}({{\ensuremath{\Bar{{\nu}}}\/}}_3)=0\text{ or }
\operatorname{Suc}_{T^2}({\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,S^1 \rangle}}){\cap}{\kappa}^0({{\ensuremath{\Bar{{\nu}}}\/}}_3) \in {{\ensuremath{\Bar{{\nu}}}\/}}_3 \right\}}},
\\
& \operatorname{Lev}_0(T^{(0)}) = \operatorname{Lev}_0(T^2),
\\
& \operatorname{Suc}_{T^{(0)}}({\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1 \rangle}})= A_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,S^1 \rangle}}} {\cap}B_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,S^1 \rangle}}},
\\
& T^{(0)}_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,{{\ensuremath{\Bar{{\nu}}}\/}}_3 \rangle}}} = {\bigcap}_
{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_2 \rangle}} \in
T^{(0)}_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1 \rangle}}}({{\ensuremath{\Bar{{\nu}}}\/}}_3)}
T^2_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,{{\ensuremath{\Bar{{\nu}}}\/}}_2,{{\ensuremath{\Bar{{\nu}}}\/}}_3 \rangle}}},
\\
& A'_{{\ensuremath{\langle \rangle}}}={\sideset{}{^0}\operatorname*{\triangle}}_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,S^1 \rangle}}}
A_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,S^1 \rangle}}},
\\
& A_{{\ensuremath{\langle \rangle}}}={\ensuremath{\left\{ {\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_3,S^3 \rangle}} \mid {\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_3,S^{3\prime} \rangle}}\in
A'_{{\ensuremath{\langle \rangle}}}\ S\text{ is generated
from }S^{3\prime}\text{ by induction} \right\}}},
\\
& B_{{\ensuremath{\langle \rangle}}}={\ensuremath{\left\{ {\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_3 \rangle}} \mid l({{\ensuremath{\Bar{{\nu}}}\/}}_3)=0\text{ or }
\operatorname{Lev}_0(T^2){\cap}{\kappa}^0({{\ensuremath{\Bar{{\nu}}}\/}}_3) \in {{\ensuremath{\Bar{{\nu}}}\/}}_3 \right\}}},
\\
& \operatorname{Lev}_0(T^{(1)}) = A_{{\ensuremath{\langle \rangle}}} {\cap}B_{{\ensuremath{\langle \rangle}}},
\\
& \operatorname{Suc}_{T^{(1)}}({\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_3 \rangle}})= {\bigcap}_
{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1, {{\ensuremath{\Bar{{\nu}}}\/}}_2 \rangle}} \in
T^{(0)}({{\ensuremath{\Bar{{\nu}}}\/}}_3)}
T^2_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,{{\ensuremath{\Bar{{\nu}}}\/}}_2,{{\ensuremath{\Bar{{\nu}}}\/}}_3 \rangle}}},
\\
& T^3 = T^2 {\cap}T^{(0)} {\cap}T^{(1)}.\end{aligned}$$ Let us assume that $$\begin{aligned}
p_2 {\mathop{{}^\frown}}p_1 {\mathop{{}^\frown}}p_0 = p \leq^3
{\ensuremath{\big\langle {\ensuremath{\langle {{\ensuremath{\bar{E}}\/}}_{\alpha},{\ensuremath{\langle \rangle}} \rangle}}, T^3 \big\rangle}}.\end{aligned}$$ There are 3 cases to consider here:
1. $p \leq^*
\big({\ensuremath{\big\langle {\ensuremath{\langle {{\ensuremath{\bar{E}}\/}}_{\alpha},{\ensuremath{\langle \rangle}} \rangle}}, T^3 \big\rangle}}
\big)_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,{{\ensuremath{\Bar{{\nu}}}\/}}_2,{{\ensuremath{\Bar{{\nu}}}\/}}_3 \rangle}}}$ where ${\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,{{\ensuremath{\Bar{{\nu}}}\/}}_2,{{\ensuremath{\Bar{{\nu}}}\/}}_3 \rangle}} \in T^3$: At once we get $\linebreak[0]{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,{{\ensuremath{\Bar{{\nu}}}\/}}_2,{{\ensuremath{\Bar{{\nu}}}\/}}_3 \rangle}}\in T$.
2. $p \leq^*
\big({\ensuremath{\big\langle {\ensuremath{\langle {{\ensuremath{\bar{E}}\/}}_{\alpha},{\ensuremath{\langle \rangle}} \rangle}}, T^3 \big\rangle}}
\big)_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,{{\ensuremath{\Bar{{\nu}}}\/}}_3,{{\ensuremath{\Bar{{\nu}}}\/}}_2 \rangle}}}$ where ${\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,{{\ensuremath{\Bar{{\nu}}}\/}}_3 \rangle}} \in T^3$, ${\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_2 \rangle}}\in T^3_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1 \rangle}}}({{\ensuremath{\Bar{{\nu}}}\/}}_3)$: In this case $\forall {\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}\rangle}}\in T^3_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1 \rangle}}}({{\ensuremath{\Bar{{\nu}}}\/}}_3)\
{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_3 \rangle}}\in
T^3_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,{{\ensuremath{\Bar{{\mu}}}\/}}\rangle}}}$. As ${\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_2 \rangle}} \in
T^3_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1 \rangle}}}({{\ensuremath{\Bar{{\nu}}}\/}}_3)$ we get ${\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_3 \rangle}} \in T_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,{{\ensuremath{\Bar{{\nu}}}\/}}_2 \rangle}}}$.
3. $p \leq^2
\big({\ensuremath{\big\langle {\ensuremath{\langle {{\ensuremath{\bar{E}}\/}}_{\alpha},{\ensuremath{\langle \rangle}} \rangle}}, T^3 \big\rangle}}
\big)_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_3 \rangle}}}$ where ${\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_3 \rangle}} \in T^3$ and $p_2 {\mathop{{}^\frown}}p_1 \leq^2
\big({\ensuremath{\big\langle {\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_3,{\ensuremath{\langle \rangle}} \rangle}}, \linebreak[0]
T^3({{\ensuremath{\Bar{{\nu}}}\/}}_3) \big\rangle}}$ By induction there is ${\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,{{\ensuremath{\Bar{{\nu}}}\/}}_2 \rangle}} \in T({{\ensuremath{\Bar{{\nu}}}\/}}_3)$ such that $$\begin{aligned}
\linebreak[0] p_2 {\mathop{{}^\frown}}p_1 \linebreak[0] \leq^* \linebreak[0]
\big({\ensuremath{\big\langle {\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_3,{\ensuremath{\langle \rangle}} \rangle}}, \linebreak[0]
T^2({{\ensuremath{\Bar{{\nu}}}\/}}_3) \big\rangle}}
\big)_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,{{\ensuremath{\Bar{{\nu}}}\/}}_2 \rangle}}}.
\end{aligned}$$ By construction $\forall {\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_1,{{\ensuremath{\Bar{{\mu}}}\/}}_2 \rangle}}\in T^2({{\ensuremath{\Bar{{\nu}}}\/}}_3)\
{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_3 \rangle}}\in
T^2_{{\ensuremath{\langle {{\ensuremath{\Bar{{\mu}}}\/}}_1,{{\ensuremath{\Bar{{\mu}}}\/}}_2 \rangle}}}$ hence ${\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_3 \rangle}}\in
T_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,{{\ensuremath{\Bar{{\nu}}}\/}}_2 \rangle}}}$.
In this way we continue to all levels.
$P_{{{\ensuremath{\bar{E}}\/}}}$-Forcing {#PEForcing}
========================================
A condition in $P_{{{\ensuremath{\bar{E}}\/}}}^*$ is of the form $${\ensuremath{\left\{ {\ensuremath{\langle {{\ensuremath{\Bar{{\gamma}}}\/}}, p^{{\ensuremath{\Bar{{\gamma}}}\/}}\rangle}} \mid {{\ensuremath{\Bar{{\gamma}}}\/}}\in g \right\}}} {\cup}{\ensuremath{\left\{ T \right\}}}$$ where
1. $g \subseteq {{\ensuremath{\bar{E}}\/}}$, ${\lvertg\rvert} \leq \kappa$,
2. $\min {{\ensuremath{\bar{E}}\/}}\in g$ and $g$ has a maximal element,
3. $p^{{\ensuremath{\Bar{{\gamma}}}\/}}\in V_{\kappa}$ is an extender sequence. We allow $p^{{\ensuremath{\Bar{{\gamma}}}\/}}=\emptyset$,
4. $p^0 = (p^{\max g})^0$.
This condition is not really needed here. It is needed in a later forcing based on this one,
5. $T$ is a $\max g$-tree such that for all $t \in T$ $p^{\max g}{\mathop{{}^\frown}}t$ is $^0$-increasing,
6. For all ${{\ensuremath{\Bar{{\gamma}}}\/}}\in g$, $p^{\max g}$ is not permitted to $p^{{\ensuremath{\Bar{{\gamma}}}\/}}$,
7. $\forall {\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}\rangle}}\in T \
{\lvert {\ensuremath{\left\{ {{\ensuremath{\Bar{{\gamma}}}\/}}\in g \mid {{\ensuremath{\Bar{{\nu}}}\/}}\text{ is permitted to } p^{{\ensuremath{\Bar{{\gamma}}}\/}}\right\}}}\rvert}
\leq {\kappa}^0({{\ensuremath{\Bar{{\nu}}}\/}})$,
8. $\forall {\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}\rangle}}\in T$ if ${{\ensuremath{\Bar{{\nu}}}\/}}$ is permitted to $p^{{\ensuremath{\Bar{{\beta}}}\/}},p^{{\ensuremath{\Bar{{\gamma}}}\/}}$ then ${\pi}_{\max g,{{\ensuremath{\Bar{{\beta}}}\/}}}({{\ensuremath{\Bar{{\nu}}}\/}}) \not =
{\pi}_{\max g, {{\ensuremath{\Bar{{\gamma}}}\/}}}({{\ensuremath{\Bar{{\nu}}}\/}})$.
We write $\operatorname{mc}(p)$, $p^{\operatorname{mc}}$, $T^p$, ${{\ensuremath{\bar{E}}\/}}(p)$, $\operatorname{supp}p$ for $\max g$, $p^{\max g}$, $T$, ${{\ensuremath{\bar{E}}\/}}$, $g$ respectively.
$$\begin{diagram}
\\
\node[4]{} \node{{{\ensuremath{\Bar{{\mu}}}\/}}_0} \node[5]{{{\ensuremath{\Bar{{\mu}}}\/}}_1} \node[5]{{{\ensuremath{\Bar{{\mu}}}\/}}_2} \node[5]{{{\ensuremath{\Bar{{\mu}}}\/}}_3} \node[5]{{{\ensuremath{\Bar{{\mu}}}\/}}_4} \node[5]{T}
\\
\node[4]{} \arrow[22]{e,-} \node[5]{}
\\
\node{\text{Support}} \node[3]{} \node{{{\ensuremath{\bar{E}}\/}}_{\kappa}} \node[5]{{{\ensuremath{\bar{E}}\/}}_{{\alpha}_1}} \node[5]{{{\ensuremath{\bar{E}}\/}}_{{\alpha}_2}} \node[5]{{{\ensuremath{\bar{E}}\/}}_{{\alpha}_3}} \node[5]{{{\ensuremath{\bar{E}}\/}}_{{\alpha}_4} = \operatorname{mc}} \node[5]{}
\end{diagram}$$
Let $p, q \in P^*_{\bar{E}}$. We say that $p$ is a Prikry extension of $q$ ($p \leq^* q$ or $p \leq^0 q$) if
1. $\operatorname{supp}p \supseteq \operatorname{supp}q$,
2. $\forall {\gamma}\in \operatorname{supp}q \ p^{\gamma}=q^{\gamma}$,
3. $T^p \leq T^q$.
We include in this definition the degenerate case $\operatorname{l}({{\ensuremath{\bar{E}}\/}})=0$. There is neither extender nor tree in this case. By $p \leq^* p$ we mean $p = q$.
$$\begin{diagram}
\node[4]{} \node{{{\ensuremath{\Bar{{\mu}}}\/}}_0} \node[2]{{{\ensuremath{\Bar{{\nu}}}\/}}_0} \node[3]{{{\ensuremath{\Bar{{\mu}}}\/}}_1} \node[2]{{{\ensuremath{\Bar{{\nu}}}\/}}_1} \node[3]{{{\ensuremath{\Bar{{\mu}}}\/}}_2} \node[5]{{{\ensuremath{\Bar{{\mu}}}\/}}_3} \node[5]{{{\ensuremath{\Bar{{\mu}}}\/}}_4} \node[5]{{{\ensuremath{\Bar{{\nu}}}\/}}_2} \node[5]{{\pi}_{{\beta}_2,{\alpha}_4}^{-1}T}
\\
\node[4]{} \arrow[27]{e,-} \node[5]{}
\\
\node{\text{Support}} \node[3]{} \node{{{\ensuremath{\bar{E}}\/}}_{{\kappa}}} \node[2]{{{\ensuremath{\bar{E}}\/}}_{{\beta}_0}} \node[3]{{{\ensuremath{\bar{E}}\/}}_{{\alpha}_1}} \node[2]{{{\ensuremath{\bar{E}}\/}}_{{\beta}_1}} \node[3]{{{\ensuremath{\bar{E}}\/}}_{{\alpha}_2}} \node[5]{{{\ensuremath{\bar{E}}\/}}_{{\alpha}_3}} \node[5]{{{\ensuremath{\bar{E}}\/}}_{{\alpha}_4}} \node[5]{{{\ensuremath{\bar{E}}\/}}_{{\beta}_2}=\operatorname{mc}}
\end{diagram}$$
A condition in $P_{{{\ensuremath{\bar{E}}\/}}}$ is of the form $$p_n {\mathop{{}^\frown}}\dotsb {\mathop{{}^\frown}}p_0$$ where
- $p_0 \in P^*_{{{\ensuremath{\bar{E}}\/}}}$,
- $p_1 \in P^*_{{{\ensuremath{\Bar{{\mu}}}\/}}_1}$,
- $\vdots$,
- $p_n \in P^*_{{{\ensuremath{\Bar{{\mu}}}\/}}_n}$,
where ${{\ensuremath{\bar{E}}\/}}, {{\ensuremath{\Bar{{\mu}}}\/}}_1, \dotsc, {{\ensuremath{\Bar{{\mu}}}\/}}_n$ are extender sequence systems satisfying $${\kappa}^0({{\ensuremath{\Bar{{\mu}}}\/}}_n) < \dotsb < {\kappa}^0({{\ensuremath{\Bar{{\mu}}}\/}}_1) < {\kappa}^0(\bar{E}).$$
$$\begin{diagram}
\node{{\tau}_0} \node[2]{{\tau}_1} \node[2]{{\tau}_2} \node[2]{{\tau}_3} \node[3]{{\tau}_4} \node[2]{R} \node[1]{}
\node{{{\ensuremath{\Bar{{\mu}}}\/}}_5} \node[2]{{{\ensuremath{\Bar{{\mu}}}\/}}_6} \node[2]{{{\ensuremath{\Bar{{\mu}}}\/}}_7} \node[2]{{{\ensuremath{\Bar{{\mu}}}\/}}_8} \node[3]{{{\ensuremath{\Bar{{\mu}}}\/}}_9} \node[2]{S} \node[1]{}
\node{{{\ensuremath{\Bar{{\nu}}}\/}}_0} \node[2]{{{\ensuremath{\Bar{{\nu}}}\/}}_1} \node[2]{{{\ensuremath{\Bar{{\nu}}}\/}}_5} \node[2]{{{\ensuremath{\Bar{{\nu}}}\/}}_6} \node[3]{{{\ensuremath{\Bar{{\nu}}}\/}}_4} \node[2]{T}
\\
\arrow[11]{e,-} \node[13]{}
\arrow[11]{e,-} \node[13]{}
\arrow[11]{e,-} \node[2]{}
\\
\node{{{\ensuremath{\Bar{{\mu}}}\/}}_0} \node[2]{{{\ensuremath{\Bar{{\mu}}}\/}}_1} \node[2]{{{\ensuremath{\Bar{{\mu}}}\/}}_2} \node[2]{{{\ensuremath{\Bar{{\mu}}}\/}}_3} \node[3]{{{\ensuremath{\Bar{{\mu}}}\/}}_4=\operatorname{mc}} \node[2]{} \node[1]{}
\node{{{\ensuremath{\Bar{{\nu}}}\/}}_0} \node[2]{{{\ensuremath{\Bar{{\nu}}}\/}}_1} \node[2]{{{\ensuremath{\Bar{{\nu}}}\/}}_2} \node[2]{{{\ensuremath{\Bar{{\nu}}}\/}}_3} \node[3]{{{\ensuremath{\Bar{{\nu}}}\/}}_4=\operatorname{mc}} \node[2]{} \node[1]{}
\node{{{\ensuremath{\bar{E}}\/}}_{{\kappa}}} \node[2]{{{\ensuremath{\bar{E}}\/}}_{{\alpha}_1}} \node[2]{{{\ensuremath{\bar{E}}\/}}_{{\alpha}_2}} \node[2]{{{\ensuremath{\bar{E}}\/}}_{{\alpha}_3}} \node[3]{{{\ensuremath{\bar{E}}\/}}_{{\alpha}_4}=\operatorname{mc}} \node[2]{}
\end{diagram}$$
Let $p,q \in P_{{{\ensuremath{\bar{E}}\/}}}$. We say that $p$ is a Prikry extension of $q$ ($p \leq^* q$ or $p \leq^0 q$) if $p,q$ are of the form $$\begin{split}
p &= p_n {\mathop{{}^\frown}}\dotsb {\mathop{{}^\frown}}p_0,
\\
q &= q_n {\mathop{{}^\frown}}\dotsb {\mathop{{}^\frown}}q_0,
\end{split}$$ and
- $p_0,q_0 \in P^*_{{{\ensuremath{\bar{E}}\/}}},\ p_0 \leq^* q_0$,
- $p_1,q_1 \in P^*_{{{\ensuremath{\Bar{{\mu}}}\/}}_1},\ p_1 \leq^* q_1$,
- $\vdots$,
- $p_n,q_n \in P^*_{{{\ensuremath{\Bar{{\mu}}}\/}}_n},\ p_n \leq^* q_n$.
\[dfn:enlarge\] Let $p \in P^*_{\bar{E}}$ and ${\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}\rangle}} \in T^p$. We =define $(p)_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}\rangle}}}$ to be $p'_1 {\mathop{{}^\frown}}p'_0$ where
1. $\operatorname{supp}p'_1=
{\ensuremath{\left\{ \pi_{\operatorname{mc}(p),{{\ensuremath{\Bar{{\gamma}}}\/}}}({{\ensuremath{\Bar{{\nu}}}\/}}) \mid {{\ensuremath{\Bar{{\gamma}}}\/}}\in \operatorname{supp}p, \, {\kappa}^0(p^{{\ensuremath{\Bar{{\gamma}}}\/}}) < {\kappa}^0({{\ensuremath{\Bar{{\nu}}}\/}}) \right\}}}$,
2. $p_1^{\prime\pi_{\operatorname{mc}(p),{{\ensuremath{\Bar{{\gamma}}}\/}}}({{\ensuremath{\Bar{{\nu}}}\/}})}=p^{{\ensuremath{\Bar{{\gamma}}}\/}}$,
3. $T^{p'_1} = T^p({{\ensuremath{\Bar{{\nu}}}\/}})$,
4. $\operatorname{supp}p'_0 = \operatorname{supp}p$,
5. $\forall {{\ensuremath{\Bar{{\gamma}}}\/}}\in \operatorname{supp}p'_0 \ p_0^{\prime{{\ensuremath{\Bar{{\gamma}}}\/}}}=
\begin{cases}
\pi_{\operatorname{mc}(p),{{\ensuremath{\Bar{{\gamma}}}\/}}}({{\ensuremath{\Bar{{\nu}}}\/}}) &
{\kappa}^0(p^{{\ensuremath{\Bar{{\gamma}}}\/}}) < {\kappa}^0({{\ensuremath{\Bar{{\nu}}}\/}})
\\
p^{{\ensuremath{\Bar{{\gamma}}}\/}}& \text{otherwise}
\end{cases}$,
6. $T^{p'_0}=T^p_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}\rangle}}}$.
Let $p,q \in P_{{{\ensuremath{\bar{E}}\/}}}$. We say that $p$ is a $1$-point extension of $q$ ($p \leq^1 q$) if $p,q$ are of the form $$\begin{split}
p &= p_{n+1} {\mathop{{}^\frown}}p_n {\mathop{{}^\frown}}\dotsb {\mathop{{}^\frown}}p_0,
\\
q &= q_n {\mathop{{}^\frown}}\dotsb {\mathop{{}^\frown}}q_0,
\end{split}$$ and there is $0 \leq k \leq n$ such that
- $p_i,q_i \in P^*_{{{\ensuremath{\Bar{{\mu}}}\/}}_i},\ p_i \leq^* q_i$ for $i=0,\dotsc,k-1$,
- $p_{i+1},q_i \in P^*_{{{\ensuremath{\Bar{{\mu}}}\/}}_i},\ p_{i+1} \leq^* q_i$ for $i=k+1,\dotsc,n$,
- There is ${\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}\rangle}} \in T^{q_k}$ such that $p_{k+1} {\mathop{{}^\frown}}p_k \leq^* (q_k)_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}\rangle}}}$.
$$\begin{diagram}
\node{{{\ensuremath{\Bar{{\mu}}}\/}}_0} \node[4]{{{\ensuremath{\Bar{{\mu}}}\/}}_1} \node[6]{{{\ensuremath{\Bar{{\mu}}}\/}}_3} \node[4]{{{\ensuremath{\Bar{{\mu}}}\/}}_4} \node[2]{T({{\ensuremath{\Bar{{\nu}}}\/}})} \node[2]{}
\node{{{\ensuremath{\Bar{{\nu}}}\/}}^0} \node[4]{{\pi}_{{\alpha}_4,{\alpha}_1}({{\ensuremath{\Bar{{\nu}}}\/}})} \node[4]{{{\ensuremath{\Bar{{\mu}}}\/}}_2}
\node[4]{{\pi}_{{\alpha}_4,{\alpha}_3}({{\ensuremath{\Bar{{\nu}}}\/}})} \node[4]{{{\ensuremath{\Bar{{\nu}}}\/}}}
\node[2]{T_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}\rangle}}}}
\\
\arrow[16]{e,-} \node[18]{}
\arrow[19]{e,-} \node[18]{}
\\
\node{{{\ensuremath{\Bar{{\nu}}}\/}}^0} \node[4]{{\pi}_{{\alpha}_4,{\alpha}_1}({{\ensuremath{\Bar{{\nu}}}\/}})} \node[6]{{\pi}_{{\alpha}_4,{\alpha}_3}({{\ensuremath{\Bar{{\nu}}}\/}})} \node[4]{{{\ensuremath{\Bar{{\nu}}}\/}}} \node[4]{}
\node{{{\ensuremath{\bar{E}}\/}}_{{\kappa}}} \node[4]{{{\ensuremath{\bar{E}}\/}}_{{\alpha}_1}} \node[4]{{{\ensuremath{\bar{E}}\/}}_{{\alpha}_2}} \node[4]{{{\ensuremath{\bar{E}}\/}}_{{\alpha}_3}} \node[4]{{{\ensuremath{\bar{E}}\/}}_{{\alpha}_4}}
\end{diagram}$$
Let $p,q \in P_{{{\ensuremath{\bar{E}}\/}}}$. We say that $p$ is an $n$-point extension of $q$ ($p \leq^n q$) if there are $p^n, \dotsc, p^0$ such that $$\begin{aligned}
p=p^n \leq^1 \dotsb \leq^1 p^0=q.\end{aligned}$$
Let $p,q \in P_{{{\ensuremath{\bar{E}}\/}}}$. We say that $p$ is an extension of $q$ ($p \leq q$) if there is n such that $p \leq^n q$.
Later on by ${{\ensuremath{P_{{{\ensuremath{\bar{E}}\/}}}\/}}}$ we mean ${\ensuremath{\langle {{\ensuremath{P_{{{\ensuremath{\bar{E}}\/}}}\/}}},\leq \rangle}}$.
When $\operatorname{l}({{\ensuremath{\bar{E}}\/}})=1$ the forcing $P_{{{\ensuremath{\bar{E}}\/}}}$ is the Gitik-Magidor forcing from section 1 of [@Moti]. When $\operatorname{l}({{\ensuremath{\bar{E}}\/}}) < {\kappa}$ the forcing $P_{{{\ensuremath{\bar{E}}\/}}}$ is similar to the forcing defined in [@Miri].
In several places we want to prevent enlargment of the support of a condition. This makes all the conditions which are stronger than some condition but with the same support resemble Radin forcing. The following definition catches the meaning of not enlarging the support. The ‘resemblence’ we look for is \[GenericForRadin\].
Let $p,q \in P_{{{\ensuremath{\bar{E}}\/}}}$. We say that $p \leq^*_R q$ if
1. $p \leq^* q$,
2. $\operatorname{supp}p = \operatorname{supp}q$.
Let $p,q \in P_{{{\ensuremath{\bar{E}}\/}}}$. We say that $p \leq^1_R q$ if
1. $p \leq^1 q$,
2. In the definition of $\leq^1$ we can substitute $\leq^*$ by $\leq^*_R$.
Let $p,q \in P_{{{\ensuremath{\bar{E}}\/}}}$. We say that $p \leq^n_R q$ if there are $p^n, \dotsc, p^0$ such that $$\begin{aligned}
p=p^n \leq^1_R \dotsb \leq^1_R p^0=q.\end{aligned}$$
Let $p,q \in P_{{{\ensuremath{\bar{E}}\/}}}$. We say that $p \leq_R q$ if there is $n$ such that $p \leq^n_R q$.
The above definitions imply that if $q \leq p$ then there is $r$ such that $q \leq^* r \leq_R p$.
Let ${{\ensuremath{\Bar{{\epsilon}}}\/}}$ be an extender sequence such that ${\kappa}^0({{\ensuremath{\Bar{{\epsilon}}}\/}})< {\kappa}^0({{\ensuremath{\bar{E}}\/}})$. $$\begin{aligned}
{{\ensuremath{P_{{{\ensuremath{\bar{E}}\/}}}\/}}}/{{\ensuremath{P_{{{\ensuremath{\Bar{{\epsilon}}}\/}}}\/}}}= {\ensuremath{\left\{ p \mid q\in {{\ensuremath{P_{{{\ensuremath{\Bar{{\epsilon}}}\/}}}\/}}}, q{\mathop{{}^\frown}}p \in {{\ensuremath{P_{{{\ensuremath{\bar{E}}\/}}}\/}}}\right\}}}.\end{aligned}$$
Basic Properties of $P_{{{\ensuremath{\bar{E}}\/}}}$ {#BasicProperties}
====================================================
$P_{{\ensuremath{\bar{E}}\/}}$ satisfies ${\kappa}^{++}$-c.c.
The usual $\Delta$-lemma argument on the support will do.
\[SubForcing\] Let $p \in {{\ensuremath{P_{{{\ensuremath{\bar{E}}\/}}}\/}}}$, $P^* = {\ensuremath{\left\{ q \leq_R p \mid p \in {{\ensuremath{P_{{{\ensuremath{\bar{E}}\/}}}\/}}}\right\}}}$. Then
1. ${\ensuremath{\langle P^*,\leq_R \rangle}}$ satisfies ${\kappa}^{+}$-c.c.,
2. ${\ensuremath{\langle P^*,\leq_R \rangle}}$ is sub-forcing of ${\ensuremath{\langle {{\ensuremath{P_{{{\ensuremath{\bar{E}}\/}}}\/}}}/p,\leq \rangle}}$.
Showing ${\kappa}^{+}$-c.c. is trivial.
Showing that $P^*$ is sub-forcing of ${{\ensuremath{P_{{{\ensuremath{\bar{E}}\/}}}\/}}}/p$ amounts to showing that any maximal anti-chain of $P^*$ is also a maximal anti-chain of ${{\ensuremath{P_{{{\ensuremath{\bar{E}}\/}}}\/}}}/p$.
Let $A$ be a maximal anti-chain of $P^*$. Let $q \in {{\ensuremath{P_{{{\ensuremath{\bar{E}}\/}}}\/}}}/p$. As $q \leq p$, there is $r' \in P^*$ such that $q \leq^* r' \leq_R p$. Assume that $r' = r'_n {\mathop{{}^\frown}}\dotsb {\mathop{{}^\frown}}r'_0$. Then also $q = q_n {\mathop{{}^\frown}}\dotsb {\mathop{{}^\frown}}q_0$. Let $r_i$ be $r'_i$ with $T^{r'_i}$ substituted by $T^{r'_i} {\cap}{\pi}_{\operatorname{mc}(q_i),\operatorname{mc}(r'_i)}(T^{q_i})$ and $r = r_n {\mathop{{}^\frown}}\dotsb {\mathop{{}^\frown}}r_0$. As $r \in P^*$ and $A$ is a maximal anti-chain there is $a \in A$ such that $a {\parallel}r$. Take $s \leq_R a, r$. Considering how we constructed $r$ from $r'$ we must have $t \leq^* s$ such that $t \leq q$. Hence $q {\parallel}a$. So we get that $A$ is a maximal anti-chain of ${{\ensuremath{P_{{{\ensuremath{\bar{E}}\/}}}\/}}}/p$.
\[GenericForRadin\] Let $p \in {{\ensuremath{P_{{{\ensuremath{\bar{E}}\/}}}\/}}}$, $P^* = {\ensuremath{\left\{ q \leq_R p \mid p \in {{\ensuremath{P_{{{\ensuremath{\bar{E}}\/}}}\/}}}\right\}}}$. Then there is $r \in R_{\operatorname{mc}(p)}$ such that $P^* \simeq R_{\operatorname{mc}(p)}/r$
For simplicity assume that $p = p_0$. Then we set $r = {\ensuremath{\langle {\ensuremath{\langle \operatorname{mc}(p_0), p_0^{\operatorname{mc}} \rangle}}, \linebreak[0] T^p \rangle}}$.
We give the isomorphism: The image of $q \in P^*$ is $s \in {\ensuremath{\langle R_{\operatorname{mc}(p)}/r, \leq \rangle}}$ such that
1. $q \leq^* s$,
2. $T^{s_i} = T^{q_i}$ where $s = s_n {\mathop{{}^\frown}}\dotsb {\mathop{{}^\frown}}s_0$, $q = q_n {\mathop{{}^\frown}}\dotsb {\mathop{{}^\frown}}q_0$.
Let $G$ be ${{\ensuremath{P_{{{\ensuremath{\bar{E}}\/}}}\/}}}$-generic.
${{\ensuremath{\bar{E}}\/}}_G$ is the enumeration of ${\ensuremath{\left\{ {{\ensuremath{\bar{E}}\/}}(p_k) \mid p_n {\mathop{{}^\frown}}\dotsb {\mathop{{}^\frown}}p_0 \in G \right\}}}$ ordered increasingly by ${\kappa}^0({{\ensuremath{\bar{E}}\/}}(p_k))$.
Let ${\zeta}< \operatorname{otp}({{\ensuremath{\bar{E}}\/}}_G)$. Then
1. $G {\mathord{\restriction}}{\zeta}= {\ensuremath{\left\{ p_n {\mathop{{}^\frown}}\dotsb {\mathop{{}^\frown}}p_k \mid p_n {\mathop{{}^\frown}}\dotsb {\mathop{{}^\frown}}p_k {\mathop{{}^\frown}}\dotsb {\mathop{{}^\frown}}p_0 \in G,
{{\ensuremath{\bar{E}}\/}}(p_k) = {{\ensuremath{\bar{E}}\/}}_G({\zeta}) \right\}}}$,
2. $G \setminus {\zeta}= {\ensuremath{\left\{ p_{k-1} {\mathop{{}^\frown}}\dotsb {\mathop{{}^\frown}}p_0 \mid p_n {\mathop{{}^\frown}}\dotsb {\mathop{{}^\frown}}p_k {\mathop{{}^\frown}}\dotsb
{\mathop{{}^\frown}}p_0 \in G, \linebreak[0]
{{\ensuremath{\bar{E}}\/}}(p_k) = {{\ensuremath{\bar{E}}\/}}_G({\zeta}) \right\}}}$.
$$\begin{aligned}
& M^{{{\ensuremath{\Bar{{\alpha}}}\/}}}_G =
\begin{cases}
{\bigcup}{\ensuremath{\left\{ M^{{{\ensuremath{\Bar{{\mu}}}\/}}}_G \mid p \in G, \, {{\ensuremath{\Bar{{\mu}}}\/}}= p^{{{\ensuremath{\Bar{{\alpha}}}\/}}} \right\}}} {\cup}{\ensuremath{\left\{ {{\ensuremath{\Bar{{\alpha}}}\/}}\right\}}} &
\exists p\in G\ {{\ensuremath{\Bar{{\alpha}}}\/}}\in \operatorname{supp}p
\\
{\ensuremath{\left\{ {{\ensuremath{\Bar{{\alpha}}}\/}}\right\}}} & \text{otherwise}
\end{cases}
\\
& C^{{{\ensuremath{\Bar{{\alpha}}}\/}}}_G = {\ensuremath{\left\{ {\kappa}({{\ensuremath{\Bar{{\mu}}}\/}}) \mid {{\ensuremath{\Bar{{\mu}}}\/}}\in M^{{{\ensuremath{\Bar{{\alpha}}}\/}}}_G \right\}}}\end{aligned}$$
1. $C^{{{\ensuremath{\bar{E}}\/}}_{\kappa}}_G\setminus{\ensuremath{\left\{ {\kappa}\right\}}}$ is a club in ${\kappa}$,
2. $C^{{{\ensuremath{\bar{E}}\/}}_{\alpha}}_G\setminus{\ensuremath{\left\{ {\alpha}\right\}}}$ is unbounded in ${\kappa}$,
3. ${{\ensuremath{\Bar{{\alpha}}}\/}}\not= {{\ensuremath{\Bar{{\beta}}}\/}}\implies C^{{{\ensuremath{\Bar{{\alpha}}}\/}}}_G \not= C^{{{\ensuremath{\Bar{{\alpha}}}\/}}}_G$.
The first two claims are immediate as these are sequences which are generated by Radin forcing.
The last is by density and noticing that when $p^{{{\ensuremath{\Bar{{\alpha}}}\/}}},p^{{{\ensuremath{\Bar{{\beta}}}\/}}}$ are permitted for ${{\ensuremath{\Bar{{\nu}}}\/}}$ we required ${\pi}_{\operatorname{mc}(p),{{\ensuremath{\Bar{{\alpha}}}\/}}}({{\ensuremath{\Bar{{\nu}}}\/}}) \not=
{\pi}_{\operatorname{mc}(p),{{\ensuremath{\Bar{{\beta}}}\/}}}({{\ensuremath{\Bar{{\nu}}}\/}})$.
Homogeneity in Dense Open Subsets {#HomogenDense}
=================================
Our aim in this section is to prove the following
\[DenseHomogen\] Let $D \subseteq {{\ensuremath{P_{{{\ensuremath{\bar{E}}\/}}}\/}}}$ be dense open and $p=p_k{\mathop{{}^\frown}}\dotsb{\mathop{{}^\frown}}p_0 \in {{\ensuremath{P_{{{\ensuremath{\bar{E}}\/}}}\/}}}$. Then there is $p^* \leq^* p$ such that $$\begin{gathered}
\exists S^k\ \exists n_k \forall {\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_{k,1},\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_{k,{n_k}} \rangle}}
\in S^k \, \dotsc
\exists S^0\ \exists n_0 \forall {\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_{0,1},\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_{0,{n_0}} \rangle}}
\in S^0 \,
\\
(p^*_k)_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_{k,1},\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_{k,n_k} \rangle}}} {\mathop{{}^\frown}}\dotsb {\mathop{{}^\frown}}(p^*_0)_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_{0,1},\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_{0,n_0} \rangle}}}
\in D\end{gathered}$$ where
1. $S^i \subseteq T^{p^*_i} {\mathord{\restriction}}[V_{\kappa}]^{n_i}$,
2. $\forall l<n_i \,\forall {\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_{l} \rangle}} \in S^i \,
\exists {\xi}\
\operatorname{Suc}_{S^i}({\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_{l} \rangle}})
\in E_{\operatorname{mc}(p^*_i)}({\xi})$.
The proof is done by a series of lemmas.
\[EnlargeEvery\] Let $p \in P^*_{{{\ensuremath{\bar{E}}\/}}}$. Let $s$ be a function such that $\operatorname{dom}s \subseteq {{\ensuremath{\bar{E}}\/}}$. and for all ${{\ensuremath{\Bar{{\alpha}}}\/}},{{\ensuremath{\Bar{{\beta}}}\/}}\in \operatorname{dom}s$, ${{\ensuremath{\Bar{{\alpha}}}\/}}\neq {{\ensuremath{\Bar{{\beta}}}\/}}$
1. $s({{\ensuremath{\Bar{{\alpha}}}\/}})$ is an extender sequence,
2. $\operatorname{l}(s({{\ensuremath{\Bar{{\alpha}}}\/}}))=\operatorname{l}(s({{\ensuremath{\Bar{{\beta}}}\/}}))$,
3. ${\kappa}^0(s({{\ensuremath{\Bar{{\alpha}}}\/}})) = {\kappa}^0(s({{\ensuremath{\Bar{{\beta}}}\/}}))$,
4. $s({{\ensuremath{\Bar{{\alpha}}}\/}}) \not= s({{\ensuremath{\Bar{{\beta}}}\/}})$.
We define $(p)_{{\ensuremath{\langle s \rangle}}}$ to be $p'_1 {\mathop{{}^\frown}}p'_0$ where
1. $\operatorname{supp}p'_1=
{\ensuremath{\left\{ s({{\ensuremath{\Bar{{\alpha}}}\/}}) \mid {{\ensuremath{\Bar{{\alpha}}}\/}}\in \operatorname{supp}p{\cap}\operatorname{dom}s, \, {\kappa}^0(p^{{\ensuremath{\Bar{{\alpha}}}\/}}) < {\kappa}^0(s({{\ensuremath{\Bar{{\alpha}}}\/}})) \right\}}}$,
2. $\forall {{\ensuremath{\Bar{{\alpha}}}\/}}\in \operatorname{supp}p'_1\ p_1^{\prime s({{\ensuremath{\Bar{{\alpha}}}\/}})}=
p^{{\ensuremath{\Bar{{\alpha}}}\/}}$,
3. If $s(\operatorname{mc}(p))\in T^{p}$ then $T^{p'_1}=T^{p}(s(\operatorname{mc}(p)))$. Otherwise we leave $T^{p'_1}$ undefined,
4. $\operatorname{supp}p'_0 = \operatorname{supp}p$,
5. $\forall {{\ensuremath{\Bar{{\alpha}}}\/}}\in \operatorname{supp}p'_0 \ p_0^{\prime{{\ensuremath{\Bar{{\alpha}}}\/}}}=
\begin{cases}
s({{\ensuremath{\Bar{{\alpha}}}\/}}) &
{{\ensuremath{\Bar{{\alpha}}}\/}}\in\operatorname{dom}s\text{ and }
{\kappa}^0(p^{{\ensuremath{\Bar{{\alpha}}}\/}}) < {\kappa}^0(s({{\ensuremath{\Bar{{\alpha}}}\/}}))
\\
p^{{\ensuremath{\Bar{{\alpha}}}\/}}& \text{otherwise}
\end{cases}$,
6. If $s(\operatorname{mc}(p))\in T^{p}$ then $T^{p'_0}=T^{p}_{{\ensuremath{\langle s(\operatorname{mc}(p)) \rangle}}}$. Otherwise we leave $T^{p'_0}$ undefined.
Let $p \in P^*_{{{\ensuremath{\bar{E}}\/}}}$. Let $s$ be a function with $\operatorname{dom}s = {1,\dotsc,n}$ such that for all $i$ $s(i)$ satisfies definition \[EnlargeEvery\]. Then we define $(p)_{{\ensuremath{\langle s \rangle}}}$ as $p^n$ where $p^n$ is defined by induction as follows: $$\begin{aligned}
& p^0 = p,
\\
& p^{i+1} = p^i_i {\mathop{{}^\frown}}\dotsb {\mathop{{}^\frown}}p^i_1 {\mathop{{}^\frown}}(p^i_0)_{{\ensuremath{\langle s(i+1) \rangle}}}.\end{aligned}$$
We note the following: If ${\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_n \rangle}} \in T^p$ and we set for all $1\leq i \leq n$ $$\begin{aligned}
s(i)={\ensuremath{\left\{ {\ensuremath{\langle {{\ensuremath{\Bar{{\alpha}}}\/}}, {\pi}_{\operatorname{mc}(p),{{\ensuremath{\Bar{{\alpha}}}\/}}}({{\ensuremath{\Bar{{\nu}}}\/}}_i) \rangle}} \mid {{\ensuremath{\Bar{{\alpha}}}\/}}\in \operatorname{supp}p \right\}}}\end{aligned}$$ then $$\begin{aligned}
(p)_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_n \rangle}}}=(p)_{{\ensuremath{\langle s \rangle}}}.\end{aligned}$$
We use this operation also in cases where $p$ is not strictly a condition. That is if $p {\cup}{\ensuremath{\left\{ T \right\}}} \in {{\ensuremath{P_{{{\ensuremath{\bar{E}}\/}}}\/}}}$ we also use $(p)_{{\ensuremath{\langle s \rangle}}}$. In this case we ignore the trees in the definition.
This definition is used in the proof of the homogeneity for the following reason: Beforehand we do not know what a legitimate extension is. By checking with all the possible ${\mu}$’s we check on all legitimate conditions which [*might*]{} be extensions.
\[canon-dense:n\] Let $D$ be dense open in ${{\ensuremath{P_{{{\ensuremath{\bar{E}}\/}}}\/}}}/{{\ensuremath{P_{{{\ensuremath{\Bar{{\epsilon}}}\/}}}\/}}}$, $p = p_0 \in {{\ensuremath{P_{{{\ensuremath{\bar{E}}\/}}}\/}}}/{{\ensuremath{P_{{{\ensuremath{\Bar{{\epsilon}}}\/}}}\/}}}$, $0<n<{\omega}$. Then there is $p^* \leq^* p$ such that one and only one of the following is true:
1. There is $S \subseteq T^{p^*}{\mathord{\restriction}}[V_{\kappa}]^{n}$ such that
1. $\forall k<n\,\exists {\xi}<\operatorname{l}({{\ensuremath{\bar{E}}\/}})\
\operatorname{Suc}_S({\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_k \rangle}})
\in E_{\operatorname{mc}(p^*)}({\xi})$,
2. $\forall {\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_{n} \rangle}} \in S$ $(p^*)_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_{n} \rangle}}} \in D$.
2. $\forall {\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_{n} \rangle}} \in
T^{p^*}
\forall q \leq^* (p^*)_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_{n} \rangle}}}\
q \notin D$.
We give the proof for $n=1$. Adapting the proof for higher $n$’s require that whenever we enumerate singletons we should enumerate $n$-tuples and when we use $j$ we should use $j_n$.
We start an induction on ${\xi}$ in which we build $$\begin{aligned}
{\ensuremath{{\ensuremath{\langle {{\ensuremath{\Bar{{\alpha}}}\/}}^{\xi}, u^{\xi}\mid {\xi}< {\kappa}\rangle}}}}.\end{aligned}$$ We start by setting $$\begin{aligned}
u^0 &= p_0 \setminus {\ensuremath{\left\{ T^{p_0} \right\}}},
\\
{{\ensuremath{\Bar{{\alpha}}}\/}}^0 &= \operatorname{mc}(p_0),
\\
T^0 &= T^{p_0} {\mathord{\restriction}}\pi^{-1}_{{{\ensuremath{\Bar{{\alpha}}}\/}}^0,0}{\ensuremath{\left\{ {{\ensuremath{\Bar{{\nu}}}\/}}\mid {\kappa}^0({{\ensuremath{\Bar{{\nu}}}\/}}) \text{ is inaccessible} \right\}}},\end{aligned}$$ and taking an increasing enumeration $$\begin{aligned}
{\ensuremath{\left\{ {\kappa}^0({{\ensuremath{\Bar{{\nu}}}\/}}) \mid {\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}\rangle}} \in T^0 \right\}}}
= {\ensuremath{{\ensuremath{\langle {\tau}_{\xi}\mid {\xi}< {\kappa}\rangle}}}}.\end{aligned}$$ Assume that we have constructed $$\begin{aligned}
{\ensuremath{{\ensuremath{\langle {{\ensuremath{\Bar{{\alpha}}}\/}}^{\xi}, u^{\xi}\mid {\xi}< {\xi}_0 \rangle}}}}.\end{aligned}$$
We have $2$ cases: ${\xi}_0$ is limit: Choose ${{\ensuremath{\Bar{{\alpha}}}\/}}^{{\xi}_0} > {{\ensuremath{\Bar{{\alpha}}}\/}}^{\xi}$ for all ${\xi}< {\xi}_0$ and set $$\begin{aligned}
u^{{\xi}_0} &= {\bigcup}_{{\xi}< {\xi}_0} u^{\xi}{\cup}{\ensuremath{\left\{ {\ensuremath{\langle {{\ensuremath{\Bar{{\alpha}}}\/}}^{{\xi}_0}, t \rangle}} \right\}}}
\text{ where }{\kappa}^0(t) = {\tau}_{{\xi}_0}.\end{aligned}$$
${\xi}_0 = {\xi}+ 1:$ For each ${{\ensuremath{\Bar{{\nu}}}\/}}$ such that ${\kappa}^0({{\ensuremath{\Bar{{\nu}}}\/}})={\tau}_{\xi}$ we set $$\begin{aligned}
S({{\ensuremath{\Bar{{\nu}}}\/}}) =
\big( \prod_{\substack{{{\ensuremath{\Bar{{\alpha}}}\/}}\in \operatorname{supp}u^{\xi}\\
{\kappa}((u^{\xi})^{{{\ensuremath{\Bar{{\alpha}}}\/}}})<{\tau}_{\xi}}}
{\ensuremath{\left\{ {{\ensuremath{\Bar{{\mu}}}\/}}\mid {\kappa}^0({{\ensuremath{\Bar{{\mu}}}\/}})={\kappa}^0({{\ensuremath{\Bar{{\nu}}}\/}}) \right\}}} \big)
\times
{\ensuremath{\left\{ {\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}\rangle}} \right\}}}.\end{aligned}$$ Let $$\begin{aligned}
S = {\bigcup}_{{\kappa}^0({{\ensuremath{\Bar{{\nu}}}\/}})={\tau}_{\xi}}
S({{\ensuremath{\Bar{{\nu}}}\/}})\end{aligned}$$ and set enumeration of $S$ $$\begin{aligned}
S={\ensuremath{{\ensuremath{\langle s^{{\xi}_0,{\rho}} \mid {\rho}< {\tau}_{{\xi}_0} \rangle}}}}.\end{aligned}$$ There are fewer than ${\tau}_{{\xi}_0}$ elements in $S$. We use ${\tau}_{{\xi}_0}$ as this is the maximum size $S$ can have which is not ‘killing’ the induction.
We do induction on ${\rho}$ which builds $$\begin{aligned}
{\ensuremath{{\ensuremath{\langle {{\ensuremath{\Bar{{\alpha}}}\/}}^{{\xi}_0,{\rho}}, u_0^{{\xi}_0,{\rho}},
T_0^{{\xi}_0,{\rho}},u_1^{{\xi}_0,{\rho}},T_1^{{\xi}_0,{\rho}} \mid {\rho}< {\tau}_{{\xi}_0} \rangle}}}}\end{aligned}$$ from which we build ${\ensuremath{\langle {{\ensuremath{\Bar{{\alpha}}}\/}}^{{\xi}_0}, u^{{\xi}_0} \rangle}}$. Set $$\begin{aligned}
& {{\ensuremath{\Bar{{\alpha}}}\/}}^{{\xi}_0,0}={{\ensuremath{\Bar{{\alpha}}}\/}}^{\xi},
\\
& u^{{\xi}_0,0}_0 = u^{\xi}_0.\end{aligned}$$ Assume we have constructed ${\ensuremath{{\ensuremath{\langle {{\ensuremath{\Bar{{\alpha}}}\/}}^{{\xi}_0,{\rho}}, u_0^{{\xi}_0,{\rho}},T_0^{{\xi}_0,{\rho}} \mid {\rho}< {\rho}_0 \rangle}}}}$.
We have $2$ cases:
${\rho}_0$ is limit: Set $$\begin{aligned}
& \forall {\rho}<{\rho}_0 \ {{\ensuremath{\Bar{{\alpha}}}\/}}^{{\xi}_0,{\rho}_0} > {{\ensuremath{\Bar{{\alpha}}}\/}}^{{\xi}_0, {\rho}},
\\
& u^{{\xi}_0,{\rho}_0}= {\bigcup}_{{\rho}< {\rho}_0} u^{{\xi}_0,{\rho}} {\cup}{\ensuremath{\left\{ {\ensuremath{\langle {{\ensuremath{\Bar{{\alpha}}}\/}}^{{\xi}_0,{\rho}_0},t \rangle}} \right\}}}
\text{ where }{\kappa}^0(t)={\tau}_{\xi}.\end{aligned}$$ We set $T^{{\xi}_0,{\rho}_0}_0$, $T^{{\xi}_0,{\rho}_0}_1$ to anything we like. We do not use them later.
${\rho}_0 = {\rho}+1$: Let ${\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}\rangle}} = s^{{\xi}_0,{\rho}}(2)$. set $$\begin{aligned}
& u'' = u''_1 {\mathop{{}^\frown}}u''_0 = (u^{{\xi}_0,{\rho}}_0)_{{\ensuremath{\langle s^{{\xi}_0,{\rho}} \rangle}}},
\\
&T''_0 = {\pi}^{-1}_{{{\ensuremath{\Bar{{\alpha}}}\/}}^{\xi},{{\ensuremath{\Bar{{\alpha}}}\/}}^0} (T^0_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}\rangle}}}),
\\
&T''_1 = {\pi}^{-1}_{\operatorname{mc}(u''_1),{{\ensuremath{\Bar{{\nu}}}\/}}} (T^0({{\ensuremath{\Bar{{\nu}}}\/}})).\end{aligned}$$ If there are $$\begin{aligned}
& q'_1 \leq^* u''_1 {\cup}{\ensuremath{\left\{ T''_1 \right\}}},
\\
& q'_0 \leq^* u''_0 {\cup}{\ensuremath{\left\{ T''_0 \right\}}},\end{aligned}$$ such that $$\begin{aligned}
q'_1 {\mathop{{}^\frown}}q'_0 \in D\end{aligned}$$ then set $$\begin{aligned}
{{\ensuremath{\Bar{{\alpha}}}\/}}^{{{\xi}_0},{\rho}_0} &= \operatorname{mc}(q'_0),
\\
u^{{{\xi}_0},{\rho}_0}_0 &= u^{{\xi}_0,{\rho}} {\cup}(q'_0 \setminus
( u''_0 {\cup}{\ensuremath{\left\{ T^{q'_0} \right\}}} )),
\\
T^{{{\xi}_0},{\rho}_0}_0 &= T^{q'_0},
\\
u^{{{\xi}_0},{\rho}_0}_1 &= q'_1 \setminus
( u'' _1 {\cup}{\ensuremath{\left\{ T^{q'_1} \right\}}} ),
\\
T^{{{\xi}_0},{\rho}_0}_1 &= T^{q'_1},\end{aligned}$$ otherwise set $$\begin{aligned}
{{\ensuremath{\Bar{{\alpha}}}\/}}^{{{\xi}_0},{\rho}_0} &= {{\ensuremath{\Bar{{\alpha}}}\/}}^{{\xi}_0, {\rho}},
\\
u^{{{\xi}_0},{\rho}_0}_0 &= u^{{\xi}_0,{\rho}},
\\
T^{{{\xi}_0},{\rho}_0}_0 &= T''_0,
\\
u^{{{\xi}_0},{\rho}_0}_1 &= \emptyset,
\\
T^{{{\xi}_0},{\rho}_0}_1 &= T''_1.\end{aligned}$$
When the induction on ${\rho}$ terminates we have ${\ensuremath{{\ensuremath{\langle {{\ensuremath{\Bar{{\alpha}}}\/}}^{{\xi}_0,{\rho}}, u^{{\xi}_0,{\rho}}_0, T^{{\xi}_0,{\rho}}_0,
u^{{\xi}_0,{\rho}}_1, T^{{\xi}_0,{\rho}}_1 \mid {\rho}< {\tau}_{{\xi}_0} \rangle}}}}$. We continue with the induction on ${\xi}$. We set $$\begin{aligned}
& \forall {\rho}< {\tau}_{{\xi}_0}\ {{\ensuremath{\Bar{{\alpha}}}\/}}^{{\xi}_0} >{{\ensuremath{\Bar{{\alpha}}}\/}}^{{\xi}_0,{\rho}},
\\
& u^{{\xi}_0}_0 = {\bigcup}_{{\rho}<{\tau}_{{\xi}_0}} u^{{\xi}_0,{\rho}}_0 {\cup}{\ensuremath{\left\{ {\ensuremath{\langle {{\ensuremath{\Bar{{\alpha}}}\/}}^{{\xi}_0},t \rangle}} \right\}}}
\text{ where }\max{\kappa}^0(t)={\tau}_{{\xi}}.\end{aligned}$$ When the induction on ${\xi}$ terminates we have ${\ensuremath{{\ensuremath{\langle {{\ensuremath{\Bar{{\alpha}}}\/}}^{{\xi}}, u^{{\xi}}_0 \mid {\xi}< {\kappa}\rangle}}}}$. Let $$\begin{aligned}
& \forall {\xi}< {\kappa}\ {{\ensuremath{\Bar{{\alpha}}}\/}}^{*\prime} >{{\ensuremath{\Bar{{\alpha}}}\/}}^{{\xi}},
\\
& p^{*\prime}_0 = {\bigcup}_{{\xi}<{\kappa}} u^{{\xi}}_0 {\cup}{\ensuremath{\left\{ {\ensuremath{\langle {{\ensuremath{\Bar{{\alpha}}}\/}}^{*\prime},t \rangle}} \right\}}}
\text{ where }\max{\kappa}^0(t)=\max p_0^0.\end{aligned}$$ We set $$\begin{aligned}
& \operatorname{Lev}_0(T^{p^{*\prime}_0}) = {\pi}^{-1}_{{{\ensuremath{\Bar{{\alpha}}}\/}}^{*\prime},{{\ensuremath{\Bar{{\alpha}}}\/}}^{0}} \operatorname{Lev}_0(T^0).\end{aligned}$$ Let us consider ${\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}\rangle}}\in \operatorname{Lev}_0(T^{p^{*\prime}_0})$. There is ${\xi}$ such that ${\kappa}^0({{\ensuremath{\Bar{{\nu}}}\/}}) = {\tau}_{\xi}$. We set $$\begin{aligned}
& s(1) = {\ensuremath{\left\{ {\ensuremath{\langle {{\ensuremath{\Bar{{\alpha}}}\/}},{\pi}_{{{\ensuremath{\Bar{{\alpha}}}\/}}^{*\prime},{{\ensuremath{\Bar{{\alpha}}}\/}}}({{\ensuremath{\Bar{{\nu}}}\/}}) \rangle}} \mid {{\ensuremath{\Bar{{\alpha}}}\/}}\in \operatorname{supp}p^*_0 \right\}}},
\\
& s(2)= {\ensuremath{\left\{ {\ensuremath{\langle {\pi}_{{{\ensuremath{\Bar{{\alpha}}}\/}}^{*\prime},{{\ensuremath{\Bar{{\alpha}}}\/}}^0}({{\ensuremath{\Bar{{\nu}}}\/}}) \rangle}} \right\}}}.\end{aligned}$$ Let ${\xi}_0={\xi}+1$. By our construction there is ${\rho}$ such that $$\begin{aligned}
& (u^{{\xi}_0,{\rho}_0}_0)_{{\ensuremath{\langle s \rangle}}} =
(u^{{\xi}_0,{\rho}_0}_0)_{{\ensuremath{\langle s^{{\xi}_0,{\rho}} \rangle}}}\end{aligned}$$ where ${\rho}_0 = {\rho}+1$. We set $$\begin{aligned}
& T^{p^{*\prime}_0}_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}\rangle}}} = {\pi}^{-1}_{{{\ensuremath{\Bar{{\alpha}}}\/}}^{*\prime},{{\ensuremath{\Bar{{\alpha}}}\/}}^{{\xi}_0,{\rho}_0}}(T^{{\xi}_0,{\rho}_0})
{\cap}{\pi}^{-1}_{{{\ensuremath{\Bar{{\alpha}}}\/}}^{*\prime},{{\ensuremath{\Bar{{\alpha}}}\/}}^{0}}
(T^0_{{\ensuremath{\langle {\pi}_{{{\ensuremath{\Bar{{\alpha}}}\/}}^*,{{\ensuremath{\Bar{{\alpha}}}\/}}^0}({{\ensuremath{\Bar{{\nu}}}\/}}) \rangle}}}),
\\
& T^{p^{* \prime}_0}({{\ensuremath{\Bar{{\nu}}}\/}}) =
\pi^{-1}_{{{\ensuremath{\Bar{{\nu}}}\/}}, \operatorname{mc}(u_1^{{\xi}_0,{\rho}_0})} T^{{\xi}_0,{\rho}_0}_1,
\\
& p'_1({{\ensuremath{\Bar{{\nu}}}\/}}) = u_1^{{\xi}_0,{\rho}_0}.\end{aligned}$$
Let us show that $p^{*\prime}_0$ approximates the $p^*$ we look for. So let ${\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}\rangle}}\in T^{p^{*\prime}_0}$ and assume
\[approx:in\] $$\begin{aligned}
& q'_1 \leq^* p'_1({{\ensuremath{\Bar{{\nu}}}\/}}) {\cup}((p^{*\prime}_0)_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}\rangle}}})_1,
\\
& q'_0 \leq^* ((p^{*\prime}_0)_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}\rangle}}})_0,
\\
&q'_1 {\mathop{{}^\frown}}q'_0 \in D.\end{aligned}$$
Let ${\xi}$ be such that ${\kappa}^0({{\ensuremath{\Bar{{\nu}}}\/}}) = {\tau}_{\xi}$. Set $s$ as $$\begin{aligned}
& s(1) = {\ensuremath{\left\{ {\ensuremath{\langle {{\ensuremath{\Bar{{\alpha}}}\/}},{\pi}_{{{\ensuremath{\Bar{{\alpha}}}\/}}^{*\prime},{{\ensuremath{\Bar{{\alpha}}}\/}}}({{\ensuremath{\Bar{{\nu}}}\/}}) \rangle}} \mid {{\ensuremath{\Bar{{\alpha}}}\/}}\in \operatorname{supp}p^*_0 \right\}}},
\\
& s(2)= {\ensuremath{\left\{ {\ensuremath{\langle {\pi}_{{{\ensuremath{\Bar{{\alpha}}}\/}}^{*\prime},{{\ensuremath{\Bar{{\alpha}}}\/}}^0}({{\ensuremath{\Bar{{\nu}}}\/}}) \rangle}} \right\}}},\end{aligned}$$ where ${\xi}_0={\xi}+1$, ${\rho}_0={\rho}+1$. By our construction there is ${\rho}$ such that $$\begin{aligned}
(u^{{\xi}_0,{\rho}_0})_{{\ensuremath{\langle s^{{\xi}_0,{\rho}} \rangle}}} = (u^{{\xi}_0,{\rho}_0})_{{\ensuremath{\langle s \rangle}}}.\end{aligned}$$ Let us set $$\begin{aligned}
& r = \big( (u^{{\xi}_0,{\rho}_0}_{{\ensuremath{\langle s^{{\xi}_0,{\rho}} \rangle}}})_1
{\cup}{\ensuremath{\left\{ T^{{\xi}_0,{\rho}_0}_1 \right\}}}
\big)
{\mathop{{}^\frown}}\big( (u^{{\xi}_0,{\rho}_0}_{{\ensuremath{\langle s^{{\xi}_0,{\rho}} \rangle}}})_0
{\cup}{\ensuremath{\left\{ T^{{\xi}_0,{\rho}_0}_0 \right\}}}
\big).\end{aligned}$$ By construction we have $$\begin{aligned}
\big( p'_1({{\ensuremath{\Bar{{\nu}}}\/}}) {\cup}((p^{*\prime}_0)_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}\rangle}}})_1 \big) {\mathop{{}^\frown}}((p^{*\prime}_0)_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}\rangle}}})_0
\leq^* r.\end{aligned}$$ So what we have is $$\begin{aligned}
D \ni q'_1 {\mathop{{}^\frown}}q'_0 \leq^* r.\end{aligned}$$ This is a positive answer to the question in the induction, hence $$\begin{aligned}
r \in D,\end{aligned}$$ which gives us, by openness of $D$, that $$\begin{aligned}
\label{approx:out}
\big( p'_1({{\ensuremath{\Bar{{\nu}}}\/}}) {\cup}((p^{*\prime}_0)_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}\rangle}}})_1 \big) {\mathop{{}^\frown}}((p^{*\prime}_0)_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}\rangle}}})_0 \in D.\end{aligned}$$ Having proved this approximation property of $p^{*\prime}_0$, let us consider the set $$\begin{aligned}
B = {\ensuremath{\left\{ {\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}\rangle}} \in T^{p^{*\prime}_0} \mid \exists q \leq^*
\big( p'_1({{\ensuremath{\Bar{{\nu}}}\/}}) {\cup}((p^{*\prime}_0)_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}\rangle}}})_1 \big) {\mathop{{}^\frown}}((p^{*\prime}_0)_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}\rangle}}})_0\
q \in D \right\}}}.\end{aligned}$$ Let ${{\ensuremath{\Bar{{\alpha}}}\/}}^{*\prime}=\operatorname{mc}(p_0^{*\prime})$. There are 2 cases here:
1. $\exists {\zeta}< \operatorname{l}({{\ensuremath{\bar{E}}\/}})\ B \in E_{{{\ensuremath{\Bar{{\alpha}}}\/}}^{*\prime}}({\zeta})$.
Let us set $$\begin{aligned}
& p^{\zeta}_1 = j(p'_1)({{\ensuremath{\bar{E}}\/}}_{{{\ensuremath{\Bar{{\alpha}}}\/}}^{*\prime}}{\mathord{\restriction}}{\zeta}),
\\
& {{\ensuremath{\Bar{{\beta}}}\/}}^{{\zeta}\prime} = \operatorname{mc}(p^{\zeta}_1),
\\
& A^{\zeta}= {\ensuremath{\left\{ {\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}\rangle}} \mid \big((p^{\zeta}_1)_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}\rangle}}}\big)_1 =
p'_1({\pi}_{{{\ensuremath{\Bar{{\beta}}}\/}}^{{\zeta}\prime},{{\ensuremath{\Bar{{\alpha}}}\/}}^{*\prime}}({{\ensuremath{\Bar{{\nu}}}\/}})) \right\}}}.\end{aligned}$$ Clearly $$\begin{aligned}
& A^{\zeta}\in E_{{{\ensuremath{\Bar{{\beta}}}\/}}^{{\zeta}\prime}}({\zeta}).\end{aligned}$$ Let ${{\ensuremath{\Bar{{\beta}}}\/}}^{\zeta}$ be the larger of ${{\ensuremath{\Bar{{\beta}}}\/}}^{{\zeta}\prime}$, ${{\ensuremath{\Bar{{\alpha}}}\/}}^{*\prime}$. Set $$\begin{aligned}
& T^{\zeta}= ({\pi}_{{{\ensuremath{\Bar{{\beta}}}\/}}^{\zeta},{{\ensuremath{\Bar{{\alpha}}}\/}}^{*\prime}})^{-1} (T^{p^{*\prime}_0}),
\\
& p^{\zeta}= p^{\zeta}_1 {\cup}p^{*\prime}_0 {\cup}{\ensuremath{\left\{ T^{\zeta}\right\}}}.\end{aligned}$$ The nice property of $p^{\zeta}$ is that when ${\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}\rangle}} \in T^{\zeta}{\mathord{\restriction}}({\pi}_{{{\ensuremath{\Bar{{\beta}}}\/}}^{\zeta},{{\ensuremath{\Bar{{\beta}}}\/}}^{{\zeta}\prime}})^{-1}(A^{\zeta})$ we get that $$\begin{aligned}
(p^{\zeta})_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}\rangle}}} \leq^*
\big( p'_1({\pi}_{{{\ensuremath{\Bar{{\beta}}}\/}}^{\zeta},{{\ensuremath{\Bar{{\beta}}}\/}}^{\prime}}({{\ensuremath{\Bar{{\nu}}}\/}})) {\cup}((p^{*\prime}_0)_{{\ensuremath{\langle {\pi}_{{{\ensuremath{\Bar{{\beta}}}\/}}^{\zeta},{{\ensuremath{\Bar{{\alpha}}}\/}}^{*\prime}}({{\ensuremath{\Bar{{\nu}}}\/}}) \rangle}}})_1
\big)
{\mathop{{}^\frown}}((p^{*\prime}_0)_{{\ensuremath{\langle {\pi}_{{{\ensuremath{\Bar{{\beta}}}\/}}^{\zeta},{{\ensuremath{\Bar{{\alpha}}}\/}}^{*\prime}}({{\ensuremath{\Bar{{\nu}}}\/}}) \rangle}}})_0.\end{aligned}$$ We set $$\begin{aligned}
& p^* = p^{\zeta},
\\
& A = \operatorname{Lev}_0(T^{\zeta}) {\cap}({\pi}_{{{\ensuremath{\Bar{{\beta}}}\/}}^{\zeta},{{\ensuremath{\Bar{{\beta}}}\/}}^{{\zeta}\prime}})^{-1} A^{\zeta},\end{aligned}$$ and show that the claim is satisfied: Assume that $$\begin{aligned}
& {\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}\rangle}} \in A,
\\
& q'_1 \leq^* ((p^*)_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}\rangle}}})_1,
\\
& q'_0 \leq^* ((p^*)_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}\rangle}}})_0,
\\
& q'_1 {\mathop{{}^\frown}}q'_0 \in D.\end{aligned}$$ Note that $$\begin{aligned}
& ((p^*)_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}\rangle}}})_1 \leq^* p'_1({\pi}_{{{\ensuremath{\Bar{{\beta}}}\/}}^{\zeta},{{\ensuremath{\Bar{{\alpha}}}\/}}^{*\prime}})
{\cup}((p^{*\prime}_0)_{{\ensuremath{\langle {\pi}_{{{\ensuremath{\Bar{{\beta}}}\/}}^{\zeta},{{\ensuremath{\Bar{{\alpha}}}\/}}^{*\prime}}({{\ensuremath{\Bar{{\nu}}}\/}}) \rangle}}})_1,
\\
& ((p^*)_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}\rangle}}})_0 \leq^*
((p^{*\prime}_0)_{{\ensuremath{\langle {\pi}_{{{\ensuremath{\Bar{{\beta}}}\/}}^{\zeta},{{\ensuremath{\Bar{{\alpha}}}\/}}^{*\prime}}({{\ensuremath{\Bar{{\nu}}}\/}}) \rangle}}})_0.\end{aligned}$$ Hence, we know that $$\begin{aligned}
& q'_1 \leq^* p'_1({\pi}_{{{\ensuremath{\Bar{{\beta}}}\/}}^{\zeta},{{\ensuremath{\Bar{{\alpha}}}\/}}^{*\prime}}({{\ensuremath{\Bar{{\nu}}}\/}}))
{\cup}((p^{*\prime}_0)_{{\ensuremath{\langle {\pi}_{{{\ensuremath{\Bar{{\beta}}}\/}}^{\zeta},{{\ensuremath{\Bar{{\alpha}}}\/}}^{*\prime}}({{\ensuremath{\Bar{{\nu}}}\/}}) \rangle}}})_1,
\\
& q'_0 \leq^*
((p^{*\prime}_0)_{{\ensuremath{\langle {\pi}_{{{\ensuremath{\Bar{{\beta}}}\/}}^{\zeta},{{\ensuremath{\Bar{{\alpha}}}\/}}^{*\prime}}({{\ensuremath{\Bar{{\nu}}}\/}}) \rangle}}})_0,
\\
& q'_1 {\mathop{{}^\frown}}q'_0 \in D.\end{aligned}$$ This is the assumption . So from we know that $$\begin{aligned}
\big( p'_1({\pi}_{{{\ensuremath{\Bar{{\beta}}}\/}}^{\zeta},{{\ensuremath{\Bar{{\alpha}}}\/}}^{*\prime}}({{\ensuremath{\Bar{{\nu}}}\/}})) {\cup}((p^{*\prime}_0)_{{\ensuremath{\langle {\pi}_{{{\ensuremath{\Bar{{\beta}}}\/}}^{\zeta},{{\ensuremath{\Bar{{\alpha}}}\/}}^{*\prime}}({{\ensuremath{\Bar{{\nu}}}\/}}) \rangle}}})_1
\big)
{\mathop{{}^\frown}}((p^{*\prime}_0)_{{\ensuremath{\langle {\pi}_{{{\ensuremath{\Bar{{\beta}}}\/}}^{\zeta},{{\ensuremath{\Bar{{\alpha}}}\/}}^{*\prime}}({{\ensuremath{\Bar{{\nu}}}\/}}) \rangle}}})_0
\in D\end{aligned}$$ hence by openness of $D$ $$\begin{aligned}
(p^*)_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}\rangle}}}
\in D,\end{aligned}$$
2. $\forall {\zeta}< \operatorname{l}({{\ensuremath{\bar{E}}\/}})\ B \notin E_{{{\ensuremath{\Bar{{\alpha}}}\/}}^{*\prime}}({\zeta})$ which is the same as saying that $$\begin{aligned}
{\ensuremath{\left\{ {\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}\rangle}} \in T^{p^{*\prime}_0} \mid \forall q \leq^* \big(p'_1({{\ensuremath{\Bar{{\nu}}}\/}}) {\cup}((p^{*\prime}_0)_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}\rangle}}})_1 \big)
{\mathop{{}^\frown}}((p^{*\prime}_0)_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}\rangle}}})_0\
q \notin D \right\}}} \in {{\ensuremath{\bar{E}}\/}}_{{{\ensuremath{\Bar{{\alpha}}}\/}}^{*\prime}}.\end{aligned}$$
In fact from the construction we can see that $$\begin{aligned}
{\ensuremath{\left\{ {\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}\rangle}} \in T^{p^{*\prime}_0} \mid p'_1({{\ensuremath{\Bar{{\nu}}}\/}}) = \emptyset \right\}}}
\in {{\ensuremath{\bar{E}}\/}}_{{{\ensuremath{\Bar{{\alpha}}}\/}}^{*\prime}}.\end{aligned}$$ So we really have $$\begin{aligned}
A = {\ensuremath{\left\{ {\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}\rangle}} \in T^{p^{*\prime}_0} \mid \forall q \leq^*
(p^{*\prime}_0)_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}\rangle}}}\
q \notin D \right\}}} \in {{\ensuremath{\bar{E}}\/}}_{{{\ensuremath{\Bar{{\alpha}}}\/}}^{*\prime}}\end{aligned}$$ and the completion is quite easy now, we set $$\begin{aligned}
& T^{p^*} = T^{p^{*\prime}} {\mathord{\restriction}}A,
\\
& p^* = p^{*\prime}_0 {\cup}{\ensuremath{\left\{ T^{p^*} \right\}}}.\end{aligned}$$
\[canon-dense\] Let $D$ be dense open in ${{\ensuremath{P_{{{\ensuremath{\bar{E}}\/}}}\/}}}/{{\ensuremath{P_{{{\ensuremath{\Bar{{\epsilon}}}\/}}}\/}}}$, $p = p_0 \in {{\ensuremath{P_{{{\ensuremath{\bar{E}}\/}}}\/}}}/{{\ensuremath{P_{{{\ensuremath{\Bar{{\epsilon}}}\/}}}\/}}}$. Then there is $p^* \leq^* p$ such that one and only one of the following is true:
1. There are $n<{\omega}$, $S \subseteq T^{p^*}{\mathord{\restriction}}[V_{\kappa}]^{n}$ such that
1. $\forall k<n\,\exists {\xi}<\operatorname{l}({{\ensuremath{\bar{E}}\/}})\
\operatorname{Suc}_S({\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_k \rangle}})
\in E_{\operatorname{mc}(p^*)}({\xi})$,
2. $\forall {\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_{n} \rangle}} \in S$ $(p^*)_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_{n} \rangle}}} \in D$,
2. $\forall n<{\omega}\ \forall {\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_{n} \rangle}} \in
T^{p^*}
\forall q \leq^* (p^*)_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_{n} \rangle}}}\
q \notin D$.
Let $p^0=p$.
Generate $p^{n+1}\leq^* p^n$ by invoking \[canon-dense:n\] for $n+1$ levels.
Take $\forall n<{\omega}\ p^* \leq p^n$.
\[DenseHomogen-0\] Let $D$ be dense open in ${{\ensuremath{P_{{{\ensuremath{\bar{E}}\/}}}\/}}}/{{\ensuremath{P_{{{\ensuremath{\Bar{{\epsilon}}}\/}}}\/}}}$, $p = p_0 \in {{\ensuremath{P_{{{\ensuremath{\bar{E}}\/}}}\/}}}/{{\ensuremath{P_{{{\ensuremath{\Bar{{\epsilon}}}\/}}}\/}}}$. Then there are $n<{\omega}$, $p^* \leq^* p$, $S \subseteq T^{p^*}{\mathord{\restriction}}[V_{\kappa}]^{n}$ such that
1. $\forall k<n\,\exists {\xi}<\operatorname{l}({{\ensuremath{\bar{E}}\/}})\
\operatorname{Suc}_S({\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_k \rangle}})
\in E_{\operatorname{mc}(p^*)}({\xi})$,
2. $\forall {\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_{n} \rangle}} \in S$ $(p^*)_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_{n} \rangle}}} \in D$.
Towards a contradiction, let us assume that the conclusion is false. That means that for all $p^* \leq^* p$, for all $n < {\omega}$, for all $S \subseteq T^{p^*} {\mathord{\restriction}}[V_{\kappa}]^n$ such that $$\begin{aligned}
\forall k<n \,\forall {\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_k \rangle}} \in S \,
\exists {\xi}< \operatorname{l}({{\ensuremath{\bar{E}}\/}})\
\operatorname{Suc}_S({\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_k \rangle}}) \in E_{\operatorname{mc}(p^*)}({\xi})\end{aligned}$$ we have $$\begin{aligned}
\exists {\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_n \rangle}} \in S \
(p^*)_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_n \rangle}}} \notin D.\end{aligned}$$ We construct a $\leq^*$-decreasing sequence as follows: We set $p^0 = p$. We construct $p^{n+1}$ from $p^n$ using \[canon-dense:n\] for $n+1$ levels. Due to our assumption we get $$\begin{aligned}
\forall {\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_n \rangle}} \in T^{p^n} \,
\forall q\leq^* (p^n)_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_n \rangle}}} \
q \notin D.\end{aligned}$$ We choose $p^{*\prime}$ such that $\forall n<{\omega}\ p^{*\prime}\leq^* p^n$ we get $$\begin{aligned}
\forall n<{\omega}\, \forall {\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_n \rangle}} \in T^{p^{*\prime}_0} \,
\forall q\leq^* (p^{*\prime})_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_n \rangle}}} \
q \notin D.\end{aligned}$$ Construct tree $T$ from $T^{p^{*\prime}}$ using \[skeleton\]. Let us call $p^{*}$ the condition $p^{*\prime}$ with $T$ substituted for $T^{p^{*\prime}}$. Now if we have $$\begin{aligned}
q \leq p^*\end{aligned}$$ then there is ${\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_n \rangle}} \in T^{p^{*\prime}}$ such that $$\begin{aligned}
q \leq^* (p^{*\prime})_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_n \rangle}}}.\end{aligned}$$ Hence $$\begin{aligned}
q \notin D.\end{aligned}$$ However, $D$ is dense. Contradiction.
\[DenseHomogen-1\] Let $D$ be dense open in ${{\ensuremath{P_{{{\ensuremath{\bar{E}}\/}}}\/}}}$, $p = p_1 {\mathop{{}^\frown}}p_0 \in {{\ensuremath{P_{{{\ensuremath{\bar{E}}\/}}}\/}}}$. Then there is $p^* \leq^* p$ such that $$\begin{gathered}
\exists S^1\ \exists n_1 \forall {\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_{1,1},\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_{1,{n_1}} \rangle}}
\in S^1 \, \dotsc
\exists S^0\ \exists n_0 \forall {\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_{0,1},\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_{0,{n_0}} \rangle}}
\in S^0 \,
\\
(p^*_1)_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_{1,1},\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_{1,n_1} \rangle}}} {\mathop{{}^\frown}}\dotsb {\mathop{{}^\frown}}(p^*_0)_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_{0,1},\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_{0,n_0} \rangle}}}
\in D\end{gathered}$$ where
1. $S^i \subseteq T^{p^*_i} {\mathord{\restriction}}[V_{\kappa}]^{n_i}$,
2. $\forall l<n_i \,\forall {\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_{l} \rangle}} \in S^i \,
\exists {\xi}\
\operatorname{Suc}_{S^i}({\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_{l} \rangle}})
\in E_{\operatorname{mc}(p^*_i)}({\xi})$.
Let ${{\ensuremath{\Bar{{\epsilon}}}\/}}$ be such that $p_1 \in {{\ensuremath{P_{{{\ensuremath{\Bar{{\epsilon}}}\/}}}\/}}}$. We prove that there are $n<{\omega}$, $p^*_0 \leq^* p_0$, $q_1 \leq p_1$, $S \subseteq T^{p^*}{\mathord{\restriction}}[V_{\kappa}]^{n}$ such that
1. $\forall k<n\,\exists {\xi}<\operatorname{l}({{\ensuremath{\bar{E}}\/}})\
\operatorname{Suc}_S({\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_k \rangle}})
\in E_{\operatorname{mc}(p^*)}({\xi})$,
2. $\forall {\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_{n} \rangle}} \in S$ $q_1{\mathop{{}^\frown}}(p^*)_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_{n} \rangle}}} \in D$.
Set $$\begin{aligned}
E = {\ensuremath{\left\{ r\in {{\ensuremath{P_{{{\ensuremath{\bar{E}}\/}}}\/}}}\mid \exists q_1\ q_1 {\mathop{{}^\frown}}r \in D, q_1 \leq p_1 \right\}}}.\end{aligned}$$ This $E$ is dense open in ${{\ensuremath{P_{{{\ensuremath{\bar{E}}\/}}}\/}}}/{{\ensuremath{P_{{{\ensuremath{\Bar{{\epsilon}}}\/}}}\/}}}$: Let $r \in {{\ensuremath{P_{{{\ensuremath{\bar{E}}\/}}}\/}}}/{{\ensuremath{P_{{{\ensuremath{\Bar{{\epsilon}}}\/}}}\/}}}$. Then $p_1 {\mathop{{}^\frown}}r \in {{\ensuremath{P_{{{\ensuremath{\bar{E}}\/}}}\/}}}$. By density of $D$, there is $q_1 {\mathop{{}^\frown}}s \in D$ such that $q_1 \leq p_1$, $s \leq r$. By the definition of $E$, $s \in E$. Hence $E$ is dense. Openness of $E$ is immediate from openness of $D$.
By \[DenseHomogen-0\] there are $p^*_0 \leq^* p_0$, $S'$, $n<{\omega}$ such that
1. $\forall k<n\,\exists {\xi}<\operatorname{l}({{\ensuremath{\bar{E}}\/}})\
\operatorname{Suc}_{S'}({\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_k \rangle}})
\in E_{\operatorname{mc}(p^*)}({\xi})$,
2. $\forall {\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_{n} \rangle}} \in S'$ $(p^*_0)_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_{n} \rangle}}} \in E$.
This means that $\forall {\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_{n} \rangle}} \in S'$ there is $q_1({{\ensuremath{\Bar{{\nu}}}\/}}_1,\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_n) \leq p_1$ such that $$\begin{aligned}
\forall {\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_n \rangle}} \in S'\
q_1({{\ensuremath{\Bar{{\nu}}}\/}}_1,\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_n) {\mathop{{}^\frown}}(p^*_0)_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_{n} \rangle}}} \in D.\end{aligned}$$ As ${\lvert{{\ensuremath{P_{{{\ensuremath{\Bar{{\epsilon}}}\/}}}\/}}}\rvert} < {\kappa}$, $q_1({{\ensuremath{\Bar{{\nu}}}\/}}_1,\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_n)$ is in fact almost always constant. Hence, by shrinking $S'$ to $S$ and letting $q_1$ be this constant value, we get $$\begin{aligned}
\forall {\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_n \rangle}} \in S\
q_1 {\mathop{{}^\frown}}(p^*_0)_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_{n} \rangle}}} \in D.\end{aligned}$$ With this, we finished the first part of the proof. We use this claim for all conditions in ${{\ensuremath{P_{{{\ensuremath{\Bar{{\epsilon}}}\/}}}\/}}}$.
Let ${{\ensuremath{P_{{{\ensuremath{\Bar{{\epsilon}}}\/}}}\/}}}= {\ensuremath{\left\{ p_1^{\zeta}\mid {\zeta}<{\lambda}\right\}}}$ where ${\lambda}< {\kappa}$.
We construct by induction a $\leq^*$-decreasing sequence ${\ensuremath{{\ensuremath{\langle p_0^{\zeta}\mid {\zeta}< {\lambda}\rangle}}}}$. Set $$\begin{aligned}
p^0_0 = p_0.\end{aligned}$$ Assume we have constructed ${\ensuremath{{\ensuremath{\langle p^{\zeta}_0 \mid {\zeta}<{\zeta}_0 \rangle}}}}$.
${\zeta}_0$ is limit: Choose $p^{{\zeta}_0}_0 \leq^* p^{\zeta}_0$ for all ${\zeta}<{\zeta}_0$.
${\zeta}_0={\zeta}+1$: Use the first part of the proof on $p^{\zeta}_1 {\mathop{{}^\frown}}p^{\zeta}_0$ to construct $p^{{\zeta}_0}_0$.
When the induction terminates we have ${\ensuremath{{\ensuremath{\langle p_0^{\zeta}\mid {\zeta}< {\lambda}\rangle}}}}$. Choose $$\begin{aligned}
\forall {\zeta}<{\lambda}\ p^*_0 \leq^* p^{\zeta}_0.\end{aligned}$$ Let $$\begin{aligned}
D_{{{\ensuremath{\Bar{{\epsilon}}}\/}}} = {\ensuremath{\left\{ q_1 \in {{\ensuremath{P_{{{\ensuremath{\Bar{{\epsilon}}}\/}}}\/}}}\mid \exists n\ \exists S\
\forall {\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_n \rangle}} \in S\
q_1 {\mathop{{}^\frown}}(p^*_0)_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_n \rangle}}} \in D \right\}}}.\end{aligned}$$ $D_{{{\ensuremath{\Bar{{\epsilon}}}\/}}}$ is dense open: Let $q_1 \in {{\ensuremath{P_{{{\ensuremath{\Bar{{\epsilon}}}\/}}}\/}}}$. Then there is ${\zeta}$ such that $q_1=p_1^{\zeta}$. By the induction we have that there are $n$, $S$, $r_1 \leq q_1$ such that $$\begin{aligned}
\forall {\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_n \rangle}} \in S\
r_1 {\mathop{{}^\frown}}(p^{{\zeta}_0}_0)_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_n \rangle}}} \in D.\end{aligned}$$ By openness of $D$ we get $$\begin{aligned}
\forall {\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_n \rangle}} \in S\
r_1 {\mathop{{}^\frown}}(p^*_0)_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_n \rangle}}} \in D.\end{aligned}$$ Hence $$\begin{aligned}
r_1 \in D_{{{\ensuremath{\Bar{{\epsilon}}}\/}}}.\end{aligned}$$ As $D_{{{\ensuremath{\Bar{{\epsilon}}}\/}}}$ is dense open we can use \[DenseHomogen-0\]. Hence there are $p^*_1 \leq p_1$, $S^1$, $n_1$ such that $$\begin{aligned}
\forall {\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_{n_1} \rangle}} \in S^1\
(p^*_1)_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_{n_1} \rangle}}} \in D_{{{\ensuremath{\Bar{{\epsilon}}}\/}}}.\end{aligned}$$ This means that $$\begin{gathered}
\forall {\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_{1,1},\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_{1,n_1} \rangle}} \in S^1\
\exists S\ \exists n\
\forall {\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_{0,1},\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_{0,n} \rangle}} \in S\
\\
(p^*_1)_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_{1,1},\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_{1,n_1} \rangle}}} {\mathop{{}^\frown}}(p^*_0)_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_{0,1},\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_{0,n} \rangle}}} \in D,\end{gathered}$$ which is what we need to prove.
Finally, we add the last touch.
The proof is done by induction on $k$. The case $k=1$ is \[DenseHomogen-1\]. We assume, then, that the theorem is proved for $k$ and prove it for $k+1$.
Then $p = p_{k+1}{\mathop{{}^\frown}}p_k {\mathop{{}^\frown}}\dotsb p_0$. Let ${{\ensuremath{\Bar{{\epsilon}}}\/}}$ be such that $p_{k+1} \in {{\ensuremath{P_{{{\ensuremath{\Bar{{\epsilon}}}\/}}}\/}}}$. We just repeat the proof of \[DenseHomogen-1\] with ${{\ensuremath{P_{{{\ensuremath{\Bar{{\epsilon}}}\/}}}\/}}}$ and use the induction hypothese to conclude the proof.
Prikry’s condition {#Prikry'sCondition}
==================
\[PrikryCondition\] Let $p \in {{\ensuremath{P_{{{\ensuremath{\bar{E}}\/}}}\/}}}$ and ${\sigma}$ a formula in the forcing language. Then there is $p^* \leq p$ such that $p^* {\mathrel\Vert}{\sigma}$.
The set ${\ensuremath{\left\{ q \in {{\ensuremath{P_{{{\ensuremath{\bar{E}}\/}}}\/}}}\mid q {\mathrel\Vert}{\sigma}\right\}}}$ is dense open. Assuming $p = p_k {\mathop{{}^\frown}}\dotsb {\mathop{{}^\frown}}p_0$ and using \[DenseHomogen\] we get that there is $q \leq^* p$ such that $$\begin{gathered}
\exists S^k\ \exists n_k \forall {\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_{k,1},\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_{k,{n_k}} \rangle}}
\in S^k \, \dotsc
\exists S^0\ \exists n_0 \forall {\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_{0,1},\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_{0,{n_0}} \rangle}}
\in S^0 \,
\\
(p^*_k)_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_{k,1},\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_{k,n_k} \rangle}}} {\mathop{{}^\frown}}\dotsb {\mathop{{}^\frown}}(p^*_0)_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_{0,1},\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_{0,n_0} \rangle}}}
{\mathrel\Vert}{\sigma}.\end{gathered}$$ Recall that we really should write $$\begin{aligned}
& S^{k-1}({{\ensuremath{\Bar{{\nu}}}\/}}_{k,1},\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_{k,n_k}),
\\
& S^{k-2}({{\ensuremath{\Bar{{\nu}}}\/}}_{k,1},\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_{k,n_k},{{\ensuremath{\Bar{{\nu}}}\/}}_{k-1,1},\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_{k-1,n_k}),
\\
& \vdots\end{aligned}$$ In order to avoid (too much) clutter, we use the following convention in the proof. When we write $$\begin{aligned}
{{\ensuremath{\Vec{{\nu}}}\/}}\in \prod_{1\leq l\leq k} S^l\end{aligned}$$ we mean that $$\begin{aligned}
&{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_{k,1},\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_{k,n_k} \rangle}} \in S^k,
\\
&\vdots
\\
&{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_{1,1},\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_{1,n_1} \rangle}} \in S^1,\end{aligned}$$ and $r({{\ensuremath{\Vec{{\nu}}}\/}})$ is $$\begin{aligned}
(q_k)_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_{k,1},\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_{k,n_k} \rangle}}}{\mathop{{}^\frown}}\dotsb{\mathop{{}^\frown}}(q_1)_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_{1,1},\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_{1,n_1} \rangle}}}.\end{aligned}$$ We start by naming $q_0$ as $q^{n_0}_0$ and $T^{q_0}$ as $T^{0,n_0}$. For ${\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_{n_0-1} \rangle}} \in S^0$ set $$\begin{aligned}
& A^0=
{\ensuremath{\left\{ {\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_{n_0} \rangle}} \in
\operatorname{Suc}_{T^{0,n_0}}({\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_{1},\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_{n-1} \rangle}}) \mid r({{\ensuremath{\Vec{{\nu}}}\/}}){\mathop{{}^\frown}}(q^{n_0}_0)_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_{1},\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_{n_0} \rangle}}} {\mathrel\Vdash}{\sigma}\right\}}},
\\
& A^1=
{\ensuremath{\left\{ {\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_{n_0} \rangle}} \in
\operatorname{Suc}_{T^{0,n_0}}({\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_{1},\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_{n_0-1} \rangle}}) \mid r({{\ensuremath{\Vec{{\nu}}}\/}}){\mathop{{}^\frown}}(q^{n_0}_0)_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_{1},\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_{n_0} \rangle}}} {\mathrel\Vdash}\lnot{\sigma}\right\}}}.\end{aligned}$$ Note that $$\begin{aligned}
& \operatorname{Suc}_{S^0}({\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_{1},\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_{n_0-1} \rangle}})
\subseteq
A^0 {\cup}A^1,
\\
& A^0{\cap}A^1 = \emptyset\end{aligned}$$ Hence, there is ${\xi}< \operatorname{l}({{\ensuremath{\bar{E}}\/}})$ such that one and only one of the following is true:
1. $A^0
\in E_{\operatorname{mc}(q_0^{n_0})}({\xi})$,
2. $A^1
\in E_{\operatorname{mc}(q_0^{n_0})}({\xi})$.
In either case, using \[fill-missing\], we can shrink $T^{0,n_0}_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_{n_0-1} \rangle}}}$ and get a condition $q'_0$ such that $r({{\ensuremath{\Vec{{\nu}}}\/}}){\mathop{{}^\frown}}q'_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_{n_0-1} \rangle}}} {\mathrel\Vert}{\sigma}$.
So we shrink now $T^{0,n_0}_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_{n_0-1} \rangle}}}$ for all ${\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_{n_0-1} \rangle}} \in S^0$ and we call this tree $T^{0,n_0-1}$. The name of the condition $q^{n_0}_0$ with $T^{0,n_0-1}$ substituted for $T^{0,n_0}$ is $q^{n_0-1}_0$.
$q^{n_0-1}_0$ satisfies $$\begin{aligned}
\forall {\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_{1},\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_{n_0-1} \rangle}} \in S^0 \
r({{\ensuremath{\Vec{{\nu}}}\/}}){\mathop{{}^\frown}}(q^{n_0-1}_0)_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_{n_0-1} \rangle}}}
{\mathrel\Vert}{\sigma}.\end{aligned}$$ We are now in the same position as we were when setting $q^{n_0}_0$. So by repeating the above arguments we get $$\begin{aligned}
p_0 \ge^* q_0=q^{n_0}_0 \ge^* q^{n_0-1}_0 \ge^* \dotsb \ge^* q^1_0 \ge^* q^0_0\end{aligned}$$ such that for each $l = n_0,n_0-1,\dotsc,1,0$ $$\begin{aligned}
\forall {\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_l \rangle}} \in S^0 \
r({{\ensuremath{\Vec{{\nu}}}\/}}){\mathop{{}^\frown}}(q^l_0)_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_l \rangle}}} {\mathrel\Vert}{\sigma}.\end{aligned}$$ Specifically we get $$\begin{aligned}
r({{\ensuremath{\Vec{{\nu}}}\/}}){\mathop{{}^\frown}}(q^0_0) {\mathrel\Vert}{\sigma}.\end{aligned}$$ Of course $q^0_0$ depends on ${{\ensuremath{\Vec{{\nu}}}\/}}$. Note that we got from $q^{n_0}_0$ to $q^0_0$ only by shrinking the trees. So, we repeat this process for all ${{\ensuremath{\Vec{{\nu}}}\/}}$ calling the resulting condition $q^0_0({{\ensuremath{\Vec{{\nu}}}\/}})$. So we have $$\begin{aligned}
\forall {{\ensuremath{\Vec{{\nu}}}\/}}\in\prod_{1\leq l\leq k}S^l\
r({{\ensuremath{\Vec{{\nu}}}\/}}){\mathop{{}^\frown}}q^0_0({{\ensuremath{\Vec{{\nu}}}\/}}) {\mathrel\Vert}{\sigma}.\end{aligned}$$ By setting $$\begin{aligned}
T^{p^*_0}={\bigcap}_{{{\ensuremath{\Vec{{\nu}}}\/}}} T^{q^0_0({{\ensuremath{\Vec{{\nu}}}\/}})}\end{aligned}$$ and letting $p^*_0$ be $q_0$ with $T^{p^*_0}$ substituted for $T^{q_0}$ we get $$\begin{aligned}
\forall {{\ensuremath{\Vec{{\nu}}}\/}}\in\prod_{1\leq l\leq k}S^l\
r({{\ensuremath{\Vec{{\nu}}}\/}}) {\mathop{{}^\frown}}p^*_0 {\mathrel\Vert}{\sigma}.\end{aligned}$$
We are in the same position as in the beginning of the proof. So we can generate in the same way $p^*_1$ from $p_1$ and so on until we have $$\begin{aligned}
p^*_k{\mathop{{}^\frown}}\dotsb{\mathop{{}^\frown}}p^*_0 {\mathrel\Vert}{\sigma}.\end{aligned}$$
Properness {#Properness}
==========
The notions ${\ensuremath{\langle N,P \rangle}}$-generic and properness are due to Shelah [@Shelah].
Let $N {\prec}H_{\chi}$ such that
1. ${\lvertN\rvert}={\kappa}$,
2. $N \supseteq V_{\kappa}$,
3. $N \supseteq N^{{\mathord{<}}{\kappa}}$,
4. $P \in N$.
Then $p \in P$ is called ${\ensuremath{\langle N,P \rangle}}$-generic if $$\begin{aligned}
p {\mathrel\Vdash}{{}^{\ulcorner} \forall D\in {\widehat{N}}\ D \text{ is dense open in }{\widehat{P}}\
\implies D {\cap}{\widetilde{G}} {\cap}{\widehat{N}} \not= \emptyset {}^{\urcorner}}.\end{aligned}$$
A forcing notion $P$ is called proper if for all $N {\prec}H_{\chi}$ such that
1. ${\lvertN\rvert}={\kappa}$,
2. $N \supseteq V_{\kappa}$,
3. $N \supseteq N^{{\mathord{<}}{\kappa}}$,
4. $P \in N$,
and for all $q \in P {\cap}N$ there is $p \leq q$ which is ${\ensuremath{\langle N, P \rangle}}$-generic.
\[NPgeneric-1\] Let $p \in {{\ensuremath{P_{{{\ensuremath{\bar{E}}\/}}}\/}}}$, $N {\prec}H_{\chi}$ such that
1. ${\lvertN\rvert}={\kappa}$,
2. $N \supseteq V_{\kappa}$,
3. $N \supseteq N^{{\mathord{<}}{\kappa}}$,
4. ${{\ensuremath{P_{{{\ensuremath{\bar{E}}\/}}}\/}}}\in N$,
5. $p \in {{\ensuremath{P_{{{\ensuremath{\bar{E}}\/}}}\/}}}{\cap}N$.
Then there is $p^* \leq^* p$ such that $p^*$ is ${\ensuremath{\langle N, {{\ensuremath{P_{{{\ensuremath{\bar{E}}\/}}}\/}}}\rangle}}$-generic
Let $p = p_{k(p)}{\mathop{{}^\frown}}\dotsb {\mathop{{}^\frown}}p_1 {\mathop{{}^\frown}}p_0$.
Let ${\ensuremath{{\ensuremath{\langle D_{\xi}\mid {\xi}< {\kappa}\rangle}}}}$ be an enumeration of all dense open subsets of ${{\ensuremath{P_{{{\ensuremath{\bar{E}}\/}}}\/}}}$ which are in $N$. Note that for ${\xi}_0 < {\kappa}$ we have that ${\ensuremath{{\ensuremath{\langle D_{\xi}\mid {\xi}< {\xi}_0 \rangle}}}} \in N$.
We start now an induction on ${\xi}$ in which we build $$\begin{aligned}
{\ensuremath{{\ensuremath{\langle {{\ensuremath{\Bar{{\alpha}}}\/}}^{\xi}, u^{\xi}\mid {\xi}< {\kappa}\rangle}}}}.\end{aligned}$$ The construction is done ensuring that ${\ensuremath{{\ensuremath{\langle {{\ensuremath{\Bar{{\alpha}}}\/}}^{\xi}, u^{\xi}\mid {\xi}< {\xi}_0 \rangle}}}} \in N$ for all ${\xi}_0 < {\kappa}$. We start by setting $$\begin{aligned}
u^0 &= p_0 \setminus {\ensuremath{\left\{ T^{p_0} \right\}}},
\\
{{\ensuremath{\Bar{{\alpha}}}\/}}^0 &= \operatorname{mc}(p_0),
\\
T^0 &= T^{p_0} {\mathord{\restriction}}\pi^{-1}_{{{\ensuremath{\Bar{{\alpha}}}\/}}^0,0}{\ensuremath{\left\{ {{\ensuremath{\Bar{{\nu}}}\/}}\mid {\kappa}^0({{\ensuremath{\Bar{{\nu}}}\/}}) \text{ is inaccessible} \right\}}},\end{aligned}$$ and taking an increasing enumeration in $N$ $$\begin{aligned}
{\ensuremath{\left\{ {\kappa}^0({{\ensuremath{\Bar{{\nu}}}\/}}) \mid {\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}\rangle}} \in T^0 \right\}}}
= {\ensuremath{{\ensuremath{\langle {\tau}_{\xi}\mid {\xi}< {\kappa}\rangle}}}}.\end{aligned}$$ Assume then that we have $$\begin{aligned}
{\ensuremath{{\ensuremath{\langle {{\ensuremath{\Bar{{\alpha}}}\/}}^{\xi}, u^{\xi}\mid {\xi}< {\xi}_0 \rangle}}}}.\end{aligned}$$ The constructions splits now according to wether ${\xi}_0$ is limit or successor. In both cases the work is done inside $N$.
${\xi}_0$ is limit: Choose ${{\ensuremath{\Bar{{\alpha}}}\/}}^{{\xi}_0} > {{\ensuremath{\Bar{{\alpha}}}\/}}^{\xi}$ for all ${\xi}< {\xi}_0$ and set $$\begin{aligned}
u^{{\xi}_0} &= {\bigcup}_{{\xi}< {\xi}_0} u^{\xi}{\cup}{\ensuremath{\left\{ {\ensuremath{\langle {{\ensuremath{\Bar{{\alpha}}}\/}}^{{\xi}_0}, t \rangle}} \right\}}}
\text{ where } {\kappa}^0(t) = {\tau}_{{\xi}_0}.\end{aligned}$$
${\xi}_0 = {\xi}+ 1:$ For each ${{\ensuremath{\Bar{{\nu}}}\/}}_1,\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_n$ such that ${\kappa}^0({{\ensuremath{\Bar{{\nu}}}\/}}_1)<\dotsb<{\kappa}^0({{\ensuremath{\Bar{{\nu}}}\/}}_n)={\tau}_{\xi}$ we set $$\begin{aligned}
S({{\ensuremath{\Bar{{\nu}}}\/}}_1,\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_n) = &
\big( \prod_{{{\ensuremath{\Bar{{\alpha}}}\/}}\in \operatorname{supp}u^{\xi}}
{\ensuremath{\left\{ {{\ensuremath{\Bar{{\mu}}}\/}}_1 \mid {\kappa}^0({{\ensuremath{\Bar{{\mu}}}\/}}_1)={\kappa}^0({{\ensuremath{\Bar{{\nu}}}\/}}_1) \right\}}} \big)
\times
\\
& \big( \prod_{{{\ensuremath{\Bar{{\alpha}}}\/}}\in \operatorname{supp}u^{\xi}}
{\ensuremath{\left\{ {{\ensuremath{\Bar{{\mu}}}\/}}_2 \mid {\kappa}^0({{\ensuremath{\Bar{{\mu}}}\/}}_2)={\kappa}^0({{\ensuremath{\Bar{{\nu}}}\/}}_2) \right\}}} \big)
\times
\\
& \vdots
\\
& \big( \prod_{{{\ensuremath{\Bar{{\alpha}}}\/}}\in \operatorname{supp}u^{\xi}}
{\ensuremath{\left\{ {{\ensuremath{\Bar{{\mu}}}\/}}_n \mid {\kappa}^0({{\ensuremath{\Bar{{\mu}}}\/}}_n)={\kappa}^0({{\ensuremath{\Bar{{\nu}}}\/}}_n) \right\}}} \big)
\times
\\
& {\ensuremath{\left\{ {\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_n \rangle}} \right\}}}.\end{aligned}$$ Let $$\begin{aligned}
S = {\bigcup}_{{\kappa}^0({{\ensuremath{\Bar{{\nu}}}\/}}_1)<\dotsb<{\kappa}^0({{\ensuremath{\Bar{{\nu}}}\/}}_n)={\tau}_{\xi}}
S({{\ensuremath{\Bar{{\nu}}}\/}}_1,\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_n) \end{aligned}$$ and set enumeration of $S$ $$\begin{aligned}
S={\ensuremath{{\ensuremath{\langle s^{{\xi}_0,{\rho}} \mid {\rho}< {\tau}_{{\xi}_0} \rangle}}}}.\end{aligned}$$ We do induction on ${\rho}$ which builds $$\begin{aligned}
{\ensuremath{{\ensuremath{\langle {{\ensuremath{\Bar{{\alpha}}}\/}}^{{\xi}_0,{\rho}}, u_0^{{\xi}_0,{\rho}},T_0^{{\xi}_0,{\rho}} \mid {\rho}< {\tau}_{{\xi}_0} \rangle}}}},\end{aligned}$$ from which we build ${\ensuremath{\langle {{\ensuremath{\Bar{{\alpha}}}\/}}^{{\xi}_0}, u^{{\xi}_0} \rangle}}$. Set $$\begin{aligned}
& {{\ensuremath{\Bar{{\alpha}}}\/}}^{{\xi}_0,0}={{\ensuremath{\Bar{{\alpha}}}\/}}^{\xi},
\\
& u^{{\xi}_0,0}_0 = u^{\xi}_0.\end{aligned}$$ Assume we have constructed ${\ensuremath{{\ensuremath{\langle {{\ensuremath{\Bar{{\alpha}}}\/}}^{{\xi}_0,{\rho}}, u_0^{{\xi}_0,{\rho}},T_0^{{\xi}_0,{\rho}} \mid {\rho}< {\rho}_0 \rangle}}}}$.
${\rho}_0$ is limit: Set $$\begin{aligned}
& \forall {\rho}<{\rho}_0 \ {{\ensuremath{\Bar{{\alpha}}}\/}}^{{\xi}_0,{\rho}_0} > {{\ensuremath{\Bar{{\alpha}}}\/}}^{{\xi}_0, {\rho}},
\\
& u^{{\xi}_0,{\rho}_0}= {\bigcup}_{{\rho}< {\rho}_0} u^{{\xi}_0,{\rho}} {\cup}{\ensuremath{\left\{ {\ensuremath{\langle {{\ensuremath{\Bar{{\alpha}}}\/}}^{{\xi}_0,{\rho}_0},t \rangle}} \right\}}}
\text{ where }{\kappa}^0(t)={\tau}_{{\xi}_0}.\end{aligned}$$ We set $T^{{\xi}_0,{\rho}_0}$ to anything we like as we do not use it later.
${\rho}_0 = {\rho}+1$: Let $s^{{\xi}_0,{\rho}}(n(s)+1)={\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_n \rangle}}$. $$\begin{aligned}
& u'' = (u^{{\xi}_0,{\rho}}_0)_{{\ensuremath{\langle s^{{\xi}_0,{\rho}} \rangle}}},
\\
&T''_0 = {\pi}^{-1}_{{{\ensuremath{\Bar{{\alpha}}}\/}}^{\xi},{{\ensuremath{\Bar{{\alpha}}}\/}}^0} (T^0_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_n \rangle}}}),
\\
&T''_1 = {\pi}^{-1}_{\operatorname{mc}(u''_1),{{\ensuremath{\Bar{{\nu}}}\/}}_n}
(T^0_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_{n-1} \rangle}}}({{\ensuremath{\Bar{{\nu}}}\/}}_n)),
\\
& \vdots
\\
&T''_{n-1} = {\pi}^{-1}_{\operatorname{mc}(u''_{n-1}),{{\ensuremath{\Bar{{\nu}}}\/}}_2} (T^0_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1 \rangle}}}({{\ensuremath{\Bar{{\nu}}}\/}}_2)),
\\
&T''_n = {\pi}^{-1}_{\operatorname{mc}(u''_n),{{\ensuremath{\Bar{{\nu}}}\/}}_1} (T^0({{\ensuremath{\Bar{{\nu}}}\/}}_1)).\end{aligned}$$ Take enumeration $$\begin{gathered}
{\ensuremath{\left\{ D_{\sigma}\mid {\sigma}< {\tau}_{{\xi}} \right\}}} \times
\\
{\ensuremath{\left\{ q \mid q \leq p_{k(p)}{\mathop{{}^\frown}}\dotsb{\mathop{{}^\frown}}p_1{\mathop{{}^\frown}}u''_{n(s)}{\cup}{\ensuremath{\left\{ T''_{n} \right\}}}{\mathop{{}^\frown}}\dotsb
{\mathop{{}^\frown}}u''_1 {\cup}{\ensuremath{\left\{ T''_1 \right\}}}
\right\}}} =
\\
{\ensuremath{{\ensuremath{\langle {\ensuremath{\langle E^{{\xi}_0,{\rho}_0,{\zeta}}, q^{{\xi}_0,{\rho}_0,{\zeta}} \rangle}} \mid {\zeta}<{\tau}_{{\xi}_0} \rangle}}}}.\end{gathered}$$
We start induction on ${\zeta}$. Set $$\begin{aligned}
& {{\ensuremath{\Bar{{\alpha}}}\/}}^{{\xi}_0,{\rho}_0,0} = {{\ensuremath{\Bar{{\alpha}}}\/}}^{{\xi}_0,{\rho}},
\\
& u^{{\xi}_0,{\rho}_0,0}_0 = u^{{\xi}_0,{\rho}}.\end{aligned}$$ Assume we have constructed ${\ensuremath{{\ensuremath{\langle {{\ensuremath{\Bar{{\alpha}}}\/}}^{{\xi}_0,{\rho}_0,{\zeta}},u^{{\xi}_0,{\rho}_0,{\zeta}}_0,T^{{\xi}_0,{\rho}_0,{\zeta}}_0 \mid {\zeta}<{\zeta}_0 \rangle}}}}$.
${\zeta}_0$ is limit: $$\begin{aligned}
& \forall {\zeta}<{\zeta}_0\ {{\ensuremath{\Bar{{\alpha}}}\/}}^{{\xi}_0,{\rho}_0,{\zeta}_0} > {{\ensuremath{\Bar{{\alpha}}}\/}}^{{\xi}_0,{\rho}_0,{\zeta}} ,
\\
& u^{{\xi}_0,{\rho}_0,{\zeta}_0}_0 = {\bigcup}_{{\zeta}< {\zeta}_0} u^{{\xi}_0,{\rho}_0,{\zeta}}_0 {\cup}{\ensuremath{\left\{ {\ensuremath{\langle {{\ensuremath{\Bar{{\alpha}}}\/}}^{{\xi}_0,{\rho}_0,{\zeta}_0},t \rangle}} \right\}}}\text{ where }
{\kappa}^0(t)={\tau}_{{\xi}_0}.\end{aligned}$$ We set $T^{{\xi}_0,{\rho}_0,{\zeta}}$ to whatever we want as no use of it is made later.
${\zeta}_0={\zeta}+1$: We set $$\begin{aligned}
& u'' = (u_0^{{\xi}_0,{\rho}_0,{\zeta}})_{s^{{\xi}_0,{\rho}}},
\\
& T''_0 = {\pi}^{-1}_{{{\ensuremath{\Bar{{\alpha}}}\/}}^{{\xi}_0,{\rho}_0,{\zeta}},{{\ensuremath{\Bar{{\alpha}}}\/}}^0} T^0_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,
\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_n \rangle}}}.\end{aligned}$$ If there is $$\begin{aligned}
& u'_0 \leq^* u''_0 {\cup}{\ensuremath{\left\{ T''_0 \right\}}}\end{aligned}$$ such that $$\begin{aligned}
q^{{\xi}_0,{\rho}_0,{\zeta}} {\cup}u'_0 \in E^{{\xi}_0,{\rho}_0,{\zeta}}\end{aligned}$$ then set $$\begin{aligned}
{{\ensuremath{\Bar{{\alpha}}}\/}}^{{\xi}_0,{\rho}_0,{\zeta}_0} &= \operatorname{mc}(u'_0),
\\
u^{{\xi}_0,{\rho}_0,{\zeta}_0}_0 &= u^{{\xi}_0,{\rho}_0,{\zeta}}_0 {\cup}(u'_0 \setminus
( u''_0 {\cup}{\ensuremath{\left\{ T^{u'_0} \right\}}} )),
\\
T^{{\xi}_0,{\rho}_0,{\zeta}_0} &= T^{u'_0},\end{aligned}$$ otherwise set $$\begin{aligned}
{{\ensuremath{\Bar{{\alpha}}}\/}}^{{\xi}_0,{\rho}_0,{\zeta}_0} &= {{\ensuremath{\Bar{{\alpha}}}\/}}^{{\xi}_0,{\rho}_0,{\zeta}},
\\
u^{{\xi}_0,{\rho}_0,{\zeta}_0}_0 &= u^{{\xi}_0,{\rho}_0,{\zeta}}_0,
\\
T^{{\xi}_0,{\rho}_0,{\zeta}_0}_0 &= T''_0.\end{aligned}$$ When the induction on ${\zeta}$ terminates we have ${\ensuremath{{\ensuremath{\langle {{\ensuremath{\Bar{{\alpha}}}\/}}^{{\xi}_0,{\rho}_0,{\zeta}}, u^{{\xi}_0,{\rho}_0,{\zeta}}_0, T^{{\xi}_0,{\rho}_0,{\zeta}} \mid {\zeta}< {\tau}_{{\xi}_0} \rangle}}}}$
We continue with the induction on ${\rho}$. We set $$\begin{aligned}
& \forall {\zeta}< {\tau}_{{\xi}_0}\ {{\ensuremath{\Bar{{\alpha}}}\/}}^{{\xi}_0,{\rho}_0} >{{\ensuremath{\Bar{{\alpha}}}\/}}^{{\xi}_0,{\rho}_0,{\zeta}} ,
\\
& u^{{\xi}_0,{\rho}_0}_0 = {\bigcup}_{{\zeta}<{\tau}_{{\xi}_0}} u^{{\xi}_0,{\rho}_0,{\zeta}}_0 {\cup}{\ensuremath{\left\{ {\ensuremath{\langle {{\ensuremath{\Bar{{\alpha}}}\/}}^{{\xi}_0,{\rho}_0},t \rangle}} \right\}}}
\text{ where }{\kappa}^0(t)={\tau}_{{\xi}_0}.\end{aligned}$$ When the induction on ${\rho}$ terminates we have ${\ensuremath{{\ensuremath{\langle {{\ensuremath{\Bar{{\alpha}}}\/}}^{{\xi}_0,{\rho}}, u^{{\xi}_0,{\rho}}_0, T^{{\xi}_0,{\rho}} \mid {\rho}< {\tau}_{{\xi}_0} \rangle}}}}$. We continue with the induction on ${\xi}$. We set $$\begin{aligned}
& \forall {\rho}< {\tau}_{{\xi}_0}\ {{\ensuremath{\Bar{{\alpha}}}\/}}^{{\xi}_0} >{{\ensuremath{\Bar{{\alpha}}}\/}}^{{\xi}_0,{\rho}} ,
\\
& u^{{\xi}_0}_0 = {\bigcup}_{{\rho}<{\tau}_{{\xi}_0}} u^{{\xi}_0,{\rho}}_0 {\cup}{\ensuremath{\left\{ {\ensuremath{\langle {{\ensuremath{\Bar{{\alpha}}}\/}}^{{\xi}_0},t \rangle}} \right\}}}
\text{ where }{\kappa}^0(t)={\tau}_{{\xi}_0}.\end{aligned}$$ When the induction on ${\xi}$ terminates we have ${\ensuremath{{\ensuremath{\langle {{\ensuremath{\Bar{{\alpha}}}\/}}^{{\xi}}, u^{{\xi}}_0 \mid {\xi}< {\kappa}\rangle}}}}$. We note that this sequence is not in $N$. Let $$\begin{aligned}
& \forall {\xi}< {\kappa}\ {{\ensuremath{\Bar{{\alpha}}}\/}}^* >{{\ensuremath{\Bar{{\alpha}}}\/}}^{{\xi}} ,
\\
& p^*_0 = {\bigcup}_{{\xi}<{\kappa}} u^{{\xi}}_0 {\cup}{\ensuremath{\left\{ {\ensuremath{\langle {{\ensuremath{\Bar{{\alpha}}}\/}}^{*},t \rangle}} \right\}}}
\text{ where } {\kappa}^0(t)=\max p_0^0.\end{aligned}$$ We construct a series of trees, $R^n$, and $T^{p^*_0}$ is ${\bigcap}_{n<{\omega}} R^n$. $$\begin{aligned}
& \operatorname{Lev}_0(R^0) = {\pi}^{-1}_{{{\ensuremath{\Bar{{\alpha}}}\/}}^{*},{{\ensuremath{\Bar{{\alpha}}}\/}}^{0}} \operatorname{Lev}_0(T^0).\end{aligned}$$ Let us consider ${\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1 \rangle}}\in \operatorname{Lev}_0(R^0)$. There is ${\xi}$ such that ${\kappa}^0({{\ensuremath{\Bar{{\nu}}}\/}}_1) = {\tau}_{\xi}$. We set $$\begin{aligned}
& s(0) = {\ensuremath{\left\{ {\ensuremath{\langle {{\ensuremath{\Bar{{\alpha}}}\/}},{\pi}_{{{\ensuremath{\Bar{{\alpha}}}\/}}^*,{{\ensuremath{\Bar{{\alpha}}}\/}}}({{\ensuremath{\Bar{{\nu}}}\/}}_1) \rangle}} \mid {{\ensuremath{\Bar{{\alpha}}}\/}}\in \operatorname{supp}p^*_0 \right\}}},
\\
& s(1)= {\ensuremath{\left\{ {\ensuremath{\langle {\pi}_{{{\ensuremath{\Bar{{\alpha}}}\/}}^*,{{\ensuremath{\Bar{{\alpha}}}\/}}^0}({{\ensuremath{\Bar{{\nu}}}\/}}_1) \rangle}} \right\}}}.\end{aligned}$$ Let ${\xi}_0={\xi}+1$. By our construction there is ${\rho}$ such that $$\begin{aligned}
& (u^{{\xi}_0,{\rho}_0}_0)_{{\ensuremath{\langle s \rangle}}} =
(u^{{\xi}_0,{\rho}_0}_0)_{{\ensuremath{\langle s^{{\xi}_0,{\rho}} \rangle}}},\end{aligned}$$ where ${\rho}_0 = {\rho}+1$. We set $$\begin{aligned}
& R^1_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1 \rangle}}} = {\pi}^{-1}_{{{\ensuremath{\Bar{{\alpha}}}\/}}^*,{{\ensuremath{\Bar{{\alpha}}}\/}}^{{\xi}_0,{\rho}_0}}(T^{{\xi}_0,{\rho}_0})
{\cap}{\pi}^{-1}_{{{\ensuremath{\Bar{{\alpha}}}\/}}^*,{{\ensuremath{\Bar{{\alpha}}}\/}}^{0}}
(T^0_{{\ensuremath{\langle {\pi}_{{{\ensuremath{\Bar{{\alpha}}}\/}}^*,{{\ensuremath{\Bar{{\alpha}}}\/}}^0}({{\ensuremath{\Bar{{\nu}}}\/}}_1) \rangle}}}).\end{aligned}$$ Assume that we have constructed $R^n$. We set the first $n$ levels of $R^{n+1}$ to be the same as the first $n$ levels of $R^n$ and we complete the tree as follows. Let us consider ${\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_n \rangle}}\in R^n$. There is ${\xi}$ such that ${\kappa}^0({{\ensuremath{\Bar{{\nu}}}\/}}_n) = {\tau}_{\xi}$. We set $s$ as folows $$\begin{aligned}
& \forall 1\leq k\leq n\
s(k) = {\ensuremath{\left\{ {\ensuremath{\langle {{\ensuremath{\Bar{{\alpha}}}\/}},{\pi}_{{{\ensuremath{\Bar{{\alpha}}}\/}}^*,{{\ensuremath{\Bar{{\alpha}}}\/}}}({{\ensuremath{\Bar{{\nu}}}\/}}_k) \rangle}} \mid {{\ensuremath{\Bar{{\alpha}}}\/}}\in \operatorname{supp}p^*_0 \right\}}},
\\
& s(n+1)= {\ensuremath{\left\{ {\ensuremath{\langle {\pi}_{{{\ensuremath{\Bar{{\alpha}}}\/}}^*,{{\ensuremath{\Bar{{\alpha}}}\/}}^0}({{\ensuremath{\Bar{{\nu}}}\/}}_1),\dotsc,
{\pi}_{{{\ensuremath{\Bar{{\alpha}}}\/}}^*,{{\ensuremath{\Bar{{\alpha}}}\/}}^0}({{\ensuremath{\Bar{{\nu}}}\/}}_n) \rangle}} \right\}}}.\end{aligned}$$ Let ${\xi}_0={\xi}+1$. By our construction there is ${\rho}$ such that $$\begin{aligned}
& (u^{{\xi}_0,{\rho}_0}_0)_{{\ensuremath{\langle s \rangle}}} = (u^{{\xi}_0,{\rho}_0}_0)_
{{\ensuremath{\langle s^{{\xi}_0,{\rho}} \rangle}}},\end{aligned}$$ where ${\rho}_0 = {\rho}+1$. We set $$\begin{aligned}
& R^{n+1}_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_n \rangle}}} =
{\pi}^{-1}_{{{\ensuremath{\Bar{{\alpha}}}\/}}^*,{{\ensuremath{\Bar{{\alpha}}}\/}}^{{\xi}_0,{\rho}_0}}(T^{{\xi}_0,{\rho}_0})
{\cap}{\pi}^{-1}_{{{\ensuremath{\Bar{{\alpha}}}\/}}^*,{{\ensuremath{\Bar{{\alpha}}}\/}}^{0}}
(T^0_{{\ensuremath{\langle {\pi}_{{{\ensuremath{\Bar{{\alpha}}}\/}}^*,{{\ensuremath{\Bar{{\alpha}}}\/}}^0}({{\ensuremath{\Bar{{\nu}}}\/}}_1),\dotsc,
{\pi}_{{{\ensuremath{\Bar{{\alpha}}}\/}}^*,{{\ensuremath{\Bar{{\alpha}}}\/}}^0}({{\ensuremath{\Bar{{\nu}}}\/}}_n) \rangle}}}).\end{aligned}$$ After ${\omega}$ stages we set $$\begin{aligned}
& T^{p^*_0} = {\bigcap}_{n<{\omega}} R^n.\end{aligned}$$ We finish the construction by setting $$\begin{aligned}
& p^* = p_{k(p)} {\mathop{{}^\frown}}\dotsb {\mathop{{}^\frown}}p_1 {\mathop{{}^\frown}}p^*_0.\end{aligned}$$ We show that $p^*$ is as required.
Let $G$ be ${{\ensuremath{P_{{{\ensuremath{\bar{E}}\/}}}\/}}}$-generic such that $p^* \in {{\ensuremath{P_{{{\ensuremath{\bar{E}}\/}}}\/}}}$. Let $D\in N$ be dense open in ${{\ensuremath{P_{{{\ensuremath{\bar{E}}\/}}}\/}}}$. We want to show that $D{\cap}G{\cap}N
\not= \emptyset$.
Choose $q {\mathop{{}^\frown}}r_0 \in D {\cap}G$ such that $$\begin{aligned}
& r_0 \leq^* p''_0,
\\
& q \leq p_{k(p)} {\mathop{{}^\frown}}\dotsb {\mathop{{}^\frown}}p_1 {\mathop{{}^\frown}}p''_{n} {\mathop{{}^\frown}}\dotsb {\mathop{{}^\frown}}p''_1,\end{aligned}$$ where $$\begin{aligned}
& {\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_n \rangle}} \in \operatorname{dom}T^{p^*_0},
\\
& {\kappa}^0({{\ensuremath{\Bar{{\nu}}}\/}}_n) = {\tau}_{\xi},
\\
& D \in {\ensuremath{{\ensuremath{\langle D_{\zeta}\mid {\zeta}<{\tau}_{\xi}\rangle}}}},
\\
& p'' = (p^*_0)_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_n \rangle}}}.
\\\end{aligned}$$ We set $s$ to be $$\begin{aligned}
& \forall 1\leq k\leq n\
s(k) = {\ensuremath{\left\{ {\ensuremath{\langle {{\ensuremath{\Bar{{\alpha}}}\/}},{\pi}_{{{\ensuremath{\Bar{{\alpha}}}\/}}^*,{{\ensuremath{\Bar{{\alpha}}}\/}}}({{\ensuremath{\Bar{{\nu}}}\/}}_k) \rangle}} \mid {{\ensuremath{\Bar{{\alpha}}}\/}}\in \operatorname{supp}p^*_0 \right\}}},
\\
& s(n+1)= {\ensuremath{\left\{ {\ensuremath{\langle {\pi}_{{{\ensuremath{\Bar{{\alpha}}}\/}}^*,{{\ensuremath{\Bar{{\alpha}}}\/}}^0}({{\ensuremath{\Bar{{\nu}}}\/}}_1),\dotsc,
{\pi}_{{{\ensuremath{\Bar{{\alpha}}}\/}}^*,{{\ensuremath{\Bar{{\alpha}}}\/}}^0}({{\ensuremath{\Bar{{\nu}}}\/}}_n) \rangle}} \right\}}}.\end{aligned}$$ We get that $$\begin{aligned}
(p^*_0)_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_n \rangle}}} = (p^*_0)_{{\ensuremath{\langle s \rangle}}}.\end{aligned}$$ We let ${\xi}_0 = {\xi}+1$. Recall the enumeration of $S$ in the construction. There is ${\rho}$ such that $$\begin{aligned}
(p^*_0\setminus {\ensuremath{\left\{ {\ensuremath{\langle \operatorname{mc}(p^*_0),(p^*_0)^{\operatorname{mc}} \rangle}} \right\}}})_{{\ensuremath{\langle s \rangle}}}
{\cup}{\ensuremath{\left\{ {\ensuremath{\langle \operatorname{mc}(p^*_0),(p^*_0)^{\operatorname{mc}} \rangle}} \right\}}} =
(p^*_0)_{{\ensuremath{\langle s^{{\xi}_0,{\rho}} \rangle}}}.\end{aligned}$$ We let ${\rho}_0 = {\rho}+1$. Considering the construction of $T^{p^*_0}$, we see that $$\begin{aligned}
& T^{p^*_0}_{{\ensuremath{\langle {{\ensuremath{\Bar{{\nu}}}\/}}_1,\dotsc,{{\ensuremath{\Bar{{\nu}}}\/}}_n \rangle}}}\leq T^{{\xi}_0,{\rho}_0}\end{aligned}$$ hence $$\begin{aligned}
(p^*_0)_{{\ensuremath{\langle s \rangle}}} \leq^* (u^{{\xi}_0,{\rho}_0}_0)_{{\ensuremath{\langle s^{{\xi}_0,{\rho}_0} \rangle}}}.\end{aligned}$$ We note that $$\begin{aligned}
\forall 1\leq k\leq n\ p''_{k} = ((p^*_0)_{{\ensuremath{\langle s \rangle}}})_k.\end{aligned}$$ Recalling that $q$ was chosen so that $$\begin{aligned}
q \leq p_{k(p)} {\mathop{{}^\frown}}\dotsb {\mathop{{}^\frown}}p_1 {\mathop{{}^\frown}}p''_{n} {\mathop{{}^\frown}}\dotsb {\mathop{{}^\frown}}p''_1\end{aligned}$$ we conclude that there is ${\zeta}$ such that $q = q^{{\xi}_0,{\rho}_0,{\zeta}}$ and $D=E^{{\xi}_0,{\rho}_0,{\zeta}}$. That is $$\begin{aligned}
E^{{\xi}_0,{\rho}_0,{\zeta}} \ni q^{{\xi}_0,{\rho}_0,{\zeta}} {\mathop{{}^\frown}}r_0 \leq^*
q^{{\xi}_0,{\rho}_0,{\zeta}} {\mathop{{}^\frown}}((u^{{\xi}_0,{\rho}_0,{\zeta}})_
{{\ensuremath{\langle s^{{\xi}_0,{\rho}} \rangle}}})_0.\end{aligned}$$ Note that this is an answer to the question we asked in the construction. Hence, due to elementarity of $N$, there was such a condition in $N$. Hence $$\begin{aligned}
q^{{\xi}_0,{\rho}_0,{\zeta}} {\mathop{{}^\frown}}((u^{{\xi}_0,{\rho}_0,{\zeta}})_{{\ensuremath{\langle s^{{\xi}_0,{\rho}} \rangle}}})_0 \in
D {\cap}N.\end{aligned}$$ The last point to note is that $$\begin{aligned}
q^{{\xi}_0,{\rho}_0,{\zeta}} {\mathop{{}^\frown}}((u^{{\xi}_0,{\rho}_0,{\zeta}})_{{\ensuremath{\langle s^{{\xi}_0,{\rho}} \rangle}}})_0 \geq^*.
q^{{\xi}_0,{\rho}_0,{\zeta}} {\mathop{{}^\frown}}p''_0 \geq^* q{\mathop{{}^\frown}}r_0 \in G\end{aligned}$$ Hence $$\begin{aligned}
q^{{\xi}_0,{\rho}_0,{\zeta}} {\mathop{{}^\frown}}((u^{{\xi}_0,{\rho}_0,{\zeta}})_{{\ensuremath{\langle s^{{\xi}_0,{\rho}} \rangle}}})_0 \in G.\end{aligned}$$
\[properness\] ${{\ensuremath{P_{{{\ensuremath{\bar{E}}\/}}}\/}}}$ is proper.
Cardinals in $V^{{{\ensuremath{P_{{{\ensuremath{\bar{E}}\/}}}\/}}}}$ {#Cardinals}
====================================================================
\[nocollapse-special\] ${\kappa}^+$ remains a cardinal in $V^{{{\ensuremath{P_{{{\ensuremath{\bar{E}}\/}}}\/}}}}$.
The proof really has no connection to the specific structure of $P_{{\ensuremath{\bar{E}}\/}}$. It is an exercise in properness.
Let $$\begin{aligned}
p {\mathrel\Vdash}{{}^{\ulcorner} {\widetilde{f}} {\mathord{:}}{\widehat{{\kappa}}} \to {\widehat{{\kappa}^+}} {}^{\urcorner}}.\end{aligned}$$ Choose ${\chi}$ large enough so that $H_{\chi}$ contains everything we are interested in. Take $N {\prec}H_{\chi}$ such that
1. $p, {{\ensuremath{P_{{{\ensuremath{\bar{E}}\/}}}\/}}}, {\widetilde{f}} \in N$,
2. ${\lvertN\rvert}= {\kappa}$,
3. $N \supseteq V_{\kappa}$,
4. $N \supseteq N^{{\mathord{<}}{\kappa}}$.
By \[properness\] there is $q \leq p$ which is ${\ensuremath{\langle N,{{\ensuremath{P_{{{\ensuremath{\bar{E}}\/}}}\/}}}\rangle}}$-generic. Let us set $$\begin{aligned}
{\lambda}= N {\cap}{\kappa}^+,\end{aligned}$$ where ${\lambda}$ is an ordinal $< {\kappa}^+$.
Let $G$ be ${{\ensuremath{P_{{{\ensuremath{\bar{E}}\/}}}\/}}}$-generic with $q\in G$. The ${\ensuremath{\langle N,{{\ensuremath{P_{{{\ensuremath{\bar{E}}\/}}}\/}}}\rangle}}$-genericity ensures us that for all ${\xi}< {\kappa}$ ${\widetilde{f}}({\xi})^{N[G]} \in N$ and ${\widetilde{f}}({\xi})^{N[G]}= {\widetilde{f}}({\xi})^{H_{\chi}[G]}$. Hence $\operatorname{ran}{\widetilde{f}}^{V[G]} \subseteq {\lambda}$. That is $$\begin{aligned}
q {\mathrel\Vdash}{{}^{\ulcorner} {\widetilde{f}} \text{ is bounded in } {\kappa}^+ {}^{\urcorner}}.\end{aligned}$$
\[nocollapse-above\] No cardinals $> {\kappa}$ are collapsed by ${{\ensuremath{P_{{{\ensuremath{\bar{E}}\/}}}\/}}}$.
${\kappa}^+$ is not collapsed by \[nocollapse-special\]. No cardinals $\geq {\kappa}^{++}$ are collapsed as ${{\ensuremath{P_{{{\ensuremath{\bar{E}}\/}}}\/}}}$ satisfies ${\kappa}^{++}$-c.c.
\[small-forcing\] Let ${\xi}< {\kappa}$ and ${\zeta}$ the ordinal such that ${\kappa}^0({{\ensuremath{\bar{E}}\/}}_G({\zeta})) \leq {\xi}< \linebreak[4] {\kappa}^0({{\ensuremath{\bar{E}}\/}}_G({\zeta}+ 1))$. Then ${{\mathcal{P}}}({\xi}) {\cap}V[G] = {{\mathcal{P}}}({\xi}) {\cap}V[G{\mathord{\restriction}}{\zeta}]$.
Take $p = p_n {\mathop{{}^\frown}}\dotsb {\mathop{{}^\frown}}p_{k+1} {\mathop{{}^\frown}}p_k {\mathop{{}^\frown}}\dotsb {\mathop{{}^\frown}}p_0 \in G$ such that ${{\ensuremath{\bar{E}}\/}}(p_{k+1}) = {{\ensuremath{\bar{E}}\/}}_G({\zeta})$, ${{\ensuremath{\bar{E}}\/}}(p_{k}) = {{\ensuremath{\bar{E}}\/}}_G({\zeta}+ 1)$. We know that $V[G] = V[G/p]$. So we work in $P_{{{\ensuremath{\bar{E}}\/}}}/p$. Set $p^l = p_n {\mathop{{}^\frown}}\dotsb {\mathop{{}^\frown}}p_{k+1}$, $p^h = {\ensuremath{\left\{ {\ensuremath{\langle {{\ensuremath{\bar{E}}\/}}_G({\zeta}), \emptyset \rangle}} \right\}}} {\mathop{{}^\frown}}p_k {\mathop{{}^\frown}}\dotsb
{\mathop{{}^\frown}}p_0$. Then $P_{{{\ensuremath{\bar{E}}\/}}}/p = P_{{{\ensuremath{\bar{E}}\/}}} {\mathord{\restriction}}p^l \times
P_{{{\ensuremath{\bar{E}}\/}}} {\mathord{\restriction}}p^h$. Note that ${\ensuremath{\langle P_{{{\ensuremath{\bar{E}}\/}}}/p^h, \leq^* \rangle}}$ is ${\kappa}^0({{\ensuremath{\bar{E}}\/}}_G({\zeta}+1))$-closed. In particular it is ${\xi}^+$-closed.
Let $A \in V[G]$, $A \subseteq {\xi}$. Choose ${\widetilde{A}}$, a canonical $P_{{{\ensuremath{\bar{E}}\/}}}/p$-name for $A$. Let $q \in P_{{{\ensuremath{\bar{E}}\/}}}/p^h$. By induction we construct ${\ensuremath{{\ensuremath{\langle q_{\tau}\mid {\tau}< {\xi}\rangle}}}}$ satisfying
1. ${\tau}_0 < {\tau}_1 \implies q_{{\tau}_1} \leq^* q_{{\tau}_0}$,
2. $q_{\tau}{\mathrel\Vert}{{}^{\ulcorner} {\widehat{{\tau}}} \in {\widetilde{A}} {}^{\urcorner}}$.
Choose $q_{\xi}\leq^* q_{\tau}$ for all ${\tau}< {\xi}$.
By density argument we can construct, ${\widetilde{B}}$, a $P{\mathord{\restriction}}p^l$-name such that $A = {\widetilde{A}}[G] = {\widetilde{B}}[G{\mathord{\restriction}}p^l]$.
No carindals $\leq {\kappa}$ are collapsed by ${{\ensuremath{P_{{{\ensuremath{\bar{E}}\/}}}\/}}}$.
Let $G \subseteq {{\ensuremath{P_{{{\ensuremath{\bar{E}}\/}}}\/}}}$ be generic. Assume ${\lambda}< {\kappa}$ is a collapsed cardinal. Let ${\mu}= {\lvert{\lambda}\rvert}^{V[G]}$. We have ${\mu}< {\lambda}$, and there is $A \in {{\mathcal{P}}}({\mu})^{V[G]}$ which codifies the order type ${\lambda}$. Let ${\zeta}$ be the unique ordinal such that ${\kappa}^0({{\ensuremath{\bar{E}}\/}}_G({\zeta})) \leq {\mu}< {\kappa}^0({{\ensuremath{\bar{E}}\/}}_G({\zeta}+1))$. By \[small-forcing\] $A \in V[G{\mathord{\restriction}}{\zeta}]$. Hence ${\lambda}$ is collapsed already in $V[G{\mathord{\restriction}}{\zeta}]$. However, by \[nocollapse-above\], $P_{{{\ensuremath{\bar{E}}\/}}_G({\zeta})}$ collapses no cardinals above ${\kappa}^0({{\ensuremath{\bar{E}}\/}}_G({\zeta}))$. Contradiction.
So, no cardinal $<{\kappa}$ is collapsed. As ${\kappa}$ is a limit of cardinals which are not collapsed, it is not collapsed.
We have just shown
No cardinals are collapsed in $V^{{{\ensuremath{P_{{{\ensuremath{\bar{E}}\/}}}\/}}}}$.
Properties of ${\kappa}$ in $V^{P_{{{\ensuremath{\bar{E}}\/}}}}$ {#PropertiesK}
================================================================
If $\operatorname{l}({{\ensuremath{\bar{E}}\/}}) = {\kappa}^+$ then $V^{P_{{{\ensuremath{\bar{E}}\/}}}} {\vDash}{{}^{\ulcorner} {\kappa}\text{ is regular} {}^{\urcorner}}$.
Let ${\lambda}< {\kappa}$, ${\widetilde{f}}$ be such that $$\begin{aligned}
{\mathrel\Vdash}_{P_{{\ensuremath{\bar{E}}\/}}} {{}^{\ulcorner} {\widetilde{f}}{\mathord{:}}{\widehat{{\lambda}}} \to {\widehat{{\kappa}}} {}^{\urcorner}}.\end{aligned}$$ Let $$\begin{aligned}
D_0 = {\ensuremath{\left\{ p \mid \exists i \ p{\mathrel\Vdash}{{}^{\ulcorner} {\widetilde{f}}(0)={\widehat{i}} {}^{\urcorner}} \right\}}}.\end{aligned}$$ As $D_0$ is a dense open set we can invoke \[DenseHomogen\] to get $p^{\prime 0}$, $n_0$, $S^0 \subset T^{p^{\prime 0}}$ such that $$\begin{aligned}
&\forall k<n_0 \,\forall {\ensuremath{\langle {\nu}_1,\dotsc,{\nu}_k \rangle}} \in S^{\prime 0} \,
\exists {\xi}< \operatorname{l}({{\ensuremath{\bar{E}}\/}})\
\operatorname{Suc}_S({\ensuremath{\langle {\nu}_1,\dotsc,{\nu}_k \rangle}}) \in E_{\operatorname{mc}(p^{\prime 0})}({\xi}),
\\
&\forall {\ensuremath{\langle {\nu}_1,\dotsc,{\nu}_{n_0} \rangle}} \in S^{\prime 0} \
(p^{\prime 0})_{{\ensuremath{\langle {\nu}_1,\dotsc,{\nu}_{n_0} \rangle}}} \in D_0.\end{aligned}$$ Let us set $$\begin{aligned}
A'_0 = {\ensuremath{\left\{ (p^{\prime 0})_{{\ensuremath{\langle {\nu}_1,\dotsc,{\nu}_{n_0} \rangle}}} \mid {\ensuremath{\langle {\nu}_1,\dotsc,{\nu}_{n_0} \rangle}} \in S^{\prime 0} \right\}}}.\end{aligned}$$ $A_0$ is an anti-chain. By shrinking $T^{p^{\prime 0}}$ as was done in the proof of \[PrikryCondition\] we can make $A_0$ into a maximal anti-chain below $p^{\prime 0}$. As ${\lambda}< {\kappa}$ and ${\ensuremath{\langle P^*_{{{\ensuremath{\bar{E}}\/}}},\leq^* \rangle}}$ is ${\kappa}$-closed we can construct a $\leq^*$-decreasing sequence $$\begin{aligned}
p^{\prime 0} \geq^* p^{\prime 1} \geq^* \dotsb \geq^*
p^{\prime {\tau}} \geq^* \dotsb \qquad {\tau}< {\lambda}.\end{aligned}$$ and $n_{\tau}$, $S^{\prime {\tau}} \subseteq T^{p^{\prime {\tau}}}$ such that $$\begin{aligned}
&\forall k<n_{\tau}\,\forall {\ensuremath{\langle {\nu}_1,\dotsc,{\nu}_k \rangle}} \in S^{\prime {\tau}} \,
\exists {\xi}< \operatorname{l}({{\ensuremath{\bar{E}}\/}})\
\operatorname{Suc}_{S^{\prime {\tau}}}({\ensuremath{\langle {\nu}_1,\dotsc,{\nu}_k \rangle}}) \in E_{\operatorname{mc}(p^{\prime {\tau}})}({\xi}),
\\
&\forall {\ensuremath{\langle {\nu}_1,\dotsc,{\nu}_{n_{\tau}} \rangle}} \in S^{\prime {\tau}} \
\exists i \
(p^{\prime {\tau}})_{{\ensuremath{\langle {\nu}_1,\dotsc,{\nu}_{n_{\tau}} \rangle}}} {\mathrel\Vdash}{{}^{\ulcorner} {\widetilde{f}}({\widehat{{\tau}}})={\widehat{i}} {}^{\urcorner}},\end{aligned}$$ and $$\begin{aligned}
A'_{\tau}= {\ensuremath{\left\{ (p^{\prime {\tau}})_{{\ensuremath{\langle {\nu}_1,\dotsc,{\nu}_{n_{\tau}} \rangle}}} \mid {\ensuremath{\langle {\nu}_1,\dotsc,{\nu}_{n_{\tau}} \rangle}} \in S^{\prime {\tau}} \right\}}}\end{aligned}$$ is a maximal anti-chain below $p^{\prime {\tau}}$.
Let $p' \leq^* p^{\prime {\tau}}$ for all ${\tau}< {\lambda}$. We set $S^{\tau}= {\pi}^{-1}_{\operatorname{mc}(p'),\operatorname{mc}(p^{\prime{\tau})}}(S^{\prime{\tau}})$ and $p^{\tau}$ to be $p'$ with ${\pi}^{-1}_{\operatorname{mc}(p'),\operatorname{mc}(p^{\prime{\tau})}}(T^{p^{\prime{\tau}}})$ substituted for $T^{p'}$ and maybe shrunken a bit so that $$\begin{aligned}
A_{\tau}= {\ensuremath{\left\{ (p^{{\tau}})_{{\ensuremath{\langle {\nu}_1,\dotsc,{\nu}_{n_{\tau}} \rangle}}} \mid {\ensuremath{\langle {\nu}_1,\dotsc,{\nu}_{n_{\tau}} \rangle}} \in S^{{\tau}} \right\}}}\end{aligned}$$ is a maximal anti-chain below $p^{\tau}$.
Let $p \leq^* p^{\tau}$ for all ${\tau}< {\lambda}$ and let ${\widetilde{g}}$ be the following ${{\ensuremath{P_{{{\ensuremath{\bar{E}}\/}}}\/}}}$-name $$\begin{aligned}
{\widetilde{g}} = {\bigcup}_{{\tau}<{\lambda}}
{\ensuremath{\left\{ {\ensuremath{\langle {\widehat{{\ensuremath{\langle {\tau}, i \rangle}}}},
(p^{\tau})_{{\ensuremath{\langle {\nu}_1,\dotsc,{\nu}_{n_{\tau}} \rangle}}} \rangle}} \mid A_{\tau}\ni (p^{\tau})_{{\ensuremath{\langle {\nu}_1,\dotsc,{\nu}_{n_{\tau}} \rangle}}}{\mathrel\Vdash}{{}^{\ulcorner} {\widetilde{f}}({\tau})= {\widehat{i}} {}^{\urcorner}}\
\right\}}}.\end{aligned}$$ Then $$\begin{aligned}
p{\mathrel\Vdash}{{}^{\ulcorner} {\widetilde{f}}={\widetilde{g}} {}^{\urcorner}}.\end{aligned}$$ Let $P^*$ be the following forcing notion: $$\begin{aligned}
P^* = {\ensuremath{\left\{ q \leq_{R} p \mid q \in {{\ensuremath{P_{{{\ensuremath{\bar{E}}\/}}}\/}}}\right\}}}.\end{aligned}$$ By \[SubForcing\] ${\ensuremath{\langle P^*, \leq_R \rangle}}$ is sub-forcing of ${\ensuremath{\langle {{\ensuremath{P_{{{\ensuremath{\bar{E}}\/}}}\/}}}/p, \leq \rangle}}$. Hence if $G$ is ${{\ensuremath{P_{{{\ensuremath{\bar{E}}\/}}}\/}}}$-generic then $G^* = G {\cap}P^*$ is $P^*$-generic. ${\widetilde{g}}$ is in fact a $P^*$-name and as can be seen from its’ definition ${\widetilde{g}}[G] =
{\widetilde{g}}[G^*] \in V[G^*]$. So in order to complete the proof it is enough to show that ${\mathrel\Vdash}_{P^*} {{}^{\ulcorner} {\widetilde{g}} \text{ is bounded} {}^{\urcorner}}$.
By \[GenericForRadin\] there is $r \in R_{\operatorname{mc}(p)}$ such that $P^* \simeq R_{\operatorname{mc}(p)}/r$. Now we use the following fact about Radin forcing: When the measure sequence is of length ${\kappa}^+$, ${\kappa}$ is regular in the generic extension. Necessarily, ${\mathrel\Vdash}_{P^*} {{}^{\ulcorner}
{\widetilde{g}} \text{ is bounded} {}^{\urcorner}}$.
We say that ${\tau}< \operatorname{l}({{\ensuremath{\bar{E}}\/}})$ is a repeat point of ${{\ensuremath{\bar{E}}\/}}$ if $P_{{\ensuremath{\bar{E}}\/}}=P_{{{\ensuremath{\bar{E}}\/}}{\mathord{\restriction}}{\tau}}$.
Note that if ${\tau}$ is a repeat point then $P_{{{\ensuremath{\bar{E}}\/}}{\mathord{\restriction}}{\tau}}\in M$.
If ${{\ensuremath{\bar{E}}\/}}$ has a repeat point, $j''{\ensuremath{\left\{ {\widetilde{A}}_{\xi}\mid {\xi}<{\lambda}\right\}}} \in M$ where $\linebreak[0]{\ensuremath{\left\{ {\widetilde{A}}_{\xi}\mid {\xi}<{\lambda}\right\}}}$ is an enumeration of all canonical ${{\ensuremath{P_{{{\ensuremath{\bar{E}}\/}}}\/}}}$-names of subsets of ${\kappa}$, then $V^{{{\ensuremath{P_{{{\ensuremath{\bar{E}}\/}}}\/}}}} {\vDash}\linebreak[0]
\ulcorner {\kappa}\text{ is}\linebreak[0] \text{ measurable} \urcorner$.
We use the usual method under these circumstances. Let ${\tau}$ be a repeat point of ${{\ensuremath{\bar{E}}\/}}$ and $G$ be ${{\ensuremath{P_{{{\ensuremath{\bar{E}}\/}}}\/}}}$-generic over $V$. For the duration of this proof let us define:
- If $p=p_0 \in P^*_{{{\ensuremath{\bar{E}}\/}}}$ then $$\begin{aligned}
p{\mathord{\restriction}}{\tau}= {\ensuremath{\left\{ {\ensuremath{\langle {{\ensuremath{\bar{E}}\/}}_{\alpha}{\mathord{\restriction}}{\tau},p^{{{\ensuremath{\bar{E}}\/}}_{\alpha}} \rangle}} \mid {{\ensuremath{\bar{E}}\/}}_{\alpha}\in \operatorname{supp}p \right\}}} {\cup}{\ensuremath{\left\{ T^p \right\}}}.
\end{aligned}$$
- If $p=p_n{\mathop{{}^\frown}}\dotsb{\mathop{{}^\frown}}p_1{\mathop{{}^\frown}}p_0$ then $$\begin{aligned}
p{\mathord{\restriction}}{\tau}= p_n{\mathop{{}^\frown}}\dotsb{\mathop{{}^\frown}}p_1{\mathop{{}^\frown}}p_0{\mathord{\restriction}}{\tau}.
\end{aligned}$$
Let us set $$\begin{aligned}
G{\mathord{\restriction}}{\tau}= {\ensuremath{\left\{ p{\mathord{\restriction}}{\tau}\mid p \in G \right\}}}.\end{aligned}$$ We note that
1. ${\lambda}< j({\kappa})$,
2. $M{\vDash}{{}^{\ulcorner} j(P^*_{{{\ensuremath{\bar{E}}\/}}})/{{\ensuremath{P_{{{\ensuremath{\bar{E}}\/}}}\/}}}\text{ is }j(k)-closed {}^{\urcorner}}$,
3. $G{\mathord{\restriction}}{\tau}$ is $P_{{{\ensuremath{\bar{E}}\/}}{\mathord{\restriction}}{\tau}}$-generic over $M$.
So, we can construct a $\leq^*$-decreasing sequence in $M[G{\mathord{\restriction}}{\tau}]$, [${\ensuremath{\langle p^{\xi}\mid {\xi}<{\lambda}\rangle}}$]{}, such that $$\begin{aligned}
p^{{\xi}+1} {\mathrel\Vert}{{}^{\ulcorner} {\kappa}\in j({\widetilde{A}}_{\xi}) {}^{\urcorner}}.\end{aligned}$$ We define an ultrafilter, $U$, in $V$ by $$\begin{aligned}
{\widetilde{A}}_{\xi}[G] \in U \iff p_{{\xi}+1}{\mathrel\Vdash}{{}^{\ulcorner} {\kappa}\in j({\widetilde{A}}_{\xi}) {}^{\urcorner}}.\end{aligned}$$
The assumptions we used in this theorem are very strong. We believe that a repeat point is enough in order to get measurability. = 1 Let ${\tau}$ be a repeat point of ${{\ensuremath{\bar{E}}\/}}$. Hence ${{\ensuremath{P_{{{\ensuremath{\bar{E}}\/}}}\/}}}= P_{{{\ensuremath{\bar{E}}\/}}{\mathord{\restriction}}{\tau}}$. For the duration of this proof let us use the following definition: For $p \in P^*_{{{\ensuremath{\bar{E}}\/}}}$ $$\begin{aligned}
p{\mathord{\restriction}}{\tau}= {\ensuremath{\left\{ {\ensuremath{\langle {{\ensuremath{\bar{E}}\/}}_{\alpha}{\mathord{\restriction}}{\tau}, p^{{{\ensuremath{\bar{E}}\/}}_{\alpha}} \rangle}} \mid {{\ensuremath{\bar{E}}\/}}_{\alpha}\in \operatorname{supp}p \right\}}}\end{aligned}$$ Note that $P_{{{\ensuremath{\bar{E}}\/}}{\mathord{\restriction}}{\tau}} \in M$ and if $p \in {{\ensuremath{P_{{{\ensuremath{\bar{E}}\/}}}\/}}}$ then $p{\mathord{\restriction}}{\tau}{\cup}{\ensuremath{\left\{ T^p \right\}}} \in P_{{{\ensuremath{\bar{E}}\/}}{\mathord{\restriction}}{\tau}}$.
Let $G$ be ${{\ensuremath{P_{{{\ensuremath{\bar{E}}\/}}}\/}}}$-generic over $V$. By setting $$\begin{aligned}
G{\mathord{\restriction}}{\tau}= {\ensuremath{\left\{ p{\mathord{\restriction}}{\tau}{\cup}{\ensuremath{\left\{ T^p \right\}}} \mid p \in G \right\}}}\end{aligned}$$ we get that $G{\mathord{\restriction}}{\tau}$ is $P_{{{\ensuremath{\bar{E}}\/}}{\mathord{\restriction}}{\tau}}$-generic over $M$.
Let $p\in P^*_{{{\ensuremath{\bar{E}}\/}}}$. Then $j(p) \in j(P^*_{{{\ensuremath{\bar{E}}\/}}})$. Due to the definition of extender sequences we get that ${\operatorname{mc}(p){\mathord{\restriction}}{\tau}} \in j(T^p)$. Hence $j(p)_{{\ensuremath{\langle {\operatorname{mc}(p){\mathord{\restriction}}{\tau}} \rangle}}} \in j({{\ensuremath{P_{{{\ensuremath{\bar{E}}\/}}}\/}}})$. Specifically we note that if we write $$\begin{aligned}
j(p)_{{\ensuremath{\langle {\operatorname{mc}(p){\mathord{\restriction}}{\tau}} \rangle}}} = p_1 {\cup}p_0\end{aligned}$$ then $p_1 = p{\mathord{\restriction}}{\tau}{\cup}{\ensuremath{\left\{ j(T^p)(\operatorname{mc}(p){\mathord{\restriction}}{\tau}) \right\}}}$. Moreover, ${j(T^p)(\operatorname{mc}(p){\mathord{\restriction}}{\tau})}$ is an $\operatorname{mc}(p)$-tree, as ${\tau}$ is a repeat point.
In what follows we need to handle differently the part of a condition which is moved by $j$ and the parts which are not moved. So, when writing $q{\mathop{{}^\frown}}p$ we mean that $q{\mathop{{}^\frown}}p = p_n{\mathop{{}^\frown}}\dotsb{\mathop{{}^\frown}}p_1{\mathop{{}^\frown}}p_0$ where $p = p_0$ and $q = p_n{\mathop{{}^\frown}}\dotsb{\mathop{{}^\frown}}p_1$. We define $U$ in $V[G]$ as follows: $$\begin{aligned}
{\widetilde{A}}[G] \in U \iff \exists q{\mathop{{}^\frown}}p\in G \ & \exists S
\\
& q{\mathop{{}^\frown}}j(p)_{{\ensuremath{\langle \operatorname{mc}(p){\mathord{\restriction}}{\tau}\rangle}}}
\setminus {\ensuremath{\left\{ j(T^p) \right\}}} {\cup}{\ensuremath{\left\{ S \right\}}}
{\mathrel\Vdash}{{}^{\ulcorner} {\widehat{{\kappa}}} \in {\widetilde{A}} {}^{\urcorner}}.\end{aligned}$$ We show that $U$ is a measure.
1. The definition makes sense: Let $q^1{\mathop{{}^\frown}}p^1,q^2{\mathop{{}^\frown}}p^2 \in G$, $S^1, S^2$ trees. Let $q{\mathop{{}^\frown}}p \in G$ such that $q{\mathop{{}^\frown}}p \leq q^1{\mathop{{}^\frown}}p^1,q^2{\mathop{{}^\frown}}p^2$. Then $$\begin{aligned}
q{\mathop{{}^\frown}}j(p)_{{\ensuremath{\langle \operatorname{mc}(p){\mathord{\restriction}}{\tau}\rangle}}}
\setminus {\ensuremath{\left\{ j(T^{p}) \right\}}} {\cup}& {\ensuremath{\left\{ S^1 {\cap}S^2 \right\}}}
\leq
\\
&q^1{\mathop{{}^\frown}}j(p^1)_{{\ensuremath{\langle \operatorname{mc}(p^1){\mathord{\restriction}}{\tau}\rangle}}}
\setminus {\ensuremath{\left\{ j(T^{p^1}) \right\}}} {\cup}{\ensuremath{\left\{ S^1 \right\}}},
\\
&q^2{\mathop{{}^\frown}}j(p^2)_{{\ensuremath{\langle \operatorname{mc}(p^2){\mathord{\restriction}}{\tau}\rangle}}}
\setminus {\ensuremath{\left\{ j(T^{p^2}) \right\}}} {\cup}{\ensuremath{\left\{ S^2 \right\}}}\end{aligned}$$
2. Completeness: Let ${\widetilde{A}}^{{\xi}}[G] \in U$ for all ${\xi}<{\kappa}$. By our definition there are $q^{\xi}{\mathop{{}^\frown}}p^{\xi}\in G$, $S^{\xi}$ such that $$\begin{aligned}
q^{\xi}{\mathop{{}^\frown}}j(p^{\xi})_{{\ensuremath{\langle \operatorname{mc}(p^{\xi}){\mathord{\restriction}}{\tau}\rangle}}}
\setminus {\ensuremath{\left\{ j(T^{p^{\xi}}) \right\}}} {\cup}{\ensuremath{\left\{ S^{\xi}\right\}}}
{\mathrel\Vdash}{{}^{\ulcorner} {\widehat{{\kappa}}} \in j({\widetilde{A}}^{\xi}) {}^{\urcorner}}\end{aligned}$$ The set $$\begin{aligned}
D = {\ensuremath{\left\{ q{\mathop{{}^\frown}}p \in {{\ensuremath{P_{{{\ensuremath{\bar{E}}\/}}}\/}}}\mid \forall {\xi}<{\kappa}\
q{\mathop{{}^\frown}}p \le q^{\xi}{\mathop{{}^\frown}}p^{\xi}\text{ or }
\exists {\xi}<{\kappa}\ q{\mathop{{}^\frown}}p {\perp}q^{\xi}{\mathop{{}^\frown}}p^{\xi}\right\}}}\end{aligned}$$ is dense, hence there is $q{\mathop{{}^\frown}}p \in G$ such that $\forall {\xi}<{\kappa}\
q{\mathop{{}^\frown}}p \leq q^{\xi}{\mathop{{}^\frown}}p^{\xi}$. Let $$\begin{aligned}
S = {\bigcap}_{{\xi}< {\kappa}} S^{\xi}\end{aligned}$$ As $M \supseteq M^{\kappa}$ we have $S\in M$. We got that for all ${\xi}<{\kappa}$ $$\begin{aligned}
q{\mathop{{}^\frown}}j(p)_{{\ensuremath{\langle \operatorname{mc}(p){\mathord{\restriction}}{\tau}\rangle}}}
\setminus {\ensuremath{\left\{ j(T^{p}) \right\}}} {\cup}& {\ensuremath{\left\{ S \right\}}}
\leq
& q^{\xi}{\mathop{{}^\frown}}j(p^{\xi})_{{\ensuremath{\langle \operatorname{mc}(p^{\xi}){\mathord{\restriction}}{\tau}\rangle}}}
\setminus {\ensuremath{\left\{ j(T^{p^{\xi}}) \right\}}} {\cup}{\ensuremath{\left\{ S^{\xi}\right\}}}\end{aligned}$$ Hence $$\begin{aligned}
q{\mathop{{}^\frown}}j(p)_{{\ensuremath{\langle \operatorname{mc}(p){\mathord{\restriction}}{\tau}\rangle}}}
\setminus {\ensuremath{\left\{ j(T^{p}) \right\}}} {\cup}& {\ensuremath{\left\{ S \right\}}}
{\mathrel\Vdash}{{}^{\ulcorner} \forall {\xi}<{\kappa}\ {\widehat{{\kappa}}}\in j({\widetilde{A}}^{\xi}) {}^{\urcorner}}\end{aligned}$$ That is $$\begin{aligned}
\operatorname*{\triangle}_{{\xi}<{\kappa}} {\widetilde{A}}^{\xi}[G] \in U\end{aligned}$$
What have we proved? {#TheTheorem}
====================
We can sum everything as follows:
We can control independently two properties of ${\kappa}$ in a generic extension. The first is the size of $2^{\kappa}$ which is controlled by ${\lvert{{\ensuremath{\bar{E}}\/}}\rvert}$. The second is how ‘big’ we want ${\kappa}$ to be which is controlled by $\operatorname{l}({{\ensuremath{\bar{E}}\/}})$.
Generic by Iteration {#ByIteration}
====================
Recall that if $R$ is Radin forcing generated from $j{\mathord{:}}V \to M$ then there is ${\tau}$ and $G \in V$ such that $G$ is $j_{0,{\tau}}(R)$-generic over $M_{\tau}$.
Our original aim was to find some form of this claim for our forcing. We have a partial result in this direction. Namely, when $\operatorname{l}({{\ensuremath{\bar{E}}\/}})=1$ we have a generic filter in $V$ over an elementary submodel in $M_{\omega}$.
In this section we assume that $\operatorname{l}({{\ensuremath{\bar{E}}\/}})=1$.
Let us take an iteration of $j=j_{0,1}$ $${\ensuremath{\langle
{\ensuremath{{\ensuremath{\langle M_n \mid n < {\omega}\rangle}}}},
{\ensuremath{{\ensuremath{\langle j_{n,m} \mid n \leq m < {\omega}\rangle}}}}
\rangle}}.$$ Choose ${\chi}$ large enough so that everything interesting is in $H_{\chi}$ (i.e. $P_{{{\ensuremath{\bar{E}}\/}}} \in H_{\chi}$) and set $$\begin{aligned}
& {\kappa}_n = j_{0,n}({\kappa}),
\\
& {{\ensuremath{\bar{E}}\/}}^n = j_{0,n}({{\ensuremath{\bar{E}}\/}}),
\\
& P^n = j_{0,n}(P_{{\ensuremath{\bar{E}}\/}})(=P_{{{\ensuremath{\bar{E}}\/}}^n}),
\\
& {\chi}_n = j_{0,n}({\chi}).\end{aligned}$$
We call [$\langle N, p \rangle$]{} a $k$-pair if
1. $M_k {\vDash}{{}^{\ulcorner} N {\prec}H_{{\chi}_k}^{M_k} {}^{\urcorner}}$,
2. $M_k {\vDash}{{}^{\ulcorner} {\lvertN\rvert} = {\kappa}_k {}^{\urcorner}}$,
3. $M_k {\vDash}{{}^{\ulcorner} N \supseteq V_{{\kappa}_k}^{M_k} {}^{\urcorner}}$,
4. $M_k {\vDash}{{}^{\ulcorner} N \supseteq N^{{\mathord{<}}{\kappa}_k} {}^{\urcorner}}$,
5. $p \in P^{k+1} {\cap}j_{k,k+1}(N) $,
6. If $D \in N$, $D$ is dense open in $P^k$ then there are $n$, $S\leq T^p$ such that $$\begin{aligned}
\forall {\ensuremath{\langle {\nu}_1,\dotsc,{\nu}_n \rangle}}\in S \
(p)_{{\ensuremath{\langle {\nu}_1,\dotsc,{\nu}_n \rangle}}} \in j_{k,k+1}(D).
\end{aligned}$$
\[find-pair-1\] Let $N \in M_k$, $k<{\omega}$ and $q \in j_{k,k+1}(N) {\cap}P^{k+1}$ such that $M_k$ satisifes:
1. $N {\prec}H_{{\chi}_k}^{M_k}$,
2. ${\lvertN\rvert} = {\kappa}_k$,
3. $N \supseteq V^{M_k}_{{\kappa}_k}$,
4. $N \supseteq N^{{\mathord{<}}{\kappa}_k}$,
5. $P^k \in N$.
Then there is $p \in j_{k,k+1}(N) {\cap}P^{k+1}$ such that
1. $p \leq^* q$,
2. ${\ensuremath{\langle N,p \rangle}}$ is a $k$-pair.
Let [${\ensuremath{\langle D^k_{{\xi}} \mid {\xi}< {\kappa}_k \rangle}}$]{} be an enumeration, in $M_k$, of all the dense open subsets of $P^k$ which are in $N$. Set $N_{k+1} = j_{k,k+1}(N)$. As we have $$\begin{aligned}
& {\ensuremath{{\ensuremath{\langle j_{k,k+1}(D^k_{{\xi}}) \mid {\xi}< {\kappa}_k \rangle}}}} \subseteq N_{k+1},
\\
& {\ensuremath{{\ensuremath{\langle j_{k,k+1}(D^k_{{\xi}}) \mid {\xi}< {\kappa}_k \rangle}}}} \in M_{k+1},
\\
&M_{k+1} {\vDash}{{}^{\ulcorner} N_{k+1} \supseteq N_{k+1}^{{\mathord{<}}{\kappa}_{k+1}}
{}^{\urcorner}},\end{aligned}$$ we get that $$\begin{aligned}
& {\ensuremath{{\ensuremath{\langle j_{k,k+1}(D^k_{{\xi}}) \mid {\xi}< {\kappa}_k \rangle}}}} \in N_{k+1}.\end{aligned}$$ Starting with $q$ we construct in $N_{k+1}$ a $\leq^*$-decreasing sequence ${\ensuremath{{\ensuremath{\langle p_{\xi}\mid {\xi}<{\kappa}_k \rangle}}}}$ using \[DenseHomogen\]. Note that we have no problem at the limit stages as $N_{k+1} {\vDash}{{}^{\ulcorner} {\ensuremath{\langle P^{k+1},\leq^* \rangle}} \text{ is } {\kappa}_{k+1}
\text{-closed} {}^{\urcorner}}$. Choose now $p \in P^{k+1} {\cap}N_{k+1}$ such that $\forall {\xi}< {\kappa}_{k}\ p \leq^* p_{\xi}$. We get that ${\ensuremath{\langle N,p \rangle}}$ is a $k$-pair.
We call ${{\ensuremath{\langle \Vec{N},\Vec{p} \rangle}}}$ a $P^{\omega}$-generic approximation sequence if $$\begin{aligned}
{{\ensuremath{\langle \Vec{N},\Vec{p} \rangle}}} = {\ensuremath{{\ensuremath{\langle {\ensuremath{\langle N_k, p^{k+1} \rangle}} \mid k_0 \leq k < {\omega}\rangle}}}}\end{aligned}$$ such that for all $k_0 \leq k < {\omega}$
1. $M_{k} {\vDash}{{}^{\ulcorner} N_k {\prec}H_{{\chi}_{k}}^{M_{k}} {}^{\urcorner}}$,
2. $M_{k} {\vDash}{{}^{\ulcorner} {\lvertN_k\rvert} = {\kappa}_{k} {}^{\urcorner}}$,
3. $M_{k} {\vDash}{{}^{\ulcorner} N_k \supseteq V^{M_{k}}_{{\kappa}_{k}} {}^{\urcorner}}$,
4. $M_{k} {\vDash}{{}^{\ulcorner} N_k \supseteq N^{{\mathord{<}}{\kappa}_{k}} {}^{\urcorner}}$,
5. $P^{k} \in N_k$,
6. ${\ensuremath{\langle N_k,p^{k+1} \rangle}}$ is a $k$-pair,
7. $j_{k_1, k_2}(N_{k_1}) = N_{k_2}$,
8. $p^{k+2} \leq^* (j_{k+1,k+2}(p^{k+1}))_{{\ensuremath{\langle \operatorname{mc}(p^{k+1}) \rangle}}}$.
Let ${{\ensuremath{\langle \Vec{N},\Vec{r} \rangle}}}$ be a $P^{\omega}$-generic approximating sequence. Then $$\begin{aligned}
G({{\ensuremath{\langle \Vec{N},\Vec{r} \rangle}}})={\ensuremath{\left\{ p \in P^{\omega}\mid \exists k\ j_{k,{\omega}}(r^k) \leq^* p \right\}}}.\end{aligned}$$
\[GenSeqComplete-1\] Let ${k_0} < {\omega}$, $q \in P^{k_0}{\cap}N_{k_0}$ and assume for all $k_0\le k < {\omega}$
1. $P^{k} \in N_{k}$,
2. $j_{k,k+1}(N_k) = N_{k+1}$,
3. $M_{k} {\vDash}{{}^{\ulcorner} N_{k} {\prec}H_{{\chi}_{k}}^{M_{k}} {}^{\urcorner}}$,
4. $M_{k} {\vDash}{{}^{\ulcorner} {\lvertN_{k}\rvert} = {\kappa}_{k} {}^{\urcorner}}$,
5. $M_{k} {\vDash}{{}^{\ulcorner} N_{k} \supseteq V^{M_{k}}_{{\kappa}_{k}} {}^{\urcorner}}$,
6. $M_{k} {\vDash}{{}^{\ulcorner} N_{k} \supseteq N_{k}^{{\mathord{<}}{\kappa}_{k}} {}^{\urcorner}}$.
Then there is a $P^{\omega}$-generic approximating sequence $$\begin{aligned}
{{\ensuremath{\langle \Vec{N},\Vec{p} \rangle}}} = {\ensuremath{{\ensuremath{\langle {\ensuremath{\langle N_k, p^{k+1} \rangle}} \mid k_0 \leq k < {\omega}\rangle}}}}\end{aligned}$$ such that $p^{{k_0}+1} \leq^* j_{k_0,k_0+1}(q)$.
We construct the $p^{k+1}$ by induction. We set $p^{k_0} = q$.
Assume that we have constructed ${\ensuremath{{\ensuremath{\langle p^{k'} \mid k_0 \leq k'\leq k \rangle}}}}$. We set $\linebreak[0] q^{k+1} = \linebreak[4] j_{k,k+1}(p^k)_{{\ensuremath{\langle \operatorname{mc}(p^k) \rangle}}}$. Invoke \[find-pair-1\] to get $p^{k+1} \leq^* q^{k+1}$ such that ${\ensuremath{\langle N_k, p^{k+1} \rangle}}$ is a $k$-pair.
When the induction terminates we have ${\ensuremath{{\ensuremath{\langle {\ensuremath{\langle N_k, p^{k+1} \rangle}} \mid k_0 \leq k < {\omega}\rangle}}}}$ as required.
\[GenSeq-1\] Let ${k_0} < {\omega}$ and assume
1. $M_{k_0} {\vDash}{{}^{\ulcorner} N_{k_0} {\prec}H_{{\chi}_{k_0}}^{M_{k_0}} {}^{\urcorner}}$,
2. $M_{k_0} {\vDash}{{}^{\ulcorner} {\lvertN_{k_0}\rvert} = {\kappa}_{k_0} {}^{\urcorner}}$,
3. $M_{k_0} {\vDash}{{}^{\ulcorner} N_{k_0} \supseteq V^{M_{k_0}}_{{\kappa}_{k_0}} {}^{\urcorner}}$,
4. $M_{k_0} {\vDash}{{}^{\ulcorner} N_{k_0} \supseteq N_{k_0}^{{\mathord{<}}{\kappa}_{k_0}} {}^{\urcorner}}$,
5. $P^{{k_0}} \in N_{k_0}$,
6. $q \in P^{k_0}{\cap}N_{k_0}$.
Then there is a $P^{\omega}$-generic approximating sequence $$\begin{aligned}
{{\ensuremath{\langle \Vec{N},\Vec{p} \rangle}}} = {\ensuremath{{\ensuremath{\langle {\ensuremath{\langle N_k, p^{k+1} \rangle}} \mid k_0 \leq k < {\omega}\rangle}}}}\end{aligned}$$ such that $p^{{k_0}+1} \leq^* j_{k_0,k_0+1}(q)$.
We set $N_k = j_{k_0, k}(N_{k_0})$ for all $k_0 < k < {\omega}$ and then we invoke \[GenSeqComplete-1\].
\[N-generic-1\] Assume
1. $M_{\omega}{\vDash}{{}^{\ulcorner} N {\prec}H_{{\chi}_{\omega}}^{M_{\omega}} {}^{\urcorner}}$,
2. $M_{\omega}{\vDash}{{}^{\ulcorner} {\lvertN\rvert} = {\kappa}_{\omega}{}^{\urcorner}}$,
3. $M_{\omega}{\vDash}{{}^{\ulcorner} N \supseteq V^{M_{\omega}}_{{\kappa}_{\omega}} {}^{\urcorner}}$,
4. $M_{\omega}{\vDash}{{}^{\ulcorner} N \supseteq N^{{\mathord{<}}{\kappa}_{\omega}} {}^{\urcorner}}$,
5. $P^{{\omega}} \in N$,
6. $q \in P^{{\omega}}{\cap}N$.
Then there is, in $V$, a filter $G \subseteq P^{\omega}$ such that
1. $q \in G$,
2. $\forall D \in N$ D is dense open in $P^{\omega}$ $\implies$ $G {\cap}D {\cap}N \not= \emptyset$.
We find, ${{\ensuremath{\langle \Vec{N},\Vec{p} \rangle}}}$, a $P^{\omega}$-generic approximating sequence. $G({{\ensuremath{\langle \Vec{N},\Vec{p} \rangle}}})$ is the required filter.
Find $k_0$ and $N_{k_0}$, $q^{k_0}$ such that $$\begin{aligned}
& P^{k_0} \in N_{k_0},
\\
& j_{k_0,{\omega}}(N_{k_0}) = N,
\\
& j_{{k_0},{\omega}}(q^{k_0}) = q.\end{aligned}$$ Invoke \[GenSeq-1\] to get from $N^{k_0}$, $q^{k_0}$ a generic approximating sequence ${{\ensuremath{\langle \Vec{N},\Vec{p} \rangle}}}$.
Let $D \in N$ be dense open in $P^{\omega}$.
Find $k \geq k_0$ and $D^k$ such that $j_{k,{\omega}}(D^k)=D$. As usual we set $D^{k+l} = j_{k,k+l}(D^k)$. By construction there is $n$ such that $$\begin{aligned}
\forall {\ensuremath{\langle {\nu}_1,\dotsc,{\nu}_n \rangle}}\in T^{p^{k+1}}\,
(p^{k+1})_{{\ensuremath{\langle {\nu}_1,\dotsc,{\nu}_n \rangle}}} \in D^{k+1} {\cap}N_{k+1},\end{aligned}$$ which means that $$\begin{aligned}
j_{k+1,k+1+n}(p^{k+1})_{{\ensuremath{\langle \operatorname{mc}(p^{k+1}),\dotsc,j_{k+1,k+n}(\operatorname{mc}(p^{k+1})) \rangle}}}
\in D^{k+1+n} {\cap}N_{k+1+n}.\end{aligned}$$ Hence $$\begin{aligned}
\label{GoverN:dense}
j_{k+1,{\omega}}(p^{k+1})_{{\ensuremath{\langle \operatorname{mc}(p^{k+1}),\dotsc,j_{k+1,k+n}(\operatorname{mc}(p^{k+1})) \rangle}}}
\in D {\cap}N.\end{aligned}$$ As ${{\ensuremath{\langle \Vec{N},\Vec{p} \rangle}}}$ is a $P^{\omega}$-generic approximation sequence it satisfies $$\begin{aligned}
p^{k+1+n} \leq^*
j_{k+1,k+1+n}(p^{k+1})_{{\ensuremath{\langle \operatorname{mc}(p^{k+1}),\dotsc,j_{k+1,k+1+n}(\operatorname{mc}(p^{k+1})) \rangle}}}.\end{aligned}$$ Hence $$\begin{gathered}
j_{k+1+n,{\omega}}(p^{k+1+n}) \leq^*
\\
j_{k+1,{\omega}}(p^{k+1})_{{\ensuremath{\langle \operatorname{mc}(p^{k+1}),\dotsc,j_{k+1,k+n}(\operatorname{mc}(p^{k+1})) \rangle}}},\end{gathered}$$ giving us that $$\begin{aligned}
\label{GoverN:generic}
j_{k+1,{\omega}}(p^{k+1})_{{\ensuremath{\langle \operatorname{mc}(p^{k+1}),\dotsc,j_{k+1,k+n}(\operatorname{mc}(p^{k+1})) \rangle}}}
\in G({{\ensuremath{\langle \Vec{N},\Vec{p} \rangle}}}),\end{aligned}$$ so $G({{{\ensuremath{\langle \Vec{N},\Vec{p} \rangle}}}}) {\cap}D {\cap}N \not= \emptyset$ by \[GoverN:dense\] and \[GoverN:generic\].
Concluding Remarks {#ConcludingRemarks}
==================
1. We believe a repeat point is enough in order to get measurability of ${\kappa}$. We use a much stronger assumption in our proof.
2. A definition of repeat point that depends only on the extender sequence and is equivalent to the one we gave (which mentions ${{\ensuremath{P_{{{\ensuremath{\bar{E}}\/}}}\/}}}$) will probably be useful.
3. It is not completly clear what $\operatorname{l}({{\ensuremath{\bar{E}}\/}})$ should be in order to make sure that ${{\ensuremath{\bar{E}}\/}}$ has a repeat point.
4. A finer analysis in the case of measurability and stronger properties is needed. For example, extending the elementary embedding to the generic extension, and not just constructing a normal ultrafilter.
5. We do not know how to get a generic by iteration when $\operatorname{l}({{\ensuremath{\bar{E}}\/}})>1$.
6. Making this forcing more ‘precise’ by adding ‘gentle’ collapses so we get a prescribed behaviour on *all* cardinals below ${\kappa}$ in the generic extension is in preparation.
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M. Magidor, [*Changing Cofinality Of Cardinals*]{}, Fundamenta Mathematicae 99 (1978), 61–71
M. Gitik and M. Magidor, [*The Singular Cardinal Hypothese Revisited*]{}, in [*Set Theory of the Continuum*]{}, H. Judah, W. Just, H. Woodin, (Eds.), Springer-Verlarg (1992), 243–278
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W. Mitchell, [*How Weak is a Closed Unbounded Ultrafilter?*]{}, in [*Logic Colloquium ’80*]{}, D. van Dalen, D. Lascar, J. Smiley, (Eds.), North-Holland Publishing Company (1982), 209–230
K. Prikry, [*Changing measurables into accessibles*]{}, Diss. Math. 68 (1970), 5–52
L. B. Radin, [*Adding Closed Cofinal Sequences To Large Cardinals*]{}, Annals of Mathematical Logic 22(1982), 243–261
M. Segal, [*On Powers Of Singular Cardinals With Cofinality $> {\omega}$*]{}, Master’s Thesis, The Hebrew University of Jerusalem (1995)
S. Shela, [*Proper Forcing*]{}, Lecture Notes in Mathematics, Vol. 940, Springer, Berlin, 1982
H. Woodin and J. Cummings, [*Chapters from an unpublished book on Radin forcing*]{}
|
---
abstract: 'We develop atmosphere models of two of the three [*Kepler*]{}-field planets that were known prior to the start of the [*Kepler*]{} mission (HAT-P-7b and TrES-2). We find that published [*Kepler*]{} and [ *Spitzer*]{} data for HAT-P-7b appear to require an extremely hot upper atmosphere on the dayside, with a strong thermal inversion and little day-night redistribution. The [*Spitzer*]{} data for TrES-2 suggest a mild thermal inversion with moderate day-night redistribution. We examine the effect of nonequilibrium chemistry on TrES-2 model atmospheres and find that methane levels must be adjusted by extreme amounts in order to cause even mild changes in atmospheric structure and emergent spectra. Our best-fit models to the [*Spitzer*]{} data for TrES-2 lead us to predict a low secondary eclipse planet-star flux ratio ($\lsim$$2\times 10^{-5}$) in the [*Kepler*]{} bandpass, which is consistent with what very recent observations have found. Finally, we consider how the [*Kepler*]{}-band optical flux from a hot exoplanet depends on the strength of a possible extra optical absorber in the upper atmosphere. We find that the optical flux is not monotonic in optical opacity, and the non-monotonicity is greater for brighter, hotter stars.'
author:
- 'David S. Spiegel, Adam Burrows'
bibliography:
- 'biblio.bib'
title: 'Atmosphere and Spectral Models of the [*Kepler*]{}-Field Planets HAT-P-7 and TrES-2'
---
Introduction {#sec:intro}
============
Extrasolar planets are being discovered at an increasingly rapid pace: roughly a quarter of the currently known exoplanets (numbering more than 450, as of June, 2010) were found since the beginning of 2009.[^1] Still, only $\sim$80 of the known planets have been seen to transit their parent stars. Transits break the degeneracy between mass and inclination, they allow direct measurement of planetary radii, and they make possible precise measurements of planetary fluxes from secondary eclipse observations. The [*Kepler*]{} mission, which is predicted to find many new transiting planets, is, therefore, particularly exciting.
Three transiting planets in the [*Kepler*]{} field were identified prior to the beginning of the [*Kepler*]{} mission – TrES-2, HAT-P-7b, and HAT-P-11b (occasionally referred to as Kepler-1b, -2b, and -3b). @spiegel_et_al2010a have already published a range of possible atmospheric models of HAT-P-11b; here, we consider HAT-P-7b and and TrES-2.
The InfraRed Array Camera (IRAC) instrument on the [*Spitzer*]{} Space Telescope has been a boon to exoplanetary science, providing observations that are diagnostic of atmospheric temperature and composition for more than a dozen planets. It had four photometric channels, centered at 3.6 $\mu$m, 4.5 $\mu$m, 5.8 $\mu$m, and 8.0 $\mu$m. Recently, @christiansen_et_al2010 and @odonovan_et_al2010 used IRAC to measure infrared fluxes from HAT-P-7b and TrES-2, respectively.
HAT-P-7b, discovered by @Pal_et_al_2008, orbits a large, hot star ($1.84R_\sun$, spectral type F8). It is one of the most highly irradiated known explanets, with a substellar flux of $\sim$$4.8\times
10^9\rm~erg~cm^{-2}~s^{-1}$. Its orbit is significantly misaligned from the stellar spin vector, indicating a possible third body in the system [@winn_et_al2009; @Narita_et_al_2009_2]. It is a particularly interesting object in part because [*Kepler’s*]{} exquisite photometry has allowed measurement of ellipsoidal variations in the star induced by the planet’s tidal field [@welsh_et_al2010]. TrES-2, by contrast, orbits a nearly solar-type star in a nearly grazing orbit [@odonovan_et_al2006; @holman_et_al2007; @sozzetti_et_al2007; @raetz_et_al2009]. @mislis+schmitt2009 find a reduction in transit duration of $\sim$3 minutes since 2006, and attribute this shortening to a change in inclination, although analysis by @rabus_et_al2009 does not corroborate such a large change.
The atmosphere modeling strategy that we employ here differs from several others that have been used. Similar to @barman_et_al2005 and @fortney_et_al2006, we calculate radiative equilibrium, chemical equlibrium models. In contrast, both @madhusudhan+seager2009 and @tinetti_et_al2005 [@tinetti_et_al2007] adjust both chemistry and thermal structure in order to find a best fit to the available data, eschewing equilibrium solutions. This latter method produces chemical and thermal profiles that are not the result of ab initio calculations, but that might reveal non-equilibrium behavior. Yet another approach is taken by @showman_et_al2009 and @burrows_et_al2010, who simulate three dimensional structure and dynamics in planetary atmospheres; so far this more sophisticated approach has not produced better fits to observations than the one-dimensional radiative models described above.
@hubeny_et_al2003 were the first to suggest that an extra optical absorber in a hot exoplanet’s upper atmosphere could lead to a thermal inversion. Observations seem to suggest such inversions [@hubeny_et_al2003; @fortney_et_al2006; @fortney_et_al2008; @burrows_et_al2007c; @burrows_et_al2008b; @richardson_et_al2007; @spiegel_et_al2009b; @madhusudhan+seager2009; @knutson_et_al2008b; @knutson_et_al2010], and the strong optical absorber titanium oxide (TiO) has frequently been suggested as the possible culprit responsible for the inferred inversions. However, in the absence of strong mixing processes, the molecular weight of TiO would tend to make it settle to the bottom of the atmosphere. Furthermore, cold traps on the nightside and below the hot upper atmosphere on the dayside can cause TiO to condense into solid grains, which necessitates even stronger macroscopic mixing to keep TiO aloft in the radiatively important upper atmosphere. Since the photospheres are above the radiative-convective boundaries, they are stably stratified; it is not obvious whether such strong mixing obtains in such a stable region [@spiegel_et_al2009b]. Various authors have tried to estimate the amount of macroscopic mixing and have found that, in some regions of the day-side atmosphere, the mixing might be vigorous enough to maintain TiO at altitude [@showman_et_al2009; @li+goodman2010; @madhusudhan+seager2010], though @youdin+mitchell2010 point out that turbulent diffusivity in excess of $10^7 \rm~cm^2~s^{-1}$ might lead to overinflation of some planets through downward transport of entropy. @zahnle_et_al2009 suggest that sulfur photochemistry provides another avenue for achieving the additional upper-atmosphere optical opacity that is needed to produce hot upper atmospheres and inversions. @knutson_et_al2010 find that the presence of thermal inversions appears to be inversely related to the host stars’ ultraviolet (UV) activity, raising the possibility that strong incident UV destroys molecules that may be responsible for thermal inversions. Here, we remain agnostic on the matter and include, for modeling purposes, an extra source of optical opacity of as-yet unknown origin.
This paper is structured as follows. In §\[sec:methods\], we describe our 1D atmosphere modeling strategy. In §\[sec:H7\] and §\[sec:T2\], we present our models of HAT-P-7b and TrES-2. We point out the effects that scattering and nonequilibrium chemistry could have on our models. In §\[sec:nonmon\], we discuss the non-monotonic relationship between the optical opacity in a planet’s upper atmosphere and the planet’s optical emergent radiation. Finally, in §\[sec:conc\], we conclude, and in an Appendix, we discuss the various methods used in the literature to represent day-night redistribution in this modeling context.
Atmosphere Modeling {#sec:methods}
===================
As in our other recent studies, we use the code [COOLTLUSTY]{} [@hubeny_et_al2003; @sudarsky_et_al2003; @burrows_et_al2006; @burrows_et_al2008b; @spiegel_et_al2009b; @spiegel_et_al2010a], a variant of the code [TLUSTY]{} [@hubeny1988; @hubeny+lanz1995], to calculate radiative equilibrium irradiated atmosphere models. Our atomic and molecular opacities are generally calculated assuming chemical equilibrium with solar elemental abundances [@sharp+burrows2007; @burrows+sharp1999; @burrows_et_al2001; @burrows_et_al2002; @burrows_et_al2005], although we also calculate a few nonequilibrium models, described in §\[sec:T2\]. The irradiating spectra in our models are taken from @kurucz1979 [@kurucz1994; @kurucz2005], interpolated to the effective temperatures and surface gravities of HAT-P-7 and the host star of TrES-2.
In addition to the observationally measured parameters of our models (orbital semimajor axis, planet and stellar radii, planet and stellar surface gravities, stellar effective temperature), and in addition to calculating equilibrium chemistry and radiative transfer, several other physical processes go into our models. These include Rayleigh scattering [@sudarsky_et_al2000; @lopez-morales+seager2007; @burrows_et_al2008; @Rowe_et_al_2008], heat redistribution, and the possible presence of an extra optical absorber that could explain the hot upper atmospheres and thermal inversions that have been inferred from infrared observations of several transiting planets. In particular, there are two key free parameters that we vary: $P_n$ and $\kappa'$.
- $P_n$ quantifies the efficiency of day-to-night heat redistribution, and is equal to the fraction of incident day-side heating that is reradiated from the nightside [@burrows_et_al2006; @burrows_et_al2008b]. In the models in this paper, the redistribution takes place between 0.01 and 0.1 bars.
- $\kappa'$ is an ad hoc extra source of optical absorption opacity in the upper atmosphere, and is motivated by the thermal inversions that have been inferred from the infrared spectra of many exoplanets. $\kappa'$ is similar to the $\kappa_e$ of some of our recent work (e.g., @burrows_et_al2008b), except that, rather than a gray optical absorber, it has the same parabolic dependence on frequency as the corresponding extra absorber in @lopez-morales_et_al2009 and @burrows_et_al2010.
The models whose properties are plotted in Figs. \[fig:H7\]–\[fig:T2n\] are summarized in Table \[ta:models\]. In addition to these, we also calculate several models with a modification to the code that allows an ad hoc extra source of optical scattering opacity, analogous to $\kappa'$, but for scattering instead of absorption.
HAT-P-7 {#sec:H7}
=======
HAT-P-7b is a $1.78M_J$, $1.36R_J$ planet in an approximately circular orbit 0.0377 AU from the 1.47 $M_\sun$, 1.84 $R_\sun$, F6 star HAT-P-7 [@Pal_et_al_2008]. In addition to being in the [*Kepler*]{} field, HAT-P-7b is interesting because Rossiter-McLaughlin measurements indicate that it is in a polar or retrograde orbit [@winn_et_al2009; @Narita_et_al_2009_2]. More relevant to modeling its atmosphere is the fact that, due to its close proximity to a large, relatively hot ($\sim$6350 K) star, HAT-P-7b experiences unusually high stellar irradiation ($\sim$$4.8\times
10^9\rm~erg~cm^{-2}~s^{-1}$ at the substellar point).
As an early confirmation that [*Kepler*]{} was performing well, @borucki_et_al2009 published 10 days of commisioning-phase data on HAT-P-7b. These data reveal a surprisingly large secondary eclipse depth in the [*Kepler*]{} band ($\sim$0.43-0.83 $\mu$m), as the corresponding planet-star flux ratio decreases by $(1.3\pm 0.1)\times
10^{-4}$ when the planet passes behind the star.[^2] @borucki_et_al2009 suggest that such a large contrast ratio could imply that the atmosphere absorbs strongly and has minimal redistribution to the night side (low $P_n$, in our language). They estimate a day-side temperature of $2650\pm 100$ K.
Infrared data for HAT-P-7b became available shortly thereafter, when @christiansen_et_al2010 presented secondary eclipse observations of the planet employing the IRAC instrument on the [ *Spitzer*]{} Space Telescope. They found planet-star flux-ratios of $(9.8\pm 1.7)\times 10^{-4}$, $(15.9\pm 0.22)\times 10^{-4}$, $(24.5\pm 3.1)\times 10^{-4}$, and $(22.5\pm 5.2)\times 10^{-4}$ at the IRAC 3.6-$\mu$m, 4.5-$\mu$m, 5.8-$\mu$m, and 8.0-$\mu$m, channels, respectively.
Although, as @christiansen_et_al2010 argue, HAT-P-7b’s atmosphere is irradiated strongly enough that it is probably too hot for the condensates that might otherwise be expected to contribute significant scattering opacity, we nevertheless tried adding an ad hoc extra scatterer to the upper atmosphere of some models. We find that the optical point can easily be matched by a model with significant optical scattering, but such models drastically underpredict the [ *Spitzer*]{} data. Since the @borucki_et_al2009 speculation of an extremely hot upper atmosphere with low albedo came before the infrared data were published, it might have been a little bit premature. On the other hand, the expectation of low albedo is well motivated by the high stellar irradiation, and seems to be confirmed by the infrared observations.
In Table \[ta:models\], we present five thermochemical equilibrium models of HAT-P-7b, four having extremely hot upper atmospheres (ranging from H2’s $\sim$3040 K to H5’s $\sim$3180 K), and a comparison model without a thermal inversion (H1). These five models span a (small) range of values of both $P_n$ (the degree of day-night redistribution) and $\kappa'$ (the strength of an ad hoc extra absorber, where the values are in cm$^2$ g$^{-1}$). Models with values of $P_n$ larger than 0.1 are not displayed in this table because such models underpredict the optical data.
The five HAT-P-7b models of Table \[ta:models\] are displayed in Fig. \[fig:H7\]. The top-left and top-right panels portray the wavelength-dependent planet-star flux ratios in the optical and the infrared, respectively, and the corresponding data [@borucki_et_al2009; @christiansen_et_al2010] are superimposed. In both panels, planet and star fluxes are integrated over the relevant bandpasses, and the resulting integrated planet-star ratios are displayed as solid filled circles. The bottom panel shows temperature-pressure profiles for these five models. Models H2-H5, with extra optical opacity, have strong thermal inversions in which the upper atmosphere is heated to temperatures $\gsim$1500 K greater than they would be in the absence of the extra absorber (as represented by model H1).[^3] The near- and mid-infrared spectra for inverted models H2-H5 are nearly indistinguishable from one another, and are all reasonable fits to the IRAC data (although the models are $\sim$1-2$\sigma$ higher than the data at 3.6 $\mu$m and 4.5 $\mu$m). Taking into account the optical data (top-left panel), model H5 is the best fit to the available data. HAT-P-7b’s real atmosphere is probably not represented by a 1D radiative equilibrium model, such as model H5, but the available data suggest that significant flux might be absorbed high in the atmosphere, where the radiative timescale is short [@iro_et_al2005; @showman_et_al2008], which would imply that day-side heat would probably be reradiated before much advective redistribution has occurred. The association of high extra opacity with low day-night redistribution in the best-fitting model, therefore, is consistent with what should be expected.
@christiansen_et_al2010 analyzed the thermochemical implications of the [*Kepler*]{}/[*Spitzer*]{} data, as well. Using the method described in @madhusudhan+seager2009 and the four IRAC data points in conjunction with the [*Kepler*]{} data, they find classes of models that fit the five data points optimally. They find that a blackbody temperature of $\sim$3175 K is needed to fit the [ *Kepler*]{} data, significantly higher than the brightness temperature inferred by @borucki_et_al2009; lower brightness temperatures are inferred for the IRAC data. Their best-fit (non-blackbody) models are (by construction) not in local radiative or chemical equilibrium, but all of the best-fitting models contain thermal inversions, as do our models H2–H5. By relaxing the fit-criterion to 1.25-$\sigma$ at each datum and allowing strongly nonequilibrium chemical abundances, they find a noninverted model with significant CH$_4$ abundance, but the high temperatures of the atmosphere favor CO and therefore favor the inverted models, consistent with our analysis.
TrES-2 {#sec:T2}
======
TrES-2 is a $1.20M_J$, $1.22R_J$ planet in nearly circular orbit 0.0356 AU from its 1.06 $M_\sun$, 1.00 $R_\sun$ solar-type (G0V) star [@odonovan_et_al2006]. At its substellar point, TrES-2 experiences an irradiating flux of $\sim$1.1$\times
10^9\rm~erg~cm^{-2}~s^{-1}$, similar to that experienced by HD 209458b.
@odonovan_et_al2010 report IRAC observations of TrES-2, together with nonequilibrium atmosphere models of the planet generated in the manner of @madhusudhan+seager2009. @odonovan_et_al2010 find that the infrared data can be reasonably well fit by a blackbody model, a model with a thermal inversion, and a model without a thermal inversion. They point out that their model without a thermal inversion requires a surprisingly low abundance of CO, given the temperature of the atmosphere ($\sim$1500 K). Therefore, they favor the model with the inversion.
Here, we consider a variety of models of TrES-2. Motivated by the analysis of @odonovan_et_al2010, we examine models both with equilibrium and with nonequilibrium chemistry, all of which are in radiative equilibrium. In Table \[ta:models\], we list five TrES-2 models with opacities defined by equilibrium chemistry at solar abundances (T1–T5) and five that are completely analogous, except with the CO abundance artificially set to 0 (T1n–T5n). In the latter group of models, the carbon that would have been in CO is instead in CH$_4$, and the excess oxygen is instead in H$_2$O. Among both the set of models with and without CO, there is a model that has no extra absorber and no inversion (T1 and T1n, respectively), and four models that do have an extra absorber and thermal inversions. After we had generated these atmosphere models, @croll_et_al2010 and @kipping+bakos2010 published [*Ks*]{}-band ($\sim$2.2 $\mu$m) and [*Kepler*]{}-band observations, respectively, of TrES-2’s secondary eclipse. The models presented herein can, therefore, be thought of as predictions, not “postdictions,” for the recent data. As a result, it was gratifying to see that the new data are consistent with our predictions, as described below.
Figures \[fig:T2\] and \[fig:T2n\] portray properties of the atmosphere models with and without CO, respectively. In these figures, similar to Fig. \[fig:H7\], the top-left panel shows the model planet-star flux ratios in the optical and the top-right panel shows the same in the infrared, while the bottom panel shows the temperature-pressure profiles.
We consider first the equilibrium chemistry models in Fig. \[fig:T2\]. The non-inverted model (T1) badly fails to reproduce the IRAC data at 4.5 $\mu$m and at 8.0 $\mu$m. The models with inversions (T2–T5) have similar temperature-pressure profiles to one another. The main difference among these models is the temperature in the region of redistribution (10$^{-2}$–10$^{-1}$ bars), where the temperature dip ranges between $\sim$200 K (T2; $P_n=0.1$) and $\sim$700 K (T4, T5; $P_n=0.3$). As a consequence of their similarity, these four profiles correspond to very similar flux ratios at 4.5 $\mu$m, 5.8 $\mu$m, and 8.0 $\mu$m. Model T5 ($\kappa'= 0.3\rm~cm^2~g^{-1}$) is the best fit at 3.6 $\mu$m, but the inverted models all overpredict the 5.8 $\mu$m-point by $\sim$2$\sigma$. Models T2–T5 all predict [ *Ks*]{}-band flux consistent with the @croll_et_al2010 observations. All five models predict very low planet-star flux ratios in the optical, in contrast to the models and the [*Kepler*]{} observation of HAT-P-7b (for which $F_p/F_*=1.3\times10^{-4}$; see §\[sec:H7\]). Aside from model T1, which is clearly disfavored by the IRAC data, the other models all predict planet-star flux ratios of $\lsim$$2\times10^{-5}$. With no extra absorber, model T1 still predicts a Kepler-band flux ratio of only $\sim$3$\times 10^{-5}$. Models T2–T5 are all consistent with the @kipping+bakos2010 observation of $(1.1\pm 0.7)\times10^{-5}$ (2-$\sigma$ errors).
The models without CO opacity (T1n–T5n), portrayed in Fig. \[fig:T2n\], are qualitatively quite similar to the equilibrium models in all our diagnostics (optical flux, infrared flux, and thermal profile). However, there are some differences in detail. The deep isothermal layers of all five models are hotter by $\sim$150 K than their equilibrium counterparts, and the upper atmosphere of the inverted models (T2n–T5n) is cooler by a comparable amount, while the upper atmosphere of the non-inverted model (T1n) is still warmer than its equlibrium analog (T1). The magnitude of the temperature dip in the redistribution range is also somewhat muted compared with the equilibrium models. These slight $T$-$P$ profile differences, in conjunction with the altered opacities, result in slightly different optical and infrared spectra. In the infrared, the inverted no-CO models produce lower flux at 5.8 $\mu$m, overpredicting the observed data by less than the equilbrium models. Among no-CO models, the IRAC data are best fit by a model with slightly less extra optical absorption (T4n, $\kappa'=0.2\rm~cm^2~g^{-1}$). These models have optical fluxes that are qualitatively similar to their equilibrium counterparts, all predicting [*Kepler*]{} band flux ratios of $\lsim$$5\times 10^{-5}$. In particular, model T1n predicts slightly higher optical flux than does model T1, but the inverted models T2n–T5n predict slightly lower optical fluxes than do models T2–T5. The predicted [*Ks*]{}-band and Kepler-band fluxes of models T3n–T5n are all consistent with observations of @croll_et_al2010 and @kipping+bakos2010, respectively.
The basic conclusion that we draw from examining models with reduced CO opacity is that even such a drastic reduction of CO abundance as entirely eliminating it does not cause dramatic changes in thermal profiles or spectra. In the analysis of @odonovan_et_al2010, changing the \[CO\]:\[CH$_4$\]:\[CO$_2$\] ratio from $10^{-6}:10^{-6}:0$ (i.e., 1:1:0) to $10^{-4}:5\times 10^{-5}:2\times 10^{-6}$ (i.e., 50:25:1), together with changing the temperature-pressure profile from a non-inverted to an inverted one, results in a modest improvement in the quality of the fit (particularly at 8-$\mu$m). We note that, in addition to having a surprisingly nonequilibrium \[CO\]:\[CH$_4$\] ratio, the @odonovan_et_al2010 noninverted model has a very low total abundance of carbon. Even though their inverted model is substantially sub-solar in carbon (by more than an order of magnitude), reducing the carbon abundance by another two orders of magnitude is nearly tantamount to removing carbon from the opacities entirely. In sum, our analysis shows that, among models with solar abundance of carbon, the presence or absence of CO in the database does not make a large difference in the thermal profiles or in the emergent spectra, and in either case way a hot upper atmosphere and thermal inversion are required in the model in order to come at all close to matching the IRAC data. Individual molecular abundances substantially different from what one would obtain for solar elemental abundances might allow for marginally improved fits to the data, but might not yet be called for by the relatively sparse data available so far.
The upper left panels of both Figs. \[fig:T2\] and \[fig:T2n\] both show that models with extra optical absorbers in the upper atmosphere have [*lower*]{} planet-star flux ratios in the [ *Kepler*]{} band than the models without an extra absorber. We revisit this point in §\[sec:nonmon\]. In light of this generic trend, and since among radiative equilbrium models the IRAC data are better fit by inverted models than by non-inverted models, we predict that [ *Kepler*]{} photometry of TrES-2 will indicate low optical flux from this planet, not more than one or a few times $10^{-5}$ of the stellar flux. If future [*Kepler*]{}-band observations reveal optical flux in excess of this amount, that might be indicative of extra optical scattering opacity that was not included in our models.
Optical Flux vs. Optical Absorber Strength {#sec:nonmon}
==========================================
Here, we point out a puzzle: the upper left panel of Fig. \[fig:H7\] shows a different trend of optical flux vs. $\kappa'$ from the analogous panels of Figs. \[fig:T2\] and \[fig:T2n\]. @lopez-morales+seager2007 noted that inverted models, with their hot upper atmospheres, might be expected to have higher optical flux than non-inverted models. But how do we explain the trend seen for TrES-2 (Figs. \[fig:T2\] and \[fig:T2n\])?
If there is an extra absorber in the optical part of the spectrum (a $\kappa_e$, in the terminology of @burrows_et_al2008, or a $\kappa'$ in the present work), the emergent optical flux is affected by two competing effects. The absorber makes the planet darker at altitude, but can also heat the upper atmosphere. The latter effect makes the upper atmosphere of the planet more emissive.
Figure \[fig:nonmon\] illustrates how the balance of these two effects depends on the opacity ($\kappa'$) of the upper-atmosphere absorber, in both HAT-P-7b and TrES-2. The [*Kepler*]{} bandpass brightness of $P_n=0.0$ models of both planets (HAT-P-7b: green; TrES-2: blue) is plotted as a function of $\kappa'$, for a series of values of $\kappa'$ between 0 and 1.1$\rm~cm^2~g^{-1}$. For both planets, a small amount of extra absorption results in reduced emergent flux in the [*Kepler*]{} band, and, for both planets, large values of $\kappa'$ ($\gsim$0.3 cm$^2$ g$^{-1}$) result in increased optical emission, as emission from the Wien tail of the hot upper atmosphere becomes more prominent in the optical part of the spectrum. There is, therefore, a generic non-monotonic character to the dependence of optical flux on $\kappa'$. The degree of the non-monotonicity, however, is greater for HAT-P-7b models than for TrES-2 models. In the former, the optical flux for $\kappa' \gsim
0.3\rm~cm^2~g^{-1}$ is greater than with no extra absorption, while in the latter, even with $\kappa'=1.1\rm~cm^2~g^{-1}$ the optical flux is just over half of what it is with no extra absorption. The origin of this difference is related to the properties of the irradiation. HAT-P-7 is an exceptionally large ($1.84R_\sun$) star that is fairly hot (F6, 6350 K), whereas TrES-2’s star is a more ordinary solar-type star ($1.003R_\sun$, G0V, 5850 K). As a result, the incident optical irradiation at HAT-P-7b is more than 4 times as great as on TrES-2. This greatly enhanced incident optical irradiation results in a much greater sensitivity to the presence of an extra optical absorber.
Conclusion {#sec:conc}
==========
We have presented atmosphere models of HAT-P-7b and TrES-2, two of the three [*Kepler*]{} field planets that were known prior to the start of the [*Kepler*]{} mission. We find that the combination of the IRAC and [*Kepler*]{} secondary eclipse data for HAT-P-7b, with a [ *Kepler*]{}-bandpass secondary eclipse ratio of $\sim$$1.3\times
10^{-4}$, appear to require an extremely hot upper atmosphere, with an extra optical absorber that creates a strong thermal inversion and with little day-night redistribution. The IRAC data for TrES-2 led us us to expect that TrES-2 has a much lower planet-star flux ratio ($\lsim$$2\times 10^{-5}$) in the [*Kepler*]{} bandpass than does the HAT-P-7 system, and indeed this is what was seen in recently published [*Kepler*]{} data of this object.
Furthermore, we find that there is a non-monotonic relationship between $\kappa'$ and a planet’s day-side emergent optical flux. This non-monotonicity highlights the need for multiwavelength observations in order to better estimate the atmospheric structure.
We thank Ivan Hubeny, Laurent Ibgui, Kevin Heng, and Jason Nordhaus for useful discussions. We also appreciate the careful reading and helpful comments from the referee, Giovanna Tinetti. This study was supported in part by NASA grant NNX07AG80G. We also acknowledge support through JPL/Spitzer Agreements 1328092, 1348668, and 1312647. The authors are pleased to acknowledge that part of this research was performed while in residence at the Kavli Institute for Theoretical Physics, and was supported in part by the National Science Foundation under Grant No. PHY05-51164.
\[sec:app\]
Parameterizing Redistribution in a 1D Model {#ssec:redist .unnumbered}
===========================================
There have been several approaches used in the literature for treating the redistribution of day-side irradiation to the nightside in the context of one-dimensional models. Here, we consider several parameterizations and the relationships among them. Early work, including that of our group prior to @burrows_et_al2006, used only a single parameter (called “$f$”) to describe redistribution. However, there are at least three effects that ought to be included in a description of how an atmosphere redistributes heat: (1) some fraction of incident energy is transported to the nightside via atmospheric motions (we call this fraction “$P_n$”); (2) this redistribution occurs at some depth between the top and the bottom of the atmosphere; and (3) the visible face of the planet (the dayside at secondary eclipse phase) has (in general) an anisotropic distribution of specific intensity in the direction of Earth. Our more recent work, including this paper, employs our attempt, however imperfect, to incorporate these physical effects that 1D models from other groups have not fully included. In particular, when other modeling efforts have implemented schemes for day-night heat transport, they have essentially taken the redistribution to occur before the incoming radiation reaches the top of the atmosphere. In contrast, we specify the range of pressures at which the redistribution occurs; while our numerical choice might be not be accurate in detail, it is a physically motivated, less ad hoc way to parameterize the physics that we know affect emergent spectra.
One of the best-known redistribution parameters is the afforementioned geometrical $f$ factor, which arises from an energy balance relation of the following form: $$\begin{aligned}
\label{eq:f} L_p & = & \frac{\pi}{f} R_p^2 \sigma T_{\rm unif}^4 \, , \\
\label{eq:f2} \mbox{or,} \quad \frac{F_p}{F_*} & = & f \left( \frac{R_p}{a} \right)^2 \, .\end{aligned}$$ In eq. (\[eq:f\]), $L_p \equiv (\pi R_p^2) L_*/ (4 \pi a^2)$ is the total stellar power intercepted by the planet (and, therefore, approximately equals the emergent radiation from the planet), $R_p$ is the planet’s radius, $L_*$ is the stellar luminosity, $a$ is the orbital separation, $\sigma$ is the Stefan-Boltzmann constant, and $T_{\rm unif}$ is the temperature of a uniform-temperature sphere of radius $R_p$ whose flux at Earth near secondary eclipse phase would be the same as that of the planet. In eq. (\[eq:f2\]), $F_p$ is the integrated flux from the planet at Earth,[^4] and $F_*$ is the integrated stellar flux. In eqs. (\[eq:f\]) and (\[eq:f2\]), and in the remainder of this section, we have ignored both scattering in the planet’s atmosphere and its own intrinsic luminosity (from its heat of formation, tidal heating, etc.). The $f$ factor has been used by a number of authors in the last decade (@burrows_et_al2003 [@burrows_et_al2005; @fortney_et_al2005; @lopez-morales+seager2007; @hansen2008]; and referred to as $\alpha$ in @barman_et_al2005).[^5] Perfect redistribution (implying that the planet is itself of uniform temperature) corresponds to $f = 1/4$. Zero redistribution, so that each annulus a given angle away from the substellar point absorbs and reradiates its local irradiation isotropically, corresponds to $f=2/3$.[^6] There is, thus, a “beaming factor” that increases $f$ by a factor of 4/3 over the value (1/2) that it would have if the dayside were of uniform temperature instead of peaked toward the center of the disk in accordance with the irradiating flux. $f$ is typically implemented in 1D atmosphere models as simply a uniform reduction of the incident flux at the top of the atmosphere (algorithmically, though not physically, equivalent to the effect of albedo).
Alternatively, we may characterize the redistribution by removing a fraction $P_n$ of the incident stellar flux on the dayside in a prescribed pressure interval (as described in §\[sec:methods\]), and inserting the energy at a similar (or different) level on the nightside. In such a prescription, $P_n$ plausibly ranges between 0 (corresponding to no redistribution) and 0.5 (corresponding to the nightside radiating as much power as the dayside, via redistributive winds). @burrows_et_al2006, @burrows_et_al2008b, and other work from our group implement this procedure. In order to calculate theoretical secondary eclipse fluxes and spectra, one must have a model of the three-dimensional distribution of temperature in a planet’s atmosphere (e.g., @showman_et_al2008 [@showman_et_al2009; @burrows_et_al2010]). In lieu of performing three-dimensional dynamical calculations, our 1D atmosphere models assume the same beaming factor for day-side emergent radiation, regardless of the total day-side radiance; and they assume uniform temperature on the nightside. In particular, on the dayside, when $P_n$ = 0.5, $f=1/3$ (which is 4/3 of the 1/4 that $f$ would equal for a uniform-temperature dayside, when half the incident flux has been redistributed to the nightside). More generally, the integrated model day-side flux from the planet can be approximately described by eq. (\[eq:f2\]), taking $f$ to be defined by $$f = \frac{2}{3}(1-P_n) \, .
\label{eq:fPn}$$ The descriptions in @cowan_et_al2007 of how “$T_{\rm day}$” and “$T_{\rm night}$” depend on $P_n$ correctly quantify the total emission from the day and night sides of the planet, but they do not include the beaming caused by an anisotropic temperature distribution on the dayside that is peaked at the substellar point; therefore, unlike what is suggested in @cowan_et_al2007, the integrated day-side flux is not described by taking the dayside to be of uniform temperature $T_{\rm day}$.
Yet a third redistribution parameter, $\varepsilon$, is suggested by @cowan+agol2010. $\varepsilon$ ranges between 0 (no redistribution) and 1 (full redistribution). It initially appears to be defined similarly to $P_n$, and is described (in part) by the following relation: $$\label{eq:epsday} L_p = \frac{\pi}{(8 - 5\varepsilon)/12} R_p^2 \sigma \widetilde{T}_{\rm day}^4 \, ,$$ where $\widetilde{T}_{\rm day}$ is the same as $T_{\rm unif}$ of eq. (\[eq:f\]), i.e., a measure of the flux in the direction of Earth. In eq. (\[eq:epsday\]), there is a tilde above the expression for day-side temperature to distinguish it from the corresponding expression in @cowan_et_al2007, which is a measure of the total day-side emission, as opposed to the flux in the direction of Earth. Comparing eqs. (\[eq:f\]) and (\[eq:epsday\]) shows that $\varepsilon$ is a rescaling of $f$: $\varepsilon = (8 - 12
f)/5$. @cowan+agol2010 suggest that their $\varepsilon=0$ limit produces a brighter dayside than does the $P_n=0$ limit, but it does not. Instead, the $P_n=0.5$ limit produces a brighter dayside than does the $\varepsilon=1$ limit (brighter by the beaming factor of 4/3).
Finally, we emphasize that while eqs. (\[eq:f2\]) and (\[eq:fPn\]) describe how the integrated planet flux depends on $P_n$, this parameter does more than simply affect the top-of-atmosphere energy budget. The real benefit of $P_n$ (which is not captured by the above equations) is that, when nonzero, instead of reducing incident irradiation, it removes energy from the dayside (and deposits it on the nightside) at more realistic levels in the atmosphere. This process contributes to a slight thermal inversion by cooling the middle atmosphere, and, therefore, affects the ratio of 3.6-$\mu$m flux to the 4.5-$\mu$m flux in a way that is not reproduced by the standard $f$ parameterizations in which incident irradiation is reduced at the top of the atmopshere.
[l|ccl]{} & & &\
& [$P_n$]{} & [$\kappa'$]{} &\
& & [(cm$^2$ g$^{-1}$)]{} &\
------------------------------------------------------------------------
H1 & 0.10 & 0.0 & equilibrium\
H2 & 0.05 & 0.4 & equilibrium\
H3 & 0.10 & 0.7 & equilibrium\
H4 & 0.10 & 1.1 & equilibrium\
H5 & 0.00 & 1.1 & equilibrium\
& & &\
T1 & 0.2 & 0.0 & equilibrium\
T2 & 0.1 & 0.2 & equilibrium\
T3 & 0.2 & 0.2 & equilibrium\
T4 & 0.3 & 0.2 & equilibrium\
T5 & 0.3 & 0.3 & equilibrium\
& & &\
T1n & 0.2 & 0.0 & no CO\
T2n & 0.1 & 0.2 & no CO\
T3n & 0.2 & 0.2 & no CO\
T4n & 0.3 & 0.2 & no CO\
T5n & 0.3 & 0.3 & no CO\
[^1]: See the catalogs at http://exoplanet.eu and http://www.exoplanets.org.
[^2]: This contrast ratio is similar to the expected dip in flux for an Earth-sized planet passing in front of a Sun-like star, and therefore indicated that [*Kepler*]{} should be capable of performing the mission for which it was designed.
[^3]: The temperature-pressure profiles for models H2-H5 are shown up to extremely low pressures ($\sim$10$^{-8}$ bars), at which point nonequilibrium processes, such as photochemical dissociation and UV opacity, would be important for determining the true thermal profiles.
[^4]: $F_p = \sigma
T_{\rm unif}^4 (R_p/d)^2$, where $d$ is the distance from the planet to Earth.
[^5]: We note that @burrows_et_al2008b also use a parameter called $f$, but this $f$ (which is typically set to 2/3) represents the direction-cosine of incident irradiation that is used in the planar atmosphere calculation, and is therefore related to distribution of temperature on the dayside, instead of to the fraction of energy that is redistributed to the nightside. Some work (e.g. that of @barman_et_al2005) integrates the contributions to the total planet-flux at Earth of a series of concentric annuli, from the substellar point to the terminator. @burrows_et_al2008b show that this integral is extremely well-approximated by taking the entire visible hemisphere as being irradiated by the ray at direction-cosine 2/3.
[^6]: In this case, the specific intensity in the direction of Earth from a annulus at direction-cosine $\mu$ away from the substellar point (at full-moon phase) is proportional to $\mu$. Specifically, $I[\mu] = \mu L_* /
(2 \pi a)^2$.
|
---
abstract: 'We present modeling and design of singly-resonant optical parametric oscillator (SR-OPO) with an intracavity idler absorber to enhance the conversion efficiency for the signal, by suppressing the back conversion of the signal and idler to the pump. Following plane wave analysis, we arrive at the optimum parameters of the OPO to achieve high conversion efficiency for the signal. For a given pump intensity, we have analyzed the effect of position and number of absorbers required for optimum performance of the device. The model is also extended to the case in which the signal is absorbed, yielding higher conversion efficiency for the idler (in mid-IR region). The magnitude of absorption and the effect of inter-crystal phase shift on the conversion efficiency are also discussed. We also present an analytical solution for twin-crystal SR-OPO with an absorber in between; taking into account the variation of signal amplitude inside the cavity, we re-affirm that the often used ‘constant signal-wave approximation’ is valid if the reflectivity of the output coupler is high for the signal.'
address:
- |
Department of Nuclear and Atomic Physics, Tata Institute of Fundamental Research,\
Mumbai, Maharashtra 400005, India.\
[email protected]
- |
Department of Physics, Indian Institute of Technology,\
Hauz Khas, New Delhi 110016, India\
[email protected]
author:
- 'TAJINDER SINGH[^1]'
- 'M. R. SHENOY'
title: 'Modeling and design of singly-resonant optical parametric oscillator with an intracavity idler absorber for enhanced conversion efficiency for the signal'
---
Introduction
============
Optical parametric oscillators (OPOs) are widely developed and used as sources of tunable light in the visible and near IR region. There is a considerable amount of literature available on the modeling, analysis and experimentation of OPOs. Most of the existing models of OPOs dealt with their performance for pump power nearly five to six times the threshold. Beyond this pump power, the conversion efficiency was limited by back conversion (annihilation of signal and idler photons back to pump photons).$^{1-6}$ If the idler wave is absorbed inside the nonlinear crystal, then it would prevent the back conversion$^7$; i.e., due to the absence of idler wave, pump cannot build up by depleting the signal wave. But the nonlinear crystals may not possess large absorption coefficient at the idler wavelength. One of the ways to increase the absorption of idler wave is to dope the nonlinear crystal with absorbing impurities. However, increasing the idler absorption coefficient by doping could affect the nonlinear properties of the crystal. Also, the absorption of idler wave inside the nonlinear crystal causes the thermal loading$^{8,9}$ of the nonlinear crystal and results in a decrease in the conversion efficiency.$^{10-12}$ There are some other OPO configurations to suppress the back conversion, for example, using a two-crystal OPO with four dichroic mirrors.$^{13}$ But such configurations have some limitations, such as longer build-up time due to the use of ring cavity and more signal loss due to the use of four mirrors.$^{14}$
In this paper, we present a plane wave analysis of the twin crystal OPO, which employs a separate (stand-alone) absorber in between the two nonlinear crystals wherein the absorption of idler wave takes place. Use of a separate absorber has some additional benefits over the configuration in which absorption takes place inside the nonlinear crystal. By varying the position of the absorber in our model, we show that the back conversion is completely suppressed if it is placed nearly mid-way between the two crystals. Putting an absorber between the crystals will introduce an additional relative inter-crystal phase shift that can be used to cancel the effect of phase mismatch in the two crystals just like in parametric amplifiers.$^{15,16}$ A brief discussion of three-crystal configuration with two idler absorbers is also presented. Further, instead of using an idler absorber, if we use a signal absorber, then we can enhance the idler conversion efficiency. To the best of our knowledge, there are very few schemes that give high conversion efficiency in mid-IR region. For single crystal configuration, constant signal-wave approximation is valid when the reflectivity of the output mirror is high.$^{17}$ We have presented a similar analysis for the twin-crystal configuration, now with an idler absorber in between the two crystals.
SR-OPO with uniform idler absorption
====================================
Under the slowly varying envelope approximation, the three coupled differential equations,$^{18}$ for optical parametric generation including the idler absorption, are given below: $$\dfrac{dE_\text{p}}{dz}=-i\kappa_\text{p}E_\text{i}E_\text{s}\exp{(i\Delta kz)}
,\label{this}$$ $$\dfrac{dE_\text{s}}{dz}=-i\kappa_\text{s}E_\text{p}E^*_\text{i}\exp{(-i\Delta kz)}
,\label{this}$$ $$\dfrac{dE_\text{i}}{dz}=-i\kappa_\text{i}E_\text{p}E^*_\text{s}\exp{(-i\Delta kz)}-\frac{\alpha_\text{i}}{2}E_\text{i}
,\label{this}$$ where $E_\text{m}$’s are the amplitudes of the three waves (m=p, s, i). $\alpha_\text{i}$ is the idler absorption coefficient and $$\kappa_\text{m}=\frac{\omega_\text{m}d_{\text{eff}}}{n_\text{m}c}
\label{this}$$ are the coupling constants for the three waves; $d_{\text{eff}}$ is the effective nonlinear coefficient of the crystal, $\omega_\text{m}$’s are the angular frequencies and $n_\text{m}$’s are the corresponding refracive indices of the interacting waves and $c$ is the speed of light in vacuum. The phase mismatch term between the pump and the generated waves is given by: $$\Delta k=k_\text{p}-(k_\text{s}+k_\text{i})
.\label{this}$$ $k_\text{m}$’s are the magnitudes of the propagation vectors of the three waves, and the energy conservation requires: $$\omega_\text{p}=\omega_\text{s}+\omega_\text{i}
.\label{this}$$ Schematic of an SR-OPO with idler absorption inside the crystal is shown in Fig. 1. M$_1$ and M$_2$ are dichroic mirrors with high reflectivity for the signal and high transmitivity for the idler and pump. $L$ is the length of the crystal.
For the case of perfect phase matching $(\Delta k=0)$ and constant signal wave intensity in the cavity (which is true if the reflectivity of the output coupler is large,$^{17}$ ($R_\text{s}>0.7$)), if we solve Eqs. (2.1)-(2.3) by standard procedure with the boundary conditions: $$E_\text{p}(z=0)=E_\text{po} \quad \text{and} \quad E_\text{i}(z=0)=0
,\label{this}$$ then we will get expressions for pump and idler amplitudes at any $z$ inside the crystal.
Since the signal is assumed to be in steady state, the number of signal photons lost per second in one round trip will be equal to the number of pump photons reduced per second in single pass through the cavity,$^{19}$ i.e., $$\frac{P_\text{s}}{\hbar \omega_\text{s}}\times \text{Signal loss}=\frac{P_\text{p}(0)-P_\text{p}(L)}{\hbar \omega_\text{p}}
,\label{this}$$ where $\hbar=h/2\pi$, $h$ is the Planck’s constant, $P_\text{m}$ (m=p, s, i) is the power of corresponding waves and the signal loss is defined as$^{19}$: $$\text{Signal loss} = 1-\exp{\left(-{\frac{2 \Gamma_\text{s} n_\text{s} L}{c}}\right)}
.\label{this}$$ $\Gamma_\text{s}$ is the damping constant for the signal wave, and is given by: $$\Gamma_\text{s}=\frac{c}{n_\text{s}}\left( \alpha_\text{s}-\frac{1}{2L} \ln(R_\text{s}) \right)
,\label{this}$$ where $\alpha_\text{s}$ is the absorption coefficient for the signal wave inside the crystal. In the practical cases $\alpha_\text{s}=0$ , and therefore neglecting $\alpha_\text{s}$ , Eqs. (2.7)-(2.10) gives: $$\frac{1}{(\beta L)^2}\left(1-\left| \frac{E^{\text{out}}_\text{p}}{E_\text{po}}\right|^2 \right)=\frac{1}{N_\text{p}}
.\label{this}$$ The conversion efficiency for the signal is given by: $$\eta_\text{s}=(1-R_\text{s})\frac{P_\text{s}}{P_\text{po}} \left( 1-\left| \frac{E^{\text{out}}_\text{p}}{E_\text{po}}\right|^2 \right)
,\label{this}$$ where $$\beta^2=\kappa_\text{p}\kappa_\text{i}|E_\text{s}|^2
.\label{this}$$ $\beta$ is the parametric gain of the OPO, $N_\text{p}$ is the input pump intensity normalized to its threshold value for $\alpha_\text{ai}=0$ , which is given by: $$I^{\text{th}}_\text{p}(\alpha_\text{i}=0)=\frac{\lambda_\text{i} \lambda_\text{s} n_\text{p} n_\text{s} n_\text{i} c \epsilon_\text{o} (1-R_\text{s})}{8\pi^2d^2_{\text{eff}}L^2}
.\label{this}$$ The threshold pump intensity is determined from the fact that near threshold $\beta L \rightarrow 0$. $$N^{\text{th}}_\text{p}=\frac{I^{\text{th}}_\text{p}(\alpha_\text{i}\neq0)}{I^{\text{th}}_\text{p}(\alpha_\text{i}=0)}=\left( \frac{4}{\alpha_\text{i}L} -\frac{8}{\alpha^2_\text{i}L^2} \left( 1-\exp \left( -\frac{\alpha_\text{i}L}{2} \right) \right) \right)^{-1}
,\label{this}$$ where $N^{\text{th}}_\text{p}$ is the normalized threshold pump intensity.
Following a slightly different approach from Lowenthal,$^7$ we have obtained the variation of conversion efficiency for the signal with normalized pump intensity for different values of $\alpha_\text{i}L$ , which is shown in Fig. 2. The numerical results are presented for the wavelengths $\lambda_\text{p}=1.064\mu m$, $\lambda_\text{s}=1.59\mu m$, and $\lambda_\text{i}=3.216\mu m$. When $\alpha_\text{i}L$ is small, then the idler wave is slightly absorbed and keeps contributing to back conversion at high pump intensities. As $\alpha_\text{i}L$ increases, idler wave gets almost completely absorbed and there is no back conversion. Our results with $\alpha_\text{i}=0$ and $\alpha_\text{i}L=1.4$ match exactly with the results of Refs. 20, 21 and 7, respectively. Also, the variation of threshold pump intensity with $\alpha_\text{i}L$ is approximately linear which is consistent with the results of Ref. 22.
The above model accurately predicts the experimental results for weak idler absorption.$^7$ However, the model does not remain valid for the strong idler absorption, as reported in Ref. 10, due to large thermal load in the device.$^9$ Thus, due to thermal (lensing) effects,$^8$ the conversion efficiency decreases. X. Zhang et. al.$^{12}$ had also reported through both numerical simulations and experiments that the absorption of idler wave inside the nonlinear crystal can strongly effect the performance of OPO. Hence, the above model is not practical. Moore and Koch$^{13}$ had suggested a two-crystal OPO with four dichroic mirrors to enable dumping the idler wave that is generated in the first crystal. But, the use of several mirrors makes the device highly unstable and more lossy; and also the proposed ring cavity has some longer optical path, and hence the build-up time is larger as compared to the linear cavity.$^{14}$ To overcome these practical limitations, we propose a linear cavity twin-crystal SR-OPO with an idler absorber or a band stop filter placed in between the two crystals. Due to the use of a localized absorber between the two crystals, OPO will be free from thermal loading and thus the thermal effects does not affect the performance of OPO.
Twin-crystal SR-OPO with an idler absorber
==========================================
Figure 3 shows a schematic of twin-crystal configuration of SR-OPO with an idler absorber placed in between the two crystals. This configuration is more compact and stable as compared to that of Ref. 13. To find the conversion efficiency for the signal we determine the evolution of the three fields inside the crystals and the absorber.
In the above analysis, the focusing effects of the three waves have been neglected. This is valid in practical cases. For example, for wavelengths in the range $\lambda =1-4\mu m$ and beam waist w$_\text{o}=1-3mm$, the angle of diffraction comes out to be few milliradians, and therefore one can assume the three beams to remain collimated and overlapping within the cavity. Further, Guha$^{23}$ had showed that under these approximations, the analytical expressions corresponding to focused beam interactions, reduced to plane wave results.
Following a similar approach, as outlined in Sec. 2, we write coupled equations$^{18}$ for the case of perfect phase matching, and under the constant signal-wave approximation, for the first crystal: $$\dfrac{dE_\text{1p}}{dz}=-i\kappa_\text{p}E_\text{1i}E_\text{1s}
,\label{this}$$ $$\dfrac{dE_\text{1s}}{dz} \approx 0
,\label{this}$$ $$\dfrac{dE_\text{1i}}{dz}=-i\kappa_\text{i}E_\text{1p}E^*_\text{1s}-\frac{\alpha_\text{i}}{2}E_\text{1i}
.\label{this}$$ $E_\text{1m}(0 \leq z \leq L/2)$’s (m=p, s, i) represent the fields inside the first crystal. Using the boundary conditions: $$E_\text{1p}(z=0)=E_\text{po} \quad \text{and} \quad E_\text{1i}(z=0)=0
,\label{this}$$ we can solve Eqs. (3.1)-(3.3) to obtain the expressions for pump and idler amplitudes inside the first crystal. Since, there is no interaction among the three waves inside the absorber, propagation through the absorber would only introduce a phase factor in the three waves and a loss factor for the idler wave. For the second crystal also, we need to solve the same set of coupled equations as given by Eqs. (3.1)-(3.3). If $E_\text{am}(t)$’s (m= p, s, i) are the fields after passing through the absorber, then by using the boundary condition: $$E_\text{2p}(z=0)=E_\text{ap}(t) \quad \text{and} \quad E_\text{2i}(z=0)=E_\text{ai}(t)
,\label{this}$$ we get the following expression for the amplitude of the pump wave at the output of the OPO: $$\begin{aligned}
E^{\text{out}}_\text{p}(z=0) &=& E_\text{po} \exp \left( i(\phi_\text{i}+\phi_\text{s}) \right) \nonumber \\
& & \left[ \exp(i\Delta \phi)\cos^2\left( \frac{\beta L}{2}\right)- \exp \left(-\frac{\alpha_\text{ai}t}{2}\right)\sin^2\left( \frac{\beta L}{2}\right) \right] ,\label{this}\end{aligned}$$ where $\beta$ is the parametric gain of the OPO defined by Eq. (2.13) and $\Delta \phi=\phi_\text{p}-\phi_\text{s}-\phi_\text{i}$ is the relative inter-crystal phase shift of the three waves. $\phi_\text{m}$’s ( m = p, s, i) are the phase accumulated on the three waves in passing through the air gap and the absorber, and $\alpha_\text{ai}$ is the absorption coefficient for the idler wave inside the absorber. For $\Delta \phi=2q\pi$ , where $q$ is an integer, the conversion efficiency is maximum, and the following results correspond to this case. The significance of this term i.e. relative intercrystal phase shift will be discussed in Sec. 7. Above expression for the pump amplitude can be used to find the conversion efficiency for the signal using Eqs. (2.11) and (2.12) as shown in Sec. 2.
Figure 4 shows the variation of conversion efficiency for the signal with normalized pump intensity for different values of $\alpha_\text{ai}t$. It is clear that increase in absorption coefficient of the idler absorber leads to the suppression of back conversion without any thermal loading of the device.
Again we can determine the threshold pump intensity from the fact that near threshold $\beta L \rightarrow 0$. $$N^{\text{th}}_\text{p}=\frac{I^{\text{th}}_\text{p}(\alpha_\text{ai}\neq 0)}{I^{\text{th}}_\text{p}(\alpha_\text{i}=0)}=\frac{2}{1+\exp{\left(-\frac{\alpha_\text{ai}t}{2} \right)}}
.\label{this}$$
 (a)  (b)
Figures 5 shows the variation of idler and pump photon flux relative to the pump photon flux at $z=0$ inside the cavity for $\alpha_\text{ai}t=0$ and $10$ ; $z=0$ corresponds to the input end of the cavity and $z=L+A$ corresponds to the output end of the cavity, where $A$ is the intercrystal gap (including the thickness of the absorber) between the two crystals. Since the back conversion is effective beyond $N_\text{p}\approx2.5$, these variations have been plotted for $N_\text{p}=6$ (say). It is clear from the figures that after absorption of the idler photons inside the absorber (for $\alpha_\text{ai}t=10$), idler wave does not build up sufficiently to induce the back conversion in the second crystal and thus the pump wave keeps on depleting.
In the above we considered the use of an absorber in between the two crystals to dump the idler wave; however, a dichroic mirror can also be used in place of an idler absorber. Figure 6 shows the schematic of twin-crystal SR-OPO with a dichroic mirror M$_3$ sandwiched between the two crystals to dump the idler wave. The analysis will remain same, except $\exp(-\alpha_\text{ai}t)$ will be replaced by $1-R_\text{i}$ , where $R_\text{i}$ is the reflectivity of the dichroic mirror at the idler wavelength.
In practice, tunable OPOs are more interesting to generate idler wave (in the mid-IR) than the signal wave. For this, one can design an idler resonating OPO, and introduce a signal absorber located mid-way between the two crystals. Following the analysis presented above, one can solve the coupled wave equations to find the variation of idler conversion efficiency with the normalized pump intensity. The variation will be similar to Fig. 4, except the scaling of y-axis.
Two-crystal SR-OPO with an idler absorber
=========================================
In the previous section we have taken the two crystals to be of equal length. In this section we will vary the position of the absorber relative to the crystal length and see its effect on the conversion efficiency for the signal. We follow the same procedure as in the previous section to solve the coupled differential equations. If $L_1$ and $L_2$ are the lengths of the first and the second crystal, respectively, with $L=L_1+L_2$, then the amplitude of the pump wave at the output of the second crystal is given by: $$\begin{aligned}
E^\text{out}_\text{p}(z=0) &=& E_\text{po} \exp \left( i(\phi_\text{i}+\phi_\text{s}) \right) \nonumber \\
& & \left[ \cos\left( \beta L_1\right) \exp(i\Delta \phi)\cos\left( \beta L_2\right)- \sin\left( \beta L_1\right) \exp \left(-\frac{\alpha_\text{ai}t}{2}\right)\sin\left( \beta L_2\right) \right]. \nonumber \\
\label{this}\end{aligned}$$
 (a)  (b)
Since near threshold $\beta L \rightarrow 0$ , normalized threshold pump intensity as a function of the position of the absorber is given by: $$N^{\text{th}}_\text{p}=\frac{L^2}{L^2_1+L^2_2}
.\label{this}$$ By considering $\Delta \phi=2q\pi$, the conversion efficiency for the signal can be determined from Eqs. (2.12), (2.13) and (4.1). Figure 7(a) shows the variation of the conversion efficiency for the signal for different positions of the absorber from which we can conclude that the back conversion is almost suppressed for the case of absorber placed nearly in mid-way between the two crystals. Therefore, it is useful to take the two crystals of equal length i.e., $L_1=L_2=L/2$.
Three-crystal SR-OPO with an idler absorber
===========================================
For the twin-crystal configuration of SR-OPO with an idler absorber in between we have seen from Fig. 7(a) that for $N_\text{p}>15$ , conversion efficiency for the signal again starts dropping down due to the rapid growth of idler wave and back conversion to the pump wave inside the first crystal. At such high pump intensities, one can use three-crystal configuration of SR-OPO with strong idler absorbers placed at $L/3$ and $2L/3$ . In order to find the conversion efficiency and normalized threshold, we find the evolution of the three waves inside the cavity by following the same procedure outlined in Sec. 3.
Figure 7(b) shows the variation of conversion efficiency for the signal with normalized pump intensity for different values of $\alpha_\text{ai}t$ . It is clear from figure that three-crystal configuration is useful in suppressing the back conversion for $N_\text{p}>15$ . (cf. Fig. 6). From Fig. 4 and 7(b), one can conclude that the range of interest of $N_\text{p}$ will decide, whether one must use a twin-crystal SR-OPO with an idler absorber, or a three-crystal SR-OPO with two-idler absorbers.
Optimum normalized pump intensity for a given value of $\alpha_\text{ai}t$
==========================================================================
In Fig. 4, one can see that as $\alpha_\text{ai}t$ increases, the pump intensity at which the conversion efficiency for the signal reaches maximum, also increases. Hence there exist an optimum normalized pump intensity $N^{\text{opt}}_\text{p}$ (at which we get maximum conversion efficiency for the signal) for a given $\alpha_\text{ai}t$ and the input pump intensities of interest are always less than the damage threshold of the nonlinear crystals.$^{24}$
Figure 8 shows the variation of $N^{\text{opt}}_\text{p}$ with $\alpha_\text{ai}t$. As $\alpha_\text{ai}t$ increases, the $N^{\text{opt}}_\text{p}$ also increases; and for $\alpha_\text{ai}t>10$, the idler wave generated inside the first crystal is completely absorbed which leads to the saturation of $N^{\text{opt}}_\text{p}$.
Relative inter-crystal phase shift
==================================
In Sec. 3, 4 and 5 while calculating the conversion efficiency for the signal, we have assumed that the relative inter-crystal phase shift, $\Delta \phi=2q\pi$ where $q$ is an integer. In this section we discuss the effect of $\Delta \phi$ on the conversion efficiency for the signal.
Relative inter-crystal phase shift ($\Delta \phi$) is an important parameter that can be used to cancel the effect of phase mismatch in the three interacting waves.$^{15,16}$ For example, if the three waves are not perfectly phase matched in the first crystal and in the second crystal they are perfectly phase matched. So the three waves leave the first crystal with a definite phase relation between them. If the three waves enter in the second crystal with such a relative phase (other than $2q\pi$), then this may also reduce the interaction among the three waves inside the second crystal. However, if we use an inter-crystal spacer that introduces an additional relative phase among the three waves in such a way that the total relative phase (due to finite phase mismatch in the first crystal and due to the spacer) is $2q\pi$, and then there is maximum interaction among the three waves in the second crystal.
Relative inter-crystal phase shift can also be used to cancel the effect of opposite signs of effective nonlinear coefficients in the two crystals.$^{15,16}$ Change in sign of effective nonlinear coefficient implies an additional relative phase of $\pi$ among the three waves. If the two crystals have opposite signs of effective nonlinear coefficients, then by using an appropriate spacer we can introduces an additional relative inter-crystal phase shift of $\pi$ among the three waves. So the total relative phase of three waves again becomes $2q\pi$ that leads to maximum interaction.
If at least one of the three interacting waves is absent at the input of second crystal then there is no effect of relative inter-crystal phase shift on the conversion efficiency for the signal.$^{13}$ Thus, if we introduce an idler absorber as the spacer with $\alpha_\text{ai}t>10$ , then the whole idler wave gets absorbed inside the absorber and at the input of second crystal there is no idler wave. So, the variation of the conversion efficiency for the signal is insensitive to the variation in the relative inter-crystal phase shift.
Figure 9 shows the variation of conversion efficiency for the signal with normalized pump intensity for different values of relative inter-crystal phase shift with $\alpha_\text{ai}t=2.5$, which reflects that for $\Delta \phi=2q\pi$ conversion efficiency for the signal is maximum.
Figure 10 shows the variation of conversion efficiency for the signal with normalized pump intensity for all values of the relative inter-crystal phase shift with $\alpha_\text{ai}t>10$. From which we concludes that the variation of conversion efficiency for the signal is insensitive to the variation of relative inter-crystal phase shift for all $\alpha_\text{ai}t>10$.
Steady state analysis of twin-crystal SR-OPO without constant signal-wave approximation
=======================================================================================
The above discussion for twin-crystal SR-OPO with an idler absorber was based on the constant signal-wave approximation, where the intensity of the signal wave is assumed to be constant inside the cavity. In this section, we present the steady state analysis for twin-crystal SR-OPO with an idler absorber, and have shown that the constant signal-wave approximation holds good if the reflectivity of the output coupler is high $R_\text{s}>0.7$. When the reflectivity of the output coupler is not too high, the variations of the three fields can be described by elliptic functions. Following Bey and Tang$^{17}$, we have obtained the numerical results for the conversion efficiency for the signal with input pump intensity for different transmission coefficients of the output coupler. In this analysis, we have used an idler absorber of infinitesimal thickness and very high idler absorption coefficient, between the two crystals each of length $L/2$. The output coupler is placed at the end of the second crystal.
Assuming perfect phase matching of the three waves inside the two crystals, coupled differential equations governing the nonlinear interaction of three waves in the first crystal are given by: $$\dfrac{dE_\text{1p}}{dz}=-i\kappa_\text{p}E_\text{1i}E_\text{1s}
,\label{this}$$ $$\dfrac{dE_\text{1s}}{dz}=-i\kappa_\text{s}E_\text{1p}E^*_\text{1i}
,\label{this}$$ $$\dfrac{dE_\text{1i}}{dz}=-i\kappa_\text{i}E_\text{1p}E^*_\text{1s}
,\label{this}$$ where $E_\text{1m}$’s (m=p, s, i) are amplitudes of the three waves inside the first crystal. Writing the amplitudes in the following form: $$E_\text{1m}(z)=\xi_\text{1m}(z)\exp{(i\phi_\text{1m}(z))}
,\label{this}$$ and, defining the normalized amplitude of the three waves as: $$u_\text{1m}=\left(\frac{n_\text{m}\epsilon_\text{o}c}{2\omega_\text{m}I}\right)^{1/2}\xi_\text{1m}
,\label{this}$$ where $I=\frac{1}{2}c\epsilon_\text{o}(n_\text{i}\xi^2_\text{1i}+n_\text{s}\xi^2_\text{1s}+n_\text{p}\xi^2_\text{1p})$ is the total intensity of the three waves which is a conserved quantity; $u^2_\text{1m}$’s are proportional to the photon flux of the three waves in the first crystal. The normalized length ($\zeta_1$) is defined as: $$\zeta_1=\frac{d_{\text{eff}}}{c}(n_\text{i}\xi^2_\text{1i}+n_\text{s}\xi^2_\text{1s}+n_\text{p}\xi^2_\text{1p})^{\frac{1}{2}}\left( \frac{\omega_\text{i}\omega_\text{s}\omega_\text{p}}{n_\text{i}n_\text{s}n_\text{p}} \right)^{\frac{1}{2}}z
,\label{this}$$ where $0\leq z\leq L/2$ and $0\leq \zeta_1 \leq l/2$ The real and imaginary parts of Eqs. (8.1)-(8.3) can now be written in the form: $$\dfrac{du_\text{1p}}{d\zeta_1}=-u_\text{1i}u_\text{1s}\sin\theta_1
,\label{this}$$ $$\dfrac{du_\text{1s}}{d\zeta_1}=u_\text{1p}u^*_\text{1i}\sin\theta_1
,\label{this}$$ $$\dfrac{du_\text{1i}}{d\zeta_1}=u_\text{1p}u^*_\text{1s}\sin\theta_1
,\label{this}$$ and $$\dfrac{d\theta_1}{d\zeta_1}=\left( \frac{\kappa_\text{i} \xi_\text{1p} \xi_\text{1s}}{\xi_\text{1i}}+\frac{\kappa_\text{s} \xi_\text{1p} \xi_\text{1i}}{\xi_\text{1s}}+\frac{\kappa_\text{p} \xi_\text{1i} \xi_\text{1s}}{\xi_\text{1p}} \right) \cos\theta_1
,\label{this}$$ respectively, where $\theta_1=\phi_\text{1p}-\phi_\text{1s}-\phi_\text{1i}$. With the boundary condition: $$u_\text{1i}(z=0)=0
.\label{this}$$ Manley-Rowe relations$^{18}$ for the first crystal can be written as: $$u^2_\text{1s}(0)+u^2_\text{1p}(0)=u^2_\text{1s}(\zeta_1)+u^2_\text{1p}(\zeta_1)
,\label{this}$$ $$u^2_\text{1p}(0)=u^2_\text{1p}(\zeta_1)+u^2_\text{1i}(\zeta_1)
,\label{this}$$ $$u^2_\text{1s}(0)=u^2_\text{1s}(\zeta_1)+u^2_\text{1i}(\zeta_1)
.\label{this}$$ Signal photon flux in the first crystal in terms of Jacobian elliptic functions$^{25,26}$ is given by the following expression: $$\begin{aligned}
u^2_\text{1s}(0 \leq \zeta_1 \leq l/2)&=&u^2_\text{1s}(0)+u^2_\text{1p}(0) \nonumber \\
& &\left[ 1-\text{sn}^2 \left[ (u^2_\text{1s}(0)+u^2_\text{1p}(0))^{\frac{1}{2}}|\zeta_1-\zeta_\text{o}| , \left(\frac{u^2_\text{1p}(0)}{u^2_\text{1s}(0)+u^2_\text{1p}(0)}\right)^{\frac{1}{2}} \right] \right] \nonumber \\\end{aligned}$$ Signal photon flux at the output of the first crystal is: $$u^2_\text{1s}\left(\frac{l}{2}\right)=u^2_\text{1s}(0)+u^2_\text{1p}(0)\left[1-A\right]
,\label{this}$$ where $$A=\text{sn}^2 \left[ (u^2_\text{1s}(0)+u^2_\text{1p}(0))^{\frac{1}{2}}\left|\frac{l}{2}-\zeta_\text{o}\right| , \left(\frac{u^2_\text{1p}(0)}{u^2_\text{1s}(0)+u^2_\text{1p}(0)}\right)^{\frac{1}{2}} \right]
.\label{this}$$ The parameter $\zeta_\text{o}$ is determined from the fact that at $\zeta_1=0$, Eq. (8.15) is also valid: $$1=\text{sn}^2 \left[ (u^2_\text{1s}(0)+u^2_\text{1p}(0))^{\frac{1}{2}}\zeta_\text{o} , \left(\frac{u^2_\text{1p}(0)}{u^2_\text{1s}(0)+u^2_\text{1p}(0)}\right)^{\frac{1}{2}} \right]
.\label{this}$$ Using Eqs. (8.12)-(8.15) we obtain the expression for the photon flux of the three waves. To find the evolution of photon flux of the three waves inside the second crystal, we follow the same approach, and solve the coupled differential equations. Due to the depletion of pump wave inside the first crystal and the presence of an idler absorber (of infinitesimal thickness) just after the first crystal that absorbs the all idler photons, boundary conditions become: $$u^2_\text{2p}(0)l'^2=u^2_\text{1p}\left(\frac{l}{2}\right)l^2
,\label{this}$$ $$u^2_\text{2s}(0)l'^2=u^2_\text{1s}\left(\frac{l}{2}\right)l^2
,\label{this}$$ $$u^2_\text{2i}(0)=0
,\label{this}$$ where $u^2_\text{2m}(0)$’s are proportional to the photon flux of the three waves at the input of second crystal and $l'$ is defined through the normalized length ($\zeta_2$) for the second crystal: $$\zeta_2=\frac{d_{\text{eff}}}{c}(n_\text{i}\xi^2_\text{2i}+n_\text{s}\xi^2_\text{2s}+n_\text{p}\xi^2_\text{2p})^{\frac{1}{2}}\left( \frac{\omega_\text{i}\omega_\text{s}\omega_\text{p}}{n_\text{i}n_\text{s}n_\text{p}} \right)^{\frac{1}{2}}z
,\label{this}$$ and $0\leq z\leq L/2$, gives $0\leq \zeta_2 \leq l'/2$. Using the Manley-Rowe relations and by solving the coupled differential equations for the second crystal, we obtain the following expression of the signal photon flux in the second crystal: $$\begin{aligned}
u^2_\text{2s}(0 \leq \zeta_2 \leq l'/2)&=&u^2_\text{2s}(0)+u^2_\text{2p}(0) \nonumber \\
& &\left[ 1-\text{sn}^2 \left[ (u^2_\text{2s}(0)+u^2_\text{2p}(0))^{\frac{1}{2}}|\zeta_2-\zeta'_\text{o}| , \left(\frac{u^2_\text{2p}(0)}{u^2_\text{2s}(0)+u^2_\text{2p}(0)}\right)^{\frac{1}{2}} \right] \right]. \nonumber \\\end{aligned}$$ Signal photon flux at the output of the second crystal is given by: $$\begin{aligned}
u^2_\text{2s}\left(\frac{l'^2}{2l}\right)^2&=&u^2_\text{1s}(0)+u^2_\text{1p}(0) \nonumber \\
& &\left[ 1-A\,\text{sn}^2 \left[ (u^2_\text{1s}(0)+u^2_\text{1p}(0))^{\frac{1}{2}}\left|\frac{l}{2}-\zeta_\text{o}''\right| , \left(\frac{u^2_\text{1p}(0)A}{u^2_\text{1s}(0)+u^2_\text{1p}(0)}\right)^{\frac{1}{2}} \right] \right], \nonumber \\\end{aligned}$$ where $$\begin{aligned}
\zeta'_\text{o}=\frac{l}{l'}\zeta_\text{o}'' , \nonumber\end{aligned}$$ where $\zeta_\text{o}''$ is the parameter, determined from the fact that the Eq. (8.23) must be valid at $\zeta_2=0$ also, that gives: $$1=\text{sn}^2 \left[ (u^2_\text{1s}(0)+u^2_\text{1p}(0))^{\frac{1}{2}}\zeta_\text{o}'' , \left(\frac{u^2_\text{1p}(0)A}{u^2_\text{1s}(0)+u^2_\text{1p}(0)}\right)^{\frac{1}{2}} \right]
.\label{this}$$ Since there is no parametric gain for the propagation of signal wave in the backward direction, for steady state, the signal photon flux at the input of the first crystal and at the output of the second crystal are related through: $$u^2_\text{1s}(0)l^2=R_\text{s}u^2_\text{2s}\left(\frac{l'}{2}\right)l'^2
.\label{this}$$ Thus, Eq. (8.24) becomes: $$\begin{aligned}
u^2_\text{1s}(0)&=&\left(\frac{R_\text{s}}{1-R_\text{s}}\right)u^2_\text{1p}(0) \nonumber \\
& &\left[ 1-A\,\text{sn}^2 \left[ (u^2_\text{1s}(0)+u^2_\text{1p}(0))^{\frac{1}{2}}\left|\frac{l}{2}-\zeta_\text{o}''\right| , \left(\frac{u^2_\text{1p}(0)A}{u^2_\text{1s}(0)+u^2_\text{1p}(0)}\right)^{\frac{1}{2}} \right] \right]. \nonumber \\\end{aligned}$$ Eqs. (8.18), (8.25) and (8.27) contain the whole information which is required to determine the general steady state behavior of the twin-crystal configuration of SR-OPO with an idler absorber. These three equations can be solved numerically to obtain $u^2_\text{1s}(0)/u^2_\text{1p}(0)$, for a given $u_\text{1p}(0)l$, which is related to the input pump intensity by the following expression: $$u_\text{1p}(0)l=\sqrt{\frac{2\omega_\text{i}\omega_\text{s}}{\epsilon_\text{o}c^3n_\text{p}n_\text{s}n_\text{i}}I_\text{p}(0)}d_{\text{eff}}L
.\label{this}$$ For the variation of threshold pump intensity with the transmission coefficient of the output coupler, one can assume the pump to be undepleted and can solve the coupled differential equations to obtain: $$(1-T)\cosh^4\left( \left[ u_\text{1p}(0)l/2 \right]_{\text{threshold}} \right)=1
.\label{this}$$ Now the conversion efficiency for the signal is given by: $$\eta_\text{s}=\left(\frac{1-R_\text{s}}{R_\text{s}}\right)\left(\frac{u^2_\text{1s}(0)}{u^2_\text{1p}(0)}\right)
.\label{this}$$
If $R_\text{s}$ is large, the number of signal photons inside the cavity are much greater than that of the pump photons. In this limit, our result for the pump amplitude at the output of OPO coincides with that obtained in Sec. 2. Figure 11 shows the variation of conversion efficiency for the signal with $u^2_\text{1p}(0)l^2$ for different values of transmission coefficients (T) of the output coupler. As can be seen, the threshold increases with increasing T , and one can also estimate the optimum output coupling for a given pump intensity. The dashed curves are obtained under the constant signal-wave approximation, while the solid curves correspond to the present analysis. It is clear from the figure that the constant signal-wave approximation is valid even for twin-crystal configuration of SR-OPO with an idler absorber, if the reflectivity of the output coupler is high (i.e., for $R_\text{s}$ approximately $0.7$ or more).
Conclusions
===========
We have presented the analysis of twin-crystal SR-OPO with an idler absorber located in between the two crystals. We have presented results for the conversion efficiency for the signal as a function of normalized pump intensity for different values of the absorption coefficient. We conclude that the introduction of an idler absorber between two crystals of equal lengths, leads to significant increase in the conversion efficiency for the signal because of the prevention of back conversion. At higher pump intensities $(N_\text{p}>15)$, one can use three-crystal SR-OPO with two absorbers to enhance the conversion efficiency. By operating the OPO at an optimum pump intensity, for a given $\alpha_\text{ai}t$, one can obtain maximum conversion efficiency. To make the conversion efficiency independent of the relative inter-crystal phase shift $(\Delta \phi)$, we can use a strong idler absorber with $\alpha_\text{ai}t>10$; otherwise, the conversion efficiency will be maximum only when $\Delta \phi$ is an integral multiple of $2\pi$. The steady state analysis of twin-crystal SR-OPO with an idler absorber also shows that the often used constant signal-wave approximation is valid if reflectivity of the output coupler is high. An idler resonating SR-OPO, with a signal absorber, would lead to relatively large idler conversion efficiency, and hence large idler power output. This should be very useful for the generation of high power mid-IR radiation.
Acknowledgments {#acknowledgments .unnumbered}
===============
Authors thank Dr. Sebabrata Mukherjee and Vikram Kamaljith for the helpful discussion.
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[^1]: Corresponding author.
|
---
abstract: 'Given an oriented surface of positive genus with finitely many punctures, we classify the finite orbits of the mapping class group action on the moduli space of semisimple complex special linear rank two representations of the fundamental group of the surface. For surfaces of genus at least two, such orbits correspond to homomorphisms with finite image. For genus one, they correspond to the finite or special dihedral representations. We also obtain an analogous result for bounded orbits in the moduli space.'
address:
- 'School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India'
- 'Department of Mathematics, Indian Institute of Science, Bangalore, India'
- 'School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India'
- 'Department of Mathematics, Massachusetts Institute of Technology, Office: Building 2 Room 2-238a, 77 Massachusetts Avenue Cambridge, MA 02139-4307, USA'
author:
- Indranil Biswas
- Subhojoy Gupta
- Mahan Mj
- Junho Peter Whang
title: 'Surface group representations in ${\rm SL}_2({\mathbb C})$ with finite mapping class orbits'
---
Introduction {#sect:1}
============
Let $\Sigma$ be an oriented surface of genus $g\geq0$ with a finite set ${{\mathcal{F}}}$ of punctures. The *${\rm SL}_2({\mathbb C})$-character variety* of $\Sigma$ $$X(\Sigma)=\operatorname{Hom}(\pi_1(\Sigma), {\rm SL}_2({\mathbb C})){\mathbin{
\mathchoice{/\mkern-6mu/} {/\mkern-6mu/} {/\mkern-5mu/} {/\mkern-5mu/}}}{\rm SL}_2({\mathbb C})$$ is an affine algebraic variety whose complex points parametrize the conjugacy classes of semisimple representations $\pi_1(\Sigma)\to\operatorname{SL}_2({{\mathbb{C}}})$ of the fundamental group of $\Sigma$. Let $\operatorname{Mod}(\Sigma)$ denote the pure mapping class group of $\Sigma$ fixing ${{\mathcal{F}}}$ pointwise. The group acts on the moduli space $X(\Sigma)$ by precomposition. This paper classifies the finite orbits of this action for surfaces of positive genus.
Our analysis divides into the cases of genus one and higher. For surfaces of genus at least two, we prove the following.
\[tha\] Let $\Sigma$ be an oriented surface of genus $g\geq2$ with $n\geq 0$ punctures. A semisimple representation $\rho:\pi_1(\Sigma)\to\operatorname{SL}_2({{\mathbb{C}}})$ has finite mapping class group orbit in the character variety $X(\Sigma)$ if and only if $\rho$ is finite.
To describe the corresponding result for surfaces of genus $1$, define an irreducible representation $\rho:\pi_1(\Sigma)\to\operatorname{SL}_2({{\mathbb{C}}})$ to be *special dihedral* if
- its image lies in the infinite dihedral group $D_\infty$ in $\operatorname{SL}_2({{\mathbb{C}}})$ (its definition is recalled in Section \[sect:2\]), and
- there is a nonseparating simple closed curve $a$ in $\Sigma$ such that the restriction of $\rho$ to the complement $\Sigma\backslash a$ is diagonal.
\[thb\] Let $\Sigma$ be an oriented surface of genus $1$ with $n\geq0$ punctures. A semisimple representation $\rho:\pi_1(\Sigma)\to\operatorname{SL}_2({{\mathbb{C}}})$ has finite mapping class group orbit in $X(\Sigma)$ if and only if $\rho$ is finite or special dihedral up to conjugacy.
Many of the technical issues in the proofs of Theorems A and B above arise while dealing with punctures. For $\Sigma$ closed (i.e. $n=0$) the proofs become considerably simplified thanks to existing results, in particular [@gkm; @px; @coopermanning]. To illustrate this, we have included a short proof of Theorem A in Section \[sect:5\] for closed surfaces of genus at least 2 using [@gkm; @px]. For surfaces of genus zero with more than three punctures, the description of the finite mapping class group orbits in the $\operatorname{SL}_2({{\mathbb{C}}})$-character variety is more complicated and in general unknown. Lisovyy-Tykhyy [@lt] completed the case of the four-punctured sphere as part of their classification of algebraic solutions to Painlevé VI differential equations, and in this case there exist finite mapping class group orbits corresponding to representations with Zariski dense image in $\operatorname{SL}_2({{\mathbb{C}}})$. We remark that the once-punctured torus case of Theorem B was essentially proved by Dubrovin-Mazzocco [@dm] (also in connection with Painlevé VI); the derivation from [@dm] is recorded in the Appendix. Our work gives a different proof in this case.
This paper pursues the theme of characterizing points on the character variety $X(\Sigma)$ with special dynamical properties. In this spirit, we also prove the result below. Given a complex algebraic variety $V$, we shall say that a subset of $V({{\mathbb{C}}})$ is *bounded* if it has compact closure in $V({{\mathbb{C}}})$ with respect to the Euclidean topology.
Let $\Sigma$ be an oriented surface of genus $g\geq1$ with $n\geq0$ punctures. A semisimple representation $\rho:\pi_1(\Sigma)\to\operatorname{SL}_2({{\mathbb{C}}})$ has bounded mapping class group orbit in the character variety $X(\Sigma)$ if and only if
1. $\rho$ is unitary up to conjugacy, or
2. $g=1$ and $\rho$ is special dihedral up to conjugacy.
Our results and methods answer some previously raised basic questions. Theorems A and B imply (Corollary \[faithful\]) that a faithful representation of a positive-genus hyperbolic surface group into $\operatorname{SL}_2({{\mathbb{C}}})$ (or $\operatorname{PSL}_2({{\mathbb{C}}})$) cannot have finite mapping class group orbit in the character variety, answering a question raised by Lubotzky. Theorems A and B also verify the $G=\operatorname{SL}_2({{\mathbb{C}}})$ case of the following conjecture of Kisin [@sinz Chapter 1]: For $\pi$ the fundamental group of a closed surface or a free group of rank $r\geq3$, the points with finite orbits for the action of the outer automorphism group $\operatorname{\textup{Out}}(\pi)$ on the character variety $\operatorname{Hom}(\pi,G){\mathbin{
\mathchoice{/\mkern-6mu/} {/\mkern-6mu/} {/\mkern-5mu/} {/\mkern-5mu/}}}G$, for reductive algebraic groups $G$, correspond to representations $\pi\to
G$ with virtually solvable image. However, there are counter-examples to this conjecture for general $G$ [@KS]. Finally, we show that, given a closed hyperbolic surface $S$ of genus $\geq2$, the energy of the harmonic map associated to a representation $\pi_1(S)\to\operatorname{SL}_2({{\mathbb{C}}})$ is bounded along the mapping class group orbit in the character variety if and only if the representation fixes a point of ${{\mathbb H}}^3$ (Theorem \[energy\]). This answers a question due to Goldman.
For a surface $\Sigma$ of negative Euler characteristic with $n\geq1$ marked punctures, the subvarieties $X_k(\Sigma)$ of $X(\Sigma)$ obtained by fixing the traces $k=(k_1,\cdots,k_n)\in{{\mathbb{C}}}^n$ of local monodromy along the punctures form a family of log Calabi-Yau varieties [@whang] with rich Diophantine structure [@whang2]. Classifying the finite mapping class group orbits (and other invariant subvarieties) forms an important step in the study of strong approximation for these varieties, undertaken in the once-punctured torus case by Bourgain-Gamburd-Sarnak [@bgs].\
[**Organization of the paper.**]{} In Section \[sect:2\], we record background on algebraic subgroups of $\operatorname{SL}_2({{\mathbb{C}}})$, character varieties, and mapping class groups. We also introduce the notion of loop configurations as a tool to keep track of subsurfaces of a given surface. In Section \[sect:3\], we study representations of surface groups whose images are contained in proper algebraic subgroups of $\operatorname{SL}_2({{\mathbb{C}}})$, and give a characterization of those with finite mapping class group orbits for surfaces of positive genus.
In Sections \[sect:4\] and \[sect:5\], we prove our main results. One of the ingredients in the proof of Theorems A and B is a theorem of Patel–Shankar–Whang [@psw Theorem 1.2], which states that a semisimple $\operatorname{SL}_2({{\mathbb{C}}})$-representation of a positive-genus surface group with finite monodromy along every simple loop must in fact be finite. (For the proof of Theorem C which runs in parallel, there is an analogous result.) Along essential curves, the requisite finiteness of monodromy can be largely obtained by studying Dehn twists, as described in Section \[sect:4\]. Finiteness of local monodromy along the punctures is more involved, and is achieved in Section \[sect:5\]. In the case where $\Sigma$ is a closed surface, we also give another proof of Theorem A relying on works of Gallo–Kapovich–Marden [@gkm] and Previte–Xia [@px].
In Section \[sect:6\], we provide applications of our work, answering earlier mentioned questions due to Lubotzky and Goldman. Finally, in the Appendix we demonstrate a derivation of the once-punctured torus case of Theorem B from Dubrovin–Mazzocco [@dm].
Background {#sect:2}
==========
Subgroups of ${\rm SL}_2({{\mathbb{C}}})$ {#sect:2.1}
-----------------------------------------
Let $G$ be a proper algebraic subgroup of $\operatorname{SL}_2({{\mathbb{C}}})$. Up to conjugation, $G$ satisfies one of the following [@vs Theorem 4.29]:
1. $G$ is a subgroup of the *standard Borel group* $$B=\left\{\begin{bmatrix}a & b\\ 0 & a^{-1}\end{bmatrix}\,\mid\, a\in{{\mathbb{C}}}^*,b\in{{\mathbb{C}}}\right\}.$$
2. $G$ is a subgroup of the *infinite dihedral group* $$D_\infty=\left\{\begin{bmatrix}c & 0 \\ 0 & c^{-1}\end{bmatrix}\,\mid\,c\in{{\mathbb{C}}}^*\right\}\cup \left\{\begin{bmatrix}0 & c \\ -c^{-1} & 0\end{bmatrix}\,\mid\,c\in{{\mathbb{C}}}^*\right\}.$$
3. $G$ is one of the finite groups $B A_4$, $B S_4$, and $B A_5$, which are the preimages in $\operatorname{SL}_2({{\mathbb{C}}})$ of the finite subgroups $A_4$ (tetrahedral group), $S_4$ (octahedral group), and $A_5$ (icosahedral group) of $\operatorname{PGL}_2({{\mathbb{C}}})$, respectively.
We refer to the Appendix (after the statement of Theorem A.3) for an explicit description of the finite groups $B A_4$, $B S_4$, and $B A_5$.
A representation $\pi\to\operatorname{SL}_2({{\mathbb{C}}})$ of a group $\pi$ is:
1. *Zariski dense* if its image is Zariski dense in $\operatorname{SL}_2({{\mathbb{C}}})$,
2. *diagonal* if it factors through the inclusion $i:{{\mathbb{C}}}^*\to\operatorname{SL}_2({{\mathbb{C}}})$ of the maximal torus consisting of diagonal matrices,
3. *dihedral* if it factors through the inclusion $j:D_\infty\to\operatorname{SL}_2({{\mathbb{C}}})$,
4. *finite* if its image is finite,
5. *unitarizable* if its image is conjugate to a subgroup of $\operatorname{SU}(2)$,
6. *reducible* if its image preserves a subspace of ${{\mathbb{C}}}^2$,
7. *irreducible* if it is not reducible,
8. *elementary* if it is unitary or reducible or dihedral, and
9. *non-elementary* if it is not elementary.
Character varieties {#sect:2.2}
-------------------
Given a finitely presented group $\pi$, let us define the *$\operatorname{SL}_2$-representation variety* $\operatorname{Rep}(\pi)$ as the complex affine scheme determined by the functor $$A\mapsto\operatorname{Hom}(\pi,\operatorname{SL}_2(A))$$ for every commutative ${{\mathbb{C}}}$-algebra $A$. Given a sequence of generators of $\pi$ with $m$ elements, we have a presentation of $\operatorname{Rep}(\pi)$ as a closed subscheme of $\operatorname{SL}_2^m$ defined by equations coming from relations among the generators. For each $a\in\pi$, let $\operatorname{tr}_a$ be the regular function on $\operatorname{Rep}(\pi)$ given by $\rho\mapsto\operatorname{tr}\rho(a)$. The *character variety* of $\pi$ over ${{\mathbb{C}}}$ is the affine invariant theoretic quotient $$X(\pi)=\operatorname{Rep}(\pi){\mathbin{
\mathchoice{/\mkern-6mu/} {/\mkern-6mu/} {/\mkern-5mu/} {/\mkern-5mu/}}}\operatorname{SL}_2=\operatorname{Spec}{{\mathbb{C}}}[\operatorname{Rep}(\pi)]^{\operatorname{SL}_2({{\mathbb{C}}})}$$ under the conjugation action of $\operatorname{SL}_2$. The complex points of $X(\pi)$ parametrize the isomorphism classes of semisimple representations $\pi\to \operatorname{SL}_2({{\mathbb{C}}})$. For each $a\,\in\,\pi$ the regular function $\operatorname{tr}_{a}$ evidently descends to a regular function on $X(\pi)$. The scheme $X(\pi)$ has a natural model over ${{\mathbb{Z}}}$. We refer to [@horowitz], [@ps], [@saito] for details.
\[exfree\] We refer to Goldman [@goldman2] for details of the examples below. Let $F_m$ denote the free group on $m\geq1$ generators $a_1,\cdots,a_m$.
1. We have $\operatorname{tr}_{a_1}:X(F_1)\simeq{{\mathbb{A}}}^1$.
2. We have $(\operatorname{tr}_{a_1},\operatorname{tr}_{a_2},\operatorname{tr}_{a_1a_2}):X(F_2)\simeq{{\mathbb{A}}}^3$ by Fricke [@goldman2 Section 2.2].
3. The coordinate ring ${{\mathbb{Z}}}[X(F_3)]$ is the quotient of the polynomial ring $${{\mathbb{Z}}}[\operatorname{tr}_{a_1},\operatorname{tr}_{a_2},\operatorname{tr}_{a_3},\operatorname{tr}_{a_1a_2},\operatorname{tr}_{a_2a_3},\operatorname{tr}_{a_1a_3},\operatorname{tr}_{a_1a_2a_3},\operatorname{tr}_{a_1a_3a_2}]$$ by the ideal generated by two elements $$\operatorname{tr}_{a_1a_2a_3}+\operatorname{tr}_{a_1a_3a_2}-(\operatorname{tr}_{a_1a_2}\operatorname{tr}_{a_3}+\operatorname{tr}_{a_1a_3}\operatorname{tr}_{a_2}+\operatorname{tr}_{a_2a_3}\operatorname{tr}_{a_1}-\operatorname{tr}_{a_1}\operatorname{tr}_{a_2}\operatorname{tr}_{a_3})$$ and $$\begin{aligned}
\operatorname{tr}_{a_1a_2a_3}\operatorname{tr}_{a_1a_3a_2}&-\{(\operatorname{tr}_{a_1}^2+\operatorname{tr}_{a_2}^2+\operatorname{tr}_{a_3}^2)+(\operatorname{tr}_{a_1a_2}^2+\operatorname{tr}_{a_2a_3}^2+\operatorname{tr}_{a_1a_3}^2)&\\
&\quad -(\operatorname{tr}_{a_1}\operatorname{tr}_{a_2}\operatorname{tr}_{a_1a_2}+\operatorname{tr}_{a_2}\operatorname{tr}_{a_3}\operatorname{tr}_{a_2a_3}+\operatorname{tr}_{a_1}\operatorname{tr}_{a_3}\operatorname{tr}_{a_1a_3})\\
&\quad +\operatorname{tr}_{a_1a_2}\operatorname{tr}_{a_2a_3}\operatorname{tr}_{a_1a_3}-4\}.\end{aligned}$$ In particular, $\operatorname{tr}_{a_1a_2a_3}$ and $\operatorname{tr}_{a_1a_3a_2}$ are integral over the polynomial subring ${{\mathbb{Z}}}[\operatorname{tr}_{a_1},\operatorname{tr}_{a_2},\operatorname{tr}_{a_3},\operatorname{tr}_{a_1a_2},\operatorname{tr}_{a_2a_3}]$.
We record the following, which is attributed by Goldman, [@goldman2], to Vogt [@vogt].
\[rellem\] Given a finitely generated group $\pi$ and $a_1,a_2,a_3,a_4\in \pi$, the following holds: $$\begin{aligned}
2{\operatorname{tr}_{a_1a_2a_3a_4}}&={\operatorname{tr}_{a_1}}{\operatorname{tr}_{a_2}}{\operatorname{tr}_{a_3}}{\operatorname{tr}_{a_4}}+{\operatorname{tr}_{a_1}}{\operatorname{tr}_{a_2a_3a_4}}+{\operatorname{tr}_{a_2}}{\operatorname{tr}_{a_3a_4a_1}}+{\operatorname{tr}_{a_3}}{\operatorname{tr}_{a_4a_1a_2}}\\
&\quad +{\operatorname{tr}_{a_4}}{\operatorname{tr}_{a_1a_2a_3}}+{\operatorname{tr}_{a_1a_2}}{\operatorname{tr}_{a_3a_4}}+{\operatorname{tr}_{a_4a_1}}{\operatorname{tr}_{a_2a_3}}-{\operatorname{tr}_{a_1a_3}}{\operatorname{tr}_{a_2a_4}}\\
&\quad -{\operatorname{tr}_{a_1}}{\operatorname{tr}_{a_2}}{\operatorname{tr}_{a_3a_4}}-{\operatorname{tr}_{a_3}}{\operatorname{tr}_{a_4}}{\operatorname{tr}_{a_1a_2}}-{\operatorname{tr}_{a_4}}{\operatorname{tr}_{a_1}}{\operatorname{tr}_{a_2a_3}}-{\operatorname{tr}_{a_2}}{\operatorname{tr}_{a_3}}{\operatorname{tr}_{a_4a_1}}.\end{aligned}$$
The above computations imply the following fact.
\[fact\] If $\pi$ is a group generated by $a_1,\cdots,a_m$, then ${{\mathbb{Q}}}[X(\pi)]$ is generated as a ${{\mathbb{Q}}}$-algebra by the collection $\{\operatorname{tr}_{a_{i_1}\cdots a_{i_k}}\,\mid\,1\leq i_1<\cdots<i_k\leq m\}_{1\leq k\leq 3}$.
The construction of $X(\pi)$ is functorial with respect to the group $\pi$. Given a homomorphism $f:\pi\to\pi'$ of finitely presented groups, the corresponding morphism $f^*:X(\pi')\to X(\pi)$ sends a representation $\rho$ to the semisimplification of $\rho\circ f$. In particular, the automorphism group $\operatorname{Aut}(\pi)$ of $\pi$ naturally acts on $X(\pi)$. This action naturally factors through the outer automorphism group $\operatorname{\textup{Out}}(\pi)$, owing to the fact that $\operatorname{tr}_{aba^{-1}}=\operatorname{tr}_{b}$ for every $a,b\in\pi$.
Let $i:{{\mathbb{C}}}^*\to\operatorname{SL}_2({{\mathbb{C}}})$ and $j:D_\infty\to\operatorname{SL}_2({{\mathbb{C}}})$ be the inclusion maps of the diagonal maximal torus and the infinite dihedral group, respectively. They induce $\operatorname{\textup{Out}}(\pi)$-equivariant maps $$\begin{aligned}
\tag{$*$}
i_*:\operatorname{Hom}(\pi,{{\mathbb{C}}}^*)\to X({{\mathbb{C}}})\quad\text{and}\quad j_*:\operatorname{Hom}(\pi,D_\infty)/D_\infty\to X({{\mathbb{C}}}).\end{aligned}$$
\[ffib\] The following two hold.
1. The map $i_*:\operatorname{Hom}(\pi,{{\mathbb{C}}}^*)\to X({{\mathbb{C}}})$ in $(*)$ has finite fibers.
2. The map $j_*:\operatorname{Hom}(\pi,D_\infty)/D_\infty\to X({{\mathbb{C}}})$ in $(*)$ has finite fibers.
\(1) Let $\rho_1,\rho_2:\pi\to{{\mathbb{C}}}^*$ be characters such that $gi_*(\rho_1)g^{-1}=i_*(\rho_2)$ for some $g\in\operatorname{SL}_2({{\mathbb{C}}})$. Without loss of generality, we may assume that $\rho_1(x)\neq\pm1$ for some $x\in\pi$, since otherwise the image of $\rho_1$ is finite and we are done. Writing $g=\left[\begin{smallmatrix}a & b\\ c &d\end{smallmatrix}\right]$ and $\rho_1(x)=\lambda$ with $\lambda\in{{\mathbb{C}}}^*\setminus\{\pm1\}$, we have $$\begin{bmatrix}a &b\\ c &d\end{bmatrix}\begin{bmatrix}\lambda & 0\\0 & \lambda^{-1}\end{bmatrix}\begin{bmatrix}d &-b\\ -c &a\end{bmatrix}=\begin{bmatrix}\lambda ad-\lambda^{-1}bc & (\lambda^{-1}-\lambda)ab\\(\lambda-\lambda^{-1})cd&\lambda^{-1}ad-\lambda bc\end{bmatrix}.$$ For the matrix on the right hand side to be diagonal, we must thus have $a=d=0$ or $b=c=0$ since $\lambda\neq\pm1$. If $a=d=0$, then $\rho_2=\rho_1^{-1}$. If $b=c=0$, then $\rho_1=\rho_2$. This proves (1).
\(2) Let $\rho_1,\rho_2:\pi\to D_\infty$ be representations such that $gj_*(\rho_1)g^{-1}=j_*(\rho_2)$ for some $g\in\operatorname{SL}_2({{\mathbb{C}}})$. Without loss of generality, we may assume that $\operatorname{tr}\rho_1(x)\notin\{0,\pm2\}$ for some $x\in\pi$, since otherwise the image of $\rho_1$ is finite and we are done. Now note that $\rho_1(x)$ must be diagonal. The equation $g\rho_1(x)g^{-1}=\rho_2(x)$ shows that, by the same computation as above, we have $g\in
D_\infty$. This proves (2).
Surfaces {#sect:2.3}
--------
Here, we set our notational convention and terminology for various topological notions. Throughout this paper, a *surface* is the complement of a finite collection of interior points in a compact oriented topological manifold of dimension 2, with or without boundary.
A *simple closed curve* on a surface is an embedded copy of an unoriented circle. We shall often refer to simple closed curves simply as curves (since immersed curves will not be important in this paper). Given a surface $\Sigma$, a curve in $\Sigma$ is *nondegenerate* if it does not bound a disk on $\Sigma$. A curve in the interior of $\Sigma$ is *essential* if it is nondegenerate, does not bound a punctured disk on $\Sigma$, and is not isotopic to a boundary curve of $\Sigma$. Given a surface $\Sigma$ and an essential curve $a\subset\Sigma$, we denote by $\Sigma|a$ the surface obtained by cutting $\Sigma$ along $a$. An essential curve $a\subset\Sigma$ is *separating* if the two boundary curves of $\Sigma|a$ corresponding to $a$ are in different connected components, and *nonseparating* otherwise.
Let $\Sigma$ be a surface of genus $g$ with $n$ punctures or boundary curves. We shall denote by $\operatorname{Mod}(\Sigma)$ the (pure) mapping class group of $\Sigma$. By definition, it is the group of isotopy classes of orientation preserving homeomorphisms of $\Sigma$ fixing the punctures and boundary points individually. Given a simple closed curve $a\subset\Sigma$, we shall denote by $\operatorname{tw}_a\in\operatorname{Mod}(\Sigma)$ the associated (left) Dehn twist.
We define the character variety of $\Sigma$ (cf. Section \[sect:2.2\]) to be $$X(\Sigma)=X(\pi_1(\Sigma)).$$ The complex points of $X(\Sigma)$ can be seen as parametrizing the isomorphism classes of semisimple $\operatorname{SL}_2({{\mathbb{C}}})$-local systems on $\Sigma$. Note that a simple closed curve $a\subset \Sigma$ unambiguously defines a function $\operatorname{tr}_a$ on $X(\Sigma)$, coinciding with $\operatorname{tr}_\alpha$ for any loop $\alpha\in\pi_1(\Sigma)$ freely homotopic to a parametrization of $a$.
Given a continuous map $f:\Sigma'\to\Sigma$ of surfaces, we have an induced morphism of character varieties $f^*:X(\Sigma)\to X(\Sigma')$ depending only on the homotopy class of $f$. In particular, the mapping class group $\operatorname{Mod}(\Sigma)$ acts naturally on $X(\Sigma)$ by precomposition. If $\Sigma'\subset\Sigma$ is a subsurface, the induced morphism on character varieties is $\operatorname{Mod}(\Sigma')$-equivariant for the induced morphism $\operatorname{Mod}(\Sigma')\to\operatorname{Mod}(\Sigma)$ of mapping class groups. In particular, if a semisimple $\operatorname{SL}_2({{\mathbb{C}}})$-representation of $\pi_1(\Sigma)$ has a finite $\operatorname{Mod}(\Sigma)$-orbit in $X(\Sigma)$, then its restriction to any subsurface $\Sigma'\subset\Sigma$ has a finite $\operatorname{Mod}(\Sigma')$-orbit in $X(\Sigma')$.
Loop configurations {#sect:2.4}
-------------------
Let $\Sigma$ be a surface of genus $g$ with $n$ punctures. We fix a base point in $\Sigma$. For convenience, we shall say that a sequence $\ell=(\ell_1,\cdots,\ell_m)$ of based loops on $\Sigma$ is *clean* if each loop is simple and the loops pairwise intersect only at the base point.
\[exgen\] Recall the standard presentation of the fundamental group $$\pi_1(\Sigma)=\langle a_1,d_1,\cdots,a_g,d_g,c_1,\cdots,c_n\,
\mid\,[a_1,d_1]\cdots[a_g,d_g]c_1\cdots c_n\rangle.$$ We can choose (the based loops representing) the generators so that the sequence of loops $(a_1,d_1,\cdots,a_g,d_g,c_1,\cdots,c_n)$ is clean. For $i=1,\cdots,g$, let $b_i$ be the based simple loop parametrizing the curve underlying $d_i$ with the opposite orientation. Note that $(a_1,b_1,\cdots,a_g,b_g,c_1,\cdots,c_n)$ is a clean sequence with the property that any product of distinct elements preserving the cyclic ordering on the sequence, such as $a_1b_g$ or $a_1a_2b_2b_g$ or $b_gc_na_1$, can be represented by a simple loop in $\Sigma$. We shall refer to $(a_1,b_1,\cdots,a_g,b_g,c_1,\cdots,c_n)$ as *an optimal sequence of generators* of $\pi_1(\Sigma)$. See Figure \[fig1\] for an illustration of the optimal generators for $(g,n)=(2,1)$.
![Optimal generators for $(g,n)=(2,1)$[]{data-label="fig1"}](1-holed-genus-2-surface)
A *loop configuration* is a planar graph consisting of a single vertex $v$ and a finite cyclically ordered sequence of directed rays, equipped with a bijection between the set of rays departing from $v$ is and the set of rays arriving at $v$. We denote by $L_{g,n}$ the loop configuration whose sequence of rays is of the form $$(a_1,b_1,\overline{a}_1,\overline{b}_1,\cdots,a_g,b_g,\overline{a}_g,\overline{b}_g,c_1,
\overline{c}_1,\cdots,c_n,\overline{c}_n),$$ where $a_i,b_i,c_i$ are the rays directed away from $v$, respectively corresponding to the rays $\overline{a}_i,\overline{b}_i, \overline{c}_i$ directed towards $v$. See Figure \[fig4\] for an illustration of $L_{2,1}$.
Given a clean sequence $\ell=(\ell_1,\cdots,\ell_m)$ of loops on $\Sigma$, we have an associated loop configuration $L(\ell)$, obtained by taking a sufficiently small open neighborhood of the base point and setting the departing and arriving ends of the loops $\ell_i$ to correspond to each other. For example, if $(a_1,b_1,\cdots,a_g,b_g,c_1,\cdots,c_n)$ is a sequence of optimal generators for $\pi_1(\Sigma)$, then $$L(a_1,b_1,\cdots,a_g,b_g,c_1\cdots,c_n)\simeq L_{g,n}.$$
![Loop configuration $L_{2,1}$[]{data-label="fig4"}](./loop-configuration-2-1)
Let $h$ and $m$ be nonnegative integers. A sequence of based loops $\ell=(\ell_1,\cdots,\ell_{2h+m})$ on $\Sigma$ is said to be *in $(h,m+1)$-position* if it is homotopic term-wise to a clean sequence $\ell'=(\ell_1',\cdots,\ell_{2h+m}')$ such that $L(\ell')\simeq L_{h,m}$. We denote by $\Sigma(\ell)\subset \Sigma$ the (isotopy class of a) subsurface of genus $h$ with $m+1$ boundary curves obtained by taking a small closed tubular neighborhood of the union of the simple curves underlying $\ell_1',\cdots,\ell_{2h+m}'$ in $\Sigma$.
Non-Zariski-dense representations {#sect:3}
=================================
Let $\Sigma$ be a surface of genus $g\geq1$ with $n\geq0$ punctures. The purpose of this section is to characterize representations $\pi_1(\Sigma)$ with non-Zariski-dense image in $\operatorname{SL}_2({{\mathbb{C}}})$ that have finite mapping class group orbit in the character variety $X(\Sigma)$. An irreducible representation $\rho:\pi_1(\Sigma)\to \operatorname{SL}_2({{\mathbb{C}}})$ will be called *special dihedral* if it factors through $D_\infty$ and there is a nonseparating essential curve $a$ in $\Sigma$ such that the restriction $\rho|(\Sigma\setminus a)$ is diagonal. The main result of this section is the following.
\[g1prop\] Let $\Sigma$ be a surface of genus $g\geq1$ with $n\geq0$ punctures. Let $\rho:\pi_1(\Sigma)\to\operatorname{SL}_2({{\mathbb{C}}})$ be a semisimple representation whose image is not Zariski-dense in $\operatorname{SL}_2({{\mathbb{C}}})$. Then $\rho$ has a finite mapping class group orbit in $X(\Sigma)$ if and only if one of the following holds:
1. $\rho$ is a finite representation.
2. $g=1$ and $\rho$ is special dihedral up to conjugation.
Proposition \[g1prop\] is evident when the image of $\rho:\pi_1(\Sigma)\to\operatorname{SL}_2({{\mathbb{C}}})$ belongs to one of the finite groups $BA_4$, $BS_4$, and $BA_5$. From the discussion in Section \[sect:2.1\] and Lemma \[ffib\], it remains to understand the finite mapping class group orbits on $\operatorname{Hom}(\pi_1(\Sigma),{{\mathbb{C}}}^\times)$ and $\operatorname{Hom}(\pi_1(\Sigma),D_\infty)/D_\infty$.
Proposition \[g1prop\] follows by combining Lemmas \[redlemm\], \[dilem2\], and \[dilem\] below.\
[**Case 1: Diagonal Representations.**]{}
\[redlemm\] Assume $\Sigma$ is a surface of genus $g\geq1$ with $n\geq0$ punctures. A representation $\rho:\pi_1(\Sigma)\to{{\mathbb{C}}}^*$ has finite (respectively, bounded) mapping class group orbit in $\operatorname{Hom}(\pi_1(\Sigma),{{\mathbb{C}}}^*)$ if and only if it has finite (respectively, bounded) image.
Let $\rho:\pi_1(\Sigma)\to{{\mathbb{C}}}^*$ be a representation with finite (respectively, bounded) mapping class group orbit in $\operatorname{Hom}(\pi_1(\Sigma),{{\mathbb{C}}}^*)$. Let $(a_1,b_1,\cdots,a_g,b_g,c_1,\cdots,c_n)$ be the optimal generators of $\pi_1(\Sigma)$. Assume first that $n=0$ or $1$. By considering the effect of Dehn twist along the curve underlying $a_1$ on the curve underlying $b_1$, we conclude that $\rho(a_1)$ must be torsion (respectively, have absolute value $1$). Applying the same argument to the other loops in the sequence of optimal generators, we conclude that $\rho$ is finite (respectively, bounded) if $n\leq1$, as desired.
Thus, only the case $n\geq2$ remains. Since $(a_1,b_1,\cdots,a_g,b_g)$ is in $(g,1)$-position, the above analysis shows that $\rho(a_i)$ and $\rho(b_i)$ are roots of unity for $i=1,\cdots,g$. Similarly, we see that the sequence $$L_i=(a_1,b_1,\cdots,a_{g-1},b_{g-1},a_g,b_gc_i)$$ is in $(g,1)$-position for every $i=1,\cdots,n$. Since $\rho$ restricted to the surface $\Sigma(L_i)$ must have finite (respectively, bounded) $\operatorname{Mod}(\Sigma(L_i))$-orbit, it follows that $\rho(c_i)$ is a root of unity (respectively, has absolute value $1$) for $i=1,\cdots,n$ as well. This shows that $\rho$ has finite (respectively, bounded) image, as desired.
[**Case 2: Dihedral Representations.**]{} We first prove Proposition \[g1prop\] for surfaces of genus $g\geq2$ in Lemma \[dilem2\] below.
\[dilem2\] Assume $\Sigma$ is a surface of genus $g\geq2$ with $n\geq0$ punctures. A representation $\rho:\pi_1(\Sigma)\to D_\infty$ has finite mapping class group orbit in $\operatorname{Hom}(\pi_1(\Sigma),D_\infty)/D_\infty$ if and only if it has finite image.
The “*if*” direction is clear, and we now prove the converse. Let $\rho:\pi_1(\Sigma)\to D_\infty$ be a representation with finite mapping class group orbit in $\operatorname{Hom}(\pi_1(\Sigma),D_\infty)/D_\infty$. We have a short exact sequence $0\to{{\mathbb{C}}}^*\to D_\infty\to {{\mathbb{Z}}}/2{{\mathbb{Z}}}\to0$, where the homomorphism $D_\infty\to {{\mathbb{Z}}}/2{{\mathbb{Z}}}$ is given by $$\begin{bmatrix}c & 0\\ 0 & c^{-1}\end{bmatrix}\mapsto 0\quad\text{and}\quad \begin{bmatrix}0 & c\\ -c^{-1} & 0\end{bmatrix}\mapsto 1\quad\text{for all $c\in{{\mathbb{C}}}^*$.}$$ This gives us a $\operatorname{Mod}(\Sigma)$-equivariant map $$\operatorname{Hom}(\pi_1(\Sigma),D_\infty)/D_\infty\to\operatorname{Hom}(\pi_1(\Sigma),{{\mathbb{Z}}}/2{{\mathbb{Z}}}).$$ The fiber of this map above the zero homomorphism consists of those points given by diagonal representations, to which Lemma \[redlemm\] applies, noting that the map $\operatorname{Hom}(\pi_1(\Sigma),{{\mathbb{C}}}^*)\to\operatorname{Hom}(\pi_1(\Sigma),D_\infty)/D_\infty$ has finite fibers. It suffices to consider the case where $\rho:\pi_1(\Sigma)\to D_\infty$ is not in the fiber over the zero homomorphism. Let $$(a_1,b_1,\cdots,a_g,b_g,c_1,\cdots,c_n)$$ be the optimal generators of $\pi_1(\Sigma)$ (see Section 2.4 for the definition). Note that $L=(a_1,b_1,\cdots,a_g,b_g)$ is in $(g,1)$-position, and we have a $\operatorname{Mod}(\Sigma(L))$-equivariant homomorphism $$\operatorname{Hom}(\pi_1(\Sigma),{{\mathbb{Z}}}/2{{\mathbb{Z}}})\to\operatorname{Hom}(\pi_1(\Sigma(L)),{{\mathbb{Z}}}/2{{\mathbb{Z}}}).$$ The action of $\operatorname{Mod}(\Sigma(L))$ on $\operatorname{Hom}(\pi_1(\Sigma(L)),{{\mathbb{Z}}}/2{{\mathbb{Z}}})$ factors through the projection $$\operatorname{Mod}(\Sigma(L))\twoheadrightarrow\operatorname{Sp}(2g,{{\mathbb{Z}}}/2{{\mathbb{Z}}}).$$ From the transitivity of $\operatorname{Sp}(2g,{{\mathbb{Z}}}/2{{\mathbb{Z}}})$ on $({{\mathbb{Z}}}/2{{\mathbb{Z}}})^{2g}$ away from the origin, it follows that, up to $\operatorname{Mod}(\Sigma)$-action, we may assume that $$\overline{\rho}(a_1)=1\quad\text{and}\quad\overline{\rho}(b_1)=\overline{\rho}(a_2)=
\cdots=\overline{\rho}(b_g)=0\, ,$$ where $\overline{\rho}$ is the image of $\rho$ in $\operatorname{Hom}(\pi_1(\Sigma),{{\mathbb{Z}}}/2{{\mathbb{Z}}})$. Up to conjugation by an element in $D_\infty$, we may moreover assume that $$\rho(a_1)=\begin{bmatrix}0 & 1\\ -1 & 0\end{bmatrix}.$$ It suffices to show that the entries of the matrices $\rho(b_1)$, $\rho(a_2)$, $\rho(b_2)$, $\cdots$, $\rho(a_g)$, $\rho(b_g)$, $\rho(c_1)$, $\cdots$, $\rho(c_n)$ are roots of unity. Let $i_1<\cdots<i_q$ be precisely the indices in $\{1,\cdots,n\}$ such that $\overline{\rho}(c_{i_j})=0$. Since $$L'=(a_2,b_2,\cdots,a_g,b_g,c_{i_1},\cdots,c_{i_q},b_1)$$ is in $(g-1,q+2)$-position, and the restriction of $\rho$ to $\Sigma(L')$ is diagonal, we see that $\rho(b_1)$, $\rho(a_2)$, $\rho(b_2)$, $\cdots$, $\rho(a_g)$, $\rho(b_g)$, $\cdots$, $\rho(c_{i_1})$, $\cdots$, $\rho(c_{i_q})$ are torsion by Lemma \[redlemm\]. Let us now take $i\in\{1,\cdots,n\}\setminus\{i_1,\cdots,i_q\}$. The restriction of $\rho$ to the subsurface $\Sigma(c_ia_1,b_1)$ of genus $1$ with $1$ boundary curve is diagonal. Writing $$\rho(c_i)=\begin{bmatrix}0 & \lambda_i\\-\lambda^{-1}&0\end{bmatrix}$$ with $\lambda_i\in{{\mathbb{C}}}^*$, we have $$\rho(c_ia_i)=\begin{bmatrix}0 & \lambda_i\\ -\lambda_i^{-1} & 0\end{bmatrix}\begin{bmatrix}0 & 1\\ -1 & 0\end{bmatrix}=\begin{bmatrix}-\lambda_i&0\\ 0 & -\lambda_i^{-1}\end{bmatrix}.$$ It follows from Lemma \[redlemm\] that $\lambda_i$ is a root of unity. This completes the proof that the entries of $\rho(b_1)$, $\rho(a_2)$, $\rho(b_2)$, $\cdots$, $\rho(a_g)$, $\rho(b_g)$, $\rho(c_1)$, $\cdots$, $\rho(c_n)$ are roots of unity, and hence the image of $\rho$ is finite, as desired.
Before proving the case $g=1$ of Proposition \[g1prop\] in Lemma \[dilem\] below, we record the following elementary lemma.
\[intlem\] Let $\Sigma$ be a surface of genus $1$ with $n\geq0$ punctures. Suppose that $\rho:\pi_1(\Sigma)\to D_\infty$ is special dihedral. Given a pair of loops $(a,b)$ in $(1,1)$-position on $\Sigma$, at least one of $\rho(a)$ and $\rho(b)$ is not diagonal.
Let $\rho\in\pi_1(\Sigma)\to D_\infty$ be special dihedral. We argue by contradiction. Assume that $(a_1,b_1)$ is a pair of loops in $(1,1)$-position on $\Sigma$ with both $\rho(a_1)$ and $\rho(b_1)$ diagonal. We can complete $(a_1,b_1)$ to a sequence of optimal generators $(a_1,b_1,c_1,\cdots,c_n)$ of $\pi_1(\Sigma)$. Since $\rho$ is special dihedral, the matrices $\rho(c_1),\cdots,\rho(c_n)$ must be diagonal (noting that the property of a matrix being diagonal is not changed under conjugation by an element in $D_\infty$). This implies that $\rho$ is in fact diagonal, contradicting the hypothesis that $\rho$ is irreducible.
\[dilem\] Let $\Sigma$ be a surface of genus $1$ with $n\geq0$ punctures. A representation $\rho:\pi_1(\Sigma)\to D_\infty$ has finite mapping class group orbit in $\operatorname{Hom}(\pi_1(\Sigma),D_\infty)/D_\infty$ if and only if it is finite or special dihedral.
The same argument as in Lemma \[dilem2\] shows that if $\rho:\pi_1(\Sigma)\to D_\infty$ has finite $\operatorname{Mod}(\Sigma)$-orbit in $\operatorname{Hom}(\pi_1(\Sigma),D_\infty)/D_\infty$, then $\rho$ is finite or special dihedral. Let now $\rho:\pi_1(\Sigma)\to D_\infty$ be a special dihedral representation. We shall show that $\rho$ has finite mapping class group orbit in $\operatorname{Hom}(\pi_1(\Sigma),D_\infty)/D_\infty$, or equivalently in the character variety $X(\Sigma)$.
The coordinate ring ${{\mathbb{C}}}[X(\Sigma)]$ of the character variety is finitely generated by the trace functions $\operatorname{tr}_{a}$ for a finite collection of essential curves $a$ in $\Sigma$ and the boundary curves, in view of Fact \[fact\] and the property of an optimal sequence of generators. Therefore, it suffices to show that the set $$\{\operatorname{tr}\rho(a)\,\mid\,a\subset\Sigma\text{ essential curve}\}\subseteq{{\mathbb{C}}}$$ is finite.
Let $(a_1,b_1,\cdots,a_g,b_g,c_1,\cdots,c_n)$ be optimal generators of $\pi_1(\Sigma)$. Since $\rho$ is special dihedral, it follows that $\rho(c_1)$, $\cdots$, $\rho(c_n)$ are diagonal matrices. Suppose $a$ is a separating simple closed curve in $\Sigma$ underlying a loop in the free homotopy class of $c_{i_1}\cdots c_{i_k}$ for some integers $i_1<\cdots< i_k$ in $\{1,\cdots,n\}$. Since $\rho(c_i)$ are diagonal for $i=1,\cdots,n$ it follows that $\operatorname{tr}\rho(a)$ lies in a finite set that only depends on the traces $\operatorname{tr}\rho(c_1)$, $\cdots$, $\operatorname{tr}\rho(c_n)$. But now, every separating simple closed curve in $\Sigma$ is sent to one of the above form by some mapping class group element. Since the mapping class group action preserves the special dihedral representations, and since it fixes the traces $\operatorname{tr}\rho(c_1)$, $\cdots$, $\operatorname{tr}\rho(c_n)$, we conclude that $\{\operatorname{tr}\rho(a)\,\mid\,a\subset\Sigma\text{ separating curve}\}$ is finite.
![Building new loops out of old ones[]{data-label="fig7"}](./new-loops)
It remains to show that $\{\operatorname{tr}\rho(a)\,\mid\,a\subset\Sigma\text{ nonseparating curve}\}$ is finite. Let $a_0$ be a nonseparating curve in $\Sigma$ such that the restriction $\rho|(\Sigma\setminus a_0)$ is diagonal; such a curve exists since $\rho$ is special dihedral. Let $b$ be a nonseparating curve in $\Sigma$. Up to isotopy, we may assume that $b$ intersects $a$ only finitely many times. If $b$ does not intersect $a_0$, then $b\subset\Sigma\setminus a_0$ and hence $\operatorname{tr}\rho(b)$ can take only finitely many values as $\rho|(\Sigma\setminus a_0)$ is diagonal.
If $b$ intersects $a$ exactly once, then we must have $\operatorname{tr}\rho(b)=0$ by Lemma \[intlem\]. Let us now assume that $b$ intersects $a_0$ more than once. Let us choose a parametrization of $b$. Since $b$ is nonseparating, there must be two neighboring points of intersection of $a_0$ and $b$ where the two segments of $b$ have the same orientation, as in Figure \[fig7\]. The operations as in Figure \[fig7\] produce for us a new simple closed curve $b'$ which is also nonseparating, as well as a pair $(c,c')$ of simple loops in $(1,1)$-position on $\Sigma$. We have the trace relation $$\operatorname{tr}\rho(b)=\operatorname{tr}\rho(c)\operatorname{tr}\rho(c')-\operatorname{tr}\rho(b').$$ By Lemma \[intlem\], we have $\operatorname{tr}\rho(c)\operatorname{tr}\rho(c')=0$, and therefore $\operatorname{tr}\rho(b)=-\operatorname{tr}\rho(b')$. Note furthermore that $b'$ intersects $a_0$ in a smaller number of points than $b$ does. Applying induction on the number of intersection points, we thus conclude that $$\{\operatorname{tr}\rho(a)\,\mid\,a\subset\Sigma\text{ nonseparating curve}\}$$ is finite. This completes the proof that special dihedral representations have finite mapping class group orbits in $X$.
Analysis of Dehn twists {#sect:4}
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Throughout this section, let $\Sigma$ be a surface of genus $g\geq1$ with $n\geq0$ punctures. For convenience of exposition, we shall denote by $(*)$ the following condition on a representation $\pi_1(\Sigma)\to\operatorname{SL}_2({{\mathbb{C}}})$:
> $(*)$ The representation is semisimple, and moreover $\Sigma$ has genus at least $2$ or the representation is not special dihedral.
Given $\rho:\pi_1(\Sigma)\to\operatorname{SL}_2({{\mathbb{C}}})$ satisfying $(*)$, Theorems A and B (respectively, Theorem C) state that $\rho$ has finite (respectively, bounded) image in $\operatorname{SL}_2({{\mathbb{C}}})$ if its mapping class group orbit in the character variety $X(\Sigma)$ is finite (respectively, bounded). Let us recall the following.
\[psw\] Let $\Sigma$ be a surface of positive genus $g\geq1$ with $n\geq0$ punctures. If a semisimple representation $\pi_1(\Sigma)\to\operatorname{SL}_2({{\mathbb{C}}})$ has finite monodromy along all simple loops on $\Sigma$, then it has finite image.
\[psw2\] Let $\Sigma$ be a surface of positive genus $g\geq1$ with $n\geq0$ punctures. If a semisimple representation $\pi_1(\Sigma)\to\operatorname{SL}_2({{\mathbb{C}}})$ has elliptic or central monodromy along all simple loops on $\Sigma$, then it is unitarizable.
Consequently, to prove Theorems A and B (respectively, Theorem C) it suffices to prove that a representation $\rho:\pi_1(\Sigma)\to\operatorname{SL}_2({{\mathbb{C}}})$ with finite (respectively, bounded) mapping class group orbit in the character variety has finite (respectively, elliptic or central) monodromy along all nondegenerate simple closed curves on $\Sigma$. Let us divide up the curves into four types:
- [Type I.]{} Nonseparating essential curves.
- [Type II.]{} Separating essential curves $a\subset\Sigma$ with each component of $\Sigma|a$ (defined in section 2.3) having genus at least one.
- [Type III.]{} Separating essential curves $a\subset\Sigma$ with one component of $\Sigma|a$ having genus zero.
- [Type IV.]{} Boundary curves.
The purpose of this section is to show that, if $\rho:\pi_1(\Sigma)\to\operatorname{SL}_2({{\mathbb{C}}})$ is a representation satisfying the hypotheses $(*)$ mentioned above with finite (respectively, bounded) mapping class group orbit in $X(\Sigma)$, then $\rho$ must have finite (respectively, elliptic or central) monodromy along all curves of type I and II. We shall also show that $\rho$ has finite (respectively, elliptic or central) monodromy along all curves of type III provided the same holds for all curves of type IV. In particular, this is enough for us to prove the following special cases of Theorems A, B and C.
\[simple-finite\] Let $\Sigma$ be a surface of genus $g\geq1$ with $n\geq1$ punctures. Suppose that $\rho:\pi_1(\Sigma)\to\operatorname{SL}_2({{\mathbb{C}}})$ is a semisimple representation with finite (respectively, elliptic or central) local monodromy around the punctures, and finite (respectively, bounded) mapping class group orbit in the character variety $X(\Sigma)$. Then one of the following holds:
1. $\rho$ is finite (respectively, unitarizable);
2. $g=1$ and $\rho$ is special dihedral up to conjugacy.
The rest of this Section proves Proposition \[simple-finite\] by dealing with curves of Types I, II and III. We shall complete the proof of our main results in Section \[sect:5\] by treating the curves of type IV.\
[**Type I.**]{} Nonseparating essential curves.
\[type1\] Let $\rho:\pi_1(\Sigma)\to\operatorname{SL}_2({{\mathbb{C}}})$ be a representation satisfying $(*)$, and let $a\subset\Sigma$ be a curve of type I. If $\rho$ has finite (respectively, bounded) orbit under $Mod(\Sigma)$ in $X(\Sigma)$, then $\rho$ has finite (respectively, elliptic or central) monodromy along $a$.
Let us choose a base point $x_0$ on $\Sigma$ lying on $a$, and let $\alpha$ be a simple based loop parametrizing $a$. Suppose $\beta$ is another simple loop such that the pair $(\alpha,\beta)$ is in $(1,1)$-position, and let $b$ denote the underlying curve. For each integer $k\in{{\mathbb{Z}}}$, the loop $\alpha^k\beta$ is homotopic to a simple loop whose underlying curve is isotopic to $\operatorname{tw}_a^k(b)$. In particular, by our hypothesis the set $\{\operatorname{tr}\rho(\alpha^k\beta)\,\mid\,k\in{{\mathbb{Z}}}\}$ is a finite (respectively, bounded) subset of ${{\mathbb{C}}}$. Up to global conjugation of $\rho$ by an element of $\operatorname{SL}_2({{\mathbb{C}}})$, we may consider two cases.
\(a) Suppose first that $$\rho(\alpha)=\begin{bmatrix}\lambda & 0 \\ 0 &\lambda^{-1}\end{bmatrix},\quad \lambda\in{{\mathbb{C}}}^\times.$$ Let $\beta$ be a simple loop on $\Sigma$ such that $(\alpha,\beta)$ is in $(1,1)$-position, and write $$\rho(\beta)=\begin{bmatrix} b_1 & b_2\\ b_3 & b_4\end{bmatrix}.$$ For each $k\in{{\mathbb{Z}}}$, we have $$\operatorname{tr}\rho(\alpha^k\beta)=\operatorname{tr}\left(\begin{bmatrix}\lambda^k &0 \\ 0 & \lambda^{-k}\end{bmatrix}\begin{bmatrix} b_1 & b_2\\ b_3 & b_4\end{bmatrix}\right)=\lambda^kb_1+\lambda^{-k}b_4.$$ The fact that $\{\operatorname{tr}\rho(\alpha^k\beta)\,\mid\,k\in{{\mathbb{Z}}}\}$ is a finite (respectively, bounded) subset of ${{\mathbb{C}}}$ then implies that $\lambda$ is a root of unity (respectively, has absolute value $1$) so that $\rho(\alpha)$ is torsion (respectively, elliptic or central), or that $b_1=b_4=0$. Since the argument applies to any loop $\beta$ such that $(\alpha,\beta)$ is in $(1,1)$-position, it remains only to consider the case where $\rho(\beta)$ has both diagonal entries zero for every such $\beta$. In this case, we see upon reflection that the restriction $\rho|(\Sigma|a)$ must be diagonal. This shows that $\rho$ is special dihedral. Since $\rho$ satisfies $(*)$ this means that $\Sigma$ moreover has genus at least $2$, so $\Sigma|a$ has genus at least $1$ and it follows from Lemma \[redlemm\] that $\rho(\alpha)$ is torsion (respectively, elliptic or central).
\(b) Suppose that $$\rho(\alpha)=s\begin{bmatrix}1 & x \\ 0 &1\end{bmatrix}$$ for some $s\in\{\pm1\}$ and $x\in{{\mathbb{C}}}$. Let us assume $s=+1$; the case $s=-1$ will follow similarly. Let $\beta$ be a simple loop on $\Sigma$ such that $(\alpha,\beta)$ is in $(1,1)$-position, and let us follow the notation for $\rho(\beta)$ from the previous case. For each $k\in{{\mathbb{Z}}}$, we have $$\operatorname{tr}\rho(\alpha^k\beta)=\operatorname{tr}\left(\begin{bmatrix}1 &kx \\ 0 & 1\end{bmatrix}\begin{bmatrix} b_1 & b_2\\ b_3 & b_4\end{bmatrix}\right)=b_1+b_4+kxb_3.$$ The fact that $\{\operatorname{tr}\rho(\alpha^k\beta)\,\mid\,k\in{{\mathbb{Z}}}\}$ is a finite (respectively, bounded) subset of ${{\mathbb{C}}}$ then implies that $x=0$ or $b_3=0$. Since the argument applies to any loops $\beta$ such that $(\alpha,\beta)$ is in $(1,1)$-position, it remains only to consider the case where $\rho(\beta)$ is upper triangular for every such $\beta$. In this case, $\rho$ is upper triangular (hence diagonal). It follows from Lemma \[redlemm\] that $\rho(\alpha)$ is torsion (respectively, elliptic or central).
The above arguments show that $\rho(\alpha)$ is torsion (respectively, elliptic or central) unless $g=1$ and $\rho$ is special dihedral, as desired.
[**Types II and III.**]{} Separating essential curves.
\[type2\] Let $\rho:\pi_1(\Sigma)\to\operatorname{SL}_2({{\mathbb{C}}})$ be a representation satisfying $(*)$, and let $a\subset\Sigma$ be a curve of type II. If $\rho$ has finite (respectively, bounded) orbit under $Mod(\Sigma)$ in $X(\Sigma)$, then $\rho$ has finite (respectively, elliptic or central) monodromy along $a$.
\[type3\] Let $\rho:\pi_1(\Sigma)\to\operatorname{SL}_2({{\mathbb{C}}})$ be a representation satisfying $(*)$, and let $a\subset\Sigma$ be a curve of type III. If $\rho$ has finite (respectively, bounded) orbit under $Mod(\Sigma)$ in $X(\Sigma)$, and moreover $\rho$ has finite (respectively, elliptic or central) monodromy along all punctures of $\Sigma$, then $\rho$ has finite (respectively, elliptic or central) monodromy along $a$.
Let us choose a base point $x_0$ on $\Sigma$ lying on $a$, and let $\alpha$ be a simple based loop parametrizing $a$. Let $\Sigma_1$ and $\Sigma_2$ be the connected components of $\Sigma|a$, and lift the base point on $\Sigma$ to $\Sigma_1$ and $\Sigma_2$. Suppose $\beta\in\pi_1(\Sigma_1)$ and $\gamma\in\pi_1(\Sigma_2)$ are simple loops such that their product $\beta\gamma$ on $\Sigma$ is homotopic to a simple loop (with underlying curve denoted $d$, say) transversely intersecting $a$ exactly twice. For convenience, in this paragraph we shall call such a pair $(\beta,\gamma)$ *good*. The loop $\beta\alpha^k\gamma\alpha^{-k}$ for each $k\in{{\mathbb{Z}}}$ is freely homotopic to a simple loop whose underlying curve belongs to the orbit $\langle\operatorname{tw}_a\rangle\cdot d$ (all curves considered up to isotopy). In particular, by our hypothesis the set $\{\operatorname{tr}\rho(\beta\alpha^k\gamma\alpha^{-k})\,\mid\,k\in{{\mathbb{Z}}}\}$ is a finite (respectively, bounded) subset of ${{\mathbb{C}}}$. Up to global conjugation of $\rho$ by an element of $\operatorname{SL}_2({{\mathbb{C}}})$, we may consider two cases.
\(a) Suppose first that $$\rho(\alpha)=\begin{bmatrix}\lambda & 0 \\ 0 &\lambda^{-1}\end{bmatrix},\quad \lambda\in{{\mathbb{C}}}^\times.$$ Let $(\beta,\gamma)$ be a good pair, and let us write $$\rho(\beta)=\begin{bmatrix} b_1 & b_2\\ b_3 & b_4\end{bmatrix}\quad\text{and}\quad \rho(\gamma)=\begin{bmatrix} c_1 & c_2\\ c_3 & c_4\end{bmatrix}.$$ For each $k\in{{\mathbb{Z}}}$, we have $$\begin{aligned}
\operatorname{tr}\rho(\beta\alpha^k\gamma\alpha^{-k})&=\operatorname{tr}\left(\begin{bmatrix}b_1 & b_2\\ b_3 & b_4\end{bmatrix}\begin{bmatrix}\lambda^k &0 \\ 0 & \lambda^{-k}\end{bmatrix}\begin{bmatrix}c_1 & c_2\\ c_3 & c_4\end{bmatrix}\begin{bmatrix}\lambda^{-k} &0 \\ 0 & \lambda^{k}\end{bmatrix}\right)\\
&=b_1c_1+\lambda^{-2k}b_2c_3+\lambda^{2k}c_2b_3+b_4c_4.
\end{aligned}$$ The fact that $\{\operatorname{tr}\rho(\beta\alpha^k\gamma\alpha^{-k})\,\mid\,k\in{{\mathbb{Z}}}\}$ is a finite (respectively, bounded) subset of ${{\mathbb{C}}}$ then implies that $\lambda$ is a root of unity (respectively, has absolute value 1) so that $\rho(\alpha)$ is torsion (respectively, elliptic or central), or that $b_2c_3=c_2b_3=0$. If the former happens, then we are done.
Suppose that the latter happens, and that at least one of $b_2$, $b_3$, $c_2$, $c_3$ is nonzero. We shall assume $b_2\neq0$; the other cases will follow similarly. As $b_2c_3=0$, we must have $c_3=0$. Applying the same argument with $\gamma$ replaced by any simple loop $\gamma'\in\Sigma_2$ such that $(\beta,\gamma')$ is good, we are reduced to the case where $\rho|\Sigma_2$ is upper triangular. If $\rho|\Sigma_1$ is also upper triangular, then $\rho$ must be upper triangular (whence diagonal), and from Lemma \[redlemm\] $\rho(\alpha)$ is torsion (respectively, elliptic or central). So suppose there is a simple loop $\beta'\in\pi_1(\Sigma)$ such that $\rho(\beta')$ is not upper triangular. By repeating the above argument with $\beta$ replaced by $\beta'$, we are reduced to the case where $\rho|\Sigma_2$ must be diagonal. If $\Sigma_2$ has genus at least $1$, then from Lemma \[redlemm\] it follows that $\rho(\alpha)$ is torsion (respectively, is elliptic or central). If $\Sigma_2$ has genus $0$, then the fact that $\rho$ has finite (respectively, elliptic or central) local monodromy along the punctures implies that $\rho(\alpha)$ has finite (respectively, elliptic or central) monodromy.
It remains to consider the case where $b_2=b_3=c_2=c_3=0$. Running through all the good pairs $(\beta,\gamma)$ and repeating the above argument, we are left with the case where $\rho$ is diagonal; Lemma \[redlemm\] then implies that $\rho(\alpha)$ is torsion (respectively, elliptic or central).
The second case:
\(b) Suppose that $$\rho(\alpha)=s\begin{bmatrix}1 & x \\ 0 &1\end{bmatrix}$$ for some $s\in\{\pm1\}$ and $x\in{{\mathbb{C}}}$. We assume $s=+1$; the case $s=-1$ will follow similarly. Let $(\beta,\gamma)$ be a good pair, and let us follow the notation for $\rho(\beta)$ and $\rho(\gamma)$ from the previous case. For each $k\in{{\mathbb{Z}}}$, we have $$\begin{aligned}
&\operatorname{tr}\rho(\beta\alpha^k\gamma\alpha^{-k})\\
&=\operatorname{tr}\left(\begin{bmatrix}b_1 & b_2\\ b_3 & b_4\end{bmatrix}\begin{bmatrix}1 &kx \\ 0 & 1\end{bmatrix}\begin{bmatrix}c_1 & c_2\\ c_3 & c_4\end{bmatrix}\begin{bmatrix}1 &-kx \\ 0 & 1\end{bmatrix}\right)\\
&=b_1c_1+b_2c_3+b_2c_3+b_4c_4+(b_1c_3-b_3c_1-b_4c_3+b_3c_4)kx-b_3c_3k^2x^2.
\end{aligned}$$ The fact that $\{\operatorname{tr}\rho(\beta\alpha^k\gamma\alpha^{-k})\,\mid\,k\in{{\mathbb{Z}}}\}$ is a finite (respectively, bounded) subset of ${{\mathbb{C}}}$ then implies that $x=0$ or $b_3c_3x^2=((b_1-b_4)c_3-b_3(c_1-c_4))x=0$. If the former happens, then we are done.
Suppose now that $x\neq0$ and that the latter happens, and that $b_3\neq0$ or $c_3\neq0$. We shall consider the case $b_3\neq0$; the other case will follow similarly. Note then we must then have $c_3=0$ and $c_1=c_4$. Applying the same argument with $\gamma$ replaced by any simple loop $\gamma'\in\Sigma_2$ such that $(\beta,\gamma')$ is good, we conclude that $\rho|\Sigma_2$ is upper triangular with image consisting of parabolic elements in $\operatorname{SL}_2({{\mathbb{C}}})$. If $\Sigma_2$ has genus at least one, then by considering $\operatorname{tr}\rho(\beta\gamma')$ for good pairs $(\beta,\gamma')$ with $\gamma'$ nonseparating we see that in fact $\rho|\Sigma_2$ must have image in $\{\pm\mathbf 1\}$. If $\Sigma_2$ has genus zero, then by our hypothesis on $\rho$ we again conclude that $\rho|\Sigma_2$ must have image in $\{\pm\mathbf 1\}$. *A fortiori*, $\rho(\alpha)$ is central in both cases, which is a contradiction.
It only remains to consider the case where $b_3=c_3=0$. Running through all the possible good pairs $(\beta,\gamma)$ and repeating the above argument, we are left with the case where $\rho$ is upper triangular (whence diagonal); then $\rho(\alpha)$ is torsion (respectively, elliptic or central). This completes the proof.
Proof of the main results {#sect:5}
=========================
The goal of this section is to complete the proof of Theorems A, B, and C. This section is organized as follows. In Section \[sec-irred\] we record an irreducibility criterion for $\operatorname{SL}_2({{\mathbb{C}}})$-representations of positive genus surface groups, to be used in subsequent parts of this section. In Section \[sec-g1n2\], we establish Theorem C for surfaces of genus one with at most two punctures. In Sections \[sec:5.3\], \[sec:5.4\], and \[sec:5.5\] we complete the proofs of our main theorems. Finally, in Section \[sec:5.6\] we give an alternative proof of Theorem A for closed surfaces using [@gkm; @px].
An irreducibility criterion {#sec-irred}
---------------------------
\[loops\] Let $(a,b,c)$ be a sequence of loops on a surface in $(1,2)$-position.
1. The following pairs are in $(1,1)$-position: $$(a,b),(a,bc),(ca,b),(ab,bc),(ca,cb),(ac,bc),(ca,ab).$$
2. The triple $(c^{-1}b^{-1}a,b,c)$ is in $(1,2)$-position.
This is seen by drawing the corresponding loop configurations of homotopic clean sequences. See Figure \[fig5\], noting that any segments not passing the central base point are not to be considered as part of each loop configuration.
![Loop configurations for Lemma \[loops\][]{data-label="fig5"}](./loop-configurations-1-1)
Let $\Sigma$ be a surface of genus $g\geq1$ with $n\geq0$ punctures. Given a pair $(a,b)$ of based loops in $(1,1)$-position on $\Sigma$, there is an embedding $\Sigma(a,b)\subset\Sigma$ of a surface of genus $1$ with $1$ boundary curve, i.e., a one-holed torus. Up to isotopy, every embedding of a one-holed torus is of the form $\Sigma(a,b)$ for some choice of $(a,b)$. The notion of loop configuration facilitates the proof of the following result, which will be used in the proof of our main theorems but may be of independent interest. (See [@coopermanning] for a proof when $n=0$.)
\[red\] Let $\Sigma$ be a surface of genus $g\geq1$ with $n\geq0$ punctures. A representation $\rho:\pi_1(\Sigma)\to\operatorname{SL}_2({{\mathbb{C}}})$ is irreducible if and only if there is a one-holed torus subsurface $\Sigma'\subset\Sigma$ such that the restriction $\rho|\Sigma'$ is irreducible.
The *if* direction is clear. To prove the converse, let us begin by fixing a representation $\rho:\pi_1(S)\to\operatorname{SL}_2({{\mathbb{C}}})$ whose restriction to every one-holed torus subsurface is reducible. In this proof, given $a\in\pi_1(\Sigma)$ we shall also denote by $a$ the matrix $\rho(a)\in\operatorname{SL}_2({{\mathbb{C}}})$ for simplicity. The statement that the restriction $\rho|\Sigma(a,b)$ is reducible for an embedding $\Sigma(a,b)\subset \Sigma$ associated to a pair $(a,b)$ of loops in $(1,1)$-position is equivalent to saying that the pair $(\rho(a),\rho(b))$ of matrices in $\operatorname{SL}_2({{\mathbb{C}}})$ has a common eigenvector in ${{\mathbb{C}}}^2$.
Throughout, we shall be using the following observation: if $a\in\operatorname{SL}_2({{\mathbb{C}}})\setminus\{\pm\mathbf 1\}$, and if $x,y,z\in{{\mathbb{C}}}^2$ are eigenvectors of $a$, then at least two of them are proportional; in notation, $x\sim y$, $x\sim z$, or $y\sim z$. First, we prove the following claim.
Any triple $(a,b,c)$ of loops on $\Sigma$ in $(1,2)$-position has a common eigenvector under the representation $\rho$.
Let $(a,b,c)$ be in $(1,2)$-position. Each of the following pairs is in $(1,1)$-position by part (1) of Lemma \[loops\], and has a common eigenvector by our hypothesis on $\rho$: $$(a,b),(a,bc),(ca,b),(ab,bc),(ca,cb),(ac,bc),(ca,ab).$$ If $ab=\pm\mathbf 1$, then since $(a,bc)$ has a common eigenvector we find that $(a,b,c)$ has a common eigenvector, as desired. Henceforth, assume that $ab\neq\pm\mathbf 1$. Now $(a,b,ab)$ has a common eigenvector, say $x$; $(ab,bc)$ has a common eigenvector, say $y$; and $(ca,ab)$ has a common eigenvector, say $z$. Since $ab\neq\pm\mathbf 1$, we must have $$x\sim y,\quad x\sim z, \quad\text{or}\quad y\sim z$$ where the relation $\sim$ indicates that the two vectors are proportional. If the first case occurs, then $(a,b,bc)$ has a common eigenvector, and hence $(a,b,c)$ does, as required. If the second case occurs, then $(a,b,ca)$ has a common eigenvector, and hence $(a,b,c)$ does, again as required. Henceforth, assume that the third case occurs, so that $(bc,ca)$ has a common eigenvector.
Now, suppose first that $\operatorname{tr}(a)=\pm2$. Up to conjugation, we have the following possibilities.
1. We have $a=\pm\mathbf 1$. Since $(b,c)$ has a common eigenvector, this implies that $(a,b,c)$ has a common eigenvector, as desired.
2. We have $$a=\pm\begin{bmatrix}1 & 1\\ 0 & 1\end{bmatrix}.$$ Since $(a,bc)$ and $(a,b)$ each have a common eigenvector, $bc$ and $b$ must be upper triangular, and hence $c$ is upper triangular. Thus $(a,b,c)$ has a common eigenvector.
It remains to treat the case $\operatorname{tr}(a)\neq\pm2$. As $(a,bc)$ has a common eigenvector, by Lemma \[tracelem\] we have $$\operatorname{tr}(a)^2+\operatorname{tr}(bc)^2+\operatorname{tr}(abc)^2-\operatorname{tr}(a)\operatorname{tr}(bc)\operatorname{tr}(abc)-2=2$$ and this implies that we cannot have $\operatorname{tr}(bc)=\operatorname{tr}(abc)=0$. After conjugation of $\rho$ by an element of $\operatorname{SL}_2({{\mathbb{C}}})$, we have one of the following three cases.
1. We have $bc=\pm\mathbf 1$. This implies that $b=\pm c^{-1}$. Since $(a,b)$ has a common eigenvector, we are done.
2. We have $$bc=\pm\begin{bmatrix}1 & 1\\ 0 & 1\end{bmatrix}.$$ Since $(a,bc)$ and $(bc,ca)$ each have a common eigenvector, $a$ and $ca$ are upper triangular, and hence $c$ is upper triangular. This in turn implies that $b$ is upper triangular, and we are done.
3. We have $$bc=\begin{bmatrix}\lambda & 0\\ 0 & \lambda^{-1}\end{bmatrix},\quad \lambda\in{{\mathbb{C}}}^*\setminus\{\pm1\}.$$ Since $(a,bc)$, $(bc,ca)$, and $(ab,bc)$ each have a common eigenvector, at least two of $a,ab,ca$ must be simultaneously upper or lower triangular. Let us assume that $a$ is upper triangular; the case where $a$ is lower triangular is dealt with similarly. If $ab$ is also upper triangular, this implies that $b$ is also upper triangular, in turn $c$ is upper triangular, and we are done. Hence, we may assume $ab$ is lower triangular. Similarly, we may assume that $ca$ is lower triangular. Since we have $$(ab)(ca)=a(bc)a$$ with left hand side lower triangular and right hand side upper triangular, $abca$ is diagonal. Writing $$a=\begin{bmatrix}\mu & x\\ 0 & \mu^{-1}\end{bmatrix},$$ we compute $$\begin{aligned}
a(bc)a=\begin{bmatrix}\mu^2\lambda & x(\mu\lambda+\lambda^{-1}\mu^{-1})\\0 & \mu^{-2}\lambda^{-1}\end{bmatrix}
\end{aligned}$$ which implies that $x=0$ or $\operatorname{tr}(abc)=\mu\lambda+\lambda^{-1}\mu^{-1}=0$. If $x=0$, that is, $a$ is diagonal, then we find that $b$ and $c$ are lower triangular, and we are done. The case $\operatorname{tr}(abc)=0$ still remains; note the hypothesis that $\operatorname{tr}(bc)\neq\pm2$ implies that $\operatorname{tr}(a)\neq0$, and hence we have $\operatorname{tr}(a),\operatorname{tr}(bc)\neq0$.
Hence, we have shown that if $(a,b,c)$ is in $(1,2)$-position then $(a,b,c)$ has a common eigenvector, except possibly if ${\operatorname{tr}_{a}},\operatorname{tr}(bc)\neq0$ and $\operatorname{tr}(abc)=0$. But in this exceptional case, since $(c^{-1}b^{-1}a,b,c)$ is in $(1,2)$-position by part (2) of Lemma \[loops\], and since $$\begin{aligned}
(\operatorname{tr}(c^{-1}b^{-1}a),\operatorname{tr}(bc),\operatorname{tr}(c^{-1}b^{-1}abc)) &=(\operatorname{tr}(a)\operatorname{tr}(bc)-\operatorname{tr}(abc),\operatorname{tr}(bc),\operatorname{tr}(a))\\
&=(\operatorname{tr}(a)\operatorname{tr}(bc),\operatorname{tr}(bc),\operatorname{tr}(a))
\end{aligned}$$ has none of the coordinates zero, running the same argument as above with $(a,b,c)$ replaced by $(c^{-1}b^{-1}a,b,c)$ we find that $(c^{-1}b^{-1}a,b,c)$ has a common eigenvector. But then $(a,b,c)$ also has a common eigenvector, as required.
Thus, we have proved our claim. To prove the proposition, we use the following inductive argument. Let $(a_1,a_2,\cdots,a_{2g-1},a_{2g},a_{2g+1},\cdots,a_{2g+n})$ be an optimal sequence of generators of $\pi_1(\Sigma)$. We show that $(a_1,\cdots,a_{2g+n})$ has a common eigenvector. For simplicity of arguments we may assume that at least one element in each pair $$(a_1,a_2),\cdots,(a_{2g-1},a_{2g})$$ is not equal to $\pm\mathbf 1$; for if some pair is of the form $(a,b)$ with $a,b\in\{\pm\mathbf 1\}$ we may simply skip over that pair in the considerations below.
If $(g,n)=(1,0)$, then we are done since every representation $\pi_1(\Sigma)\to\operatorname{SL}_2({{\mathbb{C}}})$ is abelian. We thus assume $(g,n)\neq(1,0)$. By our claim, $(a_1,a_2,a_3)$ has a common eigenvector. Assume next that $4\leq k\leq 2g+n$ and $(a_1,\cdots,a_{k-1})$ has a common eigenvector $x\in{{\mathbb{C}}}^2$. We show that $(a_1,\cdots,a_k)$ has a common eigenvector. We consider the following cases.
1. $k=5,7,\cdots,2g-1$, or $k=2g+1,2g+2,\cdots,2g+n$. The triple $(a_1,a_2,a_k)$ is in $(1,2)$-position, and hence has a common eigenvector $y$ by our claim.
Assume now first that $a_1\neq\pm\mathbf 1$. Given any $3\leq j<k$, note that $(a_1,a_2a_j,a_k)$ is in $(1,2)$-position and hence has a common eigenvector by our claim, say $y_j$. Since $a_1\neq\pm\mathbf 1$, we must have $$x\sim y,\quad x\sim y_j,\quad\text{or}\quad y_j\sim y.$$ If one of the first two occurs, then we conclude that $(a_1,\cdots,a_k)$ has a common eigenvector $x$, as required. Thus, we are left with the case $y_j\sim y$ for every $3\leq j<k$. But this implies that $(a_1,a_2,a_2a_j,a_k)$ has a common eigenvector $y$, which is thus shared also by $a_j$. Thus, $y$ is a common eigenvector of $(a_1,\cdots,a_k)$, as desired.
Now, suppose that $a_1=\pm\mathbf 1$, and hence $a_2\neq\pm\mathbf 1$. Given any $3\leq j<k$, we observe that $(a_2,a_1^{-1}a_2a_j,a_k)$ is in $(1,2)$-position and hence has a common eigenvector, say $y_j$. Since $a_2\neq\pm\mathbf 1$, we must have $$x\sim y,\quad x\sim y_j,\quad\text{or}\quad y_j\sim y.$$ If one of the first two occurs, then we conclude that $(a_1,\cdots,a_k)$ has a common eigenvector $x$, as required. Thus, we are left with the case $y_j\sim y$ for every $3\leq j<k$. But this implies that $(a_1,a_2,a_1^{-1}a_2a_j,a_k)$ has a common eigenvector $y$, which is thus shared also by $a_j$. Thus, $y$ is a common eigenvector of $(a_1,\cdots,a_k)$, as desired.
2. $k=4,6,\cdots,2g$. First, consider the case where $a_{k-1}=\pm\mathbf 1$. It then suffices to show that the sequence $(a_1,\cdots,a_{k-2},a_k)$ has a common eigenvector. This can be shown by repeating the argument the previous case (1) above. Thus, we may assume that $a_{k-1}\neq\pm\mathbf 1$. Note that, for each integer $m$ with $2m<k$, by our claim we have:
- a common eigenvector $w_m$ of $(a_{k-1},a_k, a_{2m})$, and
- a common eigenvector $z_m$ of $(a_{k-1},a_k, a_{2m-1})$.
Since $a_{k-1}\neq\pm\mathbf 1$, we must have $$x\sim w_m,\quad x\sim z_m,\quad\text{or}\quad w_m\sim z_m.$$ If one of the first two occurs, then we conclude that $(a_1,\cdots,a_k)$ has a common eigenvector $x$, and we are done. Thus, we are left with the case where $w_m\sim z_m$ for every $2m<k$. Note in this case that $w_m$ is a common eigenvector of $(a_{2m-1},a_{2m},a_{k-1},a_k)$. Comparing the vectors $x,w_m$, and $w_{m'}$ for different $m,m'$, we are in turn reduced to the case $w_m\sim w_{m'}$ for all $m,m'$, in which case $(a_1,\cdots,a_k)$ has a common eigenvector $w_1$, as desired.
Thus, $(a_1,\cdots,a_k)$ has a common eigenvector. This completes the induction, and shows that $(a_1,\cdots,a_{2g+n})$ has a common eigenvector, proving the proposition.
Genus $1$ with $1$ or $2$ punctures {#sec-g1n2}
-----------------------------------
We now give a separate proof of Theorem C for surfaces of genus one with one or two punctures. Let $\Sigma_{1,1}$ denote a surface of genus one with one puncture. We begin with the following:
\[lem-s11\] Let $\rho : \pi_1(\Sigma_{1,1}) \to \operatorname{SL}_2({{\mathbb{C}}})$ be a semisimple representation such that each nonseparating simple loop in $\Sigma_{1,1}$ maps to an elliptic or central element of $\operatorname{SL}_2({{\mathbb{C}}})$. Then $\rho$ is unitarizable, i.e., $\rho (\pi_1(\Sigma_{1,1}))$ has a global fixed point in ${{\mathbb{H}}}^3$.
This is known in the literature; we cite two sources below.
First, the lemma can be obtained as an immediate consequence of [@twz Theorem 1.2]. We sketch the connection of the setup in the current paper with that in [@twz]. Since every simple closed non-peripheral curve in $\Sigma_{1,1}$ maps to an elliptic element of $\operatorname{SL}_2({{\mathbb{C}}})$, it follows that we have $\operatorname{tr}\rho(a)\in[-2,2]$ for every $a$ in the curve complex of $\Sigma_{1,1}$. Hence, in the terminology of [@twz], the set of end-invariants is given by the set of all projective measured laminations on $\Sigma_{1,1}$. It now follows from [@twz Theorem 1.2] that $\rho (\pi_1(\Sigma_{1,1}))$ is either unitarizable or dihedral. We finish by observing that a dihedral representation $\rho:\pi_1(\Sigma)\to D_\infty\subset\operatorname{SL}_2({{\mathbb{C}}})$ sending every simple nonseparating simple loop to an elliptic or central element is unitarizable.
Second, a direct proof of what is essentially the contrapositive, based on an explicit presentation of the character variety $X(\Sigma_{1,1})$ (see the Appendix), is given as the Algebraic Lemma in [@dm Section 1.4.2].
We now turn to the twice-punctured torus $\Sigma_{1,2}$. We start with the following suggestive presentation of its fundamental group: $$\pi_1(\Sigma_{1,2}) =\langle u, x, y, p_1, p_2 \mid uxy = p_1, \, uyx = p_2 \rangle$$ where $p_1, \, p_2$ denote loops around the two different punctures of $\Sigma_{1,2}$. See Figure \[figs12\] below and section 5.3 of [@goldman2].
![Preferred generators for $\Sigma_{1,2}$[]{data-label="figs12"}](s22.png)
\[prop-s12\] Let $\rho : \pi_1(\Sigma_{1,2}) \to \operatorname{SL}_2({{\mathbb{C}}})$ be a semisimple representation such that each nonseparating simple loop in $\Sigma_{1,2}$ maps to an elliptic or central element of $\operatorname{SL}_2({{\mathbb{C}}})$. Then $\rho$ is unitarizable.
Combining with Lemma \[type1\], we note that Lemma \[lem-s11\] and Proposition \[prop-s12\] prove Theorem C for surfaces of genus one with at most two punctures. Our proof of Proposition \[prop-s12\] shall rely on the following observation. In what follows, $R_L$ denotes a hyperbolic reflection on the totally-geodesic plane $L$ in ${{\mathbb{H}}}^3$.
\[lem-plane\] Let $A$ and $B$ be two non-trivial elliptic rotations with distinct but intersecting axes lying on a common plane $K$. Then $AB$ is an elliptic rotation that has axis lying on a plane $Q$ that is at equal angles from $K$ and the plane $K^\prime$ obtained as the image of $K$ under $A$.
$A$ is a composition of two hyperbolic reflections, that is,$ A = R_Q \circ R_K$ where $Q$ is a totally-geodesic plane containing the axis of $A$, and is at half the rotation angle (of $A$) from $K$. Similarly, $B = R_K \circ R_S$ where $S$ is a totally-geodesic plane that is at half the rotation angle of $B$ from $K$. Hence the composition $AB = R_Q \circ R_S$, and its axis is the intersection of the planes $Q$ and $S$. In particular, the axis lies on the plane $Q$, proving the lemma.
Suppose that $\rho: \pi_1(\Sigma_{1,2}) \to \operatorname{SL}_2({{\mathbb{C}}})$ is a representation satisfying the hypothesis of the proposition. We may assume that $\rho$ is irreducible, since otherwise the result is clear.
Moreover, we may assume that the restriction of $\rho$ to any one-holed torus subsurface of $\Sigma_{1,2}$ is semisimple. Indeed, otherwise, we may choose an optimal sequence $(a_1,b_1,c_1,c_2)$ of generators of $\pi_1(\Sigma)$ such that the restriction of $\rho$ to the one-holed torus $\Sigma'\subset\Sigma$ associated to the pair $(a_1,b_1)$ is upper triangular, with the boundary loop $c=c_1c_2$ having non-central parabolic monodromy. The irreducibility of $\rho$ then implies that $\rho(c_1)$ and $\rho(c_2)$ cannot be upper triangular. But then an argument as in part (b) of the proof of Lemma \[type3\] shows that the restriction $\rho|\Sigma'$ is reducible and moreover $\rho(a_1)$ and $\rho(b_1)$ are parabolic, whence they must both be central and $\rho(c)$ is also central; a contradiction.
Thus, in what follows, $\rho: \pi_1(\Sigma_{1,2}) \to \operatorname{SL}_2({{\mathbb{C}}})$ is an irreducible representation satisfying the hypothesis of the proposition, with the property that its restriction to every one-holed torus subsurface is semisimple. We shall also assume that none of the nonseparating simple loops have central monodromy, since otherwise (using the preferred presentation of $\pi_1(\Sigma_{1,2})$ given above) we reduce to the case of one-holed torus, treated in Lemma \[lem-s11\].
Let us write $U = \rho (u)$, $X = \rho (x)$, and $Y = \rho (y)$ for the presentation of $\pi_1(\Sigma_{1,2})$ given above. By Lemma \[lem-s11\], the elements $U, X, Y$ have coplanar, pairwise intersecting axes. We call the common plane $P$. Let $p$ be the intersection point of the axes of $X$ and $Y$. It suffices to prove that $U$ fixes $p$ as well, that is, the axis of $U$ also passes through $p$. Assume that the axis of $U$ does not pass through $p$; we shall eventually reach a contradiction.
Consider the element $ U XY X=UT,$ say. Let $Z=X^{-1} Y X$ so that $T = X^2 Z$. Note that $T$ is elliptic as it fixes the intersection of the axes of $X, Y$. Moreover, it can be easily checked using the loop configuration diagram that the curve represented by the element $uxyx$ is simple, closed, and essential. Hence $UT$ is also an elliptic element. However, we also have:
\[claim-non-coplanar\] The axis of $T = XYX$ does not lie on the plane $P$ unless $X$ is a $\pi$–rotation.
Note that since $Z = X^{-1}YX$, the axis of $Z$ is the image of the axis of $Y$ under $X^{-1}$. Choose the plane $K$ to be the one containing the axes of $X$ and $Z$. Then $ K=X^{-1}(P)$, i.e the image of $K$ under $X$ is the plane $P$. Since $X^2$ and $Z$ have the same axes as $X$ and $Z$ respectively, Lemma \[lem-plane\] shows that the axis of $X^2Z$ will be on a plane that is at equal angles from $K$ and $P$. In particular, the axis does not lie on the plane $P$ unless $X^2$ is the identity element, that is, unless $X$ is a $\pi$–rotation.
Since we have assumed that the axis of $U$ does not pass through $p$, the plane $P$ is the unique plane that contains both $p$ and the axis of $U$. Note that the axis of $T$ contains $p$; hence if it does not lie in the plane $P$, the axes of $U$ and $T$ cannot lie on a common plane. By Lemma 3.4.1 of [@gkm], we have a contradiction to the fact that $UT$ is elliptic. Hence, $X$ must be a $\pi$–rotation. Repeating the same argument for other presentations of $\pi_1(\Sigma_{1,2})$, we are thus reduced to the case where the monodromy trace of every nonseparating simple loop on $\Sigma_{1,2}$ under $\rho$ is $0$. It is easy to see that such $\rho$ is in fact dihedral, whence the hypothesis of the Proposition implies that $\rho$ is unitarizable, a contradiction.
Proof of Theorem A {#sec:5.3}
------------------
We restate and complete the proof of Theorem A.
Let $\Sigma$ be an oriented surface of genus $g\geq2$ with $n\geq0$ punctures. A semisimple representation $\rho:\pi_1(\Sigma)\to{{\operatorname{SL}(2,\mathbb{C})}}$ has finite mapping class group orbit in the character variety $X(\Sigma)$ if and only if $\rho$ is finite.
Suppose that $\rho:\pi_1(\Sigma)\to\operatorname{SL}_2({{\mathbb{C}}})$ is a semisimple representation with finite mapping class group orbit in $X(\Sigma)$. By Lemma \[redlemm\], we are done if $\rho$ is reducible, so we may assume that $\rho$ is irreducible. It suffices to show that $\rho$ has finite monodromy along every simple closed curve around a puncture of $\Sigma$, for then we reduce to Proposition \[simple-finite\]. By Lemmas 4.4 and 4.5, we see that $\rho$ must have finite monodromy along any essential curve $a\subset\Sigma$ which is either:
- nonseparating, or
- separating with each component of $\Sigma|a=\Sigma_1\sqcup\Sigma_2$ having positive genus.
By Proposition \[red\], there is a one-holed torus subsurface $\Sigma'\subset\Sigma$ such that $\rho|\Sigma'$ is irreducible. Let $c$ be a simple closed curve around a puncture of $\Sigma$. Let $\Sigma''\subset\Sigma$ be a two-holed torus containing $\Sigma'$ and having $c$ as one of its boundary curves; let $c'$ be the other boundary curve of $\Sigma''$. (Note here that each component of $\Sigma|c'$ has positive genus, by design.) We shall prove that $\rho|\Sigma''$ is finite, so *a fortiori* $\rho$ has finite monodromy along $c$. For this, we follow the strategy of [@psw] below.
First, we know from above that $\rho$ has finite monodromy along every essential curve of $\Sigma''$, as well as along the boundary curve $c'$. In particular, the restriction of $\rho$ to the one-holed torus $\Sigma'$ has image that is conjugate to a subgroup of $\operatorname{SU}(2)$. Let $A\subset{{\mathbb{R}}}$ be the ${{\mathbb{Z}}}$-algebra generated by the set of traces of $\rho$ along the essential curves of $\Sigma''$. By considering the preferred generators for $\pi_1(\Sigma'')$ introduced in Section \[sec-g1n2\], it follows from the trace relations given in Example \[exfree\] (cf. [@goldman2 Section 5.3]) that $\operatorname{tr}\rho(c)$ satisfies a monic quadratic equation over the ring $A$, with the other root being $\operatorname{tr}\rho(c')$. Since $\operatorname{tr}\rho(c')\in{{\mathbb{R}}}$, it follows that $\operatorname{tr}\rho(c_1)\in{{\mathbb{R}}}$ as well. Applying Fact \[fact\] to a sequence of optimal generators for $\pi_1(\Sigma'')$, we deduce that the character of $\rho|\Sigma''$ is real, and since $\rho|\Sigma''$ is semisimple its image is conjugate to a subgroup of $\operatorname{SU}(2)$ or $\operatorname{SL}_2({{\mathbb{R}}})$ (see e.g. [@ms Proposition III.1.1]). The latter cannot occur, since otherwise the restriction of $\rho$ to $\Sigma'$ has image conjugate to a subgroup of $\operatorname{SO}(2)$, contradicting the fact that $\rho|\Sigma'$ is irreducible hence nonabelian. Thus, the restriction of $\rho$ to $\Sigma''$ has image conjugate to a subgroup of $\operatorname{SU}(2)$.
It also follows from the above analysis that the character of $\rho|\Sigma''$ takes values in the ring of algebraic integers in $\overline{{{\mathbb{Q}}}}$. In particular, we may assume without loss of generality that the image of $\rho|\Sigma''$ lies in $\operatorname{SL}_2(\overline{{{\mathbb{Q}}}})$. By considering conjugates of $\rho|\Sigma''$ by elements of the absolute Galois group $\operatorname{Gal}(\overline{{{\mathbb{Q}}}}/{{\mathbb{Q}}})$ of ${{\mathbb{Q}}}$ and noting that the above analysis goes through for all of these conjugates, we conclude that the eigenvalues of monodromy of $\rho$ along $c$ are algebraic integers all of whose Galois conjugates have absolute value $1$. By Kronecker’s theorem, it follows that the eigenvalues are roots of unity, i.e., $\rho$ has finite monodromy along $c$ (note that monodromy along $\rho$ cannot be unipotent). This is the desired result.
Proof of Theorem B {#sec:5.4}
------------------
We restate and complete the proof of Theorem B.
Let $\Sigma$ be an oriented surface of genus $1$ with $n\geq0$ punctures. A semisimple representation $\rho:\pi_1(\Sigma)\to\operatorname{SL}_2({{\mathbb{C}}})$ has finite mapping class group orbit in the character variety $X(\Sigma)$ if and only if $\rho$ is finite or special dihedral.
Our proof proceeds as in the proof Theorem A, with minor modifications. Suppose that $\rho:
\pi_1(\Sigma)\to\operatorname{SL}_2({{\mathbb{C}}})$ is a semisimple representation with finite mapping class group orbit in $X(\Sigma)$. By Lemma \[redlemm\], we are done if $\rho$ is reducible, so we may assume that $\rho$ is irreducible. We may also assume that $\rho$ is not special dihedral. It suffices to show that $\rho$ has finite monodromy along every simple closed curve around a puncture of $\Sigma$, for then we reduce to Proposition \[simple-finite\].
By Lemma \[type1\], $\rho$ has finite monodromy along any nonseparating essential curve $a\subset\Sigma$. Given any simple closed curve $c$ around a puncture of $\Sigma$, there is a two-holed torus subsurface of $\Sigma$ having $c$ as one of its boundary components, and by the trace relations in Example \[exfree\] (cf. [@goldman2 Section 5.3]) we see that $\operatorname{tr}\rho(c)$ is an algebraic integer. By considering an optimal sequence of generators for $\pi_1(\Sigma)$ and applying Fact \[fact\], we see that the coordinate ring of the character variety $X(\Sigma)$ is generated by the monodromy traces $\operatorname{tr}_a$ around a finite collection of curves $a$ each of which is either a nonseparating curve or a curve around a puncture of $\Sigma$. In particular, it follows that $\operatorname{tr}\rho(\alpha)$ is an algebraic integer for every $\alpha\in\pi_1(\Sigma)$. We may in particular assume that $\rho$ is a representation of $\pi_1(\Sigma)$ into $\operatorname{SL}_2(
\overline{{{\mathbb{Q}}}})$.
Now, $\rho$ is unitarizable by Theorem C proved below, and in particular the monodromy eigenvalues of $\rho$ along any curve $c$ around a puncture of $\Sigma$ have absolute value $1$. Applying this observation to every conjugate of $\rho$ by an element of the absolute Galois group $\operatorname{Gal}(\overline{{{\mathbb{Q}}}}/{{\mathbb{Q}}})$, we see that the eigenvalues of $\rho(c)$ for any curve around a puncture of $\Sigma$ are algebraic integers all of whose conjugates have absolute value $1$. By Kronecker’s theorem, it follows that the eigenvalues are roots of unity, i.e., $\rho$ has finite monodromy along $c$ (note that monodromy along $\rho$ cannot be unipotent). This is the desired result.
Proof of Theorem C {#sec:5.5}
------------------
We restate and complete the proof of Theorem C.
Let $\Sigma$ be an oriented surface of genus $g\geq1$ with $n\geq0$ punctures. A semisimple representation $\rho:\pi_1(\Sigma)\to{{\operatorname{SL}(2,\mathbb{C})}}$ has bounded mapping class group orbit in the character variety $X(\Sigma)$ if and only if:
1. $\rho$ is unitary up to conjugacy, or
2. $g=1$ and $\rho$ is special dihedral up to conjugacy.
In the case where $\Sigma$ is a surface of genus $g\geq2$, a minor modification of the argument in the proof of Theorem A proves that $\rho$ has elliptic or central local monodromy around the punctures. Proposition \[simple-finite\] then shows that $\rho$ is unitarizable, as desired.
Let us now assume that $\Sigma$ has genus $1$ with $n\geq0$ punctures. We know the case $n\leq 2$ of Theorem C by our work in Section \[sec-g1n2\]; we shall deduce the general case from it. So suppose we have a semisimple representation $\rho:\pi_1(\Sigma)\to\operatorname{SL}_2({{\mathbb{C}}})$ whose monodromy is central or elliptic along every nonseparating curve on $\Sigma$. It is clear that $\rho$ is unitarizable if it is moreover reducible, so let us assume that $\rho$ is irreducible in what follows.
By Proposition \[red\], there is a one-holed torus subsurface $\Sigma'\subset\Sigma$ such that $\rho|\Sigma'$ is irreducible. Let $c$ be a simple closed curve around a puncture of $\Sigma$. Let $\Sigma''\subset\Sigma$ be a two-holed torus containing $\Sigma'$ and having $c$ as one of its boundary curves. By Proposition \[prop-s12\], our Theorem C holds for two-holed tori, and hence we see that $\rho|\Sigma''$ is unitarizable. It follows that the monodromy of $\rho$ along $c$ is central or unitary; in particular, $\operatorname{tr}\rho(c)$ is real. By considering an optimal sequence of generators for $\pi_1(\Sigma)$ and applying Fact \[fact\], we see that the coordinate ring of the character variety $X(\Sigma)$ is generated by the monodromy traces $\operatorname{tr}_a$ around a finite collection of curves $a$ each of which is either a nonseparating curve or a curve around a puncture of $\Sigma$. In particular, it follows that $\rho$ has real character, and therefore the image of $\rho$ is conjugate to a subgroup of $\operatorname{SU}(2)$ or $\operatorname{SL}_2({{\mathbb{R}}})$. We claim that the latter cannot occur. Indeed, by Proposition \[red\] there is a one-holed torus subsurface $\Sigma'\subset\Sigma$ such that $\rho|\Sigma'$ is irreducible, and moreover $\rho|\Sigma'$ must be unitarizable by the case $n=2$ of our proposition. If the image of $\rho$ lies in $\operatorname{SL}_2({{\mathbb{R}}})$, then unitarizability implies that the image of $\rho|\Sigma'$ is conjugate to a subgroup of $\operatorname{SO}(2)$, contradicting the irreducibility of $\rho|\Sigma'$. Thus, the image of $\rho$ must be conjugate to a subgroup of $\operatorname{SU}(2)$, as desired.
Alternative proof of Theorem A for closed surfaces {#sec:5.6}
--------------------------------------------------
We give a different short proof of Theorem A in the case where $\Sigma$ is a closed surface of genus $g\geq2$, using the following result of Gallo–Kapovich–Marden.
\[gkm\] Let ${{\Sigma}}$ be a closed oriented surface of genus greater than one. If $\rho:\pi_1(\Sigma)\to\operatorname{SL}_2({{\mathbb{C}}})$ is non-elementary, then there exist simple loops $a, b$ on ${{\Sigma}}$ such that
1. the intersection number $i(a,b) = 1$,
2. The images $\rho(a), \rho(b) \in \operatorname{PSL}_2({{\mathbb{C}}})$ are loxodromic and distinct, and generate a Schottky group.
\[rgg\] Let ${{\Sigma}}$ be a closed orientable surface of genus $g\geq2$. Given a semisimple representation $\pi_1(\Sigma)\to\operatorname{SL}_2({{\mathbb{C}}})$, the orbit of $\rho$ in $X(\Sigma)$, under the action of $\operatorname{Mod}(\Sigma)$, is finite if and only if the image of $\rho$ is a finite group.
Suppose first that $\rho$ is non-elementary. By Theorem \[gkm\], there exist simple loops $a, b$ on ${{\Sigma}}$ with $i(a,b) = 1$ such that $\rho(a), \rho(b) \in \operatorname{PSL}_2({{\mathbb{C}}})$ are loxodromic and distinct. Since $\rho(a), \rho(b) \in \operatorname{PSL}_2({{\mathbb{C}}})$ are loxodromic, $\operatorname{tw}_a^n (b)$ gives an infinite sequence of curves in $S$ with $\rho$-images $\rho(a)^n \rho(b)$ whose translation length in ${{\mathbb H}}^3$ tends to infinity as $n\to \infty$, while $\operatorname{tw}_a^n (a)$ remains fixed. It follows that the $\operatorname{tw}_a^n$-orbit of $\rho$ is infinite in $X(\Sigma)$ and hence so is the $\operatorname{Mod}(\Sigma)$-orbit.
Suppose that $\rho$ is elementary. In view of the results in Section \[sect:3\], it remains only to treat the case where $\rho$ is a representation whose image is a dense subgroup of $\operatorname{SU}(2)$ in the Euclidean topology. In this case, the main theorem of [@px] states that the $\operatorname{Mod}(\Sigma)$-orbit of $\rho$ is dense in $\operatorname{Hom}(\pi_1(\Sigma),\operatorname{SU}(2))/\operatorname{SU}(2)$ in the Euclidean topology. Hence it is infinite.
Applications {#sect:6}
============
In this section, we collect applications of our results and methods developed in the previous sections. The following immediate corollary of Theorem A answers a question that was posed to us by Lubotzky:
\[faithful\] Given a surface $\Sigma$ of genus $g\geq1$ with $n\geq0$ punctures, a faithful representation $\rho:\pi_1(\Sigma)\to \operatorname{SL}_2({{\mathbb{C}}})$ (or $\operatorname{PSL}_2({{\mathbb{C}}})$) cannot have finite $\operatorname{Mod}(\Sigma)$-orbit in the character variety.
It follows from Theorems A and B that if $\rho:\pi_1(\Sigma)\to \operatorname{SL}_2({{\mathbb{C}}})$ has a finite $\operatorname{Mod}(\Sigma)$-orbit in $X(\Sigma)$, then $\rho$ cannot be faithful. The same holds when $\operatorname{SL}_2({{\mathbb{C}}})$ is replaced by $\operatorname{PSL}_2({{\mathbb{C}}})$.
Let $\Sigma$ be a closed surface of genus greater than one; we fix a a hyperbolic metric on $\Sigma$ such that $\Sigma = {{\mathbb H}}^2/\Gamma$ where $\Gamma$ is a Fuchsian group. Recall that for any semisimple representation $\rho:\Gamma \to \operatorname{PSL}_2({{\mathbb{C}}})$, there exists a $\rho$-equivariant harmonic map $\tilde{h}: {{\mathbb H}}^2\to {{\mathbb H}}^3$ from the universal cover of $\Sigma$ to the symmetric space for $\operatorname{PSL}_2({{\mathbb{C}}})$ (see, for example, [@Donaldson]). The *equivariant energy* of this harmonic map is the energy of its restriction to a fundamental domain of the $\Gamma$-action on ${{\mathbb H}}^2$. The following answers a question due to Goldman.
\[energy\] Fix a Riemann surface ${{\Sigma}}$ of genus greater than one with $\Gamma = \pi_1({{\Sigma}})$, and let $\rho:\Gamma \to \operatorname{PSL}_2({{\mathbb{C}}})$ be an semisimple representation. Suppose that the equivariant energies of the harmonic maps corresponding to the mapping class group orbit of $\rho$ in $\operatorname{Hom}(\pi_1\Sigma,\operatorname{PSL}_2({{\mathbb{C}}})){\mathbin{
\mathchoice{/\mkern-6mu/} {/\mkern-6mu/} {/\mkern-5mu/} {/\mkern-5mu/}}}\operatorname{PSL}_2({{\mathbb{C}}})$ is uniformly bounded. Then $\rho(\Gamma)$ fixes a point of ${{\mathbb H}}^3$; in particular $\rho(\Gamma)$ can be conjugated to lie in $\operatorname{PSU}(2)$.
If $\rho(\Gamma)$ is elementary, but not unitary, then $\rho(\Gamma)$ fixes a geodesic in ${{\mathbb H}}^3$, and thus the image of any equivariant harmonic map coincides with this geodesic. Let $a$ be a simple closed curve mapped to an infinite order hyperbolic element in $\rho(\Gamma)$. Let $b$ be a simple closed on $\Sigma$ with $i(a,b) > 0$. Then the translation length of $\rho(\operatorname{tw}_a^n b)$ increases to infinity as $n \to \infty$. Hence the $\operatorname{Mod}(\Sigma)-$orbit of $\rho$ cannot have bounded energy, since an uniform energy bound on the harmonic maps implies that they are uniformly Lipschitz.
Otherwise, suppose $\rho(\Gamma)$ is non-elementary. By Theorem \[gkm\], there exist simple closed curves $a, b$ on $S$ such that $i(a,b) = 1$ and $\rho(a), \rho(b) \in \operatorname{PSL}_2({{\mathbb{C}}})$ are loxodromic and distinct. It follows as in the proof of Theorem \[rgg\] that the translation lengths in ${{\mathbb H}}^3$ of $\rho(\operatorname{tw}_a^n (b))$ tend to infinity, while $\operatorname{tw}_a^n (a)$ remains fixed. Hence the ${{\mathbb{Z}}}-$orbit $\rho \circ\operatorname{tw}_a^n$ of $\rho$ under $\langle \operatorname{tw}\rangle$ cannot have bounded energy. Thus the $\operatorname{Mod}(\Sigma)-$orbit of $\rho$ cannot have bounded energy, as before.
The possibility that remains is that $\rho(\Gamma)$ is unitary.
Case of once-punctured torus {#appendix-a}
============================
In this appendix, we describe how the work of Dubrovin–Mazzocco [@dm] can be used to prove the once-punctured torus case of our Theorem B. We begin with the following well-known observation.
\[tracelem\] A pair $(a,b)$ of elements in $\operatorname{SL}_2({{\mathbb{C}}})$ has a common eigenvector in ${{\mathbb{C}}}^2$, or in other words lies in the standard Borel $B$ up to simultaneous conjugation, if and only if $\operatorname{tr}([a,b])=2$, where $[a,b]=aba^{-1}b^{-1}$.
Let $\Sigma$ be a surface of genus one with one puncture. Let $(a,b,c)$ be an optimal sequence of generators for $\pi_1(\Sigma)$, as defined in Example \[exgen\]. Let $X=X(\Sigma)$ be the character variety of $\Sigma$. Note that $\pi_1(\Sigma)$ is freely generated by $a$ and $b$. The trace functions on $X(\Sigma)$ furnish an isomorphism $$(x_1,x_2,x_3)=(\operatorname{tr}_{a},\operatorname{tr}_{b},\operatorname{tr}_{ab}):X(\Sigma)\xrightarrow{\sim}{{\mathbb{A}}}^3$$ by Fricke (see [@goldman2] for details). Observing that $\operatorname{tr}(\mathbf1)=2$ for the identity $\mathbf1\in\operatorname{SL}_2({{\mathbb{C}}})$ and that $\operatorname{tr}(A)\operatorname{tr}(B)=\operatorname{tr}(AB)+\operatorname{tr}(AB^{-1})$ for any $A,B\in\operatorname{SL}_2({{\mathbb{C}}})$, we have $$\begin{aligned}
\operatorname{tr}_c&=\operatorname{tr}_{aba^{-1}b^{-1}}=\operatorname{tr}_{aba^{-1}}\operatorname{tr}_{b^{-1}}-\operatorname{tr}_{aba^{-1}b}\\
&=\operatorname{tr}_{b}^2-\operatorname{tr}_{ab}\operatorname{tr}_{a^{-1}b}+\operatorname{tr}_{aa}=\operatorname{tr}_{b}^2-\operatorname{tr}_{ab}(\operatorname{tr}_{a^{-1}}\operatorname{tr}_{b}-\operatorname{tr}_{ab})+\operatorname{tr}_{a}^2-\operatorname{tr}_{\mathbf1}\\
&=\operatorname{tr}_{a}^2+\operatorname{tr}_{b}^2+\operatorname{tr}_{ab}^2-\operatorname{tr}_{a}\operatorname{tr}_{b}\operatorname{tr}_{ab}-2.\end{aligned}$$ In particular, under the identification $(x_1,x_2,x_3)=(\operatorname{tr}_{a},\operatorname{tr}_{b},\operatorname{tr}_{ab})$ of the coordinate functions above and Lemma \[tracelem\], we see that the locus of reducible representations in $X(\Sigma)={{\mathbb{A}}}^3$ is the cubic algebraic surface cut out by the equation $$x_1^2+x_2^2+x_3^2-x_1x_2x_3-2=2.$$ The mapping class group $\operatorname{Mod}(\Sigma)$ acts on $X(\Sigma)$ via polynomial transformations. For convenience, we shall denote the isotopy classes of simple closed curves lying in the free homotopy classes of $a$, $b$, and $ab$ by the same letters. We have the following descriptions of the associated Dehn twist actions.
\[dehnt\] The Dehn twist actions $\operatorname{tw}_a$, $\operatorname{tw}_b$, and $\operatorname{tw}_{ab}$ on $X(\Sigma)$ are given by $$\begin{aligned}
\operatorname{tw}_{a}^*&:(x_1,x_2,x_3)\mapsto (x_1,x_3,x_1x_3-x_2),\\
\operatorname{tw}_{b}^*&:(x_1,x_2,x_3)\mapsto (x_1x_2-x_3,x_2,x_1),\\
\operatorname{tw}_{ab}^*&:(x_1,x_2,x_3)\mapsto(x_2,x_2x_3-x_1,x_3).\end{aligned}$$ in terms of the above coordinates.
Note that $\operatorname{tw}_a(a)$ has the homotopy class of $\alpha$, $\operatorname{tw}_a(b)$ has the homotopy class of $\alpha\beta$, and $\operatorname{tw}_a(ab)$ has the homotopy class of $\alpha\alpha\beta$. Noting that $\operatorname{tr}_{\alpha\alpha\beta}=\operatorname{tr}_{\alpha}\operatorname{tr}_{\alpha\beta}-\operatorname{tr}_{\beta}$, we obtain the desired expression for $\operatorname{tw}_a^*$. The other Dehn twists are similar.
Let $\Pi$ be the group of polynomial automorphisms of ${{\mathbb{A}}}^3$ generated by $\operatorname{tw}_{a}^*$, $\operatorname{tw}_{b}^*$, and $\operatorname{tw}_{ab}^*$. It is precisely the image of the mapping class group $\operatorname{Mod}(\Sigma)$ in the group of polynomial automorphisms of $X(\Sigma)={{\mathbb{A}}}^3$. Let $\Pi'$ be the group generated by $\Pi$ together with the following transformations: $$\begin{aligned}
\sigma_{12}:(x_1,x_2,x_3)\mapsto(-x_1,-x_2,x_3),\\
\sigma_{23}:(x_1,x_2,x_3)\mapsto(x_1,-x_2,-x_3),\\
\sigma_{13}:(x_1,x_2,x_3)\mapsto(-x_1,x_2,-x_3).\end{aligned}$$ It is easy to see that $[\Pi':\Pi]<\infty$. Hence, a point in ${{\mathbb{A}}}^3$ has finite $\Pi$-orbit if and only if it has finite $\Pi'$-orbit. Now, the group $\Pi'$ contains a group generated by transformations $$\begin{aligned}
\beta_1=\sigma_{12}\operatorname{tw}_{ab}^*(\operatorname{tw}_{b}^*\operatorname{tw}_{a}^*)^{-1}&:(x_1,x_2,x_3)\mapsto(-x_1,x_3-x_1x_2,x_2),\\
\beta_2=\sigma_{23}\operatorname{tw}_{a}^*(\operatorname{tw}_{b}^*\operatorname{tw}_{a}^*)^{-1}&:(x_1,x_2,x_3)\mapsto(x_3,-x_2,x_1-x_2x_3).\end{aligned}$$ whose finite orbits in ${{\mathbb{A}}}^3$ were studied by Dubrovin-Mazzocco [@dm Theorem 1.6] in connection with algebraic solutions of special Painlevé VI equations. They defined a triple $(x_1,x_2,x_3)\in{{\mathbb{A}}}^3({{\mathbb{C}}})$ to be *admissible* if it has at most one coordinate zero and $x_1^2+x_2^2+x_2^2-x_1x_2x_3-2\neq2$. It is easy to verify that the admissible points are precisely those which do not correspond to reducible or special dihedral representations. The result of [@dm] we shall use is the following.
\[dubm\] The following is a complete set of representatives for the finite $\langle\beta_1,\beta_2\rangle$-orbits of admissible triples in ${{\mathbb{A}}}^3$: $$\begin{aligned}
&(0,-1,-1), (0,-1,-\sqrt{2}),(0,-1,-\varphi),(0,-1,-\varphi^{-1}),(0,-\varphi,-\varphi^{-1})\end{aligned}$$ where $\varphi=(1+\sqrt{5})/2$ is the golden ratio.
To deduce Theorem B in the once-punctured torus case from the above, we recall the following explicit description of the finite subgroups $B A_4$, $B S_4$, $B A_5$ of $\operatorname{SL}_2({{\mathbb{C}}})$. First, let us identify the group of unit quaternions $$\operatorname{Sp}(1)=\{z=(a,b,c,d)=a+bi+cj+dk\in{{\mathbb{H}}}:|z|=a^2+b^2+c^2+d^2=1\}$$ as a subgroup of $\operatorname{SL}_2({{\mathbb{C}}})$ by the map $$z=(a,b,c,d)\mapsto \begin{bmatrix}a+bi & c+di\\ -c+di & a-bi\end{bmatrix}.$$ Under the identification, the *binary tetrahedral group* $BA_4$ is given by $$B A_4=\{\pm1,\pm i,\pm j,\pm k,(\pm1\pm i\pm j\pm k)/2\}$$ with all sign combinations taken in the above. The *binary octahedral group* $B S_4$ is the union of $B A_4$ with all quaternions obtained from $(\pm1,\pm1,0,0)/\sqrt 2$ by all permutations of coordinates and all sign combinations. The *binary icosahedral group* $B A_5$ is the union of $B A_4$ with all quaternions obtained from $(0,\pm1,\pm\varphi^{-1},\pm\varphi)/2$ by an even permutation of coordinates and all possible sign combinations, where $\varphi=(1+\sqrt 5)/2$ is the golden ratio.
\[admcor\] If $(x_1,x_2,x_3)\in{{\mathbb{A}}}^3({{\mathbb{C}}})$ is an admissible triple with finite $\operatorname{Mod}(\Sigma)$-orbit, then it corresponds to a representation $\rho:\pi_1(\Sigma)\to \operatorname{SL}_2({{\mathbb{C}}})$ with finite image.
Replacing $(x_1,x_2,x_3)$ by another triple within its $\operatorname{Mod}(\Sigma)$-orbit if necessary, we may assume that $(x_1,x_2,x_3)$ is one of the triples in Theorem \[dubm\] or its image under one of the transformations $\sigma_{12}$, $\sigma_{23}$, or $\sigma_{13}$. We shall show that $$(x_1,x_2,x_3)=(\operatorname{tr}A,\operatorname{tr}B,\operatorname{tr}(AB))$$ where $A,B\in\operatorname{SL}_2({{\mathbb{C}}})$ are elements that together lie in one of the finite subgroups $B A_4$, $BS_4$, or $BA_5$ of $\operatorname{SL}_2({{\mathbb{C}}})$. Since the matrix $-\mathbf 1$ is contained in every one of these groups, it suffices to treat the case where $(x_1,x_2,x_3)$ is one of the triples in Theorem \[dubm\]. By explicit computation, we find that the triples in Theorem \[dubm\] respectively correspond to traces of the triples of matrices $$\begin{aligned}
&\left(\begin{bmatrix}0 & 1\\ -1 & 0\end{bmatrix},\begin{bmatrix}-\frac{1}{2}(1+i) & \frac{1}{2}(1-i)\\ -\frac{1}{2}(1+i) & -\frac{1}{2}(1-i)\end{bmatrix},\begin{bmatrix}-\frac{1}{2}(1+i) & -\frac{1}{2}(1-i)\\ \frac{1}{2}(1+i) & -\frac{1}{2}(1-i)\end{bmatrix}\right),\\
&\left(\begin{bmatrix}0 & \frac{1}{\sqrt{2}}(1-i)\\ \frac{1}{\sqrt 2}(1+i) & 0\end{bmatrix},\begin{bmatrix}-\frac{1}{2}(1+i) & \frac{1}{2}(1-i)\\ -\frac{1}{2}(1+i) & -\frac{1}{2}(1-i)\end{bmatrix},\begin{bmatrix}-\frac{1}{\sqrt 2} & \frac{1}{\sqrt 2}i\\ \frac{1}{\sqrt 2 }i & -\frac{1}{\sqrt 2}\end{bmatrix}\right),\\
&\left(\begin{bmatrix}0 & 1\\ -1 & 0\end{bmatrix},\begin{bmatrix}-\frac{1}{2} & \frac{1}{2}(\varphi+\varphi^{-1}i)\\ -\frac{1}{2}(\varphi-\varphi^{-1}i) & -\frac{1}{2}\end{bmatrix},\begin{bmatrix}-\frac{1}{2}(\varphi-\varphi^{-1}i) & -\frac{1}{2}\\ \frac{1}{2} & -\frac{1}{2}(\varphi+\varphi^{-1}i)\end{bmatrix}\right),\\
&\left(\begin{bmatrix}0 & 1\\ -1 & 0\end{bmatrix},\begin{bmatrix}\frac{1}{2}(1-\varphi i) & \frac{1}{2}\varphi^{-1}\\ -\frac{1}{2}\varphi^{-1} & -\frac{1}{2}(1+\varphi i)\end{bmatrix},\begin{bmatrix}-\frac{1}{2}\varphi^{-1} & -\frac{1}{2}(1+\varphi i)\\ -\frac{1}{2}(1-\varphi i) & -\frac{1}{2}\varphi^{-1}\end{bmatrix}\right),\\
&\left(\begin{bmatrix}0 & 1\\ -1 & 0\end{bmatrix},\begin{bmatrix}-\frac{\varphi}{2} &\frac{\varphi^{-1}}{2}+\frac{1}{2}i\\-\frac{\varphi^{-1}}{2}+\frac{1}{2}i & -\frac{\varphi}{2}\end{bmatrix},\begin{bmatrix}-\frac{\varphi^{-1}}{2}+\frac{1}{2}i & -\frac{\varphi}{2}\\\frac{\varphi}{2} &-\frac{\varphi^{-1}}{2}-\frac{1}{2}i\end{bmatrix}\right),\end{aligned}$$ where $\varphi=(1+\sqrt 5)/2$ is the golden ratio. The matrices for the first triple all lie in the binary tetrahedral group $BA_4$, the matrices for the second triple all lie in the binary octahedral group $BS_4$, and the matrices for the remaining three triples all lie in the binary icosahedral group $BA_5$. In each triple, the third matrix is the product of the first two. Thus, each of the triples in Theorem \[dubm\] correspond to representations $\pi_1(\Sigma)\to\operatorname{SL}_2({{\mathbb{C}}})$ with finite image, proving the corollary.
In [@dm], two proofs of Theorem \[dubm\] are given. The first proof is based on an explicit analysis of certain relevant trigonometric Diophantine equations. General equations of this type are effectively solvable by Lang’s ${{\mathbb{G}}}_m$ conjecture (proved by Laurent [@laurent]), as noted in [@bgs]. The second proof in [@dm], based on a suggestion of Vinberg, uses consideration of certain representations of Coxeter groups of reflections associated to admissible triples. Both methods use special features present in the once-punctured torus case which do not seem to generalize easily to the case of general surfaces treated in our work.
Acknowledgments {#acknowledgments .unnumbered}
===============
Theorems A and B along with parts of this paper originally appeared in Chapter 6 of JPW’s Ph.D. thesis [@whang0], but the proof had a gap which has been filled in this paper. JPW thanks Peter Sarnak and Phillip Griffiths for encouragement, and Anand Patel and Ananth Shankar for collaborative work in [@psw] which led to one of the key ingredients in this paper. IB and MM thank Bill Goldman for useful email correspondence, in particular for alerting us to dihedral representations. Research of IB partly supported by a DST JC Bose Fellowship. SG acknowledges the SERB, DST (Grant no. MT/2017/000706) and the Infosys Foundation for their support. Research of MM partly supported by a DST JC Bose Fellowship, Matrics research project grant MTR/2017/000005 and CEFIPRA project No. 5801-1. MM was also partially supported by the grant 346300 for IMPAN from the Simons Foundation and the matching 2015-2019 Polish MNiSW fund.
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abstract: 'We introduce an approach to compute reduced density matrices for local quantum unitary circuits of finite depth and infinite width. Suppose the time-evolved state under the circuit is a matrix-product state with bond dimension $D$; then the reduced density matrix of a half-infinite system has the same spectrum as an appropriate $D \times D$ matrix acting on an ancilla space. We show that reduced density matrices at different spatial cuts are related by quantum channels acting on the ancilla space. This quantum channel approach allows for efficient numerical evaluation of the entanglement spectrum and Rényi entropies and their spatial fluctuations at finite times in an infinite system. We benchmark our numerical method on random unitary circuits, where many analytic results are available, and also show how our approach analytically recovers the behaviour of the kicked Ising model at the self-dual point. We study various properties of the spectra of the reduced density matrices and their spatial fluctuations in both the random and translation-invariant cases.'
author:
- Sarang Gopalakrishnan
- Austen Lamacraft
title: Unitary circuits of finite depth and infinite width from quantum channels
---
Introduction
============
The dynamics of isolated quantum systems under generic unitary dynamics is one of the basic problems in many-body physics [@polkovnikov_review]; despite considerable recent work, many aspects of this problem are not fully understood. An isolated system, evolving under chaotic dynamics from an initial product state, becomes increasingly entangled over time. At sufficiently late times, any finite-size subsystem of an infinite system is well described by a thermal reduced density matrix, provided the system obeys the eigenstate thermalization hypothesis (ETH) [@deutsch_eth; @srednicki_eth; @rigol2008thermalization]; this approach to a thermal local density matrix is called “thermalization.” There is considerable numerical evidence that generic many-body systems obey ETH [@deutsch_review]. We are concerned with the dynamics of the reduced density matrix *before* the system has fully thermalized. To quantify the thermalization timescale more precisely, recall that the Rényi entropies of a subsystem $A$ are defined in terms of the reduced density matrix $\rho_A$ of $A$ as $$\label{eq:ren_def}
S^{(n)}_A(t) = \frac{1}{1-n}\log\operatorname{tr}\left[\rho_A(t)^n\right],$$ where $n$ is called the Rényi index. The Rényi entropies fully characterize the spectrum of $\rho_A$.
The general consensus [@calabrese2005evolution; @Kim:2013aa; @nrvh; @Keyserlingk2017; @Nahum2017] is that – unless a system experiences many-body localization [@Znidaric:2008aa] – the entropies initially obey $S^{(n)}_A(t)\sim v_n t$, increasing linearly with time with a growth rate $v_n$ that depends on the Rényi index $n$. (However, recent results suggest that for $n > 1$ the growth is sub-linear for generic initial states in the presence of conservation laws [@rpv2019; @yichen2019].) The implication for the spectrum of the reduced density matrix is as follows. Parameterizing the eigenvalues of $\rho_A$ in terms of an “entanglement energy” as $\lambda_i = e^{-\epsilon_i}$ [@ent_spectrum; @pollmann2010], and introducing the “density of states” $\varrho(\epsilon)$, we have $$S^{(n)}_A = \frac{1}{1-n}\log\left[\int d\epsilon \, \varrho(\epsilon) e^{-n\epsilon} \right].$$ The behaviour $S^{(n)}_A(t)\sim v_n t$ is then consistent with a density $\varrho(\epsilon)$ having the large deviation form $$\label{eq:large_dev}
\varrho(\epsilon) \sim \exp[t \pi(\epsilon/t)],$$ for some function $\pi(\eta)$. In the saddle point approximation we find $$\label{eq:rates}
v_n = \frac{S^{(n)}_A}{t} = \frac{\pi(\eta_n)-\eta_n}{1-n}$$ where $\eta_n$ is determined by $n=\pi'(\eta_n)$. The growth rates $v_n$ are seen to be related to the function $p(\eta)$ describing the density of states of the entanglement spectrum by Legendre transformation. Note that the numerator in Eq. vanishes at $n=1$ due to the normalization of the density matrix, yielding a finite growth rate $v_1$ for the von Neumann entropy $S^{(1)}_A$. Rényi entropies with smaller $n$ grow faster; the reduced density matrix for a subsystem becomes thermal when the slowest Rényi entropy, the so-called “min-entropy” $S_\infty \equiv \max_i(\epsilon_i)$ (i.e., the largest entanglement eigenvalue) has saturated.
As well as the rates $v_n$, the time evolution is characterized by the butterfly velocity $v_B$ at which local perturbations spread [@ho2017; @mezei2017entanglement; @nrvh; @Keyserlingk2017; @Nahum2017; @tianci]. Although the various velocities are generically separate, they coincide in exactly solvable models (such as random circuits in the limit of large local Hilbert space dimension or Clifford gates [@nrvh; @Keyserlingk2017; @Nahum2017] and the self-dual kicked Ising model [@Bertini:2018aa; @Bertini:2018fbz]), so the entanglement spectrum evolves in a trivial way. Away from these non-generic limits, little is known analytically about the entanglement spectrum. A few Rényi entropies can be explicitly computed by mapping the circuit dynamics to random classical partition functions [@Keyserlingk2017; @Nahum2017; @tianci] but these mappings do not yield the full entanglement spectrum. The picture that emerges from numerical studies is, however, that the entanglement spectrum has nontrivial structure in generic systems, such as a bandwidth that widens linearly in time; this feature is absent in the exactly solvable limits [@ccgp]. However, this structure is not fully understood at present. In the present work we develop and apply a numerical transfer-matrix approach to compute the structure of the entanglement spectrum for spatially infinite systems at early times. This approach allows us to access some aspects of entanglement for larger subsystems than were studied in previous work: for instance, we are able to compute the spatial fluctuations of entanglement in systems of $10,000$ sites. Our approach works formally with infinite systems; we assume that the system of interest has been initialized in a product state and then subjected to a finite-depth quantum circuit (i.e., evolution for a finite time) consisting of on-site or nearest-neighbor quantum gates (see Fig. \[circfig\]). We compute the spectrum of the reduced density matrix of a bipartition into two semi-infinite regions at an arbitrary point. After applying a quantum circuit of finite depth $t$, the reduced density matrix has rank $q^{t-1}$. We will see that the spectrum of the reduced density matrix can be interpreted as that of a $q^{t-1} \times q^{t-1}$ matrix $R$ acting on an ancilla space. In addition, reduced density matrices across adjacent cuts are related by quantum channels that are straightforward to construct given the circuit. These quantum channels act as transfer matrices for the entanglement spectrum.
\[circfig\] $\cdots$
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[(0.5+-4,0.5+-1.5) rectangle (-0.5+-4,-0.5+-1.5); (-4,-1.5) node [$U_7$]{};]{} [(0.5+-2,0.5+-1.5) rectangle (-0.5+-2,-0.5+-1.5); (-2,-1.5) node [$U_8$]{};]{} [(0.5+0,0.5+-1.5) rectangle (-0.5+0,-0.5+-1.5); (0,-1.5) node [$U_9$]{};]{} [(0.5+2,0.5+-1.5) rectangle (-0.5+2,-0.5+-1.5); (2,-1.5) node [$U_{10}$]{};]{} [(0.5+4,0.5+-1.5) rectangle (-0.5+4,-0.5+-1.5); (4,-1.5) node [$U_{11}$]{};]{} [(0.5+6,0.5+-1.5) rectangle (-0.5+6,-0.5+-1.5); (6,-1.5) node [$U_{12}$]{};]{}
[(0.5+-5,0.5+0) rectangle (-0.5+-5,-0.5+0); (-5,0) node [$U_{13}$]{};]{} [(0.5+-3,0.5+0) rectangle (-0.5+-3,-0.5+0); (-3,0) node [$U_{14}$]{};]{} [(0.5+-1,0.5+0) rectangle (-0.5+-1,-0.5+0); (-1,0) node [$U_{15}$]{};]{} [(0.5+1,0.5+0) rectangle (-0.5+1,-0.5+0); (1,0) node [$U_{16}$]{};]{} [(0.5+3,0.5+0) rectangle (-0.5+3,-0.5+0); (3,0) node [$U_{17}$]{};]{} [(0.5+5,0.5+0) rectangle (-0.5+5,-0.5+0); (5,0) node [$U_{18}$]{};]{}
[(0.5+-4,0.5+1.5) rectangle (-0.5+-4,-0.5+1.5); (-4,1.5) node [$U_{19}$]{};]{} [(0.5+-2,0.5+1.5) rectangle (-0.5+-2,-0.5+1.5); (-2,1.5) node [$U_{20}$]{};]{} [(0.5+0,0.5+1.5) rectangle (-0.5+0,-0.5+1.5); (0,1.5) node [$U_{21}$]{};]{} [(0.5+2,0.5+1.5) rectangle (-0.5+2,-0.5+1.5); (2,1.5) node [$U_{22}$]{};]{} [(0.5+4,0.5+1.5) rectangle (-0.5+4,-0.5+1.5); (4,1.5) node [$U_{23}$]{};]{} [(0.5+6,0.5+1.5) rectangle (-0.5+6,-0.5+1.5); (6,1.5) node [$U_{24}$]{};]{}
[(-4.5,-4) – (-4.5,2.5);]{} [(-3.5,-4) – (-3.5,2.5);]{} [(-2.5,-4) – (-2.5,2.5);]{} [(-1.5,-4) – (-1.5,2.5);]{} [(-0.5,-4) – (-0.5,2.5);]{} [(0.5,-4) – (0.5,2.5);]{} [(1.5,-4) – (1.5,2.5);]{} [(2.5,-4) – (2.5,2.5);]{} [(3.5,-4) – (3.5,2.5);]{} [(4.5,-4) – (4.5,2.5);]{} [(5.5,-4) – (5.5,2.5);]{}
[(-5.5,-4) – (-5.5,-3);]{} [(-5.5,-3) – (-5.5,-2);]{} [(-5.5,-1) – (-5.5,0);]{} [(-5.5,0) – (-5.5,1);]{}
[(6.5,-2.5) – (6.5,-0.5);]{} [(6.5,0.5) – (6.5,2.5);]{}
(0.5,-4.2) circle (.2); (1.5,-4.2) circle (.2);
(2.5,-4.2) circle (.2); (3.5,-4.2) circle (.2);
(4.5,-4.2) circle (.2); (5.5,-4.2) circle (.2);
(-1.5,-4.2) circle (.2); (-0.5,-4.2) circle (.2);
(-2.5,-4.2) circle (.2); (-3.5,-4.2) circle (.2);
(-4.5,-4.2) circle (.2); (-5.5,-4.2) circle (.2);
(6.5,2.7) circle (.2);
(0.5,2.7) circle (.2); (1.5,2.7) circle (.2);
(2.5,2.7) circle (.2); (3.5,2.7) circle (.2);
(4.5,2.7) circle (.2); (5.5,2.7) circle (.2);
(-1.5,2.7) circle (.2); (-0.5,2.7) circle (.2);
(-2.5,2.7) circle (.2); (-3.5,2.7) circle (.2);
(-4.5,2.7) circle (.2);
$\cdots$
The quantum-channel perspective is helpful for a number of reasons. First, using this method one can compute the largest few eigenvalues of the entanglement spectrum for circuits that are too deep to permit direct simulation. Second, as the transfer matrix acts on formally infinite systems, spatial fluctuations of entanglement can be directly studied. Third, standard methods from quantum optics such as the stochastic unraveling of quantum channels [@carmichael_book] can be applied to simulate the dynamics of entanglement on larger scales than direct simulation permits. Finally, for translation-invariant initial-states evolving under translation-invariant circuits, one can iterate the quantum channel until it converges, and thus extract the entanglement spectrum, free of finite-size effects, without incurring the computational overhead of time-evolving a large system.
These are the issues we explore in the present work. Our main results are as follows. First, we provide algorithms based on quantum channels for computing the entanglement spectrum and its low-energy tail, as well as for computing some Rényi entropies, by propagating the quantum channel in an ancilla space. For the exactly solvable model of Ref. [@Bertini:2018fbz] we analytically demonstrate that the entanglement spectrum is trivial, using the properties of the associated quantum channel. We compute the behavior of the “low-energy” (large Schmidt rank) tail of the entanglement spectrum for random unitary circuits, random circuits with a conservation law, and translation-invariant integrable circuits, acting on various initial states. We compute the distributions of the purity and of the min-entropy; for random circuits we find that both the second Rényi entropy and the min-entropy follow Gaussian distributions at the accessible circuit depths (the purity, therefore, follows a log-normal distribution). For circuits with conservation laws or translation-invariant circuits, the nature of this low-entanglement-energy tail is sensitive to the fluctuations in the initial state. We compute the spatial correlations of entanglement, and find that their correlation length grows sub-linearly in time. However, the correlation lengths are short at the accessible times and we are not able to identify a definite exponent.
This paper is organized as follows. In Sec. \[background\] we briefly review concepts such as unitary circuits, matrix-product states, and quantum channels, as they apply to the algorithms introduced here. In Sec. \[channeldesc\] we describe how to construct quantum channels for unitary circuits, and estimate the complexity of various exact and approximate methods for extracting the entanglement spectrum (or some of its moments). In Sec. \[kickedIsing\] we explicitly compute the transfer matrix for the self-dual kicked Ising model, and confirm that all the Rényi entropies coincide in this model, as they are known to [@Bertini:2018aa; @Bertini:2018fbz]. In Sec. \[results\] we present results for the entanglement spectrum, the distributions of purity and min-entropy, and the growth of spatial correlations in the entanglement, in random unitary circuits and some variants of these. Finally Sec. \[conclusions\] summarizes our results and discusses future directions.
Background
==========
Quantum circuits
----------------
In this work we consider systems that evolve under the application of discrete local unitary gates tiled in the pattern shown in Fig. \[circfig\]. Our approach applies both to Floquet systems in which the gates are applied periodically in time, and to random circuits where each gate is drawn independently and randomly. As with the TEBD algorithm [@vidal2007], it can be extended to continuous time evolution under strictly local Hamiltonians by discretizing the time evolution, e.g., through a Suzuki-Trotter decomposition.
Matrix Product States
---------------------
In this section we introduce some background material on matrix product states (MPS’s), which will form an essential part of the following development. More detailed expositions may be found in Refs.[@Perez-Garcia:2006aa; @Schollwock:2011aa; @Orus:2014aa].
We consider a quantum system described by $N$ identical subsystems with finite Hilbert space dimension $q$. The (pure) quantum states of the system are therefore defined by vectors in the Hilbert space $${{\mathcal H}}_N \equiv \overbrace{\mathbb{C}^q\otimes \cdots \otimes \mathbb{C}^q}^{N\text{ times}}.$$ A basis of orthonormal product states has the form $$\ket{s_{1:N}} = \ket{s_1}_q \otimes \ket{s_2}_q\cdots \otimes \ket{s_N}_q,$$ where $\ket{s}_q$ $s=1,\ldots q$ is an orthonormal basis for $\mathbb{C}^q$, and we have introduced the sequence notation $s_{1:N} \equiv s_1,s_2,\ldots s_N$. By taking components of a vector $\ket{\Psi}\in{{\mathcal H}}$ $$\Psi_{s_{1:N}} = \braket{s_{1:N}|\Psi},$$ $\ket{\Psi}$ can be regarded as a rank-$N$ tensor with components $\Psi_{s_{1:N}}$.
An MPS is a tensor of the form $$\label{eq:MPSdef}
\Psi_{s_{1:N}} = A^{(1)}_{s_1}A^{(2)}_{s_2}\cdots A^{(N)}_{s_N},$$ where $A^{(1)}_s, \ldots A^{(N)}_s \in\mathbb{C}^{D_j\times D_{j+1}}$ are matrices, and the numbers $D_j$ $j=1,\ldots N+1$ are known as the *bond dimensions*, with $D_1=D_{N+1}=1$. Thus the product Eq. has the form ‘row vector, product of matrices, column vector’, and yields a complex number for each sequence $s_1:s_N$.
An arbitrary vector $\ket{\Psi}$ may be approximated by an MPS with an error that decreases as $D\equiv\max_j D_j$ increases. We will see that the state arising from applying a unitary circuit to a product state is exactly given by an MPS with $D=q^{d-1}$, where $d$ is the depth of the circuit.
A graphical notation for MPS proves to be extremely convenient, and is discussed extensively in Ref. [@Schollwock:2011aa]. In this notation tensors – such as the matrices or vectors $A^{(j)}$ – are represented as boxes, with the number of lines or edges entering a box indicating the number of indices the object bears. An edge joining two vertices indicates the (pairwise) contraction of an index. Thus the MPS in Eq. is denoted $$\Psi_{s_{1:N}} =
\begin{tikzpicture}[baseline = (X.base),every node/.style={scale=1},scale=.85]
\draw[rounded corners] (1,2) rectangle (2,1);
\draw (1.5,1.5) node (X) {$A^{(1)}$};
\draw (1.5,.2) node {$s_1$};
\draw (2,1.5) -- (3,1.5);
\draw[rounded corners] (3,2) rectangle (4,1);
\draw (3.5,1.5) node {$A^{(2)}$};
\draw (3.5,.2) node {$s_2$};
\draw (4,1.5) -- (4.5,1.5);
\draw (1.5,1) -- (1.5,.5); \draw (3.5,1) -- (3.5,.5);
\end{tikzpicture} \dots
\begin{tikzpicture}[baseline = (X.base),every node/.style={scale=1},scale=.85]
\draw (0.5,1.5) -- (1,1.5);
\draw[rounded corners] (1,2) rectangle (2,1);
\draw (1.5,1.5) node (X) {$A^{(N)}$};
\draw (1.5,.2) node {$s_N$};
\draw (1.5,1) -- (1.5,.5);
\end{tikzpicture}$$ Here, $A^{(2)}$ has three lines attached because the collection of matrices $A^{(2)}_s\in \mathbb{C}^{D\times D}$ may be regarded as rank-3 tensor $A^{(2)}\in \mathbb{C}^{D\times D\times q}$. The vertical leg represents the indices $s_j$ that live in the ‘physical space’ while the horizontal lines represent indices in the ‘bond space’ (which we shall also refer to as the ‘ancilla space’).
The squared norm of a vector $\ket{\Psi}$ may be calculated by contracting all physical indices between $\Psi_{s_{1:N}}$ and $\bar\Psi_{s_{1:N}}$. This has the graphical representation: $$\braket{\Psi|\Psi}=\sum_{s_{1:N}} \bar\Psi_{s_{1:N}}\Psi_{s_{1:N}} =
\begin{tikzpicture}[anchor=base, baseline,every node/.style={scale=1},scale=.75]
\draw[rounded corners] (1,1.25) rectangle (2,0.25);
\draw (1.5,0.5) node (X) {$A^{(1)}$};
\draw[rounded corners] (1,-0.25) rectangle (2,-1.25);
\draw (1.5,-1) node (X) {$\bar A^{(1)}$};
\draw (1.5,0.25) -- (1.5,-0.25);
\draw (2,0.75) -- (3,0.75);
\draw (2,-0.75) -- (3,-0.75);
\draw[rounded corners] (3,1.25) rectangle (4,0.25);
\draw (3.5,0.5) node (X) {$A^{(2)}$};
\draw[rounded corners] (3,-0.25) rectangle (4,-1.25);
\draw (3.5,-1) node (X) {$\bar A^{(2)}$};
\draw (3.5,0.25) -- (3.5,-0.25);
\draw (4,0.75) -- (4.5,0.75);
\draw (4,-0.75) -- (4.5,-0.75);
\end{tikzpicture} \dots
\begin{tikzpicture}[anchor=base, baseline,every node/.style={scale=1},scale=.75]
\draw (0.5,0.75) -- (1,0.75);
\draw (0.5,-0.75) -- (1,-0.75);
\draw[rounded corners] (1,1.25) rectangle (2,0.25);
\draw (1.5,0.5) node (X) {$A^{(N)}$};
\draw[rounded corners] (1,-0.25) rectangle (2,-1.25);
\draw (1.5,-1) node (X) {$\bar A^{(N)}$};
\draw (1.5,0.25) -- (1.5,-0.25);
\end{tikzpicture}$$
More generally, the reduced density matrix for the leftmost $n$ subsystems that arises from a pure state by tracing over the remaining subsystems is denoted
$$\label{eq:rdm}
\rho_{s_{1:n},s'_{1:n}} \equiv\sum_{s_{n+1:N}} \bar\Psi_{s_{1:n}s_{n+1:N}}\Psi_{s'_{1:n}s_{n+1:N}}=
\begin{tikzpicture}[anchor=base, baseline,every node/.style={scale=0.9},scale=.75]
\draw[rounded corners] (1,1.75) rectangle (2,0.75);
\draw (1.5,1) node (X) {$1$};
\draw[rounded corners] (1,-0.75) rectangle (2,-1.75);
\draw (1.5,-1.5) node (X) {$\bar 1$};
\draw (1.5,0.75) -- (1.5,0.4);
\draw (1.5,0.2) node {$s_1$};
\draw (1.5,-0.75) -- (1.5,-0.4);
\draw (1.5,-0.3) node {$s'_1$};
\draw (2,1.25) -- (2.5,1.25);
\draw (2,-1.25) -- (2.5,-1.25);
\end{tikzpicture} \dots
\begin{tikzpicture}[anchor=base, baseline,every node/.style={scale=0.9},scale=.75]
\draw (0.5,1.25) -- (1,1.25);
\draw (0.5,-1.25) -- (1,-1.25);
\draw[rounded corners] (1,1.75) rectangle (2,0.75);
\draw (1.5,1) node (X) {$n$};
\draw[rounded corners] (1,-0.75) rectangle (2,-1.75);
\draw (1.5,-1.5) node (X) {$\bar n$};
\draw (1.5,0.75) -- (1.5,0.4);
\draw (1.5,0.2) node {$s_n$};
\draw (1.5,-0.75) -- (1.5,-0.4);
\draw (1.5,-0.3) node {$s'_n$};
\draw (2,1.25) -- (3,1.25);
\draw (2,-1.25) -- (3,-1.25);
\draw[rounded corners] (3,1.75) rectangle (4,0.75);
\draw (3.5,1) node (X) {$n+1$};
\draw[rounded corners] (3,-0.75) rectangle (4,-1.75);
\draw (3.5,-1.5) node (X) {$\overline {n+1}$};
\draw (3.5,0.75) -- (3.5,-0.75);
\draw (4,1.25) -- (4.5,1.25);
\draw (4,-1.25) -- (4.5,-1.25);
\end{tikzpicture}\cdots
\begin{tikzpicture}[anchor=base, baseline,every node/.style={scale=0.9},scale=.75]
\draw (0.5,1.25) -- (1,1.25);
\draw (0.5,-1.25) -- (1,-1.25);
\draw[rounded corners] (1,1.75) rectangle (2,0.75);
\draw (1.5,1) node (X) {$N$};
\draw[rounded corners] (1,-0.75) rectangle (2,-1.75);
\draw (1.5,-1.5) node (X) {$\bar N$};
\draw (1.5,0.75) -- (1.5,-0.75);
\end{tikzpicture},$$
where for simplicity we denote $A^{(j)}$ by $j$ and $a'^{(j)}$ by $\bar j$. The above expressions may also be written $$\label{eq:rdm_R}
\rho_{s_{1:n},s'_{1:n}} =
\begin{tikzpicture}[anchor=base, baseline,every node/.style={scale=0.9},scale=.75]
\draw[rounded corners] (1,1.75) rectangle (2,0.75);
\draw (1.5,1) node (X) {$1$};
\draw[rounded corners] (1,-0.75) rectangle (2,-1.75);
\draw (1.5,-1.5) node (X) {$\bar 1$};
\draw (1.5,0.75) -- (1.5,0.4);
\draw (1.5,0.2) node {$s_1$};
\draw (1.5,-0.75) -- (1.5,-0.4);
\draw (1.5,-0.3) node {$s'_1$};
\draw (2,1.25) -- (2.5,1.25);
\draw (2,-1.25) -- (2.5,-1.25);
\end{tikzpicture} \dots
\begin{tikzpicture}[anchor=base, baseline,every node/.style={scale=0.9},scale=.75]
\draw (0.5,1.25) -- (1,1.25);
\draw (0.5,-1.25) -- (1,-1.25);
\draw[rounded corners] (1,1.75) rectangle (2,0.75);
\draw (1.5,1) node (X) {$n$};
\draw[rounded corners] (1,-0.75) rectangle (2,-1.75);
\draw (1.5,-1.5) node (X) {$\bar n$};
\draw (1.5,0.75) -- (1.5,0.4);
\draw (1.5,0.2) node {$s_n$};
\draw (1.5,-0.75) -- (1.5,-0.4);
\draw (1.5,-0.3) node {$s'_n$};
\draw (3,0) circle (.5);
\draw (3,-0.1) node (X) {$R^{(n)}$};
\draw (3,0.5) edge[out=90,in=0] (2,1.25);
\draw (3,-0.5) edge[out=270,in=0] (2,-1.25);
\end{tikzpicture},$$ where the hermitian matrices $R^{(j)}\in \mathbb{C}^{D_{j+1}\times D_{j+1}}$ are defined by $$\label{eq:CPTP}
R^{(j-1)} = \sum_s A^{(j)\vphantom{\dagger}}_{s} R^{(j)} A^{(j)\dagger}_{s}, \qquad j=n+1,\ldots N,$$ and $R^{(N)}=1$.
### Canonical Forms
The MPS representation of $\ket{\Psi}$ has a redundancy sometimes referred to as ‘gauge freedom’. For a set $X_j$ $j=1,\ldots N-1$ of invertible matrices, the transformation $$\begin{aligned}
A^{(1)}_s &\to A^{(1)}_s X_{1}^{-1},\nonumber\\
A^{(j)}_s &\to X_{j-1}A^{(j)}_s X_{j}^{-1},\qquad j=2,\ldots N-1\nonumber\\
A^{(N)}_s &\to X_{N-1}A^{(N)}_s ,\end{aligned}$$ leaves $\Psi_{s_{1:N}}$ unchanged. Further conditions may be imposed to reduce this redundancy [@Perez-Garcia:2006aa]. We will be concerned with matrices in *left canonical* form, satisfying the condition $$\begin{aligned}
\label{eq:left-c}
\sum_s A^{(j)\dagger}_{s} A^{(j)\vphantom{\dagger}}_{s} &= \openone_{D_{j+1}}\\
\begin{tikzpicture}[baseline={([yshift=-1ex]current bounding box.center)},every node/.style={scale=0.75},scale=.55]
\draw (1,-1.5) edge[out=180,in=180] (1,1.5);
\draw[rounded corners] (1,2) rectangle (2,1);
\draw[rounded corners] (1,-1) rectangle (2,-2);
\draw (1.5,1) -- (1.5,-1);
\draw (1.5,1.5) node {$A^{(j)}$};
\draw (1.5,-1.5) node {$\bar{A}^{(j)}$};
\draw (2,1.5) -- (2.5,1.5); \draw (2,-1.5) -- (2.5,-1.5);
\end{tikzpicture}
&=
\begin{tikzpicture}[baseline={([yshift=-1ex]current bounding box.center)},every node/.style={scale=0.750},scale=.55]
\draw (1,-1.5) edge[out=180,in=180] (1,1.5);
\end{tikzpicture}.\end{aligned}$$ Note that $D_2=q$ is necessary for $A^{(1)}$ to be placed in left canonical form, but this places no restriction on the state.
Analogously, matrices in *right* canonical form satisfy $$\begin{aligned}
\label{eq:right-c}
\sum_s A^{(j)\vphantom{\dagger}}_{s}A^{(j)\dagger}_{s} &= \openone_{D_{j}}\\
\begin{tikzpicture}[baseline={([yshift=-1ex]current bounding box.center)},every node/.style={scale=0.75},scale=.55]
\draw (2,-1.5) edge[out=0,in=0] (2,1.5);
\draw[rounded corners] (1,2) rectangle (2,1);
\draw[rounded corners] (1,-1) rectangle (2,-2);
\draw (1.5,1) -- (1.5,-1);
\draw (1.5,1.5) node {$A^{(j)}$};
\draw (1.5,-1.5) node {$\bar{A}^{(j)}$};
\draw (0.5,1.5) -- (1,1.5); \draw (0.5,-1.5) -- (1,-1.5);
\end{tikzpicture}
&=
\begin{tikzpicture}[baseline={([yshift=-1ex]current bounding box.center)},every node/.style={scale=0.750},scale=.55]
\draw (1,-1.5) edge[out=0,in=0] (1,1.5);
\end{tikzpicture}.\end{aligned}$$ (requiring $D_N=q$).
One benefit of the canonical forms is that they may be contracted “automatically”. For example, in terms of an MPS in right canonical form the reduced density matrix $\rho_{s_{1:n},s'_{1:n}}$ in Eq. takes the form $$\label{eq:rdm_canon}
\rho_{s_{1:n},s'_{1:n}} =
\begin{tikzpicture}[anchor=base, baseline,every node/.style={scale=0.9},scale=.75]
\draw[rounded corners] (1,1.75) rectangle (2,0.75);
\draw (1.5,1) node (X) {$1$};
\draw[rounded corners] (1,-0.75) rectangle (2,-1.75);
\draw (1.5,-1.5) node (X) {$\bar 1$};
\draw (1.5,0.75) -- (1.5,0.4);
\draw (1.5,0.2) node {$s_1$};
\draw (1.5,-0.75) -- (1.5,-0.4);
\draw (1.5,-0.3) node {$s'_1$};
\draw (2,1.25) -- (2.5,1.25);
\draw (2,-1.25) -- (2.5,-1.25);
\end{tikzpicture} \dots
\begin{tikzpicture}[anchor=base, baseline,every node/.style={scale=0.9},scale=.75]
\draw (0.5,1.25) -- (1,1.25);
\draw (0.5,-1.25) -- (1,-1.25);
\draw[rounded corners] (1,1.75) rectangle (2,0.75);
\draw (1.5,1) node (X) {$n$};
\draw[rounded corners] (1,-0.75) rectangle (2,-1.75);
\draw (1.5,-1.5) node (X) {$\bar n$};
\draw (1.5,0.75) -- (1.5,0.4);
\draw (1.5,0.2) node {$s_n$};
\draw (1.5,-0.75) -- (1.5,-0.4);
\draw (1.5,-0.3) node {$s'_n$};
\draw (2,-1.25) edge[out=0,in=0] (2,1.25);
\end{tikzpicture}.$$ A second benefit – which is more relevant for us – is that the spectrum of the reduced density matrix $\rho_{s_{1:n},s'_{1:n}}$ coincides with the spectrum of $R^{(n)}$ for an MPS in *left* canonical form. This may be seen by introducing the spectral representation $$R^{(n)} = \sum_\alpha \lambda_\alpha r_\alpha r_\alpha^\dagger,$$ in terms of the eigenvalues $\lambda_\alpha$ and eigenvectors $r_\alpha$ of $R^{(n)}$. Upon substitution into Eq. this yields a spectral representation for $\rho_{s_{1:n},s'_{1:n}}$ $$\label{eq:rdm_spectrum}
\rho_{s_{1:n},s'_{1:n}} = \sum_\alpha \lambda_\alpha
\begin{tikzpicture}[anchor=base, baseline,every node/.style={scale=0.9},scale=.75]
\draw[rounded corners] (1,1.75) rectangle (2,0.75);
\draw (1.5,1) node (X) {$1$};
\draw[rounded corners] (1,-0.75) rectangle (2,-1.75);
\draw (1.5,-1.5) node (X) {$\bar 1$};
\draw (1.5,0.75) -- (1.5,0.4);
\draw (1.5,0.2) node {$s_1$};
\draw (1.5,-0.75) -- (1.5,-0.4);
\draw (1.5,-0.3) node {$s'_1$};
\draw (2,1.25) -- (2.5,1.25);
\draw (2,-1.25) -- (2.5,-1.25);
\end{tikzpicture} \dots
\begin{tikzpicture}[anchor=base, baseline,every node/.style={scale=0.9},scale=.75]
\draw (0.5,1.25) -- (1,1.25);
\draw (0.5,-1.25) -- (1,-1.25);
\draw[rounded corners] (1,1.75) rectangle (2,0.75);
\draw (1.5,1) node (X) {$n$};
\draw[rounded corners] (1,-0.75) rectangle (2,-1.75);
\draw (1.5,-1.5) node (X) {$\bar n$};
\draw (1.5,0.75) -- (1.5,0.4);
\draw (1.5,0.2) node {$s_n$};
\draw (1.5,-0.75) -- (1.5,-0.4);
\draw (1.5,-0.3) node {$s'_n$};
\draw (2,1.25) -- (2.5,1.25);
\draw (2,-1.25) -- (2.5,-1.25);
\draw (3,1.25) circle (.5);
\draw (3,1.1) node (X) {$r_\alpha$};
\draw (3,-1.25) circle (.5);
\draw (3,-1.3) node (X) {$\bar r_\alpha$};
\end{tikzpicture},$$ in terms of vectors in ${{\mathcal H}}_n$ that are orthonormal by the left canonical condition [@Perez-Garcia:2006aa].
A more direct way of seeing that the spectra of $R^{(n)}$ and $\rho_A$ coincide, for MPS’s in left canonical form, is shown in Fig. \[cforms\]. The moments of the spectrum are given by matrix products of the form $\rho_A^k$; one can use the left canonical property to eliminate the “boxes” pairwise and arrive at the result $\operatorname{tr}\rho_A^k = \operatorname{tr}[ (R^{(n)})^k]$. Since all moments coincide, $\rho_A$ and $R^{(n)}$ must have the same spectrum.
Quantum Channels
----------------
The above formalism has a natural interpretation in terms of quantum channels, or completely positive trace preserving (CPTP) maps [@Perez-Garcia:2006aa]. In this interpretation the matrices $R^{(j)}$ are regarded as a sequence of density matrices that represent mixed states in the bond space. The definition of $R^{(j)}$ given in Eq. guarantees that the maps from one $R^{(j)}$ to the next are completely positive (Choi’s theorem), while the left canonical condition Eq. ensures that they are trace preserving. Therefore, contracting a physical leg (i.e., moving the entanglement cut in real space) amounts to applying a CPTP map to the ancilla. In this context the $A^{(j)}_s$ are known as Kraus operators.
From now on we assume without loss of generality that all $D_j=D$, which may be achieved by padding the matrices with zeros. Square matrices $A^{(j)}_s\in \mathbb{C}^{D\times D}$ satisfying the left canonical condition may be parameterized in terms of unitary matrices $U_j\in{{\mathcal U}}(qD)$ as $$\label{eq:U_ensemble}
A^{(j)}_{s,ab} = \bra{s}_q\bra{a}_D U_j\ket{0}_q\ket{b}_D.$$ Physically, this corresponds to the amplitude for the following process: prepare the physical subsystem $j$ in a fixed state $\ket{0}_q$, and then act on the state $\ket{0}_q\ket{b}_D$ of the subsystem $j$ and ancilla with a unitary, arriving in state $\ket{s}_q\ket{a}_D$. While any CPTP map may be presented in this form [@lindblad1976generators], we will see that this is precisely how quantum channels arise in the case of unitary circuits.
Quantum channels from unitary circuits {#channeldesc}
======================================
This section is organized as follows. We first discuss how to slice up a unitary circuit into an MPS where all matrices are in canonical form. This immediately gives us a quantum channel that propagates the ancilla-space density matrix $R$ in the spatial direction. We then discuss two controlled but approximate ways of implementing the quantum channel for larger subsystems: first, an approach for computing the entanglement spectrum based on approximating the ancilla-space density matrix $R$ as a lower-rank object; and second, an approach for computing the purity, specifically, by a stochastic unraveling of the quantum channel (i.e., by sampling “quantum trajectories” in ancilla space [@carmichael_book]).
From unitary circuits to canonical-form MPS’s
---------------------------------------------
We now introduce the main idea behind our approach. A planar unitary circuit that starts from a product state may be presented as an MPS by slicing it into strips in an arbitrary way. Each slice $j$ is associated with $q^2$ matrices $A^{(j)}_{s_1,s_2}$ indexed by two physical indices. The dimension of the ancilla is $D=q^{d-1}$.For a general decomposition of a circuit, the resulting matrices $A^{(j)}_{s_1,s_2}$ will not be in the appropriate canonical form, so the spectra of $R^{(j)}$ and $\rho_A$ will not coincide. An MPS in canonical form *is* obtained for slices along the south-west to north-east diagonal $$A^{(j)}_{s_1,s_2;a_{1:3},b_{1:3}}=
\begin{tikzpicture}[baseline={([yshift=-1ex]current bounding box.center)},every node/.style={scale=1},scale=.55]
{\draw[rounded corners] (0.5+1,0.5+-3) rectangle (-0.5+1,-0.5+-3); \draw (1,-3) node {$U_1$};}
{\draw[rounded corners] (0.5+2,0.5+-1.5) rectangle (-0.5+2,-0.5+-1.5); \draw (2,-1.5) node {$U_{2}$};}
{\draw[rounded corners] (0.5+3,0.5+0) rectangle (-0.5+3,-0.5+0); \draw (3,0) node {$U_{3}$};}
{\draw[rounded corners] (0.5+4,0.5+1.5) rectangle (-0.5+4,-0.5+1.5); \draw (4,1.5) node {$U_{4}$};}
\draw (0.5,-2.75) edge[out=90,in=0] (0,-2.25);
\draw (1.5,-1.25) edge[out=90,in=0] (1,-0.75);
\draw (2.5,0.25) edge[out=90,in=0] (2,0.75);
\draw (2.5,-1.75) edge[out=-90,in=180] (3,-2.25);
\draw (3.5,-0.25) edge[out=-90,in=180] (4,-0.75);
\draw (4.5,1.25) edge[out=-90,in=180] (5,0.75);
{\draw (0.5,-4) -- (0.5,-3);}
{\draw (1.5,-4) -- (1.5,-1.5);}
{\draw (2.5,-1.5) -- (2.5,0);}
{\draw (3.5,0) -- (3.5,2.5);}
{\draw (4.5,1.5) -- (4.5,2.5);}
\draw (0.5,-4.2) circle (.2);
\draw (1.5,-4.2) circle (.2);
\draw (3.5,2.7) node (X) {$s_1$};
\draw (4.5,2.7) node (X) {$s_2$};
\draw (-0.3,-2.25) node (X) {$a_3$};
\draw (0.7,-0.75) node (X) {$a_2$};
\draw (1.7,0.75) node (X) {$a_1$};
\draw (3.3,-2.25) node (X) {$b_3$};
\draw (4.3,-0.75) node (X) {$b_2$};
\draw (5.3,0.75) node (X) {$b_1$};
\end{tikzpicture}.$$ A graphical proof is straightforward, since $$\sum_{s_1,s_2} \left(A^{(j)\dagger}_{s_1,s_2} A^{(j)}_{s_1,s_2}\right)_{b'_{1:3},b_{1:3}}=
\begin{tikzpicture}[baseline={([yshift=-1ex]current bounding box.center)},every node/.style={scale=1},scale=.55]
{\draw[rounded corners] (0.5+1,0.5+-3) rectangle (-0.5+1,-0.5+-3); \draw (1,-3) node {$U_1$};}
{\draw[rounded corners] (0.5+2,0.5+-1.5) rectangle (-0.5+2,-0.5+-1.5); \draw (2,-1.5) node {$U_{2}$};}
{\draw[rounded corners] (0.5+3,0.5+0) rectangle (-0.5+3,-0.5+0); \draw (3,0) node {$U_{3}$};}
{\draw[rounded corners] (0.5+4,0.5+1.5) rectangle (-0.5+4,-0.5+1.5); \draw (4,1.5) node {$U_{4}$};}
{\draw[rounded corners] (0.5+1,0.5+7.5) rectangle (-0.5+1,-0.5+7.5); \draw (1,7.5) node {$U^\dagger_1$};}
{\draw[rounded corners] (0.5+2,0.5+6) rectangle (-0.5+2,-0.5+6); \draw (2,6) node {$U^\dagger_{2}$};}
{\draw[rounded corners] (0.5+3,0.5+4.5) rectangle (-0.5+3,-0.5+4.5); \draw (3,4.5) node {$U^\dagger_{3}$};}
{\draw[rounded corners] (0.5+4,0.5+3) rectangle (-0.5+4,-0.5+3); \draw (4,3) node {$U^\dagger_{4}$};}
\draw (2.5,-1.75) edge[out=-90,in=180] (3,-2.25);
\draw (3.5,-0.25) edge[out=-90,in=180] (4,-0.75);
\draw (4.5,1.25) edge[out=-90,in=180] (5,0.75);
\draw (2.5,6.25) edge[out=90,in=180] (3,6.75);
\draw (3.5,4.75) edge[out=90,in=180] (4,5.25);
\draw (4.5,3.25) edge[out=90,in=180] (5,3.75);
{\draw (0.5,-4) -- (0.5,8.5);}
{\draw (1.5,-4) -- (1.5,8.5);}
{\draw (2.5,-1.5) -- (2.5,6);}
{\draw (3.5,0) -- (3.5,4.5);}
{\draw (4.5,1.5) -- (4.5,3);}
\draw (0.5,-4.2) circle (.2);
\draw (1.5,-4.2) circle (.2);
\draw (0.5,8.7) circle (.2);
\draw (1.5,8.7) circle (.2);
\draw (3.3,-2.25) node (X) {$b_3$};
\draw (4.3,-0.75) node (X) {$b_2$};
\draw (5.3,0.75) node (X) {$b_1$};
\draw (3.3,6.75) node (X) {$b'_3$};
\draw (4.3,5.25) node (X) {$b'_2$};
\draw (5.3,3.75) node (X) {$b'_1$};
\end{tikzpicture}.$$ The unitarity of the constituent blocks $$\begin{tikzpicture}[baseline={([yshift=-1ex]current bounding box.center)},every node/.style={scale=1},scale=.55]
{\draw[rounded corners] (0.5+1,0.5+0) rectangle (-0.5+1,-0.5+0); \draw (1,0) node {$U_1$};}
{\draw[rounded corners] (0.5+1,0.5+1.5) rectangle (-0.5+1,-0.5+1.5); \draw (1,1.5) node {$U^\dagger_1$};}
{\draw (0.5,-1) -- (0.5,2.5);}
{\draw (1.5,-1) -- (1.5,2.5);}
\draw (0.5,-1.5) node (X) {$b_1$};
\draw (1.5,-1.5) node (X) {$b_2$};
\draw (0.5,3) node (X) {$b'_1$};
\draw (1.5,3) node (X) {$b'_2$};
\end{tikzpicture}=
\begin{tikzpicture}[baseline={([yshift=-1ex]current bounding box.center)},every node/.style={scale=1},scale=.55]
{\draw (0.5,-1) -- (0.5,2.5);}
{\draw (1.5,-1) -- (1.5,2.5);}
\draw (0.5,-1.5) node (X) {$b_1$};
\draw (1.5,-1.5) node (X) {$b_2$};
\draw (0.5,3) node (X) {$b'_1$};
\draw (1.5,3) node (X) {$b'_2$};
\end{tikzpicture}
=\delta_{b_1,b'_1}\delta_{b_2,b'_2}$$ guarantees that the left canonical condition $$\sum_{s_1,s_2} A^{(j)\dagger}_{s_1,s_2} A^{(j)\vphantom{\dagger}}_{s_1,s_2} = \openone_{D}$$ is satisfied. Choosing the south-east to north-west diagonal gives an MPS in right canonical form.
Applying the quantum channel
----------------------------
We have shown that a unitary circuit may be represented as an MPS in canonical form. This observation may be used to efficiently compute the ancilla density matrices $R^{(j)}$, whose spectrum coincides with that of the reduced density matrix $\rho_{s_{1:j},s'_{1:j}}$.
Recall that the $R^{(j)}$ are defined by $$R^{(j-1)} = \sum_{s_1,s_2} A^{(j)\vphantom{\dagger}}_{s_1,s_2} R^{(j)} A^{(j)\dagger}_{s_1,s_2}, \qquad j=n+1,\ldots N.$$ After cutting along the SW-NE diagonal, $R^{(j-1)}_{a_{1:d-1},a'_{1:d-1}}$ has the graphical representation $$R^{(j-1)}_{a_{1:d-1},a'_{1:d-1}}=
\begin{tikzpicture}[baseline={([yshift=-1ex]current bounding box.center)},every node/.style={scale=1},scale=.55]
{\draw[rounded corners] (0.5+1,0.5+-3) rectangle (-0.5+1,-0.5+-3); \draw (1,-3) node {$U_1$};}
{\draw[rounded corners] (0.5+2,0.5+-1.5) rectangle (-0.5+2,-0.5+-1.5); \draw (2,-1.5) node {$U_{2}$};}
{\draw[rounded corners] (0.5+3,0.5+0) rectangle (-0.5+3,-0.5+0); \draw (3,0) node {$U_{3}$};}
{\draw[rounded corners] (0.5+4,0.5+1.5) rectangle (-0.5+4,-0.5+1.5); \draw (4,1.5) node {$U_{4}$};}
{\draw[rounded corners] (0.5+1,0.5+7.5) rectangle (-0.5+1,-0.5+7.5); \draw (1,7.5) node {$U^\dagger_1$};}
{\draw[rounded corners] (0.5+2,0.5+6) rectangle (-0.5+2,-0.5+6); \draw (2,6) node {$U^\dagger_{2}$};}
{\draw[rounded corners] (0.5+3,0.5+4.5) rectangle (-0.5+3,-0.5+4.5); \draw (3,4.5) node {$U^\dagger_{3}$};}
{\draw[rounded corners] (0.5+4,0.5+3) rectangle (-0.5+4,-0.5+3); \draw (4,3) node {$U^\dagger_{4}$};}
\draw (2.5,-1.75) edge[out=-90,in=180] (3,-2.25);
\draw (3.5,-0.25) edge[out=-90,in=180] (4,-0.75);
\draw (4.5,1.25) edge[out=-90,in=180] (5,0.75);
\draw (2.5,6.25) edge[out=90,in=180] (3,6.75);
\draw (3.5,4.75) edge[out=90,in=180] (4,5.25);
\draw (4.5,3.25) edge[out=90,in=180] (5,3.75);
\draw (0.5,-2.75) edge[out=90,in=0] (0,-2.25);
\draw (1.5,-1.25) edge[out=90,in=0] (1,-0.75);
\draw (2.5,0.25) edge[out=90,in=0] (2,0.75);
\draw (0.5,7.25) edge[out=-90,in=0] (0,6.75);
\draw (1.5,5.75) edge[out=-90,in=0] (1,5.25);
\draw (2.5,4.25) edge[out=-90,in=0] (2,3.75);
{\draw (0.5,-4) -- (0.5,-3);}
{\draw (1.5,-4) -- (1.5,-1.5);}
{\draw (0.5,7.5) -- (0.5,8.5);}
{\draw (1.5,6) -- (1.5,8.5);}
{\draw (2.5,-1.5) -- (2.5,0);}
{\draw (2.5,4.5) -- (2.5,6);}
{\draw (3.5,0) -- (3.5,4.5);}
{\draw (4.5,1.5) -- (4.5,3);}
\draw (0.5,-4.2) circle (.2);
\draw (1.5,-4.2) circle (.2);
\draw (0.5,8.7) circle (.2);
\draw (1.5,8.7) circle (.2);
\draw (-0.3,-2.25) node (X) {$a_3$};
\draw (0.7,-0.75) node (X) {$a_2$};
\draw (1.7,0.75) node (X) {$a_1$};
\draw (1.7,3.75) node (X) {$a'_1$};
\draw (0.7,5.25) node (X) {$a'_2$};
\draw (-0.3,6.75) node (X) {$a'_3$};
\draw (7,2.25) node (X) {$R^{(j)}$};
\draw[rounded corners] (6,3) rectangle (8,1.5);
\draw (5,0.75) edge[out=0,in=-90] (6.5,1.5);
\draw (4,-0.75) edge[out=0,in=-90] (7,1.5);
\draw (3,-2.25) edge[out=0,in=-90] (7.5,1.5);
\draw (5,3.75) edge[out=0,in=90] (6.5,3);
\draw (4,5.25) edge[out=0,in=90] (7,3);
\draw (3,6.75) edge[out=0,in=90] (7.5,3);
\end{tikzpicture}.$$ This expression may be simplified somewhat by noting that the topmost unitary ($U_4$ in the above example) may be eliminated to give $$R^{(j-1)}_{a_{1:d-1},a'_{1:d-1}}=
\begin{tikzpicture}[baseline={([yshift=-1ex]current bounding box.center)},every node/.style={scale=1},scale=.55]
{\draw[rounded corners] (0.5+1,0.5+-2) rectangle (-0.5+1,-0.5+-2); \draw (1,-2) node {$U_1$};}
{\draw[rounded corners] (0.5+2,0.5+-0.5) rectangle (-0.5+2,-0.5+-0.5); \draw (2,-0.5) node {$U_{2}$};}
{\draw[rounded corners] (0.5+3,0.5+1) rectangle (-0.5+3,-0.5+1); \draw (3,1) node {$U_{3}$};}
{\draw[rounded corners] (0.5+1,0.5+6.5) rectangle (-0.5+1,-0.5+6.5); \draw (1,6.5) node {$U^\dagger_1$};}
{\draw[rounded corners] (0.5+2,0.5+5) rectangle (-0.5+2,-0.5+5); \draw (2,5) node {$U^\dagger_{2}$};}
{\draw[rounded corners] (0.5+3,0.5+3.5) rectangle (-0.5+3,-0.5+3.5); \draw (3,3.5) node {$U^\dagger_{3}$};}
\draw (2.5,-0.75) edge[out=-90,in=180] (3,-1.25);
\draw (3.5,0.75) edge[out=-90,in=180] (4,0.25);
\draw (4.5,2.25) edge[out=-90,in=180] (5,1.75);
\draw (2.5,5.25) edge[out=90,in=180] (3,5.75);
\draw (3.5,3.75) edge[out=90,in=180] (4,4.25);
\draw (4.5,2.25) edge[out=90,in=180] (5,2.75);
\draw (0.5,-1.75) edge[out=90,in=0] (0,-1.25);
\draw (1.5,-0.25) edge[out=90,in=0] (1,0.25);
\draw (2.5,1.25) edge[out=90,in=0] (2,1.75);
\draw (0.5,6.25) edge[out=-90,in=0] (0,5.75);
\draw (1.5,4.75) edge[out=-90,in=0] (1,4.25);
\draw (2.5,3.25) edge[out=-90,in=0] (2,2.75);
{\draw (0.5,-3) -- (0.5,-2);}
{\draw (1.5,-3) -- (1.5,-0.5);}
{\draw (0.5,6.5) -- (0.5,7.5);}
{\draw (1.5,5) -- (1.5,7.5);}
{\draw (2.5,-0.5) -- (2.5,1);}
{\draw (2.5,3.5) -- (2.5,5);}
{\draw (3.5,1) -- (3.5,3.5);}
\draw (0.5,-3.2) circle (.2);
\draw (1.5,-3.2) circle (.2);
\draw (0.5,7.7) circle (.2);
\draw (1.5,7.7) circle (.2);
\draw (-0.3,-1.25) node (X) {$a_3$};
\draw (0.7,0.25) node (X) {$a_2$};
\draw (1.7,1.75) node (X) {$a_1$};
\draw (1.7,2.75) node (X) {$a'_1$};
\draw (0.7,4.25) node (X) {$a'_2$};
\draw (-0.3,5.75) node (X) {$a'_3$};
\draw (7,2.25) node (X) {$R^{(j)}$};
\draw[rounded corners] (6,3) rectangle (8,1.5);
\draw (5,1.75) edge[out=0,in=-90] (6.5,1.5);
\draw (4,0.25) edge[out=0,in=-90] (7,1.5);
\draw (3,-1.25) edge[out=0,in=-90] (7.5,1.5);
\draw (5,2.75) edge[out=0,in=90] (6.5,3);
\draw (4,4.25) edge[out=0,in=90] (7,3);
\draw (3,5.75) edge[out=0,in=90] (7.5,3);
\end{tikzpicture}.$$ The algorithm for applying the quantum channel is therefore:
1. Trace over the first index of $R^{(j)}$ $$R^{(j)}_{a_{1:D},a'_{1:D}}\rightarrow \sum_{a}R^{(j)}_{a a_{1:D-1},a a'_{1:D-1}}$$ This reduces the number of indices of $R^{(j)}$ to $2(d-2)$, or $q^{2(d-2)}$ components.
2. Apply the unitaries $U_{d-1}$ and $\bar U_{d-1}$. This increases the rank of the resulting tensor back to $2(d-1)$.
3. Continue applying unitaries from the “middle out” for $j=d-2,\ldots 2$.
4. Apply $U_1$ and $U^\dagger_1$, with the outer indices fixed.
This process involves $O(d)$ steps of matrix multiplication, where the matrices are of size $O(q^d)$. Thus if one directly applies the channel to a density matrix, the overall complexity is $O(d q^{2(d-1)})$. This is better than the naive $O(d q^{3(d-1)})$ because each of the unitaries that make up the channel is a sparse matrix. A Python implementation of the algorithm is available at <https://github.com/AustenLamacraft/ruc>.
Since our quantum channel is constructed from a diagonal cut through the unitary circuit, there will be edge effects in a rectangular circuit of finite width. Our approach is well suited to *infinite* width circuits: the channel is applied repeatedly to a random initial density matrix until a steady state density matrix is approached (for translationally invariant circuits) or a stationary distribution (for random circuits). This typically occurs on the scale of a number of steps roughly equal to the depth of the circuit.
Low-rank approximation of $R^{(n)}$ {#lra}
-----------------------------------
Away from fine-tuned points (see Sec. \[kickedIsing\]), finite-depth local unitary circuits give rise to entanglement spectra that are very broad; thus, the vast majority of the eigenstates of $\rho_A$ are close to zero and do not contribute to Rényi entropies with $n \geq 1$ [@ccgp]. This observation is implicit in the fact that different $S_n$ have different growth rates [@tianci]. This fact allows us to propagate $R^{(n)}$ with negligible error using far fewer than $q^{d-1}$ basis states. We proceed as follows. We approximate
\[equur\] R\^[(n)]{} \_[k = 1]{}\^K \_k |kk|, where $\lambda_k$ are the $K$ largest eigenvalues of $R$ and $|k\rangle$ are the associated eigenvectors. We renormalize all the eigenvalues to preserve the trace. We now evolve each $|k\rangle$ under each “leg” of the quantum channel. This evolution is efficient because each of the unitaries is a very sparse matrix. At the end of this process we have the expression
\[equus\] R\^[(n+1)]{} \_[k = 1]{}\^K \_[i = 1]{}\^4 \_k |\_[ik]{}\_[ik]{}|, where $|\phi_{ik}\rangle$ are not mutually orthogonal or normalized, but nevertheless span a $4K$-dimensional space. Eq. is the ancilla density matrix that would result from one step of the quantum channel applied to the approximate density matrix . It is a legitimate density matrix, since Eq. was. Now we can repeat this process by approximating $R^{(n+1)}$ with its top $K$ eigenvectors, renormalizing, propagating, and so on.
When $K$ is sufficiently small, the complexity of evolving the $K$ top eigenvectors scales as $O(K d q^{d-1})$, since the unitary gates are individually sparse matrices. The diagonalization step scales as $O(K^3)$, meanwhile. Benchmarking our results against exact diagonalization at small sizes (and against exact results for random unitary circuits at arbitrary sizes) we find that keeping $\alt 100$ eigenvectors suffices to capture the quantities that are of interest here—mainly, the purity and min-entropy, and their fluctuations.
Trajectory approach for computing the purity {#sec:traj}
--------------------------------------------
In this section we describe an approach based on mapping *vectors* in the ancilla space rather than density matrices. Formally, this method is equivalent to the trajectory approach developed to analyze master equations in quantum optics [@carmichael_book]. Applying unitaries to a vector is an $O(d q^{(d-1)})$ operation because the unitaries are sparse. While such an approach is evidently attractive, we will see that there is a trade-off in terms of the number of times the matrices $A^{(j)}_{s_1,s_2}$ must be applied.
The ancilla density matrix can be defined in terms of averages over trajectories in the physical indices in the following way. Starting from the Kraus form $$\label{eq:kraus}
R^{(j)} = \sum_s A^{(j)\vphantom{\dagger}}_{s} R^{(j-1)} A^{(j)\dagger}_{s}, \qquad j=n+1,\ldots N,$$ In the case of unitary circuits the index $s$ is a composite: $s=(s_1,s_2)$, and we have changed the indexing of slices so that indices increase going right to left. We see that $L$ updates correspond to summing over trajectories in the physical indices of length $L$ $$R^{(L)} = \sum_{s_1:s_L\in \{\mathbb{Z}_q\}^L} A^{(L)\vphantom{\dagger}}_{s_L}\cdots A^{(1)\vphantom{\dagger}}_{s_1} R^{(0)} A^{(1)\dagger}_{s_1}\cdots A^{(L)\dagger}_{s_L}.$$ If we start from a pure state $R^{(0)}=\ket{\psi_0}\bra{\psi_0}$ we can write this as an average over trajectories with uniform distribution $$\label{eq:exp}
R^{(L)} = q^L \operatorname*{\mathbb{E}}_{s_{1:L}\sim \text{uniform}}\left[\ket{\tilde\psi_{s_{1:L}}}\bra{\tilde\psi_{s_{1:L}}}\right],$$ where the vectors $\ket{\psi_{s_{1:L}}}$ are defined as $$\ket{\tilde \psi_{s_{1:L}}} = A^{(L)}_{s_L}\cdots A^{(1)}_{s_1}\ket{\psi_0}.$$ These vectors are unnormalized. The normalization factors $$p(s_1:s_L)\equiv\braket{\tilde\psi_{s_{1:L}}|\tilde\psi_{s_{1:L}}}$$ are a normalized probability distribution over trajectories by virtue of the left canonical condition Eq. . Denoting the normalized vectors as $\ket{\psi_{s_{1:L}}}$ we can express the ancilla density matrix as $$\label{eq:exp_traj}
R^{(L)}=\operatorname*{\mathbb{E}}_{s_{1:N}\sim p(\cdot) }\left[\ket{\psi_{s_{1:L}}}\bra{\psi_{s_{1:L}}}\right].$$ As a trajectory increases in length, the normalization factors are updated according to $$p(s_1:s_L) = p(s_1:s_{L-1})\braket{\psi_{L-1}|A^{(L)\dagger}_{s_L}A^{(L)}_{s_L}|\psi_{L-1}},$$ so that the second factor may be interpreted as a conditional probability $$\label{eq:trans}
p(s_L|s_{1:L-1})=\braket{\psi_{L-1}|A^{(L)\dagger}_{s_L}A^{(L)}_{s_L}|\psi_{L-1}},$$ Eq. expresses the ancilla density matrix in terms of vectors, but it requires an average over trajectories. The downside of this approach is that evaluating $R$ will require roughly $\gamma^{-1}$ trajectories, where $\gamma$ is the purity of a half-infinite region, which sets the approximate rank of the reduced density matrix. However, this approach lends itself to parallelization while the channel based approach does not.
For a translationally invariant system, and assuming this random process is ergodic, we can substitute an average over the length of single long trajectory in the $L\to\infty$ limit. $$\label{eq:ergodic}
R = \lim_{L\to\infty} \frac{1}{L}\sum_{l=1}^L \ket{\psi_{s_{1:l}}}\bra{\psi_{s_{1:l}}}.$$ In practice, long runs and multiple trajectories are used.
$O(D)$ evaluation of matrix products is not much use if we still need $O(D^3)$ evaluation of the spectrum of $R^{(L)}$ or $O(D^2)$ evaluation of the purity. However, we can access the purity without dealing with $R^{(L)}$ directly using $$\label{eq:pure_traj}
\gamma_L=\operatorname{tr}[R^{(L)2}]=\operatorname*{\mathbb{E}}_{s_{1:N},t_{1:N}\sim p(\cdot)}|\braket{\psi_{t_{1:L}}|\psi_{s_{1:L}}}|^2,$$ which follows from Eq. . This formula expresses the purity as the average fidelity over pairs of trajectories. In a high purity state the ancilla vectors stay close to each other as they evolve over different trajectories, whereas in a highly entangled state different trajectories explore different regions of ancilla space.
Note that the expectations discussed in this section are unrelated to any random variables that may form part of the specification of the circuit. Evaluating the average purity, for example, would require an additional average of Eq. over these variables.
Exactly solvable example: self-dual kicked Ising model {#kickedIsing}
======================================================
In order to illustrate the utility of the formalism introduced in the previous section, we now turn to an example of a unitary circuit in which the entanglement spectrum can be determined analytically. This is the kicked Ising model at the self-dual point discussed in two recent papers [@Bertini:2018aa; @Bertini:2018fbz]. The kicked Ising model describes the evolution of a system of $L$ spin-1/2 subsystems (qubits) for an integer time $t$ by the unitary operator $\left(U_\text{KI}\right)^t$, where $U_\text{KI}=K I_\mathbf{h}$ is composed of the two unitaries $$\label{eq:KIM}
I_\mathbf{h} = e^{-iH_\text{I}[\mathbf{h}]},\qquad
K = e^{-iH_\text{K}},$$ where $$\begin{aligned}
H_\text{I}[\mathbf{h}]&=\sum_{j=1}^L\left[J Z_j Z_{j+1} + h_j Z_j\right]\\
H_\text{K} &= b\sum_{j=1}^L X_j,\end{aligned}$$ and $(X_j,Y_j,Z_j)$ are the Pauli matrices for spin $j$. $H_\text{I}[\mathbf{h}]$ is the classical Ising model with arbitrary longitudinal fields $h_j$, while $H_\text{K}$ describes a transverse field.
In Ref. [@Bertini:2018fbz] the growth of the entanglement entropies was found exactly for some particular initial product states at the special ‘self-dual’ values $$\label{eq:self_dual}
|J|=|b|=\frac{\pi}{4}.$$ For a region $A$ of size $N$, the authors found that when starting from an arbitrary product state in the $Z_j$ basis the Rényi entropies to be *exactly* given by $$\label{eq:bertini_res}
\lim_{L\to\infty} S^{(n)}_A(t) =\min(2t-2,N)\log 2,$$ *independent* of Rényi index $n$. The interpretation in terms of the entanglement spectrum is striking: there are $2^{\min(2t-2,N)}$ eigenvalues equal to $2^{-\min(2t-2,N)}$ and the rest are zero.
We now show how our quantum channel approach may be used to derive the corresponding result for the case of a semi-infinite interval $$\label{eq:our_res}
\lim_{L\to\infty} S^{(n)}_A(t) =(t-1)\log 2.$$ We can present the unitary $(U_\text{KI})^t$ as a unitary circuit of depth $t$, where the layers alternate between unitaries operating between spins $2j-1$ and $2j$, and between spins $2j$ and $2j+1$. A variety of decompositions are available. For reasons that will become clear, we choose the following: $$\begin{tikzpicture}[baseline={([yshift=-1ex]current bounding box.center)},every node/.style={scale=1},scale=.75]
{\draw[rounded corners] (0.5+0,0.5+0) rectangle (-0.5+0,-0.5+0); \draw (0,0) node {$U_{12}$};}
\end{tikzpicture}=
\begin{tikzpicture}[baseline={([yshift=-1ex]current bounding box.center)},every node/.style={scale=1},scale=.75]
\draw[rounded corners] (0,2) rectangle (3,1);
\draw (1.5,1.5) node (X) {${{\mathcal I}}$};
\draw[rounded corners] (0,-2) rectangle (3,-1);
\draw (1.5,-1.5) node (X) {${{\mathcal I}}$};
{\draw[rounded corners] (0.5+0.5,0.5+0) rectangle (-0.5+0.5,-0.5+0); \draw (0.5,0) node {${{\mathcal K}}$};}
{\draw[rounded corners] (0.5+2.5,0.5+0) rectangle (-0.5+2.5,-0.5+0); \draw (2.5,0) node {${{\mathcal K}}$};}
{\draw (0.5,1) -- (0.5,0.5);}
{\draw (0.5,-1) -- (0.5,-0.5);}
{\draw (2.5,1) -- (2.5,0.5);}
{\draw (2.5,-1) -- (2.5,-0.5);}
{\draw (0.5,2) -- (0.5,2.5);}
{\draw (2.5,2) -- (2.5,2.5);}
{\draw (0.5,-2) -- (0.5,-2.5);}
{\draw (2.5,-2) -- (2.5,-2.5);}
\draw (0.5,2.8) node (X) {$a$};
\draw (2.5,2.8) node (X) {$b$};
\draw (0.5,-2.8) node (X) {$c$};
\draw (2.5,-2.8) node (X) {$d$};
\end{tikzpicture},$$ where the one qubit (${{\mathcal K}}$) and two qubit (${{\mathcal I}}$) gates have the form $$\begin{aligned}
{{\mathcal K}}&= \exp\left[-i b X\right]\\
{{\mathcal I}}&= \exp\left[-iJ Z_1 Z_2 -i \left(h_1 Z_1 + h_2 Z_2\right)/2\right].\end{aligned}$$ In the $Z$ basis, the elements of $U_{12}$ are $$\begin{gathered}
\label{eq:elem}
(U_{12})_{ab,cd} =-\frac{\sin 2b}{2} \exp\left(-iJ [ab+cd]-i\tilde J[ac+bd]\right)\\
\qquad\times \exp\left(-ih_1[a+c]/2-ih_2[b+d]/2\right)\end{gathered}$$ where $a,b,c,d\in\{1,-1\}$ and $$\label{eq:dualJ}
\tilde J = -\frac{\pi}{4}-\frac{i}{2}\log\tan b.$$
The matrix $(U_{12})_{ab,cd}$ is unitary, but the matrix $\tilde U_{12}$ with elements $(\tilde U)_{ab,cd}=(U_{12})_{ac,bd}$ is not *except* at the self-dual points Eq. . The unitarity of $\tilde U_{12}$ has the graphical representation $$\label{eq:utilde}
\begin{tikzpicture}[baseline={([yshift=-1ex]current bounding box.center)},every node/.style={scale=1},scale=.55]
{\draw[rounded corners] (0.5+1,0.5+1.5) rectangle (-0.5+1,-0.5+1.5); \draw (1,1.5) node {$U_{12}$};}
{\draw[rounded corners] (0.5+1,0.5+-1.5) rectangle (-0.5+1,-0.5+-1.5); \draw (1,-1.5) node {$U_{12}^\dagger$};}
\draw (1.5,2) edge[out=90,in=90] (2.5,2);
\draw (2.5,2) edge[out=-90,in=90] (2.5,-2);
\draw (1.5,-2) edge[out=-90,in=-90] (2.5,-2);
{\draw (1.5,2) -- (1.5,-2);}
\draw (0,2.5) edge[out=0,in=90] (0.5,2);
\draw (0,0.5) edge[out=0,in=-90] (0.5,1);
\draw (0,-2.5) edge[out=0,in=-90] (0.5,-2);
\draw (0,-0.5) edge[out=0,in=90] (0.5,-1);
{\draw (0.5,1) -- (0.5,2);}
{\draw (0.5,-1) -- (0.5,-2);}
\draw (-0.3,2.5) node (X) {$a$};
\draw (-0.3,0.5) node (X) {$b$};
\draw (-0.3,-2.5) node (X) {$a'$};
\draw (-0.3,-0.5) node (X) {$b'$};
\end{tikzpicture}=
\begin{tikzpicture}[baseline={([yshift=-1ex]current bounding box.center)},every node/.style={scale=1},scale=.55]
\draw (0,2.5) edge[out=0,in=90] (1,2);
\draw (0,0.5) edge[out=0,in=90] (0.5,0);
\draw (0,-2.5) edge[out=0,in=-90] (1,-2);
\draw (0,-0.5) edge[out=0,in=-90] (0.5,0);
{\draw (1,2) -- (1,-2);}
\draw (-0.3,2.5) node (X) {$a$};
\draw (-0.3,0.5) node (X) {$b$};
\draw (-0.3,-2.5) node (X) {$a'$};
\draw (-0.3,-0.5) node (X) {$b'$};
\end{tikzpicture}
= \delta_{aa'}\delta_{bb'}.$$ The unitarity of $\tilde U_{12}$ has the interesting consequence that the SW-NE MPS is in left *and right* canonical form, so that $$\label{eq:unital}
\sum_s A^{(j)\vphantom{\dagger}}_{s}A^{(j)\dagger}_{s} = \openone_{2^{t-1}}.$$ To see this, we first give the graphical representation of the left hand side of Eq.
$$\sum_{s_1,s_2} \left(A^{(j)}_{s_1,s_2}A^{(j)^\dagger}_{s_1,s_2}\right)_{a_{1:3},a'_{1:3}}=
\begin{tikzpicture}[baseline={([yshift=-1ex]current bounding box.center)},every node/.style={scale=1},scale=.55]
{\draw[rounded corners] (0.5+1,0.5+1.5) rectangle (-0.5+1,-0.5+1.5); \draw (1,1.5) node {$U_{12}$};}
{\draw[rounded corners] (0.5+2,0.5+3) rectangle (-0.5+2,-0.5+3); \draw (2,3) node {$U_{23}$};}
{\draw[rounded corners] (0.5+3,0.5+4.5) rectangle (-0.5+3,-0.5+4.5); \draw (3,4.5) node {$U_{34}$};}
{\draw[rounded corners] (0.5+4,0.5+6) rectangle (-0.5+4,-0.5+6); \draw (4,6) node {$U_{45}$};}
{\draw[rounded corners] (0.5+1,0.5+-1.5) rectangle (-0.5+1,-0.5+-1.5); \draw (1,-1.5) node {$U_{12}^\dagger$};}
{\draw[rounded corners] (0.5+2,0.5+-3) rectangle (-0.5+2,-0.5+-3); \draw (2,-3) node {$U_{23}^\dagger$};}
{\draw[rounded corners] (0.5+3,0.5+-4.5) rectangle (-0.5+3,-0.5+-4.5); \draw (3,-4.5) node {$U_{34}^\dagger$};}
{\draw[rounded corners] (0.5+4,0.5+-6) rectangle (-0.5+4,-0.5+-6); \draw (4,-6) node {$U_{45}^\dagger$};}
\draw (3.5,6.5) edge[out=90,in=90] (6.5,6.5);
\draw (6.5,6.5) edge[out=-90,in=90] (6.5,-6.5);
\draw (3.5,-6.5) edge[out=-90,in=-90] (6.5,-6.5);
\draw (4.5,6.5) edge[out=90,in=90] (5.5,6.5);
\draw (5.5,6.5) edge[out=-90,in=90] (5.5,-6.5);
\draw (4.5,-6.5) edge[out=-90,in=-90] (5.5,-6.5);
{\draw (4.5,6.5) -- (4.5,-6.5);}
{\draw (3.5,6.5) -- (3.5,-6.5);}
{\draw (2.5,5) -- (2.5,-5);}
{\draw (1.5,3.5) -- (1.5,0.5);}
{\draw (1.5,-3.5) -- (1.5,-0.5);}
{\draw (0.5,2) -- (0.5,0.5);}
{\draw (0.5,-2) -- (0.5,-0.5);}
\draw (0.5,0.3) circle (.2);
\draw (1.5,0.3) circle (.2);
\draw (0.5,-0.3) circle (.2);
\draw (1.5,-0.3) circle (.2);
\draw (0,2.5) edge[out=0,in=90] (0.5,2);
\draw (1,4) edge[out=0,in=90] (1.5,3.5);
\draw (2,5.5) edge[out=0,in=90] (2.5,5);
\draw (0,-2.5) edge[out=0,in=-90] (0.5,-2);
\draw (1,-4) edge[out=0,in=-90] (1.5,-3.5);
\draw (2,-5.5) edge[out=0,in=-90] (2.5,-5);
\draw (-0.3,2.5) node (X) {$a_1$};
\draw (0.7,4) node (X) {$a_2$};
\draw (1.7,5.5) node (X) {$a_3$};
\draw (-0.3,-2.5) node (X) {$a_1'$};
\draw (0.7,-4) node (X) {$a_2'$};
\draw (1.7,-5.5) node (X) {$a_3'$};
\end{tikzpicture}=
\begin{tikzpicture}[baseline={([yshift=-1ex]current bounding box.center)},every node/.style={scale=1},scale=.55]
{\draw[rounded corners] (0.5+1,0.5+1.5) rectangle (-0.5+1,-0.5+1.5); \draw (1,1.5) node {$U_{12}$};}
{\draw[rounded corners] (0.5+2,0.5+3) rectangle (-0.5+2,-0.5+3); \draw (2,3) node {$U_{23}$};}
{\draw[rounded corners] (0.5+3,0.5+4.5) rectangle (-0.5+3,-0.5+4.5); \draw (3,4.5) node {$U_{34}$};}
{\draw[rounded corners] (0.5+1,0.5+-1.5) rectangle (-0.5+1,-0.5+-1.5); \draw (1,-1.5) node {$U_{12}^\dagger$};}
{\draw[rounded corners] (0.5+2,0.5+-3) rectangle (-0.5+2,-0.5+-3); \draw (2,-3) node {$U_{23}^\dagger$};}
{\draw[rounded corners] (0.5+3,0.5+-4.5) rectangle (-0.5+3,-0.5+-4.5); \draw (3,-4.5) node {$U_{34}^\dagger$};}
\draw (3.5,5) edge[out=90,in=90] (4.5,5);
\draw (4.5,5) edge[out=-90,in=90] (4.5,-5);
\draw (3.5,-5) edge[out=-90,in=-90] (4.5,-5);
{\draw (3.5,5) -- (3.5,-5);}
{\draw (2.5,5) -- (2.5,-5);}
{\draw (1.5,3.5) -- (1.5,0.5);}
{\draw (1.5,-3.5) -- (1.5,-0.5);}
{\draw (0.5,2) -- (0.5,0.5);}
{\draw (0.5,-2) -- (0.5,-0.5);}
\draw (0.5,0.3) circle (.2);
\draw (1.5,0.3) circle (.2);
\draw (0.5,-0.3) circle (.2);
\draw (1.5,-0.3) circle (.2);
\draw (0,2.5) edge[out=0,in=90] (0.5,2);
\draw (1,4) edge[out=0,in=90] (1.5,3.5);
\draw (2,5.5) edge[out=0,in=90] (2.5,5);
\draw (0,-2.5) edge[out=0,in=-90] (0.5,-2);
\draw (1,-4) edge[out=0,in=-90] (1.5,-3.5);
\draw (2,-5.5) edge[out=0,in=-90] (2.5,-5);
\draw (-0.3,2.5) node (X) {$a_1$};
\draw (0.7,4) node (X) {$a_2$};
\draw (1.7,5.5) node (X) {$a_3$};
\draw (-0.3,-2.5) node (X) {$a_1'$};
\draw (0.7,-4) node (X) {$a_2'$};
\draw (1.7,-5.5) node (X) {$a_3'$};
\end{tikzpicture}$$
Where we used unitarity to eliminate the top and bottom gates. Using the motif Eq. we can telescope telescope the circuit until $$\label{frag58}
\sum_{s_1,s_2} \left(A^{(j)}_{s_1,s_2}A^{(j)^\dagger}_{s_1,s_2}\right)_{a_{1:3},a'_{1:3}}=
\begin{tikzpicture}[baseline={([yshift=-1ex]current bounding box.center)},every node/.style={scale=1},scale=.55]
{\draw[rounded corners] (0.5+1,0.5+1.5) rectangle (-0.5+1,-0.5+1.5); \draw (1,1.5) node {$U_{12}$};}
{\draw[rounded corners] (0.5+1,0.5+-1.5) rectangle (-0.5+1,-0.5+-1.5); \draw (1,-1.5) node {$U_{12}^\dagger$};}
\draw (1.5,2) edge[out=90,in=90] (2.5,2);
\draw (2.5,2) edge[out=-90,in=90] (2.5,-2);
\draw (1.5,-2) edge[out=-90,in=-90] (2.5,-2);
{\draw (3.5,5) -- (3.5,-5);}
{\draw (3,3.55) -- (3,-3.5);}
{\draw (1.5,2) -- (1.5,0.5);}
{\draw (1.5,-2) -- (1.5,-0.5);}
{\draw (0.5,2) -- (0.5,0.5);}
{\draw (0.5,-2) -- (0.5,-0.5);}
\draw (0.5,0.3) circle (.2);
\draw (1.5,0.3) circle (.2);
\draw (0.5,-0.3) circle (.2);
\draw (1.5,-0.3) circle (.2);
\draw (0,2.5) edge[out=0,in=90] (0.5,2);
\draw (1,4) edge[out=0,in=90] (3,3.5);
\draw (2,5.5) edge[out=0,in=90] (3.5,5);
\draw (0,-2.5) edge[out=0,in=-90] (0.5,-2);
\draw (1,-4) edge[out=0,in=-90] (3,-3.5);
\draw (2,-5.5) edge[out=0,in=-90] (3.5,-5);
\draw (-0.3,2.5) node (X) {$a_1$};
\draw (0.7,4) node (X) {$a_2$};
\draw (1.7,5.5) node (X) {$a_3$};
\draw (-0.3,-2.5) node (X) {$a_1'$};
\draw (0.7,-4) node (X) {$a_2'$};
\draw (1.7,-5.5) node (X) {$a_3'$};
\end{tikzpicture}$$ Finally, we use the explicit form of Eq. at the self-dual point to evaluate $$\begin{tikzpicture}[baseline={([yshift=-1ex]current bounding box.center)},every node/.style={scale=1},scale=.55]
{\draw[rounded corners] (0.5+1,0.5+1.5) rectangle (-0.5+1,-0.5+1.5); \draw (1,1.5) node {$U_{12}$};}
{\draw[rounded corners] (0.5+1,0.5+-1.5) rectangle (-0.5+1,-0.5+-1.5); \draw (1,-1.5) node {$U_{12}^\dagger$};}
\draw (1.5,2) edge[out=90,in=90] (2.5,2);
\draw (2.5,2) edge[out=-90,in=90] (2.5,-2);
\draw (1.5,-2) edge[out=-90,in=-90] (2.5,-2);
{\draw (1.5,2) -- (1.5,0.75);}
{\draw (1.5,-2) -- (1.5,-0.75);}
{\draw (0.5,2) -- (0.5,0.75);}
{\draw (0.5,-2) -- (0.5,-0.75);}
\draw (0.5,0.55) circle (.2);
\draw (1.5,0.55) circle (.2);
\draw (0.5,-0.55) circle (.2);
\draw (1.5,-0.55) circle (.2);
\draw (0,2.5) edge[out=0,in=90] (0.5,2);
\draw (0,-2.5) edge[out=0,in=-90] (0.5,-2);
\draw (-0.3,2.5) node (X) {$a_1$};
\draw (-0.3,-2.5) node (X) {$a_1'$};
\draw (0.5,0) node (X) {$c$};
\draw (1.5,0) node (X) {$d$};
\end{tikzpicture}=
\sum_{b} \left(U_{12}\right)_{ab,cd} \left(U^*_{12}\right)_{a'b,cd}=\delta_{aa'},$$ which corresponds to an initial state equal to a product state in the $Z_j$ basis. This verifies the condition Eq. .
An MPS in both left and right canonical form describes a *bistochastic* quantum channel: one that preserves the identity. As a result, the ancilla density matrix $R^{j}=2^{1-t}\openone_{2^{t-1}}$ for all $j$. Evaluating the Rényi entropies yields our result Eq. .
It seems likely that a similar analysis can be performed for general Clifford circuits, which also have degenerate entanglement spectra [@nrvh]. However, establishing this in general requires one to carve out several special cases (such as circuits that generate no entanglement at all from a number of initial states [@gz2018; @sg_og_2018]) and we will not pursue this here.
Numerical results: random unitary circuits {#results}
==========================================
In this section we present numerical results for the evolution of various entanglement measures, and their spatial fluctuations, for random unitary circuits. The coarse features of the evolution of the entanglement spectrum were already discussed in Ref. [@ccgp]. In particular, the bandwidth of the entanglement spectrum broadens linearly in time; as noted in that work, this broadening is a natural consequence of the wide separation between the entanglement and light-cone speeds. For circuits of depth $\geq 10$ this broadening implies that an appreciable fraction of the spectrum of the reduced density matrix is zero to within numerical precision. Our focus here is on the *large* eigenvalues of the reduced density matrix (which dominate Rényi entropies $S_n, n \geq 1$). Because the vast majority of the eigenvalues are near zero, this “low-entanglement-energy” tail can be described accurately by low-rank approximations as in Sec. \[lra\], allowing us to go to circuits of depth $L = 14$ with minor computational effort.
Benchmarking the low-rank approximation
---------------------------------------
For depths $t \leq 10$ we can compare the entanglement spectra computed by low-rank approximation with the exact ones (Fig. \[vscutoff\]). Although the rank of the reduced density matrix in this case is $256$, we find that working with the top 20 states allows us to match the low-energy behavior of the entanglement spectrum.
![Entanglement density of states (i.e., histogram of entanglement energies), at depth $t = 8$, computed both exactly and by low-rank approximation of the quantum channel (Sec. \[lra\]). For rank $\agt 20$ the low-energy behavior of the entanglement spectrum is well captured.[]{data-label="vscutoff"}](vscutoff){width="40.00000%"}
For the largest depths we have considered, exact time evolution is not feasible; however, there is an exact result for the average purity of a semi-infinte system [@Nahum2017], viz. $\bar\gamma = (4/5)^{t-1}$ [^1]. For the numerical results presented here we increase the rank of the approximation until the mean computed purity matches this exact result to within statistical error (which is about $1\%$). For the largest depth we have systematically considered ($t = 14$) we need to keep $\approx 120$ states to match the mean purity. This is only about $1\%$ of the spectrum; thus the low-rank approximation is much more efficient than direct propagation of the channel would be.
Shape of the entanglement spectrum
----------------------------------
In this section we discuss the shape of the entanglement spectrum and its relation to the evolution of the Rényi entropies for RUCs. First, let us recall that the eigenstate thermalization hypothesis predicts that the reduced density matrix of a subsystem should take the form $\mathcal{N} \exp(- \beta H)$, where $H$ is the Hamiltonian of the subsystem and $\beta$ is the inverse temperature. Therefore the entanglement Hamiltonian $-\log \rho \propto \beta H + \log \mathcal{N}$, i.e., it is a stretched and shifted version of the physical Hamiltonian. The entanglement spectrum therefore has the same shape as the physical spectrum: for a large subsystem $L_A$ with a local Hamiltonian, it will be essentially Gaussian in the bulk, with a bandwidth that increases as $\sqrt{L_A}$, although the extreme value statistics (corresponding to the shape of the spectrum near its ground state) are model-dependent. When $\beta \sqrt{L_A}$ is large, the entanglement spectrum will have a large bandwidth, and therefore (because of the Jacobian) the reduced density matrix will have a density of eigenvalues $\varrho(\lambda)$ distribution of the form $\varrho(\lambda) \sim 1/\lambda$. Infinite temperature is a singular limit, as $\beta = 0$ so the entanglement spectrum is degenerate. In practice, a typical, randomly picked state deviates form infinite temperature by an amount $\sim 1/\sqrt{L_A}$, so the entanglement “bandwidth” is $L_A$-independent (up to possible logarithmic dependences that we are not concerned with here). For Floquet systems or RUCs with no conservation laws, these arguments suggest that the entanglement spectrum of a small subsystem at very late times is degenerate up to finite size effects. Once finite size effects are included we expect a Marchenko–Pastur distribution [@Chen:2017aa].
For the case of interest to us – large subsystems at short times – the numerical evidence [@ccgp] suggests that the spectrum of the reduced density matrix has the density $\varrho(\lambda) \sim 1/\lambda$ over many decades, at any time $1 \ll t \ll l_A$; this is qualitatively unlike the (compact) Wishart distribution that obtains at very late times [@Chen:2017aa]. The bulk of the spectrum of the reduced density matrix consists of eigenvalues below machine precision whenever $t \agt 8$ (Fig. \[entedge\]). Here, we are concerned with the “low-energy” or “high-Schmidt-coefficient” edge, which governs the behavior of the Rényi entropies $S^{(n)}, n \geq 1$, which we can follow out to later times $t \approx 14$ (Fig. \[entedge\]). The evolution at early times is nontrivial, but appears to settle down into a well-defined limiting behavior for $t \agt 7$: there is a threshold in the entanglement density of states, followed by a linear increase with entanglement energy that persists out to the energies we can reliably access. The coefficient of this linear growth is approximately $2^{t/2}$ for the larger accessible $t$. This is exponentially slower than the growth of the total number of states, so the fraction of states in the tails thins out exponentially in time.
The Rényi index dependence of $S^{(n)}$ for large $n$ follows from this behavior of the limit shape, if we further assume that the entanglement spectrum is self-averaging. A simple model for the spectral density $\rho(\epsilon)$ of the entanglement energies that is consistent with the large deviation form Eq. is $$\rho(\epsilon) = \exp\left[\alpha t\Theta(E-v_\infty t)\right],$$ in which case $$v_n = \frac{\alpha - nv_\infty}{1-n}$$ A large $n$ result this implies $v_n/v_\infty \simeq 1 - 1/n$, which is consistent with our numerical observations \[Fig. \[renyidep\]\].
Although the late-time entanglement spectrum is not numerically accessible, our results allow us to comment on a few possible qualitative scenarios of the entanglement spectrum. First, it is clear numerically that the probability density of states is exponentially small near the low-energy edge of the entanglement spectrum. This turns out to be necessary for the Rényi entropies to have distinct velocities. (If one considers, e.g., a box-shaped entanglement DOS, it is simple to show that all Rényi entropies with $n > 0$ must have the same velocity, regardless of the aspect ratio of the box. Similar results hold for Marchenko-Pastur and other possible compact shapes.) Second, one might suppose the entanglement spectrum has a Gaussian shape. Matching exact results for $S_0$, the normalization of $\rho$, and $S_2$ requires the Gaussian to have a linearly growing mean and variance. Numerically, the entanglement bandwidth grows linearly in time rather than as a square root, possibly because of level repulsion. Developing a theory of how the entanglement spectrum evolves is an important question for future work: at present we do not have even a phenomenological Brownian-motion model of this growth.
Statistics and spatial correlations of entanglement
---------------------------------------------------
Out to the latest times we have considered, the sample-averaged Rényi entropies have Gaussian distributions. (Thus, quantities such as the purity are log-normally distributed.) Whether these distributions become anisotropic at much larger system sizes is unclear; however, we have not seen any sign of incipient skewness out to the times we can simulate (Fig. \[enthist\]).
We now turn to fluctuations of the entanglement across spatial cuts. The prediction of Ref. [@nrvh], based on a mapping to the KPZ equation, is that the entanglement fluctuations are spatially correlated, with a correlation length $\xi(t) \sim t^{2/3}$, and that the width of the entanglement distribution scales as $t^{1/3}$ (i.e., the entanglement “roughens”). The method used here works with an infinite system at a fixed depth, and enables one to address these spatial correlations. We find that the spatial correlations of entanglement do get longer-ranged in time, as their power spectrum clearly narrows in $k$-space (Fig. \[ent\_fourier\]). The Fourier transform has a characteristic width, from which we can extract a correlation length that clearly grows sub-linearly with $t$. However, the correlation length remains short out to the latest times we can access, so we do not have the dynamic range to extract meaningful exponents.
![Top: power spectrum of the Fourier transform of min-entropy across spatial cuts, normalized to one for all circuits. Note the narrowing of the Fourier transform with increasing $t$. Data are for a single system of length $5000$. Bottom: estimate of the correlation length $\xi$ extracted from the width of the Fourier peak.[]{data-label="ent_fourier"}](rucflucts){width="40.00000%"}
The KPZ picture also predicts that the entanglement “roughens” with time, i.e., its standard deviation grows. This is consistent with what we see, although, again, the roughening is too weak to extract meaningful exponents.
Trajectory Approach For Purity
------------------------------
Finally, we demonstrate the trajectory approach described in Section \[sec:traj\]. To evaluate the purity using Eq. \[eq:pure\_traj\] we evolve a pair of trajectories $s_{1:L}$ and $t_{1:L}$ with the transition probabilities given in Eq. . For a random unitary circuit with a purity that fluctuates with spatial position, evaluating the unaveraged purity at a point would involve averaging over many trajectories with the same set of gates. As a proof of principle we instead focus on the ensemble averaged purity, and average the fidelity $|\braket{\psi_{t_{1:l}}|\psi_{s_{1:l}}}|^2$ for $l=1,\ldots L$ with a trajectory of $L=1000$. In this case we can compare with the known exact result $\bar\gamma = (4/5)^{t-1}$ for random unitary circuits [@Nahum2017]. Since we are now evolving vectors in the ancilla space rather than density matrices we can simulate deeper circuits. Fig. \[fig:traj\] shows the average purity for depths up to 18, comparing the trajectory method with the exact result, as well as with the density matrix approach for depths up to 12. Good agreement is found in all cases.
Circuits with more structure
----------------------------
The transfer-matrix method discussed here extends directly from random unitary circuits to any other type of circuit that can be decomposed into a “brickwork” arrangement of two-site gates. We discuss two examples here: circuits with a conservation law and translation-invariant circuits.
### Number-conserving circuits
As an illustrative example we now turn to circuits with a single conservation law, which we choose to be the number of $\uparrow$ spins in the computational basis [@Khemani2017; @rpv]. For a random circuit, all classes of product states are equivalent; however, circuits with a conservation law yield very different entanglement DOS depending on the initial state. Fig. \[conslaw1\] shows the results for three classes of initial states: random product states, random bit-strings in the computational basis, and a uniform Néel state. While the Néel state behaves analogously to the random unitary circuit, at least at these depths, we see that the other two types of product states give rise to very different entanglement spectra, with substantially higher DOS at low energies. The difference can be attributed to rare states with anomalously large weight on configurations with long strings of aligned spins, which do not entangle under number-conserving dynamics [@rpv2019; @yichen2019].
### Translation-invariant circuits
Next, we consider translation-invariant circuits, in which all the gates at a given time-step are identical. Gates could be the same at different time-steps (giving a Floquet system) or random at every time-step (giving a system with perfectly spatially correlated noise). We focus here on the former case. For concreteness we focus on the integrable Trotterization of the XXZ model that was recently introduced [@Prosen_trotterization1; @Prosen_trotterization2; @Prosen_trotterization3]. This model is parameterized by two parameters $(\eta, \lambda)$, and the dynamics consists of repeated application of the two-site gate
$$U(\eta, \lambda) \equiv \left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & \frac{\sin \eta}{\sin (\eta + \lambda)} & \frac{\sin \lambda}{\sin(\eta + \lambda)} & 0 \\ 0 & \frac{\sin \lambda}{\sin(\eta + \lambda)} & \frac{\sin \eta}{\sin (\eta + \lambda)} & 0 \\ 0 & 0 & 0 & 1 \end{array} \right).$$
When $\eta$ is imaginary and $\lambda$ is real, this is a Trotterized version of the Ising phase of the XXZ chain, with larger $\eta$ corresponding to larger easy-axis anisotropy.
One could consider the dynamics of entanglement for either random or homogeneous initial states. For random states, we expect (and find) spatial fluctuations of entanglement, which have a growing correlation length as in random unitary circuits (Fig. \[xxzflucts\]). At the accessible times, we are unable to extract any clear qualitative difference between the behavior of the correlation length in these integrable circuits and the random unitary case. For translation-invariant states (specifically the Néel state), the quantum channel converges to a definite steady state after a time interval on the order of the circuit depth (Fig. \[convergence\]). Comparing the transient behavior between the XXZ circuit and a (presumably nonintegrable) circuit consisting of tiling a random two-site gate, we see that the transient behavior of the integrable case is different: the min-entropy overshoots its steady-state value in the integrable case, but not in the random case. Fig. \[xxzent\] shows the lowest 40 entanglement energies at $t = 12$ as a function of the anisotropy parameter $\eta$; as one would expect, increasing the anisotropy slows down the growth of entanglement. This manifests itself as the top eigenvalue in the Schmidt spectrum drifting toward zero, leading to a large gap in the Schmidt spectrum. However, the min-entropy, starting from the Néel state, grows linearly out to the circuit depths we can access, with no signs of curvature. This is consistent with the intuitive picture of ballistic entanglement growth in integrable systems [@alba_calabrese], since quasiparticles move ballistically although spin dynamics is diffusive [@lzp; @dbd2; @gv_superdiffusion; @Prosen_trotterization3].
An interesting quantitative difference between integrable dynamics and generic chaotic dynamics is that (for intermediate values of $\eta$) the entanglement spectrum stays “narrow” in the integrable case: Schmidt coefficients do not rapidly spread out over many decades the way they do under random unitary dynamics. This is consistent with the quasiparticle picture, which predicts that entanglement should spread with a characteristic quasiparticle velocity that does not depend on the Rényi index. How this picture extends to the case of strong anisotropy, in which the quasiparticles have a broad distribution of velocities, is an interesting open question.
Conclusions
===========
In concluding, let us summarise the technical achievements of the quantum channel approach:
1. Unitary circuits are presented directly as matrix product states in canonical form.
2. The spectrum of the reduced density matrix is related to that of the ancilla states exposed by the diagonal cut.
3. The resulting quantum channel allows us to work in the infinite width limit at finite depth.
4. Analytical results are obtained for the kicked Ising model at the self-dual point using a simple graphical calculus, simpler than the approach of Ref. [@Bertini:2018fbz].
5. The numerical evaluation of the channel may be improved by making a low rank approximation for the ancilla density matrix or by unraveling a quantum trajectory over the physical states.
In this work, we benchmarked the quantum channel approach against exact results for the purity in random unitary circuits, and used it to compute the shape and fluctuations of the entanglement spectrum at low “entanglement energies.” With relatively little computational effort we were able to get converged results for depths $t = 14$ for the entanglement spectrum and $t = 18$ for the purity. We expect that there is room to optimize the algorithm and perform more resource-intensive computations, allowing us to access somewhat later times than we have in the present work. Our results support and extend earlier work [@ccgp] showing that the entanglement spectrum has a bandwidth that grows rapidly in time, with a sharp onset. Under random unitary dynamics (with or without a conservation law), most of the Schmidt coefficients that are generated are exponentially small in circuit depth, and can therefore be truncated, allowing for efficient computation of the entanglement spectrum. We were able to compute the spatial fluctuations of entanglement for large systems, and finally for translation-invariant systems we obtained converged results for the entanglement spectrum by evolving the associated quantum channel to convergence.
A key distinction between the quantum channel approach and standard methods (such as time-evolving block decimation [@vidal2003]) for evolving a matrix-product state is that the quantum channel approach constructs the entanglement spectrum without explicitly representing the physical wavefunction at time $t$. Our truncation and sampling schemes are also conceptually somewhat different from that in time-evolving block decimation [@vidal2003; @vidal2007]. Thus the quantum channel offers benefits if one wants to compute the entanglement spectrum (since it requires storing only the object of interest); however, it is not clear how one would apply our methods to compute observables in the physical (rather than ancilla) space.
A number of avenues for future work present themselves. The thermodynamic shape of the entanglement spectrum of a semi-infinite circuit is still not well understood beyond its coarsest features. At a technical level, applying our approach to the entanglement of a finite region will involve considering the quantum channel at different ‘times’, instead of the stationary (distribution of the) ancilla density matrix. Finally, our analysis of the kicked Ising model shows that the soluble self-dual point arises simply from the relation Eq. , and suggests a criterion for searching for more models in the same class.
Acknowledgements
================
This research was supported in part by the National Science Foundation under Grant No. NSF PHY-1748958. Support of EPSRC Grant No. EP/P034616/1 (AL) and NSF Grant No. DMR-1653271 (SG) is gratefully acknowledged. S.G. thanks P.-Y. Chang, X. Chen, V. Khemani, S. Parameswaran, J. Pixley, F. Pollmann, and T. Rakovszky for helpful discussions.
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[^1]: The exponent $t-1$ here is one less than in [@Nahum2017] because our partition of the system lies between the two gates in the top layer of the circuit, which therefore leaves the entanglement unaffected.
|
---
abstract: 'A method to investigate anharmonicity, vibrational anisotropy and thermal expansion using correlated mean-square relative displacements (MSRD) parallel and perpendicular to the interatomic bonds obtained only from Extended X-ray Absorption Fine Structure (EXAFS) analysis based on cumulant expansion is suggested and applied to an amorphous Ni$_{46}$Ti$_{54}$ alloy produced by mechanical alloying. From EXAFS measurements taken on Ni and Ti K edges at several temperatures, the thermal behavior of $\text{MSRD}_\parallel$, $\text{MSRD}_\perp$ and of the cumulants $C_1^*$, $C_2^*$ and $C_3^*$ of the real distribution functions $\varrho_{ij}(r,T)$, and also the Einstein temperatures and frequencies associated with parallel and perpendicular motion were obtained, furnishing information about the anharmonicity of the interatomic potential, vibrational anisotropy and the contribution of the perpendicular motion to the total disorder and thermal expansion.'
author:
- 'K. D. Machado'
title: 'Investigation on Anharmonicity, Vibrational Anisotropy and Thermal Expansion of an Amorphous Ni$_{46}$Ti$_{54}$ Alloy Produced by Mechanical Alloying using Extended X-ray Absorption Fine Structure'
---
Introduction
============
Many physical and chemical properties of a material depend on its atomic structure. Thus, in order to understand these properties, structural studies on crystalline and amorphous materials should be carried on. These studies depend on the use of techniques able to extract structural information such as average coordination numbers, average interatomic distances, structural and thermal disorders and so on. In this case, the first approach usually is to use x-ray diffraction (XRD) [@Cullity; @Warren; @Guinier], which is the most common structural technique. XRD can be used mainly for crystals to find lattice parameters, crystallographic thermal expansion, uncorrelated mean-square displacements (MSD) and relative quantities of phases present in a sample [@klecoge; @kleCuSe]. However, XRD is a long-range probe, and information about a specific atomic species is not easy to obtain. When the material under study is amorphous, the difficulties increase, and determination of structural parameters for binary or multicomponent alloys usually require the use of experimental methods such as anomalous x-ray diffraction (AXRD), neutron diffraction (ND) with isotopic substitution (IS), or theoretical methods as molecular dynamics (MD), Monte Carlo (MC) or Reverse Monte Carlo (RMC) simulations, and sometimes a combination of two or more techniques. Unfortunately, more detailed studies, including investigations on thermal expansion, anharmonicity of interatomic potentials and vibrational anisotropy, are very difficult to be done on amorphous alloys using the cited techniques, which leads us to the extended x-ray absorption fine structure spectroscopy (EXAFS) technique. EXAFS is a powerful tool for obtaining the local atomic order around a specific atomic species [@Koningsberger; @Teo] due to its selectivity. EXAFS oscillations $\chi(k)$ obtained on an edge of an element $A$ furnish information about interactions involving only the element $A$, and the procedure used to extract such information is almost the same for crystalline or amorphous samples [@KleGaSe; @kleCoTi]. In this aspect, due to the high values of $2k$ probed by EXAFS, valuable information about the medium and mainly the short-range order can be obtained, a property very relevant for amorphous alloys since they basically exhibit structures only with this kind of order. In addition, studies on thermal properties are relatively easy to be carried on if high-quality EXAFS measurements at different temperatures were taken, which opens the possibility of more sofisticated investigations. For moderately disordered systems [@Dalba; @Tranquada], the use of the cumulant expansion analysis [@Bunker; @Koningsberger2] can take into account thermal effects and also anharmonicity. Many investigations based on this method have been done since the proposal of EXAFS as a vibrational probe in the 1970s [@Beni; @Sevillano], and information about static (or structural) disorder, thermal disorder, thermal expansion, anharmonicity effects and vibrational anisotropy were obtained [@Dalba3; @Ikemoto; @Araujo; @kleberse90s10; @Schnohr; @Purans]. The main point is that EXAFS is sensitive to the parallel and perpendicular correlated mean-square relative displacement [@Lee; @Dalba4] (MSRD$_\parallel$ and MSRD$_\perp$, respectively) and to the asymmetry of the one-dimensional effective pair potential [@Frenkel; @Yokoyama], and this feature can be exploited to furnish structural and thermal information related to the alloys under study. Some crystalline alloys were studied considering this approach but to our knowledge $\text{MSRD}_\perp$ and quantities related to it were not found for any amorphous samples. The main problem in this case is the determination of the quantity equivalent to the crystallographic distance $R_c$, which in principle is needed to obtain $\text{MSRD}_\perp$. For crystalline materials, $R_c$ is related to the lattice parameters and can be obtained, for instance, from a Rietveld refinement procedure [@Rietveld2; @Rietveld]. For amorphous samples, on the other hand, it is not easy to determine the amorphous XRD distance $R_a$ and, when it can be done, usually error bars are large and do not allow reliable values for $\text{MSRD}_\perp$. We developed a method to extract several properties such as vibrational anisotropy, anharmonicity, asymmetry of distribution functions and thermal expansion from EXAFS measurements only, and a detailed explanation of the procedure is given below. We illustrate it by investigating the structural properties of an amorphous Ni$_{46}$Ti$_{54}$ ([*a*]{}-Ni$_{46}$Ti$_{54}$) alloy produced by mechanical alloying [@MASuryanarayana]. NiTi alloys are very interesting since they can exhibit shape memory and superelastic effects, excellent ductibility and good fatigue life, good corrosion resistance and biocompatibility [@Shabalovskaya; @Duerig; @Otsuka; @Ju; @McIntosh], being candidates to use as artificial bones or teeth roots [@Lipscomb]. Other applications include the use of shape memory and superelastic NiTi bars and wires as structural elements in buildings [@DesRoches]. Amorphous NiTi alloys can be produced in a wide compositional range which extends from 20% to about 70% Ni [@Buschow], making this system very suitable for amorphization studies [@KleNiTi; @nitikleber; @kleni46ti54]. Considering EXAFS measurements on edges of Ni and Ti at several temperatures, we obtained, besides average coordination numbers and interatomic distances, $\text{MSRD}_\parallel$ and $\text{MSRD}_\perp$, structural and thermal disorder, anharmonicity of the effective interatomic pair potential, vibrational anisotropy and thermal expansion considering correlated Einstein models for the temperature dependence of cumulants $C_2^*$ and $C_3^*$.
The structure of this article is as follows. Sec. \[secteoria\] presents the theoretical fundamentals needed to the EXAFS analysis and cumulant expansion. Sec. \[secexperimental\] shows the experimental procedures used to produce the alloy and to obtain the EXAFS measurements. Results and detailed discussions are given in sec. \[secresultados\], and sec. \[secconclusoes\] summarize the conclusions obtained.
Theoretical Background {#secteoria}
======================
To obtain structural information from EXAFS we considered the well known cumulant expansion method [@Bunker; @Koningsberger; @Fornasini; @Dalba4] which is valid for small to moderate disorder. The EXAFS signal on a K absorbing edge for a coordination shell $\ell$ of an absorbing atom of type $i$ and a backscatter of type $j$ can be written as [@Fornasini; @Dalba4; @Koningsberger; @Bouldin]
$$\chi_\ell^{ij}(k,T) = \frac{S_0^2 N_{ij}}{k} \text{Im} \biggl[f_j(k) e^{2i \delta(k)}\int_0^\infty{\varrho_{ij} (r,T)
\frac{e^{-2r/\lambda}}{r^2} e^{2ikr} \, dr}\biggr] \,,
\label{chi}$$
where $S_0^2$ is the amplitude factor associated with intrinsic process that contribute to the photoabsorption but not to EXAFS, $N_{ij}$ is the average coordination number of atoms of type $j$ around atoms of type $i$, $f_j(k)$ is the complex backscattering factor, $\delta(k)$ is the phaseshift associated with the absorbing atom, $\varrho_{ij}(r,T)$ is the partial radial distribution function (RDF) [@Bouldin], $\lambda$ is the photoelectron mean free path and $T$ is the temperature. The RDF is the [*real*]{} distribution of distances, and the [*effective*]{} distribution of distances is given by
$$\Upsilon_{ij}(r,T, \lambda) = \varrho_{ij} (r,T) \frac{e^{-2r/\lambda}}{r^2}\,.
\label{effective}$$
It is important to note that the instantaneous interatomic distances $r_{ij}$ are distributed according to the real distribution $\varrho_{ij}(r,T)$, but in an EXAFS measurement the photoelectrons, which have a mean free path $\lambda$, probe the effective distribution $\Upsilon_{ij}(r,T,\lambda)$ due to the spherical photoelectron wave, to the weakening of this wave with $r$ and to the finite mean free path [@Bunker]. The function
$$\Xi_{ij}(k,T,\lambda) = \int_0^\infty{\varrho_{ij} (r,T) \frac{e^{-2r/\lambda}}{r^2} e^{2ikr} \, dr}
\label{caracteristica}$$
is called the characteristic function [@Reichl] associated with the distribution $\Upsilon_{ij}(r,T,\lambda)$. This function can be though of as being the average value of $e^{2ikr}$, which can be expanded in a power series of the moments $\langle r^n \rangle$ as
$$\Xi_{ij}(k,T,\lambda) = \langle e^{2ikr} \rangle = \biggl\langle \Bigl(1 + 2ikr + \frac{(2ik)^2r^2}{2!}
+ \frac{(2ik)^3r^3}{3!} + \cdots \Bigr)\biggr\rangle
= \langle 1 \rangle + 2ik \langle r \rangle + \frac{(2ik)^2}{2!} \langle r^2 \rangle
+ \frac{(2ik)^3}{3!} \langle r^3 \rangle + \cdots \,.
\label{xiserie}$$
The characteristic function can also be written in terms of cumulants $C_n$ through
$$\begin{gathered}
\Xi_{ij}(k,T,\lambda) = \exp{\Bigl[ \sum_{n=1}^{\infty}{\frac{(2ik)^n}{n!} C_n}\Bigr]} \\=
\exp\Bigl[2ik C_1 + \frac{(2ik)^2 C_2}{2!} + \frac{(2ik)^3 C_3}{3!} + \cdots\Bigr] \,.
\label{cumulantes}\end{gathered}$$
Expanding the right hand side of eq. \[cumulantes\] and collecting terms of same order in $2ik$ results in
$$\begin{gathered}
\Xi_{ij}(k,T,\lambda) = 1+ 2ik C_1 +
\frac{C_2 +C_1^2}{2!} (2ik)^2
\\
+ \frac{C_3 +3C_1 C_2+ C_1^3 }{3!} (2ik)^3
+ \cdots\,,
\label{cumulantes2}\end{gathered}$$
and, comparing eqs. \[xiserie\] and \[cumulantes2\], the cumulants are given by
\[cumulantes3\] $$\begin{aligned}
C_1 &= \langle r \rangle
\label{cumulantes3a}\\
C_2 &= \langle r^2 \rangle - \langle r \rangle^2 = \langle (r - \langle r \rangle)^2\rangle
\label{cumulantes3b}\\
C_3 &= \langle r^3 \rangle - 3 \langle r \rangle\langle r^2 \rangle + 2\langle r \rangle^3 =
\langle (r - \langle r \rangle)^3 \rangle
\label{cumulantes3c}\\
\vdots & \notag\\
C_n &= \langle (r - \langle r \rangle)^n \rangle \,, n \ge 2\,.
\label{cumulantes3d}\end{aligned}$$
These are the cumulants of the effective distribution $\Upsilon(r,T,\lambda)$. When the real distribution
$$g_{ij}(r,T) = \frac{\varrho_{ij}(r,T)}{r^2}
\label{g}$$
is considered, the real cumulants $C_n^*$ are obtained, and the corresponding characteristic function is
$$\Lambda(k,T) = \int_0^\infty{ \frac{\varrho_{ij}(r,T)}{r^2} \, dr}\,.$$
If this function is expanded, equations similar to eqs. \[xiserie\], \[cumulantes\] and \[cumulantes3\] are found but with cumulants $C_n^*$ instead of $C_n$. It is important to note that to analyze anharmonicity, asymmetries, vibrational anisotropy and thermal expansion we need the cumulants $C_n^*$. In particular, $C_1^*$ is the average interatomic distance, $C_2^*$ is related to the disorder and $C_3^*$ measures the asymmetry of the distribution function $g_{ij}(r,T)$. Some important quantities can be obtained directly from these three cumulants. To see that, let $\vec{u}_j$ and $\vec{u}_0$ be the instantaneous displacements of the backscatterer and absorber atom, respectively. Defining $\Delta \vec{u} = \vec{u}_j - \vec{u}_0$ as the instantaneous relative thermal displacement between the backscatterer and absorber atoms, the total MSRD is given by $\text{MSRD} = \langle (\Delta \vec{u})^2 \rangle$, which can be decomposed in a MSRD parallel to the interatomic bond (MSRD$_\parallel = \langle(\Delta u_\parallel)^2\rangle$) and in a perpendicular one (MSRD$_\perp =
\langle (\Delta u_\perp)^2\rangle$). If $\vec{R}_0$ is the relative position of the backscatter in the absence of thermal vibrations, the instantaneous relative position is
$$\vec{r} = \vec{R}_0 + \Delta \vec{u}\,,$$
and the instantaneous relative distance is [@Fornasini; @Dalba3]
$$r \simeq R_0 + \Delta u_\parallel + \frac{(\Delta u_\perp)^2}{2R_0}\,,
\label{distanciar}$$
where
$$\Delta u_\parallel = \hat{R}_0 \cdot \Delta \vec{u} = \hat{R}_0 \cdot (\vec{u}_j - \vec{u}_0)\,.$$
Then, the first cumulant $C_1^*$ is [@Dalba; @Dalba2; @Fornasini; @Fornasini2]
$$C_1^* = \langle r \rangle \simeq R_0 + \langle\Delta u_\parallel\rangle + \frac{\langle (\Delta u_\perp)^2\rangle}{2R_0}
\,,
\label{cumulantec1}$$
where
$$\begin{aligned}
r_\parallel &= \langle\Delta u_\parallel\rangle \,,& r_\perp &= \frac{\langle (\Delta u_\perp)^2\rangle}{2R_0}\,.
\label{rparaperp}\end{aligned}$$
In a crystalline sample, the crystallographic distance $R_c$ is related to the lattice parameters and can be obtained, for instance, from a Rietveld refinement procedure [@Rietveld; @Rietveld2], and it is given by
$$R_c = R_0 + \langle\Delta u_\parallel\rangle
\label{eqdefrc}$$
In this case, eq. \[cumulantec1\] becomes
$$C_1^* \simeq R_c + \frac{\langle (\Delta u_\perp)^2\rangle}{2R_0}
\label{eqrc}$$
Then, in principle, information about $\text{MSRD}_\perp $ can be obtained if $C_1^*$ and $R_c$ were known from EXAFS and XRD, respectively. The second cumulant is, to first order,
$$C_2^* = \bigl\langle (r - \langle r\rangle)^2 \bigr\rangle \simeq \langle (\Delta u_\parallel)^2\rangle =
\text{MSRD}_\parallel
= \langle [\hat{R}_0 \cdot (\vec{u}_j - \vec{u}_0)]^2\rangle =
\langle (\hat{R}_0 \cdot \vec{u}_j)^2 \rangle + \langle (\hat{R}_0 \cdot \vec{u}_0)^2 \rangle
-2 \langle (\hat{R}_0 \cdot \vec{u}_j) (\hat{R}_0 \cdot \vec{u}_0)\rangle$$
The terms $\langle (\hat{R}_0 \cdot \vec{u}_j)^2 \rangle$ and $\langle (\hat{R}_0 \cdot \vec{u}_0)^2 \rangle$ are the uncorrelated mean square displacements of the backscatterer and absorber atoms, respectively, and can be obtained from XRD measurements (for crystalline samples). The factor $\langle (\hat{R}_0 \cdot \vec{u}_j)
(\hat{R}_0 \cdot \vec{u}_0)\rangle$ is the displacement correlation function (DCF) [@Beni; @Dalba4], and XRD is not sensitive to it. An isotropic Debye crystal [@Dalba3; @Fornasini3] has $\text{MSRD}_\perp = 2 \text{MSRD}_\parallel$, so if both quantities can be found, it is possible to extract information about anisotropic vibrations, as was done recently for Cu [@Dalba3] and InP [@Schnohr]. The third cumulant, given by $C_3^*= \langle (r - \langle r\rangle)^3
\rangle$, measures the asymmetry of the [*real*]{} unidimensional distribution $\varrho(r,T)$ and can be associated with the anharmonicity of an [*effective*]{} interatomic potential
$$V(r) \simeq k_e(r-r_0)^2 - k_3 (r-r_0)^3 \,,
\label{potencialefetivo}$$
where $r_0$ is the minimum of $V(r)$, $k_e$ is the effective harmonic spring constant, and $k_3$ is the cubic anharmonicity constant.
It is usual to consider temperature dependences for $C_2^*$ and $C_3^*$ based on Einstein or Debye models [@Sevillano; @Beni; @Vaccari; @Dalba4] and, considering the correlated Einstein model, $C_2^*$ is given through
$$C_2^* = C_{2,\parallel,T}^* +C^*_{2,{\text{st}},\parallel}=
\frac{\hbar^2}{2\mu k_B \Theta_{\parallel}} \coth{\bigl(\frac{\Theta_{\parallel}}{2T} \bigr)} +
C^*_{2,{\text{st}},\parallel} \,,
\label{eqc2}$$
where $h = 2 \pi \hbar$ is the Plancks constant, $\Theta_\parallel$ is the Einstein temperature associated with vibrations parallel to the bonds, $\omega_\parallel = k_B \Theta_\parallel/\hbar$ is the parallel Einstein angular frequency, $\mu$ is the reduced mass for an absorber-scatterer pair, $k_{e,\parallel} = \mu \omega^2_\parallel$, $k_B$ is the Boltzmann’s constant and $C^*_{2,{\text{st}},\parallel}$ is the static or structural (independent of temperature) contribution to the $\text{MSRD}_\parallel$. $C_3^*$ is written as
$$C_3^* = \frac{3k_3 \hbar^6}{2\mu^3 k_B^4 \Theta_{\parallel}^4}
\biggl\{\Bigl[\coth{\bigl(\frac{\Theta_{\parallel}}{2T} \bigr)} \Bigr]^2 -1\biggr\} + C_{3,{\text{st}}}^* \,,
\label{eqc3}$$
where $C_{3,{\text{st}}}^*$ is the static or structural (independent of temperature) contribution to the asymmetry of $\varrho(r,T)$ and $k_3$ is related to the cubic anharmonic term of $V(r)$ (see eq. \[potencialefetivo\]). In a similar way, $\text{MSRD}_\perp=\langle (\Delta u_\perp)^2\rangle$ can be written as [@Vaccari; @Schnohr]
$$C_\perp^*= \text{MSRD}_\perp = \langle (\Delta u_\perp)^2\rangle=
\frac{\hbar^2}{\mu k_B \Theta_{\perp}} \coth{\bigl(\frac{\Theta_{\perp}}{2T} \bigr)} \,.
\label{eqmsrdperp}$$
Here, $\Theta_{\perp}$ is the Einstein temperature associated with vibrations perpendicular to the bonds. The temperatures $\Theta_{\perp}$ and $\Theta_{\parallel}$ should be the same only in isotropic materials. If a crystalline sample is under investigation, EXAFS measurements at some different temperatures furnish $C_1^*$, $C_2^*$ and $C_3^*$ and, considering eqs. \[eqc2\]–\[eqmsrdperp\] together with eq. \[eqrc\] and $R_c$ obtained from XRD measurements, the quantities $\Theta_\parallel$, $\Theta_\perp$, $\text{MSRD}_\parallel$, $\text{MSRD}_\perp$ and $k_3$ can in principle be obtained, furnishing information about anharmonicity, asymmetric distribution functions, anisotropic vibrations and also, from $\Delta C_1^*$, thermal expansion, which is different from the XRD thermal expansion due to the $\text{MSRD}_\perp$ in eq. \[cumulantec1\]. The problem now is how to obtain such information for an amorphous sample. The first point is that the crystallographic distance $R_c$ should be substituted for an amorphous XRD distance $R_a$, that is, from eq. \[eqdefrc\],
$$R_a = R_0 + \langle\Delta u_\parallel\rangle
\label{eqdefra}$$
which should, in principle, be obtained by XRD. However, determination of average interatomic distances for amorphous materials is not easy and, when it can be done, usually the error bars are large, making the $\text{MSRD}_\perp$ obtained from inversion of eq. \[cumulantec1\] unreliable. We have developed a method to obtain all data above using only EXAFS measurements, which could also be used to crystalline samples, but it needs several EXAFS measurements at different temperatures and on edges of both atomic species (for a binary alloy) to work. We illustrate the method in sec. \[secresultados\], by investigating an amorphous Ni$_{46}$Ti$_{54}$ alloy.
Experimental Details {#secexperimental}
====================
The Ni$_{46}$Ti$_{54}$ alloy was prepared by milling Ni (Merck, purity $>$ 99.5 %) and Ti (Alfa Aesar, purity $>$ 99.5 %) crystalline powders under argon atmosphere in a steel vial considering a ball to powder ratio of 5:1. The vial was mounted on a high energy Spex 8000 shaker mill and the powders were milled for 12 h.
EXAFS measurements on Ni and Ti K edges were taken in the transmission mode at beam line D04B-XAFS1 of the Brazilian Synchrotron Light Laboratory - LNLS. Three ionization chambers were used as detectors. [*a*]{}-Ni$_{46}$Ti$_{54}$ samples were formed by placing the powder on a porous membrane (Millipore, 0.2 $\mu$m pore size) and they were placed between the first and second chambers. Crystalline Ni and Ti foils furnished by LNLS were used as energy references and were placed between the second and third chambers. The beam size at the samples was 3 mm $\times$ 1 mm. The energy and average current of the storage ring were 1.37 GeV and 190 mA, respectively. EXAFS data were acquired at 30, 100, 200 and 300 K on Ni K edge and at 20 and 300 K on Ti K edge. The raw EXAFS data were analyzed following standard procedures using ATHENA and ARTEMIS [@athena] programs. Fourier transforms were performed considering Hanning window functions in the following ranges: 3.1–14.0 Å$^{-1}$ (Ni K edge) and 3.5–12.9 Å$^{-1}$ (Ti K edge) for the photoelectron momentum $k$ and 1.0–2.8 Å (Ni K edge) and 2.0–3.3 Å (Ti K edge) for the uncorrected phase radial distance $r$. Amplitudes and phase shifts were obtained from [*ab initio*]{} calculations using the spherical waves method [@Rehr] and FEFF8.02 software. Each measurement was fitted simultaneously with multiple $k$ weightings of 1–3 to reduce correlations between the fitting parameters. It should be noted that using the above procedure on an EXAFS analysis, the real cumulants $C_n^*$ are obtained [@Araujo; @Schnohr], not the effective ones.
Results and Discussion {#secresultados}
======================
Due to the fact that we had many EXAFS data at several temperatures an on Ni and Ti K edges, we could use several constraints during the fits. Fig. \[fig\_chi\_todas\] shows the EXAFS $k\chi(k)$ oscillations obtained from the measurements, and fig. \[figtodasft\] shows the magnitudes and imaginary parts of their Fourier transforms. It is interesting to note that the maxima of magnitudes and imaginary parts do not coincide, which is an indication of the asymmetry of the $g_{ij}(r)$ functions [@Koningsberger2]. Besides that, only the first shell is seen, as is expected for amorphous samples.
![\[fig\_chi\_todas\] EXAFS $k\chi(k)$ oscillations obtained on Ni and Ti K edges for [*a*]{}-Ni$_{46}$Ti$_{54}$.](fig_chi_todas.eps)
![\[figtodasft\] Magnitudes and imaginary parts of the Fourier transforms of the EXAFS $k\chi(k)$ data shown in fig. \[fig\_chi\_todas\] for [*a*]{}-Ni$_{46}$Ti$_{54}$.](fig_ft_todas.eps)
We made many tests considering different combinations of the experimental data and we will discuss here in details two models, defined as
Model A
: all experimental data were used without constraints related to MSRD$_\perp$.
Model B
: all experimental data were used together with the constraints related to MSRD$_\perp$.
To help the “visualization" of the possible constraints that can be introduced during the fitting process, fig. \[figdiagrama\] presents a diagram showing all relations used in the various EXAFS analyses we made. In the diagram, $c_i$ is the concentration of atoms of type $i$, where $i=1$ for Ni and $i=2$ for Ti.
![\[figdiagrama\] Diagram showing the possible constraints that can be used in the EXAFS analyses.](correlacoes.eps)
In Model A we did not consider any constraints for the first cumulant $C_1^*$ related to the MSRD$_\perp$, and we did not consider perpendicular Einstein temperatures. All other constraints shown in fig. \[figdiagrama\] were introduced in order to decrease correlations. The cumulants $C_2^{*,ij}$ and $C_3^{*,ij}$ associated with each pair $ij$ were constrained to follow eqs. \[eqc2\] and \[eqc3\], respectively. Then, we obtained the threshold energy $E_0$, $N_{ij}$ (average coordination numbers), $C^{*,ij}_{2,{\text{st}},\parallel}$ and $C^{*,ij}_{3,{\text{st}}}$, $\Theta_\parallel^{ij}$, $k_3^{ij}$ and $C_1^{*,ij} = \langle r_{ij} \rangle$. Figure \[figchitodas\] shows the real parts of the Fourier filtered first shells obtained on Ni and Ti K edges. The agreement between the simulations and the experimental data is very good, and it happens for all measurements on both edges at all temperatures. Tables \[tab1\], \[tab2\] and \[tab3\] present the values obtained for the relevant quantities above considering model A, and fig. \[figc2c3\] shows the temperature dependence of $C_2^{*,ij}$ and $C_3^{*,ij}$ (see eqs. \[eqc2\] and \[eqc3\]).
![\[figchitodas\] Real part of the Fourier filtered first shells of [*a*]{}-Ni$_{46}$Ti$_{54}$ (solid black lines) on Ni and Ti K edges at all temperatures investigated and their simulations (red squares).](figchitodas.eps)
----------- ---------------- ------------------------------------------------ ------------------------------------------------
Bond Type $N$ $C^*_{2,{\text{st}}}$ ($\times 10^{-3}$ Å$^2$) $C^*_{3,{\text{st}}}$ ($\times 10^{-4}$ Å$^3$)
Ni-Ni $6.3 \pm 1.0$ $5.3 \pm 0.2$ $-0.72 \pm 0.01$
Ni-Ti $7.1 \pm 0.6$ $15.1 \pm 0.8$ $-4.2 \pm 0.4$
Ti-Ti $5.8 \pm 0.3$ $3.1 \pm 0.4$ $-1.4 \pm 0.2$
----------- ---------------- ------------------------------------------------ ------------------------------------------------
: \[tab1\] Average coordination numbers and structural components of cumulants $C^*_2$ and $C^*_3$ obtained from the EXAFS fits shown in fig. \[figchitodas\] for [*a*]{}-Ni$_{46}$Ti$_{54}$ considering model A.
----------- ------------------------ ------------------------------ ----------------------- ------------------
Bond Type $\Theta_\parallel$ (K) $k_{e,\parallel}$ (eV/Å$^2$) $\nu_\parallel$ (THz) $k_3$ (eV/Å$^3$)
Ni-Ni $251 \pm 36$ 3.3 5.2 $6.8 \pm 1.5$
Ni-Ti $326 \pm 12$ 5.0 6.8 $144 \pm 5$
Ti-Ti $257 \pm 27$ 2.8 5.4 $6.5 \pm 2.3 $
----------- ------------------------ ------------------------------ ----------------------- ------------------
: \[tab2\] Parallel Einstein temperatures, parallel effective harmonic spring constants, parallel Einstein frequencies and cubic anharmonic force constants obtained from the EXAFS fits shown in fig. \[figchitodas\] for [*a*]{}-Ni$_{46}$Ti$_{54}$ considering model A.
--------- ---------------------------- ---------------------------- ---------------------------- --
$T$ (K) $C_1^{*,\text{Ni-Ni}}$ (Å) $C_1^{*,\text{Ni-Ti}}$ (Å) $C_1^{*,\text{Ti-Ti}}$ (Å)
20 $2.406 \pm 0.004$ $ 2.447 \pm 0.006$ $2.814 \pm 0.003$
100 $2.413 \pm 0.003$ $ 2.445 \pm 0.005$ —
200 $2.418 \pm 0.005$ $ 2.479 \pm 0.006$ —
300 $2.455 \pm 0.011$ $ 2.508 \pm 0.005$ $2.852 \pm 0.002$
--------- ---------------------------- ---------------------------- ---------------------------- --
: \[tab3\] First cumulant $C_1^{*,ij}$ obtained from the EXAFS fits shown in fig. \[figchitodas\] for [*a*]{}-Ni$_{46}$Ti$_{54}$ considering model A.
![\[figc2c3\] Cumulants $C_2^{*,ij}$ and $C_3^{*,ij}$ for [*a*]{}-Ni$_{46}$Ti$_{54}$ obtained for Ni-Ni (black squares), Ni-Ti (red circles) and Ti-Ti (blue triangles) as functions of temperature considering eqs. \[eqc2\] and \[eqc3\] respectively.](figc2c3.eps)
From Table \[tab1\] and fig. \[figc2c3\] it can be seen that the contribution of the parallel structural disorder $C_{2,\text{st},\parallel}^{*,ij}$ to the total MSRD$_\parallel$ is large, and it is the dominant factor even at 300 K for Ni-Ti pairs. We believe this fact can be associated with the fact that the sample is amorphous and also with fabrication technique used to produce the alloy. The third cumulant $C_3^{*,ij}$ indicates asymmetric distributions $\varrho_{ij}(r,T)$ at low temperatures due mainly to the contribution of the structural part since $C_{3,{\text{st}}}^{*,ij}$ is negative, but the asymmetry decreases with temperature. It is interesting to note the thermal behaviour of $C_{3}^{*,\text{Ni-Ti} }$, which is different from $C_{3}^{*,\text{Ni-Ni}}$ and $C_{3}^{*,\text{Ti-Ti}}$. This fact can be associated with the parallel Einstein temperatures obtained, since homopolar pairs have similar values and they are smaller than the value obtained for $\Theta_\parallel^{\text{Ni-Ti}}$ (see Table \[tab2\]), and also with the anharmonicity of $V(r)$, indicated by the cubic anharmonic constants $k_3^{ij}$. The average interatomic distances (first cumulant) $C_1^{*,ij}$ indicate a thermal expansion for Ni-Ni and Ti-Ti pairs and a negative (although very small) thermal expansion for Ni-Ti pairs from 20 K to 100 K followed by a positive thermal expansion. From $\Delta
C_1^{*,ij}$ the thermal expansion coefficients $\alpha_{ij}(T)$ can be estimated, and that will be done latter.
Now we can discuss model B. Since we had many constraints all structural values were obtained with enough accuracy to proceed to the next step. We fixed all values shown in Tables \[tab1\], \[tab2\] and \[tab3\] except $C_1^{*,ij}$, which were then constrained to follow eq. \[cumulantec1\] with $\langle (\Delta u_\perp^{ij})^2\rangle$ given by eq. \[eqmsrdperp\].Since we knew $C_1^{*,ij}$ from model A, we could analyze the new values obtained for $C_1^{*,ij}$ in order to avoid possible spurious results. The values obtained for the first cumulant were almost the same, and we were able to find $\Theta_\perp^{ij}$, $\text{MSRD}_\perp^{ij}$, $\langle\Delta u_\parallel^{ij} \rangle$, $R_0^{ij}$ and the corresponding amorphous distance $R_a^{ij}$ given by eq. \[eqdefra\]. The fits obtained for model B are almost identical to those shown in Fig. \[figchitodas\] and will not be shown. Fig. \[figcperp\] shows the thermal behavior of $C^{*,ij}_{\perp}$ given by eq. \[eqmsrdperp\] and Fig. \[figc1\] compares the values obtained for $C_1^{*,ij}$ using models A and B. As it can be seen from Fig. \[figc1\] and also from Tables \[tab3\] and \[tab4\], the values obtained from both models are very similar. Table \[tab5\] shows other relevant structural parameters obtained.
![\[figcperp\] Cumulants $C_\perp^{*,ij}$ for [*a*]{}-Ni$_{46}$Ti$_{54}$ obtained for Ni-Ni (black squares), Ni-Ti (red circles) and Ti-Ti (blue triangles) as a function of temperature considering eq. \[eqmsrdperp\].](figcperp.eps)
![\[figc1\] Comparison between cumulants $C_1^{*,ij}$ obtained from model A (red triangles) and model B (black squares).](figc1.eps)
--------- ---------------------------- ---------------------------- ---------------------------- --
$T$ (K) $C_1^{*,\text{Ni-Ni}}$ (Å) $C_1^{*,\text{Ni-Ti}}$ (Å) $C_1^{*,\text{Ti-Ti}}$ (Å)
20 $2.406 \pm 0.011$ $ 2.447 \pm 0.006$ $2.814 \pm 0.004$
100 $2.413 \pm 0.014$ $ 2.445 \pm 0.007$ —
200 $2.418 \pm 0.021$ $ 2.480 \pm 0.009$ —
300 $2.455 \pm 0.022$ $ 2.508 \pm 0.009$ $2.851 \pm 0.005$
--------- ---------------------------- ---------------------------- ---------------------------- --
: \[tab4\] First cumulant $C_1^{*,ij}$ obtained from the EXAFS fits shown in Fig. \[figchitodas\] for [*a*]{}-Ni$_{46}$Ti$_{54}$ considering model B.
----------- ------------------- -------------------- -------------------------- -------------------
Bond Type $R_0^{ij}$ (Å) $\Theta_\perp$ (K) $k_{e,\perp}$ (eV/Å$^2$) $\nu_\perp$ (THz)
Ni-Ni $2.400 \pm 0.006$ $230 \pm 121$ 2.7 4.8
Ni-Ti $2.438 \pm 0.002$ $269 \pm 140$ 3.4 5.6
Ti-Ti $2.810 \pm 0.001$ $233 \pm 52$ 2.3 4.9
----------- ------------------- -------------------- -------------------------- -------------------
: \[tab5\] Rest distances $R_0^{ij}$, perpendicular Einstein temperatures, perpendicular effective harmonic spring constants and perpendicular Einstein frequencies obtained from the EXAFS fits for [*a*]{}-Ni$_{46}$Ti$_{54}$ considering model B.
Due to similar $\Theta_\perp^{ij}$, the thermal behavior of $C^{*,ij}_\perp$ shown in Fig. \[figcperp\] is similar. The perpendicular Einstein temperatures are smaller than the corresponding $\Theta_{\parallel}^{ij}$, the difference being larger for Ni-Ti pairs, indicating the presence of vibrational anisotropy for all pairs specially for Ni-Ti ones. The corresponding force constants and frequencies indicate a loosening of the perpendicular bond strengths when compared to the parallel ones. So, bending vibrational modes are more easily excited than stretching modes. Fig. \[figgama\] shows the ratio $\gamma_{ij} = C_\perp^{*,ij}/C_{2,\parallel,T}^{*,ij}$ obtained using eq. \[eqc2\] and eq. \[eqmsrdperp\]. The values found are greater than 2, indicating the vibrational anisotropy, which is higher for the Ni-Ti pairs.
![\[figgama\] Ratio $\gamma_{ij} = C_\perp^{*,ij}/C_{2,\parallel,T}^{*,ij}$ obtained from EXAFS for Ni-Ni (black squares), Ni-Ti (red circles) and Ti-Ti (blue triangles) pairs in [*a*]{}-Ni$_{46}$Ti$_{54}$.](figgama.eps)
The values obtained for the first cumulants $C_1^{*,ij}$ shown in Fig. \[figc1\] and Tables \[tab3\] and \[tab4\] indicate a thermal expansion for Ni-Ni and Ti-Ti pairs for all temperatures and for Ni-Ti pairs after 100 K. However, these pairs seem to have a negative thermal expansion from 20 K to 100 K. Considering eqs. \[cumulantec1\], \[rparaperp\] and \[eqmsrdperp\], the perpendicular contribution $r_\perp^{ij}$ to $C_1^{*,ij}$ can be calculated and it is shown, together with the parallel contribution $r_\parallel^{ij}$, in Table \[tab6\].
--------- ---------------------------------- ------------------------------ ---------------------------------- ------------------------------ ---------------------------------- ------------------------------
$T$ (K) $r_\parallel^{\text{Ni-Ni}}$ (Å) $r_\perp^{\text{Ni-Ni}}$ (Å) $r_\parallel^{\text{Ni-Ti}}$ (Å) $r_\perp^{\text{Ni-Ti}}$ (Å) $r_\parallel^{\text{Ti-Ti}}$ (Å) $r_\perp^{\text{Ti-Ti}}$ (Å)
20 $0.004 \pm 0.005$ $ 0.0015 \pm 0.0008$ $0.007 \pm 0.004$ $0.0014 \pm 0.0007$ $0.002 \pm 0.002$ $0.0015 \pm 0.0003$
100 $0.011 \pm 0.007$ $ 0.002 \pm 0.001$ $0.005 \pm 0.004$ $0.002 \pm 0.001$ — $0.0019 \pm 0.0006$
200 $0.014 \pm 0.013$ $ 0.003 \pm 0.003$ $0.039 \pm 0.005$ $0.002 \pm 0.002$ — $0.003 \pm 0.001$
300 $0.051 \pm 0.012$ $ 0.004 \pm 0.004$ $0.066 \pm 0.004$ $0.003 \pm 0.003$ $0.037 \pm 0.002$ $0.004 \pm 0.002$
--------- ---------------------------------- ------------------------------ ---------------------------------- ------------------------------ ---------------------------------- ------------------------------
From Table \[tab6\] it can be seen that the perpendicular contribution $r_\perp^{ij}$ associated with the rigid shift of the potential minimum is smaller than the term $r_\parallel^{ij}$, which is related to the anharmonicity and to the shape of the effective potential, and their difference increases as the temperature is raised. This fact suggests that the thermal expansion in [*a*]{}-Ni$_{46}$Ti$_{54}$ is caused mainly by changes in the shape of the potential and the rigid shift of the potential minimum is a secondary effect.
The thermal expansion coefficient $\alpha$ can be written as
$$\alpha(T) = \frac{1}{r} \frac{dr}{dt} = \frac{1}{C_1^*} \frac{dC_1^*}{dt} =
\frac{1}{C_1^*} \frac{dr_\parallel}{dt} + \frac{1}{C_1^*} \frac{dr_\perp}{dt} =
\alpha_\parallel + \alpha_\perp$$
The contribution $\alpha_\perp$ to the thermal expansion associated with perpendicular vibrations can be found exactly and typical values are $\alpha_\perp^{\text{Ni-Ni}} = 5.1 \times 10^{-6}$ K$^{-1}$ at 300 K and $\alpha_\perp^{\text{Ni-Ti}} = 2.4 \times 10^{-6}$ K$^{-1}$ at 100 K (the largest and the smallest values, respectively). The total thermal expansion can be estimated from $\Delta C_1^{*,ij}$ and, for Ni-Ti pairs at 100 K, it is negative and has the value $\alpha^{\text{Ni-Ti}} = -9.6 \times 10^{-6}$ K$^{-1}$. All other values are positive and increase with temperature, ranging from $\alpha^{\text{Ni-Ni}} = 3.6 \times 10^{-5}$ K$^{-1}$, at 100 K, to $\alpha^{\text{Ni-Ni}} = 1.5 \times 10^{-4}$ K$^{-1}$, at 300 K. The contribution $\alpha_\perp$ to the total thermal expansion is always much smaller than $\alpha_\parallel$ except for Ni-Ti pairs at 100 K, indicating that the anharmonicity of the effective potencial is the dominant effect related to thermal expansion for the amorphous Ni$_{46}$Ti$_{54}$ alloy studied.
Conclusion {#secconclusoes}
==========
We investigated the structure, anharmonicity, asymmetry of pair distribution functions, vibrational anisotropy and thermal expansion of an amorphous Ni$_{46}$Ti$_{54}$ alloy produced by mechanical alloying considering EXAFS data only. The detailed study described here was only possible due to the experimental data available on Ni and Ti K edges and at several temperatures, which allowed us to use many constraints to obtain reliable values for the cumulants $C_1^*$, $C_2^*$ and $C_3^*$, making it possible to extract the perpendicular contributions to the MSRD and $C_1^{*,ij}$. The method should also work for crystalline samples, but we believe that EXAFS measurements on edges of all atomic species present in the sample should be used. This procedure may be the only way to extract such structural information for amorphous samples, so more tests on other alloys would be important.
Concerning the Ni$_{46}$Ti$_{54}$ alloy, it presents asymmetric $\varrho_{ij}(r,T)$ functions at low temperatures due to the structural contribution but the asymmetry decreases with temperature until 300 K. After that, the asymmetry incrases again. There is vibrational anisotropy mainly for Ni-Ti pairs, indicating that bending modes are looser than stretching ones, and Ni-Ti pairs exhibits a negative thermal expansion around 100 K. It would be very interesting to study the crystalline shape memory NiTi phase in order to associate these properties with the mechanical properties of this alloy.
We would like to thank the Brazilian agency CNPq for financial support. This study was also partially supported by LNLS (proposal XAFS1 4367/05).
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---
abstract: 'We develop an action principle for those models arising from isotropic loop quantum cosmology, and show that there is a natural conserved quantity $Q$ for the discrete difference equation arising from the Hamiltonian constraint. This quantity $Q$ relates the semi-classical limit of the wavefunction at large values of the spatial volume, but opposite triad orientations. Moreover, there is a similar quantity for generic difference equations of one parameter arising from a self-adjoint operator.'
author:
- 'D. Cartin'
title: Conserved quantities in isotropic loop quantum cosmology
---
Introduction
============
Loop quantum gravity (LQG) has been a successful research program, in the sense that it is a rigorously derived picture of the behavior of the gravitational field in the quantum regime. However, it is difficult to apply as is to physical problems – such as how quantum gravity effects in the realm of classical singularities affect cosmological models in general – because of its complexity. This has led to the development of loop quantum cosmology (LQC), a symmetry-reduced version of LQG. The Hamiltonian constraint equations of LQC are fundamentally discrete, due to the corresponding nature of quantum geometry coming from LQG, and thus are [*difference*]{} equations, rather than differential equations. The study of such difference equations has allowed the use of various analytic solution techniques, e.g. generating functions [@CarKhaBoj04], but these quickly become difficult to work with, leaving only numerical methods [@BriCarKha11]. Thus, it is fruitful to look for other analytic methods to obtain exact information about solutions to the quantum constraint equation; we do so by considering the discrete analogue of classical mechanics.
Discrete Lagrangian mechanics has been developed independently several times, in many contexts; for a list of previous work, see Marsden and West [@MarWes01]. Specifically for LQC, these ideas were developed by Shojai and Shojai [@ShoSho06], but oriented more towards finding approximate solutions to difference equations. There is also the program of consistent discretizations [@BahGamPul11], aiming to solve discretized versions of constrained mechanics, so that the constraints are preserved under evolution of the system in the model parameter $n$ (for an isotropic model). There, the emphasis is on the algebraic consistency of the discretized mechanics. In this work, we discuss a discrete version of the familiar Lagrangian mechanics, and show that for the isotropic models currently used in LQC, there is a conserved quantity $Q$ – in the sense that this function $Q_n$ of the single (discrete) parameter $n$ arising in the quantum Hamiltonian constraint equation is constant, regardless of what $n$ is evaluated at.
This paper is organized as follows. We start with an action principle for a functional ${\cal L}_n$ of a one-parameter sequence $s_n$, and derive the discrete version of the Euler-Lagrange equation which extremizes the action. Then, assuming the action is invariant under an infinitesimal transformation of $s_n$, we find a quantity $Q$ that remains constant under the action of the difference equation for $s_n$. As we will show, the generic Lagrangian for the isotropic models will be phase invariant under $s_n \to s_n' = s_n \exp(i \alpha)$; this generates a discrete version of the familiar conserved quantity for a phase invariant field $\phi(t)$. After this groundwork is laid, we turn to specific examples in isotropic LQC, and find the corresponding conserved quantities $Q$. We also consider general self-adjoint difference operators (or difference equations of the same form) and show they similarly have a conserved quantity.
An action principle for difference equations
============================================
For the rest of this work, we define the difference operator $\Delta$ acting on a sequence $s_n$ as $$\Delta s_n = s_n - s_{n-1}.$$ One particularly useful identity is the discrete version of integration by parts, namely the equivalence $$\label{IBP}
\sum_{n = -M + 1} ^M F_n \Delta G_n = \sum_{n = -M + 1} ^M [\Delta (F_{n + 1} G_n) - G_n \Delta F_{n + 1}].$$ Here the summation acts as the discrete analogue of a integral for functions of continuous variables. Much like the integral of a total derivative becomes a boundary term, the sum of a total difference gives a similar result: $$\label{bound-term}
\sum_{n = -M + 1} ^M \Delta (F_{n + 1} G_n) = F_{M+1} G_M - F_{-M+1} G_{-M}.$$ The “unbalanced" limits in these series come from the definition of $\Delta$. These limits have also been chosen to be symmetric around $n = 0$, but this is not a requirement; all of the results obtained in this work would be similar for a different choice of limits.
Now we briefly derive the Euler-Lagrange equation for a one-parameter sequence, both to keep the discussion self-contained, and to develop the notation used here. We start with the action $$S = \sum_{n = -M+1} ^M {\cal L}_n = \sum_{n = -M+1} ^M {\cal L} (n; s_n, \Delta s_n).$$ The format of ${\cal L}_n$ is to match that of “standard" continuous actions, i.e. with a coordinate variable $q$ and its time derivative ${\dot q}$. We will assume that the boundary values $s_M$ and $s_{-M}$ are fixed data, and determine the rest of the sequence. First, we seek the general Euler-Lagrange equation for the sequence $s_n$ that extremizes the value of $S$. In this vein, we consider an infinitesimal variation of the sequence $s_n$, namely, $$s_n \to s_n' = s_n + \alpha \eta_n,$$ for $|n| < M$, and find the sequence such that $$\frac{dS}{d \alpha} \biggl |_{\alpha = 0} = \sum_{n = -M+1} ^M \frac{d {\cal L} (n; s_n', \Delta s_n')}{d \alpha} \biggl |_{\alpha = 0} = 0.$$ This gives $$\label{ext-1}
\frac{d {\cal L}_n}{d \alpha} = \frac{\partial {\cal L}_n}{\partial s_n} \eta_n + \frac{\partial {\cal L}_n}{\partial (\Delta s_n)} (\Delta \eta_n).$$ When placed inside the summation, the last term in the equation (\[ext-1\]) can be integrated by parts as in (\[IBP\]), i.e. $$\begin{aligned}
\nonumber
&& \sum_{n = -M+1} ^M \frac{\partial {\cal L} (n; s_n, \Delta s_n)}{\partial (\Delta s_n)} (\Delta \eta_n) \\
\nonumber
&& = \sum_{n = -M+1} ^M \Delta \biggl[ \frac{\partial {\cal L} (n+1; s_{n + 1}, \Delta s_{n + 1})}{\partial (\Delta s_{n + 1})} \eta_n \biggl] \\
\label{parts-eqn}
&& - \sum_{n = -M+1} ^M \Delta \biggl[ \frac{\partial {\cal L} (n+1; s_{n + 1}, \Delta s_{n + 1})}{\partial (\Delta s_{n + 1})} \biggr] \eta_n.\end{aligned}$$ From equation (\[bound-term\]), the total difference in (\[parts-eqn\]) becomes $$\begin{aligned}
\sum_{n = -M+1} ^M && \hspace{-0.25in} \Delta \biggl[ \frac{\partial {\cal L} (n+1; s_{n + 1}, \Delta s_{n + 1})}{\partial (\Delta s_{n + 1})} \eta_n \biggl] \\
\nonumber
&=& \frac{\partial {\cal L} (n; s_n, \Delta s_n)}{\partial (\Delta s_n)} \biggl|_{n = M + 1} \eta_M \\
\nonumber
&-& \frac{\partial {\cal L} (n; s_n, \Delta s_n)}{\partial (\Delta s_n)} \biggl|_{n = - M + 1} \eta_{-M}
\nonumber
= 0.\end{aligned}$$ where the last equality results from the assumption that the boundary values are fixed, so $\eta_M = \eta_{-M} = 0$. Thus, $$\begin{aligned}
\frac{dS}{d \alpha} \biggl |_{\alpha = 0} &=& \sum_{n = -M+1} ^{M - 1} \biggl[ \frac{\partial {\cal L} (n; s_n, \Delta s_n)}{\partial s_n} \\
&-& \Delta \biggl( \frac{\partial {\cal L} (n+1; s_{n + 1}, \Delta s_{n + 1})}{\partial (\Delta s_{n + 1})}\biggl) \biggl] \eta_n.\end{aligned}$$ The remaining values $\eta_n$ are arbitrary, so this gives us the Euler-Lagrange equations[^1] $$\label{EL-eqn}
\frac{\partial {\cal L} (n; s_n, \Delta s_n)}{\partial s_n} - \Delta \biggl[ \frac{\partial {\cal L} (n+1; s_{n + 1}, \Delta s_{n + 1})}{\partial (\Delta s_{n + 1})} \biggl] = 0,$$ for $|n| \le M - 1$.
As an example that will be relevant later, suppose we consider the action $$\label{iso-action}
S = \sum_{n = -M + 1} ^M \bigl( f_n |\Delta s_n|^2 + g_n |s_n|^2 \bigr),$$ where $s_n$ and its complex conjugate ${\bar s}_n$ are considered independent – thus there are two equations of motion, one for each sequence – and the coefficient functions $f_n$ and $g_n$ are real. Then the Euler-Lagrange equations are $$g_n s_n - \Delta( f_{n + 1} \Delta s_{n + 1}) = 0$$ (and conjugate), or $$\label{EL-eqns}
f_{n+1} s_{n+1} - (f_{n+1} + f_n) s_n + f_n s_{n-1} = g_n s_n.$$ As we will see below, this is the same form as the equation for the eigensequences in the LQC isotropic model. For general equations of this form, there are two solutions, since the initial data can be independently selected at two values of $n$.
Conserved quantities
====================
Suppose that the Lagrangian ${\cal L} (n; s_n, \Delta s_n)$ is invariant under a transformation of the sequence $s_n$, i.e. $$s_n \to s_n + \alpha \xi_n,$$ but now including variations of the boundary values $s_M$ and $s_{-M}$; this will give us a conserved charge $Q_n$, as a special case of the discrete Noether’s theorem. Since this is essentially what we have above in equation (\[parts-eqn\]), then using the Euler-Lagrange equations, $$\label{charge-1}
\sum_{n = -M+1} ^M \Delta \biggl[ \frac{\partial {\cal L} (n+1; s_{n + 1}, \Delta s_{n + 1})}{\partial (\Delta s_{n + 1})} \xi_n \biggl] = 0.$$ One way to interpret this is that a charge $Q_n$, given by $$Q_n = \frac{\partial {\cal L} (n+1; s_{n + 1}, \Delta s_{n + 1})}{\partial (\Delta s_{n + 1})} \xi_n,$$ so that $\sum \Delta Q_n = 0$. Using the Euler-Lagrange equations, we can show that $\Delta Q_n = 0$ for all $n$, not just the sum over the entire range. Therefore we have $Q_n = Q$ = constant for all $|n| \le M$.
We turn again now to the example action (\[iso-action\]), which is invariant under $s_n \to s_n' = s_n \exp(i \alpha)$. Therefore, in the infinitesimal limit, $\xi_n = i s_n$. Since the sequences $s_n$ and ${\bar s}_n$ are considered independent, the current is $$\begin{aligned}
\nonumber
Q &=& \frac{\partial {\cal L}_{n+1}}{\partial (\Delta s_{n+1})} \xi_n + \frac{\partial {\cal L}_{n+1}}{\partial (\Delta {\bar s}_{n+1})} {\bar \xi}_n \\
\nonumber
&=& i f_{n+1} \bigl[ s_n (\Delta {\bar s}_{n+1}) - {\bar s_n} (\Delta s_{n+1}) \bigr] \\
&=& i f_{n+1} (s_n {\bar s}_{n+1} - s_{n+1} {\bar s}_n).\end{aligned}$$ Here, we make two comments. The first is that $Q$ depends only on the “kinetic" coefficient function $f_n$, and not the “potential" function $g_n$. Thus if two cosmological models have constraint equations where only the function $g_n$ differs between the two, they will have exactly the same form of conserved quantity $Q$. This occurs for Friedmann-Robertson-Walker models, where the value of the curvature constant $k$ changes only the functional form of $g_n$. Finally, we note this is a discrete version of the conserved current for a complex scalar field. In particular, in the large parameter regime $|n| \gg 1$, we can use a Taylor series expansion of $s_{n + 1}$ and ${\bar s}_{n + 1}$, so that the sequence is approximated by a continuous function $s(n)$. This limit gives $$Q \approx i f_0 (n) \biggl[ s (n) \frac{d{\bar s} (n)}{dn} - {\bar s} (n) \frac{ds (n)}{dn} \biggr],$$ where $f_0 (n)$ is the lowest order term in the series expansion of $f_{n + 1}$. In the context of a fundamentally discrete quantum cosmology, this $Q$ would be the semi-classical limit of this conserved quantity.
Before we turn to specific cases in isotropic LQC, we point out that a similar analysis is possible for difference equations with multiple parameters $n_i$, rather than the single parameter $n$ used here. However, this extension would require the difference equation to have constant lattice spacing. Such a constraint was developed for Bianchi I [@Chi06], although there is some debate whether physical considerations rule out this model in lieu of a difference equation with lattice refinement, i.e. the step sizes depend on the parameters $n_i$ themselves [@lattice-refine]. In any case, the derivations presented above cannot be extended to such lattice refined models.
Quantization of LQC isotropic model
===================================
For the first use of the general methods developed above, we consider the flat isotropic LQC model of Ashtekar, Pawlowski and Singh (APS) [@AshPawSin06b]. In this model, they consider a scalar field $\phi$ on a Friedman-Robertson-Walker space-time, and use this matter field as the internal clock for evolution. The sequences themselves are parametrized by a value $v \in \mathbb{R}$, proportional to the eigenvalue of the volume operator; the sign of $v$ gives the orientation of the spatial triad. Using the notation of APS, a sequence of values $\{\Psi (v) \}$ that solves the overall Hamiltonian constraint can be decomposed into an integral over eigensequences $e_\omega (v)$ with eigenvalues $\omega$, solving the equation $$\begin{aligned}
\nonumber
&& C^+ (v) e_\omega (v + 4) + C^0 (v) e_\omega (v) + C^- (v) e_\omega (v - 4) \\
\label{APS-cons}
&& = \omega^2 B(v) e_\omega (v),\end{aligned}$$ where
$$\begin{aligned}
C^+ (v) &=& \frac{3 \pi KG}{8} |v + 2| ||v+1| - |v+3||, \\
C^- (v) &=& C^+ (v - 4), \\
C^0 (v) &=& - C^+ (v) - C^- (v).\end{aligned}$$
and $$B(v) = \biggl( \frac{3}{2} \biggr)^3 K |v| \biggl | |v + 1|^{1/3} - |v - 1|^{1/3} \biggr |^3$$ with the constant $K = 2\sqrt{3} / (3 \sqrt{3\sqrt{3}})$. The solutions $e_\omega (v)$ of the difference equation (\[APS-cons\]) have support only on values $v = 4n + \epsilon$, where $\epsilon \in [0, 4)$. With this in mind, the relation (\[APS-cons\]) can be put into the Euler-Lagrange form (\[EL-eqns\]) if we let $$v = 4n + \epsilon,$$ for $\epsilon \in [0, 4)$ and $$s_n = e_\omega (v), \qquad
f_n = C^- (v), \qquad
g_n = \omega^2 B(v).$$ Thus, these eigensequences can be derived from a Lagrangian of the form $${\cal L}_v = C^- (v) |\Delta e_\omega (v)|^2 + \omega^2 B(v) |e_\omega (v)|^2,$$ which has a conserved quantity $$\label{improv-charge}
Q = i C^+ (v) [ e_\omega(v) {\bar e}_\omega (v + 4) - e_\omega(v + 4) {\bar e}_\omega(v)].$$ In particular, note that $C^+ (0) \ne 0$, for those wave functions having support at the classical singularity $v = 0$; this means that the conserved quantity $Q$ is not required to be zero for those solutions. This allows the use of this quantity to relate the asymptotic linear combinations of the wave function at large values of the volume, which we examine next.
In the semi-classical limit of large spatial volumes $|v| \gg 1$, we expect to recover the usual Wheeler-deWitt (WdW) theory for this space-time. Specifically, taking the Taylor series of the difference equation (\[APS-cons\]) at large $v$ gives a differential equation which matches the WdW equation at lowest order. These WdW solutions are worked out by APS, and the eigenfunctions are given by $$\label{WdW-eigen}
e_{\pm |k|} (v) = \frac{1}{\sqrt{2 \pi}} e^{\pm i |k| \ln |v|},$$ where $\omega = |k| \sqrt{12 \pi G}$. When $|v| \gg 1$, the two independent solutions $e_\omega$ to the isotropic LQC difference equation, for a given $\omega$, in the semi-classical limit should each match a linear combination of the solutions $e_{\pm |k|}$ to the corresponding WdW equation: $$\begin{aligned}
\label{eqn-limit}
e_\omega (v) &\to& A e_{|k|} (v) + B e_{-|k|} (v) \qquad v \gg 1, \\
e_\omega (v) &\to& C e_{|k|} (v) + D e_{-|k|} (v) \qquad v \ll -1,\end{aligned}$$ for constant coefficients $\{ A, B, C, D \}$. In the large volume limit, the conserved current becomes $$\label{current}
Q \approx \frac{3 i \pi KG}{16} |v| \biggl[ e_\omega (v) \frac{d {\bar e_\omega} (v)}{dv} - {\bar e_\omega} (v) \frac{d e_\omega}{dv} \biggr].$$ This asymptotic limit of $Q$ is precisely the same as the analogous conserved quantity that is obtained in WdW theory, using Noether’s theorem for the appropriate continuous action.
From (\[WdW-eigen\]) we have $$\frac{d e_\omega}{dv} = \frac{ikA}{|v|} e_{|k|} (v) - \frac{ikB}{|v|} e_{-|k|} (v).$$ In the large positive volume limit – i.e. large volume for a positively oriented triad – (\[current\]) gives $$Q_{n \to \infty} = \frac{3 \pi k KG}{16} \bigl( |A|^2 - |B|^2 \bigr),$$ while in the large negative volume limit (large volume but negative orientation of the triad), $$Q_{n \to - \infty} = \frac{3 \pi k KG}{16} \bigl( |C|^2 - |D|^2 \bigr).$$ Thus, the conservation of the current (\[current\]) gives $$\label{coeff-rel}
|A|^2 - |B|^2 = |C|^2 - |D|^2$$ as shown numerically[^2] for their quantization [@AshPawSin06a]. Note that the coefficient function $g_n$ does not enter at all into the form of the conserved quantity $Q$. Indeed, since going from the $k = 0$ isotropic model to either $k = \pm 1$ only alters this function $g_n$ [@FRW-models], the discussion here goes through without change.
Self-adjoint Hamiltonian constraint operators
=============================================
The last section dealt with a particular choice of factor-ordering for the Hamiltonian constraint of isotropic LQC, so it is interesting to see how generic these results are. The particular property we focus on here is the self-adjoint nature of the constraint operator; this is necessary in APS in order to find Dirac observables or physical states, chosen with a Hilbert space structure. Thus, in the following, we look for a conserved quantity $Q$ for a self-adjoint (gravitational) Hamiltonian constraint operator ${\hat H}_g$ acting on states $\psi$.
For the isotropic model, we have a basis of states $| \mu \rangle$ which are eigenstates of the volume operator, where $\mu \in \mathbb{R}$. There is a natural inner product for these states, namely, $$\langle \mu | \nu \rangle = \delta_{\mu, \nu}.$$ Note this is a Kronecker delta, not a Dirac delta function. Suppose we have an operator ${\hat A}$ acting on basis states $|\mu \rangle$ such that $${\hat A} |\mu \rangle = A^+ (\mu) |\mu + \delta \rangle + A^0 (\mu) |\mu \rangle + A^- (\mu) |\mu - \delta \rangle.$$ where $\delta$ is a constant value indicating the step size for the lattice used in the difference equation; the value of $\delta$ results from the holonomy operator acting on the state $| \mu \rangle$, and is related to a physical choice, such as the minimum length of the model. Then, one can show that ${\hat A}$ is self-adjoint, i.e. $\langle {\hat A} \mu | \nu \rangle = \langle \mu | {\hat A} \nu \rangle$ only if $$A^+ (\mu) = A^- (\mu + \delta).$$ If we define a physical wavefunction as $\psi_\mu = \langle \psi | \mu \rangle$, then the requirement that the gravitational Hamiltonian constraint operator ${\hat H}_g$ is self-adjoint means that the constraint $${\hat H} \psi_\mu = ({\hat H}_g + {\hat H}_{matt}) \psi_\mu = 0$$ gives rise to a difference equation of the form $$\begin{aligned}
\nonumber
&& H^+ (\mu) \psi_{\mu + \delta} (\phi) + H^0 (\mu) \psi_\mu (\phi) + H^- (\mu) \psi_{\mu - \delta} (\phi) \\
\label{generic-eqn-1}
&& = - {\hat H}_{matt} (\mu) \psi_\mu (\phi)\end{aligned}$$ with the gravitational operator ${\hat H}_g$ coefficient functions $H^0 (\mu)$ and $H^- (\mu) = H^+ (\mu - \delta)$. Here, we included the possibility of one or more matter fields $\phi$ a matter Hamiltonian operator ${\hat H}_{matt}$.
For isotropic matter, we have generically that ${\hat H}_{matt}$ acts only on the matter fields $\phi$, giving at most a pre-factor $B(\mu)$ related to the volume of the space-time with triad eigenvalue $\mu$. If we use the viewpoint of APS, we use the matter field $\phi$ as an internal clock in the model, so “evolution" means changing $\phi$. By finding solutions to the difference equation $$\begin{aligned}
\nonumber
&& H^+ (\mu) \psi_{\mu + \delta} (\phi) + H^0 (\mu) \psi_\mu (\phi) + H^- (\mu) \psi_{\mu - \delta} (\phi) \\
\label{generic-eqn-2}
&& = \omega^2 B(\mu) \psi_\mu (\phi).\end{aligned}$$ one can build up an arbitrary solution of the original constraint (\[generic-eqn-1\]) out of linear combinations of eigensequence solutions to (\[generic-eqn-2\]). To match up with the previous discussion, one can use a new parameter $n = \mu / \delta$ to get a lattice with unit spacing, then convert back into the physical parameter $\mu$. Thus, this equation is derivable from an action principle based on the Lagrangian $${\cal L}_\mu = H^- (\mu) |\Delta \psi_\mu (\phi)|^2 + F(\mu) |\psi_\mu (\phi)|^2$$ where $$F(\mu) = \omega^2 B(\mu) - H^0 (\mu) - H^+ (\mu) - H^- (\mu)$$ and conserved quantity $$Q = i H^+ (\mu) (\psi_\mu {\bar \psi}_{\mu + \delta} - \psi_{\mu + \delta} {\bar \psi_\mu})$$ As commented above, the “potential" function $F(\mu)$ is not relevant in finding the conserved quantity $Q$. Therefore, when the constraint operator ${\hat H}_g$ is self-adjoint, there is a conserved quantity $Q$ associated with the eigensequences of the operator, regardless of factor ordering issues, although specific orderings may lead to differing coefficient functions $H^+ (\mu)$.
Non-self-adjoint constraint equations in LQC
============================================
We have seen that a generic self-adjoint constraint equation is derivable from a discrete action, with a resulting conserved charge from the symmetry $s_n \to s'_n = s_n \exp(i \alpha)$. However, a conserved quantity exists even when the original Hamiltonian operator is not self-adjoint, but can be written in the form (\[generic-eqn-2\]). For example, with the earlier quantization of the flat isotropic model [@AshBojLew03], we have that $$\begin{aligned}
\label{old-cons}
\nonumber
&& (V_{\mu + 5\mu_0} - V_{\mu + 3\mu_0}) \psi_{\mu + 4\mu_0} (\phi) \\
\nonumber
&& - 2(V_{\mu + \mu_0} - V_{\mu - \mu_0}) \psi_\mu (\phi) \\
\nonumber
&& + (V_{\mu - 3\mu_0} - V_{\mu - 5\mu_0}) \psi_{\mu - 4\mu_0} (\phi) \\
&& = - \frac{8 \pi G \gamma^3 \mu_0^3 \ell_P^2}{3} {\hat H}_{matt} \psi_\mu (\phi),\end{aligned}$$ where $\mu_0$ is a constant step size, related to the minimum eigenvalue of the area operator, $\gamma$ is the Immirzi parameter, $\ell_P$ the Planck length, and $$\label{V-def}
V_\mu = \biggl( \frac{\gamma |\mu|}{6} \biggr)^{3/2} \ell_P ^3$$ are the eigenvalues of the volume operator. The precise form of the operator ${\hat H}_{matt}$ is not important, only that it acts on the matter fields $\phi$ and not directly on the triad eigenvalues $\mu$. If we write the $\phi$ dependence $\psi_\mu (\phi) = \psi_\mu \exp(i \omega \phi)$, this operator will have an action of the form $$- \frac{8 \pi G \gamma^3 \mu_0^3 \ell_P^2}{3} {\hat H}_{matt} \psi_\mu (\phi) = \omega^2 B(\mu) \psi_\mu (\phi)$$ where the function $B(\mu)$ relates to the dependence of the matter operator on the metric components. For the constraint equation (\[old-cons\]), the step size $\delta = 4\mu_0$; the parameter $\mu$ of this function can be rescaled as $n = \mu/4 \mu_0$ to give unit lattice spacing. We define the new function $$\label{s-def}
s_n (\phi) = [V_{(4n + 1) \mu_0} - V_{(4n - 1)\mu_0}] \psi_{4 n \mu_0} (\phi),$$ so that the constraint equation (\[old-cons\]) is now $$s_{n + 1} (\phi) - 2 s_n (\phi) + s_{n - 1} (\phi) = \frac{\omega^2 B(n) s_n (\phi)}{V_{(4n + 1) \mu_0} - V_{(4n - 1)\mu_0}}.$$ which is of the same form as the discrete Euler-Lagrange equation (\[EL-eqns\]) when one chooses $f_n = 1$. Note that (\[V-def\]) and (\[s-def\]) together requires that $s_0 = 0$, so there is no issue with evaluating the right-hand side of this derived difference equation at $n = 0$. This difference equation gives a conserved quantity, which when written in terms of the original parameter $\mu$ is $$\begin{aligned}
\nonumber
Q &=& i (V_{\mu + 5 \mu_0} - V_{\mu + 3 \mu_0})(V_{\mu + \mu_0} - V_{\mu - \mu_0}) \\
&& \times [\psi_\mu (\phi) {\bar \psi}_{\mu + 4 \mu_0} (\phi) - \psi_{\mu + 4 \mu_0} (\phi) {\bar \phi}_\mu (\phi) ]\end{aligned}$$ The conservation of this function can be checked directly using the original difference equation (\[old-cons\]).
A crucial difference between this conserved quantity and the charge (\[improv-charge\]) obtained from the APS quantization is what happens to the prefactor of $Q$ at the classical singularity. As stated above, for the APS model (\[APS-cons\]), at this singularity, $C^+ (0) \ne 0$, so $Q$ can take any real value; this allows the charge for wave functions passing through the classical singularity to be non-zero, and thus provide a relation between semi-classical limits of the wave function far away from the $v=0$ point. On the other hand, for the earlier quantization (\[old-cons\]), we have the difference in volumes $V_{\mu + \mu_0} - V_{\mu - \mu_0}$ at $\mu = 0$ is $$V_{\mu_0} - V_{-\mu_0} = 0$$ so that the charge $Q = 0$ for [*any*]{} wave function passing through the $\mu = 0$ classical singularity. This results from the non-self-adjoint nature of the constraint and places a strong restriction on allowable wave functions for this quantization not present for the APS model.
As mentioned previously, $s_0 = 0$ always, so the value $\psi_0$ is undetermined for the same reason $Q = 0$ for all sequences $\psi_\mu$ – the fact that the coefficient function of $\psi_\mu (\phi)$ in the constraint equation is zero when evaluated at $\mu = 0$. Thus, the equation (\[old-cons\]) puts a constraint on the other values $\psi_\mu$, and thus restricts the initial data for the wave function. The space of solutions for this equation is two-dimensional; thus, the restriction on $\psi_n$ means there is a unique (up to scaling) solution to the LQC constraint, a situation known as [*dynamical initial conditions*]{} – the evolution equation itself picks out the wave function without additional physical or theoretical input [@Boj01]. This has the bonus that there is no requirement of choosing a specific boundary condition for the wave function, a choice which may lead to differing physical results. However, in the context of discrete equations, this may lead to radically different behavior on either side of the classical singularity $\mu = 0$ [@CarKha05] (specifically, the absence of pre-classical solutions for a particular orientation of the triad), although this issue has not been explored with self-adjoint constraints.
This leads to the question of whether imposing a condition $Q = 0$ for other models may lead to a similar result; strictly speaking, this is not the same as the original idea of dynamical initial conditions, since we have added a specific input to the model beyond the constraint equation, but it is worthy to look into this choice as an alternative to various boundary conditions. The answer is negative, which we now show for the APS quantization (\[APS-cons\]). Here, the wave functions (\[WdW-eigen\]) are chosen as an orthonormal basis of the WdW solution space, and used to find the initial data for the LQC eigensequences, but these functions are certainly not the only choice. Indeed, one can pick $$\begin{aligned}
f_{1, k} (v) &\equiv& \frac{1}{\sqrt{2}} [e_{k} (v) + e_{-k} (v)] = \frac{1}{\sqrt{\pi}} \cos(|k| \ln |v|) \\
f_{2, k} (v) &\equiv& \frac{i}{\sqrt{2}} [e_{|k|} (v) - e_{-|k|} (v)] = \frac{1}{\sqrt{\pi}} \sin(|k| \ln |v|)\end{aligned}$$ For these functions, we have that $Q = 0$, so any linear combination of them also has zero charge; any eigensequence solving the quantum constraint (\[APS-cons\]), with initial data given by such a combination for $|v| \gg 1$ will have the same zero charge. The semi-classical analysis of APS can carry forward from this point, writing the generic solution to the semi-classical differential equation using the eigenfunctions $f_{1, k}$ and $f_{2, k}$, i.e. $$\Psi(v, \phi) = \int^\infty _{-\infty} dk [\psi_1 (k) f_{1, k} (v) e^{i \omega \phi} + \psi_2 (k) f_{2, k} (v) e^{-i \omega \phi}]$$ As with the APS analysis, this leads to superselection, and one can restrict solutions to only $\Psi_1$ or $\Psi_2$, where $\Psi_i$ is the solution written only in terms of the eigenfunctions $f_{i, k}$. Thus, imposing $Q = 0$ does not lead a reduction in the solution space.
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[^1]: We note here that another form of this equation has been derived in the literature. If the original discrete Lagrangian is of the form ${\cal L} (n; s_n, s_{n - 1})$ – that is, all appearances of $s_{n - 1}$ in ${\cal L}_n$ are not necessarily part of a difference $\Delta s_n$, then the Euler-Lagrange equations are of the form $$\frac{\partial {\cal L} (n; s_n, s_{n - 1})}{\partial s_n} + \frac{\partial {\cal L} (n + 1; s_{n + 1}, s_n)}{\partial s_n} = 0,$$ for $|n| \le M - 1$. Note that the position of the variables in ${\cal L}_n$ is important in the above equation, since ${\cal L}_n (u, v) \ne {\cal L}_n (v, u)$. This is the form usually used in the literature, such as by Bahr, Gambini and Pullin [@BahGamPul11], and Shojai and Shojai [@ShoSho06], although the latter allows the Lagrangian to be a function of $s_{n -2}$ as well as $s_n, s_{n - 1}$. If ${\cal L}$ is a function of $s_n$ and $\Delta s_n$ only, however, this is equivalent to the equation (\[EL-eqn\]).
[^2]: In APS, the authors choose to look only at those quantum solutions of the discrete LQC Hamiltonian constraint operator that are symmetric under the parity operator, to match with the choice of WdW eigenfunctions (\[WdW-eigen\]). However, these sequences are constructed out of linear combinations of the eigensequences $e_\omega$, which are not symmetric under the exchange $v \to -v$. Thus, the conservation law (\[coeff-rel\]) is non-trivial.
|
---
abstract: 'Response generation for task-oriented dialogues involves two basic components: dialogue planning and surface realization. These two components, however, have a discrepancy in their objectives, i.e., task completion and language quality. To deal with such discrepancy, conditioned response generation has been introduced where the generation process is factorized into action decision and language generation via explicit action representations. To obtain action representations, recent studies learn *latent actions* in an unsupervised manner based on the utterance lexical similarity. Such an action learning approach is prone to diversities of language surfaces, which may impinge task completion and language quality. To address this issue, we propose *multi-stage adaptive latent action* learning (MALA) that learns *semantic latent actions* by distinguishing the effects of utterances on dialogue progress. We model the utterance effect using the transition of dialogue states caused by the utterance and develop a semantic similarity measurement that estimates whether utterances have similar effects. For learning semantic actions on domains without dialogue states, MALA extends the semantic similarity measurement across domains progressively, i.e., from aligning shared actions to learning domain-specific actions. Experiments using multi-domain datasets, SMD and MultiWOZ, show that our proposed model achieves consistent improvements over the baselines models in terms of both task completion and language quality.'
author:
- |
Xinting Huang,^1^ Jianzhong Qi,^1^ Yu Sun,^2^ Rui Zhang^1^[^1]\
^1^[The University of Melbourne]{}, ^2^[Twitter Inc.]{}\
{xintingh@student., jianzhong.qi@, rui.zhang@}unimelb.edu.au, [email protected]
bibliography:
- 'aaai20.bib'
title: 'MALA: Cross-Domain Dialogue Generation with Action Learning'
---
Introduction
============
Task-oriented dialogue systems complete tasks for users, such as making a restaurant reservation or scheduling a meeting, in a multi-turn conversation [@gao2018neural; @sun2016contextual; @sun2017collaborative]. Recently, end-to-end approaches based on neural encoder-decoder structure have shown promising results [@wen2017network; @madotto2018mem2seq]. However, such approaches directly map plain text dialogue context to responses (i.e., utterances), and do not distinguish two basic components for response generation: *dialogue planning* and *surface realization*. Here, dialogue planning means choosing an action (e.g., to request information such as the preferred cuisine from the user, or provide a restaurant recommendation to the user), and surface realization means transforming the chosen action into natural language responses. Studies show that not distinguishing these two components can be problematic since they have a discrepancy in objectives, and optimizing decision making on choosing actions might adversely affect the generated language quality [@yarats2018hierarchical; @zhao2019rethinking].
\[!tbp\]
[L[4.0cm]{}|L[4.0cm]{}]{}\
Domain: Hotel (a). *Was there a particular section of town you were looking for?* (b). *Which area could you like the hotel to be located at?* & Domain: Attaction (c). *Did you have a particular type of attraction you were looking for?* (d). *great , what are you interested in doing or seeing ?*\
\
`Request(Area)` &[`Request(Type)`]{}\
\
(a): *\[0,0,0,1,0\]*; (b): *\[0,1,0,0,0\]* & (c): *\[0,0,0,1,0\]*; (d): *\[0,0,0,0,1\]*\
\
(a) & (b): *\[0,0,0,1,0\]* & (c) & (d): *\[0,0,0,0,1\]*\
To address this problem, conditioned response generation that relies on action representations has been introduced [@wen2015semantically; @chen2019semantically]. Specifically, each system utterance is coupled with an explicit action representation, and responses with the same action representation convey similar meaning and represent the same action. In this way, the response generation is decoupled into two consecutive steps, and each component for conditioned response generation (i.e., dialogue planning or surface realization) can optimize for different objectives without impinging the other. Obtaining action representations is critical to conditioned response generation. Recent studies adopt variational autoencoder (VAE) to obtain low-dimensional latent variables that represent system utterances in an unsupervised way. Such an auto-encoding approach cannot effectively handle various types of surface realizations, especially when these exist multiple domains (e.g., hotel and attraction). This is because the latent variables learned in this way mainly rely on the lexical similarity among utterances instead of capturing the underlying intentions of those utterances. In , for example, system utterances (a) and (c) convey different intentions (i.e., `request(area)` and `request(type)`), but may have the same auto-encoding based latent action representation since they share similar wording.
To address the above issues, we propose a multi-stage approach to learn *semantic latent actions* that encode the underlying intention of system utterances instead of surface realization. The main idea is that the system utterances with the same underlying intention (e.g., `request(area)`) will lead to similar *dialogue state transitions*. This is because dialogue states summarize the dialogue progress towards task completion, and a dialogue state transition reflect how the intention of system utterance influences the progress at this turn. To encode underlying intention into semantic latent actions, we formulate a loss based on whether the reconstructed utterances from VAE cause similar state transitions as the input utterances. To distinguish the underlying intention among utterances more effectively, we further develop a regularization based on the similarity of resulting state transitions between two system utterances.
Learning the semantic latent actions requires annotations of the dialogue states. In many domains, there are simply no such annotations because they require extensive human efforts and are expensive to obtain. We tackle this challenge by transferring the knowledge of learned semantic latent actions from state annotation rich domains (i.e., source domains) to those without state annotation (i.e., target domains). We achieve knowledge transferring in a progressive way, and start with actions that exist on both the source and target domain, e.g., `Request(Price)` in both hotel and attraction domain. We call such actions as *shared actions* and actions only exist in the target domain as *domain-specific actions*. We observe that system utterances with shared actions will lead to similar states transitions despite belonging to different domains. Following this observation, we find and align the shared actions across domains. With action-utterance pairs gathered from the above shared actions aligning, we train a network to predict the similarity of resulting dialogue state transitions by taking as input only texts of system utterances. We then use such similarity prediction as supervision to better learn semantic latent actions for all utterances with domain-specific actions.
Our contributions are summarized as follows:
- We are the first to address the problem of cross-domain conditioned response generation without requiring action annotation.
- We propose a novel latent action learning approach for conditioned response generation which captures underlying intentions of system utterances beyond surface realization.
- We propose a novel multi-stage technique to extend the latent action learning to cross-domain scenarios via shared-action aligning and domain-specific action learning.
- We conduct extensive experiments on two multi-domain human-to-human conversational datasets. The results show the proposed model outperforms the state-of-the-art on both in-domain and cross-domain response generation settings.
Related Work
============
Controlled Text Generation
--------------------------
Controlled text generation aims to generate responses with controllable attributes. Many studies focus on open-domain dialogues’ controllable attributes, e.g., style [@yang2018unsupervised], sentiment [@shen2017style], and specificity [@zhang2018learning]. Different from open-domain, the controllable attributes for task-oriented dialogues are usually *system actions*, since it is important that system utterances convey clear intentions. Based on handcrafted system actions obtained from domain ontology, action-utterance pairs are used to learn semantically conditioned language generation models [@wen2015semantically; @chen2019semantically]. Since it requires extensive efforts to build action sets and collect action labels for system utterances, recent years have seen a growing interest in learning utterance representations in an unsupervised way, i.e., *latent action learning* [@zhao2018unsupervised; @zhao2019rethinking]. Latent action learning adopts a pretraining phase to represent each utterance as a latent variable using a reconstruction based variational auto-encoder [@yarats2018hierarchical]. The obtained latent variable, however, mostly reflects lexical similarity and lacks sufficient semantics about the intention of system utterances. We utilize the dialogue state information to enhance the semantics of the learned latent actions.
Domain Adaptation for Task-oriented Dialogues
---------------------------------------------
Domain adaptation aims to adapt a trained model to a new domain with a small amount of new data. This is studied in computer vision [@saito2017asymmetric], item ranking [@wang2018joint; @huang2019carl], and multi-label classification [@wang2018kdgan; @wang2019adversarial; @sun2019internet]. For task-oriented dialogues, early studies focus on domain adaptation for individual components, e.g., intention determination [@chen2016zero], dialogue state tracking [@mrkvsic2015multi], and dialogue policy [@mo2018personalizing; @yin2018context]. Two recent studies investigate end-to-end domain adaptation. DAML [@qian2019domain] adopts model-agnostic meta-learning to learn a seq-to-seq dialogue model on target domains. ZSDG [@zhao2018zero] conducts adaptation based on action matching, and uses partial target domain system utterances as domain descriptions. These end-to-end domain adaption methods are either difficult to be adopted for conditioned generation or needing a full annotation of system actions. We aim to address these limitations in this study.
Preliminaries
=============
Let $\{d_i | 1 \leq i \leq N \} $ be a set of dialogue data, and each dialogue $d_i$ contains $n_d$ turns: $d_i = \{ (c_t, x_t) | 1\leq t \leq n_d \}$, where $c_t$ and $x_t$ are the context and system utterance at turn $t$, respectively. The context $c_t =\{u_1, x_1,...u_t \}$ consists of the dialogue history of user utterances $u$ and system utterances $x$. Latent action learning aims to map each system utterance $x$ to a representation $z_d(x)$, where utterances with the same representation express the same action. The form of the representations $z_d(x)$ can be, e.g., one-hot [@wen2015semantically], multi-way categorical, and continuous [@zhao2019rethinking]. We use the one-hot representation due to its simplicity although the proposed approach can easily extend to other representation forms. We obtain the one-hot representation via VQ-VAE, a discrete latent VAE model [@van2017neural]. Specifically, an encoder $p_{\mathcal{E}}$ encodes utterances as $z_e(x) \in \mathbb{R}^D$, and a decoder $p_{\mathcal{G}}$ reconstructs the original utterance based on inputs $z_q(x) \in \mathbb{R}^D$, where $D$ is the hidden dimension. The difference lies in that between $z_e(x)$ and $z_q(x)$, we build a discretization bottleneck using a nearest-neighbor lookup on an embedding table $e \in \mathbb{R}^{K \times D}$ and obtain $z_q(x)$ by finding the embedding vector in $e$ having the closest Euclidean distance to $z_e(x)$ i.e., $$z_q(x) =e_k \text{ where } k=\operatorname*{argmin}_{j \in \mid K \mid} {\left\lVertz_e(x)-e_j\right\rVert}_2.$$ The learned latent $z_d(x)$ is a one-hot vector that only has 1 at index $k$. All components, including $p_{\mathcal{E}}$, $p_{\mathcal{G}}$ and embedding table $e$, are jointly trained using uto-ncoding objective as $$\mathcal{L}_{\text{a-e}} =\mathbb{E}_{ x} [-\log p_{\mathcal{G}}( x|z_q( x)) + {\left\lVertz_e(x )-z_q(x)\right\rVert}_2^2]$$ The structure of VQ-VAE is illustrated in Fig. \[subfig-stage1\], where the three components are marked in grey color.
Proposed Model
==============
Overview
--------
To achieve better conditioned response generation for task-oriented dialogues, we propose *ulti-stage daptive atent ction learning* (MALA). Our proposed model works for two scenarios: (i) For domains with dialogue state annotations, we utilize these annotations to learn semantic latent actions to enhance the conditioned response generation. (ii) For domains without state annotations, we transfer the knowledge of semantic latent actions learned from the domains with rich annotations, and thus can also enhance the conditioned response generation for these domains.
The overall framework of MALA is illustrated in . The proposed model is built on VQ-VAE that contains encoder $p_{\mathcal{E}}$, embedding table $e$, and decoder $p_{\mathcal{G}}$. Besides auto-encoding based objective $\mathcal{L}_{\text{a-e}}$, we design oinwise loss $\mathcal{L}_{\text{PT}}$ and aiwise loss $\mathcal{L}_{\text{PR}}$ to enforce the latent actions to reflect underlying intentions of system utterances. For domains with state annotations (see Fig. 1a), we train $p_{\mathcal{B}}$ and $p_{\mathcal{B}}^{\text{inv}} $ to measure state transitions and develop the pointwise and pairwise loss (Sec. 4.2). For domains without state annotations (see Fig. 1b), we develop a pairwise loss $\mathcal{L}_{\text{PR}}^{\mathcal{S-T}} $ based on $p_{\mathcal{B}}$ and $p_{\mathcal{B}}^{\text{inv}} $ from annotation-rich-domains. This loss measure state transitions for a cross-domain utterance pair, and thus can find and align shared actions across domains (Sec. 4.3). We then train a similarity prediction network $p_{\text{SPN}}$ to substitute the role of state tracking models, which only taking as input raw text of utterances. We using $p_{\text{SPN}}$ predictions as supervision to form pointwise $\mathcal{L}_{\text{PT}}^{\mathcal{T-T}} $ and pairwise loss $\mathcal{L}_{\text{PR}}^{\mathcal{T-T}} $ (see Fig. 1c), and thus obtain semantic latent actions for domain without state annotations (Sec. 4.4).
The overall framework of MALA is illustrated in . We first utilize dialogue states to obtain semantic latent actions for the state-annotation-rich domains (Sec. \[stage1\]). For domains having no state annotations, we adopt an progressive strategy to transfer the knowledge of semantic latent actions. We first find and align *shared actions* across domains (Sec. \[stage2\]). We then further learn fine-grained *domain-specific actions* by extending the shared semantic latent actions to the complete action space (Sec. \[stage3\]).
The proposed model is built on VQ-VAE that contains encoder $p_{\mathcal{E}}$, embedding table $e$, and decoder $p_{\mathcal{G}}$. Besides auto-encoding based objective $\mathcal{L}_{\text{a-e}}$, we design pointwise loss $\mathcal{L}_{\text{PT}}$ and pairwise $\mathcal{L}_{\text{PR}}$ to enforce the learned discrete action representations to reflect underlying intentions of system utterances. Based on whether state annotations are available, we choose among state tracking models ($p_{\mathcal{B}}$ and $p_{\mathcal{B}}^{\text{inv}} $) or similarity prediction network ($p_{\text{SPN}}$) to provide supervision.
Stage-I: Semantic Latent Action Learning {#stage1}
----------------------------------------
We aim to learn semantic latent actions that align with the underlying intentions for system utterances. To effectively capture the underlying intention, we utilize dialogue state annotations and regard utterances that lead to similar state transition as having the same intention. We train dialogue state tracking model to measure whether any two utterance will lead to a similar state transition. We apply such measurement in (i) a pointwise manner, i.e., between a system utterance and its reconstructed counterpart from VAE, and (ii) a pairwise manner, i.e., between two system utterances.
### Dialogue State Tracking
Before presenting the proposed pointwise measure, we first briefly introduce dialogue state tracking tasks. Dialogue states (also known as dialogue belief) are in the form of predefined slot-value pairs. Dialogues with state (i.e., elief) annotations are represented as $d_i = \{ (c_t, b_t, x_t) | 1\leq t \leq n_d \} $, where $b_t \in \{0,1\}^{N_b}$ is the dialogue state at turn $t$, and $N_b$ is the number of all slot-value pairs. Dialogue state tracking (DST) is a multi-label learning process that models the conditional distribution $p(b_t|c_t)=p(b_t|u_t, x_{t-1}, c_{t-1})$. Using dialogue states annotations, we first train a state tracking model $p_{\mathcal{B}}$ with the following cross-entropy loss: $$\begin{split}
\mathcal{L} = \sum_{d_i}\sum_{t=1:n_d}-\log(b_t^{\top} \cdot p_{\mathcal{B}}(u_t, x_{t-1}, c_{t-1})) \\
\end{split}$$ $$p_{\mathcal{B}}(u_t, x_{t-1}, c_{t-1}) = \text{softmax}(h(u_t, x_{t-1}, c_{t-1}))$$ where $h(\cdot)$ is a scoring function and can be implemented in various ways, e.g., self attention models [@zhong2018global], or an encoder-decoder [@wu2019transferable].
For convenience, we refer to domains with dialogue state information as *source domains* and domains without state information as *target domains*. Note that the DST model is only available in the source domain because target domain has no labeled dialogue states.
### Pointwise Measure
Built on unsupervised latent action learning framework (i.e., VAE with discretization bottleneck), we overcome its limitation in capturing underlying intention by additionally encoding pointwise and pairwise measures beyond lexical similarity.
With the trained state tracking model $p_{\mathcal{B}}$, we now measure whether the reconstructed utterance output can lead to a similar dialogue state transition from turn $t-1$ to $t$ (i.e., orar order). We formulate such measure as a cross-entropy loss between original state $b_t$ and model $p_{\mathcal{B}}$ outputs when replacing system utterance $x_{t-1}$ in inputs with $\Tilde{x}_{t-1}$. $$\mathcal{L}_{\text{fwd}} = \mathbb{E}_{x}[-\log (b_t^{\top} \cdot p_{\mathcal{B}}(b_t|u_t, \Tilde{x}_{t-1}, c_{t-1}) )]$$ $$\Tilde{x}_{t-1} \sim p_{\mathcal{G}} (z_q(x_{t-1}))$$ where $\Tilde{x}_{t-1}$ is sampled from the decoder output. Note that once state tracking model $p_{\mathcal{B}}$ finish training, its parameters will not be updated and $\mathcal{L}_{\text{fwd}}$ is only used for training the components of VAE, i.e., the encoder, decoder and the embedding table. To get gradients for these components during back-propagation, we apply a continuous approximation trick [@yang2018unsupervised]. Specifically, instead of feeding sampled utterances as input to state tracking models, we use Gumbel-softmax [@jang2016categorical] distribution to sample instead. In this way outputs of the decoder $p_{\mathcal{G}} $ becomes a sequence of probability vectors, and we can use standard back-propagation to train the generator:
We expect the dialogue state transition in forward order can reflect the underlying intentions of system utterances. However, the state tracking model $p_{\mathcal{B}}$ heavily depends on user utterance $u_t$, meaning that shifts of system utterances intentions may not sufficiently influence the model outputs. This prevents the considered state transitions modeled from providing valid supervision for semantic latent action learning. To address this issue, inspired by inverse models in reinforcement learning [@pathak2017curiosity], we formulate a inverse state tracking to model the dialogue state transition from turn $t$ to $t-1$. Since dialogue state at turn $t$ already encodes information of user utterance $u_t$, we formulate the inverse state tracking as $p(b_{t-1}|x_{t-1}, b_{t})$. In this way the system utterance plays a more important role in determining state transition. Specifically, we use state annotations to train an inverse state tracking model $p_{\mathcal{B}}^{\text{inv}}$ using the following cross-entropy loss: $$\mathcal{L} = \sum_{d_i}\sum_{t=2:n_d}-\log(b_{t-1}^{\top} \cdot p_{\mathcal{B}}^{\text{inv}}(|x_{t-1}, b_t))$$ $$p_{\mathcal{B}}^{\text{inv}}(x_{t-1}, b_t) = \text{softmax}(g(x_{t-1}, b_{t-1}))$$ where the scoring function $g(\cdot)$ can be implemented in the same structure as $h(\cdot)$. The parameters of inverse state tracking model $p_{\mathcal{B}}^{\text{inv}}$ also remain fixed once training is finished.
We use the inverse state tracking model to measure the similarity of dialogue state transitions caused by system utterance and its reconstructed counterpart. The formulation is similar to forward order: $$\mathcal{L}_{\text{inv}} = \mathbb{E}_{x}[-\log(b_{t-1}^{\top} \cdot p_{\mathcal{B}}^{\text{inv}}(b_{t-1}| \Tilde{x} _{t-1}, b_t))]$$ $$\Tilde{x}_{t-1} \sim p_{\mathcal{G}} (z_q(x_{t-1}))$$
Thus, combining the dialogue state transitions modeled in both forward and inverse order, we get the full oinwise loss for learning semantic latent actions: $$\mathcal{L}_{\text{PT}} = \mathcal{L}_{\text{fwd}} + \mathcal{L}_{\text{inv}}$$
### Pairwise Measure
To learn semantic latent actions that can distinguish utterances with different intentions, we further develop a pairwise measure that estimates whether two utterances lead to similar dialogue state transitions.
With a slight abuse of notation, we use $x_{i}$ and $x_{j}$ to denote two system utterances. We use $u_i$, $c_i$, $b_i$ to denote the input user utterance, dialogue context, and dialogue state for dialogue state tracking models $p_{\mathcal{B}}$ and $p_{\mathcal{B}}^{\text{inv}}$, respectively. We formulate a pairwise measurement of state transitions as $$s_{i,j} = s_{\text{fwd}}(x_{i}, x_{j}) + s_{\text{inv}}(x_{i}, x_{j})$$ $$\begin{split}
s_{\text{fwd}}(x_{i}, x_{j}) &= \text{KL}(p_{\mathcal{B}}^{\text{fwd}} (u_i, x_i,c_i) \mid \mid p_{\mathcal{B}}^{\text{fwd}}(u_i,x_j,c_i) ) \\
s_{\text{inv}}(x_{i}, x_{j}) &= \text{KL}(p_{\mathcal{B}}^{\text{inv}}(x_i, b_i) \mid \mid p_{\mathcal{B}}^{\text{inv}}(x_j, b_i) )
\end{split}
\label{sim-fwd-inv}$$ where KL is the Kullback-Leibler divergence. Both $p_{\mathcal{B}}$ and $p_{\mathcal{B}}^{\text{inv}}$ take inputs related to $x_i$. We can understand $s_{i,j} $ in this way that it measures how similar the state tracking results are when replacing $x_i$ with $x_j$ as input to $p_{\mathcal{B}}$ and $p_{\mathcal{B}}^{\text{inv}} $. To encode the pairwise measure into semantic latent action learning, we first organize all system utterances in a pairwise way $ \mathcal{P} =\{\big \langle (x_i, x_j), s_{i,j} \big \rangle |1 \leq i,j \leq N_u^{\mathcal{S}} \} $ where $N_u^{\mathcal{S}}$ is the total number of system utterances in the domains with state annotations. We then develop a aiwise loss to incorporate such measure on top of the VAE learning. $$\label{pair-s1}
\mathcal{L}_{\text{PR}} = \sum_{\mathcal{P}} - s_{ij}^{\text{avg}}\log d(x_i, x_j) - (1-s_{ij}^{\text{avg}}) \log (1- d(x_i, x_j))$$ $$ d(x_i, x_j) = \sigma ( -z_e(x_{i})^{\top} z_e(x_{j}) )$$ where $\sigma$ is the sigmoid function, $s_{ij}^{\text{avg}}$ is the average of $s_{i,j}$ and $s_{j,i}$, and $z_e(x) \in \mathbb{R^{D}}$ is encoder $p_{\mathcal{E}}$ outputs. The pairwise loss $\mathcal{L}_{\text{PR}}$ trains $p_{\mathcal{E}}$ by enforcing its outputs of two system utterances to have far distances when these two utterance lead to different state transitions, and vice versa.
The overall objective function of the semantic action learning stage is: $$\mathcal{L}_{\text{S-{\uppercase\expandafter{\romannumeral 1\relax}}}} = \mathcal{L}_{\text{a-e}} + \alpha \mathcal{L}_{\text{PT}} + \beta \mathcal{L}_{\text{PR}}$$ where $\alpha$ and $\beta $ are hyper-parameters. We adopt $\mathcal{L}_{\text{S-{\uppercase\expandafter{\romannumeral 1\relax}}}}$ to train VAE with discretization bottleneck and obtain utterance-action pair (e.g., utterance (c) and its semantic latent action in Table \[toy-example\]) that encodes the underlying intentions for each system utterance in the domains with state annotations.
Stage-II: Action Alignment across Domains {#stage2}
-----------------------------------------
In order to obtain utterance-action pairs in domains having no state annotations, we propose to progressively transfer the knowledge of semantic latent actions from those domains with rich state annotations. At this stage, we first learn semantic latent actions for the utterances that have co-existing intentions (i.e., shared actions) across domains.
We use $x^{\mathcal{S}} $ and $x^{\mathcal{T}} $ to denote system utterances in the source and target domain, respectively. The set of all utterances is denoted by: $$U^{\mathcal{S}}=\{x_i^{\mathcal{S}} | 1 \leq i \leq N_u^{\mathcal{S}} \} ; U^{\mathcal{T}} =\{x_j^{\mathcal{T}} |1 \leq j \leq N_u^{\mathcal{T}} \}$$ where $N_u^{\mathcal{S}} $ and $N_u^{\mathcal{T}} $ are the total utterance number in each domain, respectively. We adopt the proposed pairwise measure to find the target domain system utterances that have shared actions with the source domain. Based on the assumption that although from different domains, utterances with the same underlying intention are expected to lead to similar state transitions, we formulate the pairwise measure of cross-domain utterance pairs as: $$s_{i,j}^{c} = s_{\text{fwd}}(x_{i}^{\mathcal{S}}, x_{j}^{\mathcal{T}} ) + s_{\text{inv}}(x_{i}^{\mathcal{S}}, x_{j}^{\mathcal{T}})$$ where $s_{\text{fwd}} $ and $s_{\text{inv}} $ are computed using the trained $p_{\mathcal{B}} $ and $p_{\mathrel{B}}^{\text{inv}}$. Since it only requires the trained dialogue state tracking models and state annotations related to $x_{i}^{\mathcal{S}}$, this pairwise measure is asymmetrical. Taking advantage of the asymmetry, this cross-domain pairwise measure can still work when we only have raw texts of dialogues in the target domain. We then utilize the cross-domain pairwise for action alignment during latent action learning on the target domain. We then formulate a loss Incorporating action alignment: $$\label{pair-s2}
\begin{split}
\mathcal{L}_{\text{PR}}^{\mathcal{S-T}} &= \sum_{x^{S}, x^{T}} - s_{i,j}^{c}\log d(x_i^{\mathcal{S}}, x_j^{\mathcal{T} })\\
& -(1-s_{i,j}^{c})\log (1-d(x_i^{\mathcal{S} },x_j^{\mathcal{T}}))
\end{split}$$ $$d(x_i^{\mathcal{S}}, x_j^{\mathcal{T}}) = \sigma( -z_e(x_{i}^{\mathcal{S}})^{\top} z_e(x_{j}^{\mathcal{T}} ) )$$ where $d(x_i^{\mathcal{S}} , x_j^{\mathcal{T} })$ is computed based on outputs of the same encoder $p_{\mathcal{E}}$ from VAE at stage-I. We also use utterances in the target domain to formulate an auto-encoding loss: $$\mathcal{L}_{\text{a-e}}^{\mathcal{T}} = \mathbb{E}_{x\in U^{\mathcal{T}}}[l_r+{\left\lVert\text{sg}(z_e(x) )-z_q(x)\right\rVert}_2]$$ The overall objective for the stage-$\text{{\uppercase\expandafter{\romannumeral 2\relax}}}$ is: $$\mathcal{L}_{\text{S-{\uppercase\expandafter{\romannumeral 2\relax}}}} = \mathcal{L}_{\text{a-e}}^{\mathcal{T}} + \beta \mathcal{L}_{\text{PR}}^{\mathcal{S-T}}$$ where $\beta$ is the hyper-parameter as the same in $\mathcal{L}_{\text{S-I}}$. With the VAE trained using $\mathcal{L}_{\text{S-{\uppercase\expandafter{\romannumeral 2\relax}}}} $, we can obtain utterance-action pairs for system utterances in the domain having no state annotations. However, for utterances having domain-specific intentions, their semantic latent actions are still unclear, which is tackled in Stage 3.
Stage-III: Domain-specific Actions Learning {#stage3}
-------------------------------------------
We aim to learn semantic latent action for utterances with domain-specific actions at this stage.
### Similarity Prediction Network (SPN)
We train an utterance-level prediction model, SPN, to predict whether two utterances lead to similar state transitions by taking as input the raw texts of system utterances only. Specifically, SPN gives a similarity score in $[0,1]$ to an utterance pair: $$p_{\text{SPN}}(x_i,x_j) = \sigma(r(x_i, x_j))$$ where $r(\cdot)$ is a scoring function (and we implement it with the same structure as $h(\cdot)$). We use the binary labels $a_{ij}$ indicating whether two utterances $x_i$ and $x_j$ have the same semantic latent action to train the SPN. Specifically, we have $a_{ij}=1$ if $z_d(x_i) = z_d(x_j)$, and otherwise $a_{ij}=0$. To facilitate effective knowledge transfer, we obtained such labels from both source and target domains. We consider all pairs of source domain utterances and obtain $$P^{\mathcal{S}} = \{ \big \langle (x_i, x_j),a_{ij} \big \rangle \mid x_i, x_j \in U^{\mathcal{S}}\}$$ We also consider pairs of target domain utterances with shared actions: we first get all target domain utterances with aligned actions $
U_{\text{shared}}^{\mathcal{T}} = \{x_j^{\mathcal{T}} | x_j^{\mathcal{T}} \in U^{\mathcal{T}}, z_d(x_j^{\mathcal{T}}) \in A^{\mathcal{S}} \}
$ where $A^{\mathcal{S}}$ represents the set of shared actions $
A^{\mathcal{S}} = \{z_d(x_i^{\mathcal{S}}) \mid x_i^{\mathcal{S}} \in U^{\mathcal{S}}\}
$ and then obtain $$P^{\mathcal{T}} = \{\big \langle (x_i, x_j ), a_{ij} \big \rangle \mid x_i, x_j \in U_{\text{shared}}^{\mathcal{T}} \}.$$ Using all the collected pairwise training instances $p =\big \langle (x_i, x_j ), a_{ij} \big \rangle $, we train SPN via the loss $$\mathcal{L}_{\text{SPN}} = \mathbb{E}_{p \in P^{\mathcal{S}} + P^{\mathcal{T}} } [\text{cross-entropy}(a_{ij}, r(x_i,x_j))].$$
We then use the trained $p_{\text{SPN}} $ to replace state tracking models in both pointwise and pairwise measure. Specifically, we formulate the following pointwise loss $$\mathcal{L}_{\text{PT}}^{\mathcal{T}} = \mathbb{E}_{x \in U^T}[-\log p_{\text{SPN}}(x^{\mathcal{T}}, \Tilde{x }^{\mathcal{T}} ) ]$$ $$\Tilde{x }^{\mathcal{T}} \sim p_{\mathcal{G}} (z_q(x^{\mathcal{T}} ))$$ which enforces the reconstructed utterances to bring similar dialogue state transitions as the original utterance. We further formulate the pairwise loss as $$\begin{split}
\mathcal{L}_{\text{PR}}^{\mathcal{T-T}} &= \sum_{x_i,x_j \in U^{\mathcal{T}}} -p_{ \text{SPN}}(x_i,x_j) \log d(x_i^{\mathcal{T}}, x_j^{\mathcal{T}}) \\
&- (1-p_{ \text{SPN}}(x_i,x_j))\log(1- d(x_i^{\mathcal{T}}, x_j^{\mathcal{T}}))
\end{split}$$ $$d(x_i^{\mathcal{T}}, x_j^{\mathcal{T}}) = \sigma( -z_e(x_{i}^{\mathcal{T}})^{\top} z_e(x_{j}^{\mathcal{T}} ) ).$$ Compared to the pairwise loss at stage-I (Eqn. \[pair-s1\]) and stage-II (Eqn. \[pair-s2\]), the main difference is that we use $p_{\text{SPN}}$ to substitute $s_{i,j}$ that relies on trained dialogue state tracking models. The overall objective function for stage- is: $$\mathcal{L}_{\text{S-{\uppercase\expandafter{\romannumeral 3\relax}}}} = \mathcal{L}_{\text{a-e}}^{\mathcal{T}} +\alpha \mathcal{L}_{\text{PT}}^{\mathcal{T}} + \beta \mathcal{L}_{\text{PR}}^{\mathcal{T-T}}$$
Conditioned Response Generation
-------------------------------
After obtaining semantic latent actions, we train the two components, dialogue planning and surface realization, for conditioned response generation. Specifically, we first train a surface ealization model $p_{r}$ that learns how to translate the semantic latent action into fluent text in context $c$ as $$\mathcal{L} = \mathbb{E}_{x}[- \log p_{r}(x|z_d(x),c ) ]$$ Then we optimize a dialogue panning model $p_{l}$ while keeping the parameters of $p_{r} $ fixed $$\mathcal{L} = \mathbb{E}_{x}\mathbb{E}_{z} [- \log p_{r}(x|z,c)p_{l}(z|c) ]$$ In this way, the response generation is factorized into $p(x|c)=p(x|z,c)p(z|c) $, where dialogue planning and surface realization are optimized without impinging the other.
Experiments
===========
To show the effectiveness of MALA, we consider two experiment settings: multi-domain joint training and cross-domain response generation (Sec. 5.1). We compare against the state-of-the-art on two multi-domain datasets in both settings (Sec. 5.2). We analyze the effectiveness of semantic latent actions and the multi-stage strategy of MALA under different supervision proportion (Sec. 5.3).
Settings
--------
### Datasets
We use two multi-domain human-human conversational datasets: (1) <span style="font-variant:small-caps;">SMD</span> dataset [@eric2017key] contains 2425 dialogues, and has three domains: *calendar*, *weather*, *navigation*; (2) <span style="font-variant:small-caps;">MultiWOZ</span> dataset [@budzianowski2018multiwoz] is the largest existing task-oriented corpus spanning over seven domains. It contains in total 8438 dialogues and each dialogue has 13.7 turns in average. We only use five out of seven domains, i.e., *restaurant, hotel, attraction, taxi, train*, since the other two domains contain much less dialogues in training set and do not appear in testing set. This setting is also adopted in the study of dialogue state tracking transferring tasks [@wu2019transferable]. Both datasets contain dialogue states annotations. We use **Entity-F1** [@eric2017key] to evaluate dialogue task completion, which computes the F1 score based on comparing entities in delexicalized forms. Compared to inform and success rate originally used on <span style="font-variant:small-caps;">MultiWOZ</span> by , Entity-F1 considers informed and requested entities at the same time and balances the recall and precision. We use **BLEU** [@papineni2002bleu] to measure the language quality of generated responses. We use a three-layer transformer [@vaswani2017attention] with a hidden size of 128 and 4 heads as base model. On <span style="font-variant:small-caps;">MultiWOZ</span>, we use Inform Rate and Success Rate as in the Dialog-Context-to-Text Generation task proposed by Budzianowski et al. (2018): **Inform** rate measures whether the system has provided an appropriate entity; **Success** rate measures whether generated responses answer all the requested attributes.
\[tbp\]
-- ----------- ---------- ----------- ---------- ----------
Entity-F1 BLEU Entity-F1 BLEU
KVRN 48.1 13.2 30.3 11.3
Mem2seq 62.6 20.5 39.2 14.8
Sequicity 81.1 21.9 57.7 17.2
LIDM 76.7 17.3 59.4 15.5
LaRL 80.4 18.2 71.3 14.8
MALA-S1 83.8 22.4 74.3 18.7
MALA-S2 84.7 21.7 76.2 20.0
MALA-S3 **85.2** **22.7** **76.8** **20.1**
-- ----------- ---------- ----------- ---------- ----------
: Multi-Domain Joint Training Results []{data-label="multi-join"}
Note that w/o and w/ Action means whether the baseline considers conditioned generation
\[tbp\]
-- ----------- ---------- ---------- ---------- ----------
Navigate Weather Schedule
Sequicity 31.7 42.6 55.7 16.0
LaRL 33.2 44.3 57.5 12.3
Sequicity 35.9 46.9 59.7 16.8
LaRL 34.7 45.0 58.6 12.1
MALA-S1 38.3 54.8 64.4 19.3
MALA-S2 39.4 57.0 65.1 18.5
MALA-S3 **41.8** **59.4** **68.1** **20.2**
-- ----------- ---------- ---------- ---------- ----------
: Cross-Domain Generation Results on <span style="font-variant:small-caps;">SMD</span> []{data-label="cross-smd"}
\[tbp\]
-- ----------- ---------- ----------- ---------- ----------- ---------- ----------- ---------- ----------- ---------- ----------
Entity-F1 BLEU Entity-F1 BLEU Entity-F1 BLEU Entity-F1 BLEU Entity-F1 BLEU
Sequicity 16.1 10.7 27.6 16.8 17.4 14.4 19.6 13.9 22.1 15.4
LaRL 17.8 10.1 30.5 12.9 24.2 11.7 19.9 9.6 28.5 11.7
Sequicity 17.3 12.3 27.0 17.6 17.9 15.8 26.0 14.5 22.4 16.9
LaRL 21.0 9.1 34.7 12.8 24.8 11.8 22.1 10.8 31.9 12.6
MALA-S1 23.3 15.5 43.5 18.1 31.5 16.2 24.7 16.5 33.6 18.0
MALA-S2 26.4 15.8 48.3 18.8 36.5 17.6 28.8 16.6 41.7 18.6
MALA-S3 **32.7** **16.7** **51.2** **19.4** **41.9** **18.1** **35.0** **17.3** **44.7** **19.0**
-- ----------- ---------- ----------- ---------- ----------- ---------- ----------- ---------- ----------- ---------- ----------
### Multi-domain Joint Training
In this setting, we train MALA and other baselines with full training set, i.e., using complete dialogue data and dialogue state annotations. We use the separation of training, validation and testing data as original <span style="font-variant:small-caps;">SMD</span> and <span style="font-variant:small-caps;">MultiWOZ</span> dataset. We compare with the following baselines that do not consider conditioned generation: (1) **KVRN** [@eric2017key]; (2) **Mem2seq** [@madotto2018mem2seq]; (3) **Sequicity** [@lei2018sequicity]; and two baselines that adopt conditioned generation: (4) **LIDM** [@wen2017latent]; (5) **LaRL** [@zhao2019rethinking]; For a thorough comparison, We include the results of the proposed model after one, two, and all three stages, denoted as **MALA-(S1/S2/S3)**, in both settings.
### Cross-domain Response Generation
In this setting, we adopt a leave-one-out approach on each dataset. Specifically we use one domain as target domain while the others as source domains. There are three and five possible configurations for <span style="font-variant:small-caps;">SMD</span> and <span style="font-variant:small-caps;">MultiWOZ</span>, respectively. For each configuration, we set that only 1% of dialogues in target domain are available for training, and these dialogues have no state annotations. We compare with Sequicity and LaRL using two types of training schemes in cross-domain response generation. [^2] (1) Target only: models are trained only using dialogues in target domain. (2) Fine tuning: model are first trained in the source domains, and we conduct fine-tuning using dialogues in target domain.
Overall Results
---------------
### Multi-Domain Joint Training
Table \[multi-join\] shows that our proposed model consistently outperforms other models in the joint training setting. MALA improves dialogue task completion (measured by Entity-F1) while maintaining a high quality of language generation (measured by BLEU). For example, MALA-S3 (76.8) outperforms LaRL (71.3) by 7.71% under Entity-F1 on <span style="font-variant:small-caps;">MultiWOZ</span>, and has the highest BLEU score. Meanwhile, we also find that MALA benefits much from and in the joint learning setting. For example, MALA-S1 and MALA-S2 achieve 9.25% and 10.43% improvements over LIDM under Entity-F1 on <span style="font-variant:small-caps;">SMD</span>. This is largely because that, having complete dialogue state annotations, MALA can learn semantic latent actions in each domain at stage-I, and the action alignment at stage-II reduce action space for learning dialogue policy more effectively by finding shared actions across domains. We further find that LIDM and LaRL perform worse than Sequicity on <span style="font-variant:small-caps;">SMD</span>. The reason is that system utterances on <span style="font-variant:small-caps;">SMD</span> have shorter length and various expressions, making it challenging to capture underlying intentions merely based on surface realization. MALA overcomes this challenge by considering dialogue state transitions beyond surface realization in semantic latent action learning.
### Cross-Domain Response Generation
The results on <span style="font-variant:small-caps;">SMD</span> and <span style="font-variant:small-caps;">MultiWOZ</span> are shown on Tables \[cross-smd\] and \[cross-woz\], respectively. We can see that MALA significantly outperforms the baselines on both datasets. For example, on , MALA-S3 outperforms LaRL by 47.5% and 55.7% under Entity-F1 using *train* and *hotel* as target domain, respectively. We also find that each stage of MALA is essential in cross-domain generation scenarios. For example, on <span style="font-variant:small-caps;">MultiWOZ</span> using *attraction* as target domain, stage-III and stage-II brings 14.7% and 15.8% improvements compared with its former stage, and MALA-S1 outperforms fine-tuned LaRL by 27.0% under Entity-F1. We further find that the contribution of each stage may vary when using different domains as target, and we will conduct a detailed discussion in the following section. By comparing fine-tuning and target only results of LaRL, we can see latent actions based on lexical similarity cannot well generalize in the cross-domain setting. For example, fine-tuned LaRL only achieves less than 3% over target-only result under Entity-F1 on MultiWOZ using *attraction* as target domain.
Discussions
-----------
We first study the effects of each stage in MALA in cross-domain dialogue generation. We compare MALA-(S1/S2/S3) with fine-tuned LaRL under different dialogue proportion in target domain. The results are shown in Fig. \[target-restau\] and \[target-taxi\]. We can see that the performance gain of MALA is largely attributed to stage-III when using *restaurant* as target domain, while attributed to stage-II using *taxi* as target. This is largely because there are many shared actions between *taxi* and *train* domains, many utterance-action pairs learned by action alignment at stage-II already capture the underlying intentions of system utterances. On the other hand, since *restaurant* does not have many shared actions across domains, MALA relies more on the similarity prediction network to provide supervision at stage-III.
Last, we study the effects of semantic latent action in both joint training and cross-domain generation settings. To investigate how pointwise measure $\mathcal{L}_{\text{PT}}$ and pairwise measure $\mathcal{L}_{\text{PR}}$ contribute to capturing utterance intentions, we compare the results of MALA without pointwise loss (MALA$\setminus$PT), and without pairwise loss (MALA$\setminus$PR) under varying size of dialogue state annotations. The results of multi-domain joint training under Entity-F1 on SMD are shown in Fig. \[state-mul\]. We can see that both pointwise and pairwise measure are both important. For example, when using 55% of state annotations, encoding pointwise and pairwise measure bring 5.9% and 8.0% improvements, respectively. For cross-domain generation results shown in Fig. \[state-target\], we can find that these two measures are essential to obtain semantic latent actions in the target domain.
Conclusion
==========
We propose multi-stage adaptive latent action learning (MALA) for better conditioned response generation. We develop a novel dialogue state transition measurement for learning semantic latent actions. We demonstrate how to effectively generalize semantic latent actions to the domains having no state annotations. The experimental results confirm that MALA achieves better task completion and language quality compared with the state-of-the-art under both in-domain and cross-domain settings. For future work, we will explore the potential of semantic action learning for zero-state annotations application.
Acknowledgement {#acknowledgement .unnumbered}
===============
We would like to thank Xiaojie Wang for his help. This work is supported by Australian Research Council (ARC) Discovery Project DP180102050, China Scholarship Council (CSC) Grant \#201808240008.
[^1]: Rui Zhang is the corresponding author.
[^2]: We also consider using DAML [@qian2019domain], but the empirical results are worse than those of target only and fine tuning.
|
---
abstract: |
Many programs allow the user to input data several times during its execution. If the program runs forever the user may input data infinitely often. A program terminates if it terminates no matter what the user does.
We discuss various ways to prove that program terminates. The proofs use well orderings, Ramsey Theory, and Matrices. These techniques are used by real program checkers.
---
[**Proving Programs Terminate using**]{}\
[**Well Orderings, Ramsey Theory, and Matrices**]{}
Exposition by William Gasarch
[**General Terms:**]{} Verification, Theory.
[**Keywords and Phrase:**]{} Proving programs terminate, Well Orderings, Ramsey Theory, Matrices.
Introduction {#se:intro}
============
We describe several ways to prove that programs terminate. By this we mean terminate on [*any*]{} sequence of inputs. The methods employed are well-founded orderings, Ramsey Theory, and Matrices. This paper is self contained; it does not require knowledge of any of these topics or of Programming Languages. The methods we describe are used by real program checkers.
Our account is based on the articles of B. Cook, Podelski, and Rybalchenko [@CPRabs; @CPRterm; @proveterm; @PRrank; @ramseypl; @PRtrans; @DBLP:conf/tacas/PodelskiR11] Lee, Jones, and Ben-Amram [@Lee:ranking; @LJA]. Termination checkers that use the methods of B. Cook, Podelski, and Rybalchenko include ARMC [@ARMC], Loopfrog [@loopfrog], and Terminator [@terminator]. Earlier Terminator checkers that used methods from [@LJA] are Terminlog [@LS:97] and Terminweb [@CT:99]. Termination checkers that use the methods of Lee, Jones, and Ben-Amram include ACL2 [@ACL2], AProVE [@aprove], and Julia [@julia].
The statement [*The Program Terminates*]{} means that it terminates no matter what the user does. The user will be supplying inputs as the program runs; hence we are saying that the user cannot come up with some (perhaps malicious) inputs that make the program run forever.
In the summary below we refer to Programs which appear later in the paper.
1. Section \[se:simpleord\]: We prove Program [3 ]{}terminates using the well founded order $({{\sf N}},\le)$. We then state Theorem \[th:useorderings\] that encapsulates this kind of proof.
2. Section \[se:compordering\]: We prove Program [4 ]{}terminates using the well founded order $({{\sf N}}\times{{\sf N}}\times{{\sf N}}\times{{\sf N}},{<_{\rm lex}})$, where ${<_{\rm lex}}$ is the lexicographic ordering, and Theorem \[th:useorderings\].
3. Section \[se:useramsey\]: We prove Program [4 ]{}terminates using Ramsey’s Theorem. We then state Theorems \[th:useramsey\] and Theorem \[th:useramseygen\] which encapsulates this kind of proof.
4. Section \[se:matrix\]: We prove Program [4 ]{}terminates using Ramsey’s Theorem (via Theorem \[th:useramsey\]) and matrices. We then state and prove Theorems \[th:matrix\] and \[th:matrixgen\] that encapsulates this kind of proof.
5. Section \[se:useramsey2\]: We prove Program [5 ]{}terminates using Ramsey’s Theorem (via Theorem \[th:useramseygen\]) and transition invariants. We then state Theorem \[th:usetrans\] that encapsulates this kind of proof. It seems difficult to obtain a proof that Program [5 ]{}terminates without using Ramsey’s Theorem.
6. Section \[se:morematrix\]: We prove Program [5 ]{}terminates using Ramsey’s Theorem and Matrices. Program [5 ]{}has some properties that make this a good illustration.
7. Section \[se:sub\]: We prove Program [6 ]{}terminates using Ramsey’s Theorem and transition invariants. Program [6 ]{}has some properties that make this a good illustration.
8. Section \[se:need\]: The proofs of Theorems \[th:prog2ramsey\], \[th:useramsey\], and \[th:useramseygen\] only used Ramsey’s Theorem for Transitive colorings. We show, in three ways, that this subcase of Ramsey’s theorem is strictly weaker than the full Ramsey’s Theorem.
9. Section \[se:dec\]: We examine some cases of program termination that are decidable and some that are undecidable.
10. Section \[se:open\]: We discuss open problems.
11. In the Appendix we present an interesting example by Ben-Amram.
Notation and Definitions
========================
1. ${{\sf N}}$ is the set $\{0,1,2,3,\ldots,\}$. All variables are quantified over ${{\sf N}}$. For example [*For all $n\ge 1$ means for all $n\in \{1,2,3,\ldots,\}$.*]{}
2. ${{\sf Z}}$ is the set of integers, $\{\ldots, -2,-1,0,1,2,\ldots \}$.
3. ${{\sf Q}}$ is the set of rationals.
4. ${{\rm R}}$ is the set of reals.
1. In a program the command
${x}= {{\bf Input}}({{\sf N}})$
means that ${x}$ gets an integer provided by the user.
2. More generally, if $A$ is any set, then
${x}= {{\bf Input}}(A)$
means that ${x}$ gets a value from A provided by the user.
3. If we represent the set A by listing it out we will write (for example)
${x}={{\bf Input}}({y},{y}+2,{y}+4,{y}+6,\ldots)$
rather than the proper but cumbersome
$x={{\bf Input}}(\{y,y+2,y+4,y+6,\ldots\}$
All of the programs we discuss do the following: initially the variables get values supplied by the user, then there is a [[**While** ]{}]{}loop. Within the [[**While** ]{}]{}loop the user can specify which one of a set of statements get executed through the use of a variable called [*control*]{}. We focus on these programs for two reasons: (1) programs of this type are a building block for more complicated programs, and (2) programs of this type can already do some things of interest. We give a very general example.
Let $n,m\in{{\sf N}}$. Let $g_{i}$ as $1\le i\le m$ be computable functions from ${{\sf Z}}^{n+1}$ to ${{\sf Z}}^n$. Program [1 ]{}is a general example of the programs we will be discussing.
Comment: $X$ is $(x[1],\ldots,x[n])$
Comment: The $g_i$ are computable functions from $\Z^{n+1}$ to $\Z^n$
$X = (\inp(\Z),\inp(\Z),\ldots,\inp(\Z))$
While $x[1]>0$ and $x[2]>0$ and $\cdots$ $x[n]>0$
control = $\inp(1,2,3,...,m)$
if control==$1$
$X = g_{1}(X,\inp(\Z))$
else
if control==$2$
$X = g_{2}(X,\inp(\Z))$
else
.
.
.
else
if control==$m$
$X =g_{m}(X,\inp(\Z))$
We define this type of program formally. We call it a [*program*]{} though it is actually a program of this restricted type. We also give intuitive comments in parenthesis.
1. A [*program*]{} is a tuple $(S,I,R)$ where the following hold.
- $S$ is a decidable set of states. (If $({x}_1,\ldots,{x}_n)$ are the variables in a program and they are of types $T_1,\ldots,T_n$ then $S=T_1\times\cdots\times T_n$.)
- $I$ is a decidable subset of $S$. ($I$ is the set of states that the program could be in initially.)
- $R\subseteq S\times S$ is a decidable set of ordered pairs. ($R(s,t)$ iff $s$ satisfies the condition of the [[**While** ]{}]{}loop and there is some choice of instruction that takes $s$ to $t$. Note that if $s$ does not satisfy the condition of the [[**While** ]{}]{}loop then there is no $t$ such that $R(s,t)$. This models the [[**While** ]{}]{}loop termination condition.)
2. A [*computation*]{} is a (finite or infinite) sequence of states $s_1,s_2,\ldots$ such that
- $s_1\in I$.
- For all $i$ such that $s_i$ and $s_{i+1}$ exist, $R(s_i,s_{i+1})$.
- If the sequence is finite and ends in $s$ then there is no pair in $R$ that begins with $s$. Such an $s$ is called [*terminal*]{}.
3. A program [*terminates*]{} if every computation of it is finite.
4. A [*computational segment*]{} is a sequence of states $s_1,s_2,\ldots,s_n$ such that, for all $1\le i\le n-1$, $R(s_i,s_{i+1})$. Note that we do not insist that $s_1\in I$ nor do we insist that $s_n$ is a terminal state.
Consider Program [2]{}.
$(x,y) = (\inp(\Z),\inp(\Z))$
While $x>0$
control = $\inp(1,2)$
if control == 1
$(x,y)=(x+10,y-1)$
else
if control == 2
$(x,y)=(y+17,x-2)$
Program [2 ]{}can be defined as follows:
- $S=I={{\sf Z}}\times{{\sf Z}}$.
- $R= \{ (x,y),(x+10,y-1){\mathrel{:}}x,y\ge 1\}\bigcup \{(x,y),(y+17,x-2){\mathrel{:}}x,y\ge 1\}.$
An ordering $T$ is [*well founded*]{} if every set has a minimal element. Note that if $T$ is well founded then there are no infinite descending sequences of elements of $T$.
A Proof Using the Ordering $({{\sf N}},\le)$ {#se:simpleord}
=============================================
To prove that every computation of Program [3 ]{}is finite we need to find a quantity that, during every iteration of the [[**While** ]{}]{}Loop, decreases. None of $x,y,z$ qualify. However, the quantity $x+y+z$ does. We use this in our proof.
$(x,y,z) = (\inp(\Z), \inp(\Z), \inp(\Z))$
While $x>0$ and $y>0$ and $z>0$
control = $\inp(1,2,3)$
if control == 1 then
$(x,y,z)=(x+1,y-1,z-1)$
else
if control == 2 then
$(x,y,z)=(x-1,y+1,z-1)$
else
if control == 3 then
$(x,y,z)=(x-1,y-1,z+1)$
\[th:prog1\] Every computation of Program [3 ]{}is finite.
Let $$f({x},{y},{z}) =
\begin{cases}
0 \text{ if any of $x,y,z$ are $\le 0$;}
\\
{x}+ {y}+{z}\text{ otherwise.}
\cr
\end{cases}$$
Before every iteration of the [[**While** ]{}]{}loop $f({x},{y},{z})>0$. After every iteration of the [[**While** ]{}]{}loop $f({x},{y},{z})$ has decreased. Eventually there will be an iteration such that, after it executes, $f({x},{y},{z})=0$. When that happens the program terminates.
The keys to the proof of Theorem \[th:prog1\] are (1) $x+y+z$ decreases with every iteration, and (2) if ever $x+y+z=0$ the the program has terminated. There is a more general theorem lurking here, which we state below without proof. Our statement uses a different notation than the original, due to Floyd [@floydpl].
\[th:useorderings\] Let $PROG=(S,I,R)$ be a program. Assume there is a well founded order $(P,<_P)$, and a map $f:S{\rightarrow}P$ such that the following occurs.
1. If $R(s,t)$ then $f(t) <_P f(s).$
2. If the program is in a state $s$ such that $f(s)$ is a minimal element of $P$, then the program terminates.
Then any computation of $PROG$ is finite.
A Proof Using the Ordering $({{\sf N}}\times{{\sf N}}\times{{\sf N}}\times{{\sf N}},{<_{\rm lex}})$ {#se:compordering}
===================================================================================================
$(w,x,y,z)= (\inp(\Z),\inp(\Z),\inp(\Z),\inp(\Z))$
While $w>0$ and $x>0$ and $y>0$ and $z>0$
control = $\inp(1,2,3)$
if control == 1 then
$x=\inp(x+1,x+2,\ldots)$
$w=w-1$
else
if control == 2 then
$y=\inp(y+1,y+2,\ldots,)$
$x=x-1$
else
if control == 3 then
$z=\inp(z+1,z+2,\ldots)$
$y=y-1$
To prove that every computation of Program [4 ]{}is finite we need to find a quantity that, during every iteration of the [[**While** ]{}]{}Loop, decreases. None of $x,y,z$ qualify. No arithmetic combination of $w,x,y,z$ qualifies.
Let $P$ be an ordering and $k\ge 1$. The [*lexicographic ordering*]{} on $P^k$ is the ordering $$(a_1,\ldots,a_k) {<_{\rm lex}}(b_1,\ldots,b_k)$$ if for the least $i$ such that $a_i\ne b_i$, $a_i<b_i$.
In the ordering $({{\sf N}}^4,{<_{\rm lex}})$ $$(1,10,10000000000,99999999999999) {<_{\rm lex}}(1,11,0,0).$$
\[th:prog2orderings\] Every computation of Program [4 ]{}is finite.
Let
$$f({w},{x},{y},{z}) =
\begin{cases}
(0,0,0,0) \text{ if any of $w,x,y,z$ are $\le$ 0;}
\\
({w},{x},{y},{z}) \text{ otherwise.}
\cr
\end{cases}$$
We will be concerned with the order $({{\sf N}}^4,{<_{\rm lex}})$. We use the term [*decrease*]{} with respect to ${<_{\rm lex}}$.
We show that both premises of Theorem \[th:useorderings\] hold.
[**Claim 1:**]{} In every iteration of the [[**While** ]{}]{}loop $f({w},{x},{y},{z})$ decreases.
[**Proof of Claim 1:**]{}
Consider an iteration of the [[**While** ]{}]{}loop. There are three cases.
1. control=1: $w$ decreases by 1, $x$ increases by an unknown amount, $y$ stays the same, $z$ stays the same. Since the order is lexicographic, and $w$ is the first coordinate, the tuple decreases no matter how much $x$ increases.
2. control=2: $w$ stays the same, $x$ decreases by 1, $y$ increases by an unknown amount, $z$ stays the same. Since the order is lexicographic, $w$ is the first coordinate and stays the same, and $x$ is the second coordinate and decreases, the tuple decreases no matter how much $y$ increases.
3. control=3: $w$ stays the same, $x$ stays the same, $y$ decreases by 1, $z$ increases by an unknown amount. This case is similar to the two other cases.
[**End of Proof of Claim 1**]{}
[**Claim 2:**]{} If $f({w},{x},{y},{z})=(0,0,0,0)$ then the program has halted.
[**Proof of Claim 2:**]{}
If $f({w},{x},{y},{z})=(0,0,0,0)$ then one of $w,x,y,z$ is $\le0$. Hence the [[**While** ]{}]{}loop condition is not satisfied and the program halts.
[**End of Proof of Claim 2**]{}
By Claim 1 and 2 both premises of Theorem \[th:useorderings\] are satisfied. Hence Program [4 ]{}terminates.
A Proof Using Ramsey’s Theorem {#se:useramsey}
==============================
In the proof of Theorem \[th:prog2orderings\] we showed that during every single step of Program [4 ]{}the quantity $(w,x,y,z)$ decreased with respect to the ordering ${<_{\rm lex}}$. The proof of termination was easy in that we only had to deal with one step but hard in that we had to deal with the lexicographic order on ${{\sf N}}\times{{\sf N}}\times{{\sf N}}\times{{\sf N}}$ rather than just the ordering ${{\sf N}}$.
In this section we will prove that Program [4 ]{}terminates in a different way. We will not need an ordering on 4-tuples. We will only deal with $w,x,y,z$ individually. However, we will need to prove that, for [*any*]{} [computational segment]{}, at least one of $w,x,y,z$ decreases.
We will use the infinite Ramsey’s Theorem [@Ramsey] (see also [@ramseynotes; @GRS; @RamseyInts]) which we state here.
1. If $n\ge 1$ then $K_n$ is the complete graph with vertex set $V=\{1,\ldots,n\}$.
2. $K_{{\sf N}}$ is the complete graph with vertex set ${{\sf N}}$.
\[de:homog\] Let $c,n\ge 1$. Let $G$ be $K_n$ or $K_{{\sf N}}$. Let $COL$ be a $c$-coloring of the edges of $G$. A set of vertices $V$ is [*homogeneous with respect to $COL$*]{} if all the edges between vertices in $V$ are the same color. We will drop the [*with respect to $COL$*]{} if the coloring is understood.
[**Infinite Ramsey’s Theorem:**]{}
\[th:ramsey\] Let $c\ge 1$. For every $c$-coloring of the the edges of $K_{{\sf N}}$ there exists an infinite homogeneous set.
\[th:prog2ramsey\] Every computation of Program [4 ]{}is finite.
We first show that for every finite [computational segment ]{}one of $w,x,y$ will decrease. There are several cases.
1. If control=1 ever occurs in the segment then $w$ will decrease. No other case makes $w$ increase, so we are done. In all later cases we can assume that control is never 1 in the segment.
2. If control=2 ever occurs in the segment then $x$ decreases. Since control=1 never occurs and control=3 does not make $x$ increase, $x$ decreases. In all later cases we can assume that control is never 1 or 2 in the segment.
3. If control=3 is the only case that occurs in the segment then $y$ decreases.
We show Program [4 ]{}terminates. Assume, by way of contradiction, that there is an infinite computation. Let this computation be
$$({w}_1,{x}_1,{y}_1,{z}_1), ({w}_2,{x}_2,{y}_2,{z}_2), \ldots.$$
Since in every [computational segment ]{}one of $w,x,y$ decrease we have that, for all $i<j$, either ${w}_i>{w}_j$ or ${x}_i>{x}_j$ or ${y}_i>{y}_j$. We use this to create a coloring of the edges of $K_{{\sf N}}$. Our colors are $W,X,Y$. In the coloring below each case assumes that the cases above it did not occur.
$$COL(i,j) =
\begin{cases}
W \text{ if ${w}_i>{w}_j$;}
\\
X \text{ if ${x}_i>{x}_j$;}
\\
Y \text{ if ${y}_i>{y}_j$.}
\cr
\end{cases}$$
By Ramsey’s Theorem there is an infinite set
$$i_1 < i_2 < i_3 < \cdots$$
such that
$$COL(i_1,i_2) = COL(i_2,i_3) = \cdots.$$
(We actually know more. We know that [*all*]{} pairs have the same color. We do not need this fact here; however, see the note after Theorem \[th:useramseygen\].)
Assume the color is $W$ (the cases for $X,Y$ are similar). Then
$${w}_{i_1} > {w}_{i_2} > {w}_{i_3} > \cdots.$$
Hence eventually ${w}$ must be less than 0. When this happens the program terminates. This contradicts the program not terminating.
The keys to the proof of Theorem \[th:prog2ramsey\] are (1) in every [computational segment ]{}one of $w,x,y$ decreases, and (2) by Ramsey’s Theorem any nonterminating computation leads to an infinite decreasing sequence in a well founded set. These ideas are from Theorem 1 of [@ramseypl], though similar ideas were in [@LJA]. The next theorem, which is a subcases of Theorem 1 of [@ramseypl], captures this proof technique.
\[th:useramsey\] Let $PROG=(S,I,R)$ be a program of the form of Program [1]{}. Note that the variables are $x[1],\ldots,x[n]$. If for all [computational segment ]{}$s_1,\ldots,s_n$ there exists $i$ such that $x[i]$ in $s_1$ is strictly less than $x[i]$ in $s_n$ then any computation of $PROG$ is finite.
To prove that a program terminates we might use some function of the variables rather than the variables themselves. The next theorem, which is a generalization of Theorem \[th:useramsey\], captures this.
\[th:useramseygen\] Let $PROG=(S,I,R)$ be a program. Assume that there exists well founded orderings ,$\ldots,$ $(P_m,<_m)$ and functions $f_1,\ldots,f_m$ such that $f_i:S{\rightarrow}P_i$. Assume the following.
1. For all [computational segment ]{}$s_i,\ldots,s_j$ there exists $a$ such that $f_a(s_i) >_a f_a(s_j)$.
2. If the program is in a state $s$ such that, for some $k$, $f_k(s)$ is a minimal element of $P_k$, then the program terminates.
Then any computation of $PROG$ is finite.
Assume, by way of contradiction, that $PROG$ does not terminate. Then there is an infinite computation. Let this computation be
$$(s_1, s_2, s_3, \ldots)$$
For all [computational segment ]{}$s_i,\ldots,s_j$ there exists $a$ such that $f_a(s_i) >_a f_a(s_j)$. We use this to create a coloring of the edges of $K_{{\sf N}}$. Our colors are $1,\ldots, m$.
$$COL(i,j) = \hbox{ the least $a$ such that $f_a(s_i) >_a f_a(s_j)$ }.$$
The rest of the proof is similar to the proof of Theorem \[th:prog2ramsey\].
The proofs of Theorems \[th:prog2ramsey\], \[th:useramsey\] and \[th:useramseygen\] do not need the full strength of Ramsey’s Theorem. Consider Theorem \[th:useramseygen\]. For any $i,j,k$ if $COL(i,j)=a$ (so $a$ is the least number such that $f_a(s_i)>_a f_a(s_j)$) and $COL(j,k)=a$ (so $a$ is the least number such that $f_a(s_j)>_a f_a(s_k)$) then one can show $COL(i,k)=a$. Such colorings are called [*transitive*]{}. Hence we only need Ramsey’s Theorem for transitive colorings. We discuss this further in Section \[se:need\].
A Proof Using Matrices and Ramsey’s Theorem {#se:matrix}
===========================================
Part of the proof of Theorem \[th:prog2ramsey\] involved showing that, for any finite [computational segment ]{}of Program [4]{}, one of $w,x,y,z$ decreases. Can such proofs be automated? Lee, Jones, and Ben-Amram [@LJA] developed a way to partially automate such proofs. They use matrices and Ramsey’s Theorem.
We use their techniques to give a proof that Program [4 ]{}terminates. We will then discuss their general technique. Cook, Podelski, Rybalchenko have also developed a way to partially automate such proofs. We discuss this in Section \[se:useramsey2\].
Let $P$ be a program with variables $x[1],\ldots,x[n]$ and control takes values in $\{1,\ldots,{{\rm m}}\}$. Let $1\le {k}\le {{\rm m}}$. Let $x[1],\ldots,x[n]$ be the values of the variables before the control=$k$ code executes and let $x[1]',\ldots,x[n]'$ be the values of the variables after. In some cases we know how $x[i]$ relates to $x[j]'$. We express this information in a matrix. The key will be that matrix multiplication (defined using $(+,\min)$ rather than $(\times,+)$) of two matrices representing pieces of code will result in a matrix that represents what happens if those pieces of code are executed one after the other.
- If it is always the case that $x[i]'\le x[j]+L$ then the $(i,j)$ entry of the matrix is $L\in {{\sf Z}}$.
- In all other cases the $(i,j)$ entry is ${\infty}$. Note that this may well be most of the cases since we often do not know how $x[i]'$ and $x[j]$ relate.
\[ex:prog2\]
We describe the matrices for Program [4]{}.
The matrix for control=1 is
$$C_1 = \left ( \begin{array}{cccc}
-1 & {\infty}& {\infty}& {\infty}\cr
{\infty}& {\infty}& {\infty}& {\infty}\cr
{\infty}& {\infty}& 0 & {\infty}\cr
{\infty}& {\infty}& {\infty}& 0 \cr
\end{array} \right )$$
The matrix for control=2 is
$$C_2= \left ( \begin{array}{cccc}
0 & {\infty}& {\infty}& {\infty}\cr
{\infty}& -1 & {\infty}& {\infty}\cr
{\infty}& {\infty}& {\infty}& {\infty}\cr
{\infty}& {\infty}& {\infty}& 0 \cr
\end{array} \right )$$
The matrix for control=3 is
$$C_3= \left ( \begin{array}{cccc}
0 & {\infty}& {\infty}& {\infty}\cr
{\infty}& 0 & {\infty}& {\infty}\cr
{\infty}& {\infty}& -1 & {\infty}\cr
{\infty}& {\infty}& {\infty}& {\infty}\cr
\end{array} \right )$$
We want to define matrix multiplication such that if $C_1$ is the matrix for control=$1$ and $C_2$ is the matrix for control=$2$ then $C_1C_2$ is the matrix for what happens if first the control=$1$ code is executed and then the control=$2$ code is executed.
Lets call the variables $x[1],\ldots,x[n]$.
1. If $C_1[i,k]=L_1$ then $x[i]\le x[k]'+L_1$. If $C_2[k,j]=L_2$ then $x[k]'\le x[j]''+L_2$. Hence we know that $x[i]\le x[j]''+(L_1+L_2)$. Therefore we want $C_1C_2[i,j]\le L_1+L_2$ Hence $$(\forall k)[C_1C_2[i,j] \le C_1[i,k] + C_2[k,j]].$$ If we define ${\infty}+L={\infty}$ and ${\infty}+{\infty}={\infty}$. then this inequality is true even if if $L_1$ or $L_2$ is infinity
2. Using the above we define $$C_1C_2[i,j]= \min_k\{ C_1[i,k]+C_2[k,j]\}.$$
The following theorem, from [@LJA], we leave for the reader.
\[le:matrix\] Let $PROG_1$ and $PROG_2$ be programs that use the variables $x[1],\ldots,x[n]$. (We think of $PROG_1$ and $PROG_2$ as being what happens in the various control cases.) Let $C_1$ be the matrix that represent what is known whenever $PROG_1$ is executed. Let $C_2$ be the matrix that represent what is known whenever $PROG_2$ is executed. Then the matrix produce $C_1C_2$ as defined above represents what is known when $PROG_1$ and then $PROG_2$ are executed.
\[th:prog2matrix\] Every computation of Program [4 ]{}is finite.
Let $C_1,C_2,C_3$ be the matrices that represent the cases of Control=1,2,3 in Program [4 ]{}. (These matrices are in Example \[ex:prog2\]). We show that the premises of Theorem \[th:useramsey\] hold. To do this we prove items 0-7 below. Item 0 is easily proven directly. Items 1,2,3,4,5,6,7 are easily proven by induction on the number of matrices being multiplied.
1. $C_1C_2=C_2C_1$, $C_1C_3=C_3C_1$, $C_2C_3=C_3C_2$.
2. For all $a\ge 1$ $$C_1^a = \left ( \begin{array}{cccc}
-a & {\infty}& {\infty}& {\infty}\cr
{\infty}& {\infty}& {\infty}& {\infty}\cr
{\infty}& {\infty}& 0 & {\infty}\cr
{\infty}& {\infty}& {\infty}& 0 \cr
\end{array} \right )$$
3. For all $b\ge 1$ $$C_2^b = \left ( \begin{array}{cccc}
0 & {\infty}& {\infty}& {\infty}\cr
{\infty}& -b & {\infty}& {\infty}\cr
{\infty}& {\infty}& {\infty}& {\infty}\cr
{\infty}& {\infty}& {\infty}& 0 \cr
\end{array} \right )$$
4. For all $c\ge 1$ $$C_3^c = \left ( \begin{array}{cccc}
0 & {\infty}& {\infty}& {\infty}\cr
{\infty}& 0 & {\infty}& {\infty}\cr
{\infty}& {\infty}& -c & {\infty}\cr
{\infty}& {\infty}& {\infty}& {\infty}\cr
\end{array} \right )$$
5. For all $a,b\ge 1$ $$C_1^aC_2^b = \left ( \begin{array}{cccc}
-a & {\infty}& {\infty}& {\infty}\cr
{\infty}& -b & {\infty}& {\infty}\cr
{\infty}& {\infty}& 0 & {\infty}\cr
{\infty}& {\infty}& {\infty}& 0 \cr
\end{array} \right )$$
6. For all $a,c\ge 1$ $$C_1^aC_3^c = \left ( \begin{array}{cccc}
-a & {\infty}& {\infty}& {\infty}\cr
{\infty}& 0 & {\infty}& {\infty}\cr
{\infty}& {\infty}& -c & {\infty}\cr
{\infty}& {\infty}& {\infty}& {\infty}\cr
\end{array} \right )$$
7. For all $b,c\ge 1$ $$C_2^bC_3^c = \left ( \begin{array}{cccc}
0 & {\infty}& {\infty}& {\infty}\cr
{\infty}& -b & {\infty}& {\infty}\cr
{\infty}& {\infty}& {\infty}& {\infty}\cr
{\infty}& {\infty}& {\infty}& {\infty}\cr
\end{array} \right )$$
8. For $a,b,c\ge 1$ $$C_1^aC_2^bC_3^c = \left ( \begin{array}{cccc}
-a & {\infty}& {\infty}& {\infty}\cr
{\infty}& {\infty}& {\infty}& {\infty}\cr
{\infty}& {\infty}& {\infty}& {\infty}\cr
{\infty}& {\infty}& {\infty}& 0 \cr
\end{array} \right )$$
We are interested in [*any*]{} sequence of executions of control=1, control=2, and control=3. Hence we are interested in [*any*]{} product of the matrices $C_1,C_2,C_3$. Since the multiplication of these matrices is commutative we need only concern ourselves with $C_1^aC_2^bC_3^c$ for $a,b,c \in {{\sf N}}$. In all of the cases below $a,b,c\ge 1$.
1. $C_1^a$: $w$ decreases.
2. $C_2^b$: $x$ decreases.
3. $C_3^c$: $y$ decreases.
4. $C_1^aC_2^b$: Both $w$ and $x$ decrease.
5. $C_1^aC_3^c$: Both $w$ and $y$ decrease.
6. $C_2^bC_3^c$: $x$ decreases.
7. $C_1^aC_2^bC_3^c$: $w$ decreases.
Hence, for any [computational segment ]{}$s_1,\ldots,s_n$ of Program [4 ]{}either $w,x,y$, or $z$ decreases. Hence by Theorem \[th:useramsey\] Program [4 ]{}terminates.
The keys to the proof of Theorem \[th:prog2matrix\] are (1) represent how the old and new variables relate after one iteration with a matrix, (2) use these matrices and a type of matrix multiplication to determine that for every [computational segment ]{}some variable decreases, (3) use Theorem \[th:useramsey\] to conclude the program terminates.
The proof technique we used above is quite general. Lee, Jones, and Ben-Amram [@LJA](Theorem 4) have noted the following folk theorem which captures it:
\[th:matrix\] Let $PROG=(S,I,R)$ be a program in the form of Program [1]{}. Let $C_1, C_2,\ldots, C_m$ be the matrices associated to control=1, $\ldots$, control=m cases. If every product of the $C_i$’s yields a matrix with a negative integer on the diagonal then the program terminates.
Consider [computational segment ]{}$s_1,\ldots,s_n$. Let the corresponding matrices be $C_{i_1},\ldots,C_{i_n}$. By the premise the product of these matrices has a negative integer on the diagonal. Hence some variable decreases. By Theorem \[th:useramsey\] the program terminates.
Theorem \[th:matrix\] leads to the following algorithm to test if a programs terminates. There is one step (alas, the important one) which we do not say how to do. If done in the obvious way it may not halt.
1. Input Program P.
2. Form matrices for all the cases of control. Let them be $C_1,\ldots,C_m$.
3. Find a finite set of types of matrices ${\cal M}$ such that that any product of the $C_i$’s (allowing repeats) is in ${\cal M}$. (If this step is implemented by looking at all possible products until a pattern emerges then this step might not terminate.)
4. If all of the elements of ${\cal M}$ have some negative diagonal element then output [*YES the program terminates!*]{}
5. If not the then output [*I DO NOT KNOW if the program terminates!*]{}
If all products of matrices fit a certain pattern, as they did in the proof of Theorem \[th:prog2matrix\], then this idea for an algorithm will terminate. Even in that case, it may output [*I DON"T KNOW if the program terminates!*]{}. However, this algorithm can be used to prove that some programs terminate, just not all. It cannot be used to prove that a program will not terminate.
Theorem \[th:matrix\] only dealt with how the variables changed. We will need a more general theorem where we look at how certain functions of the variables change. Note also the next three theorems are if-and-only-if statements.
\[th:matrixgen\] Let $PROG=(S,I,R)$ be a program. The following are equivalent:
1. There exists functions $f_1,\ldots,f_j$ such that the following occur.
1. The associated matrices are $A_1, A_2,\ldots, A_m$ describe completely how $f_i$ on the variables before the code is executed compares to $f_j$ on the variables after the code is executed, in the control=1, $\ldots$, control=m cases.
2. When one of the $f_i$ is $\le 0$ then the [[**While** ]{}]{}loop condition does not hold so the program stops.
3. Every product of the $A_i$’s yields a matrix with a negative integer on the diagonal.
2. Every computation of $PROG$ is finite.
Theorem \[th:matrixgen\] also leads to an algorithm to test if programs of the type we have been considering halt. This algorithm is similar to the one that follows Theorem \[th:matrix\] and hence we omit it. Similar to that algorithm, this one does not always terminate.
The following extensions of Theorem \[th:matrixgen\] are known. The first one is due to Ben-Amram [@BA:delta].
\[th:matrixgena\] Let $PROG=(S,I,R)$ be a program. The following are equivalent:
1. There exists functions $f_1,\ldots,f_j$ such that the following occur.
1. Items 1.a and 1.b of Theorem \[th:matrixgen\] hold.
2. For all $i$, every column of $A_i$ has at most one non-infinity value.
3. For every product of the $A_i$’s there is a power of it that has a negative integer on the diagonal.
Then every computation of $PROG$ is finite.
The condition on the columns turns out to not be necessary. Jean-Yves Moyen [@yvestocl] has shown the following.
\[th:matrixw\] Let $PROG=(S,I,R)$ be a program. The following are equivalent:
1. There exists $f_1,\ldots,f_j$, functions such that premises 1.a, 1.b, 1.d of Theorem \[th:matrixgen\] hold.
2. Every computation of $PROG$ is finite.
Is there an example of a program where the matrices have a product that has no negative on the diagonal, yet by Theorem \[th:matrixw\] terminates? Yes! Ben-Amram has provided us with an example and has allowed us to place it in the appendix of this paper.
A Proof Using Transition Invariants and Ramsey’s Theorem {#se:useramsey2}
========================================================
We proved that Program [4 ]{}terminates in three different ways. The proof in Theorem \[th:prog2orderings\] used that $({{\sf N}}\times{{\sf N}}\times{{\sf N}}\times{{\sf N}},{<_{\rm lex}})$ is a well founded order; however, the proof only had to deal with what happened during [*one*]{} step of Program [4]{}. The proofs in Theorem \[th:prog2ramsey\] and \[th:prog2matrix\] used the ordering $({{\sf N}},\le)$ and Ramsey’s Theorem; however, the proofs had to deal with [*any*]{} [computational segment ]{}of Program [4]{}. Which proof is easier? This is a matter of taste; however, all of the proofs are easy once you see them.
We present an example from [@ramseypl] of a program (Program [5 ]{}below) where the proof of termination using Ramsey’s Theorem is easy. Podelski and Rybalchenko found this proof by hand and later their termination checker found it automatically. A proof of termination using a well founded ordering seems difficult to find. Ben-Amram and Lee [@BA:mcs; @Lee:ranking] have shown that a termination proof that explicitly exhibits a well-founded order can be automatically derived when the matrices only use entries $0,-1$, and $\infty$. Alas, Program [5 ]{}is not of this type; however, using some manipulation Ben-Amram (unpublished) has used this result to show that Program [5 ]{}terminates. (The proof is in the Appendix.) Hence there is a proof that Program [5 ]{}terminates that uses a well-ordering; however, it was difficult to obtain.
$(x,y) = (\inp(\Z),\inp(\Z))$
While $x>0$ and $y>0$
control = $\inp(1,2)$
if control == 1 then
$(x,y)=(x-1,x)$
else
if control == 2 then
$(x,y)=(y-2,x+1)$
\[th:prog3ramsey\] Every computation of Program [5 ]{}is finite.
We assume that the [computational segment ]{}enters the [[**While** ]{}]{}loop, else the program has already terminated.
We could try to show that, in every [computational segment]{}, either $x$ or $y$ decreases. This statement is true but seems hard to prove directly. Instead we show that either $x$ or $y$ or $x+y$ decreases. This turns out to be much easier. Intuitively we are loading our induction hypothesis. We now proceed formally.
We show that both premises of Theorem \[th:useramseygen\] hold with $P_1=P_2=P_3={{\sf N}}$, $f_1({x},{y})={x}$, $f_2({x},{y})={y}$, and $f_3({x},{y})={x}+{y}$. It may seem as if knowing that $x+y$ decreases you know that either $x$ or $y$ decreases. However, in our proof, we will [*not*]{} know which of $x,y$ decreases. Hence we must use $x,y$, and $x+y$.
[**Claim 1:**]{} For any [computational segment]{}, one of $x,y,x+y$ decreases.
[**Proof of Claim 1:**]{}
We want to prove that, for all $n\ge 2$, for all [computational segment ]{}of length $n$ $$({x}_1,{y}_1), ({x}_2,{y}_2),\ldots,({x}_n,{y}_n),$$ either ${x}_1>{x}_n$ or ${y}_1>{y}_n$ or ${x}_1+{y}_1 > {x}_n + {y}_n$. However, we will prove something stronger. We will prove that, for all $n\ge 2$, for all [computational segment ]{}of length $n$ $$({x}_1,{y}_1), ({x}_2,{y}_2),\ldots,({x}_n,{y}_n),$$ one of the following occurs.
1. ${x}_1>0$ and ${y}_1>0$ and ${x}_n<{x}_1$ and ${y}_n \le {x}_1$ (so $x$ decreases),
2. ${x}_1>0$ and ${y}_1>0$ and ${x}_n<{y}_1-1$ and ${y}_n\le {x}_1+1$ (so $x+y$ decreases),
3. ${x}_1>0$ and ${y}_1>0$ and ${x}_n<{y}_1-1$ and ${y}_n < {y}_1$ (so $y$ decreases),
4. ${x}_1>0$ and ${y}_1>0$ and ${x}_n<{x}_1$ and ${y}_n < {y}_1$ (so $x$ and $y$ both decreases, though we just need one of them).
(In the note after the proof we refer to the OR of these four statements as [*the invariant*]{}.)
We prove this by induction on $n$.
[**Base Case:**]{} $n=2$ so we only look at one instruction.
If $({x}_{2},{y}_{2})=({x}_1-1,{x}_1)$ is executed then (1) holds.
If $({x}_{2},{y}_{2})=({y}_1-2,{x}_1+1)$ is executed then (2) holds.
[**Induction Step**]{}: We prove Claim 1 for $n+1$ assuming it for $n$. There are four cases, each with two subcases.
1. ${x}_n<{x}_1$ and ${y}_n \le {x}_1$.
1. If $({x}_{n+1},{y}_{n+1})=({x}_n-1,{x}_n)$ is executed then
- ${x}_{n+1} = {x}_n-1 < {x}_1 -1 < {x}_1$
- ${y}_{n+1} = {x}_n < {x}_1$
Hence (1) holds.
2. If $({x}_{n+1},{y}_{n+1})=({y}_n-2,{x}_n+1)$ is executed then
- ${x}_{n+1}={y}_n-2 \le {x}_1-2 < x_1$
- ${y}_{n+1} = {x}_n + 1 \le {x}_1$
Hence (1) holds.
2. ${x}_n<{y}_1-1$ and ${y}_n\le {x}_1+1$
1. If $({x}_{n+1},{y}_{n+1})=({x}_n-1,{x}_n)$ is executed then
- ${x}_{n+1} = {x}_n-1 < {y}_1-2<{y}_1-1$
- ${y}_{n+1} = {x}_n < {y}_1-1<{y}_1$
Hence (3) holds.
2. If $({x}_{n+1},{y}_{n+1})=({y}_n-2,{x}_n+1)$ is executed then
- ${x}_{n+1}={y}_n-2 \le {x}_1-1< {x}_1$
- ${y}_{n+1} = {x}_n < {y}_1$
Hence (4) holds.
3. ${x}_n<{y}_1-1$ and ${y}_n < {y}_1$
1. If $({x}_{n+1},{y}_{n+1})=({x}_n-1,{x}_n)$ is executed then
- ${x}_{n+1} = {x}_n-1 <{y}_1-2 <{y}_1-1$
- ${y}_{n+1} = {x}_n <{y}_1-1<{y}_1$.
Hence (3) holds.
2. If $({x}_{n+1},{y}_{n+1})=({y}_n-2,{x}_n+1)$ is executed then
- ${x}_{n+1}={y}_n-2 <{y}_1-2 <{y}_1-1$
- ${y}_{n+1} = {x}_n <y_1-1 <{y}_1$
Hence (3) holds.
4. ${x}_n<{x}_1$ and ${y}_n < {y}_1$
1. If $({x}_{n+1},{y}_{n+1})=({x}_n-1,{x}_n)$ is executed then
- ${x}_{n+1} = {x}_n-1 <{x}_1 -1<{x}_1$
- ${y}_{n+1} = {x}_n <{x}_1$
Hence (1) holds.
2. If $({x}_{n+1},{y}_{n+1})=({y}_n-2,{x}_n+1)$ is executed then
- ${x}_{n+1}={y}_n-2 <{y}_1-2 < {y}_1-1$.
- ${y}_{n+1} = {x}_n <{x}_1 <{x}_1+1$.
Hence (2) holds.
We now have that, for any [computational segment ]{}either $x,y$, or $x+y$ decreases.
[**End of Proof of Claim 1**]{}
The following claim is obvious.
[**Claim 2:**]{} If any of $x,y$, $x+y$ is 0 then the program terminates.
By Claims 1 and 2 the premise of Theorem \[th:useramseygen\] is satisfied. Hence Program [5 ]{}terminates.
We can state the invariant differently. Consider the following four orderings on ${{\sf N}}\times{{\sf N}}$ and their sum.
- $T_1$ is the ordering $({x}',{y}')<_1 ({x},{y})$ iff ${x}>0$ and $y>0$ and ${x}'<{x}$ and ${y}' \le {x}$.
- $T_2$ is the ordering $({x}',{y}')<_2({x},{y})$ iff ${x}>0$ and ${y}>0$ and $x'<{y}-1$ and ${y}'\le {x}+1$.
- $T_3$ is the ordering $({x}',{y}')<_3({x},{y})$ iff ${x}>0$ and ${y}>0$ and ${x}'<{y}-1$ and ${y}' <{y}$.
- $T_4$ is the ordering $({x}',{y}')<_4({x},{y})$ iff ${x}>0$ and ${y}>0$ and ${x}'<{x}$ and ${y}' < {y}$.
- $T = T_1 \cup T_2 \cup T_3 \cup T_4.$ We denote this order by $<_T$.
Note that (1) each $T_i$ is well founded, and (2) for any [computational segment ]{}
$$({x}_1,{y}_1),({x}_2,{y}_2),\ldots,({x}_n,{y}_n)$$
we have $({x}_1,{y}_1)<_T({x}_n,{y}_n)$
It is easy to see that these properties of $T$ are all we needed in the proof. This is Theorem 1 of [@ramseypl] which we state and prove.
\[de:ti\] Let $PROG=(S,I,R)$ be a program.
1. An ordering $T$, which we also denote $<_T$, on $S\times S$ is [*transition invariant*]{} if for any [computational segment ]{}$s_1,\ldots,s_n$ we have $s_n<_T s_1$.
2. An ordering $T$ is [*disjunctive well-founded*]{} if there exists well founded orderings $T_1,\ldots,T_k$ such that $T=T_1\cup\cdots\cup T_k$. Note that the $T_i$ need not be linear orderings, they need only be well founded. This will come up in the proof of Theorem \[th:prog4\].
[@ramseypl]\[th:usetrans\] Let $PROG=(S,I,R)$ be a program. Every run of $PROG$ terminates iff there exists a disjunctive well-founded transition invariant.
We prove that if there is a disjunctive well-founded transition invariant then every run terminates. The other direction we leave to the reader.
Let $T=T_1\cup\cdots\cup T_k$ be the disjunctive well-founded transition invariant for $PROG$. Let $<_c$ be the ordering for $T_c$.
Assume, by way of contradiction, that there is an infinite sequence $s_1,s_2,s_3,\ldots,$ such that each $(s_i,s_{i+1})\in R$. Define a coloring $COL$ by, for $i<j$,
$COL(i,j) = $ the least $L$ such that $s_j<_L s_i.$
By Ramsey’s Theorem there is an infinite set
$$i_1 < i_2 < i_3 < \cdots$$
such that
$$COL(i_1,i_2) = COL(i_2,i_3) = \cdots.$$
Let that color be $L$. For notational readability we denote $<_L$ by $<$ and $>_L$ by $>$. We have
$$s_{i_1} > s_{i_2} > \cdots >$$
This contradicts $<$ being well founded.
The proof of Theorem \[th:usetrans\] seems to need the full strength of Ramsey’s Theorem (unlike the proof of Theorem \[th:useramseygen\], see the note following its proof). We give an example, due to Ben-Amram, of a program with a disjunctive well-founded transition invariant where the coloring is not transitive. Consider Program not-transitive
$x = \inp(\Z)$
While $x>0$
$x = x \div 2$
It clearly terminates and you can use the transition invariant $\{({x},{x}') {\mathrel{:}}{x}> {x}' \}$ to prove it. This leads to a transitive coloring. But what if instead your transition-invariant-generator came up with the following rather odd relations instead:
1. $T_1 =\{ ({x},{x}') {\mathrel{:}}{x}> 3{x}'\}$
2. $T_2 =\{ ({x},{x}') {\mathrel{:}}{x}> {x}'+1\}$
Note that $T_1 \cup T_2$ is a disjunctive well-founded transition invariant. We show that the coloring associated to $T_1\cup T_2$ is not transitive.
- $COL(4,2)=2$. That is, $(4,2)\in T_2-T_1$.
- $COL(2,1)=2$. That is, $(2,1)\in T_2-T_1$.
- $COL((4,1)=1$. That is $(4,1)\in T_1$.
Hence $COL$ is not a transitive coloring.
If in the premise of Theorem \[th:usetrans\] all of the $T_i$’s are linear (that is, every pair of elements is comparable) then the transitive Ramsey Theorem suffices for the proof.
Finding an appropriate $T$ is the key to the proofs of termination for the termination checkers Loopfrog [@loopfrog], and Terminator [@terminator].
Another Proof Using Matrices and Ramsey’s Theorem {#se:morematrix}
=================================================
We prove Program [5 ]{}terminates using matrices. The case control=1 is represented by the matrix $$C_1 = { \begin{pmatrix} -1 & 0 \\ \infty & \infty \end{pmatrix}}.$$ The case control=2 is represented by the matrix $$C_2 = { \begin{pmatrix} \infty & -2 \\ 1 & \infty \end{pmatrix}}.$$ This will not work! Note that $C_2$ is has no negative numbers on its diagonal. Hence we cannot use these matrices in our proof! What will we do!? Instead of using $x,y$ we will use $x,y$, and $x+y$. We comment on whether or not you can somehow use $C_1$ and $C_2$ after the proof.
\[th:prog3matrix\] Every computation of Program [5 ]{}is finite.
We will use Theorem \[th:matrixgen\] with functions $x,y$, and $x+y$. Note that $x+y$ is not one of the original variables which is why we need Theorem \[th:matrixgen\] rather than Theorem \[th:matrix\].
The control=1 case of Program [5 ]{}corresponds to
$$D_1 = \left ( \begin{array}{ccc}
-1 & 0 & 1 \cr
{\infty}& {\infty}& {\infty}\cr
{\infty}& {\infty}& {\infty}\cr
\end{array} \right )$$
The control=2 case of Program [5 ]{}corresponds to
$$D_2 = \left ( \begin{array}{ccc}
{\infty}& 1 & {\infty}\cr
-2 & {\infty}& {\infty}\cr
{\infty}& {\infty}& -1 \cr
\end{array} \right )$$
We show that the premises of Theorem \[th:matrixgen\] hold. The following are true and easily proven by induction on the number of matrices being multiplied.
1. For all $a\ge 1$
$$D_1^a = \left ( \begin{array}{ccc}
-a & -a+1 & -a+2 \cr
{\infty}& {\infty}& {\infty}\cr
{\infty}& {\infty}& {\infty}\cr
\end{array} \right )$$
2. For all $b\ge 1$, $b$ odd, $b=2d-1$,
$$D_2^b = \left ( \begin{array}{ccc}
-d & {\infty}& {\infty}\cr
{\infty}& -d & {\infty}\cr
{\infty}& {\infty}& -2d \cr
\end{array} \right )$$
3. For all $b\ge 2$, $b$ even, $b=2e$,
$$D_2^b = \left ( \begin{array}{ccc}
{\infty}& -e+1 & {\infty}\cr
-e-2 & {\infty}& {\infty}\cr
{\infty}& {\infty}& -2e-1 \cr
\end{array} \right )$$
4. For all $a,b\ge 1$, $b$ odd, $b=2d-1$.
$$D_1^aD_2^b = \left ( \begin{array}{ccc}
-a-d & -a-d+1 & -a-2d+2 \cr
{\infty}& {\infty}& {\infty}\cr
{\infty}& {\infty}& {\infty}\cr
\end{array} \right )$$
5. For all $a,b\ge 1$, $b$ even, $b=2e$.
$$D_1^aD_2^b = \left ( \begin{array}{ccc}
-a-e-1 & -a-e+1 & -a-2e+1 \cr
{\infty}& {\infty}& {\infty}\cr
{\infty}& {\infty}& {\infty}\cr
\end{array} \right )$$
6. For all $a,b\ge 1$, $a$ is odd,
$$D_2^aD_1^b = \left ( \begin{array}{ccc}
{\infty}& {\infty}& {\infty}\cr
-({\left\lfloor{a/2}\right\rfloor}+b+2 & -({\left\lfloor{a/2}\right\rfloor}+b+1 & -({\left\lfloor{a/2}\right\rfloor}+b \cr
{\infty}& {\infty}& {\infty}\cr
\end{array} \right )$$
7. If $a,b\ge 1$, $a$ is even,
$$D_2^aD_1^b = \left ( \begin{array}{ccc}
-(a/2)+b & -(a/2)+b-1 & -{\left\lfloor{a/2}\right\rfloor}+b-2 \cr
{\infty}& {\infty}& {\infty}\cr
{\infty}& {\infty}& {\infty}\cr
\end{array} \right )$$
We use this information to formulate a lemma.
[**Convention:**]{} If we put $< 0$ $(\le 0$) in an entry of a matrix it means that the entry is some integer less than 0 (less than or equal to 0). We might not know what it is.
[**Claim:**]{} For all $n\ge 2$, any product of $n$ matrices all of which are $D_1$’s and $D_2$’s must be of one of the following type:
1. $$\left ( \begin{array}{ccc}
<0 & \le 0 &\le 0 \cr
{\infty}& {\infty}& {\infty}\cr
{\infty}& {\infty}& {\infty}\cr
\end{array} \right )$$
2. $$\left ( \begin{array}{ccc}
{\infty}& {\infty}& {\infty}\cr
< 0 & < 0 & < 0 \cr
{\infty}& {\infty}& {\infty}\cr
\end{array} \right )$$
3. $$\left ( \begin{array}{ccc}
<0 & {\infty}& {\infty}\cr
{\infty}& < 0 & {\infty}\cr
{\infty}& {\infty}& < 0 \cr
\end{array} \right )$$
4. $$\left ( \begin{array}{ccc}
{\infty}& < 0 & {\infty}\cr
< 0 & {\infty}& {\infty}\cr
{\infty}& {\infty}& < 0 \cr
\end{array} \right )$$
[**End of Claim**]{}
This can be proved easily by induction on $n$.
One can show that every computation of Program [5 ]{}terminates using the original matrices $2\times 2$ matrices $C_1, C_2$. This requires a more advanced theorem (Theorem \[th:matrixw\] above). Ben-Amram has done this and has allowed us to place his proof in the appendix of this paper.
Another Proof using Transition Invariants and Ramsey’s Theorem {#se:sub}
==============================================================
Showing Program [6 ]{}terminates seems easy: eventually $y$ is negative and after that point $x$ will steadily decrease until ${x}<0$. But this proof might be hard for a termination checker to find since $x$ might increases for a very long time. Instead we need to find the right disjunctive well-founded transition invariant.
$(x,y) = (\inp(\Z),\inp(\Z))$
While $x>0$
$(x,y) = (x+y,y-1)$
\[th:prog4\] Every run of Program [6 ]{}terminates.
We define orderings $T_1$ and $T_2$ which we also denote $<_1$ and $<_2$.
- $({x}',{y}')<_1({x},{y}))$ iff $0<x$ and $x' < x$.
- $({x}',{y}')<_2({x},{y}))$ iff $0\le y$ and $y' < y'$.
Let
$$T=T_1\cup T_2.$$
Clearly $T_1$ and $T_2$ are well-founded (though see note after the proof). Hence $T$ is disjunctive well-founded. We show that $T$ is a transition invariant.
We want to prove that, for all $n\ge 2$, for all [computational segment ]{}of length $n$ $$({x}_1,{y}_1), ({x}_2,{y}_2),\ldots,({x}_n,{y}_n)$$ either $T_1$ or $T_2$ holds.
We prove this by induction on $n$.
We will assume that the [computational segment ]{}enters the [[**While** ]{}]{}loop, else the program has already terminated. In particular, in the base case, $x>0$.
[**Base Case:**]{} $n=2$ so we only look at one instruction. There are two cases.
If ${y}_1\ge 0$ then $y_2=y_1-1<y_1$. Hence $(x_2,y_2)<_2 (x_1,y_1)$ whatever $x_1,x_2$ are.
If ${y}_1<0$ then $x_2=x_1+y_1< x_1$. Hence $(x_2,y_2)<_1 (x_1,y_1)$ whatever $x_1,x_2$ are.
[**Induction Step**]{}: There are four cases based on (1) ${y}\le 0$ or ${y}>0$, and (2) $<_1$ or $<_2$ holds between $({x}_1,{y}_1)$ and $({x}_n,{y}_n)$. We omit details.
$T_1$ and $T_2$ are [*partial orders*]{} not [*linear orders*]{}. In fact, for both $T_1$ and $T_2$ there are an infinite number of minimal elements. In particular
- the minimal elements for $T_1$ are $\{({x},{y}) {\mathrel{:}}{x}\le 0 \}$, and
- the minimal elements for $T_2$ are $\{({x},{y}) {\mathrel{:}}{y}< 0 \}$.
Recall that the definition of a transition invariant, Definition \[de:ti\], allows partial orders. We see here that this is useful.
How Much Ramsey Theory Do We Need? {#se:need}
==================================
As mentioned before Podelski and Rybalchenko [@DBLP:conf/tacas/PodelskiR11] noted that the proofs of Theorems \[th:prog2ramsey\], \[th:useramsey\], and \[th:useramseygen\] do not need the strength of the full Ramsey’s Theorem. In the proofs of these theorems the coloring is transitive.
A coloring of the edges of $K_n$ or $K_{{\sf N}}$ is [*transitive*]{} if, for every $i<j<k$, if $COL(i,j)=COL(j,k)$ then both equal $COL(i,k)$.
Let $c,n\ge 1$. Let $G$ be $K_n$ or $K_{{\sf N}}$. Let $COL$ be a $c$-coloring of the edges of $G$. A set of vertices $V$ is a [*monochromatic increasing path with respect to $COL$*]{} if $V=\{v_1<v_2<\cdots \}$ and $$COL(v_1,v_2)=COL(v_2,v_3)=\cdots.$$ (If $G=K_n$ then the $\cdots$ stop at some $k\le n$.) We will drop the [*with respect to $COL$*]{} if the coloring is understood. We will abbreviate [*monochromatic increasing path* ]{} by [[*MIP* ]{}]{}from now on.
Here is the theorem we really need. We will refer to it as [*the Transitive Ramsey’s Theorem*]{}.
\[th:infinitees\] Let $c\ge 1$. For every transitive $c$-coloring of $K_{{\sf N}}$ there exists an infinite [MIP]{}.
The Transitive Ramsey Theorem is weaker than Ramsey’s Theorem. We show this in three different ways: (1) Reverse Mathematics, (2) Computable Mathematics, (3) Finitary Version.
1. For all $c\ge 1$ let $RT(c)$ be Ramsey’s theorem for $c$ colors.
2. Let $RT$ be $(\forall c)[RT(c)]$.
3. For all $c\ge 1$ let $TRT(c)$ be the Transitive Ramsey’s theorem for $c$ colors.
4. Let $TRT$ be $(\forall c)[TRT(c)]$. (This is the theorem that we really need.)
[**Reverse Mathematics:**]{} Reverse Mathematics [@revmath] looks at exactly what strength of axioms is needed to prove results in mathematics. A weak axiom system called $RCA_0$ (Recursive Comprehension Axiom) is at the base. Intuitively a statement proven in $RCA_0$ is proven constructively.
Let $A$ and $B$ be statements.
- $A{\rightarrow}B$ means that one can prove $B$ from $A$ in $RCA_0$.
- $A\equiv B$ means that $A{\rightarrow}B$ and $B{\rightarrow}A$.
- $A\not{\rightarrow}B$ means that, only using the axioms in $RCA_0$, one cannot prove $B$ from $A$. It may still be the case that $A$ implies $B$ but proving this will require nonconstructive techniques.
The following are known. Items 1 and 2 indicate that the proof-theoretic complexity of $RT$ is greater than that of $TRT$.
1. $RT{\rightarrow}TRT$. The usual reasoning for this can easily be carried out in $RCA_0$.
2. Hirschfeldt and Shore [@tramsey] have shown that $TRT \not{\rightarrow}RT$.
3. For all $c$, $RT(2) \equiv RT(c)$. The usual reasoning for this can easily be carried out in $RCA_0$. Note how this contrasts to the next item.
4. Cholak, Jockusch, and Slaman [@revramsey] showed that $RT(2)\not{\rightarrow}(\forall c)[RT(c)]$.
The proof of Theorem \[th:prog2ramsey\] showed that, over $RCA_0$,
$$TRT(3) {\rightarrow}\hbox{Program~{4 }terminates}.$$
Does the following hold over $RCA_0$?
$$\hbox{Program~{4 }terminates} {\rightarrow}TRT(3).$$
We do not know.
In the spirit of the reverse mathematics program we ask the following: For each $c$ is there a program $P_c$ such that the following holds over $RCA_0$?
$$P \hbox{ terminates } \iff TRT(c).$$
The following is open: for which $i,j\ge 2$ does $TRT(i){\rightarrow}TRT(j)$?
[**Computable Mathematics:**]{} Computable Mathematics [@recmath] looks at theorems in mathematics that are proven non-effectively and questions if there is an effective (that is computable) proof. The answer is usually no. Then the question arises as to how noneffective the proof is. Ramsey’s Theorem and the Transitive Ramsey’s Theorem have been studied and compared in this light [@Gasarchcomb; @tramsey; @Hummel; @JockRamsey; @SeetSla].
\[de:ah\] Let $M_1^{(\cdots)}, M_2^{(\cdots)},\ldots$ be a standard list of oracle Turing Machines.
1. If $A$ is a set then $A'= \{ e {\mathrel{:}}M_e^A(e){\downarrow}\}$. This is also called [*the Halting problem relative to $A$*]{}. Note that ${\emptyset}'=HALT$.
2. A set $A$ is called ${\it low}$ if $A'\le_T HALT$. Note that decidable sets are low. It is known that there are undecidable sets that are low; however, they have some of the properties of decidable sets.
3. We define the levels of the arithmetic hierarchy.
- A set is in $\Sigma_0$ and $\Pi_0$ if it is decidable.
- Assume $n\ge 1$. A set $A$ is in $\Sigma_n$ if there exists a set $B\subseteq {{\sf N}}\times{{\sf N}}$ that is in $\Pi_{n-1}$ such that $$A= \{ x {\mathrel{:}}(\exists y)[(x,y)\in B]\}.$$
- Assume $n\ge 1$. A set $A$ is in $\Pi_n$ if $\overline{A}$ is in $\Sigma_n$.
- A set is in the [*Arithmetic hierarchy*]{} if it is in $\Sigma_n$ or $\Pi_n$ for some $n$.
The following are known. Items 1 and 3 indicate that the Turing degree of the infinite homogenous set induced by a coloring is greater than the Turing degree of the infinite homogenous set induced by a transitive coloring.
1. Jockusch [@JockRamsey] has shown that there exists a computable 2-coloring of the edges of $K_{{\sf N}}$ such that, for all infinite homogeneous sets $H$, $H$ is not computable in the halting set.
2. Jockusch [@JockRamsey] has shown that for every computable 2-coloring of the edges of $K_{{\sf N}}$ there exists an infinite homogeneous sets $H\in \Pi_2$.
3. For all $c$, for every computable transitive $c$-coloring of the edges of $K_{{\sf N}}$, there exists an infinite [MIP ]{}$P$ that is computable in the halting set. This is folklore.
4. There exists a computable transitive 2-coloring of the edges of $K_{{\sf N}}$ with no computable infinite [MIP ]{}. This is folklore.
5. Hirschfeldt and Shore [@tramsey] have shown that there exists a computable transitive 2-coloring of the edges of $K_{{\sf N}}$ with no infinite low infinite [MIP ]{}.
[**Finitary Version:**]{} There are finite versions of both Ramsey’s Theorem and the Transitive Ramsey’s Theorem. The finitary version of the Transitive Ramsey’s Theorem yields better upper bounds.
Let $c,k\ge 1$.
1. $R(k,c)$ is the least $n$ such that, for any $c$-coloring of the edges of $K_n$, there exists a homogeneous set of size $k$.
2. $TRT(k,c)$ is the least $n$ such that, for any transitive $c$-coloring of the edges of $K_n$, there exists a [MIP ]{}of length $k$.
It is not obvious that $R(k,c)$ and $TRT(k,c)$ exist; however, they do.
The following is well known [@ramseynotes; @GRS; @RamseyInts].
\[th:ramseyfinite\] For all $k,c\ge 1$, $c^{k/2}\le R(k,c)\le c^{ck-c+1}$,
Improving the upper and lower bounds on the $R(k,c)$ (often called [*the Ramsey Numbers*]{}) is a long standing open problem. The best known asymptotic results for the $c=2$ case are by Conlon [@ramseyupper]. For some exact values see Radziszowski’s dynamic survey [@ramseysurvey].
The following theorem is easy to prove; however, neither the statement, nor the proof, seem to be in the literature. We provide a proof for completeness.
\[th:tramseyfinite\] For all $k,c\ge 1$ $TRT(k,c)=(k-1)^c+1$.
I\) $TRT(k,c)\le (k-1)^c+1$.
Let $n=(k-1)^c+1$. Assume, by way of contradiction, that there is transitive $c$-coloring of the edges of $K_n$ that has no [MIP ]{}of length $k$.
We define a map from $\{1,\ldots,n\}$ to $\{1,\ldots,k-1\}^c$ as follows: Map $x$ to the the vector $(a_1,\ldots,a_c)$ such that the longest mono path of color $i$ that ends at $x$ has length $a_i$. Since there are no [MIP ]{}’s of length $k$ the image is a subset of $\{1,\ldots,k-1\}^c$.
It is easy to show that this map is 1-1. Since $n>(k-1)^c$ this is a contradiction.
2\) $TRT(k,c) \ge (k-1)^c+1$.
Fix $k\ge 1$. We show by induction on $c$, that, for all $c\ge 1$, there exists a transitive coloring of the edges of $K_{(k-1)^c}$ that has no [MIP ]{}of length $k$.
[**Base Case:**]{} $c=1$. We color the edges of $K_{k-1}$ all RED. Clearly there is no [MIP ]{}of length $k$.
[**Induction Step:**]{} Assume there is a transitive $(c-1)$-coloring $COL$ of the edges of $K_{(k-1)^{c-1}}$ that has no homogeneous set of size $k$. Assume that $RED$ is not used. Replace every vertex with a copy of $K_{k-1}$. Color edges between vertices in different groups as they were colored by $COL$. Color edges within a group $RED$. It is easy to see that this produces a transitive $c$-coloring of the edges of and that there are no [MIP ]{}of length $k$.
[Erd[ö]{}s ]{}and Szekeres [@ErdosSzek] showed the following:
- For all $k$, for all sequences of distinct reals of length $(k-1)^2+1$, there is either an increasing monotone subsequence of length $k$ or a decreasing monotone subsequence of length $k$.
- For all $k$, there exists a sequences of distinct reals of length $(k-1)^2$ with neither an increasing monotone subsequence of length $k$ or a decreasing monotone subsequence of length $k$.
This is equivalent to the $c=2$ case of Theorem \[th:tramseyfinite\]. For six different proofs see Steele’s article [@subseq]. Our proof of Theorem \[th:tramseyfinite\] was modeled after Hammersley’s [@seedlings] proof of the upper bound and [Erd[ö]{}s]{}-Szekeres’s proof of the lower bound.
If $c$ is small then $TRT(k,c)$ is substantially smaller than $R(k,c$). This indicates that the Transitive Ramsey’s Theorem is weaker than Ramsey’s Theorem.
Solving Subcases of the Termination Problem {#se:dec}
===========================================
The problem of determining if a program is terminating is unsolvable. This problem is [*not*]{} the traditional Halting problem since we allow the program to have a potentially infinite number of user-supplied inputs.
1. Let $M_1^{(\cdots)}, M_2^{(\cdots)},\ldots$ be a standard list of oracle Turing Machines. These Turing Machines take input in two ways: (1) the standard way, on a tape, and (2) we interpret the oracle as the user-supplied inputs.
2. If $A\subseteq {{\sf N}}$ and $s\in{{\sf N}}$ then $M_{i,s}^A{\downarrow}$ means that if you run $M_i^A$ (no input on the tape) it will halt within $s$ steps.
3. Let $M_1^{(\cdots)}, M_2^{(\cdots)},\ldots$ be a standard list of oracle Turing Machines. $$TERM = \{ i {\mathrel{:}}(\forall A)(\exists s)[M_{i,s}^A{\downarrow}]\}.$$
1. $X\in \Pi_1^1$ if there exists an oracle Turing machine $M^{(\cdots)}$ such that $$X = \{x {\mathrel{:}}(\forall A)(\exists x_1)(\forall x_2)\cdots(Q_n x_n)[M^A(x,x_1,\ldots,x_n)=1]\}.$$ ($Q_n$ is a quantifier.)
2. A set $X$ is $\Pi_1^1$-complete if $X\in \Pi_1^1$ and, for all $Y\in \Pi_1^1$, $Y\le_m X$.
The following were proven by Kleene [@kleene2; @kleene] (see also [@Rogers]).
1. $X\in \Pi_1^1$ if there exists an oracle Turing machine $M^{(\cdots)}$ such that $$X = \{x {\mathrel{:}}(\forall A)(\exists y)[M^A(x,y)=1]\}.$$
2. $TERM$ is $\Pi_1^1$-complete.
3. If $X$ is $\Pi_1^1$-complete then, for all $Y$ in the arithmetic hierarchy, $Y\le_m X$.
4. For all $Y$ in the arithmetic hierarchy $Y\le_m TERM$. This follows from (2) and (3). (See Definition \[de:ah\] for the definition of the Arithmetic Hierarchy.)
Hence $TERM$ is much harder than the halting problem. Therefore it will be very interesting to see if some subcases of it are decidable.
\[de:fun\] Let $n\in {{\sf N}}$. Let $FUN(n)$ be a set of computable functions from ${{\sf Z}}^{n+1}$ to ${{\sf Z}}^n$. Let $m\in{{\sf N}}$. An ($F(n),m)$)-[*program*]{} is a program of the form of Program [1 ]{}where the functions $g_i$ used in Program [1 ]{}are all in $FUN(n)$.
[**Open Question:**]{} For which $FUN(n),m$ is the Termination Problem restricted to $(FUN(n),m)$-programs decidable?
We list all results we know. Some are not quite in our framework. Some of the results use the [[**While** ]{}]{}loop condition $Mx \ge b$ where $M$ is a matrix and $b$ is a vector. Such programs can easily be transformed into programs of our form.
1. Tiwari [@TermLinProgs] has shown that the following problem is decidable: Given matrices $A,B$ and vector $c$, all over the rationals, is Program [7 ]{}in $TERM$. Note that the user is inputting a real.
$x = \inp(\R$)
while ($Bx > b$)
$x= Ax+c$
2. Braverman [@TermIntLinProg] has shown that the following problem is decidable: Given matrices $A,B_1,B_2$ and vectors $b_1,b_2,c$, all over the rationals, is Program [8 ]{}in $TERM$. Note that the user is inputting a real.
$x = \inp(\R$)
while ($B_1x > b_1$) and ($B_2x \ge b_2$)
$x= Ax+c$
3. Ben-Amram, Genaim, and Masud [@termintloops] have shown that the following problem is undecidable: Given matrices $A_0,A_1,B$ and vector $v$ all over the integers, and $i\in {{\sf N}}$ does Program [9 ]{}terminate.
$x =\inp(\Z)$
while ($Bx \ge b$)
if $x[i] \ge 0$
then $x=A_0x$
else
$x=A_1x$
4. Ben-Amram [@BA:delta] has shown a pair of contrasting results:
- The termination problem is undecidable for $(FUN(n),m)$-programs where $m=1$ and $FUN(n)$ is the set of all functions of the form
$f(x[1],\ldots,x[n])= \min\{ {x}[{i_{1}} ]+{c_{1}} , {x}[{i_{2}} ]+{c_{2}} , \ldots, {x}[{i_{k}} ]+{c_{k}} \}$
where $1\le {i_{1}} < \cdots < {i_{k}}$ and ${c_{1}},\ldots,{c_{k}} \in {{\sf Z}}$.
- The termination problem is decidable for $(FUN(n),m)$-programs when $m\ge 1$ and $FUN(n)$ is the set of all functions of the form
$f(x[1],\ldots,x[n])= x[i]+c$
where $1\le i \le n$ and c$\in{{\sf Z}}$. Note that Program [5 ]{}falls into this category.
Open Problems {#se:open}
=============
1. For which ($FUN(n),m$) is the Termination Problem restricted to $(FUN(n),m)$-programs decidable?
2. Find a natural example showing that Theorem \[th:usetrans\] requires the Full Ramsey Theorem.
3. Prove or disprove that Theorem \[th:usetrans\] is equivalent to Ramsey’s Theorem.
4. Classify more types of Termination problems into the classes Decidable and Undecidable. It would be of interest to get a more refined classification. Some of the undecidable problems may be equivalent to HALT while others may be complete in some level of the arithmetic hierarchy or $\Pi_1^1$ complete
5. Prove or disprove the following conjecture: for every $c$ there is a program $P_c$ such that, over $RCA_0$, $TRT(c) \iff $ every run of Program $P_c$ terminates.
Acknowledgments
===============
I would like to thank Daniel Apon, Amir Ben-Amram, Peter Cholak, Byron Cook, Denis Hirschfeldt, Jon Katz, Andreas Podelski, Brian Postow, Andrey Rybalchenko, and Richard Shore for helpful discussions. We would also like to again thank Amir Ben-Amram for patiently explaining to me many subtle points that arose in this paper. We would also like to thank Daniel Apon for a great proofreading job.
Using Theorem \[th:matrixw\] and $2\times 2$ Matrices to Prove Termination of Program [5 ]{} {#se:amir}
============================================================================================
If $\mathcal C$ is a set of square matrices of the same dimension then ${{\rm clos}}({\mathcal C})$ is the set of all finite products of elements of $\mathcal C$. For example, if ${\mathcal C} = \{C_1,C_2\}$ then $C_1^2C_2C_1^3C_2^{17} \in {{\rm clos}}(C_1,C_2)$.
This section is due to Ben-Amram and is based on a paper of his [@BA:delta]. He gives an example of a proof of termination of Program [5 ]{}where he uses the matrices $C_1,C_2$ that come out of Program [5 ]{}directly (in contrast to our proof in Theorem \[th:prog3matrix\] which used $3\times 3$ matrices by introducing $x+y$). Of more interest: there [*is*]{} an element of ${{\rm clos}}(C_1,C_2)$ that has no negative numbers on the diagonal, namely $C_2$ itself. Hence we cannot use Theorem \[th:matrix\] to prove termination. We can, however, use Theorem \[th:matrixw\].
\[th:prog3macho\] Every computation of Program [5 ]{}is finite.
The case control=1 is represented by the matrix $$C_1 = { \begin{pmatrix} -1 & 0 \\ \infty & \infty \end{pmatrix}}.$$ The case control=2 is represented by the matrix $$C_2 = { \begin{pmatrix} \infty & -2 \\ +1 & \infty \end{pmatrix}}.$$
We find a representation of a [*superset*]{} of ${{\rm clos}}(C_1,C_2)$. Let $$\mathcal E = YZ^a, \qquad \text{where\ } Y \in \{C_1, C_2, C_1C_2, C_2C_1\} \text{\ and\ } Z = { \begin{pmatrix} -1 & \infty \\ \infty & -1 \end{pmatrix}}.$$ We show that ${{\rm clos}}(C_1,C_2)\subseteq {\mathcal E}$. We will then show that every element of ${\mathcal E}={{\rm clos}}({\mathcal E})$ satisfies the premise of Theorem \[th:matrixw\]. We prove this by induction on the number of matrices are multiplied together to form the element of ${{\rm clos}}(C_1,C_2)$.
The base case is trivial since clearly $C_1,C_2 \in {\mathcal E}$.
We show the induction step by multiplying each of the four “patterns” in $\mathcal E$ *on the left* by each of the matrices $C_1,C_2$. We use the following identities: $C_1^2 = ZC_1 = C_1Z = C_1C_2$, $C_2^2 = Z$, $ZC_2 = C_2Z$.
1. $ C_1(C_1Z^a) = C_1^2 Z^a = C_1Z Z^a = C_1Z^{a+1}$
2. $ C_2(C_1Z^a) = (C_2C_1)Z^{a}$
3. $ C_1(C_2Z^a) = (C_1C_2)Z^{a}$
4. $ C_2(C_2Z^a) = C_2^2Z^{a} = Z Z^a = Z^{a+1}$
5. $ C_1(C_1C_2Z^a) = C_1^2C_2Z^{a} = C_1 Z C_2 Z^a= C_1 C_2 Z Z^a = (C_1 C_2)Z^{a+1}$
6. $ C_2(C_1C_2Z^a) = C_2(C_1Z Z^{a}= (C_2C_1)Z^{a+1}$
7. $ C_1(C_2C_1Z^a) = (C_1 C_2) C_1 Z^{a} = C_1 (Z C_1) Z^a = C_1^2 Z^{a+1} = C_1 Z^{a+2}$
8. $ C_2(C_2C_1Z^a) = ZC_1Z^{a} = C_1Z^{a+1}$
We have shown that ${{\rm clos}}(C_1,C_2)\subseteq {\mathcal E}$. Now it remains to verify that every matrix represented by $\mathcal E$, [*or some power thereof*]{}, has a negative integer on the diagonal. Note that in one of the cases, squaring is necessary.
1. $C_1Z^a = { \begin{pmatrix} -1-a & -a \\ \infty & \infty \end{pmatrix}} $
2. $(C_2Z^a)^2 = C_2^2 Z^{2a} = Z^{2a+1}$
3. $C_1C_2Z^a = C_1Z Z^{a} = C_1 Z^{a+1}$
4. $C_2C_1Z^a = { \begin{pmatrix} \infty & \infty \\ -3 & -2 \end{pmatrix}} Z^a = { \begin{pmatrix} \infty & \infty \\ -3 & -2-a \end{pmatrix}} $.
We can now apply Theorem \[th:matrixw\] and we are done.
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---
abstract: 'OSIRIS is a near-infrared (1.0–2.4 $\micron$) integral field spectrograph operating behind the adaptive optics system at Keck Observatory, and is one of the first lenslet-based integral field spectrographs. Since its commissioning in 2005, it has been a productive instrument, producing nearly half the laser guide star adaptive optics (LGS AO) papers on Keck. The complexity of its raw data format necessitated a custom data reduction pipeline (DRP) delivered with the instrument in order to iteratively assign flux in overlapping spectra to the proper spatial and spectral locations in a data cube. Other than bug fixes and updates required for hardware upgrades, the bulk of the DRP has not been updated since initial instrument commissioning. We report on the first major comprehensive characterization of the DRP using on-sky and calibration data. We also detail improvements to the DRP including characterization of the flux assignment algorithm; exploration of spatial rippling in the reduced data cubes; and improvements to several calibration files, including the rectification matrix, the bad pixel mask, and the wavelength solution. We present lessons learned from over a decade of OSIRIS data reduction that are relevant to the next generation of integral field spectrograph hardware and data reduction software design.'
author:
- 'Kelly E. Lockhart'
- Tuan Do
- 'James E. Larkin'
- Anna Boehle
- 'Randy D. Campbell'
- Samantha Chappell
- Devin Chu
- Anna Ciurlo
- Maren Cosens
- 'Michael P. Fitzgerald'
- Andrea Ghez
- 'Jessica R. Lu'
- 'Jim E. Lyke'
- Etsuko Mieda
- 'Alexander R. Rudy'
- Andrey Vayner
- Gregory Walth
- 'Shelley A. Wright'
bibliography:
- 'references.bib'
title: Characterizing and Improving the Data Reduction Pipeline for the Keck OSIRIS Integral Field Spectrograph
---
Introduction
============
![An OSIRIS observation of the Galactic center at the 35 mas plate-scale in the Kn3 filter. The NIR emission from the supermassive black hole, Sgr A\*, is labeled with a white dot. The data are taken at a position angle of 175 degrees. The observations take advantage of the Keck LGS AO system.[]{data-label="fig:gc"}](osiris_gc_image.pdf){width="\columnwidth"}
The OSIRIS [OH Suppressing InfraRed Imaging Spectrograph; @larkin2006osiris:] instrument behind the Adaptive Optics (AO) system at the W. M. Keck Observatory has proven to be an important tool for extragalactic, Galactic, and solar system observational astrophysics. OSIRIS has produced nearly one-half of the laser guide star (LGS) AO publications from Keck since its commissioning in 2005, including the majority of the extragalactic LGS papers. Results from OSIRIS observations span fields from planetary science [e.g. @laver2009the-global; @laver2009component-resolved; @brown2013salts], some of the first spectroscopic characterization of extrasolar planets, [e.g. @bowler2010near-infrared; @barman2011clouds; @konopacky2013detection], studies of crowded fields such as the center of the Milky Way galaxy [e.g. @do2013three-dimensional; @lu2013stellar; @yelda2014properties; @lockhart2018], measurement of the masses of supermassive black holes (SMBHs) in nearby galaxies [e.g. @mcconnell2011two-ten-billion-solar-mass; @medling2011mass; @walsh2012a-stellar], and studies of high redshift (1 < z < 3) galaxies [e.g. @wright2007integral; @stark2008the-formation; @law2009the-kiloparsec-scale; @jones2010resolved]. A sample observation is shown in figure \[fig:gc\], demonstrating OSIRIS’s near diffraction-limited spatial resolution.
{width="\columnwidth"}
^a^ https://github.com/Keck-DataReductionPipelines/OsirisDRP \[fig:optical\]
OSIRIS is an innovator among near-infrared (NIR, 1.0–2.4 $\micron$) integral field spectrographs (IFS) due to its unique use of a lenslet array to split the field into spatial pixels (spaxels) while preserving the high Strehl ratio point spread function (PSF) of the Keck AO system (figure \[fig:optical\]). The new planet finding spectrographs (e.g., Gemini Planet Imager, SPHERE) have a similar configuration but with a very different output data format. IFSs are also being planned for the next-generation extremely large telescopes [e.g. @2016SPIE.9908E..1WL; @2016SPIE.9908E..1YS; @2016SPIE.9908E..1XT; @2016SPIE.9913E..4AW] and forethought in the data reduction pipeline and necessary calibrations is essential for their success.
![The OSIRIS spectra are closely packed on the detector. *Top:* Unreduced 2D detector Kbb/35 sky frame. The wavelength direction increases to the left; the bright bands at the left are the rising thermal background in the red. *Bottom:* Zoom in of the panel above. Individual gray rows, alternating with the fainter background, represent the thermal continuum in individual sky spaxels. Bright spots are OH sky lines. Neighboring spectra are staggered by 32 detector columns, which translates to a stagger of 32 spectral channels in the reduced cube.[]{data-label="fig:rawdet"}](fig_rawdetector_rev.pdf){width="\columnwidth"}
The OSIRIS IFS was designed to have very stable spectra that fall at the same position on the detector, using a single, fixed grating and no moving components after the lenslet array. In order to maximize the field of view, the spectra from adjacent spaxels are only separated by 2 pixels (see figure \[fig:rawdet\]). In the current configuration, the full-width half-maximum (FWHM) of individual spectra perpendicular to the dispersion direction range from roughly 1.3 to 1.7 pixels [@boehle2016upgrade] and thus overlap each other in their wings (though the FWHM was larger and the spectra have overlapped more in previous instrument hardware configurations, as discussed further throughout this work). However, the stable format enables the use of a deconvolution process in the OSIRIS data reduction pipeline (DRP) to separate the interleaved spectra in the input 2D detector format and to assign flux to each spaxel at every wavelength in the output data cube (dimensions x, y, lambda). The DRP is essential for both real-time processing at the telescope and for post-processing of all data.
The OSIRIS data reduction pipeline {#ssec:drp}
----------------------------------
The complexity of the raw data output from the instrument necessitated a well-developed data reduction pipeline from delivery [@krabbe2002data; @2017ascl.soft10021L], the first for an instrument on Keck. The DRP, written primarily in IDL with computationally intensive processes passed into C, is modularized, such that individual modules of the reduction can be turned on or off as needed at runtime. Typical reduction modules, such as dark subtraction, cosmic ray removal, and telluric correction, are available. Also included is an implementation of the scaled sky subtraction algorithm detailed by @davies2007a-method.
The unique part of the DRP is the spatial rectification module (`spatrectif`), which separates the overlapping spectra in the 2D frame and places flux into corresponding spaxels in the output data cube. The OSIRIS spectral format allocates only 2 pixels between neighboring spectra, and there is a stagger in wavelength of about 32 pixels between lenslet neighbors. While the PSF of each lenslet in the spatial direction, perpendicular to the dispersion direction, is currently below 2 pixels in FWHM, this has not always been true, and in all cases some blending occurs which must be extracted with the deconvolution. The spatial rectification module uses empirically determined rectification matrices (§ \[ssec:recmat\]) to map the correspondence between pixels in the 2D frame and spaxels in the data cube. Using the rectification matrices as an initial estimate of the correspondence between pixels in the 2D detector and spaxels in the data cube, the Gauss-Seidel method is used to iteratively assign flux from the cleaned 2D image into individual spaxels. The algorithm is further described by @krabbe2004data and in the instrument manual[^1]. In addition to mapping the lenslet responses, the rectification matrices also serve as flat fields to determine the detector pixel response.
To run the pipeline, the user designates which modules will be used, along with setting any module-specific options or keywords, in an XML file. This XML file is saved to a queue folder that is regularly checked by the DRP. The DRP then processes any new XML files and saves the output as requested.
Hardware upgrades
-----------------
{width="6in"}
OSIRIS has been operational on Keck for over a decade. Over the course of that that time, it has been subject to repairs and upgrades which are summarized in figure \[fig:timeline\]. Two major hardware upgrades have occurred in recent years.
In December 2012, the dispersion grating was upgraded [@mieda2014efficiency], increasing the grating efficiency and flux on uncleaned raw 2D detector frames by a nearly a factor of 2. At the same time, OSIRIS was moved from Keck-II to Keck-I, to take advantage of Keck-I’s new LGS AO system. At this time, the DRP was updated to include an updated wavelength solution, to account for the different image orientation, to update the world coordinate system (WCS), and to account for differential dispersion between the Keck-II and Keck-I AO system dichroics.
In January 2016, the detector behind the spectrograph portion of OSIRIS was upgraded from a Hawaii-2 detector to a Hawaii-2RG detector [@boehle2016upgrade]. The new detector shows a lower dark current and improved throughput by up to a factor of 2 in raw 2D detector frames, due to a combination of an improved quantum efficiency and a reduction in noise sources such as crosstalk between readout channels [@boehle2016upgrade]. The DRP was updated with a new wavelength solution, revised locations of the spectra on the 2D frames, and the removal of glitch modules only needed for the old Hawaii-2 detector.
Motivation
----------
Beginning around the same time of the OSIRIS move to Keck-I and the grating upgrade in 2012, observers began to notice irregularities in their reduced data cubes: the presence of unphysically large positive and negative spikes at a few wavelength channels in a small number of spaxels in reduced data, poor performance by the cosmic ray module in successfully identifying and removing cosmic rays, and offsets in the wavelength solution. In addition, observers noticed that reduced science data of QSO observations incorrectly showed dark spaxels near the central bright compact source. Similarly, observations of bright narrow emission lines incorrectly showed absorption or negative flux at the wings of the narrow emission line. Finally, single channel maps of OH sky lines or other evenly illuminated sources were no longer flat, but instead displayed a rippling pattern unrelated to incident illumination. These issues led to an effort to better characterize the behavior of the DRP and to improve the quality of the reduced data.
Since initial delivery, the DRP has historically been maintained by Keck Observatory staff, with coding and testing performed by volunteered time from a few instrument team members. Upgrades to the software have primarily occurred after hardware upgrades that necessitated changes to the wavelength calibration, for instance. However, the scope of the necessary DRP characterization to resolve these issue was beyond the scope of usual DRP maintenance. The OSIRIS Pipeline Working Group was formed and undertook the first major comprehensive characterization of the DRP using calibration and on-sky data. We detail our efforts in the following report.
We outline the data we used for this work in § \[sec:data\], artifacts from the flux assignment algorithm in § \[sec:fluxassign\], and spatial rippling in § \[sec:spatrip\]. We describe improvements to the following calibration files in three sections: the rectification matrices (§ \[sec:crinrecmat\]), the bad pixel mask (§ \[sec:badpix\]), and the wavelength solution (§ \[sec:wavesol\]). Many of the issues and improvements we outline here are relevant to the future generation of IFSs on the next generation of telescopes; we outline our recommendations in § \[sec:rec\].
The DRP is open-source and is hosted on Github[^2] as of May 2016. Releases are made as needed, or roughly once per year. This work uses release v.4.0.0, and many of the improvements detailed here are included in v.4.1.0.
DATA {#sec:data}
====
Throughout this paper, we will use several on-sky science and engineering data sets to investigate the OSIRIS spectral extraction routine and performance. These data sets are described below and summarized in Table \[tab:obs\].
[llrcr]{}
arc & 2012 Jun 15 & 60 & Hbb & 20\
arc$^a$ & 2013 Apr 11 & 60 & Kbb & 50\
arc$^a$ & 2015 Apr 06 & 10 & Kbb & 50\
arc & 2015 Dec 17 & 32 & Kbb & 35\
arc & 2015 Dec 17 & 16 & Kbb & 50\
arc & 2015 Dec 17 & 4 & Kbb & 100\
arc & 2016 Mar 18 & 60 & Jbb & 50\
arc$^a$ & 2016 Sep 02 & 30 & Kbb & 20\
arc$^a$ & 2016 Sep 02 & 30 & Kbb & 35\
arc$^a$ & 2016 Sep 02 & 30 & Kbb & 50\
arc$^a$ & 2016 Sep 02 & 1.5 & Kbb & 100\
QSO & 2014 May 19 & 4x600 &Hn3 &100\
QSO & 2015 Aug 09 & 7x600 & Kn1&100\
sky & 2012 Jun 09 & 900 & Kn3 & 20\
sky & 2012 Jun 09 & 900 & Kn3 & 35\
sky & 2013 May 14 & 900 & Kn3 & 35\
sky & 2013 May 14 & 900 & Kn3 & 50\
sky & 2013 May 11 & 900 & Kbb & 35\
sky & 2014 May 17 & 900 & Kbb & 50\
sky & 2015 Jul 19 & 100 & Kn3 & 100\
sky & 2015 Jul 22 & 900 & Hbb & 50\
sky & 2016 Mar 21 & 600 & Jbb & 50\
sky & 2016 Mar 21 & 600 & Hbb & 50\
sky & 2016 Apr 18 & 900 & Jn2 & 35\
sky & 2016 May 14 & 900 & Kbb & 35\
sky & 2016 Jul 11 & 900 & Kbb & 35\
sky$^a$ & 2016 Sep 02 & 600 & Kbb & 35\
sky$^a$ & 2016 Sep 02 & 600 & Kbb & 50\
sky$^a$ & 2016 Sep 02 & 600 & Kbb & 100\
white light$^a$ & 2015 Sep 04 & 8$^b$ & Kbb & 50\
\[tab:obs\]
Example Science Data Sets {#sec:scidata}
-------------------------
Bright quasars and their host galaxies are a particularly useful data set for testing the DRP as they contain an extremely compact ($<$2 spaxels), bright point-like quasar with both continuum and strong emission lines, and underlying faint galaxy emission. A clean spectral extraction and high quality AO performance is needed for these data sets, since the science objective is to resolve the host galaxies of high-redshift (z$>$1) quasars, which typically span $<$1” [@vayner2016providing].
On 2014 May 19 and 2015 August 9 OSIRIS observations were taken of the quasars 3C 298 (z=1.439; R=16.0 mag; H=14.5 mag) and 3C 9 (z=2.012; R=17.4 mag; H=15.6 mag) using the 100mas plate scale with the Hn3 ($\lambda_{cen}$=1.64 $\micron$) and Kn1 ($\lambda_{cen}$=2.01 $\micron$) filters, respectively. The LGS AO system was used for both. For 3C 298 there were a total of four dithered 600 second frames, while for 3C 9, there were a total of seven dithered 600 second frames. Separate sky frames were acquired during both observations.
Initial data reduction in 2014 indicated potential issues with the DRP. The combination of the bright continuum source with strong emission lines made identification of spatial and spectral flux assignment artifacts easy. We further describe the QSO DRP results in Section \[qsodata\].
Skies
-----
NIR sky spectra are useful for testing DRP effects on emission line spectra due to the abundance of narrow OH sky lines. In addition to the OH emission lines, the sky spectrum exhibits a thermal background continuum, increasing in the K-band. Sky observations are frequently taken during the normal course of observing; however, several dedicated “deep skies” were taken in order to maximize the signal-to-noise.
Arc lamps
---------
Calibration frames using Ar, Kr, and Xe arc lamps were obtained. Emission lines were identified using a NIR line list[^3]. Arc lamp spectra are useful as a calibration source due to their high S/N emission lines, narrow line widths, and well-constrained wavelengths.
Arc lamp spectra were obtained in two ways: first, with the entire lenslet array illuminated, as for most science exposures. In addition, arc lamp spectra were observed using a single lenslet column mask. This mask consists of a slit wide enough to illuminate only one column of lenslets, which corresponds to one illuminated column of spaxels in the reduced data cube. On the detector, this produces spectra that are well-separated (i.e. separated by many times the FWHM of each spectrum) and thus flux on the detector can be assigned unambiguously to a single spaxel. The resulting data cube is expected to contain a single column of spaxels with full arc spectra, while the unilluminated lenslets correspond to dark spaxels with no spectra.
Rectification Matrices {#ssec:recmat}
----------------------
![Example single white light scan, used in the creation of a rectification matrix along with the rest of the set of white light scans. In this single scan, only one column of lenslets is illuminated, which produces well-separated traces. The x direction corresponds to the dispersion direction.[]{data-label="fig:whitelight"}](whitelightscan_Kbb_050_s150904_c003006_rev.pdf){width="\columnwidth"}
Flux on the detector is assigned to a spaxel in the data cube using an empirically determined matrix, the rectification matrix, of lenslet response curves. White light exposures are taken through the single lenslet column mask. The mask is stepped across the lenslet array, illuminating one column of lenslets at a time. The resulting frames display well-separated white light bands with no emission lines (see figure \[fig:whitelight\] for a single example scan).
As the illumination source is uniform, the resulting flux on the detector is a function of lenslet response and pixel response only. These response curves are measured and encoded in the resulting rectification matrix.
OSIRIS allows the selection of one of four spatial samplings, from 002 to 01 per spaxel, in combination with one of 23 available filters, though not all filters are designed to work with all modes. This results in up to 88 modes that each require a separate set of white light exposures and a rectification matrix.
The rectification matrix file contains three extensions. The first extension is the y position (lower and upper bounds) of each spectrum on the detector, numbered in descending order from the top of the detector. The second extension is a quality flag. The spectra from some edge spaxels do not fall completely on the detector and thus are excluded from the final data cube; these are marked with a 0 in this extension. The third extension is the three-dimensional matrix containing the lenslet response curves. The first dimension is the width of the detector along the dispersion direction (2048 pixels), the second is the extent of the vertical response of each lenslet on the detector (16 pixels), and the third is the spectrum number (dependent on the mode selected). The lenslet response curves for the spectra flagged in the second extension are set to zero here.
FLUX ASSIGNMENT ARTIFACTS {#sec:fluxassign}
=========================
The *Spectral Extraction* DRP module (`spatrectif`) works to model the spectral PSF from each lenslet (which has a one-to-one correspondence with a single spaxel in the reduced data cube) using the average 1D PSF in the spatial direction, as further described in Section \[ssec:drp\]. The current spectral extraction routine does not model the 2D structure of the PSF, which means that any asymmetries in the PSF will not be accounted for in the wavelength direction. Problems with the flux assignment algorithm were first noted beginning in December 2012, after OSIRIS moved to Keck-I and the new grating was installed. These issues manifested in reduced data cubes in both the *spatial* direction, with artifacts affecting the total flux in neighboring spaxels, and in the *spectral* direction, with artifacts appearing over a limited wavelength range in neighboring spaxels. In this section we describe the symptoms and problems seen with flux assignment artifacts with on-sky data sets and engineering data, and then follow with discussion on potential causes of the problem and future solutions.
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Spatial flux assignment artifacts from pupil misalignment {#qsodata}
---------------------------------------------------------
Spatial flux assignment artifacts are seen in QSO data taken in 2014 and 2015. The cause of these spatial artifacts is differences between the PSF of on-sky data and of the white light scans used to create the rectification matrices. This effect is apparent almost exclusively at the 100 mas plate scale. These differences are ultimately caused by misalignment between the on-sky pupil and that of the calibration unit, which feeds the Keck AO system and instruments with arc lamp and white light sources and which includes a simulated Keck pupil. Deviations between this calibration unit pupil and the on-sky pupil directly affect the shape of the PSF and the location of spectra on the detector. Data taken using the 100 mas plate scale show a small shift or translation between white light calibration spectra on the detector and that of on-sky science spectra. In addition, the PSF width of the white light calibration data is different from that of on-sky science data. We detail the investigation into the spatial flux assignment artifacts below. The on-sky data and the white light misalignment is still an ongoing issue and is not only limited to 2014 and 2015. We further discuss issues with the pupil in the 100 mas scale which may be the main cause of this misalignment.
The spatial flux assignment artifacts are evident in QSO data taken between 2014 and 2015 using the 100 mas plate scale (§ \[sec:scidata\]). The quasar, unresolved at this resolution, appears as a point source convolved by the spatial PSF in data cubes collapsed in the wavelength direction. The quasar spectra should be dominated by broad-line H$\alpha$ emission, represented by a broad Gaussian profile ($\sigma \sim$3000 km s$^{-1}$). Instead, in some individual spaxels, the broad-line H$\alpha$ emission has large discrepant dips. Other spaxels show other features that do not arise from the intrinsic science spectra. See figure \[fig:OSIRIS\_QSO\_flux\] for an example using the 3C 298 reduced data cube (Hn3/100 mas). This 2014 data set signaled that there were major issues with the spectral extraction routine and flux assignment.
We investigate this effect by comparing the quasar continuum emission location on the 2D frame to that of the white light scan for the same lenslet. A Gaussian profile is fit to both the quasar continuum spectrum on the 2D frame and to the matched white light scan in the spatial direction, perpendicular to the dispersion direction. In the 2015 data set for quasar 3C 9 (Kn1/100 mas), the peak of the quasar continuum emission is translated by 0.53 pixels in the spatial direction from the white light scan at a given wavelength. In addition, the profile fitted to the white light scan is broader by $\sim$0.2 pixels compared to the on-sky profile. A shift is applied to the quasar data to align it with the position of the white light scans, and the quasar data are convolved with a Gaussian in the spatial direction to match the width of the white light scans. We show the results of this process in Figure \[fig:OSIRIS\_QSO\_fix\].
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It is important to note that these spatial flux assignment artifacts seen in these tests with the QSO data are present almost exclusively in the 100 mas plate scale, where the on-sky pupil is slightly different than that of the calibration unit. OSIRIS’s four spatial scales (002, 0035, 005, and 01) are achieved by swapping in matched pairs of lenses that magnify the images onto the lenslet array, and have an effective pupil size matched to each beam size. This leads to differences in sensitivities and backgrounds for each of the four spatial scales and makes it less likely that spatial flux assignment artifacts will be observed in the other scales.
Indeed, tests conducted comparing single lenslet column sky frames in both the 50 and 100 mas plate scales with matched white light scans show this discrepancy between on-sky data and calibration data. A Gaussian profile is fit at each illuminated spaxel in the reduced single lenslet column sky cubes and the matching reduced white light scan cubes, perpendicular to the illuminated lenslet column. The centroid and FWHM of the profile at each spaxel is compared between the sky and the white light scans. On average, the centroid positions differ by 0.03 spaxels in the 50 mas data and 0.14 spaxels in the 100 mas data. For the 50 mas data, the average FWHM difference between the sky and white light scans is 0.01 spaxels, while the average difference for the 100 mas data is 0.30 spaxels.
The cause of this mismatch is the differing pupil masks used for different plate scales. Each of the three fine scales (002, 0035, 005) have cold pupil stops mounted within the camera wheel, while the coarse scale (01) has a fixed cold stop permanently mounted in the optical path[^4]. This has the unfortunate effect that the 01 pupil must be oversized to ensure that when using other spatial scales their beams do not vignette. This was a known issue at the time of delivery, but will cause minor mismatch between the calibration and on-sky PSFs and higher thermal noise. Indeed the 100 mas reductions were cleaner and did not have these flux assignment artifact issues before 2012 when OSIRIS was on Keck-2 with the old grating. To date, we believe that the pupil change between Keck-2 and Keck-1 calibration unit may be a significant cause of the problem.
Spectral flux assignment artifacts from PSF asymmetry {#arclamps}
-----------------------------------------------------
Spectral flux assignment artifacts are seen in bright emission line data, such as arc lamp calibrations or sky frames. These artifacts are caused by the discrepancy between the real 2D PSF of a source on the detector and the smoothed PSF assumed during the construction of the rectification matrix. The smoothed PSF effectively averages the real 2D PSF of every pixel into a one-dimensional line spread function (LSF) perpendicular to the spectral dispersion direction. For example, the 2D PSF of a bright source on the detector, such as an emission line, exhibits some asymmetries due to imperfections in the optical system. For the OSIRIS optical system, the real 2D PSF has a coma, or is flared, towards the left. However, the rectification matrix is constructed using white light scans to estimate the LSF perpendicular to the dispersion direction. It then uses this LSF estimate to assign flux iteratively from the detector into the correct spaxel, which effectively smooths or averages the PSF at each spectral channel. In effect, the white light LSF is narrower than the real PSF across the flared side and wider across the non-flared side. The width difference between the LSF and the 2D PSF introduces an error into the flux assignment algorithm and produces the flux assignment artifacts. We demonstrate this issue below.
![Example of flux assignment artifacts in a September 2016 single-column arc lamp spectrum (Kbb/35 mas). Flux is conserved in the correct spaxel, but flux assignment artifacts are visible at $\pm$32 spectral channels in adjacent spaxels. *Top panel:* Spectral channel map at 2.191 $\mu m$, the peak of a bright Kr emission line. Only one column of lenslets is illuminated. The green boxes mark the two dark spaxels whose spectra are shown at bottom, while the red box marks the shown illuminated spaxel. *Bottom panels:* Segment of a spectrum in the three adjacent spaxels indicated in the channel map. *Second panel:* Dark spaxel adjacent to an illuminated spaxel. This spaxel should show no flux, but negative artifacts are visible. The artifacts are offset by -32 spectral channels from the emission line in the middle panel. *Third panel:* Illuminated spaxel. *Bottom:* Dark spaxel adjacent to an illuminated spaxel. This spaxel should not show flux, but positive and negative artifacts are visible. The blue artifacts are at the same spectral channel as the emission line in the middle panel, while the red artifacts are offset by +32 spectral channels.[]{data-label="fig:misflux"}](plot_misflux_2016.pdf){width="\columnwidth"}
Issues with the flux assignment algorithm in the spectral direction are most evident in the presence of a bright emission line. The flux assignment artifacts manifest as increased systematic error in neighboring spaxels, shifted by 32 spectral channels away from the line. Figure \[fig:misflux\] shows an example using a single-column arc lamp spectrum.
![Example of flux assignment artifacts, shown after the raw flux has been assigned to a single spaxel but before wavelength calibration or assembly into a cube. Note that as the wavelength calibration is not yet applied, the wavelength direction is opposite from the usual sense, and increases towards the left. A section of a September 2016 reduced Kbb/50 mas single-column arc lamp frame centered on a bright emission line is shown. Each row corresponds to an individual spaxel, while each column corresponds to a spectral channel. The wavelength solution has not yet been applied so there is not a one-to-one correspondence between channel number and wavelength. Neighboring spaxels are shifted by 32 spectral channels with respect to each other. In this example, all flux should fall in spaxel 693. Residual flux above and below the bright emission line at roughly channel 990 is evidence of flux assignment artifacts. The shape of the artifacts reflects the shape of the 2D PSF.[]{data-label="fig:swapchan"}](swap_chan.pdf){width="\columnwidth"}
The 32 channel shift is due to the stagger in wavelength between neighboring spaxels; when the spectra from the 2D frame are assembled into the data cube and the wavelength calibration is applied, this translates into a shift of 32 spectral channels in neighboring spaxels. The flux assignment artifacts are thus more easily pictured without this wavelength calibration applied. Figure \[fig:swapchan\] shows an example using a segment of a single-column arc, centered on a bright emission line. Positive and negative artifacts are evident both above and below the emission line in the illuminated spaxel. These are the cause of the artifacts in the dark spaxels seen in figure \[fig:misflux\].
The positive and negative artifacts around the bright emission line show a clear asymmetry—positive artifacts are preferentially seen redward of the emission line peak while negative artifacts appear blueward of the line peak. This artifact shape corresponds with the PSF of lines on the 2D frame. Figure \[fig:PSF\] shows the supersampled PSF of a bright arc lamp emission line, which was constructed by stacking the raw images of this line in all spaxels in a single 2D frame and supersampling it by a factor of 100. Note that as there is some variation in the 2D PSF FWHM as a function of lenslet and wavelength [@boehle2016upgrade], a more robust determination of the PSF shape would come from stacking multiple matched images of a given emission line in the same lenslet to remove these effects. However, given the limitations of available calibration data, this analysis is left for future work. The PSF, shown in figure \[fig:PSF\] and effectively averaged over the detector, is asymmetric and flared towards the left, likely reflecting a coma from the instrument optics. The correspondence between the flux assignment artifacts and the average PSF reflects that the cause of the artifacts is the asymmetry of the PSF.
![PSF on the detector, supersampled by a factor of 100. Gray lines represent the detector pixel boundaries, for scale.[]{data-label="fig:PSF"}](arc2016mar_psf_cubic_super_py.pdf){width="\columnwidth"}
From testing, it appears that flux is conserved in individual spaxels in the presence of spectral flux assignment artifacts. Three tests were conducted using single-lenslet column arc lamp data from 2015 and 2016. In all tests, a circular aperture is placed on a bright emission line in the 2D detector frame, before flux assignment. The spatial rectification module is run, but the DRP is stopped before the wavelength solution is applied, creating output similar to figure \[fig:swapchan\]. A rectangular aperture is placed on just the illuminated lenslet (e.g. spaxel 693 in the example in figure \[fig:swapchan\]) to remove the effect of the artifacts. The enclosed flux is compared before and after the flux assignment. In all tests, the enclosed flux after the flux assignment matches within 3$\sigma$, and in two of the three tests, the flux matches at the 1$\sigma$ level. As these tests show that roughly the same amount of flux appears in the integrated emission line in the final data cube, the positive and negative artifacts must effectively cancel each other out.
However, this issue does induce extra error due to artifacts from both emission lines in the science data and from OH sky lines. In addition, the level of the artifacts, and the induced error, has changed with time. We create a metric to measure the level of the flux assignment artifacts with time. We measure the absolute value of the peak of the flux assignment artifacts resulting from a bright emission line as a percentage of the peak of the line itself. As the artifacts can be asymmetric, we measure the artifacts separately in the two neighboring spaxels. We calculate this metric separately for both arc lamp and sky spectra. As the sky spectra are generally much lower in S/N than the arc lamps, we calculate the metric on a median spectrum of a column of spaxels in the sky cube. For the arc lamps, the metric is calculated separately on each individual spaxel. We report the mean values and their standard deviations from our investigation for each observation type and time period in table \[tab:flux\].
[lccccc]{}
2012–2015 & arc & Hbb/20; Kbb/35, 50, 100 & 6 & 7.2$\pm$3.7% & 5.3$\pm$1.8%\
2012–2015 & sky & Hbb/50; Kn3/20, 35, 50, 100 & 6 & 2.4$\pm$0.9% & 3.0$\pm$1.0%\
2016–present & arc & Kbb/20, 35, 50, 100 & 4 & 2.2$\pm$0.6% & 5.9$\pm$0.5%\
2016–present & sky & Jn2/35; Jbb/50; Hbb/50; Kbb/35, 50, 100 & 6 & 4.8$\pm$5.1% & 6.1$\pm$2.1%\
\[tab:flux\]
Prior to December 2012, the artifacts were much smaller but this issue became more prominent after the grating upgrade. Results from the analysis detailed in this section were taken into account during the most recent hardware upgrade, the detector replacement in 2016. To mitigate both the spectral flux assignment artifacts and the spatial rippling (see next section, § \[sec:spatrip\]), the PSF elongation was rotated 90 degrees to be slightly broader in the dispersion direction. This rotation eased the discrepancy between on-sky data and the white light scans, and has qualitatively lessened the degree of the spectral flux assignment artifacts. This is also demonstrated by the reduction in the level of the arc lamp artifacts after the detector replacement (Table \[tab:flux\]). (The sky line measurements are much noisier, but are consistent with no reduction in the level of the flux artifacts.) However, it is still a stronger effect than it was prior to 2012.
For observers concerned about the spectral flux assignment artifacts, flux appears to be conserved, but additional systematic error at the level of a few percent of the peak emission line flux (Table \[tab:flux\]) may be a concern. In addition, the placement of the artifacts, if they fall on a spectral feature of interest, may impact science measurements. From our investigations, the ultimate cause is likely the method of assigning flux to spaxels (e.g. the white light scans and rectification matrices). @boehle2016upgrade show that the FWHM of individual bright emission lines in the 2D detector image vary by up to a few tenths of a pixel in both the X and the Y direction as a function of lenslet and wavelength. Thus to fully resolve this issue, an implementation of a two-dimensional PSF estimation and spectral extraction algorithm may be necessary, though the task is non-trivial. We would need a 2D estimate of the PSF on the detector for every spaxel and wavelength combination. However, as the internal structure of OSIRIS is fixed, we can currently only sample the 1D LSF. Work is underway to investigate whether existing measurements of the 2D PSF variation in OSIRIS are adequate to improve improve the spectral extraction. Alternatively, the CHARIS IFS on Subaru utilizes a tunable laser to sample the 2D PSF at all wavelengths [@brandt2017data]; something similar may need to be installed on OSIRIS to implement a 2D flux extraction algorithm.
SPATIAL RIPPLING {#sec:spatrip}
================
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Individual spectral channel maps in the reduced OSIRIS cubes show spatial rippling, or a pattern of brighter and fainter spaxels unrelated to incident flux. The spatial rippling is especially apparent for bright emission lines, is wavelength dependent, and changes rapidly with wavelength. Figure \[fig:ripphase\], left panels, shows an example of this rippling. Each panel is a 2D cut at a single wavelength channel in a reduced arc lamp cube. The three channels shown are each centered on a bright emission line in this bandpass.
The rippling pattern is roughly consistent at a single spectral channel across filters and pixel scales, with some magnification, and stays roughly consistent with time. However, the scale of the rippling, or the magnitude difference between the bright and faint spaxels in the pattern, has changed with time. In particular, it was much less evident before the grating upgrade in 2012.
The spatial rippling is caused by the subsampled PSF on the detector. Figure \[fig:ripphase\], right panels, shows the pixel phase of the bright emission lines in the corresponding left panels. These pixel phases, or the position or centeredness of the emission line PSF on a single pixel, are measured in the dispersion direction on the 2D frame, with lighter colors representing more centered emission lines.
The correspondence between the pixel phase of a given emission line and its channel map is due to the narrowness of the PSF on the detector. After the 2012 grating upgrade, the focus on the detector was sharpened along the dispersion direction, leading to a PSF that is subsampled in the dispersion direction. Emission lines that are centered on a pixel have more of their flux assigned to a single spectral channel in the reduced cube, while lines that fall at the edge of a pixel have their light split across two channels. This leads to the rippling in the reduced cube channel maps. Before the grating upgrade, the PSF was broader and was fully sampled by the given pixel scale, so this rippling effect was not present.
![The spatial rippling patterns change from channel to channel. *Top:* Channel maps across a single emission line, at the peak of the line and at 2 wavelength channels away on either side. The change in rippling pattern with wavelength is quite apparent, particularly when comparing the right sections of the top and bottom panels. *Bottom:* Vertical cuts across all three channel maps, taken between columns 50 and 60 (columns have been median combined). The cuts have been normalized for display. The vertical position of the horizontal stripes in each channel are shifted by 1–2 rows with respect to each other.[]{data-label="fig:spatripline"}](spatial_rippling_1line_wcuts.pdf){width="\columnwidth"}
The rippling pattern can affect science data, particularly because the pattern changes rapidly. Figure \[fig:spatripline\] shows three single channel maps, close in wavelength, across one arc lamp emission line. Though the maps are separated by at most 5 wavelength channels, the shift in the rippling pattern is visible by eye. This change of pattern with wavelength induces extra noise into the cube and may affect measurements such as kinematics or channel maps.
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While individual channel maps show a strong rippling pattern, integrated line flux is conserved in data from most epochs. However, data taken in the period from December 2012 to December 2015 is subject to a stronger spatial rippling pattern that does not resolve after integrating over the emission line. Figure \[fig:skyripp\] shows the peak and integrated line flux of an OH sky line in frames taken in 2013 and in 2016. In 2016 (bottom panels), while the rippling pattern is evident in single channel maps, the integrated line maps are smooth. On the other hand, integrated emission line maps from 2013 data (top panels) show spatial rippling in both the single channel and in the integrated flux maps. The cause of the persistent rippling seen in integrated flux maps from 2012 to 2015 is likely due to differences in the PSF width as compared with previous or later periods. The PSF in the dispersion direction is slightly narrower in the period between December 2012 and December 2015 than in the current configuration; during the 2016 detector upgrade, the PSF elongation was rotated so as to be broadened slightly in the dispersion direction. Correspondingly, during the period from December 2012 to December 2015, the PSF perpendicular to the dispersion direction was wider. The confluence of the flux assignment algorithm issues plus the PSF subsampling likely caused the rippling seen in the integrated flux maps.
In most data epochs, the spatial rippling issue is a nuisance but does not generally affect science data or calibrations[^5]. Integrated line maps do not contain a residual rippling pattern, though single channel maps may, if mapping a narrow emission line. If emission lines are intrinsically broadened, as for some science cases, the lines may be sampled sufficiently to produced smooth single channel maps without integration. However, caution must be used in two cases. First, if the sky emission is not subtracted using a separate sky frame, but is instead extracted from a section of the science data cube, the spatially varying rippling pattern will not subtract out and will induce artifacts at the wavelengths of the sky lines. Second, data taken between December 2012 and December 2015 containing narrow emission lines from resolved science targets will show this rippling pattern, even in integrated channel maps. Testing using the sky frames shown in figure \[fig:skyripp\] shows that the standard deviation of an integrated line map relative to the median of the frame is 15% in 2013, versus 5% for the 2016 integrated line map.
COSMIC RAYS IN THE RECTIFICATION MATRICES {#sec:crinrecmat}
=========================================
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The publicly available rectification matrices suffer from inadequate cosmic ray removal in the corresponding white light scans, which results in large flux spikes in the final reduced data cubes (figure \[fig:crspikes\], bottom panel). These spikes occur in just a few spectral channels in a few spaxels, but display flux values up to 10$^8$ DN s$^{-1}$ (both positively and negatively signed), while the maximum real flux in a single spectral channel in one spaxel of a 900 s sky frame is on the order of 1 DN s$^{-1}$. In addition, the saturation limit of the detector is 33,000 DN (pre-2016) or 65,535 DN (2016–present), so such high counts are unphysical.
Prior to 2015, the rectification matrices were created using a single white light scan at each position of the lenslet column mask, disallowing cosmic ray removal by median frame combination[^6]. To remedy the inadequate cosmic ray removal, new rectification matrices were created in 2015 using a set of three white light scans at each lenslet column position. The scans at each lenslet column position are medianed together to robustly remove cosmic rays (Randy Campbell, private communication). The matrix made with the medianed scans produces a data cube that is free from spikes unrelated to instrumental variation. Rectification matrices using the median method were created for each OSIRIS mode in 2015, covering the period between December 2012 and December 2015, and are publicly available on the OSIRIS instrument website[^7]. All new rectification matrices, including for the period beginning in January 2016, are created using this median scan method.
New white light scans and new rectification matrices are created each time the instrument is modified and are unique to its physical setup. Therefore, data taken prior to the physical configuration beginning in December 2012, when this issue was discovered, must be reduced using a rectification matrix that still suffers from cosmic ray effects. Cosmic rays are not easily removed from the original scans, as the real flux peaks can be sharply peaked and sigma-clipping routines remove too much real flux. However, the rectification matrix itself is simply a three-dimensional array and any cosmic rays are easily identified and interpolated over.
BAD PIXEL MASK {#sec:badpix}
==============
We have determined that there are two types of bad pixels that lead to artifacts in the reduced data cubes: bad pixels on the detector (dead and hot pixels) and cosmic ray hits. The number of bad pixels on the detector far outnumber the typical number of cosmic rays in a 600 to 900 s exposure.[^8] Generally about 98 to 99% of bad pixels are static and about 1 to 2% are from cosmic rays. We developed a method for identifying hot or dead pixels on the detector and have included a script in the pipeline to apply the bad pixel mask to raw data. We discuss below our method for identifying hot and dead pixels.
Hot pixels are pixels on the detector that have higher than normal counts compared to a typical pixel given the same incoming flux. These pixels can lead to large artifacts in the reduced data cube, reducing the overall signal to noise. These are a source of systematic error in spectral features.
To detect hot pixels, we created a median of 5 darks with 900 s integration time from 2017 Sep 2. Using this median dark, we identified pixels that significantly deviate from the median value of pixels in the dark frame. We developed two hot pixel masks, one with a threshold of 50$\sigma$ above the median value and one with a threshold of 15$\sigma$ above the median value. At the 50$\sigma$, about 4000 hot pixels are detected (or approximately 0.1% of total detector pixels), while at 15$\sigma$ about 10,000 hot pixels are detected (0.24%). The median dark frame and a sample hot pixel mask are shown in figure \[fig:hotpix\].
![Example of the hot pixel mask (left) generated from the median dark frame (right). These hot pixels are significantly higher compared to the median dark pixel value. Note that the discolored pixels in the bottom left of the median dark frame are dead pixels and are masked out in a separate step.[]{data-label="fig:hotpix"}](hot_pixel_example.pdf){width="\columnwidth"}
We tested this mask by applying it to raw data. After reducing the data cube, we compared it to the cube created without the mask. These tests show that the mask does indeed reduce the number of large single-pixel spikes in the final data cubes. The tests also show that the bad pixel mask introduces some artifacts. A large majority of these artifacts are at a much lower level than the pixel values that exist when the bad pixel mask is not applied. In general, the artifacts that are removed are at least 10 times larger than the artifacts that are created. In addition, using the bad pixel mask removes twice as many artifacts as it creates. Overall, using the new bad pixel mask reduces noise in reduced test data by roughly 5%. This bad pixel mask can now be optionally applied by users of the pipeline.
WAVELENGTH SOLUTION {#sec:wavesol}
===================
The wavelength solution is used to assemble the wavelength vector on a spaxel-by-spaxel basis; it converts a given x-position on the detector to a wavelength in the assembled data cube. The wavelength solution is determined empirically by calculating a third-order polynomial fit to the positions of arc lamp emission lines in each spaxel on the detector image. This fit is then used when interpolating the detector flux into wavelength space in the final reduced cube. In addition to calibration lamp measurements, corrections from instrumental effects such variations in the wavelength solution with temperature are also applied to the wavelength solution on each specific data cube.
Scientifically, accurate wavelength solutions are important for high S/N radial velocity measurements. For example, measurements of the orbits of binaries or stars at the Galactic center requires an accurate wavelength solution to derive physical properties such as the stellar mass or the mass of the supermassive black hole [e.g. @2016ApJ...830...17B]. Currently, the accuracy of the wavelength solution in the Kn3 filter, where the Br$\gamma$ line is located, is about 0.1Å, or 2 km/s at Br$\gamma$. As a function of resolution, this is a roughly 3% error.
The advantage of a Nasmyth mounted instrument like OSIRIS is that calibrations such as the wavelength solution are generally very stable, even over a period of years, if the instrument is unopened. However, when the instrument is opened for servicing, the wavelength solution generally needs to be rederived. The typical shift in the wavelength solution after a cycle of warming, opening, servicing, and cooling the instrument is 2–4 Å.
For example, in the most recent opening of OSIRIS for an upgrade in mid-2017, OH emission lines were shifted by 2.8 Å from their expected wavelengths using test data taken in the Kn3 35 mas mode. We analyzed and re-derived the wavelength solution for affected data using a standard method involving arc lamp emission line data. The new derived wavelength solution is applicable for data taken after 2017 May 9.
To verify an improvement in the new wavelength solution, we measured offsets between the new wavelength solution and the known position of OH lines in sky frames. Five filters (Kbb, Kn3, Kn1, Jn1, and Hn4) were examined as were two scales (35 and 50 mas). Sky frames in all five filters were used with the 50 mas scale, while two filters, Kbb and Kn3, were also examined with the 35 mas scale.
For each sky frame, multiple bright OH lines were selected. For each line, the measured wavelength in a given sky frame was taken as the average value for the central 182 lenslets and the error is taken as the standard deviation across the 182 lenslets. The results are shown in figure \[fig:waveoffset\] as a function of vacuum wavelength and reported in table \[tab:waveoffset\]. The resulting offsets are suggestive of a scale dependency, with the offsets measured in sky frames taken with the 35 mas scale consistent with zero, while those with a scale of 50 mas have a consistent offset between measured and vacuum wavelength of $\sim$0.4 Å. This is most evident for the two filters, Kbb and Kn3, tested in both scales.
In general, the wavelength solution has been re-derived whenever the instrument has been opened for major servicing. See table \[tab:wavesol\] for a listing of the available wavelength solutions (though note that the DRP will automatically select the appropriate wavelength solution to use based on the given observation’s date). However, it is unclear if the wavelength solution has always been verified or re-derived following an instrument opening for minor servicing. Going forward, we recommend verifying the accuracy of the wavelength solution after every instrument opening, even for minor service work, and re-deriving it if necessary.
![Wavelength dependence of the offset of measured OH lines from vacuum wavelength for five broadband filters, after applying the new wavelength solution, valid for data taken 2017 May 9 or later.[]{data-label="fig:waveoffset"}](wavlength_offset.pdf){width="\columnwidth"}
[llllllc]{}
Kbb & 50 & 1965-2381 & 2017-08-12 & 0.327 & 0.127 & 7\
Kbb & 50 & 1965-2381 & 2017-08-12 & 0.320 & 0.136 & 7\
Kbb & 35 & 1965-2381 & 2017-05-18 & 0.020 & 0.153 & 7\
Kn3 & 50 & 2121-2229 & 2017-09-02 & 0.413 & 0.112 & 5\
Kn3 & 35 & 2121-2229 & 2017-05-17 & 0.053 & 0.125 & 5\
Hn4 & 50 & 1652-1737 & 2017-07-17 & 0.357 & 0.199 & 5\
Hn4 & 50 & 1652-1737 & 2017-07-17 & 0.345 & 0.176 & 5\
Kn1 & 50 & 1955-2055 & 2017-08-12 & 0.337 & 0.194 & 3\
Jn1 & 50 & 1174-1232 & 2017-08-12 & 0.443 & 0.093 & 3\
\[tab:waveoffset\]
[ccc]{} 1 & 2005 Feb 22 & 2006 Feb 22\
2 & 2006 Feb 23 & 2009 Oct 04\
3 & 2009 Oct 05 & 2012 Jan 03\
4 & 2012 Jan 04 & 2012 Nov 09\
5 & 2012 Nov 10 & 2015 Dec 31\
6 & 2016 Jan 01 & 2017 May 08\
7 & 2017 May 09 & 2018 Mar 16\
8 & 2018 Mar 16 & present\
\[tab:wavesol\]
RECOMMENDATIONS {#sec:rec}
===============
Many of the issues we’ve outlined here for the OSIRIS DRP also have the potential to be of concern for data pipelines for the next generation IFSs on extremely large telescopes, such as IRIS [e.g. @2016SPIE.9908E..1WL; @2016SPIE.9913E..4AW] on TMT. These upcoming spectrographs are in their planning phases and both operational daytime and nightime calibrations need to be well-planned, as well as proper maintenance of the data reduction pipeline. We discuss some of the lessons learned here and how they would apply to future IFSs.
We have shown the potential risks of closely packed spectra on the detector with sufficient blending in their PSFs. The deconvolution of raw spectral data is non-trivial, especially when spectra are spaced by a width similar to the FWHM in the spatial direction. In this situation, a 1D spatial rectification algorithm may work with ideal data, but can fall short with real data. Flux may be misassigned (§ \[qsodata\]) or cause artifacts in neighboring spaxels (§ \[arclamps\]). Ultimately, a 1D estimation of the spectral PSF in the spatial direction may not be an accurate enough representation of real data, particularly for point sources. A 2D estimation of the PSF for every spaxel and wavelength may be necessary to fully resolve this issue. The ability to calibrate the 2D spectral profile of individual spaxels along the detector versus wavelength would greatly benefit post-processing. Calibration routines and analysis methods that explore this 2D spectral extraction ability in next generation IFSs should be further explored.
A similar issue involves the PSF of a point source on the detector in the spectral direction. Sharpening the instrumental PSF allows for higher spectral resolution in science observations, but can induce large-scale patterns. In § \[sec:spatrip\], we showed that sharpening the spectral PSF to the point of undersampling introduced a spatial rippling effect. This pattern is most evident in single channel maps in reduced data cubes, but when in combination with inadequate spatial rectification, the same pattern can appear in integrated line maps, leading to larger uncertainties in integrated line quantities. The optical design of the spectrograph should be optimized with both spectral width and dispersion that best matches the detector pixel size to perform a high signal-to-noise spectral extraction. This means that the optical engineers need to consider the spectral extraction routines during their layouts.
In § \[sec:crinrecmat\], we discussed the discovery of improper cosmic ray removal in the rectification matrix calibration files. These noisy spikes in the calibration files propagated through the spectral extraction routine to the reduced science data products. Care should be taken to clean the calibration files as thoroughly as science data before use.
A critical factor in our ability to diagnose and correct issues is the availability of calibration data. We find that data beyond the standard calibration data such as arc-lamp observations and white light scans were necessary in order to assess some of the effects we observed:
- *Single-column illuminated arc lamps* are needed to evaluate the efficiency of the algorithm that assigns flux to individual spaxels
- *Single-column illuminated sky observations* are essential for evaluating whether there are relative shifts between the location of the spectra on the detector from the white-light scans compared to on-sky data
- *A number of sky observations across different filters and plate scales* are necessary to characterize the wavelength solution
We recommend that observatories and instrument teams consider the value of additional calibrations to maintain and improve the data products. These calibration data are necessary whenever the instrument hardware is modified and may also be needed whenever the instrument is warmed and opened, even if the major instrument components are unchanged.
CONCLUSIONS {#sec:conc}
===========
We present the first characterization of the OSIRIS DRF using on-sky and calibration data. Our study was begun in response to issues observers had found in their reduced data. As a result of the efforts detailed in this work, the following improvements have been seen in reduced data:
- We showed that spatial flux assignment artifacts are a result of a pupil mismatch with the 01 plate scale, and demonstrated a technique to reduce these artifacts
- We found that the spectral flux assignment artifacts are due to differences between the PSF of a point source and that of the white light calibration data, and were exacerbated by the narrowness of the PSF in the dispersion direction. Due to this analysis, the dispersion-direction PSF was broadened during a subsequent hardware upgrade in 2016
- We found evidence of spatial rippling in individual channel maps, due to PSF undersampling. In addition to the spectral flux assignment artifacts, this was also used as evidence to broaden the PSF during the 2016 hardware upgrade
- We demonstrated that large spikes in reduced data cubes were due to uncorrected cosmic rays in calibration data. We retook the appropriate calibration data when possible and released new rectification matrices without cosmic ray contamination. We also implemented procedures to remove cosmic ray contamination in future rectification matrices
- We showed that the vast majority of uncorrected “cosmic rays” in reduced data cubes were due to bad detector pixels. We released a new bad pixel mask to remove these in reduced data
- We found inconsistencies in the existing wavelength solution and rederived it to better match on-sky data. We implemented procedures to check the wavelength solution more often going forward
Though OSIRIS has been a very productive instrument on both Keck-I and Keck-II for over a decade, its DRP had never been fully characterized using on-sky data. This is almost entirely due to the complexity of the necessary analysis. Our own experiences as members of the OSIRIS Pipeline Working Group showed that maintaining a pipeline of this size and complexity requires a dedicated team of people familiar with the hardware but focused on the software and on the quality of the reduced data. While it’s unlikely that observatories, focused on operations and hardware support, can maintain such dedicated software teams, it’s vital for the software team to work closely with the observatory instrument team. This eases access to calibration data for the software team, reduces duplicate efforts, and enables knowledge sharing. Finally, though the Keck Observatory Archive[^9] and similar archives offer superb access to years of on-sky data, calibration data is generally not as well archived. However, these data are necessary for this type of work, and it is especially useful to have multiple years of consistent on-sky and calibration data sets to compare.
Our work has implications for teams developing future IFSs. The complexity of IFS raw data outputs means that upcoming IFSs should consider the ease of data reduction from the beginning with their projected spectral overlap, as well as planning for calibration and processing algorithms. Incorporating the lessons learned from the OSIRIS DRP will greatly improve the ease of data reduction and resulting data quality for future IFS instrumentation.
The data presented herein were obtained at the W.M. Keck Observatory, which is operated as a scientific partnership among the California Institute of Technology, the University of California and the National Aeronautics and Space Administration. The Observatory was made possible by the generous financial support of the W.M. Keck Foundation. The authors wish to recognize and acknowledge the very significant cultural role and reverence that the summit of Maunakea has always had within the indigenous Hawaiian community.
[^1]: https://github.com/Keck-DataReductionPipelines/OsirisDRP/blob/master/OSIRIS\_Manual\_v4.2.pdf
[^2]: https://github.com/Keck-DataReductionPipelines/OsirisDRP
[^3]: http://www2.keck.hawaii.edu/inst/nirspec/lines.html
[^4]: In 2006, the 0035 and 005 scale pupil masks were redesigned, fabricated, and installed to match the mean aperture size of Keck (10.0 m) to lower the thermal background. These new pupil masks reduced the thermal background by $\sim$65% with respect to the original pupil masks. In March 2008, we conducted a servicing mission for OSIRIS to install duplicate K broadband (Kbb) and narrowband (Kn3, Kn4, Kn5) filters in the filter wheel with new smaller pupil masks attached (9 m, effective). These K-band filters with their pupil masks are optimized to only work with the 01 lenslet scale, and are referred to as Kcb, Kc3, Kc4, and Kc5 within the software, where “c” stands for “coarse.”
[^5]: Note that the wavelength solution is calculated using 2D data, as described in § \[sec:wavesol\], and is thus unaffected by this rippling effect.
[^6]: Single scans at each position were used because taking these scans is a very time-consuming process. A scan must be taken for every lenslet column for each of the 88 filter and plate scale modes. In order to achieve an adequate S/N, scans at the smaller plate scales and with some filters require exposure times of up to 30s.
[^7]: http://tkserver.keck.hawaii.edu/osiris/
[^8]: Hot pixels are expected to be less of a concern for the white light scans used for the rectification matrices. The white light scan exposures are short; the exposure time varies by mode, but is generally on order of a few seconds. The white light lamps are also bright, and thus the hot pixels are expected to be much fainter than the incident white light lamp flux. The cosmic rays are removed from the white light exposures as outlined in § \[sec:crinrecmat\].
[^9]: https://www2.keck.hawaii.edu/koa/public/koa.php
|
---
abstract: 'We analyze the dynamics of a Bose-Einstein condensate undergoing a continuous dispersive imaging by using a Lindblad operator formalism. Continuous strong measurements drive the condensate out of the coherent state description assumed within the Gross-Pitaevskii mean-field approach. Continuous weak measurements allow instead to replace, for timescales short enough, the exact problem with its mean-field approximation through a stochastic analogue of the Gross-Pitaevskii equation. The latter is used to show the unwinding of a dark soliton undergoing a continuous imaging.'
address: |
${}^1$ Los Alamos National Laboratory, Los Alamos, New Mexico 87545\
${}^2$ Intytut Fizyki Uniwersytetu Jagiellońskiego, ul. Reymonta 4, 30-059 Kraków, Poland\
${}^3$ Dipartimento di Fisica “G. Galilei”, Università di Padova, Padova 35131, Italy\
and INFM, Sezione di Roma “La Sapienza”, Roma 00185, Italy
author:
- 'Diego A. R. Dalvit${}^{1}$, Jacek Dziarmaga$^{1,2}$, and Roberto Onofrio$^{1,3}$'
date: 'December 12, 2001'
title: |
Continuous quantum measurement of a Bose-Einstein condensate:\
a stochastic Gross-Pitaevskii equation
---
Introduction
============
The interplay between quantum and classical descriptions of the physical world and the role of the measurement process are still at the heart of the understanding of quantum mechanics [@Wheeler]. The related theoretical debate has been greatly enriched in the last decades by the realization of new experimental techniques aimed at producing quantum states without classical analogue, such as entangled or squeezed states, or to explore phenomena which are intrinsically quantum mechanical, like quantum jumps. All this occurred also having in mind practical implications, such as the improvement of sensitivity of various devices operating at or near the quantum limit [@Braginsky; @Santarelli].
Recently, the production of a novel state of matter - Bose-Einstein condensates of dilute atomic gases - has opened a new road to explore macroscopic quantum phenomena with the precision characteristic of atomic physics [@Inguscio]. Bose-Einstein condensates are naturally produced by cooling down atomic gases at ultralow temperature with the phase transition occurring in the 100 nK - 1$\mu$K temperature range, [*i.e.*]{} when the thermal de Broglie wavelength becomes comparable to the average spacing between the atoms of a dilute (peak density $10^{13} - 10^{15}$ atoms/cm${}^3$) trapped gas. Usually samples of Bose-Einstein condensates are made by $10^3-10^8$ atoms in the case of alkali species [@Inguscio], and $10^9$ or more in the case of hydrogen [@Kleppner]. Their intrinsically small heat capacity does not allow for a direct manipulation and probing with material samples, such as microtips or nanostructures, since the thermal contact with the latter will induce a sudden vaporization of the Bose sample. Thus, manipulation and probing of Bose-Einstein condensates has been achieved so far only by using light beams. These probes can be classified according to the resonant or nonresonant nature of their interaction with the measured atomic sample. In the former case, the condensate interacts with a laser beam resonant (or close to resonance) with a selected atomic transition. The output beam is attenuated proportionally to the optical thickness of the condensate (also called column density), [*i.e.*]{} the condensate density integrated along the line of sight of the impinging beam. The absorption of the photons leads to recoil of the atoms then strongly perturbing the condensate. For typical values of intensity and duration of the probe light the condensate is strongly heated and a new replica has to be produced to further study its dynamics. From the viewpoint of quantum measurement theory this measurement is of type-II since it destroys the state of the observed system and forbids the study of the dynamics of a single quantum system [@Pauli]. An alternative technique, called dispersive imaging, allows for repeated measurements on a Bose-Einstein condensate and, as an extreme case, its continuous monitoring. In this measurement scheme off-resonance light is scattered by the condensate, thereby locally inducing optical phase-shifts which can be converted into light intensity modulations by homodyne or heterodyne techniques, for instance by using phase-contrast [@Andrews] or interference techniques [@Kadlecek]. Since the laser beam is off-resonant the absorption rate is small and the heating of the condensate is accordingly small. Thus, multiple shots of the same condensate can be taken - a type-I measurement in the quantum measurement theory language - allowing the study of the dynamics on the same sample. This has allowed to overcome the unavoidable shot-to-shot fluctuations always present in the production of different samples of Bose-Einstein condensates. Several phenomena whose observation rely on imaging at high accuracy the condensate, such as its formation in non-adiabatic conditions [@Miesner], short [@Andrews1] and long [@Stamper] wavelength collective excitations, superfluid dynamics [@Onofrio], vortices formation and decay [@Matthews; @Abo], can be successfully studied with this technique.
The repeated nondestructive monitoring lends itself to another question: [*is the measurement process influencing the dynamics of the condensate?*]{} Answers to this question are important also for disentangling the intrinsic condensate dynamics from the artifacts induced by the underlying measurement process. As we will see in this paper, the effect of the measurement can also be intentionally amplified to allow for unusual manipulation of the condensate itself. Dispersive imaging, in its more idealized form, is preserving the number of atoms and therefore represent a particular example of quantum nondemolition (QND) measurement [@Braginsky; @Caves; @Bocko]. We know that even quantum nondemolition measurements affect the state of the observed system, their unique feature being that the nondemolitive observable maintains the same [*average*]{} value, albeit the probability distribution of its outcome can be affected as well as the average values of all the conjugate observables. Thus, we do expect that during the nondemolitive measurement of the condensate atom number there will be a measurement-induced nontrivial dynamics for both the variance of the monitored observable and the average values of all conjugate observables. Besides gaining insight into the dynamics of the measurement process, our model allows for the description of the [*irreversible*]{} driving of the condensate toward nonclassical states.
In this paper we try to answer the question formulated above by building a realistic model for dispersive measurements of the condensate, extending the results reported in [@Dalvit] to the weak measurement regime. The plan of the paper is as follows. In Section II we introduce the dispersive coupling between atoms and light and derive the reduced master equation for the Bose condensate tracing out the variables of the electromagnetic field degrees of freedom. Under controllable approximations we obtain a Lindblad equation which allows to estabilish the rates for phase diffusion and depletion of the condensate during the dispersive imaging. In Section III we unravel the Lindblad equation in terms of a stochastic differential equation that, in the unmeasured case, corresponds to the description of a single N-body wavefunction. A solution of the stochastic N-body equation is discussed in the limit of strong continuous measurement, leading to the squeezing of number fluctuations, the main result described in [@Dalvit]. In the opposite limit of weak measurement and for an initial mean-field state this stochastic equation becomes the stochastic counterpart of the Gross-Pitaevskii equation, as discussed in Section IV. Its limit of validity is discussed in Section V by comparing its evolution for various parameters versus the exact evolution in the simple situation of a two mode system schematizing a condensate in a double well potential. This allows us to analyze, in Section VI, the effect of the measurement on the evolution of a condensate initially prepared in a soliton state. More general considerations on the potentiality of such an approach and its consequences are finally outlined in the conclusions.
Master equation for dispersive imaging of a Bose condensate
===========================================================
Our main goal in this Section is to include the atom-photon interaction present in the dispersive imaging of a Bose-Einstein condensate into its intrinsic dynamics. First attempts in this direction have been discussed in the prototypical situation of a two mode condensate in [@Walls; @Milburn]. Let us start the analysis with the effective interaction Hamiltonian between the off-resonant photons and the atoms, written as $$H_{\rm int}= {{\epsilon_0 \chi_0} \over 2}
\int d^3x~ n({\bf x}) : {\bf E}^2 : ,
\label{Intham}$$ where $n$ is the density operator of the atomic vapor, and ${\bf E}$ the electric field due to the intensity $I$ of the incoming light. The coefficient $\chi_0$ represents the effective electric susceptibility of the atoms defined as $\chi_0=\lambda^3 \delta/2 \pi^2 (1+\delta^2)$, where we have introduced the light wavelength $\lambda$ and the light detuning measured in half-linewidths $\Gamma/2$ of the atomic transition, $\delta=(\omega-\omega_{\rm at})/(\Gamma/2)$.
We express the electric field in terms of creation and annihilation operators. In the Coulomb gauge it takes the form $${\bf E}({\bf x},t) = i \sum_{{\bf k}}
\sqrt{\frac{\hbar \omega_{\bf k}}{2 \epsilon_0 L^3}}
( a_{\bf k} e^{-i c k t + i {\bf k} {\bf x} } -
a_{\bf k}^{\dagger} e^{i c k t - i {\bf k} {\bf x}}) ,$$ where $\omega_{\bf k} = c |{\bf k}|$, $[a_{\bf k},a_{\bf k'}^{\dagger}] =
\delta_{{\bf k},{\bf k'}}$, and $L^3$ is the quantization volume.
Equation (\[Intham\]) allows us to write the reduced master equation for the atomic degrees of freedom by a standard technique, [*i.e.*]{} by tracing out the photon degrees of freedom [@Carmichael]. The decoupling between the two relevant timescales for the photons (settled by the the lifetime of spontaneous emission, of order of tens ns) and for the atoms (related to the oscillation period in the trapping potential, of the order of ms), allows to use the Born-Markov approximation. Thus we get the master equation for the reduced density matrix $\rho$ of the condensate that, in the interaction picture, is written as
$$\frac{d \rho}{dt} = \frac{i}{\hbar} {\rm Tr}_{\rm R}
[ \rho(t) \otimes \rho_{\rm R}(t), H_{\rm int}] -
\frac{1}{\hbar^2} {\rm Tr}_{\rm R}
[H_{\rm int} (t), [ \int_{-\infty}^t dt' H_{\rm int}(t'),
\rho(t) \otimes \rho_{\rm R}(t) ]] .
\label{ME}$$
The last term on the righthandside contains two different contributions both of Lindblad type, $L_1 \rho $ and $L_2 \rho$. As we will see soon, the former preserves the number of atoms in the condensate and is responsible for phase diffusion phenomena, while the latter changes the number of atoms leading to its depletion.
Let us first concentrate on phase-diffusion which is a number conserving mechanism. To calculate it we insert the interaction Hamiltonian into the last term of Eq.(\[ME\]), with $n$ the condensate density operator. By introducing the Fourier transform of the density operator such that
$$n({\bf x}) = \sum_{\bf q} e^{i {\bf q}{\bf x}}
\frac{\tilde{n}({\bf q})}{\sqrt{L^3}} ,$$
we obtain $$\begin{aligned}
L_1 \rho &=& \frac{\pi \chi_0^2}{2 L^3 c}
\sum_{{\bf k}, {\bf p}} \sqrt{\omega_{{\bf k}} \omega_{{\bf p}}}
\delta(k-p) \sum_{{\bf k'}, {\bf p'}} \sqrt{\omega_{{\bf k'}}
\omega_{{\bf p'}}} e^{i c t (k'-p')} \times \nonumber \\
&&
\left[
\tilde{n}({\bf k'}-{\bf p'}) \tilde{n}({\bf k}-{\bf p}) \rho
\langle a_{\bf k'}^{\dagger} a_{\bf p'} a_{\bf k}^{\dagger}
a_{\bf p} \rangle -\tilde{n}({\bf k'}-{\bf p'}) \rho \tilde{n}
({\bf k}-{\bf p}) \langle a_{\bf k}^{\dagger} a_{\bf p}
a_{\bf k'}^{\dagger} a_{\bf p'} \rangle
\nonumber \right. \\
&& \left.
- \tilde{n}({\bf k}-{\bf p}) \rho \tilde{n}({\bf k'}-{\bf p'})
\langle a_{\bf k'}^{\dagger} a_{\bf p'} a_{\bf k}^{\dagger} a_{\bf p}
\rangle
+
\rho \tilde{n}({\bf k}-{\bf p}) \tilde{n}({\bf k'}-{\bf p'})
\langle a_{\bf k}^{\dagger} a_{\bf p} a_{\bf k'}^{\dagger}
a_{\bf p'} \rangle
\right] ,\end{aligned}$$ where $\langle \dots \rangle = {\rm Tr}_{\rm R}[\rho_{\rm R} \ldots ]$. The photons are assumed to be in a coherent plane wave state with momentum along the impinging direction, corresponding to a wavevector ${\bf k}=k_0 \hat{z}$ orthogonal to the imaging plane $x-y$. Hence
$$\langle a_{\bf k'}^{\dagger} a_{\bf p} \rangle = \frac{L^3 I} {c^2 \hbar k_0}
\delta_{{\bf k'},{\bf p}} \delta_{{\bf p},{\bf k}_0},$$
where we have written the mean numbers of photons in mode $k_0$ in terms of the intensity of the incoming beam. Expressing the expectation values in normal ordering and using the fact that phase-diffusion processes conserve the number of bosons in the condensate, it is possible to show that all normal ordered expectation values involving four operators cancel exactly, obtaining
$$L_1 \rho = \frac{ \pi \chi_0^2 I k_0 }{2 \hbar c}
\left( \frac{L}{2 \pi} \right)^3
\int d^3k\; \delta(| {\bf k} + {\bf k_0}| - k_0 )\;
[\tilde{n}(- {\bf k}), [ \tilde{n}({\bf k}), \rho]],$$
where we have used the continuum limit $\sum_{\bf k} \rightarrow (L/2
\pi)^3 \int d^3k$. Unless tomographic techniques are used as for instance in [@Andrews2], the image results from a projection of the condensate onto the $x-y$ plane, by integrating along the $z$ direction. This demands to project the dynamics of the condensate into the imaging plane. In order to write a closed 2D master equation to describe the $x-y$ dynamics we assume the condensate wavefunction to be factorizable as $\psi(x,y,z)=\phi(x,y) \Lambda(z)$. Such factorization holds if the confinement in the $z$-direction is strong enough to make the corresponding mean-field energy negligible with respect to the energy quanta of the confinement, [*i.e.*]{} $\hbar \omega_z >> g \tilde{\rho}$, where $g=4 \pi \hbar^2 a/m$, with $a$ the s-wave scattering length, $\tilde{\rho}$ the condensate density, and $\omega_z$ the angular frequency of the confinement harmonic potential along the $z$ direction, as recently experimentally demonstrated in [@Gorlitz]. We write $\tilde{n}({\bf
k}) = \tilde{n}({\bf k}_{\perp}) \tilde{n}(k_z)$ and we will use a Gaussian ansatz for the density profile along the $z$ direction, namely $$\tilde{n}(k_z) = \sqrt{\frac{2 \pi}{L}} e^{- \frac{\xi^2
k_z^2}{2}} ,$$ where $\xi$ is the lengthscale of the condensate in the $z$ direction, the width of the Gaussian state $\Lambda(z)$ under the abovementioned approximation. The effective 2D nonlinear coupling strength is $g_{2D}=g \int dz |\Lambda(z)|^4 =g
\sqrt{\pi}/\xi$. Consequently
$$L_1 \rho =
\frac{\pi \chi_0^2 I k_0}{2 \hbar c}
\left( \frac{L}{2 \pi} \right)^2
\int d^2 k_{\perp}\; [ \tilde{n}(-{\bf k}_{\perp}),
[\tilde{n}({\bf k}_{\perp}), \rho]]
\left(\frac{L}{2 \pi}\right)
\int dk_z\; \tilde{n}^2(k_z)
\delta[|{\bf k} + {\bf k}_0| - k_0] .$$
By assuming that the typical length of the BEC in the $z$ direction is much larger than the wavelength of the incoming laser, i.e., $\xi \gg
\lambda$, we can calculate the value of the last integral, and it is equal to $\exp(- \xi^2 {\bf k}_{\perp}^4 / 4 k_0^2)$. Our final result for the phase-diffusion contribution to the reduced master equation in the imaging plane is written as
$$L_1 \rho = \int d^2r_1 \int d^2r_2\; K({\bf r}_1-{\bf r}_2)\;
[ n({\bf r}_1),[n({\bf r}_2),\rho]] ,
\label{phasediff}$$
where $n({\bf r})=\Psi^{\dagger}\Psi({\bf r})$ is the 2D density operator, and $K$ is the measurement kernel $$K({\bf r})= {{\pi \chi_0^2 k_0 I} \over 2 \hbar c}
\int d^2k\; \exp(-\xi^2 k^4/4 k_0^2 + i {\bf k} {\bf r}) .
\label{KERNEL}$$ Eq. (\[phasediff\]) preserves the total number of atoms, and corresponds to a quantum nondemolition coupling between the atom and the optical fields [@Walls; @Milburn; @Onofrio1; @Li; @Leonhardt]. If the measurement kernel were a local one, $K({\bf r}_1-{\bf r}_2) \simeq \delta({\bf r}_1
-{\bf r}_2)$, Eq.(\[phasediff\]) would reduce to a Lindblad equation for the measurement of an infinite number of densities $n({\bf r})$. This assumes that no spatial correlation is established by the photon detection. However, the ultimate resolution limit in the imaging system depends on the photon wavelength, regardless of the pixel density of the detecting camera. The resolution lengthscale follows from Eq.(\[KERNEL\]) as a width of the kernel
$$\Delta r=(2\pi^2\xi/k_0)^{1/2}=(\pi \xi \lambda)^{1/2}\;,$$
the geometrical average of the light wavelength and the condensate thickness $\xi$. Eq. (\[phasediff\]) can then be rewritten as $L_1 \rho =
\gamma_1 [n,[n,\rho]]$, where $\gamma_1$ is the phase diffusion rate, given by
$$\gamma_1 = \frac{\pi \chi_0^2 k_0 I}{2 \hbar c}
\int d^2k\; e^{- \frac{\xi^2 k^4}{4 k_0^2}}
|\phi({\bf k})|^2 |\phi(-{\bf k})|^2 .$$
We estimate its magnitude assuming a Gaussian profile in the x-y plane, i.e., $|\phi({\bf k})|^2 = e^{-\alpha^2 k^2/2}$, obtaining
$$\gamma_1 = \frac{ \pi^{5/2} \chi_0^2 k_0^2 I}{2 \hbar c \xi}
e^{\frac{\alpha^4 k_0^2}{\xi^2}} \left[ 1 - {\rm Erf} \left(
\frac{\alpha^2 k_0}{\xi} \right) \right]
\approx \frac{\pi^2 \chi_0^2 k_0 I}{2 \hbar c \alpha^2} ,
\label{gamma1}$$
where the last step holds for a well localized condensate, [*i.e.*]{} $\alpha ^2 k_0 / \xi \gg 1$.
Let us now calculate the depletion contribution to the master equation for the condensate. We split the field annihilation operator into a term describing the condensate and another associated to the non-condensed particles, [*i.e.*]{}, $\psi=\psi_{\rm C}
+ \psi_{\rm NC}$. We shall assume that the non-condensed particles belong to the continuum, so that their spectrum is the one of a free-particle $\hbar \Omega_{\bf q} = \hbar^2 q^2/2 m$. Indeed, photons have large momenta with respect to the momenta of trapped atoms, so even if a small percentage of the photon momentum is absorbed by the atom, the atom is promoted into a high energy, unbounded state. Let us expand both field operators in terms of annihilation operators as
$$\begin{aligned}
\psi_{\rm C}({\bf x},t) &=& b_{\rm C} \phi_{\rm C}({\bf x}, t)
\nonumber \\
\psi_{\rm NC}({\bf x},t) &=& \frac{1}{\sqrt{L^3}}
\sum_{\bf q} e^{-i \Omega_{\bf q} t + i {\bf q} {\bf x}}\; b_{\rm q} ,\end{aligned}$$
where $\phi_{\rm C}$ is the condensate wave function. We get $$\begin{aligned}
L_2 \rho &=& \frac{\pi \chi_0^2}{8 c L^6}
\sum_{{\bf k}{\bf p}{\bf q}} \sqrt{\omega_{\bf k} \omega_{\bf p}}
\sum_{{\bf k}'{\bf p}'{\bf q'}} \sqrt{\omega_{\bf k}' \omega_{\bf p}'}
\delta(\Omega_{{\bf q}'} - c k' + c p') \times \nonumber \\
&& \left\{
e^{i t (\Omega_{{\bf q}} - c k + cp )} \tilde{\phi}_{\rm C}(q+p-k)
\tilde{\phi}^*_{\rm C}(q'+p'-k')
{\rm Tr}_{\rm R} [ a^{\dagger}_{\bf p} a_{\bf k} b^{\dagger}_{\bf q} b_c ,
[a^{\dagger}_{{\bf k}'} a_{{\bf p}'} b_{{\bf q}'} b^{\dagger}_{\rm C}, \rho]]
\right. \nonumber \\
&& \left.
+
e^{-i t (\Omega_{{\bf q}} - c k + cp )} \tilde{\phi}^*_{\rm C}(q+p-k)
\tilde{\phi}_{\rm C}(q'+p'-k')
{\rm Tr}_{\rm R}[ a^{\dagger}_{\bf k} a_{\bf p} b_{\bf q} b^{\dagger}_c ,
[a^{\dagger}_{{\bf p}'} a_{{\bf k}'} b^{\dagger}_{{\bf p}'} b_{\rm C}, \rho]]
\right\} .\end{aligned}$$ Here the trace is taken over the reservoir of the condensate, which in this case consists of the non-condensed particles and the photons. The above expression contains depletion processes, in which a photon interacts with a particle in the condensate and, as a result, that particle is kicked out of the condensate. It also contains feeding processes, in which the reverse mechanism may take place. When one assumes that in the initial state of the non-condensate plane waves are empty, only the depletion process is relevant. In this hypothesis we find $$L_2 \rho = \gamma_2 \left(- b_{\rm C} \rho b^{\dagger}_{\rm C} + \frac{1}{2}
\left\{ b^{\dagger}_{\rm C} b_{\rm C}, \rho \right\}
\right) ,$$ where $\gamma_2$ is the depletion rate $$\gamma_2 = \frac{\pi \chi_0^2 I}{4 \hbar c L^3}
\left( \frac{L}{2 \pi} \right)^6
\int d^3p d^3q \;
\omega_{\bf p} \;
| \tilde{\phi}_{\rm C}({\bf q}+{\bf p}-{\bf k}_0) |^2 \;
\delta(\Omega_{\bf q} - c k_0 + c p) .$$ We evaluate $\gamma_2$ in the thick condensate limit ($\xi \gg k_0^{-1}$) by approximating $| \tilde{\phi}_{\rm C}({\bf q}+{\bf p}-{\bf k}_0)|^2 \approx
(2 \pi/L)^3 \delta({\bf q}+{\bf p}-{\bf k}_0)$, and then, by using the fact that $\Omega_{\bf q} \ll c k_0$, we obtain $$\gamma_2 = \frac{\chi_0^2 k_0^3 I}{8 \pi \hbar c} .
\label{gamma2}$$ From Eqs.(\[gamma1\],\[gamma2\]) we see that the depletion rate is much bigger than the phase diffusion rate $\gamma_2/\gamma_1 = \alpha^2
k_0^2/4 \pi^3 \gg 1$, in accordance with Ref.[@Leonhardt]. We estimate the magnitude of both rates using the following parameters for the condensate and its imaging, relevant for the case of $^{87}{\rm Rb}$: $\lambda=780{\rm nm}$, $\chi_0=10^{-23}{\rm m}^3$, laser intensity $I=10^{-4} {\rm mW}/{\rm cm}^2$, and a typical size in the x-y plane of $\alpha = 10 \mu{\rm m}$. Then $\gamma_1 = 10^{-6} {\rm s}^{-1}$ and $\gamma_2 = 10^{-5} {\rm s}^{-1}$, corresponding to phase diffusion and depletion times of $t_1 = \gamma_1^{-1} = 10^5 {\rm s}$ and $t_2=\gamma_2^{-1} = 10^4 {\rm s}$, respectively. Although the depletion rate is larger that the phase diffusion rate, this last process can dominate because of their different scaling with the total number of condensed particles in the master equation. Indeed, the first is linear in $N$, whereas the second one is quadratic, so for large number of particles (such as $N=10^7$) phase diffusion occurs on a faster timescale than depletion. For this reason we will focus in the following on the phase diffusion contribution alone.
Strong measurement: measurement-induced number squeezing
========================================================
We will consider in the following the effect of strong measurements on the quantum state of the condensate. This has already been described in detail elsewhere [@Dalvit], thus here we only summarize the main results and their link to the following considerations. By neglecting the depletion term, the master equation for the condensate takes the form $\dot{\rho} = (-i/\hbar) [H,\rho] - L_1 \rho$. This equation preserves the total number of atoms in the condensate and corresponds to a quantum nondemolition measurement of the atomic density via the optical fields. In order to get an insight into the equation, we introduce a two-dimensional lattice with lattice constant set by the kernel resolution $\Delta r$. In this way we get
$$\frac{d \rho}{dt} = - \frac{i}{\hbar} \left[
- \hbar \omega \sum_{\langle k,l \rangle} \Psi^{\dagger}_{k}
\Psi_{k} + V, \rho \right] - S \sum_{l} [n_l,[n_l,\rho]] .
\label{lattice}$$
Here $\Psi_l$ is an annihilation operator and $n_l = \Psi_l^{\dagger}
\Psi_l$ is the number operator at a lattice site $l$. The frequency of hopping between any nearest neighbor sites $\langle k,l \rangle$ is $\omega\approx \hbar/ 2m \Delta r^2$, which is $\hbar^{-1}$ times the characteristic kinetic energy. The potential energy operator is $V=\sum_l
(U_l n_l+G n_l^2)$, where $U_l$ is the trapping potential and $G=g_{\rm
2D}/\Delta r^2$. The effective measurement strength is $S \approx \int
d^2r K({\bf r})/ \Delta r^2=(2 \pi/\Delta r)^2 (\pi \chi^2_0 k_0 I/2 \hbar
c)$. In order to solve this equation we use an unraveling in terms of pure states $|\Psi\rangle$ such that $\rho = \overline{ |\Psi\rangle \langle
\Psi|}$, the overline denoting the average over the unraveling stochastic realizations. The pointer states of Eq. (\[lattice\]) are not changed by the chosen unraveling [@DDZ]. The pure states can be expanded in a Fock basis per site, $|\Psi\rangle = \sum_{\{N_l\}} \psi_{\{N_l\}}
|\{N_l\} \rangle$ and the amplitudes $\psi_{\{N_l\}}$ satisfy the following stochastic Schrödinger equation (written in the Stratonovich convention)
$$\frac{d}{dt} \psi_{\{N_l\}} = -\frac{i}{\hbar}
\sum_{\{N'_l\}} h_{\{N_l,N'_l\}} \psi_{\{N'_l\}}
-\frac{i}{\hbar} V_{\{N_l\}} \psi_{\{N_l\}}
+ \psi_{\{N_l\}} \sum_l \left[
-S(N_l-n_l)^2+ S\sigma_l^2+
(N_l-n_l)\theta_l \right] ,$$
where the homodyne noises have averages $\overline{\theta_{l}(t)}=0$ and $\overline{\theta_{l_1}(t_1)\theta_{l_2}(t_2)}=
2S\delta_{l_1,l_2}\delta(t_1-t_2)$. Here $n_l=\sum_{\{N_l\}}N_l|\psi_{\{N_l\}}|^2$, $\sigma_l^2=\sum_{\{N_l\}}(N_l-n_l)^2|\psi_{\{N_l\}}|^2$, $h$ is the matrix element of the hopping Hamiltonian and $V_{\{N_l\}}=\sum_l(U_lN_l+GN_l^2)$ is the potential energy. When there is no hopping term ($h=0$), an exact solution to this equation as a product of Gaussian-like wavefunctions is written as $$\psi_{\{N_l\}}(t)= e^{i\varphi_{\{N_l\}}}
e^{ -\frac{i}{\hbar}V_{\{N_l\}} t}
\prod_l \frac{ e^{-\frac{[N_l-n_l(t)]^2}{4\sigma_l^2(t)}} }
{ [2\pi\sigma_l^2(t)]^{1/4} } .$$ The population mean value per site $n_l(t)$ does a random walk, while its dispersion decreases as $\sigma_l^2(t)=\sigma_l^2(0)/[1+4\sigma^2_l(0)St]$. Here $\sigma_l^2(0)$ is the initial dispersion in the number of atoms per site, and it scales as $N$ for an initial coherent state. Thus, the measurement drives the quantum state of the condensate to a Fock state. When tunneling between different lattice sites is allowed ($h \neq 0$), localization in a Fock state is inhibited due to a competition between the measurement, which drives localization, and hopping, which tries to drive the state of the condensate towards coherent states. When measurement outweighs hopping, the final state of the BEC is a number squeezed state. The timescale in which squeezing is achieved is given by
$$t_{\rm SQ} = \frac{1}{n_l S} ,$$
and the associated dispersion in the number of atoms per site for the asymptotic squeezed state is
$$\sigma_l = (\omega n_l/S)^{1/4} .$$
In a coherent state the number fluctuations per site are Poissonian, $\sigma_l=n_l^{1/2}$. The state is squeezed when $(\omega n_l/S)^{1/4}<n_l^{1/2}$, i.e. sub-Poissonian atomic number fluctuations. This condition defines the strong measurement as
$$\frac{n_l S}{\omega} \; > \; 1 \;.$$
Number squeezing of a Bose condensate has been experimentally observed in an optical lattice in [@Orzel]. Our situation has an important difference from the latter case: since the squeezing is driven by the (Lindblad) measurement term the evolution into such states is irreversible, even after removal of the imaging photon field. From this viewpoint our squeezing technique is similar to the spin squeezing through quantum nondemolition measurements proposed in [@Kuzmich] and demonstrated in [@Kuzmich1]. Of course, the system will eventually drift towards coherent, classical states due to the interaction with the external environment and the related decoherence [@Zurek], for instance due to the thermal component or residual background gas in the trapping volume.
Weak measurement: stochastic Gross-Pitaevskii equation
======================================================
Unraveling the Lindblad equation derived above leads, in the limit of weak measurements, to a stochastic equation for the condensate wavefunction. In the mean-field approximation this equation becomes the analogue of the (deterministic) Gross-Pitaevskii equation for the unmeasured system. From Eqs.(\[phasediff\],\[KERNEL\]) and ignoring the depletion term $-L_2\rho$ we obtain a continuum version of the master equation $$\frac{d\rho}{dt}=
-\frac{i}{\hbar}\left[H,\rho \right]
- \int d^2r_1 \int d^2r_2\; K({\bf r}_1-{\bf r}_2)\;
[n({\bf r}_1),[n({\bf r}_2),\rho]] ,
\label{master2D}$$ where $H$ is the self-Hamiltonian of the system of $N$ atoms in a 2D trap, $$H=\int d^2r\; \left(- \frac{\hbar^2}{2m} \nabla \Psi^{\dagger} \nabla \Psi+
U({\bf r}) \Psi^{\dagger} \Psi + \frac{g_{2D}}{2} \Psi^{\dagger}
\Psi^{\dagger}
\Psi \Psi \right)\;.$$ A nonlinear stochastic (Itô) unraveling of the master equation (\[master2D\]) is $$\begin{aligned}
d |\Psi\rangle &=& -\frac{i}{\hbar} dt H |\Psi\rangle
- dt \int d^2r_1 \int d^2r_2 \; K({\bf r}_1-{\bf r}_2)\;
\Delta n({\bf r}_1)\; \Delta n({\bf r}_2)\;
|\Psi\rangle
+ \int d^2r\; dW({\bf r})\; \Delta n({\bf r})\; |\Psi\rangle \;.
\label{un}\end{aligned}$$ Here $\Delta n({\bf r})= n({\bf r})-\langle\Psi|n({\bf r}) |\Psi\rangle$ and the Gaussian noises have correlators $\overline{dW({\bf r})}=0$ and $\overline{ dW({\bf r}_1) dW({\bf r}_2) } = 2 dt K({\bf r}_1-{\bf r}_2)$. This unravelling corresponds to phase contrast measurement of the density of the condensate. The evolution of $|\Psi\rangle$ given by Eq.(\[un\]) describes a single realization of the experiment [@Carmichael; @Wiseman; @Gatarek; @Plenio].
For a single atom, $N=1$, described by a wavefunction $\phi(t,{\bf r}) = \langle {\bf r} | \Psi \rangle$, the stochastic Schrödinger equation takes the form $$d\phi({\bf r}) \;=\;
- \frac{i}{\hbar} dt\;
\left[\;
-\frac{\hbar^2}{2m}\nabla^2\;+\;
U({\bf r})
\right]\;
\phi({\bf r})\;+\;
\left[
dW({\bf r})-
\int d^2r'\;
|\phi({\bf r}')|^2\;
dW({\bf r}')\;
\right]\;
\phi({\bf r})\;+\;{\rm C.T.}\;.
\label{SGPEN1cont}$$ The second term in brackets follows from the last term in Eq.(\[un\]), while C.T. (counterterm) comes from the second term in (\[un\]). The latter is necessary to conserve the norm $\int d^2r\;|\phi({\bf r})|^2=1$, and it is given by $${\rm C.T.}\;=\;
dt\;
\left[\;
-\;K(0)\;+\;
2\;\int d^2r_1 \;
K({\bf r}-{\bf r}_1)\;
|\phi({\bf r}_1)|^2\;-\;
\int d^2r_1 \int d^2r_2\;
|\phi({\bf r}_1)|^2\;
K({\bf r}_1-{\bf r}_2)\;
|\phi({\bf r}_2)|^2\;
\right] \phi({\bf r}).
\label{counterterm}$$ This counterterm is badly nonlocal. Fortunately, to implement the stochastic terms numerically one can use
$$\begin{aligned}
d\;|\phi({\bf r})|^2 &=&
2\;dt\;
\left[
dW({\bf r})-
\int d^2r'\;
|\phi({\bf r}')|^2\;
dW({\bf r}')\;
\right]\;
|\phi({\bf r})|^2 \nonumber \\
&& +
\overline{
\left[
dW({\bf r})-
\int d^2r'\;
|\phi({\bf r}')|^2\;
dW({\bf r}')\;
\right]^2 }\;
|\phi({\bf r})|^2+
\left[ \phi^{\star}({\bf r}) ({\rm C.T.}) + {\rm c.c.} \right]
\nonumber\\
&& =
2\;dt\;
\left[
dW({\bf r})-
\int d^2r'\;
|\phi({\bf r}')|^2\;
dW({\bf r}')\;
\right]\;
|\phi({\bf r})|^2 ,
\label{dp}\end{aligned}$$
where the C.T. is used to cancel out the average of the stochastic terms squared. The stochastic terms affect the modulus of $\phi({\bf r})$ only, so Eq.(\[dp\]) is all that one needs to implement them. Eq.(\[dp\]) manifestly conserves the norm.
For a weak measurement, when the squeezing of the quantum state is small, one can assume that the stochastic conditional state of N atoms in Eq.(\[un\]) is a product mean-field state $$|\Psi\rangle = \frac{1}{\sqrt{N!}}
\left[
\int d^2r\; \phi(t,{\bf r}) \Psi^{\dagger}({\bf r})
\right]^N |0\rangle ,$$ with all the $N$ atoms in the condensate wavefunction $\phi(t,{\bf r})$. The latter evolves according to an Itô stochastic Gross-Pitaevskii (SGP) equation $$\begin{aligned}
d\phi({\bf r}) &=&
-\frac{i}{\hbar} dt\;
\left[\;
-\frac{\hbar^2}{2m}\nabla^2\;+\;
U({\bf r})\;+\;
(N-1) \;g_{2D}\;
|\phi({\bf r})|^2\;
\right]\;
\phi({\bf r}) \nonumber \\
&& +
\left[
dW({\bf r})-
\int d^2r'\;
|\phi({\bf r}')|^2\;
dW({\bf r}')\;
\right]\;
\phi({\bf r})\;+\;{\rm C.T.}\;.
\label{SGPEcont}\end{aligned}$$ In comparison to the case of $N=1$ (see Eq.(\[SGPEN1cont\])), this SGP equation has the usual extra nonlinear term $(N-1) g_{2D}|\phi({\bf r})|^2$ but the stochastic terms containing $dW$ are the same as for a single atom described by Eq.(\[SGPEN1cont\]). This may seem strange because a state of $N\gg 1$ atoms in the same quantum state can be found much faster than a state of a single atom. One might expect the stochastic terms, which describe backaction of the measurement on the state of the system, to become stronger with increasing $N$. The backaction on the $N$-atom state is indeed stronger than on the single atom state but its effect is divided over $N> 1$ atoms instead of one. The backaction effect on the mean-field state $\phi({\bf r})$ of each of the $N$ atoms is the same as the backaction on the quantum state of a single atom. Thus the stochastic terms for $N>1$ are the same as for $N=1$.
As a final remark, we mention that a stochastic Gross-Pitaevskii equation has been proposed and studied for a quite different goal, namely to describe single trajectories through unraveling of the exact N-body quantum evolution of a Boson system [@Carusotto]. In the latter case the interpretation of the underlying stochasticity is obtained in terms of the randomness attributable to each quantum trajectory, to be confronted with the stochasticity that in our case is instead due to the opening of the condensate to a particular environment, namely the measurement apparatus.
SGP versus exact quantum evolution: measurements in a double well
=================================================================
In this Section we want to test the validity of the SGP equation in a significant but simple situation. To this end we consider the double well problem in the two mode approximation, and compare the quantum dynamics with measurement backaction with the dynamics given by the SGP equation. The Hamiltonian of the model is
$$H = \epsilon (a_1^{\dagger} a_1 + a_2^{\dagger} a_2) -
\hbar \omega (a_1^{\dagger} a_2 + a_2^{\dagger} a_1) +
\frac{G}{2} [ (a_1^{\dagger})^2 a_1^2 + (a_2^{\dagger})^2 a_2^2 ] ,$$
where $\epsilon$ is the mode frequency (assumed to be the same for both modes), $\omega$ is the tunneling angular frequency, and $G$ is the two-particle interaction strength. We perform phase-contrast imaging on each site, and we assume that the kernel resolution is much shorter than the distance between sites, so that there is no cross term due to the measurement. The Itô version of the stochastic Schrödinger equation for the state ket reads
$$d |\Psi_{\rm Q} \rangle = dt \left[ -\frac{i}{\hbar} H -
\frac{S}{2} (n_1 - \langle n_1 \rangle)^2 -
\frac{S}{2} (n_2 - \langle n_2 \rangle)^2 \right]
|\Psi_{\rm Q} \rangle
+ dW_1 (n_1 - \langle n_1 \rangle) |\Psi_{\rm Q}\rangle
+ dW_2 (n_2 - \langle n_2 \rangle) |\Psi_{\rm Q}\rangle ,$$
where the noise satisfies $\overline{dW_{\alpha}}=0$ and $\overline{dW_{\alpha} dW_{\beta}}= 2 K_{\alpha\beta} dt$, with $\alpha,\beta=1,2$, and $K_{\alpha\beta}=S \delta_{\alpha\beta}$ the measurement kernel. By assuming a total of $N$ atoms, distributed between the two sites, $N=N_1+N_2$, we can expand the state in terms of Fock states at each well
$$| \Psi_{\rm Q} \rangle = \sum_{k=0}^N \psi_k(t) |k, N-k \rangle .$$
We solve numerically the corresponding equation for the coefficients $\psi_k(t)$ starting from a mean field state with equal mean populations in each well, $|\Psi\rangle_{t=0}=\frac{1}{\sqrt{N!}}(\frac{1}{\sqrt{2}} a_1^{\dagger} +
\frac{1}{\sqrt{2}} a_2^{\dagger})^N |0\rangle $. The equation provides us with the full quantum evolution including the measurement backaction. We want to compare it with the one that results from the stochastic GP equation. The GP state is
$$| \Psi_{\rm GP} \rangle = \frac{1}{\sqrt{N!}}
[ \phi_1(t) a_1^{\dagger} + \phi_2(t) a_2^{\dagger}]^N |0 \rangle ,$$
where the wavefunctions $\phi_1$ and $\phi_2$ satisfy the following set of Itô SGP equations, that follow from Eqs.(\[counterterm\],\[SGPEcont\]) $$\begin{aligned}
d \phi_1 &=& -\frac{i}{\hbar} dt [(\epsilon + (N-1) G |\phi_1|^2)
\phi_1 - \omega \phi_2]
+ dW_1 (1-|\phi_1|^2)
+ S dt [ -1+2 |\phi_1|^2 -(|\phi_1|^4+|\phi_2|^4) ] ,
\nonumber \\
d \phi_2 &=& -\frac{i}{\hbar} dt [(\epsilon + (N-1) G |\phi_2|^2)
\phi_2 - \omega \phi_1]
+ dW_2 (1-|\phi_2|^2)
+ S dt [ -1+2 |\phi_2|^2 -(|\phi_1|^4+|\phi_2|^4) ] ,\end{aligned}$$ and we take the same initial state as in the quantum evolution, namely a coherent state $\phi_1(0)=\phi_2(0)=1/\sqrt{2}$ with balanced populations $n_1(0)=n_2(0)$. In the Fig.1 we compare the time evolution of the quantum and SGP mean populations in one well for increasing values of the measurement strength. In these simulations the total number of particles was $N=100$, and the nonlinear coupling was $G=0$. In this case, the Hamiltonian involves only one-body terms, so the mean field evolution (based on coherent states) must exactly coincide with the quantum one for the case of zero measurement ($S=0$). For small measurement strengths, $n
S/\omega \ll 1$ (here $n$ denotes the average number of particles per site), the quantum state of the condensate is still, to a high degree of accuracy, a coherent state, so the SGP evolution and the quantum one coincide. In Fig. 1a we see that the agreement is good even for times much larger than $1/n S$. This timescale is relevant for the strong measurement case of the previous Section, since it sets the time after which an asymptotic number squeezed state is reached. As we increase the measurement strength and reach $n S/\omega \geq 1$ (see Fig. 1b), the mean number of particles per well given by the SGP and the quantum evolution depart appreciably, not surprisingly since such strong measurements squeeze the quantum state of the condensate driving it outside the description in terms of coherent states, [*i.e.*]{} the associated basis for the Gross-Pitaevskii equation.
The Gross-Pitaevskii evolution can depart from the quantum one not only due to the measurement backaction but also due to the nonlinearity of the interactions. In Fig. 2a we show the SGP and quantum evolutions for $S/\omega=10^{-3}$ and $G/\omega=10^{-3}$, corresponding to the same initial state as in previous figures. We see that the inclusion of the nonlinearity causes the SGP evolution to depart from the quantum one. To gain further insight we introduce a quantity which measures the depleted fraction of atoms from the [*best*]{} mean field state, i.e. a mean field state that is the closest to the exact quantum state. Its definition is
$$D = \stackrel{{\rm min}}{\{A,\phi\}} \left(1 - \frac{1}{N}
\langle \Psi_Q | c^{\dagger} c | \Psi_Q \rangle \right)$$
where the operator $c$ is $c=\sqrt{A} a_1 + e^{i \phi} \sqrt{1-A^2} a_2$. In Fig.2b we plot this depletion for the same simulation of Fig. 2a. The depletion in Fig.2b is small, less than one atom is depleted from the condensate, but, as we see in Fig.2a, the SGP evolution departs from the exact evolution. This departure is attributable to the inclusion of the nonzero $G$.
In Fig.3a we show the GP and quantum evolution for $S=0$ and $G/\omega=10^{-3}$. Since no measurements are performed, an initial balanced population ($n_1(0)=n_2(0)$) would remain balanced for all times, both at the GP and quantum level. For this reason we take an initial unbalanced population, $n_1(0)/n_2(0)=3/2$, which due to the hopping term triggers Rabi oscillations between the two wells (just as the measurement did when we took initial balanced populations). It follows from the figure that the GP dynamics departs from the quantum one. We plot in Fig. 3b the depletion corresponding to Fig. 3a. Again, this depletion remains small, less than one atom is depleted from the condensate.
As we can see in Figs.2,3, both the SGP and GP evolutions depart from the exact evolution even in the weak measurement and interaction limits when the depletion from the condensate is small. The derivation of the backaction terms in the SGP equation requires only one assumption: all the atoms are in the condensate. In contrast, the derivation of the interaction terms (both for SGP and GP equations) not only assumes that all the atoms are in the condensate but also makes further approximations to describe the evolution of the condensate wavefunction $\phi$. This is why $G \neq 0$ causes both the SGP and GP evolutions to depart from the exact quantum evolution even for negligible depletion. However, even after the departure the SGP and GP evolutions remain qualitatively similar to the exact evolution, see Figs. 2a and 3a.
To summarize, the accuracy of the SGP equation is limited by the weak measurement condition, $nS/\omega<1$, and by the nonlinear interaction. For a weak measurement the accuracy of the SGP equation is the same as that of the GP equation. In the next Section we apply the SGP equation to show how the measurement can trigger the unwinding of a dark soliton [@DS; @DS1] in a realistic experimental setup.
Measurement-induced unwinding of a dark soliton
================================================
Once delimited the validity of the stochastic Gross-Pitaevskii equation in the simple situation of a two-mode system we can apply it to the more complex case of imaging of a condensate state with a nontrivial phase such as a soliton. Even in the limit of weak measurement, with the solution still approximable in terms of Gross-Pitaevskii coherent states, the effect of the measurement is present and affects the observable conjugate to the atom number, i.e. the phase of the condensate. As an example of application of the SGPE let us consider an isotropic harmonic 2D trapping potential $V({\bf r})=(m \Omega^2/2)
(x^2+y^2)$. The condensate is assumed to be in the Thomas-Fermi limit of strong repulsive interaction where the ground state wavefunction can be well approximated by
$$\phi_{GS}({\bf r})\;=\;
\sqrt{\frac{\mu -V({\bf r})}{Ng_{2D}}}\;.
\label{GSTF}$$
The constant $\mu=\Omega\sqrt{N g_{2D} m/\pi}$, the chemical potential, is chosen so that the wavefunction is normalized to $1$. Let us use as the initial state a dark soliton [@DS] imprinted on the Thomas-Fermi ground state
$$\phi(t=0,{\bf r})\;=\;
\tanh\left( \frac{x}{l} \right)\;
\sqrt{\frac{\mu'-V({\bf r})}{Ng_{2D}}}\;.
\label{psi0}$$
Here $l=0.6 \mu{\rm m}$ is the healing length at the peak density in the ground state (\[GSTF\]).
In our numerical simulations we assume the following parameters relevant for $^{87}$Rb: mass $m=1.4\times 10^{-25}$kg, scattering length $a=5.8$nm, $\chi_0=10^{-23}$m$^3$, wavelength $\lambda=780$nm. The width of the gaussian $\Lambda(z)$ is assumed to be $\xi=10\mu$m. With these parameters the resolution of the kernel is $\Delta r=5\mu$m. We assume a 2D harmonic trap frequency $\Omega=2\pi\times 10 {\rm s}^{-1}$. With $N=5\times 10^5$ we get a 2D Thomas-Fermi radius of $31\mu\rm{m}$. We also assume the laser intensity of $I=10^{-4} {\rm mW}/{\rm cm}^2$.
The former parameters give a weak measurement strength in the sense discussed in Sections III and IV, so that the use of the SGP equation is justified. Indeed, this continuous problem can be mapped on the lattice model Eq.(\[lattice\]), where the lattice constant is the kernel resolution $\Delta r$. Given the Thomas-Fermi radius of $31\mu\rm{m}$, and $\Delta r=5\mu$m, we estimate the lattice to be composed of $100$ sites with an average of $n=5000$ atoms per site. The measurement strength is equal to $S=7\times 10^{-5} {\rm s}^{-1}$, and the effective hopping frequency for the lattice model is $\omega=14 {\rm s}^{-1}$. Therefore $n
S/\omega \approx 2\times 10^{-2} \ll 1$, thus confirming a weak measurement regime.
To simulate the continuum SGP equation (\[SGPEcont\]) we discretized it using a lattice constant $2 \pi$ times smaller than the kernel resolution $\Delta r$. The program uses a split step method: the fast Fourier transform was used to carry out the time integration of the kinetic term and of the nonlocal terms involving the kernel, and the potential and nonlinear coupling terms were integrated in time in the position representation. A cross-section along the $x$-axis through the probability density of the initial state (\[psi0\]) is shown in (a) of Fig.4. Cross sections through probability densities at later times after the probe light beam has been sent on the condensate are shown in cases (b) and (c) of Fig.4. For comparison, the time evolution without the measurement does not result in any soliton unwinding.
The soliton unwinds after roughly $50\;$ms. This time is much shorter than the depletion time $t_2=10^4\;$s discussed at the end of Section II. The measurement induces soliton unwinding much earlier that any detectable depletion of atoms occurs. Figure \[stage0\] suggests that the unwinding will manifest itself by filling up the soliton core with atoms. Such a greying of the dark soliton can also occur through a different mechanism that involves collisions between condensate atoms. In Ref.[@DKS] it was demonstrated that the dark soliton can grey on a timescale of tens of ms because its core fills up with non-condensed atoms (quantum) depleted from the condensate as a result of atomic collisions between condensed atoms. This result is supported by Ref.[@Law] where it was shown that the quantum state with minimal depletion has depletion strongly concentrated in the soliton core. This quantum depletion process does not unwind the soliton: the condensate remains in the soliton state with a phase jump of $\pi$. The phase jump or its unwinding could be detected by interference between two condensates, one of them in a ground state and the other with a soliton [@INT]. A simpler way to verify that observed greying is due to the measurement induced unwinding is to change the measurement strength in a certain range and see if the greying time depends on the imaging laser intensity. We simulated the soliton unwinding for a range of measurement strengths. Fig. 5 shows the unwinding time $\tau$ versus the laser intensity $I$. The time $\tau$ is defined as that for which the density at $x=0$ in Fig. 4 achieves $10\%$ of the maximal density.
Conclusions
===========
Quantum measurement theory has been applied to the dispersive imaging of a Bose-Einstein condensate. In the strong measurement limit the condensate is irreversibly driven into non-classical states with reduced number fluctuations. In the opposite limit of weak measurement the condensate can be approximately described for short timescales through a stochastic counterpart of the Gross-Pitaevskii equation. The latter has been applied for the study of the dynamics induced by dispersive imaging on a condensate prepared in a soliton state. The proposed model, besides allowing to intentionally design selective manipulation of the condensate state for instance to quench vortices without introducin appreciable depletion, could also lead to a better understanding of quantum phase transitions [@Fisher; @Sachdev] in Bose condensates [@MOTTBEC].
Acknowledgements
================
We would like to thank Ivan Deutsch for calling our attention to Refs [@Kuzmich; @Kuzmich1]. The work of D.D. and J.D. was supported in part by NSA.
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abstract: '**Abstract.** We propose a scheme to implement geometric entangling gates for two logical qubits in a coupled cavity system in decoherence-free subspaces. Each logical qubit is encoded with two atoms trapped in a single cavity and the geometric entangling gates are achieved by cavity coupling and controlling the external classical laser fields. Based on the coupled cavity system, the scheme allows the scalability for quantum computing and relaxes the requirement for individually addressing atoms.'
author:
- 'Yue-Yue Chen'
- 'Xun-Li Feng'
- 'C.H. Oh'
title: 'Geometric entangling gates for coupled cavity system in decoherence-free subspaces'
---
Exploiting appropriate coherent dynamics to generate entangling gates between separate systems is of crucial importance to quantum computing and quantum communication. Several schemes have been proposed to engineer entangling gates [@SMB; @chen_song; @ZhengSB] between atoms trapped in spatially separated cavities. It is feasible and commonly used to mediate the distant optical cavities by optical fiber [@J; @S; @SC]. However, decoherence resulted from uncontrollable coupling to environment will collapse the state and impair the performance for quantum process. Thus, decoherence is the main obstacle for realizing quantum computing and quantum information processing. In order to protect the fragile quantum information and realize the promised speedup compared with classical counterpart, a wealth of strategies have been proposed to deal with decoherence. One efficient way is to construct a decoherence-free subspace (DFS) if the interaction between quantum system and its environment possesses some symmetry [@DFS]. Keeping a system inside a DFS is regarded as a passive error-prevention approach while error-correcting code, which is comprised of encoding information in a redundant way, is regarded as an active approach [@Shor]. Another promising strategy to cope with decoherence is based on the mechanism of geometric phase [@Zanardi]. Geometric phases depend only on some global geometric features of the evolution path and are insensitive to local inaccuracies and fluctuations. However, the total phases acquired during the evolution often consist of geometric phases and the concomitant dynamic phases. Dynamic phases may ruin the potential robustness of the scheme and should be removed according to conventional wisdom. Literatures [@convention] and [@LJ] proposed two simple methods to remove dynamic phases. In contrast, the so-called unconventional geometric gates, in which dynamic phases are not zero but proportional to the geometric ones, were proposed [@WangXG; @shi]. The unconventional geometric gates were suggested to be realized in cavity QED systems subsequently [@CQED1; @CQED2].
Schemes which combine the robust advantages of both DFS and the geometric phase have been presented [@LX; @X]. Reference [@LX] exploits the spin-dependent laser-ion coupling in the presence of Coulomb interactions, and then constructs a universal set of unconventional geometric quantum gates in encoded subspaces. Reference [@X] proposes to implement the geometric entangling gates in DFS by using a dispersive atom-cavity interaction in a single cavity. As is well known, the collective decoherence is often regarded as a strict requirement for DFS strategy to overcome the decoherence, however, such a requirement is largely relaxed in [@X] because only two neighboring physical qubits, which encode a logical qubit, are required to undergo collective dephasing. With this merit, in this paper we extend the idea of [@X] to a coupled cavity system where each cavity contains two atoms which encode one logical qubit. In contrast to [@X], the extension to the coupled cavity system in this work allows the realization of scalability of cavity QED based quantum computing by using the idea of the distributed quantum computing [@JA] and relaxes the requirement for individually addressing atoms.
Now let us describe our scheme more specifically. Considering two coupled cavities which are linked with an optical fiber. We suppose each cavity contains two $\Lambda$-type three-level atoms. For convenience, we label the two cavities with $j$ and $k$, respectively, the atoms in cavity $j$ ($k$) are denoted by $j_{1},j_{2}\left( k_{1},k_{2}\right) $. The atomic level configuration with couplings to the cavity modes and the driving laser fields is shown in Fig. 1: $\left\vert e\right\rangle $ is an excited state and $\left\vert 0\right\rangle $ and $\left\vert
1\right\rangle $ are two stable ground states, the latter two constitute the basis of a physical qubit. Both transitions $\left\vert 0\right\rangle \leftrightarrow\left\vert e\right\rangle
$ and $\left\vert 1\right\rangle \leftrightarrow$ $\left\vert
e\right\rangle $ are supposed to dispersively couple to the cavity mode and be driven by two classical laser fields with opposite detunings. One of the classical laser field acts on transitions $\left\vert 0\right\rangle \leftrightarrow$ $\left\vert
e\right\rangle $ and $\left\vert 1\right\rangle
\leftrightarrow\left\vert e\right\rangle $ has a frequency $\omega$ closed to the cavity frequency $\omega_{c}$. Note that, $\omega-\omega_{c}=\delta$, where $\delta$ is a small quantity. The detuning of this classical field from the transition $\left\vert
m\right\rangle \leftrightarrow\left\vert e\right\rangle $ is $\Delta_{m}=\omega_{m}-\omega$ $\left( m=0,1\right) $, where $\omega_{m}$ is the energy difference between ground state $\left\vert m\right\rangle $ and $\left\vert e\right\rangle $. The corresponding detuning for the cavity mode is $\Delta_{m}+\delta$ (see Fig. 1). Similarly, the other laser with frequency $\omega^{\prime}$ is tuned to satisfy the relation $\omega_{m}-\omega^{\prime}=-\Delta_{m}$.
To overcome the collective dephasing, we encode the logical qubit in the cavity $j$ with a pair of physical qubits in a form $\left\vert 0_{j}\right\rangle ^{L}=\left\vert 0_{j_{1}}1_{j_{2}}\right\rangle $, $\left\vert
1_{j}\right\rangle ^{L}=\left\vert 1_{j_{1}}0_{j_{2}}\right\rangle $. The subspace $\mathcal{C}_{j}^{2}=\left\{ \left\vert 0_{j}\right\rangle
^{L},\left\vert 1_{j}\right\rangle ^{L}\right\} $ constitutes a DFS for the single logical qubit $j$. Similarly, the logical qubit $k$ is encoded by the two physical qubits $k_{1}$, $k_{2}$ in the cavity $k$.
The coupling between the cavity fields and the fiber modes can be written as the interaction Hamiltonian [@SMB] $$H_{cf}=\underset{i=1}{\overset{\infty}{\sum}}\nu_{i}\left[ b_{i}\left(
a_{1}^{\dag}+\left( -1\right) e^{i\varphi}a_{2}^{\dag}\right)
+\text{H.c.}\right] ,$$ where $\nu_{i}$ is the coupling strength between fiber mode $i$ and the cavity mode, $b_{i}$ is the annihilation operator for the fiber mode $i$ while $a_{1}^{\dag}\left( a_{2}^{\dag}\right) $ is the creation operator for the cavity mode $j$($k$), and $\varphi$ is the phase induced by the propagation of the field through the fiber. In the short fiber limit, only resonant mode $b$ of the fiber interacts with the cavity mode. In this case, the Hamiltonian $H_{cf}$ can be approximately written as [@SMB] $$H_{cf}=\nu\left[ b\left( a_{1}^{\dag}+a_{2}^{\dag}\right) +H.c.\right] ,$$ where the phase $\left( -1\right) e^{i\varphi}$ in $H_{cf}$ has been absorbed into $a_{2}^{\dag}$ and $a_{2}$ [@chen_song].
To implement the geometric entangling gate, we let the classical laser fields plotted in Fig. 1 individually act on both atoms $j_{1}$ and $k_{1}$. In the interaction picture, the Hamiltonian describing the atom-field interaction takes the form $$\begin{aligned}
H_{AC} & =\underset{l=j_{1},k_{1}}{\sum}\underset{m=0,1}{\sum}\frac
{\Omega_{m}^{\prime}}{2}e^{-i\Delta_{m}t}\left\vert e\right\rangle
_{l}\left\langle m\right\vert +\frac{\Omega_{m}}{2}e^{i\Delta_{m}t}\left\vert
e\right\rangle _{l}\left\langle m\right\vert +\underset{l=j_{1},j_{2}}{\sum}\underset{m=0,1}{\sum}g_{m}\left\vert
e\right\rangle _{l}\left\langle m\right\vert a_{1}e^{i\left( \Delta
_{m}+\delta\right) t}\nonumber\\
& +\underset{l=k_{1},k_{2}}{\sum}\underset{m=0,1}{\sum}g_{m}\left\vert
e\right\rangle _{l}\left\langle m\right\vert a_{2}e^{i\left( \Delta
_{m}+\delta\right) t}+\text{H.c.}$$
Following Ref. [@SMB], we define three bosonic modes $c_{0}=\frac{1}{\sqrt{2}}\left( a_{1}-a_{2}\right) $, $c_{1}=\frac{1}{2}\left( a_{1}+a_{2}+\sqrt
{2}b\right) $, $c_{2}=\frac{1}{2}\left( a_{1}+a_{2}-\sqrt{2}b\right) $, $c_{n}(n=0,1,2)$ are linearly relative to the field modes of the cavities and fiber. Then we can rewrite the whole Hamiltonian in the interaction picture as $$H=H_{0}+H_{i},$$ where $$H_{0}=\sqrt{2}\nu c_{1}^{\dag}c_{1}-\sqrt{2}\nu c_{2}^{\dag}c_{2},$$ and $$\begin{aligned}
H_{i} & =\underset{m=0,1}{\underset{l=j_{1},k_{1}}{\sum}}\frac{\Omega
_{m}^{\prime}}{2}e^{-i\Delta_{m}t}\left\vert e\right\rangle _{l}\left\langle
m\right\vert +\frac{\Omega_{m}}{2}e^{i\Delta_{m}t}\left\vert e\right\rangle
_{l}\left\langle m\right\vert +\underset{m=0,1}{\underset{l=j_{1},j_{2}}{\sum}}g_{m}\left\vert
e\right\rangle _{l}\left\langle m\right\vert \frac{1}{2}\left( c_{1}+c_{2}+\sqrt{2}c_{0}\right) e^{i\left( \Delta_{m}+\delta\right)
t}\nonumber\\
& +\underset{m=0,1}{\underset{l=k_{1},k_{2}}{\sum}}g_{m}\left\vert
e\right\rangle _{l}\left\langle m\right\vert \frac{1}{2}\left( c_{1}+c_{2}-\sqrt{2}c_{0}\right) e^{i\left( \Delta_{m}+\delta\right)
t} +\text{H.c.}$$ $\ \ \ \ $We now perform the unitary transformation $e^{iH_{0}t}$, and obtain [@ZhengSB]
$$\begin{aligned}
H_{i}=&\underset{m=0,1}{\underset{l=j_{1},k_{1}}{\sum}}\left(\frac{\Omega
_{m}^{\prime}}{2}e^{-i\Delta_{m}t}\left\vert e\right\rangle _{l}\left\langle
m\right\vert +\frac{\Omega_{m}}{2}e^{i\Delta_{m}t}\left\vert e\right\rangle
_{l}\left\langle m\right\vert\right)+ \underset{m=0,1}{\underset{l=j_{1},j_{2}}{\sum}}g_{m}\left\vert
e\right\rangle _{l}\left\langle m\right\vert \frac{1}{2}\left( c_{1}e^{-i\sqrt{2}\nu t}+c_{2}e^{i\sqrt{2}\nu t}+\sqrt{2}c_{0}\right) e^{i\left(
\Delta_{m}+\delta\right) t}\nonumber\\
&+ \underset{m=0,1}{\underset{l=k_{1},k_{2}}{\sum}}g_{m}\left\vert
e\right\rangle _{l}\left\langle m\right\vert \frac{1}{2}\left( c_{1}e^{-i\sqrt{2}\nu t}+c_{2}e^{i\sqrt{2}\nu t}-\sqrt{2}c_{0}\right) e^{i\left(
\Delta_{m}+\delta\right) t}+ \text{H.c.}$$
Here we assume that $\Delta_{m}\gg\sqrt{2}\nu,\delta,g_{m}$ and $\Omega_{m}$ to make sure that atoms cannot exchange energy with the fiber mode, cavity modes, and classical fields on account of the large detuning. In this case, we may adiabatically eliminate the excited atomic state considering no population transferred to the this state. In order to cancel the Stark shifts caused by classical laser fields, we set $\left\vert \Omega _{m}\right\vert =\left\vert
\Omega_{m}^{\prime}\right\vert $. Assuming further $g_{m}\ll\Omega_{m}$, we can neglect the terms of $g_{m}^{2}$, which indicate the Stark shifts caused by bosonic modes. From the above, we obtain an effective Hamiltonian describing the coupling between the atoms and bosonic modes assisted by the classical fields [@JM]
$$\begin{aligned}
H_{eff} & =\left( \left\vert 0\right\rangle _{j_{1}}\left\langle
0\right\vert +\left\vert 0\right\rangle _{k_{1}}\left\langle 0\right\vert
\right) \lambda_{1}e^{-i\left( \delta-\sqrt{2}\nu\right) t}c_{1}^{\dag
} +\left( \left\vert 0\right\rangle _{j_{1}}\left\langle 0\right\vert
+\left\vert 0\right\rangle _{k_{1}}\left\langle 0\right\vert \right)
\lambda_{2}e^{-i\left( \delta+\sqrt{2}\nu\right) t}c_{2}^{\dag}\nonumber\\
& +\left( \left\vert 0\right\rangle _{j_{1}}\left\langle 0\right\vert
-\left\vert 0\right\rangle _{k_{1}}\left\langle 0\right\vert \right)
\lambda_{0}e^{-i\delta t}c_{0}^{\dag} +\left( \left\vert 1\right\rangle _{j_{1}}\left\langle 1\right\vert
+\left\vert 1\right\rangle _{k_{1}}\left\langle 1\right\vert \right)
\lambda_{1}^{\prime}e^{-i\left( \delta-\sqrt{2}\nu\right) t}c_{1}^{\dag
}\nonumber\\
& +\left( \left\vert 1\right\rangle _{j_{1}}\left\langle 1\right\vert
+\left\vert 1\right\rangle _{k_{1}}\left\langle 1\right\vert \right)
\lambda_{2}^{\prime}e^{-i\left( \delta+\sqrt{2}\nu\right) t}c_{2}^{\dag
} +\left( \left\vert 1\right\rangle _{j_{1}}\left\langle 1\right\vert
-\left\vert 1\right\rangle _{k_{1}}\left\langle 1\right\vert \right)
\lambda_{0}^{\prime}e^{-i\delta t}c_{0}^{\dag}+\text{H.c.,}$$ where
$\ \ \ \ \ \ \lambda_{0}=-\frac{\sqrt{2}\Omega_{0}g_{0}^{\ast}}{8}\left(
\frac{1}{\Delta_{0}}+\frac{1}{\Delta_{0}+\delta}\right) $, $\ \ \ \ \ \ \lambda_{1}=-\frac{\Omega_{0}g_{0}^{\ast}}{8}\left( \frac
{1}{\Delta_{0}}+\frac{1}{\Delta_{0}+\delta-\sqrt{2}\nu}\right) $, $\ \ \ \ \ \ \lambda_{2}=-\frac{\Omega_{0}g_{0}^{\ast}}{8}\left( \frac
{1}{\Delta_{0}}+\frac{1}{\Delta_{0}+\delta+\sqrt{2}\nu}\right) $,
$\ \ \ \ \ \ \lambda_{0}^{\prime}=-\frac{\sqrt{2}\Omega_{1}g_{1}^{\ast}}{8}\left( \frac{1}{\Delta_{1}}+\frac{1}{\Delta_{1}+\delta}\right) $, $\ \ \ \ \ \ \lambda_{1}^{\prime}=-\frac{\Omega_{1}g_{1}^{\ast}}{8}\left(
\frac{1}{\Delta_{1}}+\frac{1}{\Delta_{1}+\delta-\sqrt{2}\nu}\right) $, $\ \lambda_{2}=-\frac{\Omega_{1}g_{1}^{\ast}}{8}\left( \frac
{1}{\Delta_{1}}+\frac{1}{\Delta_{1}+\delta+\sqrt{2}\nu}\right) $.
Because the logical qubits $j$ and $k$ are located at different cavities, the available DFS for the whole system is constructed by $\mathcal{C}_{jk}^{4}\equiv\mathcal{C}_{j}^{2}\otimes\mathcal{C}_{k}^{2}= \left\{ \left\vert
0_{j}^{L}0_{k}^{L}\right\rangle \text{, }\left\vert 0_{j}^{L}1_{k}^{L}\right\rangle \text{, }\left\vert 1_{j}^{L}0_{k}^{L}\right\rangle \text{,
}\left\vert 1_{j}^{L}1_{k}^{L}\right\rangle \right\} ,$ and in this DFS the Hamiltonian $H_{eff}$ is diagonal and takes the form $$H_{eff}= diag\left[ H_{0_{j}0_{k}}\text{, }H_{0_{j}1_{k}}\text{, }H_{1_{j}0_{k}}\text{, }H_{1_{j}1_{k}}\right] ,$$ where the diagonal matrix elements $H_{\mu_{j}\nu_{k}}(\mu,$ $\nu=0,1)$ are of the form $$H_{\mu_{j}\nu_{k}}=\underset{n=0}{\overset{2}{\sum}}c_{n}^{\dag}\chi_{\mu
_{j}\nu_{k}}^{n}e^{-i\eta_{n}t}+\text{H.c.,}$$ where
$\ \ \ \chi_{0_{j}0_{k}}^{0}=0$, $\chi_{0_{j}0_{k}}^{1}=2\lambda_{1}$, $\chi_{0_{j}0_{k}}^{2}=2\lambda_{2}$; $\ \ \ \chi_{0_{j}1_{k}}^{0}=\lambda_{0}-\lambda_{0}^{\prime}$, $\chi
_{0_{j}1_{k}}^{1}=\lambda_{1}+\lambda_{1}^{\prime}$, $\chi_{0_{j}1_{k}}^{2}=\lambda_{2}+\lambda_{2}^{\prime}$;
$\ \ \ \chi_{1_{j}0_{k}}^{0}=\lambda_{0}^{\prime}-\lambda_{0}$, $\chi
_{1_{j}0_{k}}^{1}=\lambda_{1}+\lambda_{1}^{\prime}$, $\chi_{1_{j}0_{k}}^{2}=\lambda_{2}+\lambda_{2}^{\prime}$; $\chi_{1_{j}1_{k}}^{0}=0$, $\chi_{1_{j}1_{k}}^{1}=2\lambda_{1}^{\prime}$, $\chi_{0_{j}0_{k}}^{2}=2\lambda_{2}^{\prime}$.and
$\eta_{0}=\delta$, $\eta_{1}=\delta-\sqrt{2}\nu$, $\eta_{2}=\delta+\sqrt{2}\nu$. Obviously, in the DFS $\mathcal{C}_{jk}^{4}$, time evolution matrix $U\left(
t\right) $ also takes a diagonal form, $$U\left( t\right) =diag\left[ U_{0_{j}0_{k}}\text{, }U_{0_{j}1_{k}}\text{,
}U_{1_{j}0_{k}}\text{, }U_{1_{j}1_{k}}\right] .$$
The corresponding diagonal matrix elements$\ U_{\mu_{j}\nu_{k}}\left( t\right) $ can be derived from Eq. (10) and they are in terms of displacement operator $$\begin{aligned}
\ U_{\mu_{j}\nu_{k}}\left( t\right) & =\hat{T}\exp\left[ -i\int_{0}^{t}H_{\mu_{j}\nu_{k}}\left( \tau\right) d\tau\right] =\underset{n=0}{\overset{2}{\prod}}\exp\left( {i\phi_{\mu_{j}\nu_{k}}^{n}}\right) D\left( \int_{c}d\alpha_{\mu_{j}\nu_{k}}^{n}\right) =\exp\left[ {i}\phi_{\mu_{j}\nu_{k}}\right] \underset{n=0}{\overset
{2}{\prod}}D\left( \int_{c}d\alpha_{\mu_{j}\nu_{k}}^{n}\right) \ ,\end{aligned}$$
$\ $
with $\hat{T}$ being the time ordering operator, and $$\phi_{\mu_{j}\nu_{k}}=\overset{2}{\underset{n=0}{\sum}}\phi_{\mu_{j}\nu_{k}}^{n}=\overset{2}{\underset{n=0}{\sum}}\operatorname{Im}\left[ \int
_{c}\left( \alpha_{\mu_{j}\nu_{k}}^{n}\right) ^{\ast}d\alpha_{\mu_{j}\nu
_{k}}^{n}\right] ,$$ $$d\alpha_{\mu_{j}\nu_{k}}^{n}=-i\chi_{\mu_{j}\nu_{k}}^{n}e^{-i\eta_{n}\tau
}d\tau$$
Considering the situation, where each bosonic mode is assumed initially in vacuum state, the state of each bosonic mode evolves to coherent state at time $t_{n}>0$. The corresponding amplitude $\int_{c}d\alpha_{\mu_{j}\nu_{k}}^{n}$ is dependent on the logic computational basis state $\left\vert \mu_{j}^{L}\nu_{k}^{L}\right\rangle $. It is not difficult to obtain $\alpha_{\mu_{j}\nu_{k}}^{n}$ by integrating Eq. (14) $$\alpha_{\mu_{j}\nu_{k}}^{n}=\frac{\chi_{\mu_{j}\nu_{k}}^{n}}{\eta_{n}}\left(
e^{-i\eta_{n}t}-1\right) .$$
The above equation indicates that there is a time period $T$ fulfilling the relation $T=2\pi l_{n}/\eta_{n}$, where $l_{n}$ is a positive integer and $n=0,1,2$, in which the bosonic mode $c_{n}$ completes $l_{n}$ evolutions and returns to its initial vacuum state. During this process the system accumulates the following total phase $$\begin{aligned}
\gamma_{\mu_{j}\nu_{k}}(T) & =\phi_{\mu_{j}\nu_{k}}(T) =-\overset{2}{\underset{n=0}{\sum}}\frac{2\pi l_{n}}{\eta_{n}}\left\vert
\chi_{\mu_{j}\nu_{k}}^{n}\right\vert ^{2} =\gamma_{\mu_{j}\nu_{k}}^{g}+\gamma_{\mu_{j}\nu_{k}}^{d},\end{aligned}$$ where $\gamma_{\mu_{j}\nu_{k}}^{d}$ and $\gamma_{\mu_{j}\nu_{k}}^{g}$ stand for the dynamical and geometric phases respectively, and can be calculated by using the coherent state path integral method [@MH] $$\begin{aligned}
\gamma_{\mu_{j}\nu_{k}}^{d} & =\overset{2}{\underset{n=0}{\sum}}-\int
_{0}^{T}H_{\mu_{j}\nu_{k}}^{n}\left( \left( \alpha_{\mu_{j}\nu_{k}}^{n}\right) ^{\ast},\alpha_{\mu_{j}\nu_{k}}^{n};t\right) dt =-\overset{2}{\underset{n=0}{\sum}}\frac{4\pi l_{n}}{\eta_{n}^{2}}\left\vert \chi_{\mu_{j}\nu_{k}}^{n}\right\vert ^{2},\end{aligned}$$ $$\gamma_{\mu_{j}\nu_{k}}^{g}=\gamma_{\mu_{j}\nu_{k}}-\gamma_{\mu_{j}\nu_{k}}^{d}=\overset{2}{\underset{n=0}{\sum}}\frac{2\pi l_{n}}{\eta_{n}^{2}}\left\vert \chi_{\mu_{j}\nu_{k}}^{n}\right\vert ^{2},$$ we find $\gamma_{\mu_{j}\nu_{k}}=-\gamma_{\mu_{j}\nu_{k}}^{g}=\frac{1}{2}\gamma_{\mu_{j}\nu_{k}}^{d}$. Thus the total phase $\gamma_{\mu_{j}\nu_{k}}$ and dynamical phase $\gamma_{\mu_{j}\nu_{k}}^{d}$ possess global geometric features as does the geometric phase $\gamma_{\mu_{j}\nu_{k}}^{g}$. Therefore at time $t=T=2\pi l_{n}/\eta_{n}$ the time evolution matrix takes the form $$U\left( T\right) =diag\left[ e^{i\gamma_{0_{j}0_{k}}}\text{, }e^{i\gamma_{0_{j}1_{k}}}\text{, }e^{i\gamma_{1_{j}0_{k}}}\text{, }e^{i\gamma_{1_{j}1_{k}}}\text{ }\right] .$$ $U\left( T\right) $ is actually the geometric entangling gate operation we are targeting at and $U\left( T\right) $ is a nontrivial entangling gate when the condition $\gamma_{0_{j}0_{k}}+\gamma_{1_{j}1_{k}}\neq\gamma
_{0_{j}1_{k}}+\gamma_{1_{j}1_{k}}$ is fulfilled [@X].
We now give a brief discussion about the decoherence mechanisms of our scheme: atomic spontaneous emission, cavity decay and fiber loss. Considering none of the atoms are initially populated in the excited state since the quantum information is encoded in ground states, and atoms cannot exchange energy with the fiber mode, cavity modes and classical fields due to the large detuning, thus no population is transferred to the excited atomic state. In this sense, the spontaneous emission of the atomic excited state can be ignored.
Regarding the cavity decay and the fiber loss, the fidelity of the resulting gates will be greatly impaired by them because the geometric phases are acquired by the evolution of the optical modes. So, strictly speaking, our scheme requires ideal good cavities and fiber. However, if the mean number of photons of the optical fields is sufficiently small, the cavities and fiber are normally not excited and the moderate cavity decay and fiber loss can thus be tolerated. For a coherent state the mean number of photons is equal to the square of the amplitude of the state which is determined by Eq.(15). Thus when the condition $\frac{\chi_{\mu_{j}\nu_{k}}^{n}}{\eta_{n}}\ll1$ is fulfilled [@CQED1], the mean number of photons of the coherent state is an even smaller number and can be regarded as a sufficiently small number to ignore the effect of cavity and fiber decay. Now let us use an example for further explanation. We choose the following experimentally achievable parameters [@Kimble] $\nu/2\pi=26.72$ MHz, $g_{0}/2\pi=g_{1}/2\pi=20$ MHz, $\Omega_{0}/2\pi
=\Omega_{1}/2\pi=120$ MHz, $\Delta_{0}/2\pi=3000$ MHz, $\Delta_{1}/2\pi=600$ MHz, $\delta/2\pi=35$ MHz. These parameters satisfy the requirement $\frac{\chi_{\mu
_{j}\nu_{k}}^{n}}{\eta_{n}}\ll1$ and the approximation conditions adopted in our derivation. The resulting entangling gate corresponding to these parameters is $U(t)=diag\left\{
e^{0.1248i},e^{1.056i},e^{1.056i},e^{i\pi }\right\} $ with the gate operation time $t\approx0.3448$ $\mu$s. Obviously the gate operation time is much shorter than the photon lifetime in optical cavities [@AA]. According to Eq. (15) the amplitude of the coherent state is dependent on the atomic states, for the above parameters the amplitude corresponding to state $\left\vert
1_{j}^{L}1_{k}^{L}\right\rangle $ takes the maximal value, and the maximal mean number of photons is 0.1087. In this case, the optical modes are hardly excited and thus the moderate cavity decay and fiber loss can be tolerated.
In conclusion, we have proposed a scheme to implement geometric entangling gates for two logical qubits in a coupled cavity system in DFS. Our scheme possesses both advantages of DFS and the geometric phase. Besides, in comparison with the scheme of Ref. [@X] which works in a single cavity, the scheme proposed in this paper can easily realize the scalability of cavity QED-based quantum computing by using the idea of the distributed quantum computing[@JA] and can relax the requirement for individually addressing atoms.
The work is supported by the NSFC under Grant No. 11074079, the Ph.D. Programs Foundation of Ministry of Education of China, the Open Fund of the State Key Laboratory of High Field Laser Physics ( Shanghai Institute of Optics and Fine Mechanics), and National Research Foundation and Ministry of Education, Singapore, under research Grant No. WBS: R-710-000-008-271.
[99]{}
|
---
abstract: |
We present an analysis of spectroscopic radial velocity and photometric data of three bright Galactic Cepheids: LR Trianguli Australis (LR TrA), RZ Velorum (RZ Vel), and BG Velorum (BG Vel). Based on new radial velocity data, these Cepheids have been found to be members of spectroscopic binary systems.
The ratio of the peak-to-peak radial velocity amplitude to photometric amplitude indicates the presence of a companion for LR TrA and BG Vel. [*IUE*]{} spectra indicate that the companions of RZ Vel and BG Vel cannot be hot stars.
The analysis of all available photometric data revealed that the pulsation period of RZ Vel and BG Vel varies monotonically, due to stellar evolution. Moreover, the longest period Cepheid in this sample, RZ Vel, shows period fluctuations superimposed on the monotonic period increase. The light-time effect interpretation of the observed pattern needs long-term photometric monitoring of this Cepheid. The pulsation period of LR TrA has remained constant since the discovery of its brightness variation.
Using statistical data, it is also shown that a large number of spectroscopic binaries still remain to be discovered among bright classical Cepheids.
author:
- |
L. Szabados$^{1}$, R. I. Anderson$^2$, A. Derekas$^{1,3}$, L. L. Kiss$^{1,3,4}$, T. Szalai$^5$, P. Székely$^6$, J. L. Christiansen$^7$\
$^1$Konkoly Observatory, Research Centre for Astronomy and Earth Sciences, Hungarian Academy of Sciences, Konkoly Thege Miklós út 15-17,\
H-1121 Budapest, Hungary\
$^2$Observatoire de Genève, Université de Genève, 51 Ch. des Maillettes, CH-1290 Versoix, Switzerland\
$^3$Sydney Institute for Astronomy, School of Physics, University of Sydney, NSW 2006, Australia\
$^4$ELTE Gothard-Lendület Research Group, Szent Imre herceg út 112, H-9700 Szombathely, Hungary\
$^5$Department of Optics and Quantum Electronics, University of Szeged, Dóm tér 9, H-6720 Szeged, Hungary\
$^6$Department of Experimental Physics, University of Szeged, Szeged H-6720, Hungary\
$^7$SETI Institute/NASA Ames Research Center, M/S 244-30, Moffett Field, CA 94035, USA
date: 'Accepted Received ; in original form '
title: Discovery of the spectroscopic binary nature of three bright southern Cepheids
---
\[firstpage\]
binaries: spectroscopic – stars: variables: Cepheids
Introduction {#intro}
============
Classical Cepheid variable stars are primary distance indicators and rank among standard candles for establishing the cosmic distance scale, owing to the famous period-luminosity ($P$–$L$) relationship. Companions to Cepheids, however, complicate the situation. The contribution of the secondary star to the observed brightness has to be taken into account when involving any particular Cepheid in the calibration of the $P$–$L$ relationship.
Binaries among Cepheids are not rare at all: their frequency exceeds 50 per cent for the brightest Cepheids, while among the fainter Cepheids an observational selection effect encumbers revealing binarity [@Sz03a].
Owing to some observational projects aimed at obtaining new radial velocities (RVs) of numerous Cepheids carried out during the last decades, a part of the selection effect has been removed. This progress is visualized in Fig. \[fig-comparison\] where the current situation is compared with that 20 years ago. The data have been taken from the on-line data base on binaries among Galactic Cepheids (http://www.konkoly.hu/CEP/orbit.html). To get rid of the fluctuation at the left-hand part of the diagram, brightest Cepheids ($\langle V \rangle <5$ mag) were merged in a single bin because such stars are extremely rare among Cepheids – see the histogram in Fig. \[fig-histogram\].
In the case of pulsating variables, like Cepheids, spectroscopic binarity manifests itself in a periodic variation of the $\gamma$-velocity (i.e., the RV of the mass centre of the Cepheid). In practice, the orbital RV variation of the Cepheid component is superimposed on the RV variations of pulsational origin. To separate orbital and pulsational effects, knowledge of the accurate pulsation period is essential, especially when comparing RV data obtained at widely differing epochs. Therefore, the pulsation period and its variations have been determined with the method of the O$-$C diagram [@S05] for each target Cepheid. Use of the accurate pulsation period obtained from the photometric data is a guarantee for the correct phase matching of the (usually less precise) RV data.
![Percentage of known binaries among Galactic classical Cepheids as a function of the mean apparent visual brightness in 1993 and 2013. The decreasing influence of the observational selection effect is noticeable.[]{data-label="fig-comparison"}](szabados2013sbcepfig1.eps){height="48mm"}
![Histogram showing the number distribution of known Galactic classical Cepheids as a function of their mean apparent visual brightness.[]{data-label="fig-histogram"}](szabados2013sbcepfig2.eps){height="48mm"}
In this paper we point out spectroscopic binarity of three bright Galactic Cepheids by analysing RV data. The structure of this paper is as follows. The new observations and the equipment utilized are described in Sect. \[newdata\]. Section \[results\] is devoted to the results on the three new spectroscopic binary (SB) Cepheids: LR Trianguli Australis, RZ Velorum, and BG Velorum. Basic information on these Cepheids is given in Table \[obsprop\]. Finally, Section \[concl\] contains our conclusions.
--------- --------------------- ----------- ---------------- ----- ---------
Cepheid $\langle V \rangle$ $P$ Mode
(mag) (d) of pulsation SSO CORALIE
LR TrA 7.80 2.428289 first overtone 10 32
RZ Vel 7.13 20.398532 fundamental 30 67
BG Vel 7.69 6.923843 fundamental 27 33
--------- --------------------- ----------- ---------------- ----- ---------
: Basic data of the programme stars and the number of spectra.[]{data-label="obsprop"}
New observations {#newdata}
================
Spectra from the Siding Spring Observatory {#SSO}
------------------------------------------
We performed an RV survey of Cepheids with the 2.3 m ANU telescope located at the Siding Spring Observatory (SSO), Australia. The main aim of the project was to detect Cepheids in binary systems by measuring changes in the mean values of their RV curve which can be interpreted as the orbital motion of the Cepheid around the centre-of-mass in a binary system (change of $\gamma$-velocity). The target list was compiled to include Cepheids with a single-epoch RV phase curve or without any published RV data. Several Cepheids suspected to be members of SB systems were also put on the target list. In 64 nights between 2004 October and 2006 March we monitored 40 Cepheids with pulsation periods between 2 and 30 d. Additional spectra of some targets were obtained in 2007 August.
Medium-resolution spectra were taken with the Double Beam Spectrograph using the 1200 mm$^{-1}$ gratings in both arms of the spectrograph. The projected slit width was 2 arcsec on the sky, which was about the median seeing during our observations. The spectra covered the wavelength ranges 4200–5200 Å in the blue arm and 5700–6700 Å in the red arm. The dispersion was 0.55 Å pixel$^{-1}$, leading to a nominal resolution of about 1 Å.
All spectra were reduced with standard tasks in [iraf]{} [^1]. Reduction consisted of bias and flat-field corrections, aperture extraction, wavelength calibration, and continuum normalization. We checked the consistency of wavelength calibrations via the constant positions of strong telluric features, which proved the stability of the system. RVs were determined only for the red arm data with the task [*fxcor*]{}, applying the cross-correlation method using a well-matching theoretical template spectrum from the extensive spectral library of @Metal05. Then, we made barycentric corrections to every single RV value. This method resulted in a 1-2 km s$^{-1}$ uncertainty in the individual RVs, while further tests have shown that our absolute velocity frame was stable to within $\pm$2–3 km s$^{-1}$. This level of precision is sufficient to detect a number of Cepheid companions, as they can often cause $\gamma$-velocity changes well above 10 km s$^{-1}$.
Discovery of six SBs among the 40 target Cepheids was already reported by @Szetal13. The binarity of the three Cepheids announced here could be revealed by involving independently obtained additional data (see Section \[coralie\]). The individual RV data of the rest of the Cepheid targets will be published together with the results of the analysis of the spectra.
CORALIE observations from La Silla {#coralie}
----------------------------------
All three Cepheids were among the targets during multiple observing campaigns between 2011 April and 2012 May using the fibre-fed high-resolution ($R \sim 60000$) echelle spectrograph *CORALIE* mounted on the Swiss 1.2m Euler telescope at ESO La Silla Observatory, Chile. The instrument’s design is described in @Qetal01; recent instrumental updates can be found in @Setal10.
When it turned out that these three Cepheids have variable $\gamma$-velocities, several new spectra were obtained in 2012 December - 2013 January and 2013 April.
The spectra are reduced by the efficient online reduction pipeline that performs bias correction, cosmics removal, and flatfielding using tungsten lamps. ThAr lamps are used for the wavelength calibration. The reduction pipeline directly determines the RV via cross-correlation [@Betal96] using a mask that resembles a G2 spectral type. The RV stability of the instrument is excellent and for non-pulsating stars the RV precision is limited by photon noise; (see e.g., @Petal02). However, the precision achieved for Cepheids is lower due to line asymmetries. We estimate a typical precision of $\sim$ 0.1kms$^{-1}$ (including systematics due to pulsation) per data point for our data.
Results for individual Cepheids {#results}
===============================
LR Trianguli Australis {#lrtra}
----------------------
#### Accurate value of the pulsation period {#lrtra-period .unnumbered}
The brightness variability of LR TrA (HD137626, $\langle V \rangle
= 7.80$mag) was revealed by @Setal66 based on the Bamberg photographic patrol plates. The Cepheid nature of variability and the first values of the pulsation period was determined by @E83. This Cepheid pulsates in the first-overtone mode; therefore, it has a small pulsational amplitude and nearly-sinusoidal light and velocity curves.
In the case of Cepheids pulsating with a low amplitude, the O$-$C diagram constructed for the median brightness (the mid-point between the faintest and the brightest states) is more reliable than that based on the moments of photometric maxima [@Detal12]. Therefore we determined the accurate value of the pulsation period by constructing an O$-$C diagram for the moments of median brightness on the ascending branch of the light curve since this is the phase when the brightness variations are steepest during the whole pulsational cycle.
All published photometric observations of LR TrA covering three decades were re-analysed in a homogeneous manner to determine seasonal moments of the chosen light-curve feature. The relevant data listed in Table \[tab-lrtra-oc\] are as follows:\
Column 1: heliocentric moment of the selected light-curve feature (median brightness on the ascending branch for LR TrA, maximum brightness for both RZ Vel and BG Vel, see Tables \[tab-rzvel-oc\] and \[tab-bgvel-oc\], respectively;\
Col. 2: epoch number, $E$, as calculated from Equation (\[lrtra-ephemeris\]): $$C = 2\,453\,104.9265 + 2.428\,289{\times}E
\label{lrtra-ephemeris}$$ $\phantom{mmmmm}\pm0.0037\phantom{}\pm0.000\,003$
(this ephemeris has been obtained by the weighted least squares parabolic fit to the O$-$C differences);\
Col. 3: the corresponding O$-$C value;\
Col. 4: weight assigned to the O$-$C value (1, 2, or 3 depending on the quality of the light curve leading to the given difference);\
Col. 5: reference to the origin of data.\
The O$-$C diagram of LR TrA based on the O$-$C values listed in Table \[tab-lrtra-oc\] is plotted in Fig. \[fig-lrtra-oc\]. The plot can be approximated by a constant period by the ephemeris (\[lrtra-ephemeris\]) for the moments of median brightness on the ascending branch. The scatter of the points in Fig. \[fig-lrtra-oc\] reflects the observational error and uncertainties in the analysis of the data.
[l@r@r@c@l]{} JD$_{\odot}$ & $E\ $ & O$-$C & $W$ & Data source\
2400000 + &&&\
45018.7822 & $-$3330& 0.0581 &3 & @E83\
47633.9607 & $-$2253& $-$0.0307 & 3 & @Aetal90\
47939.9568 & $-$2127& 0.0010 &2 & [*Hipparcos*]{} [@ESA97]\
48139.0426 & $-$2045& $-$0.0329 & 3& [*Hipparcos*]{} [@ESA97]\
48440.1554 & $-$1921& $-$0.0279 & 3& [*Hipparcos*]{} [@ESA97]\
48750.9547 & $-$1793& $-$0.0496 & 3& [*Hipparcos*]{} [@ESA97]\
49814.6064 & $-$1355& 0.0115 & 3 & @B08\
50370.7115 & $-$1126& 0.0384 & 3 & @B08\
50574.6393 & $-$1042& $-$0.0101 & 3& @B08\
50909.7531 & $-$904& $-$0.0001 & 3 & @B08\
51264.2883 & $-$758& 0.0049 & 3 & @B08\
51650.4058 & $-$599& 0.0244 & 3 & @B08\
51958.8010 & $-$472& 0.0269 & 2 & @B08\
52041.3435 & $-$438& 0.0076 & 2 & ASAS [@P02]\
52366.7222 & $-$304& $-$0.0044 & 3 & @B08\
52500.2709 & $-$249& $-$0.0116 & 3 & ASAS [@P02]\
52769.8038 & $-$138& $-$0.0188 & 3 & ASAS [@P02]\
53102.5159 & $-$1& 0.0177 & 3 & @B08\
53104.9151 & 0& $-$0.0114 & 3& ASAS [@P02]\
53520.1818 & 171& 0.0179 & 3 & ASAS [@P02]\
53840.7137 & 303& 0.0156 & 3 & ASAS [@P02]\
54251.0850 & 472& 0.0061 & 3& ASAS [@P02]\
54615.3163 & 622& $-$0.0060& 3 & ASAS [@P02]\
54960.1214 & 764& $-$0.0179& 3 & ASAS [@P02]\
\[tab-lrtra-oc\]
![O$-$C diagram of LR TrA. The plot can be approximated by a constant period.[]{data-label="fig-lrtra-oc"}](szabados2013sbcepfig3.eps){height="44mm"}
#### Binarity of LR TrA {#lrtra-bin .unnumbered}
![Merged RV phase curve of LR TrA. The different symbols mean data from different years: 2005: filled triangles; 2006: empty triangles; 2007: triangular star; 2012: filled circles; 2013: empty circles. The zero phase was arbitrarily chosen at JD2400000.0 (in all phase curves in this paper).[]{data-label="fig-lrtra-vrad"}](szabados2013sbcepfig4.eps){height="48mm"}
![Temporal variation in the $\gamma$-velocity of LR TrA. The symbols for the different data sets are the same as in Fig. \[fig-lrtra-vrad\].[]{data-label="fig-lrtra-vgamma"}](szabados2013sbcepfig5.eps){height="40mm"}
[lr]{} JD$_{\odot}$ & $v_{\rm rad}$ \
2400000 + &(kms$^{-1}$)\
53599.9325 &$-$21.2\
53600.9086 &$-$32.0\
53603.9327 &$-$27.6\
53605.9290 &$-$31.0\
53805.1657 &$-$29.3\
\[tab-lrtra-data\]
[lrc]{} JD$_{\odot}$ & $v_{\rm rad}$ & $\sigma$\
2400000 + &(kms$^{-1}$) & (kms$^{-1}$)\
55938.8701 & $-$27.97 & 0.05\
55938.8718 & $-$28.10 & 0.05\
55939.8651 & $-$29.85 & 0.02\
55940.8686 & $-$22.40 & 0.03\
55941.8579 & $-$33.14 & 0.04\
\[tab-lrtra-coralie-data\]
[lccl]{} Mid-JD & $v_{\gamma}$ & $\sigma$ & Data source\
2400000+ & (kms$^{-1}$)& (kms$^{-1}$) &\
53603 & $-$25.5 & 0.5 & Present paper\
53808 & $-$24.8 & 0.5 & Present paper\
54331 & $-$29.0 & 1.0 & Present paper\
55981 & $-$27.5 & 0.1 & Present paper\
56344 & $-$26.4 & 0.1 & Present paper\
\[tab-lrtra-vgamma\]
There are no earlier RV data on this bright Cepheid. Our new data listed in Tables \[tab-lrtra-data\] and \[tab-lrtra-coralie-data\] have been folded on the accurate pulsation period given in the ephemeris (see Equation \[lrtra-ephemeris\]). The merged RV phase curve is plotted in Fig. \[fig-lrtra-vrad\]. Both individual data series could be split into seasonal subsets.
Variability in the $\gamma$-velocity is obvious. The $\gamma$-velocities (together with their uncertainties) are listed in Table \[tab-lrtra-vgamma\]. The $\gamma$-velocity in 2007 is more uncertain than in other years because this value is based on a single spectrum. Systematic errors can be excluded. Dozens of Cepheids in our sample with non-varying $\gamma$-velocities indicate stability of the equipment and reliability of the data reduction. Fig. \[fig-lrtra-vgamma\] is a better visualization of the temporal variation in the $\gamma$-velocity. The seasonal drift in the $\gamma$-velocity is compatible with both short and long orbital periods.
The photometric contribution of the companion star decreases the observable amplitude of the brightness variability as deduced from the enhanced value of the ratio of the RV and photometric amplitudes [@KSz09]. This is an additional (although slight) indication of binarity of LR TrA.
RZ Velorum {#rzvel}
----------
#### Accurate value of the pulsation period {#rzvel-period .unnumbered}
The brightness variability of RZ Vel (HD73502, $\langle V \rangle
= 7.13$mag) was revealed by Cannon [@P09]. The Cepheid nature of variability and the pulsation period were established by @H36 based on the Harvard and Johannesburg photographic plate collection which was further investigated by @Oo36.
This is the longest period Cepheid announced in this paper and it has been frequently observed from the 1950s, first photoelectrically, then in the last decades by CCD photometry. The photometric coverage of RZ Vel was almost continuous in the last 20 years thanks to observational campaigns by @B08 and his co-workers, as well as the ASAS photometry [@P02].
Long-period Cepheids are usually fundamental pulsators and they oscillate with a large amplitude resulting in a light curve with sharp maximum.
The O$-$C diagram of RZ Vel was constructed for the moments of maximum brightness based on the photoelectric and CCD photometric data (see Table \[tab-rzvel-oc\]). The weighted least squares parabolic fit to the O$-$C values resulted in the ephemeris: $$C = 2\,442\,453.6630 + 20.398\,532{\times}E + 1.397\times 10^{-6} E^2
\label{rzvel-ephemeris}$$ $\phantom{mmmmm}\pm0.0263\phantom{l}\pm 0.000\,080 \phantom{mm}
\pm 0.191\times 10^{-6}$
[l@r@r@c@l]{} JD$_{\odot}$ & $E\ $ & O$-$C & $W$ & Data source\
2400000 + &&&\
33784.5646 &$-$425 & 0.2777 & 1 & @Eetal57\
34804.5174 &$-$375 & 0.3039 & 1 & @Wetal58\
34845.2119 &$-$373 & 0.2013 & 3 & @Eetal57\
35192.0024 &$-$356 & 0.2168 & 1 & @I61\
40760.8647 &$-$83 & 0.2799 & 3 & @P76\
41719.0924 &$-$36 &$-$0.2234& 3& @M75\
41862.1249 &$-$29 & 0.0193 & 3 & @Detal77\
42453.6330 & 0 &$-$0.0030 & 3 & @Detal77\
44371.0472 & 94 &$-$0.0778 & 3 & @CC85\
44391.3842 & 95 &$-$0.1393 & 2 & @E82\
45003.2906 & 125 &$-$0.1889 & 3 & @CC85\
48226.4369 & 283 &$-$0.0107 & 3 & [*Hipparcos*]{} [@ESA97]\
48797.5877 & 311 &$-$0.0188 & 3 & [*Hipparcos*]{} [@ESA97]\
49185.1653 & 330 &$-$0.0133 & 1 & Walker & Williams (unpublished)\
49817.8011 & 361 & 0.2680 &3 & @B08\
50144.1979 & 377 & 0.2883 & 2 & @B02\
50389.0443 & 389 & 0.3524 & 3 & @B08\
50511.3662 & 395 & 0.2831 & 3 & @B02\
50572.4468 & 398 & 0.1681 & 3 & @B08\
50899.0581 & 414 & 0.4029 & 3 & @B08\
51266.1488 & 432 & 0.3200 & 3 & @B08\
51653.7650 & 451 & 0.3641 & 3 & @B08\
51939.2846 & 465 & 0.3042 & 2 & ASAS [@P02]\
51959.7692 & 466 & 0.3903 & 3 & @B08\
52347.4262 & 485 & 0.4752 & 3 & @B08\
52653.3896 & 500 & 0.4606 & 3 & ASAS [@P02]\
52653.4100 & 500 & 0.4810 & 3 & @B08\
53000.1794 & 517 & 0.4754 & 3 & ASAS [@P02]\
53000.2610 & 517 & 0.5570 & 3 & @B08\
53428.4384 & 538 & 0.3652 & 3 & ASAS [@P02]\
53754.8864 & 554 & 0.4367 & 3 & ASAS [@P02]\
54183.1657 & 575 & 0.3468 & 3 & ASAS [@P02]\
54509.5729 & 591 & 0.3775 & 3 & ASAS [@P02]\
54815.4343 & 606 & 0.2609 & 3 & ASAS [@P02]\
55121.3569 & 621 & 0.2055 & 2 & ASAS [@P02]\
\[tab-rzvel-oc\]
The O$-$C diagram of RZ Vel plotted in Fig. \[fig-rzvel-oc\] indicates a continuously increasing pulsation period with a period jitter superimposed. This secular period increase has been caused by stellar evolution: while the Cepheid crosses the instability region towards lower temperatures in the Hertzsprung–Russell diagram, its pulsation period is increasing. Continuous period variations (of either sign) often occur in the pulsation of long-period Cepheids [@Sz83].
Fig. \[fig-rzvel-oc2\] shows the O$-$C residuals after subtracting the parabolic fit defined by Equation (\[rzvel-ephemeris\]). If the wave-like fluctuation seen in this $\Delta (O-C)$ diagram turns out to be periodic, it would correspond to a light-time effect in a binary system. In line with the recent shortening in the pulsation period, the current value of the pulsation period is $20.396671 \pm 0.000200$ days (after JD 2452300).
![O$-$C diagram of RZ Vel. The plot can be approximated by a parabola indicating a continuously increasing period.[]{data-label="fig-rzvel-oc"}](szabados2013sbcepfig6.eps){height="55mm"}
![$\Delta(O-C)$ diagram of RZ Vel.[]{data-label="fig-rzvel-oc2"}](szabados2013sbcepfig7.eps){height="44mm"}
#### Binarity of RZ Vel {#rzvel-bin .unnumbered}
![RV phase curve of RZ Vel. Data obtained between 1996 and 2013 are included in this plot. The meaning of various symbols is explained in the text.[]{data-label="fig-rzvel-vrad"}](szabados2013sbcepfig8.eps){height="55mm"}
![$\gamma$-velocities of RZ Velorum. The symbols for the different data sets are the same as in Fig. \[fig-rzvel-vrad\].[]{data-label="fig-rzvel-vgamma"}](szabados2013sbcepfig9.eps){height="42mm"}
[lr]{} JD$_{\odot}$ & $v_{\rm rad}$ \
2400000 + &(kms$^{-1}$)\
53307.2698 &4.2\
53310.2504 &1.4\
53312.2073 &9.0\
53364.2062 &49.6\
53367.1823 &27.5\
\[tab-rzvel-data\]
[lrc]{} JD$_{\odot}$ & $v_{\rm rad}$ & $\sigma$\
2400000 + &(kms$^{-1}$) & (kms$^{-1}$)\
55654.5528 & $-$3.08 & 0.02\
55656.6626 & 5.23 & 0.01\
55657.6721 & 9.86 & 0.02\
55659.6585 & 18.85 & 0.03\
55662.5137 & 31.50 & 0.01\
\[tab-rzvel-coralie-data\]
[lccl]{} Mid-JD & $v_{\gamma}$ & $\sigma$ & Data source\
2400000+ & (kms$^{-1}$)& (kms$^{-1}$) &\
34009 &25.5 &1.5& @S55\
40328 &22.1 &1.5& @LE68 [@LE80]\
42186 &29.2 &1.0& @CC85\
44186 &22.6 &1.0& @CC85\
44736 &24.4 &1.0& @CC85\
50317 &25.1 &0.2& @B02\
53184 &24.0 &0.5& @Netal06\
53444 &26.9 &0.6& Present paper\
53783 &28.8 &1.0& Present paper\
55709 &25.6 &0.1& Present paper\
56038 &25.3 &0.1& Present paper\
\[tab-rzvel-vgamma\]
There are several data sets of RV observations available in the literature for RZ Vel: those published by @S55, @LE68 [@LE80], @CC85, @B02, and @Netal06. Our individual RV data are listed in Tables \[tab-rzvel-data\] and \[tab-rzvel-coralie-data\].
Based on these data, the RV phase curve has been constructed using the 20.398532 d pulsation period appearing in Equation (\[rzvel-ephemeris\]). In view of the complicated pattern of the O$-$C diagram the RV data have been folded on by taking into account the proper phase correction for different data series. The merged RV phase curve is plotted in Fig. \[fig-rzvel-vrad\]. For the sake of clarity, RV data obtained before JD2450000 have not been plotted here because of the wider scatter of these early RV data but the $\gamma$-velocities were determined for each data set. The individual data series are denoted by different symbols: filled squares mean data by @B02, empty squares those by @Netal06, and our 2005, 2006, 2012 and 2013 data are denoted by filled triangles, empty triangles, filled circles and empty circles, respectively. The wide scatter in this merged RV phase curve plotted in Fig. \[fig-rzvel-vrad\] is due to a variable $\gamma$-velocity.
The $\gamma$-velocities determined from each data set (including the earlier ones) are listed in Table \[tab-rzvel-vgamma\] and are plotted in Fig. \[fig-rzvel-vgamma\]. The plot implies that RZ Vel is really an SB as suspected by @B02 based on a much poorer observational material (before JD 2450500). An orbital period of about 5600-5700 d is compatible with the data pattern in both Fig. \[fig-rzvel-oc2\] and Fig. \[fig-rzvel-vgamma\] but the phase relation between the light-time effect fit to the $\Delta (O-C)$ curve and the orbital RV variation phase curve obtained with this formal period is not satisfactory.
BG Velorum {#bgvel}
----------
#### Accurate value of the pulsation period {#bgvel-period .unnumbered}
The brightness variability of BG Vel (HD78801, $\langle V \rangle
= 7.69$mag) was revealed by Cannon [@P09]. Much later @OL37 independently discovered its light variations but he also revealed the Cepheid nature and determined the pulsation period based on photographic plates obtained at the Riverview College Observatory. @vH50 also observed this Cepheid photographically in Johannesburg but these early data are unavailable, therefore we only mention their studies for historical reasons.
This Cepheid is a fundamental-mode pulsator. The O$-$C differences of BG Vel calculated for brightness maxima are listed in Table \[tab-bgvel-oc\]. These values have been obtained by taking into account the constant and linear terms of the following weighted parabolic fit: $$C = 2\,453\,031.4706 + 6.923\,843{\times}E + 2.58\times 10^{-8} E^2
\label{bgvel-ephemeris}$$ $\phantom{mmmmm}\pm0.0020\phantom{}\pm 0.000\,007 \phantom{ml}
\pm 0.27\times 10^{-8}$
The parabolic nature of the O$-$C diagram, i.e., the continuous increase in the pulsation period, is clearly seen in Fig. \[fig-bgvel-oc\]. This parabolic trend corresponds to a continuous period increase of $(5.16 \pm 0.54)\times 10^{-8}$ dcycle$^{-1}$, i.e., $\Delta P = 0.000272$ d/century. This tiny period increase has been also caused by stellar evolution as in the case of RZ Vel.
The fluctuations around the fitted parabola in Fig. \[fig-bgvel-oc\] do not show any definite pattern: see the $\Delta(O-C)$ diagram in Fig. \[fig-bgvel-oc2\].
[l@r@r@c@l]{} JD$_{\odot}$ & $E\ $ & O$-$C & $W$ & Data source\
2400000 + &&&\
34856.5526 & $-$2625 & 0.1699 & 3 & @Wetal58\
35237.3813 & $-$2570 & 0.1872 & 3 & @I61\
40748.6592 & $-$1774 & 0.0861 & 3 & @P76\
42853.4433 & $-$1470 & 0.0219 & 3 & @D77\
44300.5426 & $-$1261 & 0.0380 & 3 & @B08\
48136.3167 & $-$707 & 0.0031 & 3 & [*Hipparcos*]{} [@ESA97]\
48627.9239 & $-$636 & 0.0174 & 3 & [*Hipparcos*]{} [@ESA97]\
50379.6329 & $-$383 & $-$0.0058 & 3 & @B08\
50573.4987 & $-$355 & $-$0.0076 & 3 & @B08\
50905.8549 & $-$307 & 0.0041 & 3 & @B08\
51265.9127 & $-$255 & 0.0221 & 3 & @B08\
51646.7345 & $-$200 & 0.0325 & 3 & @B08\
51937.5210 & $-$158 & 0.0176 & 3 & ASAS [@P02]\
51958.2712 & $-$155 & $-$0.0038 & 3 & @B08\
52359.8640 & $-$97 & 0.0062 & 3 & ASAS [@P02]\
52359.8778 & $-$97 & 0.0200 & 3 & @B08\
52650.6575 & $-$55 & $-$0.0017 & 3 & @B08\
52726.8212 & $-$44 & $-$0.0003 & 3 & ASAS [@P02]\
53003.7916 & $-$4 & 0.0164 & 3 & @B08\
53031.4758 & 0 & 0.0052 & 3 & ASAS [@P02]\
53336.1201 & 44 &0.0004 & 1 & [*INTEGRAL*]{} OMC\
53460.7390 & 62 & $-$0.0099 & 3 & ASAS [@P02]\
53779.2202 & 108 & $-$0.0254 & 3 & ASAS [@P02]\
54180.8337 & 166 & 0.0052 & 3 & ASAS [@P02]\
54540.8499 & 218 & $-$0.0185 & 3 & ASAS [@P02]\
54838.5810 & 261 & $-$0.0126 & 3 & ASAS [@P02]\
55143.2425 & 305 & $-$0.0002 & 2 & ASAS [@P02]\
\[tab-bgvel-oc\]
#### Binarity of BG Vel {#bgvel-bin .unnumbered}
![O$-$C diagram of BG Vel. The plot can be approximated by a parabola indicating a continuously increasing pulsation period.[]{data-label="fig-bgvel-oc"}](szabados2013sbcepfig10.eps){height="44mm"}
![$\Delta(O-C)$ diagram of BG Vel.[]{data-label="fig-bgvel-oc2"}](szabados2013sbcepfig11.eps){height="44mm"}
![Merged RV phase curve of BG Vel. There is an obvious shift between the $\gamma$-velocities valid for the epoch of our data obtained in 2005-2006 and 2012-2013 (empty and filled circles, respectively). The other symbols are explained in the text.[]{data-label="fig-bgvel-vrad"}](szabados2013sbcepfig12.eps){height="49mm"}
![$\gamma$-velocities of BG Vel. The symbols for the different data sets are the same as in Fig. \[fig-bgvel-vrad\].[]{data-label="fig-bgvel-vgamma"}](szabados2013sbcepfig13.eps){height="42mm"}
[lr]{} JD$_{\odot}$ & $v_{\rm rad}$ \
2400000 + &(kms$^{-1}$)\
53312.2372 &17.3\
53364.2219 &$-$0.2\
53367.1992 &20.5\
53451.0000 &20.0\
53452.0021 &23.8\
\[tab-bgvel-data\]
[lrc]{} JD$_{\odot}$ & $v_{\rm rad}$ & $\sigma$\
2400000 + &(kms$^{-1}$) & (kms$^{-1}$)\
55937.7555 & 24.13 & 0.02\
55938.6241 & 7.77 & 0.02\
55939.6522 & $-$1.25 & 0.01\
55941.6474 & 7.99 & 0.10\
55942.6917 & 11.78 & 0.03\
\[tab-bgvel-coralie-data\]
There are earlier RV data of this Cepheid obtained by @S55 and @LE80. Variability in the $\gamma$-velocity is seen in the merged phase diagram of all RV data of BG Velorum plotted in Fig. \[fig-bgvel-vrad\]. In this diagram, our 2005–2006 data (listed in Table \[tab-bgvel-data\]) are represented with the empty circles, while 2012–2013 data (listed in Table \[tab-bgvel-coralie-data\]) are denoted by the filled circles, the triangles represent Stibbs’ data, and the $\times$ symbols refer to Lloyd Evans’ data. Our RV data have been folded with the period given in the ephemeris Equation (\[bgvel-ephemeris\]) omitting the quadratic term. Data obtained by Stibbs and Lloyd Evans have been phased with the same period but a proper correction has been applied to allow for the phase shift due to the parabolic O$-$C graph.
The $\gamma$-velocities determined from the individual data sets are listed in Table \[tab-bgvel-vgamma\] and plotted in Fig. \[fig-bgvel-vgamma\]. Since no annual shift is seen in the $\gamma$-velocities between two consecutive years (2005–2006 and 2012–2013), the orbital period cannot be short, probably it exceeds a thousand days.
Similarly to the case of LR TrA, BG Vel is also characterized by an excessive value for the ratio of RV and photometric amplitudes indicating the possible presence of a companion (see Fig. \[fig-ampratio\]).
[lccl]{} Mid-JD & $v_{\gamma}$ & $\sigma$ & Data source\
2400000+ & (kms$^{-1}$)& (kms$^{-1}$) &\
34096 &11.4 &1.5& @S55\
40545 & 8.4 &1.5& @LE80\
53572 &12.6 &0.6& Present paper\
56043 &10.3 &0.1& Present paper\
\[tab-bgvel-vgamma\]
Conclusions {#concl}
===========
We pointed out that three bright southern Galactic Cepheids, LR TrA, RZ Vel and BG Vel, have a variable $\gamma$-velocity implying their membership in SB systems. RV values of other target Cepheids observed with the same equipment in 2005–2006 and 2012 testify that this variability in the $\gamma$-velocity is not of instrumental origin, nor an artefact caused by the analysis.
The available RV data are insufficient to determine the orbital period and other elements of the orbits. However, some inferences can be made from the temporal variations of the $\gamma$-velocity. An orbital period of 5600–5700 d of the RZ Vel system is compatible with the data pattern. In the case of BG Vel, short orbital periodicity can be ruled out. For LR TrA, even the range of the possible orbital periods remains uncertain.
The value of the orbital period for SB systems involving a Cepheid component is often unknown: according to the on-line data base [@Sz03a] the orbital period has been determined for about 20% of the known SB Cepheids. The majority of known orbital periods exceeds a thousand days.
A companion star may have various effects on the observable photometric properties of the Cepheid component. Various pieces of evidence of duplicity based on the photometric criteria are discussed by @Sz03b and @KSz09. As to our targets, there is no obvious sign of a companion from optical multicolour photometry. This indicates that the companion star cannot be much hotter than any of the Cepheids discussed here. There is, however, a phenomenological parameter, viz. the ratio of RV to photometric amplitudes [@KSz09] whose excessive value is a further hint at the probable existence of a companion for both LR TrA and BG Vel (see Fig. \[fig-ampratio\]). Moreover, the [*IUE*]{} spectra of bright Cepheids analysed by @E92 gave a constraint on the temperature of a companion to remain undetected in the ultraviolet spectra: in the case of RZ Vel, the spectral type of the companion cannot be earlier than A7, while for BG Vel this limiting spectral type is A0. Further spectroscopic observations are necessary to characterize these newly detected SB systems.
![The slightly excessive value of the $A_{V_{\rm RAD}}/A_B$ amplitude ratio of LR TrA and BG Vel (large circles) with respect to the average value characteristic at the given pulsation period is an independent indication of the presence of a companion star. This is a modified version of fig. 4f of @KSz09. The open symbols in the original figure correspond to known binaries and the filled symbols to Cepheids without known binarity. For the meaning of various symbols, see @KSz09. []{data-label="fig-ampratio"}](szabados2013sbcepfig14.eps){height="54mm"}
Our findings confirm the previous statement by @Sz03a about the high percentage of binaries among classical Cepheids and the observational selection effect hindering the discovery of new cases (see also Fig. \[fig-comparison\]).
Regular monitoring of the RVs of a large number of Cepheids will be instrumental in finding more SBs among Cepheids. RV data to be obtained with the [*Gaia*]{} astrometric space probe (expected launch: 2013 September) will certainly result in revealing new SBs among Cepheids brighter than the 13–14th magnitude [@Eyetal12]. In this manner, the ‘missing’ SBs among Cepheids inferred from Fig. \[fig-comparison\] can be successfully revealed within few years.
Acknowledgments {#acknowledgments .unnumbered}
===============
This project has been supported by the ESTEC Contract No.4000106398/12/NL/KML, the Hungarian OTKA Grants K76816, K83790, K104607, and MB08C 81013, as well as the European Community’s Seventh Framework Program (FP7/2007-2013) under grant agreement no.269194, and the “Lendület-2009” Young Researchers Program of the Hungarian Academy of Sciences. AD was supported by the Hungarian Eötvös Fellowship. AD has also been supported by a János Bolyai Research Scholarship of the Hungarian Academy of Sciences. AD is very thankful to the staff at The Lodge in the Siding Spring Observatory for their hospitality and very nice food, making the time spent there lovely and special. Part of the research leading to these results has received funding from the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007–2013)/ERC grant agreement no.227224 (PROSPERITY). The [*INTEGRAL*]{} photometric data, pre-processed by ISDC, have been retrieved from the OMC Archive at CAB (INTA-CSIC). We are indebted to Stanley Walker for sending us some unpublished photoelectric observational data. Our thanks are also due to the referee and Dr. Mária Kun for their critical remarks leading to a considerable improvement in the presentation of the results.
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\[lastpage\]
[^1]: [iraf]{} is distributed by the National Optical Astronomy Observatories, which are operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation.
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abstract: 'This paper studies a problem of controlling trajectories of a platoon of vehicles on a highway section with advanced connected and automated vehicle technologies. This problem is very complex because each vehicle trajectory is essentially an infinite-dimensional object and neighboring trajectories have complex interactions (e.g., car-following behavior). A parsimonious shooting heuristic algorithm is proposed to construct vehicle trajectories on a signalized highway section that comply with the boundary condition for vehicle arrivals, vehicle mechanical limits, traffic lights and vehicle following safety. This algorithm breaks each vehicle trajectory into a few segments that each is analytically solvable. This essentially decomposes the original hard trajectory control problem to a simple constructive heuristic. Then we slightly adapt this shooting heuristic algorithm to one that can efficiently solve the leading vehicle problem on an uninterrupted freeway. To study theoretical properties of the proposed algorithms, the time geography theory is generalized by considering finite accelerations. With this generalized theory, it is found that under mild conditions, these algorithms can always obtain a feasible solution to the original complex trajectory control problem. Further, we discover that the shooting heuristic solution is a generalization of the solution to the classic kinematic wave theory by incorporating finite accelerations. We identify the theoretical bounds to the difference between the shooting heuristic solution and the kinematic wave solution. Numerical experiments are conducted to verify the theoretical results and to draw additional managerial insights into the potential of trajectory control in improving traffic performance. In summary, this paper provides a methodological and theoretical foundation for advanced traffic control by optimizing the trajectories of connected and automated vehicles. Built upon this foundation, an optimization framework will be presented in a following paper as Part II of this study.'
author:
- |
Fang Zhou$^{\mbox{a}}$, Xiaopeng Li$^{\mbox{a}}$[^1], Jiaqi Ma$^{\mbox{b}}$\
a. Department of Civil and Environmental Engineering,\
Mississippi State University, MS 39762, USA\
b. Transportation Solutions and Technology Applications Division,\
Leidos, Inc., Reston, VA 20190, USA
bibliography:
- 'Literature.bib'
title: 'Parsimonious shooting heuristic for trajectory control of connected automated traffic part I: Theoretical analysis with generalized time geography'
---
Introduction
============
Background
----------
As illustrated by the trajectories in the time-space diagram in Figure \[fig:motivating\_example\](a), traffic on a signalized arterial is usually forced to decelerate and accelerate abruptly as a result of alternating green and red lights. When traffic density is relatively high, stop-and-go traffic patterns will be formed and propagated backwards along so-called shock waves. Similar stop-and-go traffic also occurs frequently on freeways even without explicit signal interruptions. Such stop-and-go traffic imposes a number an adverse impacts to highway performance. Obviously, vehicles engaged in abrupt stop-and-go movements are exposed to a high crash risk [@hoffmann1994drivers], not to mention extra discomfort to drivers [@beard2013discomfort]. Also, frequent decelerations and accelerations cause excessive fuel consumption and emissions [@li2014stop], which pose a severe threat to the urban environment. Further, when vehicles slow down or stop, the corresponding traffic throughput decreases and the highway capacity drops [@Cassidy1999], which can cause excessive travel delay.
![Vehicle trajectories along a highway section upstream of an signalized intersection: (a) benchmark manual vehicle trajectories; (b) smoothed automated vehicle trajectories. \[fig:motivating\_example\]](Manual_traj_illustr "fig:")![Vehicle trajectories along a highway section upstream of an signalized intersection: (a) benchmark manual vehicle trajectories; (b) smoothed automated vehicle trajectories. \[fig:motivating\_example\]](Auto_traj_illustr "fig:")
(a)(b)
Although stop-and-go traffic has been intensively studied in the context of freeway traffic with either theoretical models (e.g., @Herman1958 [@Bando1995; @Li09c]) or empirical observations (e.g., @Kuhne1987 [@Kerner1996; @Mauch2002; @ahn2005; @Li09c; @Laval11]), few studies had investigated how to smooth traffic and alleviate corresponding adverse consequences on signalized highways until the advent of vehicle-based communication (e.g., connected vehicles or CV) and control (e.g., automated vehicles or AV) technologies. CV basically enables real-time information sharing and communications among individual vehicles and infrastructure control units[^2]. AV aims to replacing a human driver with a robot that constantly receives environmental information via various sensor technologies (as compared to human eyes and ears) and consequently determines vehicle control decisions (e.g., acceleration and braking) with proper computer algorithms (as compared to human brains) and vehicle control mechanics (as compared to human limbs)[^3]. The combination of these two technologies, which is referred as connected and automated vehicles (CAV), essentially enables disaggregated control (or coordination) of individual vehicles with real-time vehicle-to-vehicle and vehicle-to-infrastructure communications. Before these technologies, highway vehicle dynamics was essentially determined by microscopic human driving behavior. However, there is not even a universally accepted formulation of human driving behavior [@treiber2010three] due to the unpredictable nature of humans [@Kerner1996] and limited empirical data to comprehensively describe such behavior [@daganzo1999possible]. Therefore, it was very challenging, if not completely impossible, to perfectly smooth vehicle trajectories with traditional infrastructure-based controls (e.g., traffic signals) that are designed to accommodate human behavior. Whereas CAV enables replacing (at least partially) human drivers with programmable robots whose driving algorithms can be flexibly customized and accurately executed. This opens up a range of opportunities to control individual vehicle trajectories in ways that cooperate with aggregated infrastructure-based controls so as to optimize both individual drivers’ experience and overall traffic performance. These opportunities inspired several pioneering studies to explore how to utilize CAV to improve mobility and safety at intersections [@dresner2008; @Lee2012] and reduce environmental impacts along highway segments [@ahn2013ecodrive; @yang2014control]. However, these limited studies mostly focus on controlling one or very few vehicles at a particular highway facility (e.g., either an intersection or a segment) to achieve a certain specific objective (e.g., stability, safety or fuel consumption) rather than smoothing a stream of vehicles to improve its overall traffic performance. Mos of the developed control algorithms require sophisticated numerical computations and their real-time applications might be hindered by excessive computational complexities.
This study aims to propose a new CAV-based traffic control framework that controls detailed trajectory shapes of a stream of vehicles on a stretch of highway combining a one-lane section and a signalized intersection. As illustrated in Figure \[fig:motivating\_example\](b), the very basic idea of this study is smoothing vehicle trajectories and clustering them to platoons that can just properly occupy the green light windows and pass the intersection at a high speed. Note that a higher passing speed indicates a larger intersection capacity, and thus we see that the CAVs in Figure \[fig:motivating\_example\](b) not only have much smoother trajectories but also spend much less travel times compared with the benchmark manual vehicles in Figure \[fig:motivating\_example\](a). Further, smoother trajectories imply safer traffic, less fuel consumption, milder emissions, and better driver experience. While the research idea is intuitive, the technical development is quite sophisticated, because this study needs to manipulate continuous trajectories that not only individually have infinite control points but also have complex interactions between one another due to the shared rights of way. In order to overcome these modeling challenges, we first partition each trajectory into a few parabolic segments that each is analytically solvable. This essentially reduces an infinite-dimensional trajectory into a few set of parabolic function parameters. Further, we only use four acceleration and deceleration variables that are nonetheless able to control the overall smoothness of the whole stream of vehicle trajectories while assuring the exceptional parsimony and simplicity of the proposed algorithm. With these treatments, we propose an efficient shooting heuristic algorithm that can generate a stream of smooth and properly platooned trajectories that can pass the intersection efficiently and safely yielding minimum environmental impacts. Also, note that this algorithm can be easily adapted to freeway speed harmonization as well because the freeway trajectory control problem is essentially a special case of the investigated problem with an infinite green time. We investigate a lead vehicle problem to study this freeway adaption. After generalizing the concept of time geography [@miller2005measurement] by allowing finite acceleration and deceleration, we discover a number of elegant properties of these algorithms in the feasibility to the original trajectory control problem and the connection with classic traffic flow models.
This Part I paper focuses construction of parsimonious feasible algorithms and analysis of related theoretical properties. We want to note that the ultimate goal of this whole study is to establish a methodology framework that determines the best trajectory vectors under several traffic performance measures, such as travel time, fuel consumption, emission and safety surrogate measures. While this paper also qualitatively discusses optimality issues with visual patterns in trajectory plots, we leave the detailed computational issues and the overall optimization framework to the Part II paper [@Ma2015].
Literature Review
-----------------
Freeway traffic smoothing has drawn numerous attentions from both academia and industry in the past several decades. Numerous studies have been conducted in attempts to characterize stop-and-go traffic on freeway [@Herman1958; @Chandler1958; @Kuhne1987; @Bando1995; @Kerner1997; @Bando1998; @Kerner1998; @Mauch2002; @ahn2005; @Li09c; @Laval11] @Herman1958 [@Bando1995; @Li09c]. However, probably due to the lack of high resolution trajectory data [@daganzo1999possible], no consensus has been formed on fundamental mechanisms of stop-and-go traffic formation and propagation, particularly at the microscopic level [@treiber2010three]. To harmonize freeway traffic speed, scholars and practitioners have proposed and tested a number of infrastructure-based control methods mostly targeting at aggregated traffic (rather than individual vehicles), including variable speed limits [@lu2014review], ramp metering[@hegyi2005model], and merging traffic control [@spiliopoulou2009toll]. While theoretical results show that these speed harmonization methods can drastically improve traffic performance in all major performance measures, e.g., safety, mobility and environmental impacts [@islam2013assessing; @yang2014control], field studies show the performances of these methods exhibit quite some discrepancies [@bham2010evaluation]. Probably due to limited understandings of microscopic behavior of highway traffic, these field practices of speed harmonization are mostly based on empirical experience and trial-and-error approaches without taking full advantage of theoretical models. Also, drivers may not fully comply with the speed harmonization control and their individual responses may be highly stochastic, which further comprises the actual performance of these control strategies. Therefore, these aggregated infrastructure-based traffic smoothing measures may not perform as ideally as theoretical model predictions.
With the advance of vehicle-based communication (i.e., CV) and control (i.e., AV) technologies, researcher started exploring ways of freeway traffic smoothing by controlling individual vehicles. @schwarzkopf1977control analytically solved the optimal trajectory of a single vehicle on certain grade profiles with simple assumptions of vehicle characteristics based on Pontryagin’s minimum principle. @hooker1988optimal instead proposed a simulation approach that capture more realistic vehicle characteristics. @Van_Arem2006 finds that traffic flow stability and efficiency at a merge point can be improved by cooperative adapted cruise control (a longitudinal control strategy of CAV) that smooths car-following movements. @liu2012reducing solved an optimal trajectory for one single vehicle and used this trajectory as a template to control multiple vehicles with variable speed limits. @ahn2013ecodrive proposed an rolling-horizon individual CAV control strategy that minimizes fuel consumption and emission considering roadway geometries (e.g., grades). @yang2014control proposed a vehicle speed control strategy to mitigate traffic oscillation and reduce vehicle fuel consumption and emission based on connected vehicle technologies. They found with only a 5 percent compliance rate, this control strategy can reduce traffic fuel consumption by up to 15 percent. Wang et. al. [-@wang2014rolling-non-coop; -@wang2014rolling-coop] proposed optimal control models based on dynamic programming and Pontryagin’s minimum principle that determine accelerations of a platoon of AVs or CAVs to minimize a variety of objective cost functions. @li2014stop revised a classic manual car-following model into one for CAV following by incorporating CAV features such as faster responding time and shared information. They found the CAV following rules can significantly reduce magnitudes of traffic oscillation, emissions and travel time. Despite relatively homogenous settings and complex algorithms, these adventurous developments have demonstrated a great potential of these advanced technologies in improving freeway mobility, safety and environment.
Despite these fruitful developments on the freeway side, traffic smoothing on interrupted highways (i.e., with at-grade intersections) is a relative recent concept. This concept is probably motivated by recent CAV technologies that allow vehicles paths to be coordinated with signal controls. The existing traffic smooth studies for interrupted highways can be in general categorized into two types. The first type assumes that CAVs can communicate with each other to pass an intersection in a self-organized manner (e.g., like a school of fish) even without conventional traffic signals. For example, @dresner2008 proposes a heuristic control algorithm that process vehicles as a queuing system. While this development probably performs excellently when traffic is light or moderate, its performance under dense traffic is yet to be investigated. Further, @lee2012development proposes a nonlinear optimization model to test the limits of a non-stop intersection control scheme. They show that ideally, the optimal non-stop intersection control can significantly outperform classic signalized control in both mobility and environmental impacts at different congestion levels. @zohdy2014intersection integrates an embedded car-following rule and an intersection communication protocol into an nonlinear optimization model that manages a non-stop intersection. This model considers different weather conditions, heterogeneous vehicle characteristics and varying market penetrations. Overall, these developments on non-stop unsignalized control usually only focus on the operations of vehicles in the vicinity of an intersection and require complicated control algorithms and simulation. How to implement this complex mathematical programming model in real-time application might need further investigations.
The second type of studies for interrupted traffic smoothing consider how to design vehicles trajectories in compliance with existing traffic signal controls at intersections. The basic ideal is that a vehicle shall slow down from a distance when it is approaching to a red light so that this vehicle might be able to pass the next green light following a relatively smooth trajectory without an abrupt stop. Trayford et al. [-@trayford1984fuel2; -@trayford1984fuel1] tested using speed advice to vehicles approaching to an intersection so as to reduce fuel consumption with computer simulation. Later studies extend the speed advice approach to car-following dynamics [@sanchez2006predicting], in-vehicle traffic light assistance [@iglesias2008i2v; @wu2010energy] multi-intersection corridors [@mandava2009arterial; @guan2013predictive; @de2013eco], scaled-up simulation [@tielert2010impact], and electric vehicles [@wu2015Energy]. These approaches mainly focused on the bulky part of a vehicle’s trajectory with constant cruise speeds without much tuning its microscopic acceleration. However, acceleration detail actually largely affects a vehicle’s fuel consumption and emissions [@rakha2011eco]. To address this issue, @kamalanathsharma2013multi proposes an optimization model that considers a more realistic yet more sophisticated fuel-consumption objective function in smoothing a single vehicle trajectory at an signalized intersection. While such a model captures the advantage from microscopically tuning vehicle acceleration, it requires a complicated numerical solution algorithm that takes quite some computation resources even for a single trajectory.
In summary, there have been increasing interests in vehicle smoothing using advanced vehicle-based technologies in recent years. However, most relevant studies only focus on controlling one or a few individual vehicles. Most studies either ignore acceleration detail and allow speed jumps to assure the model computational tractability or capture acceleration in very sophisticated algorithms that are difficult to be simply implemented in real time. Further, few studies investigated theoretical properties of the proposed controls and their relationships with classic traffic flow theories. Without such theoretical insights, we would miss the great opportunity of transferring the vast elegant developments on existing manual traffic in the past few decades to future CAV traffic.
This proposed trajectory optimization framework aims to fill these research gaps. We investigate a general trajectory control problem that optimizes individual trajectories of a long stream of interactive CAVs on a signalized highway section. This problem is general such that the lead vehicle problem on a freeway can be represented as its special case (e.g., by setting the red light time to zero). This problem is a very challenging infinite-dimension nonlinear optimization problem, and thus it is very hard to solve its exact optimal solution. We instead propose a heuristic shooting algorithm to solve an near-optimum solution to this problem. This algorithm can be easily extended to the freeway lead vehicle problem. While the proposed algorithms can flexibly control trajectory shapes by tuning acceleration across a broad range, it is extremely parsimonious: it compresses a trajectory into a very few number of analytical segments, and includes only a few acceleration levels as control variables. With such parsimony and simplicity, these algorithms expect to be quite suitable for real-time applications and further adaptations. The simple structure of these algorithms also allows us to analyze its theoretical properties. By extending the traditional time-geography theory to a second-order version that considers finite acceleration, we are able to analytically investigate the feasibility of the proposed algorithms and the implication to the feasibility of the original problem. This novel extension also allows us to relate the trajectories solved by the lead vehicle problem algorithm to those generated by classic traffic flow models. This helps us reveal the fundamental commonalities between these two highway traffic management paradigms based on completely different technologies.
This paper is organized as follows. Section \[sec:Problem-Statement\] states the studied CAV trajectory optimization problem on a signalized highway section and its variant for the lead vehicle problem on a freeway segment. Section \[sec:Shooting-Heuristic-Algorithms\] describes the proposed shooting heuristic algorithms for the original problem and its variant, respectively. Section \[sec:Theoretical-Properties-of\] analyzes the theoretical properties of the proposed algorithms based on an extended time-geography theory. Section \[sec:Numerical-Examples\] demonstrates the proposed algorithms and their properties with a few illustrative examples. Section \[sec:Conclusion\] concludes this paper and briefly discusses future research directions.
Problem Statement\[sec:Problem-Statement\]
==========================================
Primary Problem
---------------
This section describes the primary problem of trajectory construction on a signalized one-lane highway segment, as illustrated in Figure \[fig:problem\_statement\]. The problem setting is described below.
![Illustration of the studied problem. \[fig:problem\_statement\]](problem_statement){width="60.00000%"}
[Roadway]{}
: **Geometry**: We consider a single-lane highway section of length $L$. Location on this segment starts from 0 at the upstream and ends at $L$ at the downstream. We use location set $[0,L]$ to denote this highway section. Traffic goes from location $0$ to $L$ on this section. A fixed-time traffic light is installed at location $L.$ The effective green phase starts at time $0$ with an duration of $G$, followed by an effective red phase of duration $R$, and this pattern continues all the way. This indicates the signal cycle time is always $C:=R+G$. We denote the set of green time intervals as $\mathcal{G}:=\left\{ \left[g_{m}:=mC,r_{m}:=mC+G\right)\right\} _{m\in\mathbb{Z}^{+}}$ where $\mathbb{Z}^{+}$ is the non-negative integer set. We define function $$G(t)=\min\{t'\in\mathcal{G},t'>t\},\forall t\in[-\infty,\infty],$$ which identifies the next closest green time to $t$. Note that $G(t)=t$ if $t\in\mathcal{G}$ or $G(t)>t$ if $t\notin\mathcal{G}$.
[Vehicle]{}
: **Characteristics**: We consider a stream of $N$ identical automated vehicles indexed as $n\in\mathcal{N}:=\left\{ 1,2,\cdots,N\right\} $. Each vehicle’s acceleration at any time is no less than deceleration limit $\underline{a}<0$ and no greater than acceleration limit $\overline{a}>0$. The speed limit on this segment is $\bar{v}$, and we don’t allow a vehicle to back up, thus a vehicle’s speed range is $[0,\overline{v}]$.
We want to design a set of trajectories for these vehicles to follow on this highway section. A trajectory is formally defined below.
A *trajectory* is defined as a second-order semi-differentiable function $p(t),\forall t\in(-\infty,\infty)$ such that its first order differential (or *velocity*) $\dot{p}(t)$ is absolutely continuous and its second-order right-differential $\ddot{p}(t)$ (or *acceleration*) is Riemann integrable over any $t\in(-\infty,\infty)$. We denote the set of all trajectories by $\bar{\mathcal{T}}$. We call the subsection of $p$ between times $t^{-}$ and $t^{+}$ ($-\infty\le t^{-}<t^{+}\le\infty$) a *trajectory section*, denoted by $p\left(t^{-}:t^{+}\right)$.
We let function $p_{n}$ denote the trajectory of vehicle $n,\forall n\in\mathcal{N}$. At any time $t$, $p_{n}(t)$ essentially denotes the location of vehicle $n$’s front bumper. Collectively we denote all vehicle trajectories by *trajectory vector* $\mathbf{p}:=\left[p_{n}(t)\right]_{n\in\mathcal{N}}$. The trajectories in vector $\mathbf{p}$ shall satisfy the following constraints.
[Kinematic]{}
: **Constraints**: Trajectory $p$ is *kinetically feasible* if $\dot{p}(t)\in[0,\bar{v}]$ and $\ddot{p}(t)\in\left[\underline{a},\bar{a}\right],\forall t\in(-\infty,\infty)$. We denote the set of all kinetically feasible trajectories as $$\mathcal{T}:=\left\{ p\in\mathcal{\bar{T}}\left|0\le\dot{p}(t)\le\bar{v},\underline{a}\le\ddot{p}(t)\le\bar{a},\forall t\in\left(-\infty,\infty\right)\right.\right\} .\label{eq:set-kinematic-constraints}$$
[Entry]{}
: **Boundary Condition**: Let $t_{n}^{-}$ and $v_{n}^{-}$ denote the time and the speed when vehicle $n$ arrives at the entry of this segment (or location 0), $\forall n\in\mathcal{N}$. We require $t_{1}^{-}<t_{2}^{-}<\cdots<t_{N}^{-}$ and separation between $t_{n-1}$ and $t_{n}$ is sufficient for the safety requirement, $\forall n\in\mathcal{N}\backslash\{1\}$. Define the subset of trajectories in $\mathcal{T}$ that are consistent with vehicle $n$’s entry boundary condition as $$\mathcal{T}_{n}^{-}:=\left\{ p\in\mathcal{T}\left|p(t_{n}^{-})=0,\dot{p}(t_{n}^{-})=v_{n}^{-}\right.\right\} ,\forall n\in\mathcal{N}.\label{eq: set-entry-boundary}$$
[Exit]{}
: **Boundary Condition**: Due to the traffic signal at location $L$, vehicles can only exit this section during a green light. Denote the set of trajectories in $\mathcal{T}$ that pass location $L$ during a green signal phase by $$\mathcal{T}^{+}:=\left\{ p\in\mathcal{T}\left|p^{-1}(L)\in\mathcal{G}\right.\right\} ,\forall n\in\mathcal{N},\label{eq: set-exit-boundary}$$ where the generalized inverse function is defined as $p^{-1}(l):=\inf\left\{ t\left|p(t)\ge l\right.\right\} ,\forall l\in\left[0,\infty\right),p\in\mathcal{T}$. Note that function $p^{-1}(\cdot)$ shall satisfy the following properties:
**P1**: Function $p^{-1}(l)$ is increasing with $l\in\left(-\infty,\infty\right)$;
**P2**: Due to speed limit $\bar{v}$, $p^{-1}(l+\delta)\ge p^{-1}(l)+\delta/\bar{v},\forall l\in\left(-\infty,\infty\right),\delta\in[0,\infty)$.
Further, let $\mathcal{T}_{n}$ denote the set of trajectories that satisfy both vehicle $n$’s entry and exit boundary conditions, i.e., $\mathcal{T}_{n}=\mathcal{T}_{n}^{-}\bigcap\mathcal{T}^{+}$.
[Car-following]{}
: **Safety**: We require that the separation between vehicle $n$’s location at any time and its preceding vehicle $(n-1$)’s location a communication delay $\tau$ ago is no less than a jam spacing $s$ (which usually includes the vehicle length and a safety buffer), $\forall n\in\mathcal{N}\backslash\{1\}$. For a genetic trajectory $p\in\mathcal{T}$ , a *safely following trajectory* of $p$ is a trajectory $p'\in\mathcal{T}$ such that $p'(t-\tau)-p(t)\ge s,\forall t\in(-\infty,\infty)$. We denote the set of all safely following trajectories of $p$ as $\mathcal{F}(p)$, i.e., $$\begin{gathered}
\mathcal{F}(p):=\left\{ p'\left|p'(t-\tau)-p(t)\ge s,\forall t\in(-\infty,\infty)\right.\right\} ,\forall p\in\mathcal{T}.\label{eq: set-safety-constraints}\end{gathered}$$ With this, we define $\mathcal{T}_{n}(p_{n-1}):=\mathcal{F}(p_{n-1})\bigcap\mathcal{T}_{n}$ that denotes the set of feasible trajectories for vehicle $n$ that are safely following given vehicle $(n-1)$’s trajectory $p_{n-1}$.
In summary, a feasible lead vehicle’s trajectory $p_{1}$ has to fall in $\mathcal{T}_{1}$, and any feasible following vehicle trajectory $p_{n}$ has to belong to $\mathcal{T}_{n}(p_{n-1}),\forall n\in\mathcal{N}\backslash\{1\}$ . We say a trajectory vector $\mathbf{p}$ is *feasible* if it satisfies all above-defined constraints. Let $\mathcal{P}$ denote the set of all ** feasible trajectory vectors, i.e., $$\mathcal{P}:=\left\{ \mathbf{p}:=\left[p_{n}\right]{}_{n\in\mathcal{N}}\left|p_{1}\in\mathcal{T}_{1},p_{n}\in\mathcal{T}_{n}\left(p_{n-1}\right),\forall n\in\mathcal{N}\backslash{1}\right.\right\} .\label{eq:P_feasible_platoon}$$ The primary problem (PP) investigated in this paper is finding and analyzing feasible solutions to $\mathcal{P}$.
Although a realistic vehicle trajectory only has a limited length, we set a trajectory’s time horizon to $\left(-\infty,\infty\right)$ to make the mathematical presentation convenient without loss of generality. In our study, we are only interested in trajectory sections between locations $0$ and $L$, i.e., $p_{n}\left(t_{n}^{-},p_{n}^{-1}(L)\right).$ Therefore, we can just view $p_{n}(-\infty,t_{n}^{-})$ as the given trajectory history that leads to the entry boundary condition, and $p_{n}\left(p_{n}^{-1}(L),\infty\right)$ as some feasible yet trivial projection above location $L$ (e.g., accelerating to $\bar{v}$ with rate $\bar{a}$ and then cruising at $\bar{v}$). Further, with this extension, safety constraint also ensures that vehicles did not collide before arriving location $0$ and will not collide after exiting location $L$.
This study can be trivially extended to static yet time-variant signal timing; i.e., the signal timing plan is pre-determined, yet different cycles could have different green and red durations, e.g., alternating like $G_{1},R_{1},G_{2},R_{2},\cdots$. In this case, define signal timing switch points $g_{m}=\sum_{i=1}^{m}\left(G_{i}+R_{i}\right),$ $r_{m}=\sum_{i=1}^{m}\left(G_{i}+R_{i}\right)+G_{i},$ $\forall m\in\mathbb{Z}^{+}$ and $\mathcal{G}$ becomes $\left\{ \left[g_{m},r_{m}\right)\right\} _{\forall m=1,2}$ where $r_{0}:=0$, and all the following results shall remain valid.
Problem Variation: Lead-Vehicle Problem
---------------------------------------
In the classic traffic flow theory, the lead-vehicle problem (LVP) is a well-known fundamental problem that predicts traffic flow dynamics on one-lane freeway given the lead vehicle’s trajectory and the following vehicles’ initial states [@Daganzo2006]. We notice that PP can be easily adapted to LVP by relaxing exit boundary condition yet fixing trajectory $p_{1}$. The LVP is officially formulated as follows. Given lead vehicle’s trajectory $p_{1}\in\mathcal{T}$, the set of feasible trajectories for LVP is
$$\mathcal{P}^{\mbox{LVP}}\left(p_{1}\right):=\left\{ \mathbf{p}:=\left[p_{n}\right]{}_{n\in\mathcal{N}}\left|p_{n}\in\mathcal{F}(p_{n-1})\bigcap\mathcal{T}_{n}^{-},\forall n\in\mathcal{N}\backslash{1}\right.\right\} .\label{eq:P_feasible_platoon_LVP}$$
The LVP investigated in this paper is finding and analyzing feasible solutions to $\mathcal{P}^{\mbox{LVP}}$.
Shooting Heuristic Algorithms \[sec:Shooting-Heuristic-Algorithms\]
===================================================================
This section proposes customized heuristic algorithms to solve feasible trajectory vectors to PP and LVP. Although a trajectory is defined over the entire time horizon $(-\infty,\infty)$, these algorithms only focus on the trajectory sections from the entry time $t_{n}^{-}$ for each vehicle $n\in\mathcal{N},$ because the trajectory sections before $t_{n}^{-}$ should be trivial given history and do not affect the algorithm results. Therefore, in the following presentation, we view a trajectory and the corresponding trajectory section over time $[t_{n}^{-},\infty)$ the same.
Shooting Heuristic for PP
-------------------------
This section presents a shooting heuristic (SH) algorithm that is able to construct a smooth and feasible trajectory vector to solve $\mathcal{P}$ in PP very efficiently. Traditional methods for trajectory optimization include analytical approaches that can only solve simple problems with special structures and numerical approaches that can accommodate more complex settings yet may demand enormous computation resources [@von1992direct]. Since a vehicle trajectory is essentially an infinite-dimensional object along which the state (e.g. location, speed, acceleration) at every point can be varied, it is challenging to even construct one single trajectory, particularly under nonlinear constraints. Note that our problem deals with a large number of trajectories for vehicles in a traffic stream that constantly interact with each other and are subject to complex nonlinear constraints -. Therefore we deem that it is very complex and time-consuming to tackle this problem with a traditional approach. Therefore, we opt to devise a new approach that circumvents the need for formulating high-dimensional objects or complex system constraints. This leads to the development of a shooting heuristic (SH) algorithm that can efficiently construct a smooth feasible trajectory vector with only a few control parameters.
Figure \[fig:shooting-process\] illustrates the components in the proposed SH algorithm. Basically, for each vehicle $n\in\mathcal{N}$, SH first constructs a trajectory, denoted by $p_{n}^{\mbox{f}}$ , with a forward shooting process that conforms with kinematic constraint , entry boundary constraint and safety constraint (if $n>1$). As illustrated in Figure \[fig:shooting-process\](a), if trajectory $p_{n}^{\mbox{f}}$ (dashed blue curve) turns out far enough from preceding trajectory $p_{n-1}$ (red solid curve) such that safety constraint is even not activated (or if $n=1$ and $p_{n}^{\mbox{f}}$ is already the lead trajectory), $p_{n}^{\mbox{f}}$ basically accelerates from its entry boundary condition $\left(t_{n}^{-},v_{n}^{-}\right)$ at location 0 with a forward acceleration rate of $\bar{a}^{\mbox{f}}\in(0,\bar{a}]$ until reaching speed limit $\bar{v}$ and then cruises at constant speed $\bar{v}$. Otherwise, as illustrated in Figure \[fig:shooting-process\](b), if trajectory $p_{n}^{\mbox{f}}$ is blocked by $p_{n-1}$ due to safety constraint , we just let $p_{n}^{\mbox{f}}$ smoothly merge into a safety bound (the red dotted curve) translated from $p_{n-1}$ that just keeps spatial separation $s$ and temporal separation $\tau$ from $p_{n}^{\mbox{f}}$. The transitional segment connecting $p_{n}^{\mbox{f}}$ with the safety bound decelerates at a forward deceleration rate of $\underline{a}^{\mbox{f}}\in[\underline{a},0)$. If trajectory $p_{n}^{\mbox{f}}$ from the forward shooting process is found to violate exit boundary constraint (or run into the red light), as illustrated in Figure \[fig:shooting-process\](c), a backward shooting process is activated to revise $p_{n}^{\mbox{f}}$ to comply with constraint . The backward shooting process first shifts the section of $p_{n}^{\mbox{f}}$ above location $L$ rightwards to the start of the next green phase to be a backward shooting trajectory $p_{n}^{\mbox{b}}$ . Then $p_{n}^{\mbox{b}}$ shoots backwards from this start point at a backward acceleration rate $\bar{a}^{\mbox{b}}\in(0,\bar{a}]$ until getting close enough to merge into $p_{n}^{\mbox{f}}$ , which may require $p_{n}^{\mbox{b}}$ stops for some time if the separation between the backward shooting start point and $p_{n}^{\mbox{f}}$ is long relative to acceleration rate $\bar{a}^{\mbox{b}}$ . Then, $p_{n}^{\mbox{b}}$ shoots backwards a merging segment at a backward deceleration rate of $\underline{a}^{\mbox{b}}\in[\underline{a},0)$ until getting tangent to $p_{n}^{\mbox{f}}$. Finally, merging $p_{n}^{\mbox{f}}$ and $p_{n}^{\mbox{b}}$ yields a feasible trajectory $p_{n}$ for vehicle $n$. Such forward and backward shooting processes are executed from vehicle $1$ through vehicle $N$ consecutively, and then SH concludes with a feasible trajectory vector $\mathbf{p}=\left[p_{n}\right]{}_{n\in\mathcal{N}}\in\mathcal{P}.$ Note that SH only uses four control variables $\left\{ \bar{a}^{\mbox{f}},\underline{a}^{\mbox{f}},\bar{a}^{\mbox{b}},\underline{a}^{\mbox{b}}\right\} $ that are yet able to control the overall smoothness of all trajectories (so as to achieve certain desired objectives). Further, the constructed trajectories are composed of only a few quadratic (or linear) segments that are all analytically solvable. Therefore, the proposed SH algorithm is parsimonious and simple to implement.
![Components in the proposed shooting heuristic: (a) forward shooting process without activating safety constraint ; (b) forward shooting process with activating safety constraint ; and (c) backward shooting process. \[fig:shooting-process\]. ](FSP_no_block_illur "fig:"){width="33.00000%"}![Components in the proposed shooting heuristic: (a) forward shooting process without activating safety constraint ; (b) forward shooting process with activating safety constraint ; and (c) backward shooting process. \[fig:shooting-process\]. ](FSP_block_illur "fig:"){width="33.00000%"}![Components in the proposed shooting heuristic: (a) forward shooting process without activating safety constraint ; (b) forward shooting process with activating safety constraint ; and (c) backward shooting process. \[fig:shooting-process\]. ](BSP_illur "fig:"){width="33.00000%"}
(a)(b)(c)
To formally state SH, we first define the following terminologies in Definitions \[def: state\_point\]-\[def: elemental\_segment\] with respect to a single trajectory, as illustrated in Figure \[fig:seg\_def\](a).
We define a *state point* by a three-element tuple $(l,v,t')$ , which represents that at time $t'$, the vehicle is at location $l$ and operates at speed $v$. A *feasible state point* $(l,v,t')$ should satisfy $v\in[0,\bar{v}]$.\[def: state\_point\]
We use a four-element tuple $(l,v,a,t')$ to denote the *quadratic function* that passes location $l$ at time $t'$ with velocity $v$ and acceleration $a$; i.e., $0.5a(t-t')^{2}+v(t-t')+l$ with respect to time variable $t\in(-\infty,\infty)$. Note that this quadratic function definition also includes a linear function (i.e., $a=0$). For simplicity, we can use a boldface letter to denote a quadratic function, e.g., $\mathbf{f}:=(l,v,a,t')$ and $\mathbf{f}(t):=0.5a(t-t')^{2}+v(t-t')+l$.
We use a five-element tuple $\mathbf{s}:=(l,v,a,t',t'')$ to denote a segment of quadratic function $\mathbf{f}=(l,v,a,t')$ between time $\min\left\{ t',t''\right\} $ and $\max\left\{ t',t''\right\} $. We call this tuple a *quadratic segment*. Define $\mathbf{s}(t):=\mathbf{f}(t),\forall t\in\left[\min\left\{ t',t''\right\} ,\max\left\{ t',t''\right\} \right]$. If the speed and the acceleration on every point along this segment satisfy constraint , we call it a *feasible quadratic segment*. \[def: elemental\_segment\]
In Definitions \[def: state\_point\]-\[def: elemental\_segment\], if one or more elements in a tuple are unknown or variable, we use “$\cdot$” to hold their places (e.g., $(l,\cdot,t')$, $(l,v,\cdot,t',\cdot)$ ).
![Illustrations of definitions ($x$-axis for time elapsing rightwards and $y$-axis for location increasing upwards): (a) state point, quadratic function and segment; (b) shadow trajectory and shadow segment; and (c) segment distance. \[fig:seg\_def\]. ](seg_def "fig:"){width="33.00000%"}![Illustrations of definitions ($x$-axis for time elapsing rightwards and $y$-axis for location increasing upwards): (a) state point, quadratic function and segment; (b) shadow trajectory and shadow segment; and (c) segment distance. \[fig:seg\_def\]. ](shadow "fig:"){width="33.00000%"}![Illustrations of definitions ($x$-axis for time elapsing rightwards and $y$-axis for location increasing upwards): (a) state point, quadratic function and segment; (b) shadow trajectory and shadow segment; and (c) segment distance. \[fig:seg\_def\]. ](seg_dist "fig:"){width="33.00000%"}
(a)(b)(c)
For a trajectory $p\in\bar{\mathcal{T}}$ composed by a vector of consecutive quadratic segments $\mathbf{s}_{k}:=$ $[(l_{k},v_{k},a_{k},t_{k},t_{k+1})]_{k=1,2,\cdots,\bar{k}}$ with $-\infty=t_{1}<t_{2}<\cdots<t_{\bar{k}+1}=\infty$, we denote $p=\left[\mathbf{s}_{k}\right]_{k=1,2,\cdots,\bar{k}}$. Without loss of generality, we assume that any trajectory in this study can be decomposed into a vector of quadratic segments.
During the forward shooting process, we need to check safety constraints at every move for any vehicle $n\ge2$. As illustrated in Figure \[fig:seg\_def\](b), we basically create a shadow (or safety bound) of the preceding trajectory $p_{n-1}$ by shifting $p_{n-1}$ downwards by $s$ and rightwards by $\tau$. Then safety constraint is simply equivalent to that $p_{n}$ does not exceed this shadow trajectory at any time. The following definition specifies the shadow trajectory and its elemental segments.
For a trajectory $p_{n}(t),\forall t\in(-\infty,\infty)$, we define its *shadow trajectory* $p_{n}^{\mbox{s}}$ as $p_{n}^{\mbox{s}}(t):=p_{n}(t-\tau)-s,\forall t\in(-\infty,\infty)$. It is obvious that $t^{-}\left(p_{n}^{\mbox{s}}\right)=t_{n}^{-}+\tau$ and $t^{+}\left(p_{n}^{\mbox{s}}\right)=\infty$. Note that if the following trajectory $p_{n+1}(t)$ initiated at time $t_{n+1}^{-}$ satisfies $p_{n+1}(t)\le p_{n}^{\mbox{s}},\forall t\in(-\infty,\infty)$, then $p_{n}$ and $p_{n+1}$ satisfies safety constraint . Further, a *shadow segment* of $\mathbf{s}:=(l,v,a,t',t'')$ is simply $\mathbf{s}^{\mbox{s}}:=(l-s,v,a,t'+\tau,t''+\tau)$. We also generalize this definition to the $m^{\mbox{th}}$*-order shadow trajectory* of $p_{n}$ as $p_{n}^{\mbox{s}^{m}}(t):=p_{n}(t-m\tau)-sm,\forall t\in(-\infty,\infty)$ and $m^{\mbox{th}}$*-order shadow segment* $\mathbf{s}^{\mbox{s}^{m}}:=(l-ms,v,a,t'+m\tau,t''+m\tau)$, which are essentially the results of repeating the shadow operation by $m$ times.
The following definitions specify an analytical function that checks the distance between two segments (e.g., the current segment to be constructed and a reference shadow segment in forward shooting), as illustrated in Figure \[fig:seg\_def\](c).
\[def:trajectory\_distance\] Given two segments $\mathbf{s}_{1}:=\left(l_{1},v_{1},a_{1},t'_{1},t''_{1}\right)$, $\mathbf{s}_{2}:=\left(l_{2},v_{2},a_{2},t'_{2},t''_{2}\right)$, define *segment distance* from $\mathbf{s}_{1}$ to $\mathbf{s}_{2}$ as
$$D\left(\mathbf{s}_{1}-\mathbf{s}_{2}\right):=\begin{cases}
\min_{t\in\left[t^{-},t^{+}\right]}\mathbf{s}_{1}(t)-\mathbf{s}_{2}(t), & \mbox{if }t^{-}\le t^{+};\\
\infty, & \mbox{otherwise,}
\end{cases}$$ where $t^{-}:=\max\left\{ \min\{t'_{1},t''_{1}\},\min\{t'_{2},t''_{2}\}\right\} $ and $t^{+}:=\min\left\{ \max\{t'_{1},t''_{1}\},\max\{t'_{2},t''_{2}\}\right\} $. If $t^{-}\le t^{+}$, $D\left(\mathbf{s}_{1}-\mathbf{s}_{2}\right)$ can be solved analytically as
$$D\left(\mathbf{s}_{1}-\mathbf{s}_{2}\right):=\begin{cases}
\mathbf{s}_{1}\left(t^{*}\left(\mathbf{s}_{1}-\mathbf{s}_{2}\right)\right)-\mathbf{s}_{2}\left(t^{*}\left(\mathbf{s}_{1}-\mathbf{s}_{2}\right)\right),\mbox{ if }a_{1}-a_{2}>0\mbox{ and }t^{*}\left(\mathbf{s}_{1}-\mathbf{s}_{2}\right)\in(t^{-},t^{+})\\
\min\left\{ \mathbf{s}_{1}\left(t^{-}\right)-\mathbf{s}_{2}\left(t^{-}\right),\mathbf{s}_{1}\left(t^{+}\right)-\mathbf{s}_{2}\left(t^{+}\right)\right\} , & \mbox{otherwise}.
\end{cases}$$ where $$t^{*}\left(\mathbf{s}_{1}-\mathbf{s}_{2}\right):=\frac{v_{1}-a_{1}t'_{1}-v_{2}+a_{2}t'_{2}}{a_{2}-a_{1}}.\label{eq:t_min_dist}$$ We also extend the distance definition to trajectory sections and trajectories below. Given two trajectory sections $p(t^{-}:t^{+})$ and $p'(t'^{-}:t'^{+}),$ the ** distance from $p(t^{-}:t^{+})$ to $p'(t'^{-}:t'^{+})$ is defined as $$D(p(t^{-}:t^{+})-p'(t'^{-}:t'^{+})):=\min_{t\in[\max(t^{-},t'^{-}),\min(t^{+},t'^{+})]}p(t)-p'(t)$$ if $\max(t^{-},t'^{-})\le\min(t^{+},t'^{+})$ or $D(p(t^{-}:t^{+})-p'(t'^{-}:t'^{+})):=\infty$. Suppose these two sections can be partitioned into quadratic segments, i.e., $p(t^{-}:t^{+})=\left[\mathbf{s}_{k}\right]_{k=1,2,\cdots,\bar{k}}$ and $p'(t'^{-}:t'^{+})=\left[\mathbf{s}'_{k'}\right]_{k'=1,2,\cdots,\bar{k}'}$, then
$$D\left(p(t^{-}:t^{+})-p'(t'^{-}:t'^{+})\right)=\min_{k=1,\cdots,\bar{k},k'=1,\cdots,\bar{k}',}D(\mathbf{s}_{k}-\mathbf{s}'_{k'}).$$ Note that function $D(\cdot)$ has a transitive relationship; i.e., $D(A-B)\ge D_{AB}$ and $D(B-C)\ge D_{BC}$ indicates $D(A-C)\ge D_{AB}+D_{BC}$.
\[def: forward\_shooting \] Next, we define an analytical operation that determines how we take a move in the forward shooting process, as illustrated in Figure \[fig:shooting\_operation\](a). Given a quadratic segment $\mathbf{s}':=\left(l',v',a',t'^{-},t'^{+}\right)$ with $t'^{-}<t'^{+}$, a feasible state point $\left(l,v,t^{-}\right)$ with $t^{-}<t'^{+}$, acceleration rate $a^{+}\ge0$ and deceleration rate $a^{-}<0$ (and $a^{-}\le a'$), we want to construct a *forward shooting segment* $\mathbf{s}:=\left(l,v,a^{+},t^{-},t^{\mbox{m}}\right)$ followed by a *forward merging segment* $\mathbf{s}^{\mbox{m}}:=\left(l^{\mbox{m}},v^{\mbox{m}},a^{-},t^{\mbox{m}},t^{+}\right)$ where $v^{\mbox{m}}:=v+a^{+}(t^{\mbox{m}}-t^{-})$, $l^{\mbox{m}}:=p+v(t^{\mbox{m}}-t^{-})+0.5a^{+}(t^{\mbox{m}}-t^{-})^{2}$ with $t^{+}\ge t^{\mbox{m}}\ge t^{-}$ in the following way. We basically select $t^{-}$ and $t^{m}$ values to make $\mbox{\ensuremath{\mathbf{s}}}$ and $\mathbf{s}^{\mbox{m}}$ satisfy the following conditions. First, we want to keep $\mathbf{s}'$ above $\mathbf{s}$ and the segment extended from $\mathbf{s}^{\mbox{m}}$ to time $\infty$, i.e., $$D\left[\mathbf{s}'-\mathbf{s}\right]\ge0,\mbox{ and }D\left[\mathbf{s}'-\left(p^{\mbox{m}},v^{\mbox{m}},a^{-},t^{\mbox{m}},\infty\right)\right]\ge0.\label{eq:non_cross_condition}$$ Further, the exact values of $t^{\mbox{m}}$ and $t^{+}$ shall be determined in the following three cases: (I) if no $t^{\mbox{m}}\in[t^{-},\infty)$ can be found to satisfy constraint , this shooting operation is infeasible and return $t^{m}=t^{+}=-\infty$; (II) otherwise, we try to find $t^{+}\in\left[\max\left\{ t'^{-},t^{-}\right\} ,t'^{+}\right]$ and $t^{\mbox{m}}\in[t^{-},t^{+}]$ such that $\mathbf{s}'$ and $\mathbf{s}^{\mbox{m}}$ get tangent at time $t^{+}$; and (III) if this trial fails, set $t^{\mbox{m}}=t^{+}=\infty$. Fortunately, since these segments are all simple quadratic segments, $t^{\mbox{m}}$ and $t^{+}$ can be solved analytically in the following *forward shooting operation (FSO)* algorithm, where the final solutions to $t^{\mbox{m}}$ and $t^{+}$ are denoted by $t^{\mbox{mf}}\left(\mathbf{s}',\left(l,v,t^{-},a^{+}\right),a^{-}\right)$ and $t^{\mbox{+f}}\left(\mathbf{s}',\left(l,v,t^{-},a^{+}\right),a^{-}\right)$, respectively.
[FSO-1:]{}
: If $D\left(\mathbf{s}'-\bar{\mathbf{s}}\right)<0$ where $\bar{\mathbf{s}}:=\left(l,v,a^{-},t^{-},\infty\right)$, there is no feasible solution, and we just set $t^{\mbox{m}}=t^{+}=-\infty$ (Case I). Go to Step FSO-3.
[FSO-2:]{}
: Shift the origin to time $t^{-}$ and denote $\hat{t}'^{-}:=t'^{-}-t^{-}$, $\hat{t}'^{+}:=t'^{+}-t^{-}$, $\hat{t}^{\mbox{m}}:=t^{\mbox{m}}-t^{-}$, $\hat{t}^{\mbox{+}}:=t^{\mbox{+}}-t^{-}$ and $\hat{t}^{-}:=\max\{\hat{t}'^{-},0\}$. Then get a quadratic function $\mathbf{q}$ by subtracting $\left(l^{\mbox{m}},v^{\mbox{m}},a^{-},t^{\mbox{m}}\right)$ from $(l',v',a',t'^{-})$, i.e., $\mathbf{q}:=\left(\hat{l},\hat{v},a'-a^{-},0\right)$ where $$\hat{l}:=0.5a'\left(\hat{t}'^{-}\right)^{2}+0.5\left(a^{+}-a^{-}\right)\left(\hat{t}^{\mbox{m}}\right)^{2}-a'\hat{t}'^{-}+l'-l,$$ and $$\hat{v}=v'-v-a'\hat{t}'^{-}-\left(a^{+}-a^{-}\right)\hat{t}^{\mbox{m}}.$$
[FSO-2-1:]{}
: If $a'=a^{-}$, test whether we can make $\mathbf{q}(t)=0,\forall t\in[\hat{t}^{-},\infty)$, i.e., whether $\hat{l}=0$ with $\hat{t}^{\mbox{m}}=\left(v'-v-a'\hat{t}'^{-}\right)/\left(a^{+}-a^{-}\right)$. If yes and $\hat{t}^{\mbox{m}}\in\left[\hat{t}^{-},\hat{t}'^{+}\right]$ (Case II), set $t^{\mbox{m}}=t^{-}+\hat{t}^{\mbox{m}}$ and $t^{\mbox{+}}=t'^{+}$. Otherwise (Case III), set $t^{\mbox{+}}=t^{\mbox{m}}=\infty$. Go to Step FSO-3.
[FSO-2-2:]{}
: If $a^{'}>a^{-}$,then $\mathbf{q}$ is a convex quadratic function, and we need to solve $\alpha\left(\hat{t}^{\mbox{m}}\right)^{2}+\beta\hat{t}^{\mbox{m}}+\gamma=0$ where $\alpha:=\left(a^{+}-a^{-}\right)\left(a^{+}-a'\right)$, $\beta:=-2\left(a^{+}-a^{-}\right)\left(v'-v-a'\hat{t}'^{-}\right)$ and $\gamma:=\left(v'-v-a'\hat{t}'^{-}\right)^{2}-(a'-a^{-})\left(a'\left(\hat{t}'^{-}\right)^{2}-2v'\hat{t}'^{-}+2l'-2l\right)$ . In case of $\alpha=\beta=0$ (Case III), set $t^{\mbox{+}}=t^{\mbox{m}}=\infty$, and go to Step FSO-3. Otherwise, we need to try candidate solutions to $\hat{t}^{\mbox{m}}$ and $\hat{t}^{\mbox{+}}$, denoted by $\hat{t}^{\mbox{mc}}$ and $\hat{t}^{+\mbox{c}}$, respectively. In case of $\alpha=0$ but $\beta\neq0$, solve $\hat{t}^{\mbox{mc}}=-\gamma/\beta$ and $$\hat{t}^{+\mbox{c}}=\frac{v'-v-a'\hat{t}'^{-}-\left(a^{+}-a^{-}\right)\hat{t}^{\mbox{mc}}}{a^{-}-a'}.\label{eq:t_hat_plus}$$ Otherwise, we should have $\alpha\neq0$, and then solve both candidate solutions $$\hat{t}^{\mbox{mc}}=\frac{-\beta\pm\sqrt{\beta^{2}-4\alpha\gamma}}{2\alpha}\label{eq:t_hat_m}$$ and the corresponding $\hat{t}^{+\mbox{c}}$ with equation. If the candidate solutions are not real numbers, then we just set $\hat{t}^{\mbox{+c}}=\hat{t}^{\mbox{mc}}=\infty$. Otherwise, try both sets of solutions and select the set satisfying $\hat{t}^{+\mbox{c}}\ge\hat{t}^{\mbox{mc}}\ge0$. For either of these two cases, if $\hat{t}^{+\mbox{c}}\in\left[\hat{t}^{-},\hat{t}'^{+}\right]$ (Case II) , we set $t^{\mbox{m}}=t^{-}+\hat{t}^{\mbox{mc}}$ and $\hat{t}^{\mbox{+}}=\hat{t}^{\mbox{+c}}$. Otherwise, set $t^{\mbox{+}}=t^{\mbox{m}}=\infty$ (Case III). Go to Step FSO-3.
[FSO-3:]{}
: Finally, we return $t^{\mbox{mf}}\left(\mathbf{s}',\left(l,v,t^{-},a^{+}\right),a^{-}\right)=t^{\mbox{m}}$ and $t^{\mbox{+f}}\left(\mathbf{s}',\left(l,v,t^{-},a^{+}\right),a^{-}\right)=t^{\mbox{+}}$.
![Illustrations of shooting operations: (a) a forward shooting operation, and (b) a backward shooting operation. \[fig:shooting\_operation\]. ](forward_shooting "fig:"){width="33.00000%"}![Illustrations of shooting operations: (a) a forward shooting operation, and (b) a backward shooting operation. \[fig:shooting\_operation\]. ](backward_shooting "fig:"){width="33.00000%"}
(a)(b)
\[def: FSP\]We extend one forward shooting operation to the following *forward shooting process (FSP)* that generates a whole trajectory. Given a shadow trajectory $p^{\mbox{s}}:=\left[\mathbf{s}_{h}^{\mbox{s}}:=\left(l_{h}^{\mbox{s}},v_{h}^{\mbox{s}},a_{h}^{\mbox{s}},t_{h}^{\mbox{s}},t_{h+1}^{\mbox{s}}\right)\right]{}_{h=1,\cdots,\bar{h}}.$ (with $t_{1}^{\mbox{s}}=-\infty$ and $t_{\bar{h}+1}^{\mbox{s}}=\infty$) and a feasible entry state point $(l,v,t^{-})$, we basically want to construct a *forward shooting trajectory* $p^{\mbox{f}}((l,v,t^{-}),p^{\mbox{s}})$ that starts from $(l,v,t^{-})$ and maintains acceleration $\bar{a}^{\mbox{f}}$ or speed $\bar{v}$ until being bounded by $p^{\mbox{s}}$. We consider a template trajectory starting at $(l,v,t^{-})$ and composed by these two candidate segments $\mathbf{s}^{\mbox{a}}:=(l,v,\bar{a}^{\mbox{f}},t^{-},t^{\mbox{a}}:=t^{-}+(\bar{v}-v)/\bar{a}^{\mbox{f}})$ (which accelerates from $v$ to $\bar{v}$ given $v<\bar{v}$) and $\mathbf{s}^{\mbox{\ensuremath{\infty}}}:=\left(l^{\mbox{a}}:=l+0.5\left(\bar{v}^{2}-v^{2}\right)/\bar{a}^{\mbox{f}},\bar{v},0,t^{\mbox{a}},\infty\right)$ (which maintains maximum speed $\bar{v}$ all the way), i.e.,
$$p^{\mbox{t}}:=\begin{cases}
\left[\mathbf{s}^{\mbox{a}},\mathbf{s}^{\infty}\right], & \mbox{if }v<\bar{v};\\
\left[\mathbf{s}^{\infty}\right], & \mbox{if }v=\bar{v}.
\end{cases}$$ Then $p^{\mbox{f}}((l,v,t^{-}),p^{\mbox{s}})$, if bounded by $p^{\mbox{s}}$, shall first follow $p^{\mbox{t}}$ and then merge into $p^{\mbox{s}}$ with a merging segment $\left(p^{\mbox{t}}(t^{\mbox{m}}),\dot{p}^{\mbox{t}}(t^{\mbox{m}}),\underline{a}^{\mbox{f}},t^{\mbox{m}},t^{\mbox{+}}\right)$. This can be solved analytically with the following FSP algorithm.
[FSP-1:]{}
: Initiate $h=1$, $t^{+}=t^{\mbox{m}}=\infty$, $p^{\mbox{f}}=\emptyset$ and iterate through the segments in $p^{\mbox{s}}$ as follows.
[FSP-2:]{}
: If $v<\bar{v}$, apply the FSO algorithm to solve candidate time points $t^{\mbox{mc}}:=t^{\mbox{mf}}\left(\mathbf{s}_{h}^{\mbox{s}},(l,v,\bar{a}^{\mbox{f}},t^{-}),\underline{a}^{\mbox{f}}\right)$ and $t^{+\mbox{c}}:=t^{\mbox{+f}}\left(\mathbf{s}_{h}^{\mbox{s}},(l,v,\bar{a}^{\mbox{f}},t^{-}),\underline{a}^{\mbox{f}}\right)$. If $t^{\mbox{mc}}>t^{\mbox{a}}$, revise $t^{\mbox{mc}}:=(t^{\mbox{mf}}\mathbf{s}_{h}^{\mbox{s}},(l^{\mbox{a}},\bar{v},0,t^{\mbox{a}}),\underline{a}^{\mbox{f}})$ and $t^{\mbox{+c}}:=t^{\mbox{+f}}(\mathbf{s}_{h}^{\mbox{s}},(l^{\mbox{a}},\bar{v},0,t^{\mbox{a}}),\underline{a}^{\mbox{f}})$. If $v=\bar{v}$, solve $t^{\mbox{mc}}:=t^{\mbox{mf}}(\mathbf{s}_{h}^{\mbox{s}},(l,\bar{v},0,t^{-}),\underline{a}^{\mbox{f}})$ and $t^{\mbox{+c}}:=(t^{\mbox{+f}}\mathbf{s}_{h}^{\mbox{s}},(l,\bar{v},0,t^{-}),\underline{a}^{\mbox{f}})$. If $t^{\mbox{mc}}=-\infty$, the algorithm cannot find a feasible solution and return $p^{\mbox{f}}((l,v,t^{-}),p^{\mbox{s}})=\emptyset$, $t^{\mbox{mf}}((l,v,t^{-}),p^{\mbox{s}})=-\infty$ and $t^{+\mbox{f}}((p,v,t^{-}),p^{\mbox{s}})=-\infty$. If $t^{\mbox{+c}}\in[t_{h}^{\mbox{s}},t_{h+1}^{\mbox{s}}]$, set $t^{\mbox{m}}=t^{\mbox{mc}}$ and $t^{+}=t^{+\mbox{c}}$, and go to the Step FSP-3. Otherwise, $t^{\mbox{+c}}$ shall be $\infty$. Then if $h<\bar{h}$, set $h=h+1$ and repeat this step.
[FSP-3:]{}
: If $v<\bar{v}$ and $t^{\mbox{a}}>t^{-}$, append segment $(l,v,\bar{a}^{\mbox{f}},t^{-},\min(t^{\mbox{m}},t^{\mbox{a}}))$ to $p^{\mbox{f}}$ (appending means adding this segment as the last element of $p^{\mbox{f}}$). If $t^{\mbox{m}}>t^{\mbox{a}},$ append $(l^{\mbox{a}},\bar{v},0,t^{\mbox{a}},t^{\mbox{m}})$ to $p^{\mbox{f}}$. If $t^{+}<t_{\bar{h}+1}^{\mbox{s}}$, which indicates $t^{+}\in[t_{h}^{\mbox{s}},t_{h+1}^{\mbox{s}}]$, we first append segment $(l_{h}^{\mbox{s}}+v_{h}^{\mbox{s}}(t^{+}-t_{h}^{\mbox{s}})+0.5a_{h}^{\mbox{s}}(t^{+}-t_{h}^{\mbox{s}})^{2},v_{h}^{\mbox{s}}+a_{h}^{\mbox{s}}(t^{+}-t_{h}^{\mbox{s}}),a_{h}^{\mbox{s}},t^{+},t_{h+1}^{\mbox{s}})$ to $p^{\mbox{f}}$, and then append all segments $[\mathbf{s}_{h'}^{\mbox{s}}]_{h'=h+1,\cdots,\bar{h}}$ to $p^{\mbox{f}}$.
[FSP-4:]{}
: Finally, return $p^{\mbox{f}}((l,v,t^{-}),p^{\mbox{s}})=p^{\mbox{f}}$, $t^{\mbox{mf}}((l,v,t^{-}),p^{\mbox{s}})=t^{\mbox{m}}$ and $t^{+\mbox{f}}((p,v,t^{-}),p^{\mbox{s}})=t^{+}$.
\[def:backward\_shooting\] This definition further specifies how we take a symmetric move in the backward shooting, as illustrated in Figure \[fig:shooting\_operation\](b). Given a quadratic segment $\mathbf{s}':=\left(l',v',a',t'^{-},t'^{+}\right)$ with $t'^{-}<t'^{+}$ (e.g., a segment generated from the forward shooting process), a feasible state point $\left(l,v,t^{-}\right)$ with $t^{-}>t'^{-}$, acceleration rate $a^{+}\ge0$ and deceleration rate $a^{-}<0$ (and $a^{-}\le a'$), we construct a *backward shooting segment* $\mathbf{s}:=\left(l,v,a^{+},t^{-},t^{\mbox{m}}\right)$ preceded by a *backward merging segment* $\mathbf{s}^{\mbox{m}}:=\left(l^{\mbox{m}},v^{\mbox{m}},a^{-},t^{\mbox{m}},t^{+}\right)$ where again $v^{\mbox{m}}:=v+a^{+}(t^{\mbox{m}}-t^{-})$, $l^{\mbox{m}}:=l+v(t^{\mbox{m}}-t^{-})+0.5a^{+}(t^{\mbox{m}}-t^{-})^{2}$, and $t^{+}\le t^{\mbox{m}}\le t^{-}$, such that again condition is satisfied (and thus $\mathbf{s}'$ is above $\mathbf{s}$ and $\mathbf{s}{}^{\mbox{m}}$). And again there are three cases in determining $t^{-}$ and $t^{\mbox{m}}$ values: (I) if no $t^{\mbox{m}}\in(-\infty,t^{-}]$ can be found to satisfy constraint , this shooting operation is infeasible and return $t^{m}=t^{+}=\infty$; (II ) otherwise we try to find $t^{+}\in\left[t'^{+},\min\left\{ t^{-},t'^{-}\right\} \right]$ and $t^{\mbox{m}}\in[t^{+},t^{-}]$ such that $\mathbf{s}'$ and $\mathbf{s}^{\mbox{m}}$ get tangent at time $t^{+}$ (as Figure \[fig:shooting\_operation\](b) indicates); and (III) if no such $t^{+}$ is found, set $t^{m}=t^{+}=-\infty$. Solutions $t^{\mbox{m}}$ and $t^{-}$ are denoted as functions $t^{\mbox{mb}}\left(\mathbf{s}',\left(l,v,t^{-}\right),a^{+},a^{-}\right)$ and $t^{\mbox{-b}}\left(\mathbf{s}',\left(l,v,t^{-}\right),a^{+},a^{-}\right)$, respectively, and they can be solved analytically in the following *backward shooting operation* ($BSO$) algorithm.
[BSO-1:]{}
: If $D[\mathbf{s}'-\left(l,v,a^{-},t^{-},-\infty\right)]<0$, there is no feasible solution, and we just return $t^{\mbox{m}}=t^{+}=-\infty$ (Case I). Go to Step BSO-3.
[BSO-2:]{}
: Again shift the origin point to $t^{-}$ and denote $\hat{t}'^{-}:=t'^{-}-t^{-}$, $\hat{t}'^{+}:=t'^{+}-t^{-}$, $\hat{t}^{\mbox{m}}:=t^{\mbox{m}}-t^{-}$, $\hat{t}^{\mbox{+}}:=t^{\mbox{+}}-t^{-}$and $\hat{t}^{-}:=\min\{\hat{t}'^{+},0\}$. And obtain $\mathbf{q}$ by subtracting $\left(l^{\mbox{m}},v^{\mbox{m}},a^{-},t^{\mbox{m}}\right)$ from $(l',v',a',t'^{-})$, which is formulated the same as that in Step FSO-2.
[BSO-2-1:]{}
: If $a^{'}=a^{-}$, test whether $\hat{l}=0$ with $\hat{t}^{\mbox{m}}=\left(v'-v-a'\hat{t}'^{-}\right)/\left(a^{+}-a^{-}\right)$. If yes and $\hat{t}^{\mbox{m}}\in\left[\hat{t}^{-},\hat{t}'^{-}\right]$ (Case II),
: $t^{\mbox{m}}=t^{-}+\hat{t}^{\mbox{m}}$ and $t^{\mbox{+}}=t'^{-}$; Otherwise, return $\hat{t}^{\mbox{m}}=\hat{t}{}^{+}=-\infty$ (Case III). Go to BSO-3.
[BSO-2-2:]{}
: If $a^{'}>a^{-}$, we need to again solve $\alpha\left(\hat{t}^{\mbox{m}}\right)^{2}+\beta\hat{t}^{\mbox{m}}+\gamma=0$ formulated in Step FSO-2-2. In case of $\alpha=\beta=0$ (Case III), set $t^{\mbox{+}}=t^{\mbox{m}}=-\infty$, and go to Step FSO-3. Otherwise, we again try candidate solutions $\hat{t}^{\mbox{mc}}$ and $\hat{t}^{+\mbox{c}}$. In case of $\alpha=0$ but $\beta\neq0$, solve $\hat{t}^{\mbox{mc}}=-\gamma/\beta$ and $\hat{t}^{+\mbox{c}}$ with equation . In case of $\alpha\neq0$, then we solve two sets of solutions $\hat{t}^{\mbox{mc}}$ and $\hat{t}^{\mbox{+c}}$ with equations and respectively. if the candidate solutions are not real numbers, then we just set $\hat{t}^{\mbox{+c}}=\hat{t}^{\mbox{mc}}=\infty$. Otherwise, try both sets of solutions and select the set satisfying $\hat{t}^{+\mbox{c}}\le\hat{t}^{\mbox{mc}}\le0$. With this, if we obtain $\hat{t}^{+\mbox{c}}\in\left[\hat{t}^{-},\hat{t}'^{-}\right]$ (Case II), we set $t^{\mbox{m}}=t^{-}+\hat{t}^{\mbox{mc}}$ and $\hat{t}^{\mbox{+}}=\hat{t}^{\mbox{+c}}$. Otherwise, set $t^{\mbox{+}}=t^{\mbox{m}}=\infty$. Go to Step BSO-3.
[BSO-3:]{}
: Finally, we return $t^{\mbox{mb}}\left(\mathbf{s}',\left(l,v,t^{-},a^{+}\right),a^{-}\right)=t^{m}$ and $t^{\mbox{+b}}\left(\mathbf{s}',\left(l,v,t^{-},a^{+}\right),a^{-}\right)=t^{+}$.
Symmetric to Definition \[def: FSP\], we extend one backward move BSO to the following *backward shooting process* *(BSP)*. We consider *a backward shooting template trajectory* starting at a feasible entry state point $(l,v,t^{-})$ composed by one or both of $\mathbf{s}^{-\mbox{a}}:=(l,v,\bar{a}^{\mbox{f}},t^{-},t^{-\mbox{a}}:=t^{-}-v/\bar{a}^{\mbox{b}})$ (which decelerates backward from $v$ to $0$ given $v>0$) and $\mathbf{s}^{-\mbox{\ensuremath{\infty}}}:=\left(l^{\mbox{-a}}:=l-0.5v^{2}/\bar{a}^{\mbox{b}},0,0,t^{\mbox{-a}},-\infty\right)$ (which denotes the vehicle is stopped prior to time $t^{\mbox{-a}}$), i.e., $$p^{\mbox{t}}:=\begin{cases}
\left[\mathbf{s}^{-\infty},\mathbf{s}^{\mbox{-a}}\right], & \mbox{if }v>0;\\
\left[\mathbf{s}^{-\infty}\right], & \mbox{if }v=0.
\end{cases}$$ Further, we are given the original trajectory (e.g., those generated from FSP) $$p^{\mbox{f}}:=\left\{ \mathbf{s}_{h}:=\{l_{h},v_{h},a_{h},t_{h},t_{h+1}\}\right\} {}_{h=1,\cdots,\bar{h}}.$$ We will find a *backward shooting trajectory section* $p^{\mbox{b}}((l,v,t^{-}),p)$ that is to merge into $p^{\mbox{f}}$ with a merging segment $\left(p^{\mbox{t}}(t^{\mbox{m}}),\dot{p}(t^{\mbox{m}}),\underline{a}^{\mbox{b}},t^{\mbox{m}},t^{\mbox{+}}\right)$ and does not exceed (i.e., going left of) $p^{\mbox{f}}$ at any time. This can be solved analytically with the following BSP algorithm.
[BSP-1:]{}
: Initiate $h$ being the largest segment index of $p^{\mbox{f}}$ such that $t_{h}<t^{-}$, set $t^{+}=t^{\mbox{m}}=-\infty$, $p^{\mbox{b}}=\emptyset$ and iterate through the segments in $p^{\mbox{f}}$.
[BSP-2:]{}
: If $v>0$, apply BSO to solve $t^{\mbox{mc}}:=t^{\mbox{mb}}\left(\mathbf{s}_{h},(l,v,\bar{a}^{\mbox{b}},t^{-}),\underline{a}^{\mbox{b}}\right)$ and $t^{+\mbox{c}}:=t^{\mbox{mf}}\left(\mathbf{s}_{h},(l,v,\bar{a}^{\mbox{b}},t^{-}),\underline{a}^{\mbox{b}}\right)$. If $t^{\mbox{mc}}<t^{-\mbox{a}}$, revise $t^{\mbox{mc}}:=t^{\mbox{mb}}\left(\mathbf{s}_{h},(l^{\mbox{-a}},0,0,t^{\mbox{-a}}),\underline{a}^{\mbox{b}}\right)$ and $t^{\mbox{+c}}:=t^{\mbox{+b}}\left(\mathbf{s}_{h},(l^{\mbox{-a}},0,0,t^{\mbox{-a}}),\underline{a}^{\mbox{b}}\right)$ with BSO. If $v=0$, directly solve $t^{\mbox{mc}}:=t^{\mbox{mb}}\left(\mathbf{s}_{h},(l,0,0,t^{-}),\underline{a}^{\mbox{b}}\right)$ and $t^{\mbox{+c}}:=t^{\mbox{+b}}\left(\mathbf{s}_{h},(l,0,0,t^{-}),\underline{a}^{\mbox{b}}\right)$ with BSO. If $t^{\mbox{mc}}=\infty$, the algorithm cannot find a feasible solution (because any trajectory through $(l,v,t^{-})$ will go above $p^{\mbox{f}}$) and thus returns $p^{\mbox{b}}((l,v,t^{-}),p)=p^{\mbox{eb}}\left((l,v,t^{-}),p^{\mbox{f}}\right)=\emptyset$. If $t^{\mbox{+c}}=-\infty$ and $h=1$, the algorithm cannot find a feasible backward trajectory that can touch $p^{\mbox{f}}$ and thus return $p^{\mbox{b}}((l,v,t^{-}),p)=p^{\mbox{eb}}\left((l,v,t^{-}),p^{\mbox{f}}\right)=\emptyset$. If $t^{\mbox{+c}}\in[t_{h}^{\mbox{s}},t_{h+1}^{\mbox{s}}]$, set $t^{\mbox{m}}=t^{\mbox{mc}}$ and $t^{+}=t^{+\mbox{c}}$ and go to Step BSP-3. Otherwise, if $h>1$, set $h=h-1$ and repeat this step.
[BSP-3:]{}
: If $v>0$ and $t^{\mbox{-a}}<t^{-}$, insert segment $\left(l^{-\mbox{ma}},v^{-\mbox{ma}},\bar{a}^{\mbox{b}},t^{-\mbox{ma}},t^{-}\right)$ to $p^{\mbox{b}}$ (inserting means adding this segment before the head of $p^{\mbox{b}}$), where $t^{-\mbox{ma}}:=\max(t^{\mbox{m}},t^{\mbox{-a}})$, $v^{-\mbox{ma}}:=v-\bar{a}^{\mbox{b}}(t^{-}-t^{-\mbox{ma}})$ and $l^{-\mbox{ma}}:=l-v(t^{-}-t^{-\mbox{ma}})+0.5\bar{a}^{\mbox{b}}(t^{-}-t^{-\mbox{ma}})^{2}$. If $t^{\mbox{m}}<t^{-\mbox{a}},$ insert $(l^{\mbox{-a}},0,0,t^{\mbox{m}},t^{\mbox{-a}})$ to $p^{\mbox{b}}$. Then we insert merging segment $(l^{\mbox{m}},v^{\mbox{m}}\bar{a}^{\mbox{b}},t^{+},t^{\mbox{m}})$ to $p^{\mbox{b}}$ where $l^{\mbox{m}}:=l_{h}+v_{h}(t^{+}-t_{h}^{\mbox{s}})+0.5a_{h}(t^{+}-t_{h}^{\mbox{s}})^{2},$ and $v^{\mbox{m}}:=v_{h}+a_{h}(t^{+}-t_{h})$.
[BSP-4:]{}
: Now we have obtained a *backward shooting trajectory section* $p^{\mbox{b}}\left((l,v,t^{-}),p^{\mbox{f}}\right)=p^{\mbox{b}}\left(t^{\mbox{m}}:t^{-}\right)$ (or $p^{\mbox{b}}$ for simplicity). We further extend $p^{\mbox{b}}\left(t^{\mbox{m}}:t^{-}\right)$ by inserting $p^{\mbox{f}}(t_{1}:t^{\mbox{m}})$ and appending $p^{\mbox{f}}\left(\left(l,v,t^{-}\right),p^{\mbox{f}}\right)$ generated from an *auxiliary FSP*, and construct the *extended backward shooting trajectory* $p^{\mbox{eb}}\left((l,v,t^{-}),p^{\mbox{f}}\right)$ $:=\left[p^{\mbox{f}}(t_{1}:t^{\mbox{m}}),p^{\mbox{b}}\left(t^{\mbox{m}}:t^{-}\right),p^{\mbox{f}}\left(\left(l,v,t^{-}\right),p^{\mbox{f}}\right)\right]$. Return $p^{\mbox{b}}\left((l,v,t^{-}),p^{\mbox{f}}\right)$ and $p^{\mbox{eb}}\left((l,v,t^{-}),p^{\mbox{f}}\right)$.
:
Now we are ready to present the proposed shooting algorithm that yields a trajectory vector $P\left(\bar{a}^{\mbox{f}},\underline{a}^{\mbox{f}},\bar{a}^{\mbox{b}},\bar{a}^{b}\right)$ as a functional of these four acceleration rates.
[SH-1:]{}
: Initialize control variables $\bar{a}^{\mbox{f}}$, $\underline{a}^{\mbox{f}}$, $\bar{a}^{\mbox{b}}$ and $\underline{a}^{\mbox{b}}$. Set $n=1$, trajectory vector $P=\emptyset$.
[SH-2:]{}
: Apply the FSP to obtain $p_{n}^{\mbox{f}}=\left[\mathbf{s}_{nk}=\left(l_{nk},v_{nk},a_{nk},t_{nk},t_{n(k+1)}\right)\right]{}_{k=1,\cdots,\bar{k}_{n}}=p^{\mbox{f}}((0,v_{n}^{-},t_{n}^{-}),p_{n-1}^{\mbox{s}})$ (define $p_{0}^{\mbox{s}}:=\emptyset$). We call this process the *primary FSP* (to differentiate from the auxiliary FSP in the BSP). If $p_{n}^{\mbox{f}}=\emptyset$, which means that this algorithm cannot find a feasible solution for trajectory platoon $P$, set $P\left(\bar{a}^{\mbox{f}},\underline{a}^{\mbox{f}},\bar{a}^{\mbox{b}},\bar{a}^{b}\right):=P=\emptyset$ and return.
[SH-3:]{}
: This steps checks the need for the BSP. Find the segment index $k_{n}^{L}$ such that $L\in\left[l_{nk^{L}},l_{n\left(k^{L}+1\right)}\right)$ and solve the time $t_{n}^{L}$ when vehicle $n$ passes location $L$ as follows $$t_{n}^{L}:=t_{nk^{L}}+\begin{cases}
\frac{-v_{nk^{L}}+\sqrt{v_{nk^{L}}^{2}+2a_{nk^{L}}(L-l_{nk^{L}})}}{a_{nk^{L}}}, & \mbox{if }a_{nk^{L}}\neq0;\\
\frac{L-l_{nk^{L}}}{v_{nk^{L}}}, & \mbox{if }a_{nk^{L}}=0.
\end{cases}$$ If $t_{n}^{L}=G\left(t_{n}^{L}\right)$, which means that $p_{n}^{\mbox{f}}$ does not violate the exit boundary constraint (or does not run into the red light), we set $p_{n}=p_{n}^{\mbox{f}}$ and go to SH4. Otherwise, $p_{n}^{\mbox{f}}$ violates constraint and we need to apply BSP to revise it in the following step. Set $v_{n}^{L}:=v_{nk^{L}}+a_{nk^{L}}\left(t_{n}^{L}-t_{nk^{L}}\right),$ apply the BSP to solve $p_{n}^{\mbox{b}}=p^{\mbox{b}}\left(\left(L,v_{n}^{L},G\left(t_{n}^{L}\right)\right),p_{n}^{\mbox{f}}\right)$. If the start location of obtain $p_{n}^{\mbox{b}}$ is no less than 0, set $p_{n}=p^{\mbox{eb}}\left(\left(L,v_{n}^{L},G\left(t_{n}^{L}\right)\right),p_{n}^{\mbox{f}}\right).$ Otherwise, $p_{n}^{\mbox{b}}$ can not meet $p_{n}^{\mbox{f}}$ on this highway section, and return $P\left(\bar{a}^{\mbox{f}},\underline{a}^{\mbox{f}},\bar{a}^{\mbox{b}},\bar{a}^{b}\right):=\emptyset$ .
[SH-4:]{}
: Append $p_{n}$ to $P$. Return $P^{\mbox{SH}}\left(\underline{a}^{\mbox{f}},\bar{a}^{\mbox{f}},\underline{a}^{\mbox{b}},\bar{a}^{\mbox{b}}\right):=P$ if $n=N$, or otherwise set $n=n+1$ and go to SH-2.
Although FSO (Definition \[def: forward\_shooting \]) and BSO (Definition \[def:backward\_shooting\]) do not explicitly impose speed limits to the generated trajectory segments, as long as a non-empty trajectory vector $P$ is returned by the SH algorithm, $P$ shall satisfy all constraints defined in Section \[sec:Problem-Statement\] (or $P\in\mathcal{P}$), as proven in the following proposition.
\[prop: SH\_feas\_necessity\]If the SH algorithm successfully generates a vector of trajectories $P\left(\underline{a}^{\mbox{f}},\bar{a}^{\mbox{f}},\underline{a}^{\mbox{b}},\bar{a}^{\mbox{b}}\right)$ with $\underline{a}^{\mbox{f}},\underline{a}^{\mbox{b}}\in\left[\underline{a},0\right)$ and $\bar{a}^{\mbox{f}},\bar{a}^{\mbox{b}}\in\left(0,\bar{a}\right]$, they shall fall in the feasible trajectory vector set $\mathcal{P}$ defined in .
Since the acceleration of each segment generated from the SH algorithm is either explicitly specified within $[\underline{a},\bar{a}]$ (i.e., one of $\underline{a}^{\mbox{f}},\bar{a}^{\mbox{f}},\underline{a}^{\mbox{b}},\bar{a}^{\mbox{b}}$ and 0) or just following a shadow trajectory’s acceleration that shall fall in $[\underline{a},\bar{a}]$ as well. So the constraint with respect to acceleration in is satisfied.
Then, we will use mathematical induction to exam the remaining constraints in -. First for vehicle 1, FSP can generate $p_{1}$ with at maximum 2 segments, which apparently falls in $\mathcal{T}_{1}^{-}$ (and thus both constraints -. are satisfied). If BSP is not needed, exit constraint is automatically satisfied and thus $p_{1}\in\mathcal{T}_{1}$. Otherwise, the new segments generated from BSP below $L$ start from a speed no greater than $\bar{v}$ and decelerate backwards to a value no less than 0 and then increase the speed and merge into the forward shooting trajectory at a speed no greater than $\bar{v}$. During this process, the speed shall always stay within $[0,\bar{v}]$ and therefore constraint remains valid. The auxiliary FSP is similar to the primary FSP and thus will not violate constraint as well. Further, the BSP step SH3 does not affect the entry boundary condition and makes exit condition feasible in addition. Therefore we obtain $p_{1}\in\mathcal{T}_{1}$.
Then we assume that $p_{n-1}\in\mathcal{T}_{n-1},\forall n=2,\cdots,N$, and we will prove that $p_{n}\in\mathcal{T}_{n}(p_{n-1})$. If $p_{n}$ is not blocked by $p_{n-1}$ during the FSP, then the construction of $p_{n}$ is similar to that of $p_{1}$ and thus $p_{n}$ should automatically satisfy constraints - and thus $p_{n}\in\mathcal{T}_{n}$. Further, $p_{n}$ shall be always below $p_{n-1}^{\mbox{s}}$ and therefore $p_{n}$ shall satisfy safety constraint , i.e., $p_{n}\in\mathcal{T}_{n}(p_{n-1})$. Otherwise, if $p_{n}$ is blocked by $p_{n-1}$, the construction of $p_{n}$ would generate some more segments that merge the forward trajectory into $p_{n-1}^{\mbox{s}}$ (e.g., as Figure \[fig:shooting\_operation\](a) illustrates) and then follow $p_{n-1}^{\mbox{s}}$, as compared with the construction of $p_{1}$. Due to the induction assumption, the segments following $p_{n-1}^{\mbox{s}}$ shall satisfy kinematic constraint the same as the corresponding segments in $p_{n-1}$. For the merging segment, since it starts from a forward shooting segment and ends at a shadow segment and therefore its speed range should be bounded by $[0,\bar{v}]$. Therefore, we obtain $p_{n}\in\mathcal{T}$. Again, the primary FSP ensures that $p_{n}$ satisfies the entry boundary and BSP ensures that $p_{n}$ satisfy exit constraint . This yields $p_{n}\in\mathcal{T}_{n}$. Further we see that any segment generated from FSP shall be either below or on $p_{n-1}^{\mbox{s}}$. If the BSP generates new segments, they shall be strictly right to the forward shooting trajectory. Therefore, $p_{n}$ shall fall in $\mathcal{T}_{n}\left(p_{n-1}\right)$. This proves that $P\left(\underline{a}^{\mbox{f}},\bar{a}^{\mbox{f}},\underline{a}^{\mbox{b}},\bar{a}^{\mbox{b}}\right)\in\mathcal{P}$.
Shooting Heuristic for the Lead Vehicle Problem\[sub:SH\_LVP\]
---------------------------------------------------------------
The above proposed SH algorithm can be easily adapted to solve LVP . The only modifications are to fix $p_{1}$ and to set $\mathcal{G}=[-\infty,\infty]$ (and therefore no BSP is needed). We denote this adapted SH for LVP by SHL. More importantly, we further find that SHL can be alliteratively implemented in a parallel manner. Each trajectory can be calculated directly from the input parameters without its preceding trajectory’s information. We denote this parallel alternative of SHL with PSHL. Apparently, PSHL allows further improved computational efficiency with parallel computing. This section describes PSHL and validates that PSHL indeed solves LVP. We first introduce an *extended forward shooting operation* (EFSO) that merges two feasible trajectories.
[EFSO-1:]{}
: Given two feasible trajectories $p:=\left[\mathbf{s}_{j}:=\left(l_{j},v_{j},a_{j},t_{j},t_{j+1}\right)\right]_{j=1,2,\cdots,J}$ and $p'=\mathbf{s}'_{k}:=\left[\left(l'_{k},v'_{k},a'_{k},\right.\right.$ $\left.\left.t'_{k},t'_{k+1}\right)\right]_{k=1,2,\cdots,K}\in\mathcal{T}$, and deceleration rate $a^{-}<0$. Set iterators $j=1$ and $k=1$ .
[EFSO-2:]{}
: Solve $t_{jk}^{\mbox{m}}:=t^{\mbox{mf}}\left(\mathbf{s}'_{k},\left(l_{j},v_{j},t_{j},a_{j}\right),a^{-}\right)$ and $t_{jk}^{+}=t^{\mbox{+f}}\left(\mathbf{s}'_{k},\left(l_{j},v_{j},t_{j},a_{j}\right),a^{-}\right)$. If $t_{jk}^{\mbox{m}}=-\infty$, return $t^{\mbox{m}}(p',p,a^{-})=t^{+}(p',p,a^{-})=-\infty$. If $t_{jk}^{\mbox{m}}<\infty$ and $t_{jk}^{\mbox{m}}\in\left[t_{j},t_{j+1}\right]$, return $t^{\mbox{m}}(p',p,a^{-})=t_{jk}^{\mbox{m}}$, $t^{\mbox{+}}(p',p,a^{-})=t_{jk}^{+}$. If $k<K$, set $k=k+1$ and repeat this step; otherwise, go to the next step.
[EFSO-2:]{}
: If $j<J$, set $j=j+1$ and $k=1$ and go to Step EFSO-2. Otherwise, return $t^{\mbox{m}}(p',p,a^{-})=t^{+}(p',p,a^{-})=\infty$.
Based on EFSO, we devise the following *extended forward shooting process* (EFSP) that generates a forward shooting trajectory constrained by a series of upper bound trajectories.
[EFSP-1:]{}
: Given a feasible state point $(l,v,t)$, and a set of trajectories $\left\{ p_{m}\in\mathcal{T}\right\} _{m=1,\cdots,M}$. Initiate $m=1$ and $p=p^{\mbox{f}}\left((l,v,t),\emptyset\right)$ with FSP.
[EFSP-2:]{}
: Call EFSO to solve $t^{\mbox{m}}:=t^{\mbox{m}}\left(p_{m},p,\underline{a}^{\mathbf{\mbox{f}}}\right)$ and $t^{\mbox{+}}:=t^{\mbox{+}}\left(p_{m},p,\underline{a}^{\mathbf{\mbox{f}}}\right)$. If $t^{\mbox{m}}=-\infty$, return $p^{\mbox{f}}\left((l,v,t),\left\{ p_{m}\right\} _{m=1,\cdots,M}\right)=\emptyset$. If $t^{\mbox{m}}<\infty$, revise $p:=\left[p\left(t^{-}\left(p\right):t^{\mbox{m}}\right),\left(p\left(t^{\mbox{m}}\right),\dot{p}\left(t^{\mbox{m}}\right),\underline{a}^{\mathbf{\mbox{f}}},t^{\mbox{m}},t^{+}\right),\right.$ $\left.p_{m}\left(t^{+}:\infty\right)\right]$.
[EFSP-3:]{}
: If $m<M$, set $m=m+1$ and go to PSFP-2; otherwise return $p^{\mbox{f}}\left((l,v,t),\left\{ p_{m}\right\} _{m=1,\cdots,M}\right)=p$.
Then we are ready to present the PSHL algorithm as follows.
:
[PSHL-1:]{}
: Given lead trajectory $p_{1}\in\mathcal{T}$ and boundary condition $\left[v_{n}^{-},t_{n}^{-}\right]_{n\in\mathcal{N}\backslash\{1\}}$. Initialize acceleration parameters $\bar{a}^{\mbox{f}}$ and $\underline{a}^{\mbox{f}}$. Set $\bar{p}_{1}:=p_{1},\bar{p}_{n}:=p^{\mbox{f}}\left((0,v_{n}^{-},t_{n}^{-}),\emptyset\right),\forall n\in\mathcal{N}\backslash\{1\}.$ Set initial trajectory platoon $P=[p_{1}]$,$n=2$.
[PSHL-2:]{}
: Solve $p_{n}:=p^{\mbox{f}}\left(\left(0,v_{n}^{-},t_{n}^{-}\right),\left\{ \bar{p}_{m}^{\mbox{s}^{n-m}}\right\} _{m=1,\cdots,n-1}\right)$ with EFSP.
[PSHL-3:]{}
: If $p_{n}=\emptyset$, return $P^{\mbox{PSH}}\left(\underline{a}^{\mbox{f}},\bar{a}^{\mbox{f}}\right):=\emptyset$. Otherwise, append $p_{n}$ to $P$. If $n=N$, return $P^{\mbox{PSH}}\left(\underline{a}^{\mbox{f}},\bar{a}^{\mbox{f}}\right):=P$; otherwise, set $n=n+1$ and go to PSHL-2.
$P^{\mbox{PSH}}\left(\underline{a}^{\mbox{f}},\bar{a}^{\mbox{f}}\right)$ obtained from PSHL when $p_{1}\in\mathcal{T}$ is fixed and $\mathcal{G}=\left(-\infty,\infty\right)$.\[prop: PSHL=00003DSHL\]
We first consider the cases that $P^{\mbox{PSH}}\left(\underline{a}^{\mbox{f}},\bar{a}^{\mbox{f}}\right)\neq\emptyset$. We will prove this proposition via induction with the iterator being vehicle index $n$. Let $\left[p_{1,}p_{2},\cdots,p_{N}\right]$ denote the trajectories in $P^{\mbox{PSH}}\left(\underline{a}^{\mbox{f}},\bar{a}^{\mbox{f}}\right)$ and $\left[p'_{1,}p'_{2},\cdots,p'_{N}\right]$ denote the trajectories in $P^{\mbox{SH}}\left(\underline{a}^{\mbox{f}},\bar{a}^{\mbox{f}},\underline{a}^{\mbox{b}},\bar{a}^{\mbox{b}}\right)$. Apparently, when $n=1$,$p'_{1}=p_{1}$ since the lead vehicle’s trajectory is fixed. Assume that when $n=k$, $p'_{k}=p_{k}$. Based on the definition of SH, $p'_{k+1}=\left[\bar{p}_{k+1}\left(t_{k+1}^{-}:t^{\mbox{m}}\right),\left(\bar{p}_{k+1}\left(t^{\mbox{m}}\right),\dot{\bar{p}}_{k+1}\left(t^{\mbox{m}}\right),\underline{a}^{\mathbf{\mbox{f}}},t^{\mbox{m}},t^{+}\right),p_{k}^{\mathbf{'\mbox{s}}}\left(t^{+}:\infty\right)\right]$ where $\bar{p}_{k+1}:=p^{\mbox{f}}\left((0,v_{k+1}^{-},t_{k+1}^{-}),\emptyset\right)$, $p_{k}^{\mathbf{'\mbox{s}}}(t):=p'_{k}(t-\tau)-s$, $t^{\mbox{m}}:=t^{\mbox{m}}\left(p_{k}^{\mathbf{'\mbox{s}}},\bar{p}_{k+1},\underline{a}^{\mathbf{\mbox{f}}}\right)$, $t^{\mbox{+}}:=t^{\mbox{+}}\left(p_{k}^{\mathbf{'\mbox{s}}},\bar{p}_{k+1},\underline{a}^{\mathbf{\mbox{f}}}\right)$. Based on the induction assumption, $p_{k}^{\mathbf{'\mbox{s}}}=p_{k}^{\mathbf{\mbox{s}}}$ can be obtained by repeatedly calling EFSO to merge $\left\{ \bar{p}_{m}^{\mbox{s}^{k+1-m}}\right\} _{m=1,\cdots,k}$ as in PSHL-2. Therefore, $p'_{k+1}$ is obtained by merging $\left\{ \bar{p}_{m}^{\mbox{s}^{k+1-m}}\right\} _{m=1,\cdots,k+1}$ and thus $p'_{k+1}=p_{k+1}$.
When $P^{\mbox{PSH}}\left(\underline{a}^{\mbox{f}},\bar{a}^{\mbox{f}}\right)=\emptyset$, from the above discussion that shows the equivalence of generating $p_{k}$ and $p'_{k}$, it is obvious that $P^{\mbox{SH}}\left(\underline{a}^{\mbox{f}},\bar{a}^{\mbox{f}}\right)=\emptyset$ as well. This completes the proof.
Given lead trajectory $p_{1}\in\mathcal{T}$ , PSHL yields $P^{\mbox{PSH}}\left(\underline{a}^{\mbox{f}},\bar{a}^{\mbox{f}}\right)\in\mathcal{P}^{\mbox{LVP}}\left(p_{1}\right)$ if $\underline{a}^{\mbox{f}},\bar{a}^{\mbox{f}}\in\left[\underline{a},\bar{a}\right]$.
Theoretical Property Analysis \[sec:Theoretical-Properties-of\]
===============================================================
This section analyze theoretical properties of the proposed SH algorithms, including their solution feasibility and relationship with the classic traffic flow theory. It is actually quite challenging to analyze such properties because the original problem defined in Section \[sec:Problem-Statement\] involves infinite-dimensional trajectory variables and highly nonlinear constraints. Fortunately, the concept of time geography [@miller2005measurement] is found related to the bounds to feasible trajectory ranges. We generalize this concept to enable the following theoretical analysis.
Quadratic Time Geography
-------------------------
We first generalize the time geography theory considering acceleration range $[\bar{a},\underline{a}]$ in additional to speed range $[0,\bar{v}]$. These generalized theory, which we call the *quadratic time geography* (QTG) theory, are illustrated in Figure \[fig:gen\_time\_geography\] and discussed in detail in this subsection.
![Illustrations of the generalized time geography theory in : (a) quadratic cone; and (b) quadratic prism. \[fig:gen\_time\_geography\]](cone "fig:"){width="50.00000%"}![Illustrations of the generalized time geography theory in : (a) quadratic cone; and (b) quadratic prism. \[fig:gen\_time\_geography\]](prism "fig:"){width="50.00000%"}
(a)(b)
We call the set of feasible trajectories (i.e., in $\mathcal{T}$) passing a common feasible state point *$\left(l,v,t^{-}\right)$ the quadratic cone* of $\left(l,v,t^{-}\right)$, denoted by $\mathcal{C}_{lvt^{-}}$, illustrated as the shaded area in Figure \[fig:gen\_time\_geography\](a) and formulated below: $$\mathcal{C}_{lvt^{-}}=\left\{ p\left|p\in\mathcal{T},\,p(t^{-})=l,\,\dot{p}(t^{-})=v,\forall t\in\left(-\infty,\infty\right)\right.\right\} ,$$ where *the upper bound* *trajectory* $\bar{p}_{lvt^{-}}$ of $\left(l,v,t^{-}\right)$, illustrated as the top boundary of the shade in Figure \[fig:gen\_time\_geography\](a), is formulated as
$$\bar{p}_{lvt^{-}}(t):=\begin{cases}
l-\frac{v^{2}}{2\bar{a}}, & \mbox{if }t\in\left(-\infty,t^{-}-\frac{v}{\bar{a}}\right];\\
l+v(t-t^{-})+0.5\bar{a}(t-t^{-})^{2}, & \mbox{if }t\in\left[t^{-}-\frac{v}{\bar{a}},t^{-}+\frac{\bar{v}-v}{\bar{a}}\right];\\
l+\bar{v}(t-t^{-})-\frac{\left(\bar{v}-v\right)^{2}}{2\bar{a}}, & \mbox{if }t\in\left[t^{-}+\frac{\bar{v}-v}{\bar{a}},\infty\right),
\end{cases}$$ and *the lower bound* *trajectory $\underline{p}_{lvt^{-}}$*of $\left(l,v,t^{-}\right)$, illustrated as the bottom boundary of the shade in Figure \[fig:gen\_time\_geography\](a), is formulated as
$$\underline{p}_{lvt^{-}}(t)=\begin{cases}
l+\bar{v}(t-t^{-})-\frac{\left(\bar{v}-v\right)^{2}}{2\underline{a}}, & \mbox{if }t\in\left(-\infty,t^{-}-\frac{v-\bar{v}}{\underline{a}}\right];\\
l+v(t-t^{-})+0.5\underline{a}(t-t^{-})^{2}, & \mbox{if }t\in\left[t^{-}-\frac{v-\bar{v}}{\underline{a}},t^{-}+\frac{-v}{\underline{a}}\right];\\
l+\frac{-v^{2}}{2\underline{a}}, & \mbox{if }t\in\left[t^{-}+\frac{-v}{\underline{a}},\infty\right).
\end{cases}$$ In other words, $\bar{p}_{lvt^{-}}$ is composed of quadratic segments $(l-\frac{v^{2}}{2\bar{a}},0,0,t^{-}-\frac{v}{\bar{a}},-\infty)$, $(l-\frac{v^{2}}{2\bar{a}},0,\bar{a},t^{-}-\frac{v}{\bar{a}},t^{-}+\frac{\bar{v}-v}{\bar{a}})$ and $(l+\frac{\bar{v}^{2}-v^{2}}{2\bar{a}},\bar{v},0,t^{-}+\frac{\bar{v}-v}{\bar{a}},\infty)$, and $\bar{p}_{lvt^{-}}$ is comprised of quadratic segments $(l-\frac{v^{2}-\bar{v}^{2}}{2\underline{a}},\bar{v},0,t^{-}-\frac{v-\bar{v}}{\underline{a}},-\infty)$, $(l-\frac{v^{2}-\bar{v}^{2}}{2\underline{a}},\bar{v},\bar{a},t^{-}-\frac{v-\bar{v}}{\underline{a}},t^{-}+\frac{-v}{\underline{a}})$ and $(l+\frac{-v^{2}}{2\underline{a}},0,0,t^{-}+\frac{-v}{\underline{a}},\infty)$. Note that $\bar{p}_{lvt^{-}}=p^{\mbox{f}}\left(\left(l,v,t^{-}\right),\emptyset\right)$ from the FSP with $\bar{a}^{\mbox{f}}=\bar{a}$ and $\underline{a}^{\mbox{f}}=\underline{a}$, $\bar{p}_{lvt^{-}}(t^{-})=\underline{p}_{lvt^{-}}(t^{-})=l$, and $\bar{p}_{lvt^{-}}(t)\ge\underline{p}_{lvt^{-}}(t),\forall t\neq t^{-}$. Further, $\mathcal{C}_{lvt^{-}}$ is always non-empty as long as $(l,v,t^{-})$ is feasible.
We call the set of trajectories in $\mathcal{T}$ passing two feasible state points $(l^{-},v^{-},t^{-})$ and $(l^{+},v^{+},t^{+})$ with $t^{-}<t^{+}$, $l^{-}\le l^{+}$ *a quadratic prism*, denoted by $\mathcal{P}_{l^{-}v^{-}t^{-}}^{l^{+}v^{+}t^{+}}$, as illustrated in Figure \[fig:gen\_time\_geography\](b) and formulated below:
$$\mathcal{P}_{l^{-}v^{-}t^{-}}^{l^{+}v^{+}t^{+}}:=\left\{ p\in\mathcal{T}\left|p(t^{-})=l^{-},\dot{p}(t^{-})=v^{-},p(t^{+})=l^{+},\dot{p}(t^{+})=v^{+}\right.\right\} =\mathcal{C}^{p^{-}v^{-}t^{-}}\bigcap\mathcal{C}^{p^{+}v^{+}t^{+}}.$$ The upper bound of this $\mathcal{P}_{l^{-}v^{-}t^{-}}^{l^{+}v^{+}t^{+}}$ (as the top boundary of the shade in \[fig:gen\_time\_geography\](b)), denoted by $\bar{p}_{l^{-}v^{-}t^{-}}^{l^{+}v^{+}t^{+}}$ , should be composed by $\bar{p}_{l^{-}v^{-}t^{-}}\left(\infty:\bar{t}^{\mbox{m}}\right)$, merging segment $\bar{\mathbf{s}}^{\mbox{m}}\left(\bar{p}_{l^{-}v^{-}t^{-}}\left(\bar{t}^{\mbox{m}}\right),\dot{\bar{p}}_{l^{-}v^{-}t^{-}}\left(\bar{t}^{\mbox{m}}\right),\bar{a},\bar{t}^{\mbox{m}},\bar{t}^{\mbox{+}}\right)$ and $\bar{p}_{l^{+}v^{+}t^{+}}\left(\bar{t}^{+}:\infty\right)$, where we can actually apply FSP with $\bar{a}^{\mbox{f}}=\bar{a}$ and $\underline{a}^{\mbox{f}}=\underline{a}$ to obtain connection points $\bar{t}^{\mbox{m}}:=t^{\mbox{mf}}\left(\left(l^{-},v^{-},t^{-}\right),\bar{p}_{l^{+}v^{+}t^{+}}\right)$ and $\bar{t}^{\mbox{+}}:=t^{\mbox{+f}}\left(\left(l^{-},v^{-},t^{-}\right),\bar{p}_{l^{+}v^{+}t^{+}}\right)$. The lower bound of this prism (as the bottom boundary of the shade in \[fig:gen\_time\_geography\](b)), denoted by $\underline{p}_{l^{-}v^{-}t^{-}}^{l^{+}v^{+}t^{+}},$ is composed by section $\underline{p}_{l^{-}v^{-}t^{-}}(\infty:\underline{t}^{\mbox{m}})$, merging segment $\bar{\mathbf{s}}^{\mbox{m}}\left(\underline{p}_{l^{-}v^{-}t^{-}}(\underline{t}^{\mbox{m}}),\dot{\underline{p}}_{l^{-}v^{-}t^{-}}(\underline{t}^{\mbox{m}}),\bar{a},\underline{t}^{\mbox{m}},\underline{t}^{\mbox{+}}\right)$ and $\underline{p}_{l^{+}v^{+}t^{+}}\left(\underline{t}^{+}:\infty\right)$, where we can apply FSP in a transformed coordinate system to solve $\underline{t}^{\mbox{m}}$ and $\underline{t}^{\mbox{+}}.$ We shift the first shift the origin to $(l^{+},t^{+})$ and rotate the whole coordinate system by 180 degree. Then state points $(l^{+},v^{+},t^{+})$ and $(l^{-},v^{-},t^{-})$ transfer into $(0,v^{+},0)$ and $(l^{+}-l^{-},v^{-},t^{+}-t^{-})$, respectively, and $\underline{p}_{l^{-}v^{-}t^{-}}$ transfers into $\hat{\underline{p}}_{l^{-}v^{-}t^{-}}$ defined as $\hat{\underline{p}}_{l^{-}v^{-}t^{-}}(t):=l^{+}-\underline{p}_{l^{-}v^{-}t^{-}}\left(2t^{+}-\left(t^{-}+t\right)\right)$. Then we solve $\hat{\underline{t}}^{\mbox{m}}:=t^{\mbox{mf}}\left(\left(0,v^{+},0\right),\hat{\underline{p}}_{l^{-}v^{-}t^{-}}\right)$ and $\hat{\underline{t}}^{\mbox{+}}:=t^{\mbox{+f}}\left(\left(0,v^{+},0\right),\hat{\underline{p}}_{l^{-}v^{-}t^{-}}\right)$ with FSP with $a^{\mbox{f}}=-\underline{a}$ and $\underline{a}^{\mbox{f}}=-\bar{a}$. Then we obtain $\underline{t}^{+}=t^{+}-\hat{\underline{t}}^{\mbox{m}}$ and $\underline{t}^{\mbox{m}}=t^{+}-\hat{\underline{t}}^{\mbox{+}}$.
Note that the feasibility of $\mathcal{P}_{l^{-}v^{-}t^{-}}^{l^{+}v^{+}t^{+}}$ depends on the values of $(l^{-},v^{-},t^{-})$ and $(l^{+},v^{+},t^{+})$, as discussed in the following propositions.
\[prop:prism\_feasibility\]Given two feasible state points $(l^{-},v^{-},t^{-})$ and $(l^{+},v^{+},t^{+})$, quadratic cone $\mathcal{P}_{l^{-}v^{-}t^{-}}^{l^{+}v^{+}t^{+}}$is not empty if and only if $D\left(\bar{p}_{l^{+}v^{+}t^{+}}-\underline{p}_{l^{-}v^{-}t^{-}}\right)\ge0$ and $D\left(\bar{p}_{l^{-}v^{-}t^{-}}-\underline{p}_{l^{+}v^{+}t^{+}}\right)\ge0$ .
We first prove the necessity. If there exists a feasible trajectory $p\in\mathcal{P}_{l^{-}v^{-}t^{-}}^{l^{+}v^{+}t^{+}}$, then we know the $D\left(\bar{p}_{l^{+}v^{+}t^{+}}-p\right)\ge0$ and $D\left(p-\underline{p}_{l^{-}v^{-}t^{-}}\right)\ge0$, which indicates $D\left(\bar{p}_{l^{+}v^{+}t^{+}}-\underline{p}_{l^{-}v^{-}t^{-}}\right)\ge0$. Symmetrically, $D\left(\bar{p}_{l^{-}v^{-}t^{-}}-p\right)\ge0$ and $D\left(p-\underline{p}_{l^{+}v^{+}t^{+}}\right)\ge0$ indicates $D\left(\bar{p}_{l^{-}v^{-}t^{-}}-\underline{p}_{l^{+}v^{+}t^{+}}\right)\ge0$.
Then we investigate the sufficiency. Given $D\left(\bar{p}_{l^{+}v^{+}t^{+}}-\underline{p}_{l^{-}v^{-}t^{-}}\right)\ge0$ and $D\left(\bar{p}_{l^{-}v^{-}t^{-}}-\underline{p}_{l^{+}v^{+}t^{+}}\right)\ge0$, we can first obtain that $l^{-}<l^{+}$ and $t^{-}<t^{+}$. Further we know that there exists a point $\bar{t}^{*}\in[t^{-},t^{+}]$ such that $\bar{p}_{l^{+}v^{+}t^{+}}(t)\ge\bar{p}_{l^{-}v^{-}t^{-}}(t),\forall t\in\left[-\infty,\bar{t}^{*}\right]$, $\bar{p}_{l^{+}v^{+}t^{+}}\left(\bar{t}^{*}\right)=\bar{p}_{l^{-}v^{-}t^{-}}\left(\bar{t}^{*}\right)$ and $\bar{p}_{l^{+}v^{+}t^{+}}(t)\le\bar{p}_{l^{-}v^{-}t^{-}}(t),\forall t\in\left[\bar{t}^{*},\infty\right]$. Then we can obtain a trajectory $p^{\mbox{m}}\in\mathcal{C}_{l^{-}v^{-}t^{-}}$ composed by $\bar{p}_{lvt^{-}}\left(t^{-}:\hat{t}^{\mbox{m}}\right)$, $\mathbf{s}^{\mbox{m}}:=\left(\bar{p}_{lvt^{-}}\left(\hat{t}^{\mbox{m}}\right),\dot{\bar{p}}_{lvt^{-}}\left(\hat{t}^{\mbox{m}}\right),\underline{a},\hat{t}^{\mbox{m}},\bar{t}^{\mbox{m}}\right)$, $\bar{p}_{l^{+}v^{+}t^{+}}\left(\bar{t}^{\mbox{m}}:\infty\right)$ satisfying $t^{-}\le\hat{t}^{\mbox{m}}\le\bar{t}^{\mbox{m}}<\infty$ and $D\left(\bar{p}_{l^{+}v^{+}t^{+}}-p^{\mbox{m}}\right)\ge0$. Next, we will prove $\bar{t}^{\mbox{m}}\le t^{+}$ by contradiction. If $\bar{t}^{\mbox{m}}>t^{+}$, then $\bar{p}_{l^{+}v^{+}t^{+}}\left(\bar{t}^{\mbox{m}}\right)>l^{+}$ and $\dot{\bar{p}}_{l^{+}v^{+}t^{+}}\left(\bar{t}^{\mbox{m}}\right)>v^{+}$. Since segment $\mathbf{s}^{\mbox{m}}$ decelerates at $\underline{a}$, then $D\left(\underline{p}_{l^{+}v^{+}t^{+}}-\mathbf{s}^{\mbox{m}}\right)>0$, which however is contradictory to $D\left(\bar{p}_{l^{-}v^{-}t^{-}}-\underline{p}_{l^{+}v^{+}t^{+}}\right)\ge0$ because the start point of $\mathbf{s}^{\mbox{m}}$ is on $\bar{p}_{l^{-}v^{-}t^{-}}$. This proves that $t^{-}\le\hat{t}^{\mbox{m}}\le\bar{t}^{\mbox{m}}\le t^{+}$. Therefore, $p^{\mbox{m}}\in\mathcal{C}_{l^{-}v^{-}t^{-}}\cap\mathcal{C}_{l^{+}v^{+}t^{+}}=\mathcal{P}_{l^{-}v^{-}t^{-}}^{l^{+}v^{+}t^{+}}$. This completes the proof.
\[prop: shift\_prism\_feasibility\]Given any $\delta\ge0$ and two feasible state points $(l^{-},v^{-},t^{-})$ and $(l^{+},v^{+},t^{+})$ with $D(\bar{p}_{l^{-},v^{-},t^{-}}-\underline{p}_{l^{+},v^{+},t^{+}})\ge0$, if quadratic prism $\mathcal{P}_{l^{-}v^{-}t^{-}}^{l^{+}v^{+}\left(t^{+}+\delta\right)}$ is not empty, then $\mathcal{P}_{l^{-}v^{-}t^{-}}^{l^{+}v^{+}t^{+}}$ is not empty and $D\left(\bar{p}_{l^{-}v^{-}t^{-}}^{l^{+}v^{+}t^{+}}-p\right)\ge0,\forall p\in\mathcal{P}_{l^{-}v^{-}t^{-}}^{l^{+}v^{+}\left(t^{+}+\delta\right)}$.
If $\mathcal{P}_{l^{-}v^{-}t^{-}}^{l^{+}v^{+}\left(t^{+}+\delta\right)}$ is not empty, Proposition \[prop:prism\_feasibility\] indicates that $D\left(\bar{p}_{l^{+}v^{+}\left(t^{+}+\delta\right)}-\underline{p}_{l^{-}v^{-}t^{-}}\right)\ge0$. Further, apparently $D\left(\bar{p}_{l^{+}v^{+}t^{+}}-\bar{p}_{l^{+}v^{+}\left(t^{+}+\delta\right)}\right)\ge0$ and thus due to the transitive property of function $D$, we obtain $D\left(\bar{p}_{l^{+}v^{+}t^{+}}-\underline{p}_{l^{-}v^{-}t^{-}}\right)\ge0$, which combined with the given condition $D\left(\bar{p}_{l^{-},v^{-},t^{-}}-\underline{p}_{l^{+},v^{+},t^{+}}\right)\ge0$ indicates that $\mathcal{P}_{l^{-}v^{-}t^{-}}^{l^{+}v^{+}t^{+}}$ is not empty based on Proposition \[prop:prism\_feasibility\].
Further, for any $p\in\mathcal{P}_{l^{-}v^{-}t^{-}}^{l^{+}v^{+}\left(t^{+}+\delta\right)}$, we have $D\left(\bar{p}_{l^{-}v^{-}t^{-}}^{l^{+}v^{+}\left(t^{+}+\delta\right)}-p\right)>0$. Since $D\left(\bar{p}_{l^{+}v^{+}t^{+}}-\bar{p}_{l^{+}v^{+}\left(t^{+}+\delta\right)}\right)\ge0$, we also have $D\left(\bar{p}_{l^{+}v^{+}t^{+}}-p\right)\ge0$ due to the transitive property of function $D$. This implies that $p\in\mathcal{P}_{l^{-}v^{-}t^{-}}^{l^{+}v^{+}t^{+}}$ and thus $D\left(\bar{p}_{l^{-}v^{-}t^{-}}^{l^{+}v^{+}t^{+}}-p\right)\ge0.$ This completes the proof.
Note that as $\bar{a}\rightarrow\infty$ and $\underline{a}\rightarrow-\infty$, every smooth speed transition segment on the borders of a quartic cone or prism reduces into a vertex, and the QTG concept converges to the classic time geography [@miller2005measurement]. Besides, when the spatiotemporal range of the studied problem is far greater than that where acceleration $\bar{a}$ and deceleration $\underline{a}$ is discernible, neither is QTG much different from the classic time geography.
Relationship Between QTG and SH
--------------------------------
As preparing for the investigation to the feasibility and optimality of the SH solution, we now discuss the relationships between trajectories generated from FSP and BSP and the borders of the corresponding quadratic cone and prism. For uniformity, this subsection only considers FSP and BSP with the extreme acceleration control variable values, i.e., $\left[\bar{a}^{\mbox{f}},\underline{a}^{\mbox{f}},\bar{a}^{\mbox{b}},\underline{a}^{\mbox{b}}\right]=\left[\bar{a},\underline{a},\bar{a},\underline{a}\right]$.
\[prop: FSP-p\_bar\]The forward shooting trajectory $p^{\mbox{f}}\left(\left(l,v,t^{-}\right),\emptyset\right)$ generated from FSP overlaps
\[prop: BSP-p\_bar\]Given two feasible state points $\left(l^{-},v^{-},t^{-}\right)$ and $\left(l^{+},v^{+},t^{+}\right)$ with $l^{+}>l^{-}$ and $t^{+}>t^{-}$ such that quadratic prism $\mathcal{P}_{l^{-}v^{-}t^{-}}^{l^{+}v^{+}t^{+}}\neq\emptyset$, the extended backward shooting trajectory $p^{\mbox{eb}}((l^{+},v^{+},t^{+}),p^{\mbox{f}}((l^{-},v^{-},t^{-}),\emptyset))$ generated from BSP overlaps
These two propositions obviously hold based on the definitions of FSP and BSP and thus we omit the proofs. These properties can be extended to cases where the current trajectory is bounded by one or multiple preceding trajectories from the top.
Given a set of trajectories $\mathbf{q}=\left\{ q_{1},q_{2},\cdots,q_{M}\in\mathcal{T}\right\} $, we define $u(\mathbf{q},t):=\min_{m\in\left\{ 1,\cdots,M\right\} }q_{m}(t),\forall t$ and we call function $u(\mathbf{q},\cdot)$ a *quasi-trajectory* and denote it with $u(\mathbf{q})$ for simplicity. Let $\mathcal{U}$ denote the set of all quasi-trajectories. Note that distance function $D$ can be easily extended to $\mathcal{U}$, i.e., $D\left(u-u'\right):=\min_{t\in(\infty,\infty)}\left(u(t)-u'(t)\right),\forall u,u'\in\mathcal{U}$. We can also denote EFSP result $p^{\mbox{f}}\left(\left(l,v,t\right),\mathbf{q}\right)$ as $p^{\mbox{f}}\left(\left(l,v,t\right),u(\mathbf{q})\right)$. Further, let $p(\mathbf{q})$ denote the trajectory generated by merging all trajectories in $\mathbf{q}$ with EFSO, and we can also denote BSP result $p^{\mbox{b}}\left(\left(l,v,t\right),p(\mathbf{q})\right)$ and $p^{\mbox{eb}}\left(\left(l,v,t\right),p(\mathbf{q})\right)$ as $p^{\mbox{b}}\left(\left(l,v,t\right),u(\mathbf{q})\right)$ and $p^{\mbox{eb}}\left(\left(l,v,t\right),u(\mathbf{q})\right)$, respectively, for simplicity.
Given a feasible state point $(l,v,t^{-})$ and a quasi-trajectory $u\in\mathcal{U}$ , we define $\mathcal{C}_{lvt^{-}}^{u}:=\left\{ p\left|p\in\mathcal{C}_{lvt^{-}},D\left(u-p\right)\ge0\right.\right\} $, which we call a *bounded cone* of $(l,v,t^{-})$ with respect to $u$.
Given two feasible state points $(l^{-},v^{-},t^{-})$, $(l^{+},v^{+},t^{+})$ and a quasi-trajectory $u\in\mathcal{U}$, we define $\mathcal{P}_{l^{-}v^{-}t^{-}}^{l^{+}v^{+}t^{+},u}:=\left\{ p\left|p\in\mathcal{P}_{l^{-}v^{-}t^{-}}^{l^{+}v^{+}t^{+}},D\left(u-p\right)\ge0\right.\right\} $, which we call a *bounded prism* of $(l^{-},v^{-},t^{-})$ and $(l^{+},v^{+},t^{+})$ with respect to $u$.
Apparently, bounded cones and prisms shall satisfy the following properties.
\[prop:bounded\_transitive\]Given any two feasible state points $(l^{-},v^{-},t^{-})$, $(l^{+},v^{+},t^{+})$ and two quasi-trajectories $u,u'\in\mathcal{T}$ with $D(u'-u)\ge0$, then $\mathcal{C}_{l^{-}v^{-}t^{-}}^{u}\subseteq\mathcal{C}_{l^{-}v^{-}t^{-}}^{u'}$ and $\mathcal{P}_{l^{-}v^{-}t^{-}}^{l^{+}v^{+}t^{+},u}\subseteq\mathcal{P}_{l^{-}v^{-}t^{-}}^{l^{+}v^{+}t^{+},u'}$.
Then we will prove less intuitive properties for bounded cones and prisms.
\[prop: FSP-feasibility\]Given a feasible state point $(l,v,t^{-})$ and a trajectory $p'\in\mathcal{T}$, then trajectory $p^{\mbox{f}}((l,v,t^{-}),p')$ obtained from FSP is not empty $\Leftrightarrow$ $\mathcal{C}_{lvt^{-}}^{p'}\neq\emptyset$ $\Leftrightarrow$ $D\left(p'-\underline{p}_{lvt^{-}}\right)\ge0$.
It is apparent that $p^{\mbox{f}}:=p^{\mbox{f}}((l,v,t^{-}),p')\neq\emptyset$ leads to $\mathcal{C}_{lvt^{-}}^{p'}\neq\emptyset$ because $p^{\mbox{f}}\in\mathcal{C}_{lvt^{-}}^{p'}$. Further if we can find a feasible trajectory in $p\in\mathcal{C}_{lvt^{-}}^{p'}$, we know that $D\left(p'-p\right)\ge0$ and $D\left(p-\underline{p}_{lvt^{-}}\right)\ge0$, which yields $D\left(p'-\underline{p}_{lvt^{-}}\right)\ge0$ since $D$ is transitive. Now we only need to prove that $D\left(p'-\underline{p}_{lvt^{-}}\right)\ge0$ leads to $p^{\mbox{f}}((l,v,t^{-}),p')\neq\emptyset$. Now we are given $p'(t)\ge\underline{p}_{lvt^{-}}(t),\forall t\in[t^{-},\infty)$. If $p'(t)\ge\bar{p}_{lvt^{-}}(t),\forall t\in[t^{-},\infty)$, then Proposition \[prop: FSP-p\_bar\] indicates $p^{\mbox{f}}((l,v,t^{-}),p')=p^{\mbox{f}}((l,v,t^{-}),\emptyset)$ that overlaps $\bar{p}_{lvt^{-}}(t^{-}:\emptyset)$. Therefore $p^{\mbox{f}}((l,v,t^{-}),p')$ should be always non-empty. Otherwise, it should be that $p'(t)\ge\bar{p}_{lvt^{-}}(t),\forall t\in[t^{-},t^{*})$, $p'(t^{*})=\bar{p}_{lvt^{-}}(t^{*})$ and $p'(t)<\bar{p}_{lvt^{-}}(t),\forall t\in\left(t^{*},\infty\right)$ for some $t^{*}\in[t^{-},\infty)$. We first define a continuous function of time $\hat{t}\in[t^{-},\infty)$ as follows. We construct a trajectory denoted by $\underline{p}_{\hat{t}}$ composed by maximally accelerating section $\bar{p}_{lvt^{-}}(t^{-}:\hat{t})$ and a maximally decelerating section $\underline{p}_{\underline{p}_{lvt^{-}}(\hat{t})\dot{\underline{p}}_{lvt^{-}}(\hat{t})t^{-}}(t^{-}:\infty)$. Then we define function $$d(\hat{t}):=D\left(p'-\underline{p}_{\hat{t}}\right)=\min_{t\in[t^{-},\infty)}p'(t)-\underline{p}_{\hat{t}}(t).$$ Note that as $\hat{t}$ increases continuously, $\underline{p}_{\hat{t}}(t)$ increases continuously at every $t\in[t^{-},\infty)$. Then we can see that function $d(\hat{t})$ shall continuously decrease with $\hat{t}$. Note that $\underline{p}_{\hat{t}}$ is identical to $\underline{p}_{lvt^{-}}$ when $\hat{t}=0$. Then since $p'(t)\ge\underline{p}_{lvt^{-}}(t),\forall t\in[t^{-},\infty)$, we obtain $d(0)\ge0$. Further, as $\hat{t}$ increases to $t^{*}$, then $\underline{p}_{\hat{t}}(t)$ and $p'(t)$ shall intersection at $t^{*}$, which indicates that $d(t^{*})\le0$. Due to Bolzano’s Theorem [@apostol1969calculus], we can always find a $\hat{t}^{\mbox{m}}\in[t^{-},t^{*}]$ such that $d(\hat{t}^{\mbox{m}})=0$, which indicates that $p'$ and $\underline{p}_{\hat{t}^{\mbox{m}}}$ get tangent at a time $\bar{t}{}^{\mbox{m}}\in[\hat{t}^{\mbox{m}},\infty).$ Then a trajectory $p^{\mbox{f}}$ can be obtained by concatenating $\bar{p}_{lvt^{-}}\left(t^{-}:\hat{t}^{\mbox{m}}\right)$, $\left(\bar{p}_{lvt^{-}}\left(\hat{t}^{\mbox{m}}\right),\dot{\bar{p}}_{lvt^{-}}\left(\hat{t}^{\mbox{m}}\right),\underline{a},\hat{t}^{\mbox{m}},\bar{t}^{\mbox{m}}\right)$, $p'\left(\bar{t}^{\mbox{m}}:\infty\right)$. Note that $p^{\mbox{f}}$ is exactly $p^{\mbox{f}}((l,v,t^{-}),p')$ and thus $p^{\mbox{f}}((l,v,t^{-}),p')$ is not empty. This completes the proof.
We can further extend this result to a quasi-trajectory as a upper bound.
\[cor:FSP-Feasibility\]Given a feasible state point $(l,v,t^{-})$ and a quasi-trajectory $u\in\mathcal{U}$, then trajectory $p^{\mbox{f}}((l,v,t^{-}),u)$ obtained from the EFSP is not empty $\Leftrightarrow$ $\mathcal{C}_{lvt^{-}}^{u}\neq\emptyset$ $\Leftrightarrow$ $D\left(u-\underline{p}_{lvt^{-}}\right)\ge0$.
\[prop: FSP-shift-bound\]Given a feasible state point $(l,v,t^{-})$, a quasi-trajectory $u\in\mathcal{U}$ and scalars if $\mathcal{C}_{lvt}^{u}\neq\emptyset$, then $\mathcal{C}_{(l-\delta)v\left(t^{-}+\delta'\right)}^{u}\neq\emptyset$ and we obtain $D\left(p^{\mbox{f}}((l,v,t^{-}),u)-p\right)\ge0,\forall p\in\mathcal{C}_{(l-\delta)v\left(t^{-}+\delta'\right)}^{u}$.
We first prove if $\mathcal{C}_{lvt}^{u}\neq\emptyset$, then $\mathcal{C}_{(l-\delta)v\left(t^{-}+\delta'\right)}^{u}\neq\emptyset$. Due to Corollary \[cor:FSP-Feasibility\], we know $D\left(u-\underline{p}_{lvt^{-}}\right)\ge0$. Since apparently $D\left(\underline{p}_{lvt^{-}}-\underline{p}_{(l-\delta)v\left(t^{-}+\delta'\right)}\right)\ge0$, we obtain $D\left(u-\underline{p}_{(l-\delta)v\left(t^{-}+\delta'\right)}\right)\ge0$ due to the transitive property of operator $D$. This yields $\mathcal{C}_{(l-\delta)v\left(t^{-}+\delta'\right)}^{u}\neq\emptyset$ based on Corollary \[cor:FSP-Feasibility\].
The we prove the remaining part of this proposition. In case that $D\left(u-\bar{p}_{lvt^{-}}\right)\ge0$, $p^{\mbox{f}}:=p^{\mbox{f}}((l,v,t^{-}),u)$ shall be identical to $\bar{p}_{lvt^{-}}$ and thus $D\left(p^{\mbox{f}}-p\right)\ge0$ obviously holds $\forall p\in\mathcal{C}_{lvt^{-}}^{u}=\mathcal{C}_{lvt^{-}}.$ Otherwise, then we know that $p^{\mbox{f}}=$$\left[\bar{p}_{lvt^{-}}\left(t^{-}:t^{\mbox{m}}\right),\mathbf{s}^{\mbox{m}}:=\left(\bar{p}_{lvt^{-}}\left(\hat{t}^{\mbox{m}}\right),\dot{\bar{p}}_{lvt^{-}}\left(\hat{t}^{\mbox{m}}\right),\underline{a},\hat{t}^{\mbox{m}},\bar{t}^{\mbox{m}}\right),p'\left(t^{\mbox{+}}:\infty\right)\right]$ for some $t^{\mbox{m}}$ and $t^{+}$ satisfying $t^{-}\le t^{\mbox{m}}\le t^{\mbox{+}}\le\infty$, where $p'$ is the lower-bound trajectory merged by all elements in $u$ with EFSO. If $\exists p\in\mathcal{C}_{lvt^{-}}^{u}$ such that $D\left(p^{\mbox{f}}-p\right)<0$, then there much exist a $\tilde{t}\in(t^{\mbox{m}},t^{\mbox{+}})$ such that $p\left(\tilde{t}\right)$ is strictly above $\mathbf{s}^{\mbox{m}}$. Since $D$$\left(\bar{p}_{lvt^{-}}-p\right)\ge0$ and $D$$\left(p-\underline{p}_{p\left(\tilde{t}\right)\dot{p}\left(\tilde{t}\right)\tilde{t}}\right)\ge0$, thus we have $D$$\left(\bar{p}_{lvt^{-}}-\underline{p}_{p\left(\tilde{t}\right)\dot{p}\left(\tilde{t}\right)\tilde{t}}\right)\ge0$, which indicates $\underline{p}_{p\left(\tilde{t}\right)\dot{p}\left(\tilde{t}\right)\tilde{t}}$ and $\mathbf{s}^{\mbox{m}}$ have to intersect at a time $t'\in\left[\hat{t}^{\mbox{m}},\tilde{t}\right)$. However, sine $D\left(u-p\right)\ge0$, thus $\underline{p}_{p\left(\tilde{t}\right)\dot{p}\left(\tilde{t}\right)\tilde{t}}$ needs to intersect with $\mathbf{s}^{\mbox{m}}$ at another time $t"\in\left(\tilde{t},\bar{t}^{\mbox{m}}\right]$. This is contradictory to the fact that $\mathbf{s}^{\mbox{m}}$ has already decelerated at the extreme deceleration rate $\underline{a}$. This contradiction proves that $D\left(p^{\mbox{f}}-p\right)\ge0,\forall p\in\mathcal{C}_{lvt^{-}}^{p'}$. Further, given $\delta,\delta'\ge0$, for any $p\in\mathcal{C}_{(l-\delta)v\left(t^{-}+\delta'\right)}^{u},$we can find a $p'\in\mathcal{C}_{lvt^{-}}^{p'}$satisfying $D(p'-p)\ge0$ with a similar argument. This proves that $D\left(p^{\mbox{f}}-p'\right)\ge0,\forall p'\in\mathcal{C}_{(l-\delta)v\left(t^{-}+\delta'\right)}^{u},\delta,\delta'\ge0.$
Symmetrically, we can obtain similar properties for BSP as well.
\[cor:BSP-Feasibility\]Given a feasible state point $(l,v,t^{+})$ and a quasi-trajectory $u\in\mathcal{U}$,then trajectory $p^{\mbox{eb}}((l,v,t^{+}),u)$ obtained from BSP is not empty $\Leftrightarrow$ $\mathcal{C}_{lvt^{+}}^{u}\neq\emptyset$ $\Leftrightarrow$ $D\left(u-\underline{p}_{pvt^{+}}\right)\ge0$. If $\mathcal{C}_{lvt^{+}}^{u}\neq\emptyset$, we obtain $\mathcal{C}_{(l-\delta)v\left(t^{+}+\delta'\right)}^{u}\neq\emptyset$ and $D\left(p^{\mbox{eb}}-p\right)\ge0,\forall p\in\mathcal{C}_{(l-\delta)v\left(t^{+}+\delta'\right)}^{u}.$
Combining Corollaries \[cor:FSP-Feasibility\]and \[cor:BSP-Feasibility\] leads to the following property with respect to a quadratic prism.
\[cor:BSP-Feasibility-Prism\]Given two feasible state point $(l^{-},v^{-},t^{-})$, $(l^{+},v^{+},t^{+})$ satisfying $D\left(\bar{p}_{l^{-}v^{-}t^{-}}-\underline{p}_{l^{+}v^{+}t^{+}}\right)\ge0$ and $D\left(\bar{p}_{l^{+}v^{+}t^{+}}-\underline{p}_{l^{-}v^{-}t^{-}}\right)\ge0$, and a quasi-trajectory $u\in\mathcal{U}$, then $p^{\mbox{f}}((l^{-},v^{-},t^{-}),u)\neq\emptyset$ and $p^{\mbox{eb}}((l^{+},v^{+},t^{+}),u)\neq\emptyset$ $\Leftrightarrow$ $\mathcal{P}_{l^{-}v^{-}t^{-}}^{l^{+}v^{+}t^{+},u}\neq\emptyset$ $\Leftrightarrow$ $D(u-\underline{p}_{l^{-}v^{-}t^{-}})\ge0$ and $D(u-\underline{p}_{l^{+}v^{+}t^{+}})\ge0$. Whenever $\mathcal{P}_{l^{-}v^{-}t^{-}}^{l^{+}v^{+}t^{+},u}\neq\emptyset$, we obtain $D\left(p^{\mbox{eb}}-p\right)\ge0,$
Feasibility Properties\[sub:Feasibility-Properties\]
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Although the proposed heuristic algorithms are essentially heuristics that may not explore the entire feasible region of the original problem, it can be used as a touchstone for the feasibility of the original problems under certain mild conditions. This section will discuss the relationship between the feasibility of the SH solutions and that of the original problems. Note that with different acceleration values, the SH algorithms yields different solutions. For uniformity, this section only investigates the representative SH algorithms with $\bar{a}^{\mbox{f}}=\bar{a}^{\mbox{b}}=\bar{a}$ and $\underline{a}^{\mbox{f}}=\underline{a}^{\mbox{b}}=\underline{a}$.
The following analysis investigates LVP, i.e., the feasibility of $\mathcal{P}^{\mbox{LVP}}$ and that of $P^{\mbox{PSH}}\left(\underline{a},\bar{a}\right)$ .
Entry boundary condition $\left[v_{n}^{-},t_{n}^{-}\right]_{n\in\mathcal{N}}$ is *proper* if $v_{n}^{-}\in\left[0,\bar{v}\right],\forall n\in\mathcal{N}$ and $$D\left(\bar{p}_{(ms-ns)v_{m}^{-}(t_{m}^{-}+n\tau-m\tau)}-\underline{p}_{0v_{n}^{-}t_{n}^{-}}\right)\ge0,\forall m<n\in\mathcal{N}.$$
\[prop:proper\_condition\]$\mathcal{P}\neq\emptyset$ $\Rightarrow$$\left[v_{n}^{-},t_{n}^{-}\right]_{n\in\mathcal{N}}$ is proper.
When $\mathcal{P}\neq\emptyset,$ let $P=\left[p_{n}\right]_{n\in\mathcal{N}}$ denote a generic feasible trajectory vector in $\mathcal{P}$. For any given $m<n\in\mathcal{N}$, based on safety constraint , we obtain $D\left(p_{m}^{s}-p_{m+1}\right)\ge0,$ $D\left(p_{m+1}^{s}-p_{m+2}\right)\ge0,\cdots,$ $D\left(p_{n-1}^{s}-p_{n}\right)\ge0$. This can be translated as $D\left(p_{m}^{s^{n-m}}-p_{m+1}^{s^{n-m-1}}\right)\ge0,$ $D\left(p_{m+1}^{s^{n-m-1}}-p_{m+2}^{s^{n-m-2}}\right)\ge0,\cdots,$ $D\left(p_{n-1}^{s}-p_{n}\right)\ge0$, which indicates that $D\left(p_{m}^{s^{n-m}}-p_{n}\right)\ge0$ due to the transitive property of function $D\left(\cdot\right)$. Further, since $D\left(\bar{p}_{(ms-ns)v_{m}^{-}\left[t_{m}^{-}+(n-m)\tau\right]}-p_{m}^{s^{n-m}}\right)\ge0$ and $D\left(p_{n}-\underline{p}_{0v_{n}^{-}t_{n}^{-}}\right)\ge0$ , we obtain $D\left(\bar{p}_{(ms-ns)v_{m}^{-}\left[t_{m}^{-}+(n-m)\tau\right]}-\underline{p}_{0v_{n}^{-}t_{n}^{-}}\right)\ge0,\forall m<n\in\mathcal{N}$. This completes the proof.
\[theo: feasibility\_all\_green\] $\mathcal{P}^{\mbox{LVP}}\neq\emptyset$ $\Leftrightarrow$ $\Leftrightarrow$ $\left[v_{n}^{-},t_{n}^{-}\right]_{n\in\mathcal{N}}$ is proper.
When $\left[v_{n}^{-},t_{n}^{-}\right]_{n\in\mathcal{N}}$ is proper, based on Corollary \[cor:BSP-Feasibility\], we know that every EFSP step will generate a feasible trajectory. Thus $P^{\mbox{PSH}}\left(\underline{a},\bar{a}\right)$ is feasible. Since $P^{\mbox{PSH}}\left(\underline{a},\bar{a}\right)\in\mathcal{P}^{\mbox{LVP}}$, we see that $\mathcal{P}^{\mbox{LVP}}$ is feasible, too. Further, Proposition \[prop:proper\_condition\] indicates that when $\mathcal{P}^{\mbox{LVP}}$ is feasible, which means $\mathcal{P}$ too is feasible, $\left[v_{n}^{-},t_{n}^{-}\right]_{n\in\mathcal{N}}$ is proper. This completes the proof.
Now we add back the signal control and investigate the feasibility of $\mathcal{P}$ under milder conditions with the SH solution. We consider a special subset of $\mathcal{P}$ where every trajectory has the maximum speed of $\bar{v}$ at the exit location $L$, i.e, $$\hat{\mathcal{P}}:=\left\{ \left[p_{n}\right]{}_{n\in\mathcal{N}}\in\mathcal{P}\left|\dot{p}_{n}\left(p_{n}^{-1}\left(L\right)\right)=\bar{v},\forall n\in\mathcal{N}\right.\right\} .$$ This subset is not too restrictive, because in order to assure a high traffic throughput rate, the exist speed of each vehicle should be high. The following analysis investigates the relationship between $\hat{\mathcal{P}}$ and $P^{\mbox{SH}}\left(\underline{a},\bar{a},\underline{a},\bar{a}\right)$.
\[prop: exit\_maximum\_speed\]When $L\ge\bar{v}^{2}/(2\bar{a})$, if $P^{\mbox{SH}}\left(\underline{a},\bar{a},\underline{a},\bar{a}\right)=\left[p_{n}\right]{}_{n\in\mathcal{N}}$ is feasible, then $\dot{p}_{n}\left(t\right)=\bar{v},\forall t\ge p_{n}^{-1}(L),n\in\mathcal{N}$ .
We use induction to prove this proposition. The induction assumption is $\dot{,p}_{n}\left(t\right)=\bar{v},\forall t\ge p_{n}^{-1}(L)$. If the BSP is not needed, Proposition \[prop: FSP-p\_bar\] indicates that $p_{1}=p_{1}^{\mbox{f}}=\bar{p}_{0v_{1}^{-}t_{1}^{-}}\left(t_{1}^{-}:\infty\right)$ and thus since $L\ge\bar{v}^{2}/(2\bar{a})$, we obtain $\dot{p}_{1}\left(t\right)=\dot{\bar{p}}_{0v_{1}^{-}t_{1}^{-}}\left(t\right)=\bar{v},\forall t\ge p_{1}^{-1}(L)=\bar{p}_{0v_{1}^{-}t_{1}^{-}}^{-1}(L)$. Otherwise, if BSP is activated, it shoots backwards from the state point $\left(L,\bar{p}_{0v_{1}^{-}t_{1}^{-}}\left(\bar{p}_{0v_{1}^{-}t_{1}^{-}}^{-1}(L)\right)=\bar{v},G\left(\bar{p}_{0v_{1}^{-}t_{1}^{-}}^{-1}(L)\right)\right)$, and thus Proposition \[prop: BSP-p\_bar\] indicates that $p_{1}=\bar{p}_{0v_{1}^{-}t_{1}^{-}}^{L\bar{v}G\left(\bar{p}_{0v_{1}^{-}t_{1}^{-}}^{-1}(L)\right)}$. Therefore, $\dot{p}_{1}\left(t\right)=\bar{v},\forall t\ge p_{1}^{-1}(L)$ holds again since $L\ge\bar{v}^{2}/(2\bar{a})$. Then assume that this assumption holds for $n=k-1$, and we will investigate whether it holds for $n=k,\forall k\in\mathcal{N}\backslash\{1\}.$ If the primary FSP is not blocked by $p_{k-1}^{\mbox{s}}$, apparently $p_{k}=\bar{p}_{0v_{k}^{-}t_{k}^{-}}$ if the BSP is not needed, or $p_{k}=\bar{p}_{0v_{k}^{-}t_{k}^{-}}^{L\bar{v}G\left(\bar{p}_{0v_{k}^{-}t_{k}^{-}}^{-1}(L)\right)}$ if the BSP is activated. Either way $\dot{p}_{k}\left(t\right)=\bar{v},\forall t\ge p_{k}^{-1}(L)$ holds since $L\ge\bar{v}^{2}/(2\bar{a})$. Otherwise, $p_{k}^{\mbox{f}}$ merges into $p_{k-1}^{\mbox{s}}$ before reaching $L$, and thus based on the induction assumption we obtain $\dot{p}_{k}^{\mbox{f}}\left(p_{k}^{\mbox{f}-1}(L)\right)=\dot{p}_{k-1}^{\mbox{s}}\left(p_{k-1}^{\mbox{s}-1}(L)\right)=\dot{p}_{k-1}\left(p_{k-1}^{-1}(L+s)\right)=\bar{v}.$ Note that again the BSP could only shift segments in parallel and thus $\dot{p}_{k}\left(p_{k}^{-1}(L)\right)=\dot{p}_{k}^{\mbox{f}}\left(p_{k}^{\mbox{f}-1}(L)\right)=\bar{v}$. The induction assumption also indicates that the segments of $p_{k-1}$ used in the auxiliary FSP for $p_{k}$ is at constant speed $\bar{v}$. This means that the auxiliary FSP for $p_{k}$ will not be blocked by $p_{k-1}$, and thus we obtain $\dot{p}_{k}\left(t\right)=\bar{v},\forall t\ge p_{k}^{-1}(L)$. This completes the proof.
When $L\ge\bar{v}^{2}/(2\bar{a})$ , $\hat{\mathcal{P}}\neq\emptyset$ $\Leftrightarrow$ $P^{\mbox{SH}}\left(\underline{a},\bar{a},\underline{a},\bar{a}\right)\neq\emptyset$.\[theo:P\_hat\_feasibility\]
The proof of the sufficiency is simple. Again, we can write $P^{\mbox{SH}}\left(\underline{a},\bar{a},\underline{a},\bar{a}\right)$ as $\left[p_{n}\right]{}_{n\in\mathcal{N}}$. When $P^{\mbox{SH}}\left(\underline{a},\bar{a},\underline{a},\bar{a}\right)\neq\emptyset$, Proposition \[prop: exit\_maximum\_speed\] indicates that $P\in\hat{\mathcal{P}}$ and thus $\hat{\mathcal{P}}\neq\emptyset.$
Then we only need to prove the necessity. When there exists $P'=\left[p'_{n}\right]{}_{n\in\mathcal{N}}\in\hat{\mathcal{P}}$ that is not empty, we will show that $\left[p_{n}\right]{}_{n\in\mathcal{N}}$ too is not empty with the following induction. For the notation convenience, we denote $t{}_{n}^{'+}=\left(p'_{n}\right)^{-1}(L)$ and $t{}_{n}^{+}:=p{}_{n}^{-1}(L)$. The induction assumption is that $p_{n}$ exists and $D\left(p_{n}-p'_{n}\right)\ge0$. When $n=1,$ if BSP is not activated in constructing $p_{1}$, then $p_{1}=p^{\mbox{f}}\left(\left(0,v_{1}^{-},t_{1}^{-}\right),\emptyset\right)=\bar{p}_{0v_{1}^{-}t_{1}^{-}}(t^{-}:\infty)$ based on Proposition \[prop: FSP-p\_bar\]. Therefore, $p_{1}$ exists and $D\left(p_{1}-p'_{1}\right)\ge0$. Otherwise, denote $\bar{t}_{1}^{+}=\bar{p}_{0v_{1}^{-}t_{1}^{-}}^{-1}(L)$ and $t_{1}^{+}:=G\left(\bar{t}_{1}^{+}\right)$. Note that since $L\ge\bar{v}^{2}/(2\bar{a})$, $\dot{\bar{p}}_{0v_{1}^{-}t_{1}^{-}}\left(\bar{p}_{0v_{1}^{-}t_{1}^{-}}^{-1}(L)\right)=\bar{v}$. Apparently, $D\left(\bar{p}_{0v_{1}^{-}t_{1}^{-}}-\underline{p}_{L\bar{v}\bar{t}_{1}^{+}}\right)\ge0$ and $D\left(\underline{p}_{L\bar{v}\bar{t}_{1}^{+}}-\underline{p}_{L\bar{v}t_{1}^{+}}\right)\ge0$, which indicates $D\left(\bar{p}_{0v_{1}^{-}t_{1}^{-}}-\underline{p}_{L\bar{v}t_{1}^{+}}\right)\ge0$. Further since $L\ge\bar{v}^{2}/(2\bar{a})$, we know $D\left(\bar{p}_{L\bar{v}t_{1}^{+}}-\underline{p}_{0v_{1}^{-}t_{1}^{-}}\right)\ge0$. Then we obtain $p_{1}=\bar{p}_{0v_{1}^{-}t_{1}^{-}}^{L\bar{v}t_{1}^{+}}$ based on Proposition \[prop: BSP-p\_bar\]. Further, apparently $t{}_{1}^{'+}\ge\bar{t}_{1}^{+}$, and then $t{}_{1}^{'+}=G\left(t{}_{1}^{'+}\right)\ge G\left(\bar{t}_{1}^{+}\right)=t_{1}^{+}$ since function $G(\cdot)$ is increasing. Then Proposition \[prop: shift\_prism\_feasibility\] indicates that $D\left(p_{1}-p'_{1}\right)\ge0$.
Then assume that the induction assumption holds for $n=k-1,\forall k\in\mathcal{N}\backslash\{1\}.$ Then for $n=k$, Since $p'\in\mathcal{P}_{0v_{k}^{-}t_{k}^{-}}^{L\bar{v}t_{k}^{'+},p_{k-1}^{\mbox{'s}}}$, then we obtain from Corollary \[cor:BSP-Feasibility-Prism\] that $D\left(p_{k-1}^{\mbox{'s}}-\underline{p}_{0v_{k}^{-}t_{k}^{-}}\right)\ge0$ where $p_{k-1}^{\mbox{'s}}$ is the shadow trajectory of $p'_{k-1}$. And the induction assumption tells that $D\left(p_{k-1}^{\mbox{s}}-p_{k-1}^{\mbox{'s}}\right)\ge0$ where $p_{k-1}^{\mbox{s}}$ is the shadow trajectory of $p_{k-1}$, which further indicates that $D\left(p_{k-1}^{\mbox{s}}-\underline{p}_{0v_{k}^{-}t_{k}^{-}}\right)\ge0$ based on the transitive property of $D$. Then Proposition \[prop: FSP-feasibility\] indicate that $p^{\mbox{f}}\left((0,v_{k}^{-},t_{k}^{-}),p_{k-1}^{\mbox{s}}\right)\neq\emptyset$. For the simplicity of presentation, we denote $p^{\mbox{f}}\left((0,v_{k}^{-},t_{k}^{-}),p_{k-1}^{\mbox{s}}\right)$ by $p_{k}^{\mbox{f}}$. If BSP is not used in constructing $p_{k}$, then $p_{k}=p_{k}^{\mbox{f}}$ exists and Proposition \[prop: FSP-shift-bound\] indicates $D\left(p_{k}-p'_{k}\right)\ge0$. Otherwise, let $\bar{t}_{k}^{+}=p_{k}^{\mbox{f}-1}\left(L\right)$ and $t_{k}^{+}=G\left(\bar{t}_{k}^{+}\right)$. Then if $p_{k}$ exists, then $p_{k}^{-1}(L)=t_{k}^{+}$ and $\dot{p}_{k}\left(t_{k}^{+}\right)=\bar{v}$ since $L\ge\bar{v}^{2}/(2\bar{a})$. Again, it is easy to see that $D\left(\bar{p}_{0v_{k}^{-}t_{k}^{-}}-\underline{p}_{L\bar{v}t_{k}^{+}}\right)\ge0$ and $D\left(\bar{p}_{L\bar{v}t_{k}^{+}}-\underline{p}_{0v_{k}^{-}t_{k}^{-}}\right)\ge0$, and therefore Corollary \[cor:BSP-Feasibility-Prism\] indicates that $p_{k}$ exists. Also, apparently $t{}_{k}^{'+}\ge\bar{t}_{k}^{+}$, and then $t{}_{k}^{'+}=G\left(t{}_{k}^{'+}\right)\ge G\left(\bar{t}_{k}^{+}\right)=t_{k}^{+}$ since function $G(\cdot)$ is increasing. Note that $p_{k}=p^{\mbox{eb}}\left((L,\bar{v},t_{k}^{+}),p_{k}^{\mbox{f}}\right)$ from BSP with $\bar{a}^{\mbox{f}}=\bar{a}^{\mbox{b}}=\bar{a}$ and $\underline{a}^{\mbox{f}}=\underline{a}^{\mbox{b}}=\underline{a}$. Then Corollary \[cor:BSP-Feasibility-Prism\] further indicates that $D\left(p_{k}-p'_{k}\right)\ge0$. This completes the proof.
Further, we will show that when $L$ is sufficiently long, the feasibility of $P^{\mbox{SH}}\left(\underline{a},\bar{a},\underline{a},\bar{a}\right)$ is equivalent to the feasibility of $\mathcal{P}$.
When $L\ge\frac{\bar{v}^{2}}{2\bar{a}}+\frac{\bar{v}^{2}}{-2\underline{a}^{\mbox{b}}}+\frac{\bar{v}^{2}}{-2\underline{a}^{\mbox{b}}}+s(N-1)$ , $P^{\mbox{SH}}\left(\underline{a},\bar{a},\underline{a},\bar{a}\right)\neq\emptyset$ $\Leftrightarrow$ $\mathcal{P}\neq\emptyset$ $\Leftrightarrow$ $\left[v_{n}^{-},t_{n}^{-}\right]_{n\in\mathcal{N}}$ is proper.\[theo:P\_feasibility\]
When $P^{\mbox{SH}}\left(\underline{a},\bar{a},\underline{a},\bar{a}\right)\neq\emptyset$, it is trivial to see that $\mathcal{P}\neq\emptyset$ since $P^{\mbox{SH}}\left(\underline{a},\bar{a},\underline{a},\bar{a}\right)\in\mathcal{P}$. Proposition \[prop:proper\_condition\] indicates that $\mathcal{P}\neq\emptyset$ leads to that $\left[v_{n}^{-},t_{n}^{-}\right]_{n\in\mathcal{N}}$ is proper. Then we only need to prove that when $\left[v_{n}^{-},t_{n}^{-}\right]_{n\in\mathcal{N}}$ is proper, $P^{\mbox{SH}}\left(\underline{a},\bar{a},\underline{a},\bar{a}\right)\neq\emptyset$. We denote the trajectories in $P^{\mbox{SH}}\left(\underline{a},\bar{a},\underline{a},\bar{a}\right)$ with $\left[p_{n}\right]{}_{n\in\mathcal{N}}$. Note that for each trajectory $p_{n}$, if BSP in SH3 is activated, then it shall be always successfully completed within highway segment $\left[\hat{L}:=L-\bar{v}^{2}/\left(-2\bar{a}^{\mbox{b}}\right)-\bar{v}^{2}/\left(-2\bar{a}^{\mbox{b}}\right),L\right]$. Therefore, BSP neither causes infeasibility nor affects the shape of $p_{\mbox{n}}$ within $\left[0,\hat{L}_{\mbox{n}}:=L-\bar{v}^{2}/\left(-2\bar{a}^{\mbox{b}}\right)-\bar{v}^{2}/\left(-2\bar{a}^{\mbox{b}}\right)-s(n-1)\right]$. Even if we consider the backward wave propagation caused by the previous trajectories. This indicates $p_{n}\left(t_{n}^{-}:\dot{p}_{n}^{-1}\left(\hat{L}_{\mbox{n}}\right)\right)=p_{n}^{\mbox{f}}\left(t_{n}^{-}:\dot{p}_{n}^{-1}\left(\hat{L}_{\mbox{n}}\right)\right)$, and if the feasibility check of $p_{n}$ fails in SH, it must happens in this section. Note that Theorem \[theo: feasibility\_all\_green\] shows that if $\left[v_{n}^{-},t_{n}^{-}\right]_{n\in\mathcal{N}}$ is proper, $P^{\mbox{PSH}}\left(\underline{a},\bar{a}\right)$ is not empty. We denote $P^{\mbox{PSH}}\left(\underline{a},\bar{a}\right)$ with $\left[\hat{p}_{n}\right]{}_{n\in\mathcal{N}}$. Then to prove this theorem, we only need to show that $p_{n}\left(t_{n}^{-}:p_{n}^{-1}\left(\hat{L}_{n}\right)\right)=\hat{p}_{n}\left(t_{n}^{-}:p_{n}^{-1}\left(\hat{L}_{n}\right)\right),\forall n\in\mathcal{N}$ from the following induction.
The induction assumption is $p_{n}\left(t_{n}^{-}:p_{n}^{-1}\left(\hat{L}_{n}\right)\right)=\hat{p}_{n}\left(t_{n}^{-}:p_{n}^{-1}\left(\hat{L}_{n}\right)\right)$ and $\dot{p}_{n}(t)=\bar{v},\forall t\in[p_{n}^{-1}(\hat{L}_{n}),p_{n}^{-1}(\hat{L})]$. When $n=1$, the trajectory generated from FSP satisfies $p_{1}^{\mbox{f}}:=p_{1}^{\mbox{f}}\left((0,v_{1}^{-},t_{1}^{-}),\emptyset\right)=\hat{p}_{n}$. Since BSP will not affect the shape of $p_{1}$ below $\hat{L}_{1}$, thus $p_{1}\left(t_{1}^{-}:p_{1}^{-1}\left(\hat{L}_{1}\right)\right)=\hat{p}_{1}\left(t_{1}^{-}:p_{1}^{-1}\left(\hat{L}_{1}\right)\right)$. Since $p_{1}^{\mbox{f}}$ shall finish accelerating to $\bar{v}$ before reaching location $\hat{L}$ based on the value of $\hat{L}$, then $\dot{p}_{1}(t)=\dot{\hat{p}}_{1}(t)=\bar{v},\forall t\in[p_{n}^{-1}(\hat{L}_{n}),p_{n}^{-1}(\hat{L})]$. Then we assume that the induction assumption holds for $n=k-1,\forall k\in\mathcal{N}\backslash\{1\}.$ When $n=k$, the forward shooting trajectory is $p_{k}^{\mbox{f}}:=p^{\mbox{f}}\left(\left(0,v_{k}^{-},t_{k}^{-}\right),p_{k-1}^{\mbox{s}}\right)$. If $D\left(p_{k-1}^{\mbox{s}}-\bar{p}_{0v_{k}^{-}t_{k}^{-}}\right)\ge0$, then apparently $p_{k}^{\mbox{f}}=\hat{p}_{k}=p^{\mbox{f}}\left(\left(0,v_{k}^{-},t_{k}^{-}\right),\emptyset\right),$ and thus $p_{k}(t_{k}^{-}:p_{k}^{-1}(\hat{L}_{k}))=p_{k}^{\mbox{f}}(t_{k}^{-}:p_{k}^{-1}(\hat{L}_{k}))=\hat{p}_{k}(t_{k}^{-}:p_{k}^{-1}(\hat{L}_{k}))$ and $\dot{p}_{k}(t)=\bar{v},\forall t\in[p_{k}^{-1}(\hat{L}_{k}),p_{k}^{-1}(\hat{L})]$. Otherwise, note that $p_{k}^{\mbox{f}}=p^{\mbox{f}}((0,v_{k}^{-},t_{k}^{-}),p_{k-1}^{s})=p^{\mbox{f}}((0,v_{k}^{-},t_{k}^{-}),\hat{p}_{k-1}^{s})=\hat{p}_{k}$ and thus the existence of $\hat{p}_{k}$ indicates that $p_{k}^{\mbox{f}}$ exists as well. The induction assumption of $\dot{p}_{k-1}(t)=\bar{v},\forall t\in[p_{k-1}^{-1}(\hat{L}_{k-1}),p_{k-1}^{-1}(\hat{L})]$ indicates that $\dot{p}_{k-1}^{\mbox{s}}(t)=\bar{v},\forall t\in[p_{k-1}^{\mbox{s}-1}(\hat{L}_{k}=\hat{L}_{k-1}-s),p_{k-1}^{\mbox{s}-1}(\hat{L}-s)]$, and therefore, $p_{k}^{\mbox{f}}$ shall merge with $p_{k-1}^{\mbox{s}}$ at a location before $\hat{L}$ because the merging speed has to be less than $\bar{v}$, and therefore $\dot{p}_{k}(t)=\dot{p}_{k}^{\mbox{f}}(t)=\bar{v},\forall t\in[p_{k}^{-1}(\hat{L}_{k}),p_{k}^{-1}(\hat{L})]$. Therefore, $p_{k}$ and $\hat{p}_{k}$ overlap over highway segment $\left[0,\hat{L}_{n}\right]$, or $p_{k}\left(t_{k}^{-}:p_{k}^{-1}\left(\hat{L}_{k}\right)\right)=\hat{p}_{k}\left(t_{k}^{-}:p_{k}^{-1}\left(\hat{L}_{k}\right)\right)$ . This completes the proof.
Relationship to Classic Traffic Flow Models
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Evolution of highway traffic has been traditionally investigated with various microscopic models (e.g., car following [@Brackstone1999181], cellular automata [@Nagel1992]) and macroscopic kinematic models (kinemetic models [@Lighthill1955; @Richards1956] and cell transmission [@Daganzo1994]). @Daganzo2006 proves the equivalence between the kinematic wave model with the triangular fundamental diagram (KWT) [@newell1993simplified], Newell’s lower-order model [@Newell2002] and the linear cellular automata model [@Nagel1992]. We will just show the relevance of the the SHL solution to the KWT model, and this relevance can be easily transferred to other models based on their equivalence. Given the first vehicle’s trajectory $q_{1}=p_{1}\in\mathcal{T}$, the KWT model specifies a rule to construct vehicle $n$’s trajectory, denoted by $q_{n}$, with vehicle $n's$ entry condition $(0,\cdot,t_{n}^{-})$ and preceding trajectory $q_{n-1}$, as formulated below $$q_{n}(t)=\min\left\{ \bar{v}(t-t_{n}^{-}),q_{n-1}(t-\tau)-s\right\} ,\forall n\in\mathcal{N}\backslash\{1\},t\in[t_{n}^{-},\infty).\label{eq:KWT-rule}$$ For notation convenience, we denote this equation with $q_{n}=q^{\mbox{KWT}}\left((0,\cdot,t_{n}^{-}),q_{n-1}\right)$. This section analyzes the relationship between the SHL solution with the KWT solution for the same LVP setting. We denote the SHL trajectory vector with $P^{\mbox{LVP}}\left(\underline{a}^{\mbox{f}},\bar{a}^{\mbox{f}}\right)=\left[p_{n}\right]_{n\in\mathcal{N}}$ and that from KWT by $Q=\left[q_{n}\right]_{n\in\mathcal{N}}$ . Without loss of generality, we only investigate the case when $\left[v_{n}^{-},t_{n}^{-}\right]_{n\in\mathcal{N}}$ is proper or $P^{\mbox{LVP}}\left(\underline{a}^{\mbox{f}},\bar{a}^{\mbox{f}}\right)$ is feasible.
@Daganzo2006 showed that KWT has a contraction property; i.e., the result of KWT is insensitive to small input errors, as stated in the following proposition.
Given two trajectories $q,q'\in\bar{\mathcal{T}}$ satisfying $\max_{t\in(-\infty,\infty)}\left|q(t)-q'(t)\right|\le\epsilon$ for some $\epsilon>0$, then $\max_{t\in(-\infty,\infty)}\left|q^{\mbox{KWT}}\left((0,\cdot,t^{-}),q^{\mbox{s}}\right)-q^{\mbox{KWT}}\left((\delta,\cdot,t^{-}),q^{'\mbox{s}}\right)\right|\le\epsilon$for any $\left|\delta\right|<\epsilon$ andyielding and
We now show that $P^{\mbox{LVP}}\left(\underline{a}^{\mbox{f}},\bar{a}^{\mbox{f}}\right)$ has the same contraction property.
\[Theo: contraction\]Given two feasible trajectories $p,p'\in\mathcal{T}$ satisfying $\max_{t\in(-\infty,\infty)}\left|p(t)-p'(t)\right|\le\epsilon$, then $\max_{t\in(-\infty,\infty)}\left|p^{\mbox{f}}((0,v^{-},t^{-}),p^{\mbox{s}})-p^{\mbox{f}}((\delta,v{}^{-},t^{-}),p{}^{'\mbox{s}})\right|\le\epsilon$ for any $\left|\delta\right|<\epsilon$ and any yielding $p^{\mbox{f}}((0,v^{-},t^{-}),$ $p^{\mbox{s}})\neq\emptyset$ and $p^{\mbox{f}}((\delta,v^{-},t^{-}),p{}^{'\mbox{s}})\neq\emptyset$.
Since $\max_{t\in(-\infty,\infty)}\left|p(t)-p'(t)\right|\le\epsilon$, we obtain $\max_{t\in(-\infty,\infty)}\left|p^{\mbox{s}}(t)-p^{'\mbox{s}}(t)\right|\le\epsilon.$ Further, define $p^{\mbox{s}+\epsilon}(t):=p^{\mbox{s}}(t)+\epsilon,$ $p^{\mbox{s}-\epsilon}(t):=p^{\mbox{s}}(t)-\epsilon$, $p^{\mbox{f}+\epsilon}:=p^{\mbox{f}}\left(\left(\epsilon,v^{-},t^{-}\right),p^{\mbox{s}+\epsilon}\right)$ and $p^{\mbox{f}-\epsilon}:=p^{\mbox{f}}\left(\left(-\epsilon,v^{-},t^{-}\right),p^{\mbox{s}-\epsilon}\right)$. Note that $p(t)=p^{+\epsilon}(t)-\epsilon=p^{-\epsilon}(t)+\epsilon,\forall t\in(-\infty,\infty)$. Since $D\left(p^{s+\epsilon}-p^{\mbox{s}}\right)=\epsilon$ and $\max_{t\in(-\infty,\infty)}\left|p(t)-p'(t)\right|\le\epsilon$, we obtain $D\left(p^{s+\epsilon}-p^{\mbox{'s}}\right)\ge0$. Then Proposition \[prop:bounded\_transitive\] indicates that $\mathcal{C}_{\delta vt^{-}}^{p^{\mbox{'s}}}\subseteq\mathcal{C}_{\delta vt^{-}}^{p^{\mbox{s}+\epsilon}}$. Further, Proposition \[prop: FSP-shift-bound\] indicates $D\left(p^{\mbox{f}+\epsilon}-\hat{p}\right)\ge0,\forall\hat{p}\in\mathcal{C}_{\delta vt^{-}}^{p^{\mbox{s}+\epsilon}}$. We denote $p^{\mbox{f}}\left((0,v^{-},t^{-}),p^{\mbox{s}}\right)$ and $p^{\mbox{f}}\left((\delta,v{}^{-},t^{-}),p{}^{'\mbox{s}}\right)$ as $p^{\mbox{f}}$ and $p^{'\mbox{f}}$ for short, respectively. Since $p^{'\mbox{f}}\in\mathcal{C}_{\delta vt^{-}}^{p^{\mbox{'s}}}\subseteq\mathcal{C}_{\delta vt^{-}}^{p^{\mbox{s}+\epsilon}}$, we obtain $D\left(p^{\mbox{f}+\epsilon}-p^{'\mbox{f}}\right)\ge0$. On the other hand, we obtain $D\left(p^{\mbox{'s}}-p^{s-\epsilon}\right)\ge0$. Then Proposition \[prop:bounded\_transitive\] indicates $\mathcal{C}_{-\epsilon vt^{-}}^{p^{\mbox{s}-\epsilon}}\subseteq\mathcal{C}_{-\epsilon vt^{-}}^{p^{\mbox{'s}}}$. Further, Proposition \[prop: FSP-shift-bound\] indicates $D\left(p'-\hat{p}\right)\ge0,\forall\hat{p}\subseteq\mathcal{C}_{-\epsilon vt^{-}}^{p^{\mbox{'s}}}$. Since $p^{\mbox{f}-\epsilon}\in\mathcal{C}_{-\epsilon vt^{-}}^{p^{\mbox{s}-\epsilon}}\subseteq\mathcal{C}_{-\epsilon vt^{-}}^{p^{\mbox{'s}}}$, we obtain $D\left(p^{'\mbox{f}}-p^{\mbox{f}-\epsilon}\right)\ge0$. Combining $D\left(p^{\mbox{f}+\epsilon}-p^{'\mbox{f}}\right)\ge0$ and $D\left(p^{'\mbox{f}}-p^{\mbox{f}-\epsilon}\right)\ge0$ yields $\max_{t\in(-\infty,\infty)}\left|p^{\mbox{f}}(t)-p^{'\mbox{f}}(t)\right|\le\epsilon$ .
Theorem \[Theo: contraction\] impliesthat $P^{\mbox{LVP}}\left(\underline{a}^{\mbox{f}},\bar{a}^{\mbox{f}}\right)$ is not sensitive to small input errors as well. However, we shall note that the solution to the SH algorithm for a signalized section may be sensitive to small errors because the exit time of a trajectory, if close to the start of a red phase, could be pushed back to the next green phase due to a small input perturbation. Nonetheless, this kind of “jump” only affects a limited number of trajectories that are close to a red phase, and most other trajectories will not be much affected.
The following analysis investigates the difference between $P^{\mbox{LVP}}\left(\underline{a}^{\mbox{f}},\bar{a}^{\mbox{f}}\right)$ and $Q$.
\[prop:Parallel\_KWT\]$Q=\left\{ q_{n}\right\} _{n\in\mathcal{N}}$ formulated in can be solved as $$q_{n}(t)=\min\left\{ p_{1}^{\mbox{s}^{n-1}}(t),\bar{v}\left(t-t_{n}^{-}\right)\right\} ,\forall t\in[t_{n}^{-},\infty),n\in\mathcal{N}.\label{eq:Parallel-KWT}$$ where $p_{1}^{\mbox{s}^{k}}(t):=p_{1}\left(t-k\tau\right)-ks,\forall k.$
We prove this proposition with induction. Apparently, equation holds for $n=1$ since $p_{1}^{\mbox{s}^{0}}=p_{1}\in\mathcal{T}$ and thus $\bar{v}\left(t-t_{1}^{-}\right)\ge p_{1}^{\mbox{s}^{0}}(t),\forall t\in[t_{1}^{-},\infty)$. Assume that equation holds for $n=k-1$. When $n=k$, equation indicates $q_{k}(t)=\min\{\bar{v}(t-t_{k}^{-}),q_{k-1}(t-\tau)-s\}$. Then plug the induction assumption into the above equation and we obtain $q_{k}(t)=\min\{\bar{v}(t-t_{k}^{-}),p_{1}^{\mbox{s}^{k-2}}(t-\tau)-s,\bar{v}\left(t-t_{k-1}^{-}-\tau\right)-s\}$$=\min\{p_{1}^{\mbox{s}^{k-1}}(t),\bar{v}(t-\max\{t_{k}^{-},t_{k-1}^{-}+\tau+\bar{v}/s\})\}.$ Since $\left[v_{n}^{-},t_{n}^{-}\right]_{n\in\mathcal{N}}$ is proper, we obtain $\max\left\{ t_{k}^{-},t_{k-1}^{-}+\tau+\bar{v}/s\right\} =t_{k}^{-}$. This completes the proof.
\[theo:KWT\_SHL\_bounds\] If $P^{\mbox{LVP}}\left(\underline{a}^{\mbox{f}},\bar{a}^{\mbox{f}}\right)\neq\emptyset$ and $Q\neq\emptyset$, then $D\left(q_{n}-p_{n}\right)=0$ and $$D\left(p_{n}-q_{n}\right)\ge\min\left\{ -0.5\bar{v}^{2}/\bar{a}^{\mbox{f}},0.5\bar{v}^{2}/\underline{a}^{\mbox{f}}\right\} ,\forall n\in\mathcal{N}\backslash\left\{ 1\right\} .\label{eq:SH-KWT lower bound}$$
We compare parallel formulations from $P^{\mbox{LVP}}\left(\underline{a}^{\mbox{f}},\bar{a}^{\mbox{f}}\right)$ from PSHL in Section \[sub:SH\_LVP\] with equation for $Q$. We only need to prove this theorem for a generic $n\in\mathcal{N}\backslash\left\{ 1\right\} $. From PSHL, we see that $p_{n}$ is essentially obtained by smoothing quasi-trajectory $u_{n}:=u\left(\left\{ \bar{p}_{m}^{\mbox{s}^{n-m}}\right\} _{m=1,\cdots,n}\right)$ with tangent segments at constant decelerating rate $\underline{a}$. Define $\bar{q}_{n}(t):=\bar{v}\left(t-t_{n}^{-}\right),\forall t\in\left[t_{n}^{-},+\infty\right),\underline{q}_{n}:=p^{\mbox{f}}\left(\left(0,0,\underline{t}_{n}\right),\emptyset\right)$. Note that $D\left(\underline{q}_{n}-\bar{q}_{n}\right)=-0.5\bar{v}^{2}/\bar{a}^{\mbox{f}}$, and $q_{n}(t)=\min\left\{ \bar{q}_{n}(t),p_{1}^{\mbox{s}^{n-1}}(t)\right\} .$ Note that $D\left(q_{n}-u_{n}\right)\ge0$ since $\bar{q}_{n}\ge\bar{p}_{n}$. Also, $D\left(u_{n}-p_{n}\right)\ge0$ since the merging segments generated from EFSO-2 are all below $u_{n}$. This indicates $D\left(q_{n}-p_{n}\right)\ge0$. Further since $q_{n}$ and $p_{n}$ always meet at the initial point $(0,t_{n}^{-})$, we obtain $D\left(q_{n}-p_{n}\right)=0$.
Then we will prove $D\left(p_{n}-q_{n}\right)\ge\min\left\{ -0.5\bar{v}^{2}/\bar{a}^{\mbox{f}},0.5\bar{v}^{2}/\underline{a}^{\mbox{f}}\right\} $. We first examine $p'_{n}:=p^{\mbox{f}}((0,v_{n}^{-},t_{n}^{-}),$ $\{\bar{p}_{m}^{\mbox{s}^{n-m}}\}_{m=2,\cdots,n-1}),\forall n\in\mathcal{N}\backslash\{1\}$, and claim $D\left(p'_{n}-\underline{q}_{n}\right)\ge0$. Basically, $p'_{n}$ is composed by some shooting segments from $\left\{ \bar{p}_{m}^{\mbox{s}^{n-m}}\right\} _{m=2,\cdots,n-1}$and intermediate tangent merging segments. Apparently, $D\left(\bar{p}_{m}^{\mbox{s}^{n-m}}-\underline{q}_{n}\right)\ge0$. Note that every merging segment shall be above $\underline{p}_{n}$ or it will not catch up with the next shooting segment that accelerates at the maximum rate $\bar{a}^{\mbox{f}}$ before reaching speed $\bar{v}$. Thus this claim holds and thus $D\left(p'_{n}-\bar{q}_{n}\right)\ge-0.5\bar{v}^{2}/\bar{a}^{\mbox{f}}$. Therefore, $D\left(p'_{n}-q_{n}\right)\ge-0.5\bar{v}^{2}/\bar{a}^{\mbox{f}}$. Note that $p_{n}$ is obtained by merging $p'_{n}$ and $\bar{p}_{1}^{\mbox{s}^{n-1}}$with a tangent segment denoted by $\hat{s}_{n}$. Apparently, $D\left(\bar{p}_{1}^{\mbox{s}^{n-1}}-q_{n}\right)\ge0$ since $q_{n}(t)$ is defined as $\min\left\{ \bar{q}_{n}(t),p_{1}^{\mbox{s}^{n-1}}(t)\right\} $. Further note that $D\left(\hat{s}_{n}-\bar{p}_{1}^{\mbox{s}^{n-1}}\right)\ge0.5\bar{v}^{2}/\underline{a}^{\mbox{f}}$, and since $D\left(\bar{p}_{1}^{\mbox{s}^{n-1}}-q_{n}\right)\ge0$, we obtain $D\left(\hat{s}_{n}-q_{n}\right)\ge0.5\bar{v}^{2}/\underline{a}^{\mbox{f}}$. Therefore $D(p_{n}-q_{n})$$\ge\min\left\{ D\left(p'_{n}-q_{n}\right),D\left(\bar{p}_{1}^{\mbox{s}^{n-1}}-q_{n}\right),D\left(\hat{s}_{n}-q_{n}\right)\right\} \ge\min\left\{ -0.5\bar{v}^{2}/\bar{a}^{\mbox{f}},0.5\bar{v}^{2}/\underline{a}^{\mbox{f}}\right\} $. This completes the proof.
The above theorem reveals that a trajectory generated from$P^{\mbox{LVP}}\left(\underline{a}^{\mbox{f}},\bar{a}^{\mbox{f}}\right)$ should be always below the counterpart trajectory in $Q$ and the difference is attributed to smoothed accelerations instead of speed jumps. One elegant finding from this theorem is that this difference does not accumulate much across vehicles but is bounded by a constant. Further, the lower bound to this difference is actually tight and we can find instances where $D\left(p_{n}-q_{n}\right)$ for every vehicle $n\in\mathcal{N}\backslash\left\{ 1\right\} $ is exactly identical to $\min\left\{ -0.5\bar{v}^{2}/\bar{a}^{\mbox{f}},0.5\bar{v}^{2}/\underline{a}^{\mbox{f}}\right\} $. One such instance can be specified by setting $p_{1}(t)=L,\forall t$ for a sufficiently long $L$, and$v_{n}^{-}=0,\forall n\in\mathcal{N}\backslash\left\{ 1\right\} $. This theorem also leads to the following asymptotic relationship.
If $\bar{a}^{\mbox{f}}\rightarrow\infty$ and $\underline{a}^{\mbox{f}}\rightarrow-\infty$, then we have $P^{\mbox{LVP}}\left(\underline{a}^{\mbox{f}},\bar{a}^{\mbox{f}}\right)\rightarrow Q.$\[cor\_KWT\_SH\_asymptotic\]
This relationship indicates that SHL can be viewed as a generalization of KWT as well as other equivalent models, including Newell’s lower order model and the linear cellular automata model. Essentially, the SH solution can be viewed as a smoothed version of these classic models that circumvents a common issue of these classic models, i.e., infinite acceleration/deceleration or “speed jumps”. Such speed jumps would cause unrealistic evaluation of traffic performance measures, particularly those on safety and environmental impacts. Further, SHL inherits the simple structure of these models and thus can be solved very efficiently. One commonality between SHL and KWT is that at stationary states, i.e., when each trajectory move at a constant speed, both the SHL solution and the KWT solution are consistent with a triangular fundamental diagram @newell1993simplified, as stated below.
\[theo: fundamental\] If solutions $P^{\mbox{LVP}}\left(\underline{a}^{\mbox{f}},\bar{a}^{\mbox{f}}\right)$ and $Q$ are both feasible and each $\dot{p}_{n}(t)$ or $\dot{q}_{n}(t)$ remains constant $\forall t\ge t_{n}^{-}$, then there exists some $V\in[0,\bar{v}]$ such that
$$p_{n}(t)=q_{n}(t)=V\left(t-t_{n}^{-}\right),\dot{p}_{n}(t)=\dot{q}_{n}(t)=V,\forall t\in\left[t_{n}^{-},\infty\right),n\in\mathcal{N}.\label{eq:p_q_equal}$$
Further, if $V=\bar{v}$, $$t_{n}^{-}-t_{n-1}^{-}\ge\tau+V/s,\forall n\in\mathcal{N},\label{eq:t_n_minus_sep_less}$$ If $V<\bar{v},$ then
$$t_{n}^{-}=(n-1)(\tau+V/s)-t_{1}^{-},\forall n\in\mathcal{N}\backslash\left\{ 1\right\} .\label{eq:t_n_minus_sep_equal}$$
Further, define the traffic density as $K:=\frac{N-1}{\sum_{n=2}^{N}\left(p_{n-1}(t)-p_{n}(t)\right)}=\frac{N-1}{\sum_{n=2}^{N}\left(q_{n-1}(t)-q_{n}(t)\right)}$ for any $t\ge t_{n}^{-}$ and traffic volume as $O:=\frac{N-1}{p_{N}^{-1}(l)-p_{1}^{-1}(l)}=\frac{N-1}{q_{N}^{-1}(l)-q_{1}^{-1}(l)}$ for any $l\ge0$, then we will always have $$\begin{cases}
K\le1/(s+V\tau), & \mbox{if }V=\bar{v};\\
K=1/(s+V\tau) & \mbox{if }V<\bar{v}.
\end{cases}\label{eq:K-V}$$
$$O=KV=\min\left\{ K\bar{v},(1-sK)/\tau\right\} \in\left[0,\bar{v}/(s+\bar{v}\tau)\right].\label{eq:O-K}$$
We fist investigate the case when every $\dot{p}_{n}(t)$ is constant. If there exist two consecutive trajectories not parallel with each other, then they will either intersect or depart from each other to an infinite spacing since they both are straight lines. The former is impossible because of safety constraints . Neither is the latter possible since the following trajectory has to accelerate when their spacing exceeds the shadow spacing and runs at a speed lower than the preceding vehicle. Therefore $\dot{p}_{n}(t)$ values are identical to a $V\in[0,\bar{v}]$ across $n\in\mathcal{N}$. Therefore, $p_{n}(t)=V\left(t-t_{n}^{-}\right),\dot{p}_{n}(t)=V,\forall t\in\left[t_{n}^{-},\infty\right),n\in\mathcal{N}.$
Since $P^{\mbox{LVP}}\left(\underline{a}^{\mbox{f}},\bar{a}^{\mbox{f}}\right)$ is feasible, then we know from Theorem \[theo: feasibility\_all\_green\] that $\left[v_{n}^{-},t_{n}^{-}\right]_{n\in\mathcal{N}}$ is proper, which indicates that equation holds either $V=\bar{v}$ or $V<\bar{v}$. Further, when $V<\bar{v}$, equation becomes a strict equality (or otherwise the following trajectory needs to accelerate). Thus equation holds in this case.
Then we will show that $q_{n}=p_{n}$. Since all trajectories in $P^{\mbox{LVP}}\left(\underline{a}^{\mbox{f}},\bar{a}^{\mbox{f}}\right)$ are at speed $V$, we know that $v_{n}^{-}=V,\forall n\in\mathcal{N}$ and the lead trajectory is given as $p_{1}(t)=V\left(t-t_{1}^{-}\right)$. If $V=\bar{v}$, apparently no trajectory blocks its following trajectory, and thus $q_{n}(t)=V\left(t-t_{n}^{-}\right)=p_{n}(t),\forall t\in\left[t_{n}^{-},\infty\right),n\in\mathcal{N}.$ Otherwise if $V<\bar{v}$, due to equation , $q_{n}$ has no room to accelerate and thus has to stay at speed $V$. Therefore, equation holds. Then we will prove equations and , which quantify the triangular fundamental diagram. First, since $\left[v_{n}^{-},t_{n}^{-}\right]_{n\in\mathcal{N}}$ is proper, then we obtain $p_{n-1}\left(t_{n}^{-}\right)-p_{n}\left(t_{n}^{-}\right)\ge V\tau+s,\forall n\in\mathcal{N}\backslash\left\{ 1\right\} $, and thus $K\ge1/(V\tau+s)$ holds. When $V<\bar{v}$, due to , then $p_{n-1}\left(t_{n}^{-}\right)-p_{n}\left(t_{n}^{-}\right)=V\tau+s,\forall n\in\mathcal{N}\backslash\left\{ 1\right\} $, and thus $K=1/(V\tau+s)$ holds. This proves equation . Further note that $p_{n}^{-1}(0)-p_{n-1}^{-1}(0)=\left(p_{n-1}\left(t_{n}^{-}\right)-p_{n}\left(t_{n}^{-}\right)\right)/V,\forall n\in\mathcal{N}\backslash\left\{ 1\right\} $, then $O=\frac{N-1}{\sum_{n=2}^{N}\left(p_{n}^{-1}(0)-p_{n-1}^{-1}(0)\right)}=\frac{N-1}{\sum_{n=2}^{N}\left(p_{n-1}\left(t_{n}^{-}\right)-p_{n}\left(t_{n}^{-}\right)\right)/V}=KV$. When $V=\bar{v}$, plugging inequality $K\le1/(s+V\tau)$ in equation into the previous equation, we obtain $KV=K\bar{v}\le\bar{v}/(s+\bar{v}\tau)$, and $K\le1/(s+V\tau)$ can be rearranged into to $(1-sK)/\tau\ge KV$. Therefore, equation holds in this case. When $V<\bar{v}$, apparently, $KV<K\bar{v}$. Further, equality $K=1/(s+V\tau)$ in equation can be rearranged as $KV=(1-sK)/\tau$. Note that $K=1/(s+V\tau)>1/(s+\bar{v}\tau)$, and plugging this inequality into $KV=(1-sK)/\tau$ yields $KV=(1-sK)/\tau<(1-s/(s+\bar{v}\tau))/\tau=\bar{v}/(s+\bar{v}\tau)$. This indicates that equation holds in the case too. This completes the proof.
Illustrative Examples \[sec:Numerical-Examples\]
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This section presents a few illustrative examples that help visualize results of the algorithms and related theoretical properties. In the following examples, we set the parameters to their default values unless explicitly stated otherwise. These default parameter values are $L=1000$ m, $N=50$, $\underline{a}=-5\mbox{ m/s}^{2}$, $\overline{a}=2\mbox{ m/s}^{2}$, $\bar{v}=25$ m/s, $G=R=25\mbox{ s}$, $s=7$ m, and the default boundary condition is generated as follows. We first set the arrival times as $$t_{1}^{-}=0,\,t_{n}^{-}=t_{n-1}^{-}+(\tau+s/\bar{v})\left[1+\left(\frac{C}{Gf^{\mbox{s}}}-1\right)\left(1-\alpha+\alpha\epsilon_{n}\right)\right],\forall n\in\mathcal{N}\backslash\{1\},\label{eq:t_dist}$$ where $f^{\mbox{s}}\in\left(0,C/G\right]$ is the traffic saturation rate (or the ratio of the arrival traffic volume to the intersection’s maximum capacity), $\alpha\in[0,1]$ is the dispersion factor of the headway distribution, and $\left\{ \epsilon_{n}\right\} _{n\in\mathcal{N}\backslash\{1\}}$ are uniformly distributed non-negative random numbers that satisfy $\sum_{n\in\mathcal{N}\backslash\{1\}}\epsilon_{n}=N-1$. The default values of $f^{\mbox{s}}$ and $\alpha$ are both set to 1. We set the arrival times in this way such that the average time headway equals $(\tau+s/\bar{v})C/\left(Gf^{\mbox{s}}\right)$ yet the individual arrival times can be stochastic. The stochasticity of the arrival times increases with headway dispersion factor $\alpha$. Note that we always maintain every headway no less than $(\tau+s/\bar{v})$ because the boundary condition would be infeasible otherwise. Next, for each $n\in\mathcal{N}$, the initial speed $v_{n}^{-}$ is consecutively drawn as a random number uniformly distributed over the corresponding lower bound and upper bound. These two bounds assure that $v_{n}^{-}$ satisfies $v_{n}^{-}\in[0,\bar{v}]$, $D\left(\bar{p}_{(ms-ns)v_{m}^{-}\left[t_{m}^{-}+(n-m)\tau\right]}-\underline{p}_{0v_{n}^{-}t_{n}^{-}}\right)\ge0,\forall m<n\in\mathcal{N}$ and $D\left(\bar{p}_{(ns-ms)v_{n}^{-}\left[t_{n}^{-}+(m-n)\tau\right]}-\underline{p}_{0v_{m}^{-}t_{m}^{-}}\right)\ge0,\exists v_{m}^{-}\in[0,\bar{v}],\forall m>n\in\mathcal{N}$ with the maximum acceleration and the minimum deceleration for QTG downscaled to $\bar{a}/3$ and $\underline{a}/3$, respectively. The downscale of the acceleration limits assures that the boundary condition is feasible for any forward acceleration $\bar{a}^{\mbox{f}}\ge\bar{a}/3$ and any forward deceleration $\underline{a}^{\mbox{f}}\ge\underline{a}/3$, which allows to test a range of $\bar{a}^{\mbox{f}}$ and $\underline{a}^{\mbox{f}}$ values. If a proper boundary condition cannot be found, we will regenerate random parameters $\left\{ \epsilon_{n}\right\} _{n\in\mathcal{N}\backslash\{1\}}$ with a different random seed and repeat this process until finding a proper boundary condition.
Section \[sub:Manual-v.s.-Automated\] compares the SH solutions with a benchmark instance that simulates the manually-driven traffic counterpart. This comparison aims to qualitatively show the advantaged of the proposed CAV control strategies over manual driving. Section \[sub:Leading-Vehicle-Problem\] compares SHL and PSHL results and shows they produce the identical results for the same LVP input. Section \[sub:Feasibility-Tests\] tests the feasibility of SH and SHL solutions with different boundary conditions to verify some theory predictions and reveal insights into how parameter changes affect the solution feasibility. Section \[sub:Comparison-with-Classic\] compares SHL and LWK results and measures their difference to check the theoretical error bounds.
Manual v.s. Automated Trajectories\[sub:Manual-v.s.-Automated\]
---------------------------------------------------------------
To illustrated the advantage of results, we construct a benchmark instance that simulates the manually-driven traffic counterpart. We adapt the Intelligent Driver model [@Treiber2000b] as the manual-driving rule for every vehicle:
$$\ddot{p}_{n}(t)=\max\left\{ \min\left\{ \bar{a}\left(1-\frac{s^{*}}{f{}_{n-1}(t-\tau)-p_{n}(t-\tau)-l^{0}}\right),\bar{a}'\right\} ,\underline{a}'\right\} ,\label{eq:IDM}$$
where vehicle length $l^{0}=5$m, acceleration bounds $\bar{a}'=\begin{cases}
0, & \mbox{if }\dot{p}_{n}(t)\ge\bar{v};\\
\bar{a}, & \mbox{otherwise},
\end{cases}$ and $\underline{a}'=\begin{cases}
0, & \mbox{if }\dot{p}_{n}(t)\le0;\\
\underline{a}, & \mbox{otherwise},
\end{cases}$ , comfort deceleration $b=1.67$m/s$^{2}$, and the desired spacing $s^{*}=(s-l^{0})+\dot{p}_{n}(t-\tau)\tau+\dot{p}_{n}(t-\tau)\frac{\dot{p}_{n}(t-\tau)-\dot{p}'_{n-1}(t-\tau)}{\sqrt{\bar{a}b}}$. The effect of traffic lights is emulated with a bounding frontier $f{}_{n-1}$ that is $p{}_{n-1}$ if $p_{n}$ is not blocked by a red light or a virtual vehicle parked at $L+s$ otherwise. We define the yellow time prior to the beginning of a red phase as $y=3s$ and $f{}_{n-1}$ can be formulated as $$f{}_{n-1}(t)=\begin{cases}
L+s, & \mbox{if }L-p_{n}(t-\tau)<\bar{v}^{2}/(2b),t\in[mC-y,mC+G]\mbox{ and }\bar{p}_{p_{n}(t-\tau)\dot{p}_{n}(t-\tau)(t-\tau)}^{-1}(L)\notin\mathcal{G};\\
p{}_{n-1}(t), & \mbox{otherwise},
\end{cases},$$ To accommodate the lead vehicle that does not have a preceding trajectory, without loss of generality, we define a virtual preceding vehicle $p_{0}(t)=\infty$ and $\dot{p}_{0}(t)=\bar{v},\forall t$. There are two reasons to select this model. First, the trajectories produced from this model appear to be consistent with our driving experience at a signalized intersection: we tend to slow down and make a stop only when we get close the intersection at a yellow or red light. Secondly, it is easy to verify that at a stationary state, the same macroscopic relationship between density and flow volume defined in Theorem \[theo: fundamental\] holds. This way, the comparison between the SH solution and this benchmark will only focus on the “trajectory smoothing” effect rather than improvement of stationary traffic characteristics, which has been investigated in other studies (e.g., @Shladover2009).
Figure \[fig:traj\_results\] compares the benchmark result with the SH output. Figure \[fig:traj\_results\](a) plots the benchmark manual-driving trajectories generated with car-following model . We see that due to abrupt accelerations and decelerations in the vicinity of the traffic lights, a number of consecutive stop-and-go waves are formed and propagated backwards from the intersection. These stop-and-go waves slow down the passing speed of the vehicles at the intersections, and thus decrease the traffic throughput and increase the travel delay. As a result, the total travel time (i.e., the time duration between the first vehicle’s entry at location 0 and the last vehicle’s exist at location $L$) is over 300 seconds for the benchmark case. Further, it is intuitive that these stop-and-go waves adversely impact fuel consumptions and emissions and amplify collision risks. Figure \[fig:traj\_results\](b) plots the SH result with $\left(\bar{a}^{\mbox{f}},\underline{a}^{\mbox{f}},\bar{a}^{\mbox{b}},\underline{a}^{\mbox{b}}\right)$ identical to their bounding values $\left(\bar{a},\underline{a},\bar{a},\underline{a}\right)$. We see that despite some sharp accelerations and decelerations, all vehicles can pass the intersection at the maximum speed and thus the traffic throughput gets maximized. The total travel time now is only around 170 seconds. Figure \[fig:traj\_results\](c) plots the SH result with $\left(\bar{a}^{\mbox{f}},\underline{a}^{\mbox{f}},\bar{a}^{\mbox{b}},\underline{a}^{\mbox{b}}\right)$ downscaled to $\left(\bar{a}/3,\underline{a}/3,\bar{a}/3,\underline{a}/9\right)$. We see that with the acceleration/deceleration magnitudes reduces, trajectories become much smoother while the total travel time keeps low around 170 seconds. This will further reduce the traffic’s environmental impacts and enhance its safety.
![(a) Benchmark manual-driving trajectories, (b) SH result $P\left(\bar{a},\underline{a},\bar{a},\underline{a}\right),$ and (c) SH result $P\left(\bar{a}/3,\underline{a}/3,\bar{a}/3,\underline{a}/9\right).$ \[fig:traj\_results\]](manual_traj_result "fig:")![(a) Benchmark manual-driving trajectories, (b) SH result $P\left(\bar{a},\underline{a},\bar{a},\underline{a}\right),$ and (c) SH result $P\left(\bar{a}/3,\underline{a}/3,\bar{a}/3,\underline{a}/9\right).$ \[fig:traj\_results\]](SH_traj_result_accrate_1_1_1_1 "fig:")![(a) Benchmark manual-driving trajectories, (b) SH result $P\left(\bar{a},\underline{a},\bar{a},\underline{a}\right),$ and (c) SH result $P\left(\bar{a}/3,\underline{a}/3,\bar{a}/3,\underline{a}/9\right).$ \[fig:traj\_results\]](SH_traj_result_accrate_3_3_3_9 "fig:")
(a)(b)(c)
To inspect the differences between the benchmark and the SH result from a macroscopic point of view, we measure the macroscopic traffic characteristics, including density and flow volumes for the trajectory sets in Figure \[fig:traj\_results\] with the measuring method proposed by @Laval2011. Basically, we roll a parallelogram with a length of 100m and a time interval of 5s along the shock wave direction (at a speed of $-s/\tau$) across the trajectories in every plot in Figure \[fig:traj\_results\] by a 100m$\times$5s step size. We measure the flow volume and density at each parallelogram and plot the measurements as circles in the corresponding diagrams in Figure \[fig:fund\_measurements\], where the solid curves represent the stationary flow-density relationships specified in equation \[eq:O-K\]. We see that in Figure \[fig:fund\_measurements\](a) for the bench mark trajectories, many measurements are distributed on the congested side of this diagram and most of them are below the stationary curve, which explains why the performance of the benchmark case is the worst. This is probably because traffic the stop and go waves (or traffic oscillations) result in a lower traffic throughput even at the same density, which is known as the capacity drop phenomenon [@Cassidy1999; @Ma2015]. In Figure \[fig:fund\_measurements\](b) for the SH result $P\left(\bar{a},\underline{a},\bar{a},\underline{a}\right)$, there are much fewer measurements falling in the congested branch, and these measurements are closer to the stationary curve. In Figure \[fig:fund\_measurements\](c) for the smoothed SH result $P\left(\bar{a}/3,\underline{a}/3,\bar{a}/3,\underline{a}/9\right)$, even more measurements lie in the free-flow branch, and these measurements become consistent with the stationary curve. This suggests that the proposed SH algorithm with proper parameter values can counteract the capacity drop phenomenon and bring macroscopic traffic characteristics toward the free-flow branch of the stationary curve.
![Macroscopic characteristics for (a) the benchmark trajectories, (b) $P\left(\bar{a},\underline{a},\bar{a},\underline{a}\right)$, and (c) $P\left(\bar{a}/3,\underline{a}/3,\bar{a}/3,\underline{a}/9\right).$ \[fig:fund\_measurements\]](manual_fund_accrate "fig:"){width="33.00000%"}![Macroscopic characteristics for (a) the benchmark trajectories, (b) $P\left(\bar{a},\underline{a},\bar{a},\underline{a}\right)$, and (c) $P\left(\bar{a}/3,\underline{a}/3,\bar{a}/3,\underline{a}/9\right).$ \[fig:fund\_measurements\]](SH_fund_accrate_1_1_1_1 "fig:"){width="33.00000%"}![Macroscopic characteristics for (a) the benchmark trajectories, (b) $P\left(\bar{a},\underline{a},\bar{a},\underline{a}\right)$, and (c) $P\left(\bar{a}/3,\underline{a}/3,\bar{a}/3,\underline{a}/9\right).$ \[fig:fund\_measurements\]](SH_fund_accrate_3_3_3_9 "fig:"){width="33.00000%"}
(a)(b)(c)
Overall, the results in this Subsection show that the proposed SH algorithm can much improve the highway traffic performance in mobility, environment and safety. To realize the full utility of the SH algorithm, quantitative optimization needs to be conducted, which will be detailed in Part II of this study.
Lead Vehicle Problem \[sub:Leading-Vehicle-Problem\]
----------------------------------------------------
This subsection presents LVP results from manual driving law and the proposed CAV driving algorithms. In the LVP, we set $G=\infty$ and $R=0$, and we update saturation rate to $f_{s}=0.5$ in generating the boundary condition (so that the average headway remains $0.5(\tau+s/\bar{v})$). The lead trajectory is set to initially cruise at speed $\bar{v}$ for 20 seconds, then deceleration to the zero speed with a decelerating rate of $\underline{a}/3$, then keep stopped for 20 seconds, then accelerate to $\bar{v}$ with a rate of $\bar{a}/3$, and finally keep cruising at this speed, i.e., $$\begin{aligned}
p_{1}: & = & \left[\left(0,\bar{v},0,0,20\right),\left(20\bar{v},\bar{v},\frac{\underline{a}}{3},20,20-\frac{3\bar{v}}{\underline{a}}\right),\left(20\bar{v}-\frac{3\bar{v}^{2}}{2\underline{a}},0,0,20-\frac{3\bar{v}}{\underline{a}},40-\frac{3\bar{v}}{\underline{a}}\right),\right.\nonumber \\
& & \left.\left(20\bar{v}-\frac{3\bar{v}^{2}}{2\underline{a}},0,\frac{\underline{a}}{3},40-\frac{3\bar{v}}{\underline{a}},40-\frac{3\bar{v}}{\underline{a}}+\frac{3\bar{v}}{\bar{a}}\right),\left(20\bar{v}-\frac{3\bar{v}^{2}}{2\underline{a}}+\frac{3\bar{v}^{2}}{2\bar{a}},\bar{v},0,40-\frac{3\bar{v}}{\underline{a}}+\frac{3\bar{v}}{\bar{a}},\infty\right)\right].\label{eq:leading_traj_gen}\end{aligned}$$ This way, $p_{1}$ triggers a stopping wave and we can examine its propagation under different driving conditions. Figure \[fig:traj\_results-LVP\] shows the trajectory comparison results. We see that first, in Figures \[fig:traj\_results-LVP\] (b) and (c), the PSHL results (solid lines) exactly overlap with the SHL results (crosses), which verifies Proposition \[prop: PSHL=00003DSHL\] that states the equivalence between PSHL and SHL. Compared with the manual-driving case in Figure \[fig:traj\_results-LVP\](a), the automated-driving case in Figure \[fig:traj\_results-LVP\](b), even with the extreme acceleration and deceleration rates $\left(\bar{a},\underline{a}\right)$, relatively better absorbs the backward stopping wave within a fewer number of vehicles. Reducing the acceleration and deceleration magnitudes to $\left(\bar{a}/3,\underline{a}/3\right)$ in Figure \[fig:traj\_results-LVP\](c) can further smooth vehicle trajectories and dampen the impact from the stopping wave. These results imply that proper CAV controls can effectively smooth stop-and-go traffic and reduce backward shock wave propagation on a freeway.
![(a) Benchmark manual-driving trajectories, (b) SHL and PSHL results with $\left(\bar{a}^{\mbox{f}},\underline{a}^{\mbox{f}}\right)=\left(\bar{a},\underline{a}\right),$ and (c) SHL and PSHL results with $\left(\bar{a}^{\mbox{f}},\underline{a}^{\mbox{f}}\right)=\left(\bar{a}/3,\underline{a}/3\right).$ \[fig:traj\_results-LVP\]](manual_LVP_traj "fig:"){width="0.33\linewidth"}![(a) Benchmark manual-driving trajectories, (b) SHL and PSHL results with $\left(\bar{a}^{\mbox{f}},\underline{a}^{\mbox{f}}\right)=\left(\bar{a},\underline{a}\right),$ and (c) SHL and PSHL results with $\left(\bar{a}^{\mbox{f}},\underline{a}^{\mbox{f}}\right)=\left(\bar{a}/3,\underline{a}/3\right).$ \[fig:traj\_results-LVP\]](SHL_traj_accrate_1_1 "fig:"){width="33.00000%"}![(a) Benchmark manual-driving trajectories, (b) SHL and PSHL results with $\left(\bar{a}^{\mbox{f}},\underline{a}^{\mbox{f}}\right)=\left(\bar{a},\underline{a}\right),$ and (c) SHL and PSHL results with $\left(\bar{a}^{\mbox{f}},\underline{a}^{\mbox{f}}\right)=\left(\bar{a}/3,\underline{a}/3\right).$ \[fig:traj\_results-LVP\]](SHL_traj_accrate_3_3 "fig:"){width="33.00000%"}
(a)(b)(c)
Feasibility Tests\[sub:Feasibility-Tests\]
------------------------------------------
This section conducts some numerical tests to test the feasibility of the proposed algorithms with different input settings. We first investigate SHL (or PSHL) for LVP with $\left(\bar{a}^{\mbox{f}},\underline{a}^{\mbox{f}}\right)=\left(\bar{a},\underline{a}\right).$ Theorem \[theo: feasibility\_all\_green\] proves that SHL is feasible if and only if boundary condition $\left[v_{n}^{-},t_{n}^{-}\right]_{n\in\mathcal{N}}$ is proper. Thus we investigate how the feasibility of SHL changes with the distribution of $v_{n}^{-}$ and $t_{n}^{-}$ . Since the default boundary condition generation method always assures that $\left[v_{n}^{-},t_{n}^{-}\right]_{n\in\mathcal{N}}$ is proper, this subsection uses a different generation method to allow $\left[v_{n}^{-},t_{n}^{-}\right]_{n\in\mathcal{N}}$ to be non-proper. We still use equation to generate $t_{n}^{-}$, and thus the dispersion of time headway is controlled by $\alpha$. In the next step, each $v_{n}^{-}$ is instead randomly pulled along a uniform interval $\left[(1-\beta)\bar{v},\bar{v}\right]$ where $\beta\in[0,1]$ is the speed dispersion factor and increases with the dispersion of the $v_{n}^{-}$ distribution. Note that the final values of $t_{n}^{-}$ and $v_{n}^{-}$ are randomly generated. We generate 20 boundary condition instances with the same $\alpha$ and $\beta$ values yet different random seeds. Then we feed each boundary condition instance to the SHL algorithm and record the feasibility of the result. We call the percentage of feasible solutions over all 20 boundary conditions the *feasibility rate* with regard to this specific parameter setting. Figure \[fig:feasibility\_PSHL\] plots hot maps for the feasibility rate over $\alpha\times\beta\in[0,1]\times[0,1]$ with different $f^{\mbox{s}}$ values. We can see that overall, as $\alpha$ and $\beta$ increase, the feasibility rate decreases, and more instances are infeasible as $f^{\mbox{s}}$ increases. This is because higher dispersion of $t_{n}^{-}$ and $v_{n}^{-}$ is more likely to cause conflicts between trajectories that cannot be reconciled under safety constraint , and such conflicts may increase as traffic gets denser. Note that in all maps in Figure \[fig:feasibility\_PSHL\], the transition band between 100% feasibility rate (the white color) and 0% feasibility rate (the black color) is very narrow. This indicates that SHL’s feasibility is dichotomous. With this observation, the parameters could be partitioned into only two phases (feasible and infeasible) to facilitate relevant analysis.
![Feasibility rate of PSHL with (a) $f^{\mbox{s}}=0.2$, (b) $f^{\mbox{s}}=0.5$, and (c) $f^{\mbox{s}}=0.8.$ \[fig:feasibility\_PSHL\]](feasibility_map_fs_0\lyxdot 2 "fig:"){height="30.00000%"}![Feasibility rate of PSHL with (a) $f^{\mbox{s}}=0.2$, (b) $f^{\mbox{s}}=0.5$, and (c) $f^{\mbox{s}}=0.8.$ \[fig:feasibility\_PSHL\]](feasibility_map_fs_0\lyxdot 5 "fig:"){height="30.00000%"}![Feasibility rate of PSHL with (a) $f^{\mbox{s}}=0.2$, (b) $f^{\mbox{s}}=0.5$, and (c) $f^{\mbox{s}}=0.8.$ \[fig:feasibility\_PSHL\]](feasibility_map_fs_0\lyxdot 8 "fig:"){height="30.00000%"}
(a)(b)(c)
Next we investigate the SH algorithm considering traffic lights with $\left(\bar{a}^{\mbox{f}},\underline{a}^{\mbox{f}},\bar{a}^{\mbox{b}},\underline{a}^{\mbox{b}}\right)=\left(\bar{a},\underline{a},\bar{a},\underline{a}\right)$. We conduct similar experiments as those for Figure \[fig:feasibility\_PSHL\] with the same adapted boundary condition generation method. Theorems \[theo:P\_hat\_feasibility\] and \[theo:P\_feasibility\] suggest that the feasibility of SH is related to segment length $L$. Traffic congestion $f^{\mbox{s}}$ and platoon size $N$ shall also affect SH’s feasibility. This time, we fix $\alpha=\beta=0.5$ and analyze how the feasibility rate varies with $L$ and $f^{\mbox{s}}$ over different $N$ values, and the results are shown in Figure \[fig:feasibility\_SH\]. We see that again the increase of $f^{\mbox{s}}$ raises the chance of infeasibility. Further, as $L$ decreases, the likelihood of feasibility diminishes. This is because a short section may not be sufficient to store enough stopping or slowly moving vehicles to both comply with the signal phases and allow the following vehicles to enter the section at their due times. Also, more instances are infeasible as $N$ increases. This is because more vehicles shall bare a higher chance of producing an irreconcilable conflict between two consecutive vehicles against safety constraint . Similarly, the SH feasibility is dichotomous and a two-phase representation might be applicable.
![Feasibility rate of PSHL with (a) $N=25$; (b) $N=50$; and (c) $N=100.$ \[fig:feasibility\_SH\]](feasibility_SH_map_N_25 "fig:"){height="30.00000%"}![Feasibility rate of PSHL with (a) $N=25$; (b) $N=50$; and (c) $N=100.$ \[fig:feasibility\_SH\]](feasibility_SH_map_N_50 "fig:"){height="30.00000%"}![Feasibility rate of PSHL with (a) $N=25$; (b) $N=50$; and (c) $N=100.$ \[fig:feasibility\_SH\]](feasibility_SH_map_N_100 "fig:"){height="30.00000%"}
(a)(b)(c)
Comparison with Classic Traffic Flow Models\[sub:Comparison-with-Classic\]
--------------------------------------------------------------------------
This section compares KWT solution $Q$ and SHL solution $P^{\mbox{LVP}}\left(\underline{a}^{\mbox{f}},\bar{a}^{\mbox{f}}\right)$ for LVP. Again, we set $G=\infty$ and $R=0$, $f^{\mbox{s}}=0.5$. The lead trajectory is generated with equation , and the boundary condition is generated with the default method associated with equation to assure the feasibility. Figure \[fig:KWT\_SHL\_Traj\] compares these trajectories with different $\left(\bar{a}^{\mbox{f}},\underline{a}^{\mbox{f}}\right)$ values. In this section we allow $\left(\bar{a}^{\mbox{f}},\underline{a}^{\mbox{f}}\right)$ go beyond $\left(\bar{a},\underline{a}\right)$ so as to investigate the asymptotic properties of SHL. We see that in general, trajectories in $Q$ have abrupt turns while those in $P^{\mbox{LVP}}\left(\underline{a}^{\mbox{f}},\bar{a}^{\mbox{f}}\right)$ are relatively smooth. All trajectories in $P^{\mbox{LVP}}\left(\underline{a}^{\mbox{f}},\bar{a}^{\mbox{f}}\right)$ is below those in $P^{\mbox{LVP}}\left(\underline{a}^{\mbox{f}},\bar{a}^{\mbox{f}}\right)$, which is consistent with the upper bound property stated in Theorem \[theo:KWT\_SHL\_bounds\]. As $\left(\bar{a}^{\mbox{f}},\underline{a}^{\mbox{f}}\right)$ amplifies from $\left(\bar{a}/3,\underline{a}/3\right)$ to $\left(3\bar{a},3\underline{a}\right)$, we see that accelerations and decelerations in $P^{\mbox{LVP}}\left(\underline{a}^{\mbox{f}},\bar{a}^{\mbox{f}}\right)$ become sharper and trajectories in $P^{\mbox{LVP}}\left(\underline{a}^{\mbox{f}},\bar{a}^{\mbox{f}}\right)$ get closer to those in $Q$, which is consistent with the asymptotic property stated in Corollary \[cor\_KWT\_SH\_asymptotic\].
![Comparison between KWT solution $Q$ and SHL solution $P^{\mbox{LVP}}(\bar{a}^{\mbox{f}},\underline{a}^{\mbox{f}})$ with (a) $\left(\bar{a}^{\mbox{f}},\underline{a}^{\mbox{f}}\right)=\left(\bar{a}/3,\underline{a}/3\right)$, (b) $\left(\bar{a}^{\mbox{f}},\underline{a}^{\mbox{f}}\right)=\left(\bar{a},\underline{a}\right)$, and (c) $\left(\bar{a}^{\mbox{f}},\underline{a}^{\mbox{f}}\right)=\left(3\bar{a},3\underline{a}\right)$.\[fig:KWT\_SHL\_Traj\].](SH_KWT_compare_0\lyxdot 33333_ "fig:"){width="33.00000%"}![Comparison between KWT solution $Q$ and SHL solution $P^{\mbox{LVP}}(\bar{a}^{\mbox{f}},\underline{a}^{\mbox{f}})$ with (a) $\left(\bar{a}^{\mbox{f}},\underline{a}^{\mbox{f}}\right)=\left(\bar{a}/3,\underline{a}/3\right)$, (b) $\left(\bar{a}^{\mbox{f}},\underline{a}^{\mbox{f}}\right)=\left(\bar{a},\underline{a}\right)$, and (c) $\left(\bar{a}^{\mbox{f}},\underline{a}^{\mbox{f}}\right)=\left(3\bar{a},3\underline{a}\right)$.\[fig:KWT\_SHL\_Traj\].](SH_KWT_compare_1_ "fig:"){width="0.33\linewidth"}![Comparison between KWT solution $Q$ and SHL solution $P^{\mbox{LVP}}(\bar{a}^{\mbox{f}},\underline{a}^{\mbox{f}})$ with (a) $\left(\bar{a}^{\mbox{f}},\underline{a}^{\mbox{f}}\right)=\left(\bar{a}/3,\underline{a}/3\right)$, (b) $\left(\bar{a}^{\mbox{f}},\underline{a}^{\mbox{f}}\right)=\left(\bar{a},\underline{a}\right)$, and (c) $\left(\bar{a}^{\mbox{f}},\underline{a}^{\mbox{f}}\right)=\left(3\bar{a},3\underline{a}\right)$.\[fig:KWT\_SHL\_Traj\].](SH_KWT_compare_3_ "fig:"){width="0.33\linewidth"}
(a)(b)(c)
Figure \[fig:KWT\_SHL\_Error\] plots the errors between $Q$ and $P^{\mbox{LVP}}\left(\gamma\bar{a},\gamma\underline{a}\right)$ and their bounds (defined in Theorem \[theo:KWT\_SHL\_bounds\]), where acceleration factor $\gamma$ increases from $1/3$ to $3$. We see that $D\left(q_{n}-p_{n}\right)$ is always identical to 0 (or the upper bound) and $D\left(p_{n}-q_{n}\right)$ is always above the lower bound for all $\gamma$ values. As $\gamma$ increases, the errors and their bounds all converge to 0, which again confirms Corollary \[cor\_KWT\_SH\_asymptotic\]. In summary, these experiments show that SHL can be viewed as a smoothed version of KWT that replaces speed jumps in KWT with smooth accelerations and decelerations.
![Measured errors between KWT solution $Q$ and SHL solution $P^{\mbox{LVP}}\left(\gamma\bar{a},\gamma\underline{a}\right)$ v.s. their bounds \[fig:KWT\_SHL\_Error\].](SH_KWT_error)
Conclusion\[sec:Conclusion\]
============================
This paper investigates the problem of controlling multiple vehicle trajectories on a highway with CAV technologies. We propose a shooting heuristic to efficiently construct vehicle trajectories that follow one another under a number of constraints, including the boundary condition, physical limits, following safety, and traffic signals. With slight adaptation, this heuristic is applicable to not only highway arterials with interrupted traffic but also uninterrupted freeway traffic. We generalize the time geography theory to consider finite accelerations. This allows us to study the behavior of the proposed algorithms. We find that the proposed algorithms can always find a feasible solution to the original complex multi-trajectory control problem under certain mild conditions. We further point out that the shooting heuristic solution to the lead vehicle problem can be viewed as a smoothed version of the classic kinematic theory’s result. We find that the kinematic wave theory is essentially a special case of the proposed shooting heuristic with infinite accelerations. Further, the difference between the shooting heuristic solution and the kinematic wave solution is found to be limited within two theoretical bounds independent of the size of the vehicles in the studied traffic stream. Numerical experiments are conducted to illustrate some theoretical results and draw additional insights into how the proposed algorithms can improve highway traffic.
This paper provides a methodological and theoretical foundation for management of future CAV traffic. The following part II paper @Ma2015 of this study will apply the proposed constructive heuristic with given acceleration rates to a trajectory optimization framework that optimizes the overall performance of CAV traffic (e.g., in terms of mobility, environment impacts and safety) by finding the best acceleration rates. Theoretical results and numerical examples on this optimization framework will be presented. This study overall expects to provide both theoretical principles and application guidance for upgrading the existing highway traffic management systems with emerging CAV technologies. It can be extended in a number of directions to address practical challenges and emerging opportunities in deploying CAV technologies, such as calibration with field experiments, more complex geometries, heterogeneous vehicles, and mixed manual and automated traffic.
Acknowledgments {#acknowledgments .unnumbered}
===============
This research is supported in part by the U.S. National Science Foundation through Grants CMMI CAREER\#1453949, CMMI \#1234936 and CMMI \#1541130 and by the U.S. Federal Highway Administration through Grant DTFH61-12-D-00020.
[^1]: Corresponding author. Tel: 662-325-7196, E-mail: [email protected].
[^2]: http://www.its.dot.gov/connected\_vehicle/connected\_vehicle.htm.
[^3]: http://en.wikipedia.org/wiki/Autonomous\_car
|
---
abstract: 'We calculate flavour dependent lepton asymmetries within the $E_6$ inspired Supersymmetric Standard Model ($\rm E_6SSM$). Our analysis reveals that the substantial lepton CP asymmetries can be induced even if $M_1\simeq 10^6\,\mbox{GeV}$.'
author:
- 'S. F. King'
- 'R. Luo, D. J. Miller, R. Nevzorov'
title: 'Generation of Flavour Dependent Lepton Asymmetries in the E$_6$SSM'
---
INTRODUCTION
============
In models with heavy right–handed neutrinos lepton asymmetry can be dynamically generated via the out–of equilibrium decay of the lightest right–handed neutrino $N_1$. This asymmetry subsequently gets converted into the baryon asymmetry due to sphaleron interactions. In the standard model and its minimal supersymmetric extension the appropriate amount of baryon asymmetry can be induced only if the mass of the lightest right–handed neutrino $M_1\gtrsim 10^9\,\mbox{GeV}$. In this case reheat temperature has to be relatively high, i.e. $T_R> 10^9\,\mbox{GeV}$, that results in an overproduction of gravitinos. To avoid gravitino problem $T_R$ should be low enough $T_R\lesssim 10^{6-7}\,\mbox{GeV}$. Here we study the generation of lepton CP asymmetries within the the Exceptional Supersymmetric Standard Model (E$_6$SSM) [@e6ssm].
THE E$_6$SSM
============
The E$_6$SSM is based on the $SU(3)_C\times SU(2)_W\times U(1)_Y \times U(1)_N$ gauge group which is a subgroup of $E_6$. By definition $
U(1)_N=\frac{1}{4}\, U(1)_{\chi}+\frac{\sqrt{15}}{4} U(1)_{\psi}\,,
$ where $E_6\to SO(10)\times U(1)_{\psi}$, $SO(10)\to SU(5)\times U(1)_{\chi}$. To ensure anomaly cancellation the particle content of the E$_6$SSM is extended to include three complete [**$27$**]{} representations of $E_6$ [@e6ssm]. The [**$27_i$**]{} multiplets contain SM family of quarks and leptons, right–handed neutrino $N^c_i$, SM singlet field $S_i$ which carry non–zero $U(1)_{N}$ charge, a pair of $SU(2)_W$–doublets $H^d_{i}$ and $H^u_{i}$ which have the quantum numbers of Higgs doublets and a pair of colour triplets of exotic quarks $\overline{D}_i$ and $D_i$ which can be either diquarks (Model I) or leptoquarks (Model II). $H^d_{i}$ and $H^u_{i}$ form either Higgs or inert Higgs multiplets. In addition to the complete $27_i$ multiplets the low energy particle spectrum of the E$_6$SSM is supplemented by $SU(2)_W$ doublet $L_4$ and anti-doublet $\overline{L}_4$ from extra $27'$ and $\overline{27'}$ to preserve gauge coupling unification [@e6ssm]. In the E$_6$SSM the right-handed neutrinos do not participate in the gauge interactions and therefore are expected to gain masses at some intermediate scale, while the remaining matter survives down to the electroweak scale. The part of the E$_6$SSM superpotential describing the interactions of $N_i^c$ with other fields can be written as $$\label{1}
W_N= h^N_{kxj}(H^u_{k} L_x)N_j^c + g^{N}_{kij}D_{k}d^{c}_{i}N^{c}_{j}\,,$$ where $x=1,2,3,4$ while $k,i,j=1,2,3$. In the Model I $g^{N}_{kij}=0$ whereas in the Model II all terms in Eq. (\[1\]) can be present.
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CP ASYMMETRIES
==============
The process of the lepton asymmetry generation is controlled by the flavour CP (decay) asymmetries. The CP asymmetries associated with the decays of $N_1$ and $\widetilde{N}_1$ into leptons and sleptons are given by $$\label{2}
\varepsilon^k_{1,\,f}=\frac{\Gamma^k_{1f}-\Gamma^k_{1\bar{f}}}
{\sum_{m,f'} \left(\Gamma^{m}_{1f'}+\Gamma^{m}_{1\bar{f}'}\right)},\qquad\qquad
\varepsilon^k_{\widetilde{1},\,f}=\frac{\Gamma^k_{\widetilde{1} f}-\Gamma^k_{\widetilde{1} \bar{f}}}
{\sum_{m,\,f'} \left(\Gamma^m_{\widetilde{1} f'}+\Gamma^m_{\widetilde{1} \bar{f}'}\right)},$$ where $f$ and $f'$ correspond to either lepton or slepton field while $\Gamma^k_{1f}$ and $\Gamma^k_{\widetilde{1}f}$ are partial decay widths of $N_1\to f +H^u_k$ and $\widetilde{N}_1\to f+H^{u}_k$. At the tree level CP asymmetries vanish. The non–zero contributions to the CP asymmetries arise from the interference between the tree–level amplitudes of the lightest right–handed neutrino decays and one–loop corrections to them. Supersymmetry ensures that $
\varepsilon^k_{1,\,\ell_k}=\varepsilon^k_{1,\,\widetilde{\ell}_k}=\varepsilon^k_{\widetilde{1},\,\ell_k}=
\varepsilon^k_{\widetilde{1},\,\widetilde{\ell}_k}\,.
$ The tree–level and one–loop diagrams that contribute to the $\varepsilon^k_{1,\,\ell_k}$ are shown in Fig. 1. When $M_1\ll M_2, M_3$ we get (see [@King:2008qb]) $$\label{4}
\varepsilon^{k}_{1,\,\ell_x}=-\frac{1}{8\pi A_1}\sum_{j=2,3}\mbox{Im}\biggl[2 A_j h^{N*}_{kx1} h^{N}_{kxj}+
\sum_{m,\,y} h^{N*}_{my1} h^{N}_{mxj} h^{N}_{kyj} h^{N*}_{kx1}\biggr]
\left(\frac{M_1}{M_j}\right)\,,$$ where $A_j=\sum_{m,y} h^{N*}_{my1} h^{N}_{myj} + \frac{3}{2}\sum_{m,n} g^{N*}_{mn1} g^{N}_{mnj}$ if $j=2,3$ while $A_1=\sum_{m,y} h^{N*}_{my1} h^{N}_{my1}$.
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The CP asymmetries associated with the decays of $N_1$ and $\widetilde{N}_1$ into exotic quarks (squarks) can be defined similarly to the lepton ones. The interference of the tree–level decay amplitude with the one–loop corrections (see Fig. 2) yields $$\label{5}
\varepsilon^{i}_{1,\,D_k}=-\frac{1}{8\pi A_0}\sum_{j=2,3}\mbox{Im}\biggl[2 A_j g^{N}_{kij} g^{N*}_{ki1}+
\sum_{m,\,n} g^{N*}_{mn1} g^{N}_{mij} g^{N}_{knj} g^{N*}_{ki1}\biggr]\left(\frac{M_1}{M_j}\right)\,,$$ where $A_0=\sum_{k,\,i}g^{N}_{ki1} g^{N*}_{ki1}$.
RESULTS AND CONCLUSIONS
=======================
To simplify our analysis we impose $Z^H_2$ symmetry under which all superfields except $H_d\equiv H^d_{3}$, $H_u\equiv H^u_{3}$ and $S\equiv S_3$ are odd. The $Z^H_2$ symmetry allows to avoid unacceptably large non-diagonal flavour transitions and reduces the part of the E$_6$SSM superpotential that describes the interactions of $N_i^c$ with other superfields $$\label{6}
W_N\to h^N_{3xj}(H_{u} L_x)N_j^c.$$ We also assume that the right–handed neutrino mass scale is relatively low ($M_1\sim 10^6\,\mbox{GeV}$), so that $h^N_{3ij}$ are negligibly small ($i=1,2,3$), and $M_3\gg M_1, M_2$. In this approximation $\varepsilon^{3}_{1,\,L_4}$ is much larger than other CP asymmetries. In the considered case the maximal absolute value of $\varepsilon^{3}_{1,\,L_4}$ is given by $$\label{7}
|\varepsilon^{k}_{1,\,\ell_4}|= \frac{3 M_1}{8\pi M_2} |h^{N}_{342}|^2,$$ where $h^{N}_{342}$ is a coupling of the second lightest right–handed neutrino to $H_u$ and $L_4$. From Eq. (\[7\]) one can see that in the considered case the substantial lepton decay asymmetry $\varepsilon^{k}_{1,\,\ell_4}$ ($\sim 10^{-6}-10^{-4}$) can be induced even for $M_1\simeq 10^6\,\mbox{GeV}$ if $|h^{N}_{342}|$ vary from $0.01$ to $0.1$. At low energies the asymmetry generated in the $L_4$ sector gets converted into the ordinary lepton asymmetries via the decays of $L_4$. This suggests that in the E$_6$SSM successful thermal leptogenesis can be achieved without encountering the gravitino problem.
We would like to thank S. Antusch, C. D. Froggatt, L. B. Okun and D. Sutherland for fruitful discussions.
RN acknowledges support from the SHEFC grant HR03020 SUPA 36878.
[9]{} S.F. King, S. Moretti, R. Nevzorov, “Theory and phenomenology of an exceptional supersymmetric standard model”, Phys. Rev. D [**73**]{} (2006) 035009; S.F. King, S. Moretti, R. Nevzorov, “Exceptional supersymmetric standard model”, Phys. Lett. B [**634**]{} (2006) 278; S.F. King, S. Moretti, R. Nevzorov, “Gauge Coupling Unification in the Exceptional Supersymmetric Standard Model,” Phys. Lett. B [**650**]{} (2007) 57.
S.F. King, R. Luo, D.J. Miller, R. Nevzorov, “Leptogenesis in the Exceptional Supersymmetric Standard Model: flavour dependent lepton asymmetries”, arXiv:0806.0330 \[hep-ph\].
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---
abstract: 'The core idea of stochastic stability is that thermodynamic observables must be robust under small (random) perturbations of the quenched Gibbs measure. Combining this idea with the cavity field technique, which aims to measure the free energy increment under addition of a spin to the system, we sketch how to write a stochastic stability approach to diluted mean field spin glasses which explicitly gives overlap constraints as the outcome. We then show that, under minimal mathematical assumptions and for gauge invariant systems (namely those with even Ising interactions), it is possible to “reverse” the idea of stochastic stability and use it to derive a broad class of constraints on the unperturbed quenched Gibbs measure. This paper extends a previous study where we showed how to derive (linear) polynomial identities from the “energy” contribution to the free energy, while here we focus on the consequences of “entropic” constraints. Interestingly, in diluted spin glasses, the entropic approach generates more identities than those found by the energy route or other techniques. The two sets of identities become identical on a fully connected topology, where they reduce to the ones derived by Aizenman and Contucci.'
author:
- 'Peter Sollich[^1], Adriano Barra[^2]'
title: Spin glass polynomial identities from entropic constraints
---
Introduction
============
Polynomial identities have a long history in spin glass theory, from the early development by Ghirlanda and Guerra (GG) [@gg] and by Aizenman and Contucci (AC) [@ac; @guerra2] at the end of the 1990s. The link with the peculiar organization of states (in the low temperature phase) discovered by Parisi [@MPV] was guessed immediately; however, it is only in the past few years – and only for fully connected mean field systems, namely the Sherrington-Kirkpatrick model (SK) [@sk] – that Panchenko has been able to show the deep connection between polynomial identities and ultrametricity [@panchenko1; @panchenko2; @panchenko3].
Following the seminal approaches, the former based on checking the stability of states by adding all possible $p$-spin terms [@gardner; @derrida; @grem] and then sending their strength to zero, the latter using a property of robustness of the quenched Gibbs measure with respect to small stochastic perturbation [@ac; @pierluz; @contucci], identities for the SK model have by now been obtained with a number of different techniques, e.g. via smooth cavity field expansion [@barra1], linear response stability [@claudio1], random overlap structures [@peter] or even as Noether invariants [@BG1]. In the diluted counterpart of the SK model, which is the Viana-Bray model (VB) [@vb], a similar research effort has produced classes of identities that naturally generalize the ones obtained earlier by AC (see for instance [@barra4; @franz1]) and GG (see for instance [@franz2; @t1]). It has since been possible to show to validity of these polynomial identities even in short-range, finite-dimensional models [@boviernuovo; @claudio2; @t1; @T], and novel techniques to obtain other identities, with the aim of finding a set of constraints on the overlap probability distribution strong enough to enforce the replica symmetry breaking scheme, are still of great interest, especially beyond the SK framework (see e.g. [@claudio3; @parisitala]).
In this paper, to complement the analysis begun in [@peter] (where we showed how to obtain polynomial identities in spin glasses by considering the energy contribution of random overlap structures developed in [@ass]), we show how to derive AC-like polynomial constraints for the Viana-Bray model even from the entropic contribution. Namely, we add a (random) perturbation term in the Boltzmann factor – close in spirit to the cavity field approach – and then show that, in the thermodynamic limit, this is irrelevant on average as it coincides with a negligible shift in the connectivity. However, the introduction of this “innocent” perturbation, within the standard stochastic stability framework, enables us to derive linear combinations of the constraints on the perturbed Boltzmann measure. The latter converges to the unperturbed measure and returns the identities (all together, combined into an infinite series) as a consequence.
To obtain the constraints as separate identities, we go further and “reverse” the idea of stochastic stability. We introduce the random perturbation only as an overlap generator via derivatives; once we then get the desired polynomials, we evaluate all their averages within the original unperturbed quenched Gibbs measure. Remarkably this procedure, subject to minimal mathematical assumptions, produces separately all the AC-like identities (which reduce to the standard AC constraints in the SK model limit of high connectivity), generalizing all known results. Our arguments do not amount to a rigorous proof, but we hope that they may serve as inspiration for future work in this direction.
Model, notations, cavity perspective and preliminaries
======================================================
In this section we provide a streamlined summary of previous results to make the paper self-contained. In particular, after introducing the model (and the associated standard statistical mechanics definitions), we explain in two further subsections the cavity field and stochastic stability perspectives as applied to diluted spin glasses, with the aim of showing the deep link between these two approaches. In the last subsection we discuss a decomposition of the free energy that highlights the synergy among the cavity field and stochastic stability points of view, and provides a suitable starting point for our investigation of polynomial overlap constraints.
The diluted spin glass
----------------------
To introduce the model (originally studied by Viana and Bray in [@vb]), let us consider $N$ Ising spins $\sigma_i\in\{+1,-1\}$, with $i$ running from 1 to $N$; $\sigma=(\sigma_1,\ldots,\sigma_N)$ will denote the complete spin configuration. Let $P_\zeta$ be a Poisson random variable of mean $\zeta$, and let the $\{J_\nu\}$ be independent and identically distributed copies of a random coupling strength variable $J$ with symmetric distribution. For the sake of simplicity, and without undue loss of generality [@gt2], we will assume $J=\pm 1$. The coupling strengths $J_\nu$ will determine binary interactions between spins at sites $\{i_\nu\},\{j_\nu\}$; the latter are independent identically distributed random variables, with uniform distribution over $1,\ldots,N$. If there is no external field, the Hamiltonian of the Viana-Bray (VB) model for dilute mean field spin glass is then $$\label{ham}
H_N(\sigma, \alpha; \mathcal{J})=
-\sum_{\nu=1}^{P_{\alpha N}} J_\nu \sigma_{i_\nu}\sigma_{j_\nu}\ ,\
\alpha\in\mathbb{R}_+\ .$$ The non-negative parameter $\alpha$ is called [*degree of connectivity*]{}: if the sites $i$ are regarded as vertices of a graph, and the pairs $(i_\nu,j_\nu)$ define the edges of this graph, then each vertex is the endpoint of on average $2\alpha$ edges as explained below. The Hamiltonian (\[ham\]) as written has the advantage that it is the sum of (a random number of) i.i.d. terms. To see the connection to the original VB-Hamiltonian, note that the Poisson-distributed total number of bonds obeys $P_{\alpha
N}=\alpha N + O(\sqrt{N})$ for large $N$. As there are $N^2$ ordered spin pairs $(i,j)$, each gets a bond with probability $\sim \alpha/N$ for large $N$. The probabilities of getting two, three (and so on) bonds scale as $1/N^2,1/N^3,\ldots$ so can be neglected. The probability of having a bond between any unordered pair of spins is twice as large, i.e. $2\alpha/N$. For large $N$ each site therefore has on average $2\alpha$ bonds connecting to it, and more precisely this number of bonds to each site has a Poisson distribution with mean $2\alpha$. The self-loops that we have allowed just add $\sigma$-independent constant to the Hamiltonian so are irrelevant. We will denote by $\mathbb{E}$ the expectation with respect to all the (quenched) variables, i.e. all the random variables except the spins, collectively denoted by $\mathcal{J}$. The Gibbs measure $\omega$ is defined by $$\omega(\varphi)=\frac{1}{Z_N(\alpha,\beta)}\sum_{\sigma}
\varphi(\sigma) e^{-\beta H_N(\sigma, \alpha; \mathcal{J})}$$ for any observable $\varphi:\{-1,+1\}^{N}\to \mathbb{R}$, where $Z_N(\alpha,\beta)= \sum_{\sigma}\exp(-\beta H_{N}(\sigma,\alpha;\mathcal{J}))$ is the partition function for a finite number of spins $N$. When dealing with more than one configuration, the product Gibbs measure will be denoted by $\Omega$, and spin configurations taken from each space in such a product are called “replicas”. We use the symbol $\langle . \rangle$ to mean $\langle . \rangle = \mathbb{E}\Omega(.)$.
We will often omit the dependence on the size of the system $N$ of various quantities, a convention already deployed above. In general, we will allow slight abuses of notation to lighten the expressions as long as there is no risk of confusion. The pressure $P_N(\alpha,\beta)$ and the free energy density $f_{N}(\alpha,\beta)$ for given system size $N$ are defined by $$P_N(\alpha,\beta)= -\beta f_{N}(\alpha,\beta)=\frac1N\mathbb{E}\ln Z_{N}(\alpha,\beta),$$ and we assume that the limit $\lim_{N \to \infty}P_N(\alpha,\beta)=P(\alpha,\beta)$ exists.
The entire physical behavior of the model is encoded by the distribution of the (even) multi-overlaps $q_{1\ldots 2n}$, which are functions of several configurations $\sigma^{(1)},\sigma^{(2)},\ldots,\sigma^{(2n)}$ defined by $$q_{1\ldots 2n}=\frac1N \sum_{i=1}^N
\sigma_i^{(1)}\cdots\sigma_i^{(2n)}\ .$$ By studying the behavior of these order parameters it is possible to obtain a phase diagram for diluted spin glasses in the $(\alpha,\beta)$ plane which consists of an ergodic phase (where all overlaps vanish in the thermodynamic limit of large $N$) and a spin glass phase (where the overlaps are positive), separated by a second order critical line given by 2()=1.
The cavity perspective
----------------------
Following the idea at the heart of the cavity approach (namely, measuring the effect on the free energy of the addition of one spin to the system; see [@ass2; @peter] for a summary), we write, in distribution, $$\label{step}
H_{N+1}(\sigma, \sigma_{N+1}, \alpha; \mathcal{J}) =
-\sum_{\nu=1}^{P_{\alpha \frac{N^{2}}{N+1}}}
J_{\nu}\sigma_{i_{\nu}}\sigma_{j_{\nu}}
-\sum_{\nu=1}^{P_{\alpha \frac{2N}{N+1}}}
J'_{\nu}\sigma_{i'_{\nu}}\sigma_{N+1}
-\sum_{\nu=1}^{P_{\frac{\alpha}{N+1}}}
J''_{\nu}\sigma_{N+1}^2 \ ,$$ where $\sigma_{N+1}$ is the added spin. The $\{J'_{\nu},J''_\nu\}$ are independent copies of $J$, and $\{i_{\nu}\}$, $\{j_{\nu}\}$, $\{i'_{\nu}\}$ are independent random variables all uniformly distributed over $\{1,\ldots,
N\}$. The last term in (\[step\]) does not contribute when $N$ is large, and at any rate is a constant which cancels from the Boltzmann measure.
Note that we can equivalently write the above decomposition as $$\label{acca1}
H_{N+1}(\sigma, \sigma_{N+1}, \alpha; \mathcal{J}) = H_{N}(\sigma,\alpha^{\prime};\mathcal{J})
+h_{N+1}\sigma_{N+1}$$ where $$\alpha^{\prime}=\alpha\frac{N}{N+1}\ ,\
h_{N+1}=-\sum_{\nu=1}^{P_{2
\alpha^{\prime}}}J'_{\nu}\sigma_{i'_{\nu}}\ .$$ Exploiting the additivity property of Poisson variables, we can also decompose the Hamiltonian for an $N$-spin system so that it shares the first term with $H_{N+1}$: $$\label{acca2}
H_{N}(\sigma,\alpha;\mathcal{J})=H_{N}(\sigma,\alpha^{\prime}; \mathcal{J})
+H_{N}(\sigma,\alpha^{\prime}/N; \mathcal{\hat{J}})\ ,$$ where the two Hamiltonians on the right hand side have independent quenched random variables $\mathcal{J}$ and $\mathcal{\hat{J}}$. Hence, if we call $$H_{N}(\sigma;\alpha^{\prime}/N; \mathcal{\hat{J}})
=\hat{H}_N(\sigma,\alpha^{\prime};\mathcal{\hat{J}})=
-\sum_{\nu=1}^{P_{\alpha^{\prime}}}\hat{J}_{\nu}
\sigma_{\hat{i}_{\nu}}\sigma_{\hat{j}_{\nu}}\ ,$$ then $$\mathbb{E}\ln\frac{Z_{N+1}(\alpha,\beta)}{Z_{N}(\alpha,\beta)}
=\mathbb{E}\ln\frac{\sum_{\sigma,\sigma_{N+1}}
\xi_{\sigma}\exp(-\beta h_{N+1}\sigma_{N+1})}{\sum_{\sigma}
\xi_{\sigma}\exp(-\beta \hat{H}(\sigma,\alpha^{\prime};\mathcal{\hat{J}}))}\ ,$$ with $$\xi_{\sigma}=\exp(-\beta H_N(\sigma,\alpha^{\prime};\mathcal{J}))\ .$$ As elegantly explained in [@ass2], and discussed in detail in [@peter], this equation expresses the incremental contribution to the free energy in terms of the mean free energy of a spin added to a reservoir whose internal state is described by $(\sigma, \xi_{\sigma})$, corrected by an inverse-fugacity term $\hat{H}$, which encodes a connectivity shift. The former may be thought of as the [*cavity*]{} into which the $(N + 1)$ particle is added: for $N \gg 1$, the value of the added spin, $\sigma_{N+1}$, does not significantly affect the field that would act for the next increment in $N$. Hence, for the next addition of a particle we may continue to regard the state of the reservoir as given by just the configuration $\sigma$. However, the weight of the configuration (which is still to be normalized to yield the probability of the configuration) changes according to $$\xi_{\sigma}\to\xi_{\sigma}e^{-\beta h_{N+1}\sigma_{N+1}}\ .$$ This transformation is called [*cavity technique*]{}.
The link to stochastic stability
--------------------------------
The addition of a new spin can, because of the randomness of the couplings, effectively be regarded as an external random field that vanishes in the thermodynamic limit. This is essentially the perspective of the stochastic stability approach [@pierluz; @contucci].
By an interpolation method [@barra1] the ($N+1$)-th spin can be added to the $N$-spin system smoothly via an appropriately defined cavity function $\Psi(\alpha,\beta,t)$, $t \in [0,1]$, which reads $$\label{psi2}
\Psi(\alpha,\beta,t) =\mathbb{E}\ln\omega(e^{\beta \sum_{\nu=1}^{P_{2\alpha t}}
J'_{\nu}\sigma_{i'_{\nu}}}).$$ Due to the gauge symmetry of the VB model, namely the symmetry $\sigma_{i_{\nu}}\to \sigma_{i_{\nu}}\sigma_{N+1}$ (whose action leaves the VB Hamiltonian invariant), the above cavity function turns out to contain an effective two body interaction, as in the original Hamiltonian, and the sum over $\sigma_{N+1}=\pm 1$ in the partition function gives a trivial factor two because $\sigma_{N+1}$ plays the role of a hidden variable; this factor two yields, once the logarithm is taken, just the high temperature entropy.
Inspired by the cavity perspective, taking $\varphi$ as a generic function of the spin configuration, we can define a generalized Boltzmann measure (denoted by the subscript $\langle
. \rangle_t$) as $$\omega_t(\varphi)=\frac{\omega(\varphi(\sigma) e^{\beta\sum_{\nu=1}^{P_{2\alpha
t}}J'_{\nu}\sigma_{i'_{\nu}}})}{\omega(e^{\beta\sum_{\nu=1}^{P_{2\alpha
t}}J'_{\nu}\sigma_{i'_{\nu}}})}.$$ Note that in the $t =0$ case we always recover the unperturbed Boltzmann measure of an $N$-spin system and in the $t =1$ case we recover the unperturbed Boltzmann measure of an $N+1$-spin system, with a small shift in the connectivity that becomes negligible in the thermodynamic limit.
Let us now briefly describe the stochastic stability properties for averaged overlap correlation functions (OCFs); these will become useful shortly. We split OCFs into two categories: filled OCFs, showing [*robustness*]{} with respect to the stochastic perturbation, and fillable OCFs, showing [*saturability*]{} with respect to the same perturbation.
- Filled OCFs are monomials in overlaps among $s$ replicas such that each replica appears an even number of times. Examples are $q_{12}^2$, $q_{1234}^2$ or $q_{12}q_{23}q_{13}$.
- Fillable OCFs are overlap monomials among $s$ replicas which become filled when multiplied by a single overlap among exactly those replicas appearing only an odd number of times. Examples are $q_{12}$, $q_{1234}$ or $q_{12}q_{13}$.
It should be pointed out that all monomial OCFs are either filled or fillable, because one can always find a multioverlap to fill any (monomial) OCF that is not filled. This contrasts with the case of the SK model, where only overlaps among two replicas can be used to fill an OCF [@barra1]. The division into filled and fillable OCFs is made because of differences in how their averages react to the perturbing field induced by the cavity function [@barra1; @barra4; @peter]. In the thermodynamic limit, the averages of the filled OCFs become independent of $t$, i.e. $$\lim_{N \to \infty}\partial_t \langle \mbox{filled\ OCF}\rangle_t = 0.$$ We refer to this property as [*robustness*]{}.
On the other hand, using the gauge symmetry, one has in the thermodynamic limit that the averages of fillable OCFs become filled at $t=1$, namely [@barra1; @barra4; @peter] $$\lim_{N \rightarrow \infty} \langle \mbox{fillable\ OCF}
\rangle_{t=1} = \lim_{N \rightarrow \infty} \langle \mbox{filled\ OCF} \rangle_{t=1}= \lim_{N \rightarrow \infty} \langle \mbox{filled\ OCF} \rangle.$$ We refer to this last property as [*saturability*]{}. Note that we have dropped the subscript $t$ in the last equality because of the robustness of filled OCFs. Examples of saturability are $\langle q_{12} \rangle_{t=1}
= \langle q_{12}^2 \rangle$, $\langle q_{1234}\rangle_{t=1} = \langle q_{1234}^2 \rangle$ and $\langle q_{12}q_{13} \rangle_{t=1} = \langle q_{12}q_{13}q_{23} \rangle$, with the limit $N\to\infty$ always understood.
We only sketch the proof of the above propositions and refer the reader to [@barra1; @barra2; @barra4; @peter] for a detailed discussion and proofs. Let us show how the fillable OCFs turn out to become filled OCFs in the $N\rightarrow \infty$ limit at $t=1$. The stability of the filled OCFs will then be a straightforward consequence of their gauge invariance, which is heavily used in the proof. Consider the simplest case of a monomial $Q_{ab}$ that is fillable by multiplying by $q_{ab}$, with replicas $a$ and $b$ each appearing only once in $Q_{ab}$. Then we can write $$\langle Q_{ab} \rangle_t
= \langle
\sum_{ij}(\sigma_i^a\sigma_j^b/N^2)Q_{ij}(\sigma)\rangle_t$$ where $Q_{ij}$ contains all factors that do not depend on replicas $a$ or $b$. Factorizing the state $\Omega_t$ we obtain $$\begin{aligned}
\langle
Q_{ab} \rangle_t =
\frac{1}{N^2}\mathbb{E}\Big( \sum_{ij}
\omega_t(\sigma_i^a)\omega_t(\sigma_j^b)\Omega_t(Q_{ij}) \Big).\end{aligned}$$ Now rewrite the last expression for $t=1$: by applying the gauge transformation $\sigma_i \rightarrow \sigma_i\sigma_{N+1}$, the states acting on the replicas $a$ and $b$ are $\omega_{t=1}(\sigma_i^a)\rightarrow
\omega(\sigma_i^a\sigma_{N+1}^a)+ O(N^{-1})$ while the remaining product state $\Omega_t$ continues to act on a even number of occurrences of each replica and is not modified (in a manner directly analogous to the robustness of averages of filled OCFs). Putting all the replicas back into a single product state, we have: (\_i\^a\_[N+1]{}\^a)(\_i\^b\_[N+1]{}\^b)(Q\_[ij]{}) = (\_i\^a\_j\^b\_[N+1]{}\^a\_[N+1]{}\^bQ\_[ij]{}). Now the index $N+1$ can be replaced by a dummy index $k$ that is averaged according to $1=N^{-1}\sum_{k=1}^N$; this does not change the result except for $O(N^{-1})$ corrections – from values of $k$ that coincide with $i$, $j$, or further summation indices in $Q_{ij}$ – that vanish in the thermodynamic limit. Since $N^{-1}\sum_{k=1}^N
\sigma_{k}^a\sigma_{k}^b = q_{ab}$, this gives the desired result. $\Box$
The free energy decomposition
-----------------------------
In this subsection we want to show that the free energy density can be written in terms of an “energy-like” contribution and an “entropy-like” one. As a consequence of this decomposition, and given that we have previously investigated the constraints deriving from the energy-like term we will then restrict our investigation to the entropy-like contribution, which (as we are going to show) is encoded in the cavity function.
It is in fact always possible, via the fundamental theorem of calculus, to relate the free energy to its derivative with respect to a chosen parameter, here the connectivity $\alpha$. Clearly the result is a relation between the free energy and its $\alpha$-derivative where, interestingly, the missing term is exactly the cavity function. In the thermodynamic limit this decomposition takes the form $$\label{sumrule}
P(\alpha,\beta) + \alpha \partial_\alpha P(\alpha,\beta) = \ln
2 + \Psi(\alpha,\beta,t=1).$$ We emphasize that the equation above, which we are going to prove using continuity and the fundamental theorem of calculus, can be thought of as a generalized thermodynamic definition of the free energy. In this approach, the cavity function naturally acts as the thermodynamic entropy, which is why its investigation suggested the title of the paper.
To see briefly how (\[sumrule\]) arises, let us write down the partition function of a system of $N+1$ spins at connectivity $\alpha^*=\alpha(N+1)/N$, using the decomposition (\[step\]) of the relevant Hamiltonian: $$\begin{aligned}
Z_{N+1}(\alpha^*,\beta) &=& e^{\beta \sum_{\nu=1}^{P_{\alpha/N}}
J''_\nu} \sum_{\sigma,\,\sigma_{N+1}= \pm 1} e^{\beta
\sum_{\nu=1}^{P_{\alpha N}}J_{\nu}\sigma_{i_\nu}\sigma_{j_{\nu}}
+\beta\sum_{\nu=1}^{P_{2\alpha}} J'_\nu
\sigma_{i'_\nu}\sigma_{N+1}}
\\
&=& e^{\beta \sum_{\nu=1}^{P_{\alpha/N}} J''_\nu}
\sum_{\sigma,\,\sigma_{N+1}= \pm 1} e^{\beta
\sum_{\nu=1}^{P_{\alpha N}}J_{\nu}\sigma_{i_\nu}\sigma_{j_{\nu}}
+\beta\sum_{\nu=1}^{P_{2\alpha}} J'_\nu \sigma_{i'_\nu}}
\\
&=&2e^{\beta \sum_{\nu=1}^{P_{\alpha/N}} J''_\nu} \sum_{\sigma}
e^{-\beta H_N(\sigma,\alpha;\mathcal{J}) +\beta\sum_{\nu=1}^{P_{2\alpha}}
J'_\nu \sigma_{i'_\nu}},
$$ where in going from the first to the second line we have gauge transformed $\sigma_i \to \sigma_i \sigma_{N+1}$. Multiplying and dividing by $Z_N(\alpha,\beta)$ and taking logs we get: $$\begin{aligned}
\ln Z_{N+1}(\alpha^*,\beta)= \ln 2 + \beta
\sum_{\nu=1}^{P_{\alpha/N}} J''_\nu +\ln Z_N(\alpha,\beta)
+\ln\omega(e^{\beta\sum_{\nu=1}^{P_{2\alpha}}
J'_\nu\sigma_{i'_{\nu}}}). \nonumber\end{aligned}$$ Averaging over the disorder removes the second term and transforms the last one into the cavity function at $t=1$. Rearranging slightly, the result reads $$[\mathbb{E}\ln Z_{N+1}(\alpha^*,\beta)-\mathbb{E}\ln
Z_{N+1}(\alpha,\beta)]+ [\mathbb{E}\ln Z_{N+1}(\alpha,\beta)
-\mathbb{E}\ln Z_N (\alpha,\beta)] =\ln 2 + \Psi_N(\alpha,\beta,t=1).$$ Now $\alpha^*-\alpha=\alpha/N$ becomes small as $N$ grows so we can Taylor expand the first difference on the l.h.s. as $(\alpha/N)\partial_\alpha\mathbb{E} \ln
Z_{N+1}(\alpha,\beta)+O(1/N)$. The second difference on the l.h.s., on the other hand, gives the pressure as $N\to\infty$, and so the decomposition (\[sumrule\]) follows in the limit.
Identities from “direct” stochastic stability {#barra}
=============================================
Now that the theoretical framework has been outlined, we can turn to the polynomial identities themselves. First, in this section, we review the application of a standard approach from fully connected models to the diluted case [@barra1]. To get constraints on averages of overlap polynomials from this method, one needs to heuristically separate term in a power series. In the next section, after deriving the general form of the required cavity streaming equation, we present a modification of this approach that automatically provides such a separation. We stress that, in the original SK contest, constraints are usually derived in $\beta$-average [@barra1; @contucci; @gg], namely one can prove that they are zero in the thermodynamic limit whenever one takes an average over a (however small) $\beta$ interval but not point by point. Here the same results are obtained for the diluted counterpart by tuning $\alpha$ instead of $\beta$, hence constraints are obtained in $\alpha$-average.
The standard approach referred to above studies the family of linear polynomial constraints (identities) on the distribution of the overlaps which can be obtained by perturbing the original Gibbs measure defined by the Hamiltonian (\[ham\]) with a random term suggested by the cavity technique, where we use as a probe for stochastic stability a generalization of the random perturbation given by the connectivity shift from Eq. (\[step\]).
Specifically, we consider the quenched expectation of a generic function of $s$ replicas, with respect to a perturbed measure defined by the following Boltzmann factor $$\label{pesi}
B(\alpha,\beta,\alpha^{\prime},\beta^{\prime},t) = \exp\left(-\beta H_{N}(\sigma;\alpha;\mathcal{J})+\beta^{\prime}
\sum_{\nu=1}^{P_{2\alpha^{\prime} t}}
\tilde{J}'_{\nu}\sigma_{i'_{\nu}}\right)\ ,
$$ whose use will be indicated with a subscript $\alpha',\beta',t$ in the expectations $\Omega_{\alpha',\beta',t}$ and $\langle
. \rangle_{\alpha',\beta',t}$. In this way the stochastic perturbation coincides with the one from the cavity technique if we choose $t=1$ and $\alpha'=\alpha$, $\beta'=\beta$, while for $t=0$ we recover the unperturbed Gibbs measure.
Loosely speaking, after a gauge transformation one can interpret Eq. (\[pesi\]) as the Boltzmann factor of a system of $N+1$ spins, as in the decomposition (\[acca1\]). The generalization consists in letting the additional spin $\sigma_{N+1}$ experience a different inverse temperature, $\beta'$ rather than $\beta$, and similarly allow for its connectivity to the other $N$ spins to be set by a parameter $\alpha'\neq \alpha$. The new variables $\beta^{\prime}$and $\alpha^{\prime}$ have been introduced for mathematical convenience; in the end, we will then evaluate everything at $\beta^{\prime}=\beta$ and $\alpha^{\prime}=\alpha$.
In the following equations we will require powers of $\tanh(\beta'J)$. Abbreviating $\theta =
\tanh(\beta^{\prime})$ and exploiting that $J=\pm 1$, one has $\tanh^{2n}(\beta^{\prime} J)=\theta^{2n}$ and $\tanh^{2n+1}(\beta^{\prime}J)=J\theta^{2n+1}$ $\forall\ n \in \mathbb{N}$.
To see how one can attempt to obtain constraints from the above generalized stochastic perturbation, consider a generic function $F_s$ of $s$ replicas. Its change with $t$ is given by a “streaming equation” of the following form: $$\begin{gathered}
\label{stream}
\partial_t\langle F_s \rangle_{\alpha',\beta',t} =
-2\alpha^{\prime}\langle F_s
\rangle_{\alpha',\beta',t} +2\alpha^{\prime} \mathbb{E}\biggl[
\Omega_{\alpha',\beta',t}\biggl( F_s \{ 1 + J\sum_{a}
\sigma^{a}_{i_{1}}\theta
+ \sum_{a < b}
\sigma^{a}_{i_{1}}\sigma^{b}_{i_{1}} \theta^2 \\
{}+ J\sum_{a < b < c}
\sigma^{a}_{i_{1}}\sigma^{b}_{i_{1}}\sigma^{c}_{i_{1}}
\theta^3 + \cdots \} \{ 1 - s
J\theta \omega_{\alpha',\beta',t}(\sigma) + \frac{s(s+1)}{2!}\theta^2
\omega_{\alpha',\beta',t}^{2}(\sigma)\\
{}-\frac{s(s+1)(s+2)}{3!}J\theta^3\omega_{\alpha',\beta',t}^{3}(\sigma)
+ \cdots \}\biggr)\biggr]\ .
\end{gathered}$$ Here the replica indices $a,b,c$ all run from 1 to $s$. The above result can be checked by direct calculation and is shown for instance in [@barra4]. We will derive a more general form below.
If one chooses for $F_s$ a function whose average does not depend on $t$ (for instance a filled OCF, which is robust), the left hand side of (\[stream\]) is zero. In other words, one uses as the generator of constraints on the distribution of the overlaps the robustness property $$\lim_{N\to\infty}
\partial_t \langle F_s \rangle_{\alpha,\beta,t} = 0$$ where $F_s$ is filled and $\alpha^{\prime}=\alpha$, $\beta^{\prime}=\beta$. For $F_s$ as above, one has on the r.h.s. of (\[stream\]) averages of fillable OCFs. The streaming equation is a power series in $\theta$ (and hence in $\beta^{\prime}$). Heuristically, driven by the critical behavior of the OCF (as the general multi-overlap $q^2_{2n}$ scales as $(2\alpha\theta-1)^{2n}$ [@barra2]), one can argue that all coefficients of this power series should vanish if the l.h.s. of (\[stream\]) vanishes. However, one has to bear in mind that, before setting $\alpha'=\alpha$ and $\beta'=\beta$, the averages on the r.h.s. will still be dependent on $\beta'$. The r.h.s. is not, therefore, a standard power series with constant coefficients.
The simplest example of this reasoning is provided by $F_s=q_{12}^{2}$ with $s=2$. The streaming equation is $$\lim_{N\to\infty}
\partial_{t}\langle q_{12}^2 \rangle_{\alpha',\beta',t} =
\lim_{N\to\infty}
\langle q_{12}^3 - 4 q_{12}^2q_{23} + 3 q^2_{12}q_{34}
\rangle_{\alpha',\beta',t}\theta^{2} +O(\theta^{4})=
0$$ If one now assumes that the coefficients of the powers of $\theta$ are separately zero, then by setting $t=1$, $\alpha'=\alpha$ and $\beta'=\beta$ one can transform the fillable average into a filled one to obtain $$\lim_{N\to\infty} \langle q_{12}^4 - 4 q_{12}^2q_{23}^2 + 3 q_{12}^2
q_{34}^2 \rangle=0$$ This is the well-known Aizenman-Contucci relation.
Choosing instead instead $F_s=q^{2}_{1234}$ ($s=4$), and limiting the expansion to the first two orders $\theta$ of the streaming equation, one obtains (again for $N\to\infty$) $$\begin{gathered}
\partial_{t}\langle q_{1234}^2 \rangle_{\alpha',\beta',t} =
\theta^2 \langle 3q_{1234}^2q_{12} - 8q_{1234}^2q_{15} + 5
q_{1234}^2q_{56}\rangle_{\alpha',\beta',t} \\
{} + \theta^4\langle q_{1234}^3-16q_{1234}^2q_{1235}+60q_{1234}^2q_{1256}
-80q_{1234}^2q_{1567}+35q^2_{1234}q_{5678}\rangle_{\alpha',\beta',t}
+O(\theta^{6})=0\end{gathered}$$ We re-emphasize that one cannot deduce that each term in this expansion vanishes separately, as the Boltzmann factor inside the averages is a function of $\beta^{\prime}$ and hence $\theta$, and therefore so are the averages. If one nevertheless proceeds and sets the coefficient of the second power of $\theta$ to zero, one obtains at $\alpha'=\alpha$, $\beta'=\beta$ and $t=1$ a relation between filled OCFs: $$\langle q_{1234}^2q_{15}^2
\rangle = \frac{3}{8}\langle q_{1234}^2 q_{12}^2\rangle + \frac{5}{8}
\langle q_{1234}^2q_{56}^2 \rangle\ ,$$ and similarly from the fourth order in $\theta$ $$\langle q_{1234}^4
\rangle = \langle 16q_{1234}^2q_{1235}^2 -60q_{1234}^2q_{1256}^2
+80q_{1234}^2q_{1567}^2 -35q^2_{1234}q^2_{5678} \rangle\ .$$ These relations are in perfect agreement with previous investigations [@franz2] and generalize the standard identities to diluted systems. Clearly when the connectivity diverges the multi-overlaps go to zero and the equations then no longer provide any non-trivial information.
We note briefly here that a similar heuristic step, where coefficients of a power series are taken to be zero even though they initially depend on the variable that one is expanding in, is used implicitly in the derivation of the AC-like identities in [@franz2]. The same comment applies to the arguments leading to the GG-like relations in [@franz1].
“Reversing” stochastic stability
================================
General streaming equation
--------------------------
To go beyond the heuristic arguments reviewed in the previous section, we need to write down first the general form of the streaming equation (\[stream\]). The $t$-dependence in the Boltzmann factor (\[pesi\]) arises only from the Poisson variable $P_{2\alpha't}$. For a generic function of such a variable, leaving off the factor $2\alpha'$ initially, one has $
\mathbb{E}f(P_t) = e^{-t}\sum_{k=0}^\infty f(k) t^k/k!
$ and so $$\partial_t \mathbb{E}f(P_t) = -e^{-t}\sum_{k=0}^\infty f(k) t^k/k! + e^{-t} \sum_{k=1}^\infty f(k) t^{k-1}/(k-1)! = -\mathbb{E}f(P_t) + \mathbb{E}f(1+P_t)$$ Applying this to $\langle F_s\rangle_{\alpha',\beta',t}$ gives $$\frac{\partial_t \langle F_s\rangle_{\alpha',\beta',t}}{2\alpha'} = - \langle F_s\rangle_{\alpha',\beta',t} +
\mathbb{E} \frac{\Omega_{\alpha',\beta',t}(F_s e^{\beta' J \sum_{a=1}^s\sigma_i^a})}{\Omega_{\alpha',\beta',t}(e^{\beta' J \sum_{a=1}^s\sigma_i^a})}$$ where $J$ is the random interaction strength for the additional coupling and $i$, drawn uniformly from $\{1,\ldots,N\}$, is the associated spin index. The resulting exponential can be written as $\prod_{a=1}^s[\cosh(\beta') +\sinh(\beta')J\sigma_i^a]$ and cancelling the common factor $[\cosh(\beta')]^s$ yields $$\frac{\partial_t \langle F_s\rangle_{\alpha',\beta',t}}{2\alpha'} + \langle F_s\rangle_{\alpha',\beta',t} =
\mathbb{E} \frac{\Omega_{\alpha',\beta',t}(F_s \prod_{a=1}^s[1+J\theta\sigma_i^a])}
{\Omega_{\alpha',\beta',t}(\prod_{a=1}^s[1+J\theta\sigma_i^a])}$$ where $\theta=\tanh(\beta')$ as before. The denominator factorizes across replicas, giving a factor $[1+J\theta\Omega_{\alpha',\beta',t}(\sigma_i)]^{-s}$. One expands this and also the numerator in powers of $\theta$ to get for the r.h.s. of the last equation $$\mathbb{E} \Omega_{\alpha',\beta',t}\left(F_s
\sum_{k=0}^s(J\theta)^k \sum_{1\leq a_1<\ldots<a_k\leq s}
\sigma_i^{a_1} \cdots \sigma_i^{a_k}\right) \sum_{l=0}^\infty
\frac{(s+l-1)!}{l!(s-1)!}(-1)^l (J\theta)^l [\Omega_{\alpha',\beta',t}(\sigma_i)]^l$$ To cast each term as an average over a replicated measure again, one can write $[\Omega_{\alpha',\beta',t}(\sigma_i)]^l
= \Omega_{\alpha',\beta',t}(\sigma_i^{s+1}\cdots\sigma_i^{s+l})$. This gives as the general streaming equation = F\_swhere for brevity we have dropped the subscript $\alpha',\beta',t$ on all averages. Now we gather terms according to equal powers $m=k+l$ of $\theta$, and note that the $m=0$ term cancels with the $-1$: = F\_s\_[m=1]{}\^(J)\^[m]{} \_[l=0]{}\^[m]{} \_[1a\_1<…<a\_[m-l]{}s]{} \_i\^[a\_1]{} \_i\^[a\_[m-l]{}]{} (-1)\^[l]{} \_i\^[s+1]{}\_i\^[s+l]{}. Here and below it is understood that the sum over $a_1$, …, $a_{m-l}$ vanishes when $m-l>s$, because it is then not possible to satisfy the constraint $1\leq a_1<\ldots<a_{m-l}\leq s$.
Performing now the expectation over $J$ cancels all odd orders $m$, and the expectation over $i$ produces a multi-overlap, giving = F\_s\_[m=2,4,…]{} \^[m]{} \_[l=0]{}\^[m]{} (-1)\^[l]{} \_[1a\_1<…<a\_[m-l]{}s]{} q\_[a\_1…a\_[m-l]{},s+1…s+l]{}. To state this result in a compact form, we define C\^[(m,n)]{}\_s = \_[l=0]{}\^[m]{} (-1)\^[l]{} \_[1a\_1<…<a\_[m-l]{}s]{} q\^n\_[a\_1…a\_[m-l]{},s+1…s+l]{}. \[C\_def\] The superscript $m$ indicates how many replicas are involved in each of the overlaps in this expression. Each overlap is taken to the power $n$, a generalization which will be useful shortly. Then after re-instating the subscripts on the averages, the streaming equation can be written simply as = F\_s\_[m=2,4,…]{} \^[m]{} C\_s\^[(m,1)]{}\_[’,’,t]{}. \[stream\_general\] We can now state the arguments of the previous sections in more general form. If $F_s$ is filled then the derivative on the l.h.s. of (\[stream\_general\]) must vanish. If one evaluates at $\alpha'=\alpha$, $\beta'=\beta$, $t=1$ and for $N\to\infty$, all the overlaps in $C_s^{(m)}$ become squared – the fillable averages become filled – so each factor $C^{(m,1)}_s$ turns into a “higher order AC factor” $C_s^{(m,2)}$. The identities from directed stochastic stability are therefore (for filled $F_s$ and in the limit $N\to\infty$) F\_s\_[ø=2,4,…]{} \^[m]{} C\_s\^[(m,2)]{}= 0. As explained above, one cannot necessarily separate the different orders in $\theta$ in this result. This is possible by “reversing” the approach, as we will now see.
Identities from reversed stochastic stability
---------------------------------------------
Progress in demonstrating that each term of the above expression must vanish separately, i.e.$\left\langle F_s C_s^{(m,2)}\right\rangle = 0$ for each $s$ and even $m$, can be made by considering not $t=1$ but derivatives at $t=0$ and assuming that, in general, stochastic stability holds [@pierluz; @contucci]. Let us take a generic overlap polynomial $F_s$, and consider as above the “smooth cavity” perturbation with a modified temperature $\beta'$ and corresponding $\theta=\tanh(\beta')$. Then the first derivative w.r.t. $t$ is (writing $m=2n$ now) = \_[n=1]{}\^\^[2n]{} F\_s C\_s\^[(2n,1)]{}\_[’,’,t]{}. Pulling the infinite sum out of the expectation and differentiating again (assuming that we can interchange differentiation with the infinite sum over $n$) we get = \_[n=1]{}\^\_[m=1]{}\^\^[2(m+n)]{} F\_s C\_s\^[(2n,1)]{}C\_[s+2n]{}\^[(2m,1)]{}\_[’,’,t]{}. \[t0\_aux\] From now on, take $F_s$ to be filled and set $t=0$ instead of $t=1$: we can then drop the subscripts on the average on the r.h.s. because at $t=0$ we have an unperturbed Boltzmann average. Then because of gauge invariance of this unperturbed state, all terms with $m\neq n$ give vanishing averages: $F_s$ is filled, $C_s^{(2n,1)}$ consists of a sum of $n$-th order overlaps of $n$ distinct replicas, and $C_{s+2n}^{(2m,1)}$ of a sum of $m$-th order overlaps of $m$ distinct replicas. At least $|m-n|\geq 1$ replicas therefore occur an odd number of times in all possible combinations of terms. Hence we can collapse the sum to .|\_[t=0]{} = \_[n=1]{}\^\^[2n]{} F\_s C\_s\^[(2n,1)]{}C\_[s+2n]{}\^[(2n,1)]{}. \[t0\_aux2\] One can simplify further: $C_s^{(2n,1)}$ is as function only of replicas $1,\ldots, s+2n$. This means that in the sum (\[C\_def\]) defining $C_{s+2n}^{(2n,1)}$, only the term with $l=0$ can give non-vanishing averages, because all other terms depend on some of the replicas $s+2n+1,\ldots, s+4n$ and these would remain unpaired. In other words, F\_s C\_s\^[(2n,1)]{}C\_[s+2n]{}\^[(2n,1)]{}= F\_s C\_s\^[(2n,1)]{} \_[1b\_1<…<b\_[2n]{} s+2n]{} q\_[b\_1…b\_[2n]{}]{} . But now for each term $q_{a_1,\ldots,a_{2n}}$ in $C_s^{(2n,1)}$ (with $1\leq a_1<\ldots<a_{2n}\leq s+2n$) there is exactly one entry in the sum over $1\leq b_1<\ldots<b_{2n}\leq s+2n$ which fills this term and gives a nonzero average, namely $b_1=a_1$, …, $b_{2n}=a_{2n}$. The multiplication by the sum over $b_1,\ldots,b_{2n}$ therefore just has the effect of squaring all the overlaps in $C_s^{(2n,1)}$ and we get F\_s C\_s\^[(2n,1)]{}C\_[s+2n]{}\^[(2n,1)]{}= F\_s C\_s\^[(2n,2)]{}. Inserting into (\[t0\_aux2\]) gives then .|\_[t=0]{} = \_[n=1]{}\^\^[2n]{} F\_s C\_s\^[(2n,2)]{}. Now we assume that we have stochastic stability for generic $\theta$ [@pierluz; @contucci] and $t$, i.e. that $\langle F_s\rangle_{\alpha',\beta',t}$ is independent of $\beta'$ and $t$ in the limit $N\to\infty$. Then the last expression vanishes and hence so must all coefficients of different powers of $\theta$. We thus obtain the relations (for $s\geq 2$ and $n\geq 1$) \_[N]{} F\_s C\_s\^[(2n,2)]{}= 0. \[general\_identity\] This includes for $n=1$ all the standard SK AC-identities, but also all their higher-order generalizations.
We now compare with closely related identities obtained by de Sanctis and Franz [@franz1], and Franz, Leone and Toninelli [@franz2]. It is easy to check that the identities obtained in these papers can be written in our notation as $$\lim_{N\to\infty} \langle q_{1\ldots s}^{2r} C_s^{(2n,2p)}\rangle = 0.$$ for arbitrary positive integers $r$ and $p$. The identities (\[general\_identity\]) relate to $p=1$ but are then rather more general because they allow arbitrary filled overlap polynomials for $F_s$. For example our identities for $F_3=q_{12} q_{23} q_{13}$ are not contained in the set of identities from [@franz1; @franz2].
Generalization to fields with multiple spins
--------------------------------------------
The generalization of the identities (\[general\_identity\]) to exponents greater than two can be achieved relatively simply by a standard approach, allowing not just fields but $p$-spin interactions in the perturbing term. Mirroring the technique developed for two-body interactions, starting with a gauge invariant $P$-spin Hamiltonian (hence built trough even $P$ only), namely \[pspin\] H\_N(,; ) = -\_[=1]{}\^[P\_[N]{}]{}J\_\_[i\_\^1]{}...\_[i\_\^P]{}, the way to obtain the proper fields perturbing the Boltzmann measure is trough gauge symmetry $\sigma_{i_{\nu}^k} \to \sigma_{i_{\nu}^k}\sigma_{N+1}$ (which is clearly a symmetry of Hamiltonian (\[pspin\])). The perturbation term from (\[pesi\]) would then be generalized to $$\beta^{\prime} \sum_{\nu=1}^{P_{2\alpha^{\prime} t}} \tilde{J'}_{\nu}\sigma_{i^1_{\nu}}
\cdots \sigma_{i^{p-1}_{\nu}},$$ and reduces to the latter for $p=2$ as it should.
In the streaming equation w.r.t. $t$, factors like $\sigma_i^a$ are then consistently replaced by $\sigma_{i^1}^a \cdots \sigma_{i^{p-1}}^a$, and accordingly one obtains in the end = F\_s\_[m=2,4,…]{} \^[m]{} C\_s\^[(m,p-1)]{}\_[’,’,t]{}. \[general\_1st\] The calculation of the second derivative at $t=0$ generalizes in the same way, giving $$\left.\frac{\partial_t^2 \langle
F_s\rangle_{\alpha',\beta',t}}{(2\alpha')^2}\right|_{t=0} =
\sum_{n=1}^\infty \theta{}^{2n}
\left\langle F_s C_s^{(2n,2(p-1))}\right\rangle.
\label{general_2nd}$$ Based again on the assumption of stochastic stability under a small perturbation of the Boltzmann measure caused by the introduction of $O(N^0)$ random $(p-1)$-spin interaction terms into a system of $N$ spins, one then deduces the identities \_[N]{} F\_s C\_s\^[(2n,2(p-1))]{}= 0. \[most\_general\_identity\] With the overlap exponent now generalized from $2$ to $2(p-1)$, these form a strict superset of the identities from [@franz2; @franz1].
We have been somewhat casual above in not distinguishing between odd and even order $p-1$ of the perturbing Hamiltonian. Specifically, the reasoning that leads to the simplified form of the second derivative (\[general\_2nd\]) works, by analogy with the case $p=2$, only when $p-1$ is odd. For even $p-1$ a more complicated expression analogous to (\[t0\_aux\]) would result. However, in this case one can exploit the vanishing of the first derivative (\[general\_1st\]). Writing $p-1=2(p'-1)$ because $p-1$ is even, one then obtains again the identities (\[most\_general\_identity\]), with $p$ replaced by $p'$ which is now an arbitary integer.
As far as $P$ is even, hence gauge-invariance in the Hamiltonian \[pspin\] is preserved, everything works fine straightforwardly; however, for odd $P$, similar argument still apply (as only the true Hamiltonian (\[ham\]) needs the gauge-invariance) as it is enough to consider the first rather than the second $t$-derivative to get AC-relations with exponent $p-1$.
While we have assumed throughout Poissonian graphs, where each pair interaction is present independently of the others, one would expect that the identities (\[most\_general\_identity\]) hold also for spin glass model on other graphs with finite connectivity. The treatment in [@franz2] is more general in this regard, and it may be possible to adapt the methods used there to generalize (\[most\_general\_identity\]) to this broader range of settings.
Our reasoning leading to the general higher-order AC-identitites (\[most\_general\_identity\]) is, like the one in Ref. [@franz2; @franz1], not rigorous. The main assumption here, as in the other papers, is stochastic stability for general values of $\beta'\neq \beta$. More technically, we have also been somewhat cavalier in our treatment of infinite sums, interchanging e.g. differentation w.r.t. $t$ with summation.
Conclusion
==========
The phenomenon of full replica symmetry breaking, and its probably best known consequence, namely ultrametricity, have deep implications in physics. It is for this reason that the Sherrington-Kirkpatrick model, which is the fully connected limit of the Viana-Bray diluted spin glass discussed here, is sometimes described as the harmonic oscillator of complex systems.
Parisi ultrametricity is a strong constraint on overlap probability distributions, and entails peculiar constraints for averages of polynomials of these. These linear (in the averages) polynomial identities were the subject of this paper. They are of interest in themselves, but also with regards to the question of how far results from the fully connected mean field framework can be extended to other scenarios.
Linear identities in diluted spin glasses have already been obtained with standard techniques, mainly intensivity of the internal energy [@franz2]. In our own work we have shown how they can be embedded within the framework of random overlap structures, and have analyzed in particular what identities can be obtained from the “energy” contribution to the free energy in this context [@peter]. In this paper we showed how to recover all known linear identities by focussing attention on the entropy instead of the internal energy and then we highlighted how our method allowed us to further enlarge the set of identities.
We reviewed first the results of a classical stochastic stability analysis, which contains within it, via a gauge transformation, the physics of the cavity approach. Linear identities for multi-overlaps can be argued for within this method, but require a heuristic separation into the different orders in $\theta=\tanh(\beta')$, where $\beta'$ is the inverse temperature associated with the stochastic perturbation.
The main contribution of this paper was then to go beyond this. We started from a general streaming equation describing the effect of a stochastic perturbation on the Gibbs measure. Instead of perturbing the Gibbs measure and then using the thermodynamic limit to make the effect of the perturbation vanish, we evaluated averages directly in the unperturbed Gibbs state, using second derivatives with respect to the perturbation parameter $t$. This gives stronger results, in that identities from different orders in $\theta$ can be cleanly separated; it also yields a larger number of identities compared to those obtained previously using other techniques.
Conceptually, it is interesting that this new approach does not directly exploit the cavity nature of the perturbation, i.e. the mapping to a system of $N+1$ spins when $t=1$ and $\beta'=\beta$. But it certainly does use the gauge invariance of the unperturbed Boltzmann state.
As we have emphasized, our arguments are not rigorous, requiring as the key assumption stochastic stability under general perturbations as well as some more technical conditions. We would hope, however, that our reasoning might in the future form the basis for a rigorous proof of all linear polynomial identities in diluted spin glasses.
Acknowledgements {#acknowledgements .unnumbered}
================
AB acknowledges the FIRB grant RBFR08EKEV and Sapienza Università di Roma for partial financial support. The authors are grateful to Francesco Guerra, Pierluigi Contucci and Cristian Giardinà for enlightening conversations.
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[^1]: King’s College London, Department of Mathematics, Strand, London WC2R 2LS, U.K., [email protected]
[^2]: Dipartimento di Fisica, Sapienza Universita’ di Roma, P.le Aldo Moro 5, 00185, Roma, Italy, [email protected]
|
---
abstract: 'Using a large number of weights, deep neural networks have achieved remarkable performance on computer vision and natural language processing tasks. However deep neural networks usually use lots of parameters and suffer from overfitting. Dropout is a widely use method to deal with overfitting. Although dropout can significantly regularize fully connected layers in neural networks, it usually leads to suboptimal results when used for convolutional layers. To tackle this problem, we propose DropFilter, a dropout method for convolutional layers. DropFilter considers the filters rather than the neural units as the basic data drop units. Because it is observed that co-adaptations are more likely to occur inter rather than intra filters in convolutional layers. Using DropFilter, we remarkably improve the performance of convolutional networks on CIFAR and ImageNet.'
author:
- 'Zhengsu Chen Jianwei Niu Qi Tian\'
bibliography:
- 'egbib.bib'
title: 'DropFilter: Dropout for Convolutions'
---
Introduction
============
Recently, deep neural networks have enabled breakthroughs on computer vision and natural language processing tasks [@girshick2014rich; @long2015fully; @graves2013speech; @Vinyals2015Show]. Convolutional neural networks (CNNs) are the widely used models on computer vision tasks. Since AlexNet [@krizhevsky2012imagenet] significantly improved the classification performance on ImageNet, many new deep convolutional network architectures (VGG [@simonyan2014very], Inception [@szegedy2015going; @szegedy2016rethinking; @szegedy2016inception], ResNet [@he2016deep], DenseNet [@huang2016densely] ) are proposed. These networks achieve remarkable success on recognition tasks and usually employ lots of parameters.
Deep neural networks usually suffer from overfitting. To avoid overfitting, dropout is used in many network architectures [@srivastava2014dropout; @simonyan2014very]. Dropout randomly suppresses the outputs of neural units to reduce the co-adaptations between them, which sometimes can significantly improve the networks. The co-adaptive neural units are similar with each other. It is a waste of parameters and may cause overfitting. Dropout is usually used for fully connected layers. When dropout is used for convolutional layers, the network is slightly improved or even gets worse sometimes. Two reasons may account for this problem. First, compared to fully connected layers, convolutional layers use much fewer parameters. Overfitting and co-adaptations are not so serious in convolutional layers. Second, the co-adaptations in fully connected layers are different from that in convolutional layers.
Dropout can avoid the co-adaptations between neural units. For fully connected layers, every unit is equivalent in position, which is the main reason why co-adaptations occur. To decrease the co-adaptations, dropout randomly suppresses the units. During the backward pass, the gradients of these units are also suppressed. Consequently, even if the units have similar weights, they will experience different gradients. Their weights will be updated to different directions. The units will become different to each other and co-adaptations will be reduced.
In convolutional networks, the outputs of convolutional layers are a set of feature maps. The outputs in the same position of different feature maps are equivalent in position. However, because of the different receptive field on the input image, the outputs in the same feature map are not equivalent. **Co-adaptations are more likely to occur between units that are equivalent in position**. Therefore, for convolutional neural networks, the co-adaptations should be reduced inter rather than intra filters. Standard dropout focuses on the co-adaptations between neural units. For the units that are unlikely to produce co-adaptations, using dropout for them may introduce unnecessary noise.
To reduce the co-adaptations between filters, we propose DropFilter in this paper. DropFilter randomly drops some feature maps of convolutional layers. In other words, DropFilter randomly suppresses the outputs of some convolutional filters and that is why it is called DropFilter. Using DropFilter, the filters in convolutional layers will experience different inputs and gradients, which will reduce the co-adaptations between filters.
DropFilter is similar to DropPath to some extent. DropPath is a widely used regularization method for multi-path networks [@larsson2016fractalnet; @huang2016deep; @zoph2017learning]. DropPath randomly drops some paths during training. Every path is a set of feature maps. Thus both DropFilter and DropPath will drop some feature maps during training. DropPath can be seen as a special case of DropFilter. DropPath can only be used for multi-path networks, while DropFilter can be used for all convolutional neural networks.
DropFilter and dropout randomly suppress the outputs of convolutional layer. Deep convolutional neural networks usually contain many convolutional layers. If data drop methods are used for every convolutional layer with the same data retaining rate, directly suppressing the outputs is a little coarse for very deep CNNs. The networks may be very sensitive to the data retaining rate and hard to be tuned. To tackle this problem, we propose ScaleFilter. Unlike DropFilter, ScaleFilter scales the feature maps of the outputs with random weights rather than directly sets some of them to zeros. ScaleFilter is similar to DropFilter but is more stable and easier to be tuned.
DropFilter and ScaleFilter are tested on ResNets, WRNs and non-residual networks in this paper. It is observed that DropFilter and ScaleFilter consistently outperform standard dropout. Sometimes standard dropout slightly improve the networks, while DropFilter and ScaleFilter can still bring significant improvement.
Related Work
============
Dropout
-------
Dropout is proposed by [@srivastava2014dropout] to regularize the networks during training. Dropout randomly suppresses the outputs of networks and is widely used for fully connected layers [@krizhevsky2012imagenet; @simonyan2014very]. Afterwards, @Wan2013Regularization present DropConnect, which randomly suppresses the connections rather than the outputs of neural units. DropConnect performs better in generalization. @Wang2013Fast use Gaussian noise for dropout. In their opinions, Gaussian noise is more stable and faster than Bernoulli noise for dropout. As for the dropout retaining rate, instead of using fixed retaining rate, @Morerio2017Curriculum linearly decrease the data retaining rate during training. They propose that co-adaptations will not occur in the beginning. Thus they use big data retaining rate to avoid introducing unnecessary noise in the beginning. In this paper, we first concentrate on the particular co-adaptations in convolutional layers and propose DropFilter and ScaleFilter to regularize CNNs.
DropPath
--------
There are some networks that have many paths [@larsson2016fractalnet; @huang2016deep; @zoph2017learning]. Residual networks can also be seen as a multi-path networks [@veit2016residual]. DropPath drops some paths to regularize the networks during training, which can only be used for multi-path networks. Stochastic depth ResNets [@huang2016deep] randomly drops the residual block of ResNets. ResNets with stochastic depth is faster during training and outperform constant depth ResNets on some datasets. FractalNet [@larsson2016fractalnet] is a multi-path network which is designed with self-similarity. Using DropPath, FractalNet produces shallow and deep subnetworks that have different speed and accuracy. Thus these subentworks can be used for various tasks. Shake-shake regularization [@gastaldi2017shake] combines the paths in networks with random weights. Based on the their multi-path networks, shake-shake networks achieve state-of-the-art results on CIFAR. The architecture of NASNets [@zoph2017learning] is learned by a reinforcement learning system. NASNets are constructed by multi-path substructures. Using DropPath, the performance of NASNets is significantly improved. Like DropPath, DropFilter randomly drops some feature maps of the networks. However, DropFilter can be used for any convolutional neural network architecture while DropPath can only be used for specially designed multi-path networks.
Methods
=======
\[fig:path\_blocks\]
DropFilter
----------
Given an input image, $\textbf{x}$ is the input of a convolutional layer for a CNN. $c$ is the filter number of this layer. $Y$ is the output feature maps and $W =\{ \textbf{w}_{i} | 1\leq i \leq c\}$ is the weights of the filters in this layer. We have:
$$\begin{aligned}
\textbf{y}_{i} & = f(\textbf{w}_{i} \textbf{x} + b_{i}) \\
Y &= \{ \textbf{y}_{i} | 1\leq i \leq c \} \nonumber \label{FilterDrop1}\end{aligned}$$
where $i$ indexes the filters and the output feature maps in the layer. $f$ is the activate function. $b_{i}$ is the bias. Here, the structure of $\textbf{x}$ is reconstructed according to the filters. Consequently the convolution can be presented with multiplication.
Using standard dropout, the outputs will be multiplied by a Bernoulli mask: $$\begin{aligned}
\textbf{y}^{dropout}_{i} &= \textbf{r}_{i}*\textbf{y}_{i} / p \\
\textbf{r}_{i} & \sim Bernoulli(p) \nonumber\\
p & \in [0,1] \nonumber\\
Y_{dropout} &= \{ \textbf{y}^{dropout}_{i} | 1\leq i \leq c\nonumber \}\label{Dropout}\end{aligned}$$ where $p$ is the data retaining rate. The shape of $\textbf{r}_{i}$ is same as $\textbf{y}_{i}$. To compensate the variance shift of dropout, the outputs are scaled by $1/p$ during training [@abadi2016tensorflow]. An alternative choice is removing $1/p$ during training and scaling the outputs by $p$ during testing [@srivastava2014dropout].
Using Filter drop, the outputs: $$\begin{aligned}
\textbf{y}^{DropFilter}_{i} & = r_{i}\textbf{y}_{i}/p \\
r_{i} & \sim Bernoulli(p) \nonumber\\
p & \in [0,1] \nonumber\\
Y_{DropFilter} &= \{ \textbf{y}^{DropFilter}_{i} | 1\leq i \leq c \}\nonumber\label{FilterDrop2}\end{aligned}$$ where $r_{i}$ is a weight to determinate whether to drop the feature map $\textbf{y}_{i}$. Like standard dropout, DropFilter is also scaled by $1/p$ during training and is removed during testing.
Dropout aims to avoid the co-adaptations between units. It randomly suppresses the neural units in a convolutional layer equally. DropFilter focuses on the co-adaptations between filters. It randomly drops whole feature maps to avoid the co-adaptations between filters. Unlike fully connected layers, convolutional layers have more complex structure. The co-adaptations is also different from that in fully connected layers. We believe that this is the main reason why dropout does not work well for convolutional layers. According to structure of convolutions, DropFilter is designed to reduce the co-adaptations in covolutional layers. DropFilter regards the filter rather than the neural unit as the basic co-adaptive unit. This is the main difference between dropout and DropFilter.
\[fig:diffpq\]
\[fig:path\_blocks\]
DropPath and DropFilter
-----------------------
DropPath is an optional regularization method only for multi-path CNNs. Multi-path networks with DropPath have achieved remarkable results on CIFAR and ImageNet [@zoph2017learning; @gastaldi2017shake]. DropPath ramdomly suppresses some paths in multi-path networks. Each path is a set of feature maps. DropPath can be seen as reducing the co-adaptations between paths. DropFilter also drops feature maps. However, DropFilter can be used for all convolutional neural network architecture. In DropFilter, the basic drop unit is the filter. In DropPath, the basic drop unit is the path.
DropFilter is similar to DropPath to same extent. We can understand DropFilter through DropPath. As shown in Fig. \[fig:path\_blocks:a\] and Fig. \[fig:path\_blocks:b\], each convolutional filter can be considered as a path for the convolutional layer. Afterwards , the outputs of every filter are concatenated as the output of this multi-path structure. When applying DropFilter to a convolutional layer, it is equal to randomly drop some tiny paths for the multi-path structure in Fig. \[fig:path\_blocks:b\], which is very similar with DropPath in multi-path networks.
ScaleFilter
-----------
DropFilter directly suppresses the outputs of a filter. For some very deep convolutional networks, if DropFilter is applied to every convolutional layer, the network may be very sensitive to the retaining rate $p$. Therefore we propose ScaleFilter, which randomly scales the outputs of the filters rather than directly suppresses them. Using ScaleFilter: $$\begin{aligned}
\textbf{y}^{ScaleFilter}_{i} & = r_{i}\textbf{y}_{i} \\
r_{i} & \sim Uniform(1-q,1+q) \nonumber\\
q & \in [0,1]\nonumber\\
Y_{ScaleFilter} &= \{ \textbf{y}^{ScaleFilter}_{i} | 1\leq i \leq c \}\nonumber\label{FilterDrop3}\end{aligned}$$ The outputs do not need to be scaled, because the mean of $r_{i}$ is $1$. During testing, ScaleFilter is removed.
Both DropFilter and ScaleFilter focus on changing the outputs to reduce the co-adaptations between filters. DropFilter directly suppresses the outputs. ScaleFilter randomly scales the outputs instead. ScaleFilter is weaker but more stable than DropFilter. It changes the output data modestly. For some very deep CNNs, the super parameter $q$ for ScaleFilter is easier to tune than $p$ for DropFilter. ScaleFilter sometime can achieve better results.
Experiments
===========
Datasets and Settings
---------------------
### CIFAR-10 and CIFAR-100
CIFAR-10 and CIFAR-100 are colored $32\times32$ natural image datasets that consist of 50,000 images in training set and 10,000 images in testing set. Following the common practice [@he2016deep; @zagoruyko2016wide], these tiny images are padded with four pixels on each side and are randomly cropped with a $32\times32$ window afterwards. Before being fed into the networks, the images are subtracted by channel means and divided by $128$.
### Subset of ImageNet
DropFilter is further tested on ILSVRC2012 datasets which is a subset of ImageNet [@russakovsky2015imagenet] database. These dataset contains 1.3M image on training set and 50K images on validation set. ILSVRC2012 is a quite large dataset and requires many resources to run very deep networks upon it. Like miniImageNet [@vinyals2016matching], we randomly select 100 classes in ILSVRC2012 and randomly select 600 images in each class. Unlike miniImageNet, the images are not resized. During training the images and their horizontal flips are cropped by $224\times224$ window after the short edge being resized to $256$ following [@simonyan2014very]. The trained models are test on the validation set in which the images that do not belong to the selected 100 classes are removed.
Implementation Details
----------------------
The methods in this paper is test on ResNets [@he2016identity], WRNs [@zagoruyko2016wide], ResNeXt [@xie2016aggregated] and plain networks. We use the template in [@he2016deep] to construct the ResNets on CIFAR. The ResNets have three stages. Each stage consists of $n$ residual blocks. Each block contains two $3\times3$ layers. The filter numbers in three stage are $\{16,32,64\}$ respectively. We use different $n$ to construct the networks with different depths. The ResNets used in this paper is the pre-activation ResNets, if not specified.
Like [@zagoruyko2016wide], WRNs are constructed by widening the ResNets in this paper. WRN-4-20 represents the network with four times more filters in each layer than ResNet-20.
If not specified, the networks for CIFAR are trained for 200 epochs with 0.1 initial learning rate. The learning rate drops by 0.2 at 60,120 and 160 epochs following [@zagoruyko2016wide]. The batch size is 128 and the weight decay is 0.0005. The data drop method is applied to every convolutional layer for ResNets and WRNs. For ResNeXt, data drop is used before and after the $3\times3$ group convolutions.
On ImageNet, the batch size is 256 and the weight decay is 0.0005. The model is trained for 110 epochs with 0.1 initial learning rate. The learning rate drops by 0.1 at 30, 60, 85, 95 and 105 epochs according to [@wu2016tensorpack]. Data drop regularization is used before and after the $3\times3$ convolutions in bottleneck residual blocks.
All the networks are implemented by tensorflow [@abadi2016tensorflow] and tensorpack [@wu2016tensorpack]. The codes will be released.
Network without data drop(%) standard dropout(%) DropFilter(%) ScaleFilter(%)
------------ ---------------------- --------------------- --------------- ----------------
ResNet-56 6.20 6.08 [6.07]{} [ 5.65]{}
ResNet-110 5.88 5.78 [5.42]{} [ 5.40]{}
WRN-4-20 5.04 4.84 [ 4.55]{} [4.64]{}
WRN-8-20 4.93 4.37 [4.21]{} [ 4.14]{}
Network without data drop(%) standard dropout(%) DropFilter(%) ScaleFilter(%)
------------ ---------------------- --------------------- --------------- ----------------
ResNet-56 28.07 27.74 [ 25.83]{} [ 25.58]{}
ResNet-110 26.16 25.85 [24.12]{} [24.12]{}
WRN-4-20 22.90 22.53 [21.24]{} [21.40]{}
WRN-8-20 20.99 20.78 [19.70]{} [19.50]{}
Different Data Retaining Rate
-----------------------------
The networks are regularized by DropFilter and ScaleFilter with different $p$ and $q$. Fig. \[fig:diffpq:a\] shows the results of standard dropout, DropFilter and ScaleFilter on CIFAR-10 with ResNet-110. The retaining rate 1 means that no data drop method is used. Standard dropout can slightly improve the networks with 0.9 retaining rate. DropFilter consistently performs better than standard dropout and significantly reduces the test error with the retaining rate of 0.9.
However, the network is very sensitive to the retaining rate of standard dropout and DropFilter, because data drop methods are applied to every convolutional layer in ResNet-110. This problem can be solved by ScaleFilter. As can be seen in Fig. \[fig:diffpq:a\], ScaleFilter supplies modest and stable regularization.
The results on CIFAR-100 are shown in Fig. \[fig:diffpq:b\]. As can be seen, DropFilter and ScaleFilter outperform standard dropout, which is similar with the results on CIFAR-10. On CIFAR-100, the improvement of DropFilter and ScaleFilter is more convincing according to error bars of five runs.
Network without data drop(%) standard dropout(%) DropFilter(%) ScaleFilter(%)
------------ ---------------------- --------------------- --------------- ----------------
plain-4-8 7.56 6.86 [6.51]{} [6.60]{}
plain-8-14 5.47 5.46 [5.41]{} [5.36]{}
Network without data drop(%) standard dropout(%) DropFilter(%) ScaleFilter(%)
------------ ---------------------- --------------------- --------------- ----------------
plain-4-8 28.70 27.83 [26.12]{} [26.54]{}
plain-8-14 24.91 24.58 [24.38]{} [24.21]{}
Results on ResNets and WRNs
---------------------------
In this section, DropFilter is tested on ResNets and WRNs on CIFAR-10 and CIFAR-100. We apply data drop to every convolutional layer. The networks are tested using different retaining rate and the best retaining rate results are shown in Table \[resnet10\] and Table \[resnet100\].
As shown in Table \[resnet10\], DropFilter and ScaleFilter consistently outperform standard dropout for networks with different depths and widths. On CIFAR-10, WRN-4-20 with DropFilter outperforms WRN-8-20 without data drop. Note that WRN-8-20 use about four times more parameters than WRN-4-20. When DropFilter and ScaleFilter are used for WRN-8-20, its performance is further improved. For ResNet-56, DropFilter get a poor result, because it is too coarse for this network. ScaleFilter achieves a better result with a relatively low $q$ (0.6).
On CIFAR-100, the improvement of DropFilter and ScaleFilter is more remarkable. As shown in Table \[resnet100\], standard drop can only provide modest improvement. DropFilter and ScaleFilter reduce the test error by more than 1% for all the tested networks.
On CIFAR-100, a little bigger retaining rate can be used. WRN-8-20 uses $q=0.4$ for ScaleFilter on CIFAR-10. On CIFAR-100, $q=0.2$ is the best for this network. For ResNet-56, $p=0.9$ is a little too big for DropFilter on CIFAR-10. But on CIFAR-100, ResNet-56 performs well with the same retaining rate.
Results on Plain Networks
-------------------------
ResNets and WRNs are networks with residual connections. To demonstrate that DropFilter and ScaleFilter are useful for general convolutional networks, these two data drop methods are tested on plain networks. The plain networks are constructed by the same template as WRNs except that the residual connections are removed. Plain-14-4 represents the 14 layer network with four times more filters in each layer than ResNet template on CIFAR. Very deep plain networks suffer from the problem of vanishing/exploding gradients. Data drop methods are useless and only increase the test error for these networks. Therefore we test our methods on relatively shallow and wide plain nets(plain-8-4, plain-14-4).
As shown in Table \[plain10\] and Table \[plain100\], for plain-8-4, DropFilter and ScaleFilter still perform better than standard dropout. Compared to plain-8-4, the margins for plain-14-4 are lower. We guess that data drop methods will deteriorate the problem of gradients. The gradients problem of deeper plain nets will be enlarged when using data drop method. Thus the margins of using data drop methods are very low for plain-14-4. That also helps to explain why data drop methods are useless for very deep plain networks.
DropFilter with Other Methods
-----------------------------
In this section, more experiments are conducted demonstrate that DropFilter does not conflict with new residual networks achitecture and other training methods. The methods used in the experiments are followings:
- **New Residual Architecture**. ResNeXt is a new residual network architecture that using group convolutions for bottleneck residual blocks [@xie2016aggregated]. It introduces cardinality to residual networks and achieves better results than ResNets upon many visual recognition tasks. ResNeXt-29(16x64d) is used to test our data drop method in this paper. ResNeXt-29(16x64) uses group convolutions for the $3\times3$ convolutions. The group number is 16 and each group consists of 64 filters.
- **Cosine Annealing for Learning Rate**. Cosine annealing for learning rate is introduced by [@loshchilov2016sgdr]. Instead of dropping the learning rate with a factor after each training stage, they drop the learning rate more smoothly by cosine annealing. Afterwards, this method is used in many researches [@gastaldi2017shake; @zoph2017learning].
- **Curriculum Dropout**. Curriculum dropout is introduced by [@Morerio2017Curriculum]. In their opinions, the co-adaptations will not occur in the beginning of training, because the networks are initialized randomly. Thus they linearly decrease the data retaining rate during training. We linearly decrease the data retaining rate of standard dropout and DropFilter as well. The retaining rate is decreased from 1.0 to 0.6.
- **More Training Epochs**. Training the network for more epochs usually will improve the network performance on CIFAR-10[@loshchilov2016sgdr; @gastaldi2017shake; @zoph2017learning]. The training epochs are increased from 200 to 600 in our experiments.
The results on CIFAR-10 is shown in Table \[resnext\]. As can be seen, because of introducing too much unnecessary noise, standard dropout increases the test error. DropFilter focuses on reducing the co-adaptations between filters. It does not introducing too much useless noise and can still improve the networks.
The results of competitive methods on CIFAR-10 is shown in Table \[state-of-the-art\]. As can be seen, improving the results on CIFAR-10 is much difficult. ResNeXt-29 (16x64d) only outperforms ResNeXt-29 (8x64d) by 0.7%. DropFilter further improves the result nearly without introducing extra computation. Only Shake-Shake-26(2x96d) outperforms our model. But Shake-Shake-26(2x96d) needs to be trained for three times more epochs(1800 epochs).
method test error(%)
------------------- ---------------
without data drop 3.42
standard dropout 3.61
DropFilter 3.25
: The test error of ResNeXt-29(16x64d) with data drop regularization on CIFAR-10.[]{data-label="resnext"}
Method parameters(M) CIFAR-10(%)
-------------------------------------------------- --------------- -------------
original-ResNet [@he2016deep] 10.2 7.93
stoc-depth-110 [@huang2016deep] 1.7 5.23
stoc-depth-1202 10.2 4.91
FractalNet [@larsson2016fractalnet] 38.6 5.22
with Dropout/Drop-path 38.6 4.60
pre-ResNet [@he2016identity] 10.2 4.62
WRN-16-8 [@zagoruyko2016wide] 36.5 4.00
WRN (Dropout) 36.5 3.80
ResNeXt-29 (8x64d) [@xie2016aggregated] 34.4 3.65
ResNeXt-29 (16x64d) 68.1 3.58
DenseNet(L = 100,k = 24) [@huang2016densely] 27.2 3.74
DenseNet-BC (L = 100,k = 40) [@huang2017densely] 25.6 3.46
NASNet-A 3.3 3.41
ResNeXt-29 (16x64d) DropFilter(ours) 68.1 [3.25]{}
Shake-Shake-26 (2x96d) 26.2 [2.86]{}
Results on the Subset of ImageNet
---------------------------------
The methods in this paper are tested on the Subset of ImageNet. The tested network is ResNet-152. Data drop methods are used before and after the $3\times3$ convolutional layers in residual blocks. For SclaeFilter, $q=0.4$ is used. The data retaining rate for DropFilter and standard dropout is 0.9.
As shown in Table \[imagenet\], standard dropout reduces the test error by about 0.3%. DropFilter reduce the test error by 0.8%. ScaleFilter performs best and improve the performance by about 1.1%. DropFilter and ScaleFilter can not only work well on tiny image datasets, but also remarkably improve the results of CNNs on bigger image dataset.
method top 5 test error(%)
------------------- ---------------------
without data drop 11.43
standard dropout 11.12
DropFilter 10.63
ScaleFilter 10.34
: The top 5 test error(%) of ResNet152 on the subset of ImageNet. The retaining rate for standard dropout and DropFilter is 0.9. ScaleFilter uses $q =0.4$ for this network.[]{data-label="imagenet"}
Discussions
===========
The Data Retaining Rate
-----------------------
For data drop regularization methods, the retaining rate is very important. According to our experiments, deeper and wider networks need smaller data retaining rate. If the networks are trained with curriculum in [@Morerio2017Curriculum], smaller data retaining rate can be used. For DropFilter, if 0.9 or 0.95 is still too small for the networks, we suggest using ScaleFilter rather than using DropFilter with bigger retaining rate like 0.98. DropFilter randomly drops the feature maps with the retaining rate. At the first stage of ResNet-56, there are only 16 feature maps which is too few to reveal the difference between 0.95 and 0.98 for DropFilter.
ScaleFilter can be seen as the weakened version of DropFilter. But they are a little different sometimes. Scaling the data is different from dropping the data after all. That is why DropFilter outperform ScaleFilter sometimes.
Network Structure and Data Drop Regularization
----------------------------------------------
We propose that co-adaptations are more likely to occur between units that are equivalent in position. According to this, we present DropFilter and ScaleFilter to regularize convolutions. Note that this suggestion can be used for regularizing different structure networks and constructing new networks. If there is a network that should be regularized with data drop methods, we could construct new data drop method according to the co-adaptations. When we construct new network structure, we should take care of the co-adaptations that will occur. For example, when we construct multi-path networks, the networks may perform better if the paths are different from each other. That is a perspective to explain why inception [@szegedy2015going; @szegedy2016rethinking; @szegedy2016inception] and NASNets [@zoph2017learning] perform well.
Conclusions
===========
In this paper, we propose DropFilter which is a new regularization method for convolutions. Unlike standard dropout, DropFilter only aims to reduce the co-adaptations inter filters. ScaleFilter randomly scales the outputs of filters. CNNs can be regularized by ScaleFilter when DropFilter is too coarse for the networks. Using the methods in this paper, we significantly improve the performance of different architecture CNNs on CIFAR and the subset of ImageNet. In addition, DropFilter nearly does not introduce extra computation like standard dropout.
Multi-path networks have achieved state-of-the-art results [@gastaldi2017shake; @zoph2017learning] on CIFAR and ImageNet with DropPath, which demonstrates that data drop regularization methods are promising for improving deep neural networks. We bridge the gaps not only between multi-path networks and single path networks but also between standard dropout and DropPath. We hope that more concerns could be paid to data drop regularization methods. We believe that this is a promising perspective to understand and improve deep neural networks.
|
---
abstract: 'Continuing the program of [@DS] and [@Usher], we introduce refinements of the Donaldson-Smith standard surface count which are designed to count nodal pseudoholomorphic curves and curves with a prescribed decomposition into reducible components. In cases where a corresponding analogue of the Gromov-Taubes invariant is easy to define, our invariants agree with those analogues. We also prove a vanishing result for some of the invariants that count nodal curves.'
author:
- Michael Usher
title: 'Standard surfaces and nodal curves in symplectic 4-manifolds'
---
Introduction
============
Let $(X,\omega)$ be a closed symplectic 4-manifold. We assume that $[\omega]\in H^2(X,\mathbb{Z})$; however, the main theorems in this paper concern Gromov invariants, which are unchanged under deformations of the symplectic form, so since any symplectic form is deformation equivalent to an integral form there is no real loss of generality here. According to [@Don], if $k$ is large enough, taking a suitable pair of sections of a line bundle $L^{\otimes k}$ where $L$ has Chern class $[\omega]$ and blowing $X$ up at the common vanishing locus of these sections to obtain the new manifold $X'$ gives rise to a symplectic Lefschetz fibration $f{\colon\thinspace}X'\to \mathbb{C}P^1$ (the exceptional curves of the blowup $\pi{\colon\thinspace}X'\to X$ appear as sections of $f$, while at other points $x'\in X'$, $f(x')\in\mathbb{C}\cup\{\infty\}$ is the ratio of the two chosen sections of $L^{\otimes k}$ at $\pi
(x')\in X$). In other words, $f$ is a fibration by Riemann surfaces over the complement of a finite set of critical values in $S^2$, while near its critical points $f$ is given in smooth local complex coordinates by $f(z,w)=zw$. Results of [@Smith2] show that the critical points of $f$ may be assumed to lie in separate fibers, and all fibers of $f$ may be assumed irreducible. Once we choose a metric on $X'$, Donaldson’s construction thus presents a suitable blowup of $X$ as a smoothly $\mathbb{C}P^1$-parametrized family of Riemann surfaces, all but finitely of which are smooth and all of which are irreducible with at worst one ordinary double point. Where $\kappa_X=c_1 (T^* X)$ is the canonical class of $X$, note that the adjunction formula gives the arithmetic genus of the fibers as $g=1+(k^2[\omega]^2+k\kappa_X\cdot\omega)/2$.
Beginning with the work of S. Donaldson and I. Smith in [@DS], some efforts have recently been made toward determining whether such a Lefschetz fibration can shed light on any questions concerning pseudoholomorphic curves in $X$. More specifically, for any natural number $r$ Donaldson and Smith construct the *relative Hilbert scheme*, which is a smooth symplectic manifold $X_r (f)$ with a map $F{\colon\thinspace}X_r (f)\to \mathbb{C}P^1$ whose fiber over a regular value $t$ of $f$ is the symmetric product $S^r f^{-1}(t)$. If we choose an almost complex structure $j$ on $X'$ with respect to which $f$ is a pseudoholomorphic map, a $j$-holomorphic curve $C$ in $X'$ which contains no fiber components will, by the positivity of intersections between $j$-holomorphic curves, meet each fiber in $r:=[C]\cdot [fiber]$ points, counted with multiplicities. In other words, $C\cap
f^{-1}(t)\in S^{r}f^{-1}(t)$, so that, letting $t$ vary, $C$ gives rise to a section $s_C$ of $X_r (f)$. Conversely, a section $s$ of $X_{r}(f)$ gives rise to a subset $C_s$ of $X'$ (namely the union of all the points appearing in the divisors $s(t)$ as $t$ varies), and from $j$ one may construct a (nongeneric and generally not even $C^1$) almost complex structure $\mathbb{J}_j$ with the property that $C$ is a (possibly disconnected) $j$-holomorphic curve in $X'$ if and only if $s_C$ is a $\mathbb{J}_j$-holomorphic section of $X_{r}(f)$.
Accordingly, it seems reasonable to study pseudoholomorphic curves in $X'$ by studying pseudoholomorphic sections of $X_{r}(f)$. If $\alpha\in H^2 (X';\mathbb{Z})$, the *standard surface count* $\mathcal{DS}_f (\alpha)$ is defined in [@Smith] (and earlier in [@DS] for $\alpha=\kappa_{X'}$) as the Gromov-Witten invariant which counts $J$-holomorphic sections $s$ whose corresponding sets $C_s$ are Poincaré dual to the class $\alpha$ and pass through a generic set of $d(\alpha)=\frac{1}{2}(\alpha^2-\kappa_{X'}\cdot \alpha)$ points of $X'$, where $J$ is a generic almost complex structure on $X_r
(f)$. Smith shows in [@Smith] that there is at most one homotopy class $c_{\alpha}$ of sections $s$ such that $C_s$ is Poincaré dual to $\alpha$, and moreover that the complex dimension of the space of $J$-holomorphic sections in this homotopy class is, for generic $J$, the aforementioned $d(\alpha)$, which the reader may recognize as the same as the expected dimension of $j$-holomorphic submanifolds of $X$ Poincaré dual to $\alpha$. Further, the moduli space of $J$-holomorphic sections in the homotopy class $c_{\alpha}$ is compact for generic $J$ if $k$ is taken large enough. The moduli space in the definition of $\mathcal{DS}_f$ is therefore a finite set, and $\mathcal{DS}_f$ simply counts the members of this set with sign according to the usual (spectral-flow-based) prescription.
Donaldson and Smith have proven various results about $\mathcal{DS}$, perhaps the most notable of which is the main theorem of [@Smith], which asserts that if $\alpha\in
H^{2}(X;\mathbb{Z})$, if $b^{+}(X)>b_1 (X)+1$, and if the degree $k$ of the Lefschetz fibration is high enough, then $$\label{SD} \mathcal{DS}_f (\pi^* \alpha)=\pm \mathcal{DS}_f (\pi^*
(\kappa_X - \alpha)).$$ Their work has led to new, more symplectic proofs of various results in 4-dimensional symplectic topology which had previously been accessible only by Seiberg-Witten theory (as an example we mention the main theorem of [@DS], according to which $X$ admits a symplectic surface Poincaré dual to $\kappa_X$, again assuming $b^{+}(X)>b_1
(X)+1$). In [@Usher] it was shown that the invariant $\mathcal{DS}_f$ agrees with the Gromov invariant $Gr$ which was introduced by C. Taubes in [@Taubes] and which counts possibly-disconnected pseudoholomorphic submanifolds of $X'$ Poincaré dual to a given cohomology class. This in particular shows that $\mathcal{DS}_f$ is independent of the choice of Lefschetz fibration structure, and, in combination with Smith’s duality theorem (\[SD\]) and the fact that under a blowup $\pi$ one has $Gr(\pi^* \alpha)=Gr(\alpha)$, yields a new proof of the relation $$Gr(\alpha)=\pm Gr(\kappa_X-\alpha)$$ if $b^{+}(X)>b_1 (X)+1$, a result which had previously only been known as a shadow of the charge conjugation symmetry in Seiberg-Witten theory.
The information contained in the Gromov invariants comprises only a part of the data that might be extracted from pseudoholomorphic curves in $X$. The present paper aims to show that many of these additional data can also be captured by Donaldson-Smith-type invariants. For instance, $Gr(\alpha)$ counts all of the curves with any decomposition into connected components whose homology classes add up (counted with multiplicities) to $\alpha$. It is natural to wish to keep track of the decompositions of our curves into reducible components; accordingly we make the following:
\[invts\] Let $\alpha\in H^{2}(X;\mathbb{Z})$. Let $$\alpha=\beta_1+\cdots +\beta_m+c_1 \tau_1+\cdots +c_n \tau_n$$ be a decomposition of $\alpha$ into distinct summands, where none of the $\beta_i$ satisfies $\beta_{i}^{2}=\kappa_X \cdot\beta_i =0$, while the $\tau_i$ are distinct classes which are primitive in the lattice $H^{2}(X;\mathbb{Z})$ and all satisfy $\tau_{i}^2=\kappa_X
\cdot \tau_i =0$. Then $$Gr(\alpha;\beta_1,\ldots,\beta_m,c_1\tau_1,\cdots,c_n\tau_n)$$ is the invariant counting ordered $(m+n)$-tuples $(C_1,\ldots,C_{m+n})$ of transversely intersecting smooth pseudoholomorphic curves in $X$, where
- for $1\leq i\leq m$, $C_i$ is a connected curve Poincaré dual to $\beta_i$ which passes through some prescribed generic set of $d(\beta_i)$ points;
- for $m+1\leq k\leq m+n$, $C_k$ is a union of connected curves Poincaré dual to classes $l_{k,1}\tau_k,\cdots,l_{k,p}\tau_k$ decorated with positive integer multiplicities $m_{k,q}$ with the property that $$\sum_q
m_{k,q}l_{k,q}=c_k.$$
The weight of each component of each such curve is to be determined according to the prescription given in the definition of the Gromov invariant in [@Taubes] (in particular, the components $C_{k,q}$ in class $l_{k,q}\tau_k$ are given the weight $r(C_{k,q},m_{k,q})$ specified in Section 3 of [@Taubes]), and the contribution of the entire curve is the product of the weights of its components.
The objects counted by $Gr(\alpha;\alpha_1,\ldots,\alpha_n)$ will then be reducible curves with smooth irreducible components and a total of $\sum \alpha_i\cdot \alpha_j$ nodes arising from intersections between these components. $Gr(\alpha)$ is the sum over all decompositions of $\alpha$ into classes which are pairwise orthogonal under the cup product of the $$\frac{d(\alpha)!}{\prod (d(\alpha_i)!)}Gr(\alpha;\alpha_1,\ldots,\alpha_n);$$ in turn, one has $$Gr(\alpha;\alpha_1,\ldots,\alpha_n)=\prod_{i=1}^{n}Gr(\alpha_i;\alpha_i).$$
In Section \[red\], given a symplectic Lefschetz fibration $f{\colon\thinspace}X\to S^2$ with sufficiently large fibers, by counting sections of a relative Hilbert scheme we construct a corresponding invariant $\widetilde{\mathcal{DS}}_f(\alpha;\alpha_1,\ldots,\alpha_n)$ provided that none of the $\alpha_i$ can be written as $m\beta$ where $m>1$ and $\beta$ is Poincaré dual to either a symplectic square-zero torus or a symplectic $(-1)$-sphere. Further:
$\frac{(\sum
d(\alpha_i))!}{\prod
(d(\alpha_i)!)}Gr(\alpha;\alpha_1,\ldots,\alpha_n)=\widetilde{\mathcal{DS}_f}(\alpha;\alpha_1,\ldots,\alpha_n)$ provided that the degree of the fibration is large enough that $\langle [\omega_{X}],fiber\rangle>[\omega_{X}]\cdot
\alpha$.
The sections $s$ counted by $\widetilde{\mathcal{DS}}_f(\alpha;\alpha_1,\ldots,\alpha_n)$ correspond tautologically to curves $C_s=\cup C^{i}_{s}$ in $X$ with each $C^{i}_{s}$ Poincaré dual to $\alpha_i$. The $C_{s}^{i}$ will be symplectic, and Proposition \[posints\] guarantees that they will intersect each other positively, so there will exist an almost complex structure making $C_s$ holomorphic. However, if $s_1$ and $s_2$ are two different sections in the moduli space enumerated by $\widetilde{\mathcal{DS}}_f(\alpha;\alpha_1,\ldots,\alpha_n)$, it is unclear whether there will exist a single almost complex structure on $X$ making both $C_{s_1}$ and $C_{s_2}$ holomorphic.
The almost complex structures on $X_r(f)$ used in the definition of $\widetilde{\mathcal{DS}}$ are, quite crucially, required to preserve the tangent space to the *diagonal stratum* consisting of divisors with one or more points repeated. One might hope to define analogous invariants which agree with $Gr(\alpha;\alpha_1,\ldots,\alpha_n)$ using arbitrary almost complex structures on $X_r(f)$. If one could do this, though, the arguments reviewed in Section \[review\] would rather quickly enable one to conclude that $Gr(\alpha;\alpha_1,\ldots,\alpha_n)=0$ whenever $\alpha$ has larger pairing with the symplectic form than does the canonical class and $\alpha_i\cdot\alpha_j=0$ for $i\neq j$. However, this is not the case: the manifold considered in [@MT] admits a symplectic form such that, for certain primitive, orthogonal, square-zero classes $\alpha$, $\beta$, $\gamma$, and $\delta$ each with positive symplectic area, the canonical class is $2(\alpha+\beta+\gamma)$ but the invariant $Gr(2(\alpha+\beta+\gamma)+\delta;\alpha,\beta,\gamma,\alpha+\beta+\gamma+\delta)$ is nonzero.
While the Gromov–Taubes invariant restricts attention to curves whose components are all covers of embedded curves which do not intersect each other, it is natural to hope for information about curves Poincaré dual to $\alpha$ having some number $n$ of transverse self-intersections. One might like to define an analogue $Gr_n(\alpha)$ of the Gromov–Taubes invariant counting such curves, but as we review in Section \[secfam\], owing to issues relating to multiple covers it is somewhat unclear what the definition of such an invariant should be in general. If one imposes some rather stringent conditions on $\alpha$ ($\alpha$ should be “$n$-semisimple” in the sense of Definition \[sem\]), there is however a natural such choice.
Note that for arbitrary $\alpha$ and $n$, following [@RT] one may define an invariant $RT_{n}(\alpha)$ which might naively be viewed as a count of *connected* pseudoholomorphic curves Poincaré dual to $\alpha$ with $n$ self-intersections by enumerating solutions $u{\colon\thinspace}\Sigma_g\to X$ of the equation $(\operatorname{\bar{\partial}}_ju)=\nu (x,u(x))$ for generic $j$ and “inhomogeneous term” $\nu$, where the genus $g$ of the source curve is given in accordance with the adjunction formula by $2g-2=\alpha^2+\kappa_X\cdot\alpha-2n$. (Note that the nontrivial dependence of $\nu$ on $x$ prevents multiple cover problems from arising.) In the case $n=0$, the main theorem of [@IP2] provides a universal formula equating $Gr(\alpha)$ with a certain combination of the Ruan–Tian invariants $RT_0$. The proof of that theorem goes through easily to show that in the case when $\alpha$ is $n$-semisimple, there exists a similar formula equating $Gr_n(\alpha)$ with a combination of Ruan–Tian invariants. We mention also that, again as an artifact of the multiple cover problem, the Ruan–Tian invariants are obliged to take values in $\mathbb{Q}$ rather than $\mathbb{Z}$. $Gr(\alpha)$, on the other hand, is an integer-valued invariant.
By combining the approaches of [@DS] and [@Liu], in the presence of a Lefschetz fibration $f{\colon\thinspace}X\to S^2$ we construct in Section \[secfam\] an integer-valued invariant $\mathcal{FDS}_{f}^{n}(\alpha-2\sum e_i)$ which we conjecture to be an appropriate candidate for a “nodal version” $Gr_n(\alpha)$ of the Gromov invariant for general classes $\alpha$ (after perhaps dividing by $n!$ to account for a symmetry in the construction). Pleasingly, the technical difficulties that often arise in defining invariants like $Gr_n (\alpha)$ do not affect $\mathcal{FDS}$: since $\mathcal{FDS}$ counts sections of a (singular) fibration, which of course necessarily represent a primitive homology class in the total space, we need not worry about multiple covers; further, the fact that any bubbles that form in the limit of a sequence of holomorphic sections must be contained in the fibers of the fibration turns out (via an easy elaboration of a dimension computation from [@DS]) to generically rule out bubbling as well. In principle, though, $\mathcal{FDS}^{n}_{f}$ might depend on the choice of Lefschetz fibration $f$.
Note that if $\pi{\colon\thinspace}X'\to X$ is a blowup with exceptional divisor Poincaré dual to $\epsilon$, whenever $Gr_n(\beta)$ is defined we will have $(Gr_n)_{X'}(\beta +\epsilon)=(Gr_n)_X(\beta)$ (here and elsewhere we use the same notation for $\beta\in
H^2(X;\mathbb{Z})$ and $\pi^{*}\beta\in H^2(X';\mathbb{Z})$), as the curves contributing to $(Gr_n)_X(\beta)$ generically miss the point being blown up, and so the unions of their proper transforms with the exceptional divisor will be precisely the curves contributing to $(Gr_n)_{X'}(\beta +\epsilon)$. With this said, we formulate the:
\[conj\] Let $(X,\omega)$ be a symplectic 4-manifold and $\alpha\in H^2(X;\mathbb{Z})$, and $f{\colon\thinspace}X'\to S^2$ a Lefschetz fibration obtained from a sufficiently high-degree Lefschetz pencil on $X$, with the exceptional divisors of the blowup $X'\to X$ Poincaré dual to the classes $\epsilon_1,\ldots,\epsilon_N$. Then the family Donaldson–Smith invariants $$\mathcal{FDS}^{n}_{f}\left(\alpha+\sum_{i=1}^{N}\epsilon_i-2\sum_{k=1}^{n}e_k\right)$$ are independent of the choice of $f$, and have a general expression in terms of the Ruan–Tian invariants of $X$.
Note that this conjectural general expression would then produce an integer by taking appropriate combinations of the (*a priori* only rational) Ruan–Tian invariants, similarly to the formula of [@IP2]
In light of the behavior of $Gr_n$ under blowups, Theorem \[famsame\] amounts to the statement that:
If $\alpha$ is strongly $n$-semisimple, then Conjecture \[conj\] holds for $\alpha$; more specifically, we have $$\mathcal{FDS}^{n}_{f}\left(\alpha+\sum_{i=1}^{N}\epsilon_i-2\sum_{k=1}^{n}e_k\right)=n!Gr_n(\alpha)$$ if $f$ has sufficiently high degree.
We also prove that $\mathcal{FDS}$ vanishes under certain circumstances. This result depends heavily on the constructions used by Smith in [@Smith] to prove his duality theorem, and so we review these constructions in Section \[review\]. Section \[vanproof\] is then devoted to a proof of the following theorem.
\[famvan\] If $b^+(X)>b_1(X)+4n+1$, then for all $\alpha \in
H^2(X;\mathbb{Z})$ such that $r=\langle \alpha,[\Phi]\rangle$ satisfies $r>\max\{g(\Phi)+3n+d(\alpha),(4g(\Phi)-11)/3\}$, either $\mathcal{FDS}^{n}_{f}(\alpha-2\sum e_i)=0$ or there exists an almost complex structure $j$ on $X$ compatible with the fibration $f{\colon\thinspace}X\to S^2$ which simultaneously admits holomorphic curves $C$ and $D$ Poincaré dual to the classes $\alpha$ and $\kappa_X-\alpha$. In particular, $\mathcal{FDS}^{n}_{f}(\alpha-2\sum e_i)=0$ if $\alpha$ has larger pairing with the symplectic form than does $\kappa_X$.
Note that in the Lefschetz fibrations obtained from degree-$k$ Lefschetz pencils on some fixed symplectic manifold $(X,\omega)$, the number $N$ of exceptional sections is $k^2[\omega]^2$ while the number $2g(\Phi)-2$ is asymptotic to $k^2[\omega]^2$, so the invariants $$\mathcal{FDS}^{n}_{f}\left(\alpha+\sum_{i=1}^{N}\epsilon_i-2\sum_k
e_k\right)$$ considered in Conjecture \[conj\] all eventually satisfy the restriction on $r$ in Theorem \[famvan\].
The almost complex structure in the second alternative in Theorem \[famvan\] cannot be taken to be regular (in the sense that the moduli spaces $\mathcal{M}^{j}_{X}(\beta)$ of $j$-holomorphic curves Poincaré dual to $\beta$ are of the expected dimension); the most we can say appears to be that it can be taken to be a member of a regular $4n$-real-dimensional family of almost complex structures, $i.e.$, a family of almost complex structures $\{j_b\}$ parametrized by elements $b$ of an open set in $\mathbb{R}^{4n}$ such that the spaces $\{(b,C)| C\in
\mathcal{M}^{j_b}_{X}(\beta)\}$ are of the expected real dimension $2d(\beta)+4n$ near each $(b,C)$ such that $C$ has no multiply-covered components. Also, if $X$ is in fact Kähler and admits a compatible *integrable* complex structure $j_0$ with respect to which the fibration $f$ is holomorphic, then we can take the $j$ in Theorem \[famvan\] equal to $j_0$.
In fact, if we could take $j$ to be regular, then we could rule out the second alternative in Theorem \[famvan\] entirely (when $n>0$) using the following argument: the invariant vanishes trivially when $d(\alpha)< n$, so we can assume\
$d(\alpha)=-\frac{1}{2}\alpha\cdot (\kappa-\alpha)>0$. But then our curves Poincaré dual to $\alpha$ and $\kappa-\alpha$ have negative intersection number, which is only possible if they share one or more components of negative square. For generic $j$, a virtual dimension computation shows that the only irreducible $j$-holomorphic curves of negative square are $(-1)$-spheres. Moreover whatever $(-1)$-spheres appear in $X$ must be disjoint, since if they were not, blowing one of two intersecting $(-1)$-spheres down would cause the image of the other to be a symplectic sphere of nonnegative self-intersection, which (by a result of [@McD]) would force $X$ to have $b^+=1$, which we assumed it did not. Ignoring all the $(-1)$-spheres in $C$ and $D$ and taking the union of what is left over would then give a $j$-holomorphic curve Poincaré dual to a class $\kappa_X-\sum a_i e_i$ where the $e_i$ are classes of $(-1)$-spheres with $e_i\cdot e_k=0$ for $i\neq k$ and where at least one $a_i\geq 2$. But one easily finds $d(\kappa_X-\sum a_i e_i)<0$, so this too is impossible for generic $j$. For nongeneric $j$, this argument breaks down because of the possibility that $C$ and $D$ might share components of negative square and negative expected dimension, and there is a wider diversity of possible homology classes of such curves.
The final section of the paper contains proofs of two technical results that are used in the proofs of the main theorems. First, we show that the operation of blowing up a point can be performed in the almost complex category, a fact which does not seem to appear in the literature and whose proof is perhaps more subtle than one might anticipate. The paper then closes with a proof of the following result, which is necessary for the compactness argument that we use to justify the definition of our invariant $\mathcal{FDS}$:
Let $F{\colon\thinspace}\mathcal{H}_r\to D^2$ denote the $r$-fold relative Hilbert scheme of the map $(z,w)\mapsto zw$, $\phi_0$ the partial resolution map $F^{-1}(0)\to Sym^r\{zw=0\}$, and $\Delta\subset \mathcal{H}_r$ the diagonal stratum. At any point $p\in \Delta\cap F^{-1}(0)$ with $\phi_0(p)=\{(0,0),\ldots,(0,0)\}$, where $T_p\Delta$ is the tangent cone to $\Delta$ at $p$, we have $T_p\Delta\subset T_p
F^{-1}(0)$.
We end the introduction with some remarks on the possible relation of $\mathcal{FDS}$ to (family) Seiberg–Witten theory. In [@Sa] it was shown that where $X$ is the product of $\mathbb{R}$ and a fibered three-manifold, so that $X$ fibers over a cylinder, if one examines the Seiberg–Witten equations on $X$ using a family of metrics for which the size of the fibers shrinks to zero, then one obtains in the adiabatic limit the equations for a holomorphic family of solutions to the symplectic vortex equations on the fibers. In turn, there is a natural isomorphism between the space of solutions to the vortex equations on a Riemann surface and the symmetric product of the surface. In other words, in this simple context the adiabatic limit of the Seiberg–Witten equations is the equation for a holomorphic family of elements of the symmetric products of the fibers of the fibration $X\to \mathbb{R}\times S^1$. As was noted in [@DS], since for a Lefschetz fibration $f{\colon\thinspace}X\to S^2$ $\mathcal{DS}_f$ precisely counts pseudoholomorphic families of elements of the symmetric products of the fibers of $f$, one might take inspiration from Salamon’s example and hope to obtain the equivalence between $\mathcal{DS}_f$ and the Seiberg–Witten invariant by considering the Seiberg–Witten equations on $X$ for a family of metrics with respect to which the size of the fibers shrinks to zero.
Now our invariant $\mathcal{FDS}^{n}_{f}$ is constructed by counting pseudoholomorphic families of elements of the symmetric products of the fibers of a family of Lefschetz fibrations $f^b$ obtained by restricting a map $f_n{\colon\thinspace}X_{n+1}\to S^2\times X_n$ to the preimage $X^b$ of $S^{2}\times\{b\}$ as $b$ ranges over the complement $X'_n$ of a set of codimension 4 in $X_n$. In the above vein, one might hope to relate the family Seiberg–Witten invariants $FSW$ for the family of $4$-manifolds $X_{n+1}\to X_n$ (which enumerate Seiberg–Witten monopoles in the various $X^b$ as $b$ ranges over $X_n$; see, *e.g.*, [@LL]) to $\mathcal{FDS}^{n}_{f}$ via an adiabatic limit argument. This would in particular yield a proof of the independence of $\mathcal{FDS}^{n}_{f}$ from $f$ in Conjecture \[conj\], and indeed may well be the most promising way to establish this independence in the absence of a suitable invariant $Gr_n$ (or of a “family Gromov–Taubes invariant” $FGr$) with which $\mathcal{FDS}^{n}_{f}$ might be directly equated.
As was shown in [@Liu2], when $X$ is an algebraic surface and $b^{+}(X)=1$ the family Seiberg–Witten invariants agree with certain curve counts in algebraic geometry. For larger values of $b^+$, though, the family Seiberg–Witten invariants that are hoped to correspond to nodal curve counts are expected to vanish due to the fact that symplectic manifolds have Seiberg–Witten simple type; note that Theorem \[famvan\] suggests that $\mathcal{FDS}^{n}_{f}$ also tends to vanish for large $b^+$. By contrast, there are plenty of nontrivial nodal curve counts in algebraic surfaces with $b^+>1$ (see [@Liu] for a review of some of these); these counts correspond to Liu’s “algebraic Seiberg–Witten invariants” $\mathcal{ASW}$ and differ from $FSW$ when $b^+>1$.
Acknowledgements {#acknowledgements .unnumbered}
----------------
Section \[red\] of this paper appeared in my thesis [@thesis]; I would like to thank my advisor Gang Tian for suggesting that I attempt to study nodal curves using the Donaldson–Smith approach. Thanks also to Cliff Taubes for helping me identify an error in an earlier version of this paper, to Dusa McDuff for making me aware of the need for Section \[app2\], and to Ivan Smith for helpful remarks. This work was partially supported by the National Science Foundation.
Refining the standard surface count {#red}
===================================
Throughout this section, $X_r(f)$ will denote the relative Hilbert scheme constructed from some high-degree but fixed Lefschetz fibration $f{\colon\thinspace}X\to S^2$ obtained by Donaldson’s construction applied to the fixed symplectic 4-manifold $(X,\omega)$. The fiber of $f$ over $t\in S^2$ will occasionally be denoted by $\Sigma_t$, and the homology class of the fiber by $[\Phi]$.
As has been mentioned earlier, $\mathcal{DS}_f (\alpha)$ is a count of holomorphic sections of the relative Hilbert scheme $X_r
(f)$ in a certain homotopy class $c_{\alpha}$ characterized by the property that if $s$ is a section in the class $c_{\alpha}$ then the closed set $C_s\subset X$ “swept out” by $s$ (that is, the union over all $t$ of the divisors $s(t)\in\Sigma_t$) is Poincaré dual to $\alpha$ (note that points of $C_s$ in this interpretation may have multiplicity greater than 1). That $c_{\alpha}$ is the unique homotopy class with this property is seen in Lemma 4.1 of [@Smith]; in particular, for instance, we note that sections which descend to *connected* standard surfaces Poincaré dual to $\alpha$ are not distinguished at the level of homotopy from those which descend to disjoint unions of several standard surfaces which combine to represent $PD(\alpha)$.
Of course, in studying standard surfaces it is natural to wish to know their connected component decompositions, so we will presently attempt to shed light on this. Suppose that we have a decomposition $$\alpha=\alpha_1 +\cdots +\alpha_n$$ with $$\langle \alpha,[\Phi]\rangle=r, \quad \langle
\alpha_i,[\Phi]\rangle =r_i.$$ Over each $t\in S^2$ we have an obvious “divisor addition map” $$\begin{aligned}
+{\colon\thinspace}\prod_{i=1}^{n}S^{r_i}\Sigma_t&\to S^r\Sigma_t \nonumber \\
(D_1,\ldots,D_n)&\mapsto D_1+\cdots+D_n; \nonumber \end{aligned}$$ allowing $t$ to vary we obtain from this a map on sections: $$\begin{aligned}
+{\colon\thinspace}\prod_{i=1}^{n}\Gamma(X_{r_i}(f))&\to \Gamma(X_r (f))\nonumber \\
(s_1,\ldots,s_n)&\mapsto \sum_{i=1}^n s_i. \nonumber \end{aligned}$$ As should be clear, one has $$+(c_{\alpha_1}\times\cdots\times
c_{\alpha_n})\subset c_\alpha$$ if $\alpha =\sum \alpha_i$, since $C_{\sum \alpha_i}$ is the union of the standard surfaces $C_{s_i}$ and hence is Poincaré dual to $\alpha$ if each $C_{s_i}$ is Poincaré dual to $\alpha_i$. Further, we readily observe:
\[closed\] The image $+(c_{\alpha_1}\times\cdots\times c_{\alpha_n})\subset c_{\alpha}$ is closed with respect to the $C^0$ norm.
Suppose we have a sequence $(s^{m}_{1},\ldots,s^{m}_{n})_{m=1}^{\infty}$ in $c_{\alpha_1}\times\cdots\times c_{\alpha_n}$ such that $\sum
s_{i}^{m}\to s\in c_{\alpha}$. Now each $S^{r_i}\Sigma_t$ is compact, so at each $t$, each of the sequences $s_{i}^{m}(t)$ must have subsequences converging to some $s_{i}^{0}(t)$. But then necessarily each $\sum s_{i}^{0}(t)=s(t)$, and then we can see by, for any $l$, fixing the subsequence used for all $i\neq l$ and varying that used for $i=l$ that in fact every subsequence of $s_{l}^{m}(t)$ must converge to $s_{l}^{0}(t)$. Letting $t$ vary then gives sections $s_{i}^{0}$ such that every $s_{i}^{m}\to
s_{i}^{0}$ and $\sum s_{i}^{0}=s$; the continuity of $s$ is readily seen to imply that of the $s_{i}^{0}$.
At this point it is useful to record an elementary fact about the linearization of the divisor addition map.
\[plus\] Let $\Sigma$ be a Riemann surface and $r=\sum r_i$. The linearization $+_*$ of the addition map $$+{\colon\thinspace}\prod_{i=1}^{n}S^{r_i}\Sigma\to S^r\Sigma$$ at $(D_1,\ldots ,D_n)$ is an isomorphism if and only if $D_i\cap D_j=\varnothing$ for $i\neq j$. If two or more of the $D_i$ have a point in common, then the image of $+_*$ at $(D_1,\ldots,D_n)$ is contained in $T_{\sum D_i}\Delta$, where $\Delta\subset S^r\Sigma$ is the diagonal stratum consisting of divisors with a repeated point.
By factoring $+$ as a composition $$S^{r_1}\Sigma\times
S^{r_2}\Sigma\times\cdots\times S^{r_n}\Sigma\to
S^{r_1+r_2}\Sigma\times\cdots\times S^{r_n}\Sigma\to\cdots\to
S^{r}\Sigma$$ in the obvious way we reduce to the case $n=2$. Now in general for a divisor $D=\sum a_i p_i \in S^d\Sigma$ where the $p_i$ are distinct, a chart for $S^d \Sigma$ is given by $\prod S^{a_i}U_i$, where the $U_i$ are holomorphic coordinate charts around $p_i$ and the $S^{a_i}U_i$ use as coordinates the elementary symmetric polynomials $\sigma_1,\ldots,\sigma_{a_i}$ in the coordinates of $U_{i}^{a_i}$. As such, if $D_1$ and $D_2$ are disjoint, a chart around $D_1 +D_2 \in S^{r_1 +r_2}\Sigma$ is simply the Cartesian product of charts around $D_1\in
S^{r_1}\Sigma$ and $D_2\in S^{r_2}\Sigma$, and the map $+$ takes the latter diffeomorphically (indeed, biholomorphically) onto the former, so that $(+_*)_{(D_1,D_2)}$ is an isomorphism.
On the other hand, note that $$+{\colon\thinspace}S^a\mathbb{C}\times
S^b\mathbb{C}\to S^{a+b}\mathbb{C}$$ is given in terms of the local elementary symmetric polynomial coordinates around the origin by $$(\sigma_1,\ldots,\sigma_a,\tau_1,\ldots,\tau_b)\mapsto
(\sigma_1+\tau_1,\sigma_2+\sigma_1\tau_1+\tau_2,\ldots,\sigma_a\tau_b),$$ and so has linearization $$(+_*)_{(\sigma_1,\ldots,\tau_b)}(\eta_1,\ldots,\eta_a,\zeta_1,\ldots,\zeta_b)=
(\eta_1+\zeta_1,\eta_2+\sigma_1\zeta_1+\tau_1\eta_1+\zeta_2,\ldots,\sigma_a\zeta_b+\tau_b\eta_a).$$ We thus see that $Im(+_*)_{(0,\ldots,0)}$ only has dimension $\max\{a,b\}$ and is contained in the image of the linearization of the smooth model $$\begin{aligned}
\mathbb{C}\times S^{a+b-2}\mathbb{C}&\to S^{a+b}\mathbb{C} \nonumber \\ (z,D)&\mapsto 2z+D \nonumber \end{aligned}$$ for the diagonal stratum at $(0,0+\cdots +0)$. Suppose now that $D_1$ and $D_2$ contain a common point $p$; write $D_i =a_i p+D_i
'$ where $D_i\in S^{r_i-a_i}\Sigma$ are divisors which do not contain $p$. Then from the commutative diagram $$\begin{CD}
{S^{a_1}\Sigma\times S^{r_1-a_1}\Sigma\times S^{a_2}\Sigma\times S^{r_2-a_2}\Sigma}@>>>{S^{r_1}\Sigma\times S^{r_2}\Sigma}\\
@VVV @ VV{+}V\\
{S^{a_1+a_2}\Sigma\times S^{r_1+r_2-a_1-a_2}\Sigma}@>>>{S^{r_1+r_2}\Sigma}\\
\end{CD}$$ and the fact that the linearization of the top arrow at $(a_1 p,D_{1}',a_2 p,D_{2}')$ is an isomorphism (by what we showed earlier, since the $D_{i}'$ do not contain $p$), while the linearization of the composition of the left and bottom arrows at $(a_1 p,D_{1}',a_2 p,D_{2}')$ has image contained in $T_{D_1+D_2}\Delta$, it follows that $(+_*)_{(D_1,D_2)}$ has image contained in $T_{D_1+D_2}\Delta$ as well, which suffices to prove the proposition.
\[tgt\] If $s_i \in \Gamma (X_{r_i}(f))$ are differentiable sections such that $C_{s_i}\cap C_{s_j} \neq\varnothing$ for some $i\neq j$, then $s=\sum s_i \in \Gamma(X_r(f))$ is tangent to the diagonal stratum of $X_r (f)$.
Indeed, if $C_{s_i}\cap C_{s_j} \neq\varnothing$, then there is $x\in S^2$ such that the divisors $s_i (x)$ and $s_j (x)$ contain a point in common, and so for $v\in T_x S^2$ we have $$s_*
v=(+\circ(s_i,s_j))_* v=+_*(s_{1*}v,s_{2*}v)\in T_{s(t)}\Delta$$ by Proposition \[plus\].
Note that it is straightforward to find cases in which the $s_i$ are only continuous with some $C_{s_i}\cap C_{s_j}$ nonempty and the sum $s=\sum s_i$ is smooth but not tangent to the diagonal. For example, let $r=2$, and in local coordinates let $s_1$ be a square root of the function $z\mapsto Re(z)$ and $s_2=-s_1$. Then in the standard coordinates on the symmetric product we have $s(z)=(0,-Re(z))$, so that $T(Im s)$ shares only one dimension with $T\Delta$ at $z=0$. If $s$ is *transverse* to $\Delta$, one can easily check that a similar situation cannot arise.
We now bring pseudoholomorphicity in the picture. Throughout this treatment, all almost complex structures on $X_r (f)$ will be assumed to agree with the standard structures on the symmetric product fibers, to make the map $F{\colon\thinspace}X_r (f)\to S^2$ pseudoholomorphic, and, on some (*not* fixed) neighborhood of the critical fibers of $F$, to agree with the holomorphic model for the relative Hilbert scheme over a disc around a critical value for $f$ provided in Section 3 of [@Smith]. Let $\mathcal{J}$ denote the space of these almost complex structures. It follows by standard arguments (see Proposition 3.4.1 of [@MS] for the general scheme of these arguments and Section 4 of [@DS] for their application in the present context) that for generic $J\in\mathcal{J}$ the space $\mathcal{M}^{J}(c_{\alpha})$ is a smooth manifold of (real) dimension $2d(\alpha)=\alpha^2-\kappa_{X}\cdot\alpha$ (the dimension computation comprises Lemma 4.3 of [@Smith]); this manifold is compact, for bubbling is precluded by the arguments of Section 4 of [@Smith] assuming we have taken a sufficiently high-degree Lefschetz fibration.
Inside $\mathcal{M}^{J}(c_{\alpha})$ we have the set $\mathcal{M}^{J}(c_{\alpha_1}\times\cdots\times c_{\alpha_n})$ consisting of holomorphic sections which lie in the image $+(c_{\alpha_1}\times\cdots\times c_{\alpha_n})$. By Lemma \[closed\] and the compactness of $\mathcal{M}^{J}(c_{\alpha})$, $\mathcal{M}^{J}(c_{\alpha_1}\times\cdots\times c_{\alpha_n})$ is evidently compact; however, the question of its dimension or even whether it is a manifold appears to be a more subtle issue in general.
Let us pause to consider what we would like the dimension of $\mathcal{M}^{J}(c_{\alpha_1}\times\cdots\times c_{\alpha_n})$ to be. The objects in $\mathcal{M}^{J}(c_{\alpha_1}\times\cdots\times c_{\alpha_n})$ are expected to correspond in some way to unions of holomorphic curves Poincaré dual to $\alpha_i$. Accordingly, assume we have chosen the $\alpha_i$ so that $d(\alpha_i)=\frac{1}{2}(\alpha_{i}^{2}-\kappa_{X}\cdot\alpha_i)\geq
0$ (for otherwise we would expect $\mathcal{M}^{J}(c_{\alpha_1}\times\cdots\times c_{\alpha_n})$ to be empty). Holomorphic curves in these classes will intersect positively as long as they do not share any components of negative square; for a generic almost complex structure the only such components that can arise are $(-1)$-spheres, so if we choose the $\alpha_i$ to not share any $(-1)$-sphere components (*i.e.*, if the $\alpha_i$ are chosen so that there is no class $E$ represented by a symplectic $(-1)$-sphere such that $\langle
\alpha_i , E\rangle<0$ for more than one $\alpha_i$), then it would also be sensible to assume that $\alpha_i\cdot\alpha_j\geq
0$ for $i\neq j$.
The above naive interpretation of $\mathcal{M}^{J}(c_{\alpha_1}\times\cdots\times c_{\alpha_n})$ would suggest that its dimension ought to be $\sum d(\alpha_i)$. Note that $$d(\alpha)=d(\sum\alpha_i)=\sum
d(\alpha_i)+\sum_{i>j}\alpha_i\cdot\alpha_j,$$ so under the assumptions on the $\alpha_i$ from the last paragraph we have that the expected dimension of $\mathcal{M}^{J}(c_{\alpha_1}\times\cdots\times c_{\alpha_n})$ is at most the actual dimension of $\mathcal{M}^J(c_{\alpha})$ (as we would hope, given that the former is a subset of the latter), with equality if and only if $\alpha_i\cdot\alpha_j=0$ whenever $i\neq
j$.
As usual, we will find it convenient to cut down the dimensions of our moduli spaces by imposing incidence conditions, so we shall fix a set $\Omega$ of points $z\in X$ and consider the space $\mathcal{M}^{J,\Omega}(c_{\alpha_1}\times\cdots\times
c_{\alpha_n})$ of elements $s\in\mathcal{M}^{J}(c_{\alpha_1}\times\cdots\times c_{\alpha_n})$ such that $C_s$ passes through each of the points $z$ (or, working more explicitly in $X_r(f)$, such that $s$ meets each divisor $z+S^{r-1}\Sigma_t$, $\Sigma_t$ being the fiber which contains $z$). $\mathcal{M}^{J,\Omega}(c_{\alpha})$ is defined similarly, and standard arguments show that for generic choices of $\Omega$ $\mathcal{M}^{J,\Omega}(c_{\alpha})$ will be a compact manifold of dimension $$2(d(\alpha)-\#\Omega).$$
We wish to count $J$-holomorphic sections $s$ of $X_r(f)$ such that the reducible components of $C_s$ are Poincaré dual to the $\alpha_i$. If we impose $\sum d(\alpha_i)$ incidence conditions, then according to the above discussion $\mathcal{M}^{J,\Omega}(c_{\alpha})$ will be a smooth manifold of dimension $2\sum_{i>j}\alpha_i\cdot\alpha_j$. A section $\sum
s_i\in +(c_{\alpha_1}\times\cdots c_{\alpha_n})$ whose summands are all differentiable would then, by Corollary \[tgt\], have one tangency to the diagonal $\Delta$ for each of the intersections between the $C_{s_i}$, of which the total expected number is $\sum_{i>j}\alpha_i\cdot\alpha_j$. This suggests that the sections we wish to count should be found among those elements of $\mathcal{M}^{J,\Omega}(c_{\alpha})$ which have $\sum_{i>j}\alpha_i\cdot\alpha_j$ tangencies to $\Delta$, where $\Omega$ is a set of $\sum d(\alpha_i)$ points.
To count pseudoholomorphic curves tangent to a symplectic subvariety it is necessary to restrict to almost complex structures which preserve the tangent space to the subvariety (see [@IP] for the general theory when the subvariety is a submanifold). Accordingly, we shall restrict attention to the class of almost complex structures $J$ on $X_r (f)$ which are *compatible with the strata* in the sense to be explained presently (for more details, see Section 6 of [@DS], in which the notion was introduced).
Within $\Delta$, there are various strata $\chi_{\pi}$ indexed by partitions $\pi: r=\sum a_i n_i$ with at least one $a_i >1$; these strata are the images of the maps $$\begin{aligned}
p_{\chi}{\colon\thinspace}X_{n_1}(f)\times_{S^2}\cdots\times_{S^2}X_{n_k}(f)&\to X_r (f) \nonumber\\
(D_1,\ldots ,D_k)&\mapsto \sum a_i D_i; \nonumber \end{aligned}$$ in particular, $\Delta = \chi_{r=2\cdot 1+1\cdot (r-2)}$. An almost complex structure $J$ on $X_r (f)$ is said to be compatible with the strata if the maps $p_{\chi}$ are $(J',J)$-holomorphic for suitable almost complex structures $J'$ on their domains.
Denoting by $Y_{\chi}$ the domain of $p_{\chi}$, Lemma 7.4 of [@DS] and the discussion preceding it show:
\[DS\] For almost complex structures $J$ on $X_r (f)$ which are compatible with the strata, each $J$-holomorphic section $s$ of $X_r (f)$ lies in some unique minimal stratum $\chi$ and meets all strata contained in $\chi$ in isolated points. In this case, there is a $J'$-holomorphic section $s'$ of $Y_{\chi}$ such that $s=p_{\chi}\circ s'$. Furthermore, for generic $J$ among those compatible with the strata, the actual dimension of the space of all such sections $s$ is equal to the expected dimension of the space of $J'$-holomorphic sections $s'$ lying over $s$.
We note the following analogue for standard surfaces of the positivity of intersections of pseudoholomorphic curves.
\[posints\] Let $s=m_1 s_1+\cdots +m_k s_k$ be a $J$-holomorphic section of $X_r (f)$, where the $s_i\in c_{\alpha_i}\subset \Gamma
(X_{r_i}(f))$ are each not contained in the diagonal stratum of $X_{r_i}(f)$, and where the almost complex structure $J$ on $X_r
(f)$ is compatible with the strata. Assume that the $s_i$ are all differentiable. Then all isolated intersection points of $C_{s_i}$ and $C_{s_j}$ contribute positively to the intersection number $\alpha_i\cdot \alpha_j$.
We shall prove the lemma for the case $k=2$, the general case being only notationally more complicated. The analysis is somewhat easier if the points of $C_{s_1}\cap C_{s_2}\subset X$ at issue only lie over $t\in S^2$ for which $s_1 (t)$ and $s_2 (t)$ both miss the diagonal of $X_{r_1}(f)$ and $X_{r_2}(f)$, respectively, so we first argue that we can reduce to this case. Let $\chi$ be the minimal stratum (possibly all of $X_r (f)$) in which $s=m_1s_1+m_2s_2$ is contained, so that all intersections of $s$ with lower strata are isolated. Let $p\in X$ be an isolated intersection point of $C_{s_1}$ and $C_{s_2}$ lying over $0\in
S^2$, and let $\delta
>0$ be small enough that there are no other intersections of $s$ with any substrata of $\chi$ (and so in particular no other points of $C_{s_1}\cap C_{s_2}$) lying over $D_{2\delta}(0)\subset S^2$. We may then perturb $s=m_1s_1+m_2s_2$ to $\tilde{s}=m_1\tilde{s_1}+m_2\tilde{s_2}$, still lying in $\chi$, such that\
- Over $D_{\delta}(0)$, $\tilde{s}$ is $J$-holomorphic and disjoint from all substrata having real codimension larger than 2 in $\chi$, and the divisors $\tilde{s_1}(0)$ and $\tilde{s_2}(0)$ both still contain $p$;
- Over the complement of $D_{2\delta}(0)$, $\tilde{s}$ agrees with $s$; and
- Over $D_{2\delta}(0)\setminus D_{\delta}(0)$, $\tilde{s}$ need not be $J$-holomorphic but is connected to $s$ by a family of sections $s_t$ contained in $\chi$ which miss all substrata of $\chi$
(it may be necessary to decrease $\delta$ to find such $\tilde{s}$, but after doing so such $\tilde{s}$ will exist by virtue of the abundance of $J$-holomorphic sections over the small disc $D_{\delta}(0)$ which are close to $s|_{D_{\delta}(0)}$). The contribution of $p$ to the intersection number $\alpha_1\cdot\alpha_2$ will then be equal to the total contribution of all the intersections of $C_{\tilde{s_1}}$ and $C_{\tilde{s_2}}$ lying over $D_{\delta}(0)$, and the fact that $\tilde{s}$ misses all substrata with codimension larger than 2 in $\chi$ is easily seen to imply that these intersections (of which there is at least one, at $p$) are all at points where $\tilde{s_1}$ and $\tilde{s_2}$ miss the diagonals in $X_{r_1}(f)$ and $X_{r_2}(f)$.
As such, it suffices to prove the lemma for intersection points at which $s_1$ and $s_2$ both miss the diagonal. In this case, in a coordinate neighborhood $U$ around $p$, the $C_{s_i}$ can be written as graphs $C_{s_i}\cap U=\{w=g_i (z)\}$, where $w$ is the holomorphic coordinate on the fibers of $X$, $z$ is the pullback of the holomorphic coordinate on $S^2$, and $g_i$ is a differentiable complex-valued function which vanishes at $z=0$. Suppose first that $m_1=m_2=1$. Then near $s(0)$, we may use coordinates $(z,\sigma_1,\sigma_2,y_3,\ldots,y_r)$ for $X_r (f)$ obtained from the splitting $T_0S^2\oplus T_{2p}S^2\Sigma_0\oplus
T_{s(t)-2p}S^{r-2}\Sigma_0$, and the first two vertical coordinates of $s(z)=(s_1+s_2)(z)$ with respect to this splitting are $(g_1(z)+g_2(z),g_1(z)g_2(z))$. Now $s$ is $J$-holomorphic and meets the $J$-holomorphic diagonal stratum $\Delta$ at $(0,s(0))$, and at this point $\Delta$ is tangent to the hyperplane $\sigma_2=0$, so it follows from Lemma 3.4 of [@IP] that the Taylor expansion of $g_1 (z)g_2(z)$ has form $a_0 z^{d}
+O(d+1)$. But then the Taylor expansions of $g_1 (z)$ and $g_2
(z)$ begin, respectively, $a_1 z^{d_1}+O(d_1+1)$ and $a_2
z^{d_2}+O(d_2+1)$, with $d_1+d_2=d$. Then since $C_{s_i}\cap
U=\{w=g_i (z)\}$, it follows immediately that the $C_{s_i}$ have intersection multiplicity $\max\{d_1,d_2\}>0$ at $p$.
There remains the case where one or both of the $m_i$ is larger than 1. In this case, where $Y_{\chi}=X_{r_1}(f)\times_{S^2}X_{r_2}(f)$ is the smooth model for $\chi$, because $J$ is compatible with the strata, $(s_1,s_2)$ is a $J'$-holomorphic section of $Y_{\chi}$ for an almost complex structure $J'$ such that $p_{\chi}{\colon\thinspace}Y_{\chi}\to X_r (f)$ is $(J',J)$-holomorphic. Now where $\tilde{\Delta}=\{(D_1,D_2)\in
Y_{\chi}|D_1\cap D_2\neq \varnothing\}$, compatibility with the strata implies that $\tilde{\Delta}$ will be $J'$-holomorphic. In a neighborhood $V$ around $(s_1 (z),s_2(z))$, we have, in appropriate coordinates, $\tilde{\Delta}\cap
V=\{(z,w,w,D_1,D_2)|w\in \Sigma_z\}$,while $(s_1 (z),s_2 (z))$ has first three coordinates $(z,g_1 (z),g_2 (z))$. From this it follows by Lemma 3.4 of [@IP] that $$g_1 (z)-g_2 (z)=a_0 z^d
+O(d+1)$$ for some $d$, in which case $C_{s_1}$ and $C_{s_2}$ have intersection multiplicity $d>0$ at $p$.
Let $\Omega$ be a set of $\sum d(\alpha_i)$ points and let $J$ be an almost complex structure compatible with the strata. $\mathcal{M}^{J,\Omega}_{0}(\alpha_1,\ldots,\alpha_n)$ shall denote the set of $J$-holomorphic sections $s\in c_{\alpha}$ with $\Omega\subset C_s$ such that there exist $C^1$ sections $s_i\in
c_{\alpha_i}$ with $s=\sum s_i$, while the $s_i$ themselves do not admit nontrivial decompositions as sums of $C^1$ sections.
We would like to assert that for generic $J$ and $\Omega$, the space $\mathcal{M}^{J,\Omega}_{0}(\alpha_1,\ldots,\alpha_n)$ does not include any sections contained within the strata. This is not true in full generality; rather we need the following assumption in order to rule out the effects of multiple covers of square-zero tori and $(-1)$-spheres in $X$.
\[ass\] None of the $\alpha_i$ can be written as $\alpha_i =m\beta$ where $m>1$ and either $\beta^2=\kappa_{X}\cdot\beta=0$ or $\beta^2=\kappa_X\cdot\beta=-1.$
Under this assumption, we note that if $s=\sum s_i\in
\mathcal{M}^{J,\Omega}_{0}(\alpha_1,\ldots,\alpha_n)$ were contained in $\Delta$, then since the $\alpha_i$ and hence the $s_i$ are distinct we can write each $s_i$ as $s_i =
m_i\tilde{s_i}$ with at least one $m_i >1$. The minimal stratum of $s$ will then be $\chi_{\pi}$ where $\pi = \left\{r=\sum
m_i\left(\frac{r_i}{m_i}\right)\right\}$ and $s'=(\tilde{s_1},\ldots,\tilde{s_n})$ will be a $J'$-holomorphic section of $Y_{\chi}$ with $s=p_{\chi}\circ s'$, in the homotopy class $[c_{\alpha_1/m_1}\times\cdots\times c_{\alpha_n/m_n}]$.
If any of the $d(\alpha_i/m_i)<0$, then Lemma \[DS\] implies that there will be no such sections $s'$ at all; otherwise (again by Lemma \[DS\]) the real dimension of the space of such sections (taking into account the incidence conditions) will be $$\label{dim} 2\left(\sum d(\alpha_i/m_i)-\sum d(\alpha_i)\right).$$ But an easy manipulation of the general formula for $d(\beta)$ and the adjunction formula (which applies here because the standard surface corresponding to a section of $X_r (f)$ which meets $\Delta$ positively will be symplectic; c.f. Lemma 2.8 of [@DS]) shows that if $d(\beta)\geq 0$ and $m\geq 2$ then $d(m\beta)> d(\beta)$ unless either $\beta^2=\kappa_{X}\cdot\beta=0$ or $\beta^2=\kappa_{X}\cdot\beta=-1$, and these are ruled out in this context by (i) and (ii) above, respectively. So Assumption \[ass\] implies that the dimension in Equation \[dim\] is negative, so no such $s'$ will exist for generic $J$. This proves part of the following:
\[cpct\] Under Assumption \[ass\], for generic pairs $(J,\Omega)$ where $J$ is compatible with the strata and $\#\Omega =\sum d(\alpha_i)$, $\mathcal{M}^{J,\Omega}_{0}(\alpha_1,\ldots,\alpha_n)$ is a finite set consisting only of sections not contained in $\Delta$.
That no member of $\mathcal{M}^{J,\Omega}_{0}(\alpha_1,\ldots,\alpha_n)$ is contained in $\Delta$ follows from the above discussion. As for the dimension of our moduli space, note that any $s=\sum
s_i\in\mathcal{M}^{J,\Omega}_{0}(\alpha_1,\ldots,\alpha_n)$ has one tangency (counted with multiplicity) to $\Delta$ for each of the intersections of the $C_{s_i}$, of which there are $\sum
\alpha_i\cdot\alpha_j$ (counted with multiplicity; this multiplicity will always be positive by Proposition \[posints\]). By the results of section 6 of [@IP], the space $\mathcal{M}^{J,\Omega}_{\delta,\Delta}(c_{\alpha})$ of $J$-holomorphic sections in the class $c_{\alpha}$ having $\delta$ tangencies to $\Delta$ and whose descendant surfaces pass through $\Omega$ will, for generic $(J,\Omega)$, be a manifold of dimension $$2(d(\alpha)-\sum d(\alpha_i)-\delta)=2(\sum
\alpha_i\cdot \alpha_j-\delta),$$ which is equal to zero in the case $\delta=\sum \alpha_i\cdot\alpha_j$ of present relevance to us.
Let us now show that $\mathcal{M}^{J,\Omega}_{0}(\alpha_1,\ldots,\alpha_n)$ is compact. Now since $+(c_{\alpha_1}\times\cdots \times c_{\alpha_n})$ is $C^0$-closed in $c_{\alpha}$, by Gromov compactness any sequence $s^{(m)}=\sum_{i=1}^{n} s^{(m)}_{i}$ in $\mathcal{M}^{J,\Omega}_{0}(\alpha_1,\ldots,\alpha_n)$ has (after passing to a subsequence) a $J$-holomorphic limit $s=\sum s_i$ where the $s_i\in c_{\alpha_i}$ are at least continuous. We claim that, at least for generic $(J,\Omega)$, we can guarantee the $s_i$ to be $C^1$. In light of Proposition \[plus\], the differentiability of the $s_i$ is obvious at all points where $s$ misses the diagonal, since $s$ is smooth by elliptic regularity and the divisor addition map induces an isomorphism on the tangent spaces away from the diagonal. Now each $s^{(m)}$ has $\sum
\alpha_i\cdot\alpha_j$ tangencies to the diagonal, corresponding to points $t\in S^2$ at which some pair of the divisors $s^{(m)}_{i}(t)$ share a point in common. The limit $s$ will then likewise have $n$ tangencies to the diagonal; the dimension formulas in [@IP] ensure that for generic $(J,\Omega)$ no two of the tangencies will coalesce into a higher order tangency to the smooth part of $\Delta$ in the limit, and all of the intersecions of $Im\, s$ with the smooth part of the diagonal other than these $n$ tangencies will be transverse. Furthermore, one may easily show (using for instance an argument similar to the one used in Lemma 2.1 of [@Usher] to preclude generic 0-dimensional moduli spaces of pseudoholomorphic curves in a Lefschetz fibration from meeting the critical points) that since the singular locus of $\Delta$ has codimension 4 in $X_r(f)$, if $J$ has been chosen generically then $s$ will not meet $\Delta^{sing}$, and so no $s(t)$ will contain more than one repeated point (and that point cannot appear with multiplicity larger than two). In light of this, each tangency of $s$ to $\Delta$ will occur at a point $s(t)$ where some pair $s_i(t)$ and $s_j(t)$ have some point $p$ in common, and all other points contained in any $s_k(t)$ are distinct from each other and from $p$. Thanks to Proposition \[plus\], this effectively reduces us to the case $r=2$, with $s=s_1+s_2$ a sum of continuous sections with $s_1(0)=s_2(0)=0$ which is holomorphic with respect to an almost complex structure which preserves the diagonal stratum $\Delta$ in $D^2\times Sym^2D^2$, such that $s$ is tangent to $\Delta$. Then letting $\delta(t)=(s_1+s_2)^2(t)-4s_1(z)s_2(t)$ be the discriminant, that $s$ is tangent to the diagonal stratum implies, using Lemma 3.4 of [@IP], that $\delta(t)=at^2+O(3)$ for some constant $a$; in particular $\delta(t)$ has two $C^1$ square roots $\pm r(t)$. Since $s$ is smooth, so is its first coordinate $t\mapsto s_1(t)+s_2(t)$; adding this smooth function to the $C^1$ functions $\pm r(t)$ and dividing by two then recovers the functions $s_1(t)$ and $s_2(t)$ and verifies that they are $C^1$ at $t=0$.
We have thus shown that the $s_i$ are all $C^1$ at the points where $s=\sum s_i$ is tangent to $\Delta$. Where $s$ is transverse to $\Delta$, one sees easily that the $s_i$ are pairwise disjoint, with one $s_i$ transverse to the diagonal in $X_{r_i}(f)$ and all others missing their diagonals, so the differentiability of the $s_i$ is clear. This indeed verifies that the limit $s=\sum s_i$ is a sum of $C^1$ sections $s_i$, since our generic choice of $J$ is such that the only intersections of $Im
\, s$ with $\Delta$ only are either transverse or of second order.
Now each of the $C_{s^{(m)}_{i}}$ is connected, so $C_{s_i}$ is connected as well. *A priori*, it is possible that $s$ might not lie in $\mathcal{M}^{J,\Omega}_{0}(\alpha_1,\ldots,\alpha_n)$ because some of the $s_i$ might decompose further, say as $s_i=m_1
u_{i_1}+\cdots+m_lu_{i_l}$ where $u_{i_j}\in c_{\beta_{i_j}}$ are $C^1$. But since $C_{s_i}$ is connected, the $C_{u_{i_j}}$ cannot all be disjoint, and by Corollary \[tgt\] any intersection between two of them would give rise to an additional tangency of $s$ to $\Delta$, over and above the $n$ tangencies arising from the intersections between distinct $C_{s_i}$. Once again, this is ruled out for generic $J$ by the dimension formulas of [@IP]. This proves that (for generic $J$) the summands $s_i$ in a sequence $s=\sum
s_i$ occurring as a limit point of $\mathcal{M}^{J,\Omega}_{0}(\alpha_1,\ldots,\alpha_n)$ cannot decompose further and hence themselves lie in $\mathcal{M}^{J,\Omega}_{0}(\alpha_1,\ldots,\alpha_n)$, so that $\mathcal{M}^{J,\Omega}_{0}(\alpha_1,\ldots,\alpha_n)$ is compact.
Since we have already shown that $\mathcal{M}^{J,\Omega}_{0}(\alpha_1,\ldots,\alpha_n)$ is zero-dimensional, the proposition follows.
For generic $(J_0,\Omega_0)$ and $(J_1,\Omega_1)$ as in Proposition \[cpct\] and generic paths $(J_t,\Omega_t)$ connecting them, the space $$\mathcal{PM}_{0}(\alpha_1,\ldots,\alpha_n)=\{(t,s)|s\in\mathcal{M}^{J_t,\Omega_t}_{0}(\alpha_1,\ldots,\alpha_n)\}$$ is a compact one-dimensional manifold.
This follows immediately from the above discussion, noting that in the proof of Proposition \[cpct\] we saw that any possible boundary components of $\mathcal{M}^{J}_{0}(\alpha_1,\ldots,\alpha_n)$ have real codimension 2 and so will not appear in our one-dimensional parametrized moduli space.
Note that we can orient these moduli spaces by using the spectral flow of the linearization of the $\overline{\partial}$ operator at an element $s\in\mathcal{M}^{J,\Omega}_{0}(\alpha_1,\ldots,\alpha_n)$ acting on sections of $s^{*}T^{vt}X_r (f)$ which preserve the incidence conditions and the tangencies to $\Delta$; $\mathcal{PM}_{0}(\alpha_1,\ldots,\alpha_n)$ will then be an oriented cobordism between $\mathcal{M}^{J_0,\Omega_0}_{0}(\alpha_1,\ldots,\alpha_n)$ and $\mathcal{M}^{J_1,\Omega_1}_{0}(\alpha_1,\ldots,\alpha_n)$. Accordingly, we may make the following definition.
Let $\alpha=\alpha_1 +\cdots +\alpha_n$ be a decomposition of $\alpha\in H^2(X,\mathbb{Z})$ which satisfies Assumption \[ass\]. Then $$\widetilde{\mathcal{DS}_f}(\alpha;\alpha_1,\ldots,\alpha_n)$$ is defined as the number of points, counted with sign according to orientation, in the space $\mathcal{M}^{J,\Omega}_{0}(\alpha_1,\ldots,\alpha_n)$ for generic $(J,\Omega)$ as in Proposition \[cpct\].
\[tildesame\] If $\alpha=\alpha_1+\cdots\alpha_n$ is a decomposition satisfying Assumption \[ass\] then $$\frac{(\sum d(\alpha_i))!}{\prod
(d(\alpha_i)!)}Gr(\alpha;\alpha_1,\ldots,\alpha_n)=\widetilde{\mathcal{DS}_f}(\alpha;\alpha_1,\ldots,\alpha_n)$$ provided that the degree of the fibration is large enough that $\langle [\omega_{X}],[\Phi]\rangle>[\omega_{X}]\cdot \alpha$.
Let $j$ be an almost complex structure on $X$ generic among those compatible with the fibration $f{\colon\thinspace}X\to S^2$, and $\Omega$ a generic set of $\sum d(\alpha_i)$ points. The curves in $X$ contributing to $Gr(\alpha;\alpha_1,\cdots,\alpha_n)$ are unions $$C=\bigcup_{i=1}^{n} C^i$$ of embedded $j$-holomorphic curves $C^i$ which are Poincaré dual to $\alpha_i$ (note that Assumption \[ass\] implies that none of these curves will be multiple covers) with $\Omega_i\subset C^i$ for some fixed generic sets $\Omega_i$ of $d(\alpha_i)$ points. In Section 3 of [@Usher] it was shown that there is no loss of generality in assuming that $j$ is integrable near $\cup_i Crit(f|_{C^i})$, so let us assume that this is the case. Where $s_C$ is the section of $X_r (f)$ tautologically corresponding to $C$, in the context of [@Usher] this local integrability condition was enough to ensure that the almost complex structure $\mathbb{J}_j$ on $X_r(f)$ constructed from $j$ was smooth on a neighborhood of $s_C$. Here that is not quite the case, for $\mathbb{J}_j$ might only be Hölder continuous at the points of $Im(s_C)$ tautologically corresponding to the intersection points of the various $C^i$.
However, just as in Section 5 of [@Usher], we can still define the contribution $r'(C)$ to $\widetilde{DS}_f
(\alpha_1,\ldots,\alpha_n)$ by perturbing $\mathbb{J}_j$ to a generic almost complex structure $J$ which is compatible with the strata and Hölder-close to $\mathbb{J}_j$, and then counting with sign the elements of $\mathcal{M}^{J,\Omega}_{0}(\alpha_1,\ldots,\alpha_n)$ which lie near $s_C$; since the curves $C$ which contribute to $Gr(\alpha_1,\ldots,\alpha_n)$ are isolated, and since the members of $\mathcal{M}^{\mathbb{J}_j,\Omega}_{0}(\alpha_1,\ldots,\alpha_n)$ are precisely the $s_C$ corresponding to the curves $C$, it follows from Gromov compactness that for sufficiently small perturbations $J$ of $\mathbb{J}_j$ all elements of $\mathcal{M}^{J,\Omega}_{0}(\alpha_1,\ldots,\alpha_n)$ will be close to one and only one of the $s_C$. Thus $$\widetilde{\mathcal{DS}}_f (\alpha_1,\ldots,\alpha_n)=\sum_{\pi\in
p(\Omega)}\sum_{C\in\mathcal{M}^{j,\Omega,\pi}(\alpha_1,\ldots,\alpha_n)}r'(C)$$ where $p(\Omega)$ is the set of partitions of $\Omega$ into subsets $\Omega_i$ of cardinality $d(\alpha_i)$ and, writing $\pi=(\Omega_1,\ldots,\Omega_n)$, $\mathcal{M}^{j,\Omega,\pi}(\alpha_1,\ldots,\alpha_n)$ is the space of curves $C=\cup C^i$ contributing to $Gr(\alpha;\alpha_1,\ldots,\alpha_n)$ with $C^i$ passing through $\Omega_i$. Meanwhile, for any $\pi$, we have $$Gr(\alpha;\alpha_1,\ldots,\alpha_n)=\sum_{C\in
\mathcal{M}^{j,\Omega,\pi}(\alpha_1,\ldots,\alpha_n)} r(C),$$ $r(C)$ being the product of the spectral flows of the linearizations of $\overline{\partial}_j$ at the embeddings of the $C^i$ where $C=\cup C^i$. The theorem will thus be proven if we show that $r'(C)=r(C)$, which we now set about doing.
So let $C=\cup C^i\in
\mathcal{M}^{j,\Omega,\pi}(\alpha_1,\ldots,\alpha_n)$. Taking $j$ generically, we may assume that all intersections of the $C^i$ are transverse and occur away from $crit(f|_{C^i})$ (this follows from the arguments of Lemma 2.1 of [@Usher]). Let $p\in C^i\cap
C^k$. In a coordinate neighborhood $U$ around $p$, where $w$ is a holomorphic coordinate on the fibers and $z$ the pullback of the coordinate on $S^2$, we may write $$C^i\cap U=\{w=g(z)\} \qquad
C^k\cap U=\{w=h(z)\}.$$ If the almost complex structure $j$ is given in $U$ by $$\label{T01} T^{0,1}_j =\langle
\partial_{\bar{z}}+b(z,w)\partial_w,\partial_{\bar{w}}\rangle$$ (note that we may choose the horizontal tangent space so that $b(0,0)=0$), that $C^i$ and $C^k$ are $j$-holomorphic amounts to the statement that $$\partial_{\bar{z}}g(z)=b(z,g(z)) \qquad \partial_{\bar{z}}h(z)=b(z,h(z));$$ in particular, we have $g_{\bar{z}}(0)=h_{\bar{z}}(0)=0$. Since $C^i\pitchfork C^k$, we have $(g-h)_z (0)\neq 0$, and by the inverse function theorem $(g-h){\colon\thinspace}\mathbb{C}\to\mathbb{C}$ is invertible on some disc $D_{2\delta}(0)$. Let $g_t$ and $h_t$ $(t\in [0,1])$ be one-parameter families of functions satisfying
- $g_0=g$, $h_0=h$;
- On $D_{2\delta}(0)$, $g_t-h_t$ is invertible as a complex-valued smooth function, with inverse $p_t$;
- $g_t$ and $h_t$ agree with $g$ and $h$, respectively, outside $D_{2\delta}(0)$;
- $g_t(0)=h_t(0)=\partial_{\bar{z}}g_t
(0)=\partial_{\bar{z}}h_t (0)=0$; and
- $g_1(z)$ and $h_1(z)$ are both holomorphic on $D_{\delta}(0)$.
Let $$C_{t}^{i}=(C^i\cap (X\setminus U))\cup\{w=g_t(z)\} \mbox{ and
} C_{t}^{k}=(C^k\cap (X\setminus U))\cup\{w=h_t(z)\}.$$ Now set $$b_t
(z,w)=(\partial_{\bar{z}}h_t)(z)+\partial_{\bar{z}}\left(g_t-h_t\right)\left(p_t(w-h_t(z))\right).$$
Then, since $p_t=(g_t-h_t)^{-1}$, $$b_t
(z,h_t(z))=\partial_{\bar{z}}h_t(z)+\partial_{\bar{z}}(g_t-h_t)(0)=\partial_{\bar{z}}h_t
(z)$$ while $$b_t (z,g_t(z))=\partial_{\bar{z}}h_t(z)+\partial_{\bar{z}}(g_t-h_t)(z)=\partial_{\bar{z}}g_t(z).$$
Let $b'_t$ agree with $b_t$ near $\{(z,w)\in C_{t}^i\cup
C_{t}^k|z\in D_{2\delta}(0)\}$ and with $b$ sufficiently far from the origin in $U$. Then defining $j'_t$ by $T^{0,1}_{j'_t}=\langle
\partial_{\bar{z}}+b'_t\partial_w,\partial_{\bar{w}}\rangle$, $j'_t$ agrees with $j$ near $\partial U$ and makes $C^{i}_{t}\cup
C^{k}_{t}$ holomorphic. Further, we see that $b_1(z,w)\equiv 0$ for $z\in D_{\delta}(0)$, from which a Nijenhuis tensor computation shows that $j'_1$ is integrable on a neighborhood of the unique point $p$ of $C^{i}_{1}\cap C^{k}_{1}\cap U$.
Carrying out this construction near all intersection points of the $C^i$, we obtain curves $C_t=\cup C^{i}_{t}$ and almost complex structures $j'_t$ on $X$ such that $j'_1$ is integrable near all intersection points of the $C^{i}_{1}$. Since $j'_1$ agrees with $j$ and $C^{i}_{1}$ with $C^i$ away from small neighborhoods of these intersection points, $j'_1$ is also integrable on a neighborhood of $crit(f|_{C^{1}_i})$ for each $i$.
If $p$ is a point of $C_1$ near which $j'_1$ is not already integrable, then in a neighborhood $U$ of $p$ we have $C_1 \cap
U=\{w=g(z)\}$, and so the condition for an almost complex structure $j'$ given by $T^{0,1}_{j'}=\langle\partial_{\bar{z}}+b\partial_w,\partial_{\bar{w}}\rangle$ to make $C_1$ holomorphic near $p$ is just that $\partial_{\bar{z}}g(z)=b(z,g(z))$, while the condition for $j'$ to be integrable in the neighborhood is that $\partial_{\bar{w}}b(z,w)=0$. As in Lemmas 4.1 and 4.4 of [@Usher], then, we may easily find a path of almost complex structures $j'_t$ ($1\leq t\leq 2$) such that each $j'_t$ makes $C_1$ holomorphic and $j'_2$ is integrable on a neighborhood of $C_1$. So, changing notation slightly, we have proven:
\[isotopy\] There exists an isotopy $(C_t,j_t)$ of pairs consisting of almost complex structures $j_t$ compatible with the fibration $f{\colon\thinspace}X\to S^2$ and $j_t$-holomorphic curves $C_t$ such that $(C_0,j_0)=(C,j)$ and $j_1$ is integrable on a neighborhood of $C_1$.
In the situation of the above lemma, $\mathbb{J}_{j_1}$ is not only smooth but also integrable on a neighborhood of $C_1$; Lemma 4.2 of [@Usher] shows that if $j_1$ is chosen generically among almost complex structures which make both $C_1$ and $f$ pseudoholomorphic and are integrable near $C_1$ the linearization of $\bar{\partial}_{\mathbb{J}_{j_1}}$ at $s_C$ will be surjective, as will the linearizations of $\bar{\partial}_{j_1}$ at the embeddings of each of the $C_{1}^{i}$. We now fix the isotopy $C_t$ and the almost complex structure $j_1$ which is nondegenerate in the above sense; Lemma \[isotopy\] then gives a path $j_t$ from $j=j_0$ to $j_1$ such that each $C_t$ is $j_t$-holomorphic. We may then define $r'_{j_t}(C_t)$ in the same way as $r'(C)$, by counting $J$-holomorphic sections close to $s_{C_t}$ for some $J$ Hölder-close to $\mathbb{J}_{j_t}$. Meanwhile, if the linearization $D\bar{\partial}_{j_t}$ is surjective at the embeddings of the $C^{i}_{t}$, its spectral flow gives a number $r_{j_t}(C_t)$, and our goal is to show that $r_{j_0}(C_0)=r'_{j_0}(C_0)$. To this end, we see from Lemma 5.5, Corollary 5.6, and their proofs in [@Usher] that:
For generic paths $j_t$ from $j_0$ to $j_1$ as above such that $C_t$ is $j_t$-holomorphic, the following statements hold. $D\bar{\partial}_{j_t}$ is surjective at the embeddings of the $C^{i}_{t}$ for all but finitely many values of $t$. For $t$ near any value $t_0$ for which $D\bar{\partial}_{j_{t_0}}$ fails to be surjective, the set of elements of $\mathcal{M}^{j_t,\Omega}(\alpha_1,\ldots,\alpha_n)$ in a tubular neighborhood of $C_t$ is given by $\{C_t,\tilde{C}_t\}$ for a smooth family of curves $\tilde{C}_t$ with $\tilde{C}_{t_0}=C_{t_0}$. Further, for small $\epsilon >0$, we have $$r'_{j_{t_0+\epsilon}}(C_{t_0+\epsilon})=r'_{j_{t_0-\epsilon}}(\tilde{C}_{t_0-\epsilon})=-r'_{j_{t_0-\epsilon}}(C_{t_0-\epsilon})$$ and $$r_{j_{t_0+\epsilon}}(C_{t_0+\epsilon})=r_{j_{t_0-\epsilon}}(\tilde{C}_{t_0-\epsilon})=-r_{j_{t_0-\epsilon}}(C_{t_0-\epsilon}).$$ Moreover, on intervals not containing any $t_0$ for which $j_{t_0}$ has a non-surjective linearization, $r'_{j_t}(C_t)$ and $r_{j_t}(C_t)$ both remain constant.
Since (for generic paths $j_t$), $r'_{j_t}(C_t)$ and $r_{j_t}(C_t)$ stay constant except for finitely many points at which they both change sign, to show that $r'_{j_0}(C_0)=r_{j_0}(C_0)$ it is enough to see that $r'_{j_1}(C_1)=r_{j_1}(C_1)$. But since $j_1$ is *integrable* and nondegenerate near $C_1$, as is $\mathbb{J}_{j_1}$ near $s_{C_1}$, we immediately see that $r'_{j_1}(C_1)=r_{j_1}(C_1)=1$, and the theorem follows.
The above proof suggests a simplification of the proof that $\mathcal{DS}=Gr$ in [@Usher]. As mentioned above, in Section 3 of [@Usher] it is shown that we can take the almost complex structure $j$ to be integrable on neighborhoods of the critical points of the various $f|_C$ for $C$ contributing to $Gr(\alpha)$. Given arbitrary generic fibration-compatible $j$, however, as in the proof of Theorem \[tildesame\], the arguments of Sections 4 and 5 of [@Usher] go through as long as we can find an isotopy $(C_t,j_t)$ of pairs consisting of almost complex structures $j_t$ compatible with the fibration $f{\colon\thinspace}X\to S^2$ and $j_t$-holomorphic curves $C_t$ such that $(C_0,j_0)=(C,j)$ and $j_1$ is integrable on a neighborhood of $C_1$. This is indeed possible; if near a critical point of $f|_C$ $C$ has the form $\{z=w^n+O(n+1)\}$, we can take $C_t$ such that $C_t$ agrees with $C$ away from a neighborhood of $Crit(f|_C)$ and $C_1$ has the form $\{z=w^n\}$ on a smaller neighborhood of the critical point, and then we can choose $j_t$ to make $C_t$ holomorphic. (The easiest approach to this seems to be to have $C_t$ be constant for $t\leq 1/2$ and arrange the function $b_{1/2}(z,w)$ in the notation (\[T01\]) to depend only on $w$ near the critical points; then for $t>1/2$, the form of $C_t$ determines uniquely a $z$-independent function $b_t$ which causes $C_t$ to be $j_t$-holomorphic, and we will have $b_1(z,w)=0$ near the critical point. Details are left to the reader.)
The family standard surface count {#secfam}
=================================
While much is known about the structure the Gromov–Taubes invariants, which count embedded holomorphic curves in symplectic 4-manifolds, we know comparatively little about invariants counting singular curves. We explain here an approach to nodal curves using Donaldson and Smith’s constructions.
We should mention first of all that whereas Taubes’ work gives us a natural invariant $Gr(\alpha)$ counting all embedded curves (regardless of their connected-component decomposition) Poincaré dual to some class $\alpha$, if we instead wish to assemble all of the possibly-reducible curves Poincaré dual to $\alpha$ and having some number $n>0$ of ordinary double points into an invariant $Gr_n(\alpha)$, it is somewhat unclear how we should proceed in many cases. Just as with the difficulties surrounding the Gromov–Taubes invariant, this stems from the multiple-cover problem: if for some class $\beta\in H^2(X,\mathbb{Z})$ and $m>1$ we have $d(\beta)\geq \max\{0, d(m\beta)-n\}$, then for generic almost complex structures $j$ there will arise the possibility of a sequence of curves Poincaré dual to $m\beta$ which have $n$ double points converging to an $m$-fold cover of a curve Poincaré dual to $\beta$. When $n=0$, as was noted in the previous section the formula for $d(\beta)$ and the adjunction formula imply that this only arises when $\beta$ is Poincaré dual to a square-zero torus, and Taubes’ work shows how to incorporate multiple covers into the definition of $Gr$ in the correct way. When $n>0$, the equation $d(\beta)\geq d(m\beta)-n$ becomes easier to satisfy and it is less clear how multiple covers should be dealt with, especially in the case of a strict inequality $d(\beta)>d(m\beta)-n$, when the multiple covers form a space of larger dimension than that of the space we are interested in.
Of course, there will typically be at least some classes for which this issue does not arise:
\[sem\] A class $\alpha\in H^2(X,\mathbb{Z})$ is called *strongly* $n$-*semisimple* if there exist *no* decompositions $\alpha= \alpha_1+\cdots+\alpha_l$ into nonnegatively-intersecting classes $\alpha_i$ such that each $\alpha_i$ has $d(\alpha_i)\geq 0$ and is Poincaré dual to the image of a symplectic immersion, and $\alpha_1$ is equal to $m\beta$ ($m>1$) where $\beta$ satisfies $d(\beta)\geq
\max\{0,d(\alpha_1)-n+\alpha_1\cdot(\alpha-\alpha_1)\}$. $\alpha$ is called *weakly $n$-semisimple* if the only decompositions $\alpha=\alpha_1+\cdots +\alpha_n$ as above which exist have $\alpha_{1}^{2}=\kappa_X\cdot\alpha_1=0$.
For instance, every class is weakly $0$-semisimple, while the only classes which are not weakly $1$-semisimple are those classes $\alpha$ such that there exists a class $\beta\in
H^2(X;\mathbb{Z})$ such that $\beta\cdot(\alpha-2\beta)=0$ and $\beta$ is Poincaré dual either to a symplectic sphere of square 0 or a symplectic genus-two curve of square 1, while $\alpha-2\beta$ is Poincaré dual to some embedded (and possibly disconnected) symplectic submanifold. For strong semisimplicity, one needs to add the assumption that $\alpha$ is not Poincaré dual to a symplectic immersion having a component which is a square-zero torus in a non-primitive homology class.
For a weakly- or strongly-$n$-semisimple classes $\alpha$, there is an obvious analogue of the Gromov–Taubes invariant $Gr_{n}(\alpha)$, defined by counting $j$-holomorphic curves $C$ which are unions of curves $C_i$ Poincaré classes $\alpha_i$ carrying multiplicities $m_i$ which are equal to 1 unless $C_i$ is a square-zero torus with $\sum m_i\alpha_i=\alpha$, such that $C$ has $n$ transverse double points and passes through a generic set of $d(\alpha)-n$ points of $X$; each such $C$ contributes the product of the Taubes weights $r(C_i,m_i)$ to the count $Gr_n(\alpha)$. Since the condition of $n$-semisimplicity is engineered to rule out the only additional possible source of noncompactness of the relevant moduli spaces, the proof that $Gr(\alpha)$ is independent of the choice of almost complex structure used to define it goes through to show the same result for $Gr_n(\alpha)$.
For that matter, if $\alpha$ is weakly $n$-semisimple and we have $n_i\geq 0$ and $\alpha_i$ with $\sum \alpha_i=\alpha$ and $\sum
n_i=n-\sum_{i<j}\alpha_i\cdot\alpha_j$, we can form a refinement $Gr_{(n_1,\ldots,n_k)}(\alpha;\alpha_1,\ldots,\alpha_k)$ along the lines of Definition \[invts\] which counts (modulo the usual square-zero torus issues) curves with reducible components which are Poincaré dual to the $\alpha_i$ and have $n_i$ transverse self-intersections. In this case, under Assumption \[ass\] it is also straightforward to modify the constructions of the previous section to produce an invariant $\widetilde{\mathcal{DS}}_{(n_1,\ldots,n_k)}(\alpha;\alpha_1,\ldots,\alpha_k)$ which counts holomorphic sections $s$ of $X_r(f)$ in the homotopy class $c_{\alpha}$ which decompose into a sum of $C^1$ sections $s_i\in c_{\alpha_i}$ such that each $s_i$ has $n_i$ tangencies to the diagonal stratum of $X_{r_i}(f)$ and does not itself decompose as a nontrivial sum of $C^1$ sections. Furthermore, the proof of Theorem \[tildesame\] goes through unchanged to show that $$Gr_{(n_1,\ldots,n_k)}(\alpha;\alpha_1,\ldots,\alpha_k)=\widetilde{\mathcal{DS}}_{(n_1,\ldots,n_k)}(\alpha;\alpha_1,\ldots,\alpha_k).$$
Instead, though, we aim to produce an invariant similar to $Gr_{n}(\alpha)$ which does not require $\alpha$ to be $n$-semisimple. For general $\alpha$, the multiple cover problem discussed above has its mirror on the side of $\widetilde{\mathcal{DS}}$ in the fact that the moduli spaces for the latter will tend to have undesirably-large components consisting of sections which are mapped entirely into the diagonal stratum, so $\widetilde{\mathcal{DS}}$ will not be much help toward this goal. Instead, we take a hint from the approach used by A.K. Liu in [@Liu] and construct family versions of the standard surface count. These new invariants will use almost complex structures which generally do not make the diagonal stratum pseudoholomorphic, and so we will not encounter moduli spaces with unexpectedly large components consisting of sections mapped into $\Delta$.
Be given a symplectic Lefschetz fibration $f{\colon\thinspace}X\to S^2$. Write $f_0=f$, $X_0=\{pt\}$, $X_1=X$, and let $g_0{\colon\thinspace}X_1\to X_0$ be the map of $X$ to a point. As in [@Liu], for $n\geq 1$ form $X^{0}_{n+1}=X_{n}\times_{g_{n-1}}X_{n}$, and let $X_{n+1}$ be the blowup of the relative diagonal in $X_{n+1}^{0}$. Let $g_{n}{\colon\thinspace}X_{n+1}\to X_{n}$ be the projection onto the first factor. Each $X^b:=g_{n}^{-1}(b)$ ($b\in X_{n}$) is then an $n$-fold blowup of $X$, with the parameter $b$ indicating which points have been blown up. Composing the maps $g_n$ gives a map $X_{n+1}\to X_1=X$; let $f_n{\colon\thinspace}X_{n+1}\to S^2$ be the composition of this map with the Lefschetz fibration $f$. (Equivalently, on each $n$-fold blowup $X^b=g_{n}^{-1}(b)$, $f_n|_{X^b}$ is the composition of the blowdown map with the Lefschetz fibration $f$.)
Write $f^{b}=f_{n}|_{X^b}$. $f^b{\colon\thinspace}X\#n\overline{\mathbb{C}P^2}\to S^2$ then has the same structure as $f$, except that if $k$ points on some fiber (in class $[\Phi]$) are among the blown up points, that (initially irreducible) fiber has been replaced by a reducible curve with components in classes $[\Phi]-E_1-\cdots -E_k$, $E_1,\ldots,E_k$, where the $E_i$ are classes of exceptional spheres. Straightforward local coordinate calculations show that, if none of the blown-up points are critical points of any of the $f_i$ ($i<n$), then the only intersection points between components are ordinary double points, and that near the double points $f^b$ has form $(z,w)\mapsto zw$. In particular, each $f^b=f_n|_{X^b}$ is still a Lefschetz fibration provided that no critical points of any of the intermediate fibrations are blown up in forming $X^b$.
Denote a point $b\in X_n$ by $(p_1,\ldots,p_n)$, where each $p_{j+1}\in X^{(p_1,\ldots,p_{j})}$. Let:
1. $X'_{n}$ be the set of $(p_1,\ldots,p_n)\in X_n$ such that no $p_{j+1}$ is a critical point of $f^{(p_1,\ldots,p_{j})}{\colon\thinspace}X^{(p_1,\ldots,p_j)}\to S^2$.
2. $X''_{n}$ be the set of $(p_1,\ldots,p_n)\in X_n$ such that no $p_{j+1}$ lies in a singular fiber of $f^{(p_1,\ldots,p_{j})}{\colon\thinspace}X^{(p_1,\ldots,p_j)}\to S^2$.
If $b\in X'_n$, then, our above remarks show that $f^b{\colon\thinspace}X^b\to
S^2$ is a Lefschetz fibration; if moreover $b\in X''_n$, then no fiber of $f^b$ will contain more than one critical point (and also none of the $n$ blowups involved in the creation of $X^b$ will be at a point on an exceptional divisor of a previous blowup).
*(i)* For any $b\in X'_n$, $F^b{\colon\thinspace}X^{b}_{r}(f^b)\to S^2$ shall denote the relative Hilbert scheme constructed from $f^b$ as in the Appendix of [@DS] and Section 3 of [@Smith].*(ii)* $\mathcal{X}^{n}_{r}(f)=\{(D,b): b\in X'_n,\,D\in
X^{b}_{r}(f^b)\}$. In particular we have a map $\mathcal{F}^n{\colon\thinspace}\mathcal{X}^{n}_{r}(f)\to S^2\times X'_n$.
For $b\in X''_n$, $X^b$ contains disjoint exceptional divisors $E_1,\ldots,E_n$, and our intention is to define an invariant counting sections of the various $X^{b}_{r}(f^b)$ which descend to curves Poincaré dual to $\alpha-2\sum PD(E_i)$, as $b$ ranges over $X''_n$. We have to be somewhat careful in the definition of this invariant, though, since our parameter space $X''_n$ is noncompact.
For $b\in X''_n$, $X^{b}_{r}(f^b)$ is a smooth symplectic manifold, as is the total space of $\mathcal{X}^{n}_{r}(f)\to S^2\times X'_n$.
That the relative Hilbert scheme constructed from any Lefschetz fibration (such as $f^b$ when $b\in X''_n$) in which there is at most one critical point per fiber is smooth is shown in Theorem 3.4 of [@Smith] (as noted in Remark 3.5 of [@Smith], Smith’s provision of a local coordinate description for the relative Hilbert scheme makes irrelevant his assumption that all of the fibers of the Lefschetz fibration are irreducible). When $b\in X'_{n}\setminus X''_n$, so that $f^b$, while still a Lefschetz fibration, may have more than one critical point per fiber, the individual $X^{b}_{r}(f^b)$ will tend not to be smooth near points on the Hilbert scheme of the singular fibers $\Sigma_0$ which are sent by the map $Hilb^{[r]}\Sigma_0\to
S^r\Sigma_0$ to divisors which contain more than one of the nodes of $\Sigma_0$. We will show presently, though, that the freedom to vary $b\in X'_{n}$ results in the total space $\mathcal{X}^{n}_{r}(f)$ still being smooth at these points.
To see this, note that Donaldson and Smith show (c.f. the proof of Proposition A.8 of [@DS]) that when $f$ only has one node per fiber, at a singular point of a fiber of $X_s(f)$ (corresponding to a divisor with points near the node of a fiber) the behavior of $F{\colon\thinspace}X_s(f)\to S^2$ is modeled by $(z_1,\ldots,z_{s+1})\mapsto
z_1z_2$. When there are multiple nodes in a fiber, then, the relative Hilbert scheme will be modeled near a point corresponding to a divisor containing $s_i$ copies of the nodes $p_i$ ($i=1,\ldots,l$) by the fiber product of the various maps $(z^{(i)}_1,\ldots,z^{(i)}_{s_i+1})\mapsto
z^{(i)}_{1}z^{(i)}_{2}$. This fiber product is the common vanishing locus of the various $z^{(i)}_{1}z^{(i)}_{2}-z^{(j)}_{1}z^{(j)}_{2}$ (which is of course singular where $z^{(i)}_{1}=z^{(i)}_{2}=0$ for all $i$).
More generally, though, if $p_i$ is a node lying near the fiber over zero, $X_{s}(f)\to S^2$ is modeled near points corresponding to divisors with points near $p_i$ by\
$(z^{(i)}_1,\ldots,z^{(i)}_{s})\mapsto
z^{(i)}_{1}z^{(i)}_{2}+f(p_i)$. In our present context the fibration map is $f^b$; say for notational simplicity that $b=(p_1,\ldots,p_n)$ gives rise to an $n$-fold blowup with all exceptional divisors in the same fiber (of course if some exceptional divisors are in different fibers we can work fiber-by-fiber and reduce to this case). The space $\mathcal{X}^{n}_{r}(f)$ is then, at worst, modeled locally by $$\label{model}
\{(\vec{z}^{(0)},\vec{z}^{(1)},\ldots,\vec{z}^{(n)},q_1,\ldots,q_n):
z^{(0)}_{1}z^{(0)}_{2}=z^{(i)}_{1}z^{(i)}_{2}+f^{(p_1,\ldots,p_{i-1})}(q_i)\}.$$ Here $\vec{z}^{(0)}$ are the coordinates on the relative Hilbert scheme corresponding to divisors which contain any nodes that may have existed in our fiber before blowing up (and we are of course assuming throughout that the original $f$ was chosen so that there is at most one such). The $q_i$ are elements of a coordinate chart centered on $p_i\in
X^{(p_1,\ldots,p_{i-1})}$. But (\[model\]) defines a smooth manifold at any point with $q_i=p_i$ as long as none of the $p_i$ are critical points for $f^{(p_1,\ldots,p_{i-1})}$, and this latter condition is precisely ensured by the fact that $b\in
X'_n$.
This shows that $\mathcal{X}^{n}_{r}(f)$ is smooth; the existence of a symplectic structure on it then follows exactly as in the proof of the existence of a symplectic structure on $X_r(f)$ in [@DS]: where $\mathcal{X}^{n}_{r}(f)$ fails to be a fibration we have a local Kähler model for it, and we can extend the resulting form to the entire manifold by the usual methods of Gompf and Thurston.
We consider almost complex structures $J$ on the $X^{b}_{r}(f^b)$ which make the fibration maps $F^b{\colon\thinspace}X^{b}_{r}(f^b)\to S^2$ pseudoholomorphic and have the following special type: for each reducible fiber of $X^b$, letting $E$ denote the union of the spherical components of that fiber, we require that there exist neighborhoods $U\supset V$ of $E$ with $f^b(U)=f^b(V)=W\subset
S^2$ and almost complex structures $J_{1}^{q}$ and $J_{2}^{q}$ on the restricted relative Hilbert schemes $X_q(f^b|_U)$ and $X_{r-q}(f^b|_{(f^b)^{-1}(W)-V})$ such that the natural “addition map” $X^{b}_{q}(f^b|_U)\times_{F^b}
X^{b}_{r-q}(f^b|_{(f^b)^{-1}(W)-V})\to X^{b}_{r}(f^b)$ is $(J_{1}^{q}\times_{F^b}J_{2}^{q}, J)$-holomorphic; moreover, we require that $J_{1}^{q}$ agree with the complex structure induced (via the algebro-geometric description for the relative Hilbert scheme given in Section 3 of [@Smith]) by an integrable complex structure on $U\supset E$ with respect to which $f^b$ is holomorphic. Note that one way of forming such a $J$ is by taking any almost complex structure on $X^{b}_{r}(f^b)$ which agrees near the singular fibers with the almost complex structure $\mathbb{J}_j$ tautologically corresponding to a structure $j$ on $X^{b}$ which is integrable near the singular fibers of $X^b$. If $j$ is instead integrable only on the neighborhood $U$ of the exceptional spheres, we still obtain a Hölder almost complex structure satisfying the requirement, which may then be Hölder-approximated by smooth almost complex structures also satisfying the requirement by smoothing the almost complex structures $J_{2}^{q}$ in a coherent way at points of the $X_{r-q}(f^b|_{f^{-1}(W)-V})$ corresponding to divisors having points missing $U$.
Let $\mathcal{J}$ denote the space of smooth tame almost complex structures on $\mathcal{X}^{n}_{r}(f)$ which restrict to each $X^{b}_{r}(f^b)=(\mathcal{F}^n)^{-1}(S^2\times\{b\})$ as a $J$ of the above form. For each $b$, the blowdown map $\pi^b{\colon\thinspace}X^b\to X$ naturally induces a generically injective map $\Pi^b{\colon\thinspace}X^{b}_{r}(f^b)\to X_r(f)$ on relative Hilbert schemes. For $J\in
\mathcal{J}$ we obtain commutative diagrams $$\begin{CD}
{X^{b}_{q}(f^b|_U)\times_{F^b}
X^{b}_{r-q}(f^b|_{(f^b)^{-1}(W)-V})}@>>>{X^{b}_{r}(f^b)}\\
@VVV @ VV{\Pi^b}V\\
{X_{q}(f|_{\pi^b(U)})\times_{F}
X_{r-q}(f^b|_{f^{-1}(W)-\pi^b(V)})}@>>>{X_r(f)}\\
\end{CD}$$ in which $\Pi^b$ pushes $J$ forward to a smooth almost complex structure $J_b$ on $X_r(f)$. The $J_b$ vary smoothly in $b$, and indeed extend by continuity to a smoothly $X_n$-parametrized family of almost complex structures on $X_{r}(f)$ (rather than just an $X'_n$-parametrized family). Since our sections of the $F^b{\colon\thinspace}X^{b}_{r}(f^b)\to S^2$ pass through all of the fibers of $F^b$, restricting our almost complex structures to behave in this way near the special fibers of $F^b$ will not prevent moduli spaces of $J$-holomorphic sections of the $X^{b}_{r}(f^b)$ from being of the expected dimension for generic $J\in \mathcal{J}$.
For $\alpha\in H^2 (X;\mathbb{Z})$, $b\in X''_n$, and $e_i$ ($i=1,\ldots,n$) the Poincaré duals to the exceptional divisors of the blowups which form $X^{b}$, note that the expected complex dimension of the space of curves Poincaré dual to $\alpha-2\sum
e_i$ is $d(\alpha-2\sum e_i)=d(\alpha)-3n$, so since the the *real* dimension of $X''_n$ is $4n$ we would expect the space of such curves appearing in any $X^b$ as $b$ ranges over $X''_n$ to have complex dimension $d(\alpha)-n$.
\[cpct2\] Let $\alpha\in H^2(X;\mathbb{Z})$, and choose a generic set $\Omega$ of $d(\alpha)-n$ points in $X$. For generic $J\in \mathcal{J}$, and also for generic paths $J_t$ in $\mathcal{J}$ connecting two such generic $J$, the spaces $$\mathcal{M}^{n}_{J,\Omega}(\alpha-2\sum e_i)=\{(s,b): b\in
X''_n,\, s\in c_{\alpha-2\sum e_i}\subset
\Gamma(X^{b}_{r}(f^b)),\, \overline{\partial}_J
s=0,\,\Omega\subset C_s\}$$ and $$\mathcal{PM}^{n}_{(J_t),\Omega}(\alpha-2\sum e_i)=\{(s,b,t): b\in
X''_n,\, s\in c_{\alpha-2\sum e_i}\subset
\Gamma(X^{b}_{r}(f^b)),\, \overline{\partial}_{J_t}
s=0,\,\Omega\subset C_s\}$$ are compact manifolds of real dimensions zero and one, respectively, provided that $r=\langle
\alpha,[\Phi]\rangle \geq g+3n$ where $g$ is the genus of the generic fiber of $f{\colon\thinspace}X\to S^2$.
That the dimensions will generically be as expected is a standard result (for the general theory of “parametrized Gromov–Witten invariants” of the sort that we are in the process of defining see [@Ruan], though the compactness result proved presently makes much of Ruan’s machinery unnecessary for our purposes), so we only concern ourselves with compactness.
Let $(s^m,b^m)$ be a sequence of $J$-holomorphic sections (or $J_{t_m}$-holomorphic sections with $J_{t_m}\to J$) from either of the sets at issue. *A priori*, there are two possible sources of noncompactness: the $b^m$ might have a limit in $X_n\setminus X''_n$, or the $b^m$ might converge to $b\in X''_n$ with the $s^m$ converging to a bubble tree. As usual for section-counting invariants, we can eliminate the second possibility: because $J|_{X^{b}_{r}(f)}$ makes $X^{b}_{r}(f)\to
S^2$ holomorphic, any bubbles must be contained in the fibers, and so the section component of the resulting bubble tree would descend to a set Poincaré dual to $\alpha-2\sum e_i-PD(i_*B)$, where $B$ is some class in one of the fibers $(f^b)^{-1}(t)$ of the fibration $f^b:X^b\to S^2$. If $(f^b)^{-1}(t)$ is irreducible, $B$ will necessarily be a positive multiple of the fundamental class of the fiber, and just as in Section 4 of [@Smith] we will have $d(\alpha-2\sum e_i-PD(i_*B))\leq
d(\alpha-2\sum e_i)-(r-g+1)$, which rules such bubble trees out for generic one-parameter families of $J$. If $(f^b)^{-1}(t)$ is reducible, with components in classes $[\Phi]-E$ and $E$, then $B$ will have form $m([\Phi]-E)+pE$ where $m,p\geq 0$ and at least one is positive, and a routine computation then yields that $$d(\alpha-2\sum
e_i-PD(i_*B))-d(\alpha-2\sum
e_i)=-m(r-g+1)-\frac{5}{2}(p-m)-\frac{1}{2}(p-m)^2,$$ which, since we have assumed that $r\geq g+3$, will always be negative when $m,p\geq 0$ and are not both zero. Thus for generic $J$ or $J_t$, none of the possible bubble trees appear.
There remains the issue that the $b^m$ might converge to some $b\notin X''_n$. We rule this out in two steps: first, we prove:
If $b\in X_n\setminus X'_n$ then $b^m$ cannot converge to $b$.
Let $\pi^{b^m}{\colon\thinspace}X^{b^m}\to X$ be the blowdown map, and\
$\Pi^{b^m}{\colon\thinspace}X^{b^m}_{r}(f^{b^m})\to X_r(f)$ the map that it induces on relative Hilbert schemes. By the definition of our space $\mathcal{J}$ of almost complex structures, the $\Pi^{b^m}\circ
s^m$ are $J_{b^m}$-holomorphic sections of $X_r(f)$ in the class $c_{\alpha}$, and so converge modulo bubbling to a $J^b$-holomorphic section $\bar{s}$ of $X_r(f)$. In fact, we can rule out bubbling, since we can assume that the family $J_b$ is regular as a $4n$-real-dimensional family of almost complex structures on $X_r(f)$, and so as above no bubbles can form in the limit thanks to the fact that all fibers of $f$ are irreducible and $$\begin{aligned}
2n+d(\alpha-mPD[\Phi])&=d(\alpha)+2n-m(r-g+1)\\&\leq
d(\alpha)-n-(r-g+1-3n)<d(\alpha)-n \end{aligned}$$ by the hypothesis of the lemma.
Since $b\notin X'_n$, where $b=(p_1,\ldots,p_n)$ there will be some minimal $l$ such that $p_{l+1}$ is a critical point of $f^{(p_1,\ldots,p_{l})}{\colon\thinspace}X^{(p_1,\ldots,p_l)}\to S^2$. Suppose first that $p_{l+1}$ lies on just one irreducible component of its fiber (so that it is a double point of that component). Write $t^m=f^{(p^{m}_{1},\ldots,p^{m}_{l})}(p^{m}_{l+1})$ and $T=f^{(p_1,\ldots,p_l)}(p_{l+1})$. Now since $C_{s^m}\subset X^b$ meets the exceptional divisor formed by blowing up $p^{m}_{l+1}$ transversely exactly twice, we deduce that $\Pi\circ s^m\in
\Gamma(X_r(f))$ acquires a tangency to the diagonal at a divisor containing two copies of $\pi^{b^m}(p_{l+1}^{m})$; more specifically, assuming that $\bar{s}(T)$ corresponds to a divisor containing $p_{l+1}$ with multiplicity $q$, for large $m$ in a neighborhood $U$ around $T,t^m\in S^2$ we have a decomposition $\Pi\circ s^m|_U=+(s^{m}_{1},s^{m}_{2})$ into disjoint summands $s^{m}_{1}\in \Gamma(X_q(f)|_U)$ and $s^{m}_{2}\in
\Gamma(X_{r-q}(f)|_U)$, with $s^{m}_{1}$ tangent to the diagonal at a point of form $\{p^{m}_{l+1},p^{m}_{l+1},x_3,\ldots,x_q\}$. Since the divisors $s^{m}_{1}(t)$ and $s^{m}_{2}(t)$ are disjoint for $t\in U$, the smoothness of the $\Pi\circ s^m$ implies the smoothness of $s^{m}_{1}$ and $s^{m}_{2}$ over $U$. Similarly, where $V$ is a neighborhood of $p_{l+1}$ with $f(V)\subset U$ $\bar{s}$ splits near $T$ into disjoint sections $\bar{s}_1$ of $\mathcal{H}_q\cong X_{q}(f|_V)$ and $\bar{s}_2$ of $X_{r-q}(f|_{f^{-1}(f(V))-V})$; here $\mathcal{H}_q$ is the $q$-fold relative Hilbert scheme of the map $(z,w)\mapsto zw$. Moreover, we have $s^{m}_{1}\to \bar{s}_1$. But then since $p_{l+1}^{m}\to p_{l+1}$, $\bar{s}_1$ must then be tangent to the diagonal in $\mathcal{H}_q$ at a point corresponding to $\{(0,0),\ldots,(0,0)\}\in Sym^{q}\{zw=0\}$. This, however, is impossible, since $\bar{s}_1$ is a *section* of $\mathcal{H}_q$, so that $Im(d\bar{s}_1)_T$ cannot be tangent to the fiber, whereas according to Theorem \[appmain\] in Section \[app1\] the tangent cone to $\Delta\subset\mathcal{H}_q$ is contained in the tangent space to the fiber at $\bar{s}_1(T)$.
The other possibility is that $p_{l+1}$ is an intersection point between two irreducible components of its fiber, in which case one of those components is the exceptional sphere $E$ formed by a previous blowup (say at $p_a$). Where again $t^m=f^{(p^{m}_{1},\ldots,p^{m}_{l})}(p^{m}_{l+1})$, in local coordinate systems $U^m$ around $t^m$ (which may be shrinking but are scale-invariant) we have $$\Pi\circ s^m=\{c_m z,d_m z\}+s^{m}_{2}(z)$$ where $s^{m}_{2}$ is a local section of $X_{r-2}(f)$ which does not meet $z\mapsto\{c_m z,d_m z\}$. Now the fact that $p^{l+1}_{m}\to
p_{l+1}$ which is an intersection point between the fiber containing $p_{l+1}\in X^{(p_1,\ldots,p_l)}$ and the exceptional sphere of one of the blowups implies that, in $X$ (where the blowup has not yet taken place), the two branches $c_mz$ and $d_mz$ of $\Pi\circ s^m$ near $\pi^{b^m}(p^{l+1})$ both tend toward the vertical, so that $c_m,d_m\to \infty$. But then this implies that $|d(\Pi\circ s^m)_{t^m}|\to \infty$, which is impossible by elliptic regularity since $\Pi\circ s^m\to \bar{s}$.
Finally we show that, generically, if $b^m\to b\in X'_n$ then in fact $b\in X''_n$. Indeed, since $b\in X'_n$, so that $X^{b}_{r}(f^b)\subset \mathcal{X}^{n}_{r}(f)$, Gromov compactness on the symplectic manifold $\mathcal{X}^{n}_{r}(f)$ implies that after passing to a subsequence the sections $s^m$ will converge to some smooth section $\bar{s}$ of $X^{b}_{r}(f^b)$. Just as above, the fact that $\bar{s}$ is a smooth section implies that it misses the critical locus of $F^b{\colon\thinspace}X^{b}_{r}(f^b)\to S^2$; in particular, if $b\in X'_n\setminus X''_n$, $Im(\bar{s})$ is contained in the smooth part of the relative Hilbert scheme $X^{b}_{r}(f^b)$. But then a neighborhood of $Im(\bar{s})$ in $X^{b}_{r}(f^b)$ will be diffeomorphic to a neighborhood of $Im(s^m)$ in $X^{b^m}_{r}(f^{b_m})$ for large m, and so the index of the Cauchy-Riemann operator acting on perturbations of the former will be the same as the index of the Cauchy-Riemann operator acting on perturbations of the latter, namely $d(\alpha)-3n$. Hence since the real dimension of $X'_n\setminus
X''_n$ is $4n-2$, the expected complex dimension of the space of possible limits $\bar{s}$ with $b\in X'_n\setminus X''_n$ is $d(\alpha)-n-1$, so for generic $J$, and also for generic one-real-parameter families $J_t$, on $\mathcal{X}^{n}_{r}(f)$, no such limits $\bar{s}$ with $C_{\bar{s}}$ satisfying our $d(\alpha)-n$ incidence conditions will exist.
Given this compactness result, the standard cobordism argument permits us to make the following definition.
\[fds\] Let $\alpha$ be as in Lemma \[cpct2\]. $\mathcal{FDS}^{n}_{f}(\alpha-2\sum e_i)$ is then defined as the number of elements, counted with sign according to the spectral flow, in the moduli space $\mathcal{M}^{n}_{J,\Omega}(\alpha-2\sum
e_i)$ for generic $J$ and $\Omega$ as in Lemma \[cpct2\].
\[famsame\] Suppose that $\alpha$ is as in Lemma \[cpct2\] and is strongly $n$-semisimple. Then $$n!Gr_n(\alpha)=\mathcal{FDS}^{n}_{f}\left(\alpha-2\sum
e_i\right),$$ provided that $\langle \omega_{X},[\Phi]\rangle>\omega_{X}\cdot \alpha\geq g(\Phi)+3n$.
As in the proof of Theorem \[tildesame\], we may evaluate $Gr_n(\alpha)$ using an almost complex structure $j$ which makes the Lefschetz fibration $f$ pseudoholomorphic and which has the property that, for any of the curves $C=\bigcup_i C^i$ being counted by $Gr_n(\alpha)$, $j$ is integrable on a neighborhood of $\bigcup_i Crit(f|_{C^i})$; each intersection point between the $C^i$ occurs away from $\bigcup_i Crit(f|_{C^i})$; and $C$ misses the critical locus of the fibration $f$. For each $b\in X_n$, let $j_b$ be pullback of $j$ via the blowup $\pi^b{\colon\thinspace}X^b\to X$ (see Section \[app2\] for the proof that $j_b$ exists and is Lipschitz), so that $X^b\to X$ is $(j_b,j)$-holomorphic. Then, for any of the $n!$ elements $b$ of $X'_n$ corresponding to the $n!$ different orders in which the nodes of $C$ may be blown up, the proper transform $\tilde{C}$ of $C$ will be a curve in $X^b$ (with $b\in X'_n$ as a result of the fact that $C$ misses the critical points of $f$) Poincaré dual to $\alpha-2\sum e_i$. In fact, we claim that for a generic initial choice of $j$ these proper transforms $\tilde{C}$ are guaranteed to be the only $j_b$-holomorphic curves Poincaré dual to $\alpha-2\sum e_i$ in any $X^b$ which have no components contained in the fibers of $f^b{\colon\thinspace}X^b\to S^2$.
Indeed, suppose that $\tilde{C}=\cup_i\tilde{C}_i$ is a $j_b$-holomorphic curve in one of the $X^b$ Poincaré dual to $\alpha-2\sum e_i$, with the (possibly-multiply-covered) components $\tilde{C}_i$ Poincaré dual to $\beta_i-\sum
c_{ik}e_k$. We need to show that, where $\pi^b{\colon\thinspace}X^b\to X$ is the blowup, $\pi^b(\tilde{C})$ has $n$ nodes, located at the points $p_i,\ldots,p_n$ which were blown up to form $X^b$ (as $\pi^b(\tilde{C})$ is obviously a $j$-holomorphic curve Poincaré dual to $\alpha$). Now for each $k$, $\sum_i c_{ik}=-2$, while by positivity of intersections in $X^b$, we have each $c_{ik}\leq 0$. If $k$ is such that there are distinct $q$ and $s$ with $c_{qk}=c_{sk}=-1$, then the curves $\pi(\tilde{C}_q)$ and $\pi(\tilde{C}_s)$ intersect transversely at the point $p_k$, contributing the desired node. On the other hand, if $k$ is such that the only nonzero $c_{ik}$ is some $c_{qk}=-2$, then $\pi^b(C_q)$ might *a priori* be either a singly-covered curve Poincaré dual to $\beta_q$ which has a self-intersection at $p_k$, or a double cover of a curve in class $\beta_q/2$ which passes through $p_k$. However, the $n$-semisimplicity condition rules the second possibility out for generic choices of $j$, since we will have either $d(\beta_q/2)<0$ or $d(\beta_q/2)<d(\beta_q)-n\leq d(\alpha)-n$, and so no such curves satisfying our incidence conditions will exist.
We conclude, then, that the only $j_b$-holomorphic curves $\tilde{C}$ in any $X^b$ Poincaré dual to $\alpha-2\sum e_i$ are proper transforms of $j$-holomorphic curves which contribute to $Gr_n(\alpha)$. With this established, the proof of the theorem becomes almost just an application of our usual methods. Since the restriction of $j_b$ to the exceptional spheres is standard, we can choose smooth almost complex structures $j'_b$ which are integrable near the exceptional spheres and are $C^0$-close to the $j^b$. By Gromov compactness for $C^0$ convergence of almost complex structures [@IS] and the fact that $d(\alpha-2\sum
e_i)=-n$, we deduce as usual that for generic choices of these perturbed $j'_b$ each $\tilde{C}$ will have finitely many $j'_{b_i}$-holomorphic curves $\tilde{C}_1,\ldots,\tilde{C}_N$ near it (for various $b_i$ near $b$). On the relative Hilbert schemes we have almost complex structures $\mathbb{J}_{j'_b}$. If $\tilde{C}_i$ is one of the curves above with the intersections of its components resolved by the blowup $X^{b_i}\to X$, we define $r''(\tilde{C}_i)$ as the signed count of $J_{b'}$ holomorphic sections of $X^{b'}_{r}(f^{b'})$ near $s_{\tilde{C}_i}$ for $b'$ near $b_i$ and $J_{b'}$ a generic family of smooth almost complex structures Hölder-close to the $\mathbb{J}_{j_{b_i}}$.
For $C$ a curve contributing to the Gromov invariant with nodes resolved by $X^b\to X$ and proper transform $\tilde{C}$, we define the contribution $r'(C)$ of $C$ to $\mathcal{FDS}$ as $\sum_{i=1}^{n} r''(\tilde{C}_i)$ where the $\tilde{C}_i$ are obtained as above. When $j$ is integrable near $C$, each $j_{b'}$ will be integrable near $\tilde{C}$ and near the exceptional spheres of $X^{b'}$ for $b'$ near $b$,, so that the first perturbation of the $j_{b'}$ to $j'_{b'}$ is not necessary and the only $\tilde{C}_i$ is $\tilde{C}$ itself. Moreover, each $\mathbb{J}_{j_{b'}}$ will be integrable near $s_{\tilde{C}}$ for $b'$ near $b$, and so (under suitable nondegeneracy assumptions) both contributions will be 1. Further, exactly as in the proof of Theorem \[tildesame\], the contributions transform under variations in $j$ in the same way by virtue of the fact that $\mathcal{FDS}$ is independent of the almost complex structure used to define it. The agreement of the invariants then follows.
If $\alpha$ is only weakly $n$-semisimple, then if $C\in
PD(\alpha)$ is the disjoint union of a double cover of a square-zero torus with a curve having $n-1$ nodes, then the proper transform of $C$ under blowup at the nodes of $C$ and at any point on the torus gives a curve in some $X^b$ Poincaré dual to $\alpha-2\sum e_i$, even though $C$ does not contribute to $Gr_n(\alpha)$. On perturbing the family $(\mathbb{J}_{j_b})$ on $\mathcal{X}^{n}_{r}(f)$ to a generic family $(J_b)$, we might find that the sections corresponding to these curves contribute to $\mathcal{FDS}^{n}_{f}(\alpha-2\sum e_i)$. It seems reasonable, though, to believe that these additional contributions could be expressed in terms of the various other Gromov invariants of $X$, consistently with Conjecture \[conj\].
A review of Smith’s constructions {#review}
=================================
Our vanishing theorem for $\mathcal{FDS}$ will follow by adapting the constructions found in Section 6 of [@Smith] to the family context. Let us review these.
In addition to the relative Hilbert scheme, Donaldson and Smith constructed from the Lefschetz fibration $f{\colon\thinspace}X\to S^2$ a *relative Picard scheme* $P_r(f)$ whose fiber over a regular value $t\in S^2$ is naturally identified with the Picard variety $Pic^r\Sigma_t$ of degree-$r$ line bundles on $\Sigma_t$. Over each $\Sigma_t$, we have an Abel–Jacobi map $S^r\Sigma_t\to
Pic^r\Sigma_t$ mapping a divisor $D$ to its associated line bundle $\mathcal{O}(D)$; letting $t$ vary over $S^2$, we then get a map $$AJ{\colon\thinspace}X_r(f)\to P_r(f)$$ (that all of these constructions extend smoothly over the critical values of $f{\colon\thinspace}X\to S^2$ is seen in the Appendix of [@DS]). Meanwhile, by composing the Abel–Jacobi map for effective divisors of degree $2g-2-r$ with the Serre duality map $\mathcal{L}\mapsto\kappa_{\Sigma_t} \otimes
\mathcal{L}^{\vee}$, we obtain a map $$\begin{aligned}
i{\colon\thinspace}X_{2g-2-r}(f)&\to P_r (f) \nonumber\\ D&\mapsto
\mathcal{O}(\kappa-D). \end{aligned}$$ Moreover, using a result from Brill-Noether theory due to Eisenbud and Harris [@EH], Smith obtains that (cf. Theorem 6.1 and Proposition 6.2 of [@Smith]):
([@Smith])\[BN\] For a generic choice of fiberwise complex structures on $X$, if $3r>4g-11$ where $g$ is the genus of the fibers of $f{\colon\thinspace}X\to
S^2$, then $i{\colon\thinspace}X_{2g-2-r}(f)\to P_r (f)$ is an embedding. Further, $AJ{\colon\thinspace}X_r (f)\to P_r (f)$ restricts to $AJ^{-1}(i(X_{2g-2-r}(f)))$ as a $\mathbb{P}^{r-g+1}$-bundle, and is a $\mathbb{P}^{r-g}$-bundle over the complement of $i(X_{2g-2-r}(f))$.
The reason for this is that in general $AJ^{-1}(\mathcal{L})=\mathbb{P}H^0(\mathcal{L})$, which by Riemann-Roch is a projective space of dimension $r-g+h^1(\mathcal{L})$. The result of [@EH] ensures that for $r>(4g-11)/3$ and for generic families of complex structures on the $\Sigma_t$, none of the fibers of $f$ admit any line bundles $\mathcal{L}$ with degree $r$ and $h^1(\mathcal{L})>1$; then $Im(i)\subset P_r(f)$ consists of those bundles for which $h^1(\mathcal{L})=h^0 (\kappa\otimes \mathcal{L}^{\vee})=1$. To see the bundle structure, rather than just set-theoretically identifying the fibers, note that on any $\Sigma_t$, when we identify the tangent space to $Pic^r\Sigma_t$ with $H^0(\kappa_{\Sigma_t})$, the orthogonal complement of the linearization $(AJ_*)_D$ at $D\in S^r\Sigma_t$ consists of those elements of $H^0(\kappa_{\Sigma_t})$ which vanish along $D$ (this follows immediately from the fact that, after choosing a basepoint $p_0\in \Sigma_t$ and a basis $\{\phi_1,\ldots,\phi_g\}$ for $H^0(\kappa_{\Sigma_t})$ in order to identify $Pic^{r}(\Sigma_t)$ with $\mathbb{C}^g/H^1(\Sigma_t,\mathbb{Z})$, $AJ$ is given by $AJ(\sum
p_i)=\left(\sum\int_{p_0}^{p_i}\phi_1,\ldots,\sum\int_{p_0}^{p_i}\phi_g\right)$). If $AJ(D)\notin Im (i)$, so that $H^0(\kappa-D)=0$, this shows that $(AJ_*)_D$ is surjective, so that $AJ$ is indeed a submersion away from $AJ^{-1}(Im\, i)$. Meanwhile, if $\mathcal{L}=i(D')\in
Im (i)$, the above description shows that the only directions in the orthogonal complement of any $Im(AJ_*)_D$ with $AJ(D)=\mathcal{L}$ are those 1-forms which vanish at $D$, but since $AJ(D)=i(D')$ such 1-forms also vanish at $D'$ and so are also orthogonal to $Im(i_*)_{D'}$. So if $AJ(D)=i(D')$, $Im
(AJ_*)_D$ contains $T_{i(D')}(Im\, i)$, implying that $AJ$ does in fact restrict to $AJ^{-1}(Im\, i)$ as a submersion and hence as a $\mathbb{P}^{r-g+1}$ bundle.
Smith’s duality theorem, and also the vanishing result in this paper, depend on the construction of almost complex structures which are especially well-behaved with respect to the Abel-Jacobi map. From now on, we will fix complex structures on the fibers of $X$ satisfying the conditions of Lemma \[BN\]; these induce complex structures on the fibers of the $X_r(f)$ and $P_r(f)$, but on all of our spaces (including $X$) we still have the freedom to vary the “horizontal-to-vertical” parts of the almost complex structures. Almost complex structures agreeing with these fixed structures on the fibers will be called “compatible.”
The following is established in the discussion leading to Definition 6.4 of [@Smith].
([@Smith]) \[dual\] In the situation of Lemma \[BN\], for any compatible almost complex structure $J_1$ on $X_{2g-2-r}(f)$ and any compatible $J_2$ on $P_{r}(f)$ such that $J_2|_{T(Im\,i)}=i_*J_1$, there exist compatible almost complex structures $J$ on $X_r (f)$ with respect to which $AJ{\colon\thinspace}X_r(f)\to P_r(f)$ is $(J,J_2)$-holomorphic.
We outline the construction of $J$: Since $AJ{\colon\thinspace}AJ^{-1}(Im
\,i)\to X_{2g-2-r}(f)$ is a $\mathbb{P}^{r-g+1}$-bundle, given the natural complex structure on $\mathbb{P}^{r-g+1}$ and the structure $J_1$, the structures on $ AJ^{-1}(Im \,i)$ making this fibration pseudoholomorphic correspond precisely to connections on the bundle; since this bundle is the projectivization of the vector bundle with fiber $H^0(\kappa-D)$ over $D$, a suitable connection on the latter gives rise to a connection on our projective-space bundle and thence to an almost complex structure $J$ on $AJ^{-1}(Im \,i)$ making the restriction of $AJ$ pseudoholomorphic.
To extend $J$ to all of $X_r (f)$, we first use the fact that, as in Lemma 3.4 of [@DS], $$AJ_*{\colon\thinspace}\left(N_{ AJ^{-1}(Im
\,i)}X_r (f)\right)|_{AJ^{-1}(i(D))}\to (N_{Im\,i}P_r
(f))_{i(D)}$$ is modeled by the map $$\begin{aligned}
\{(\theta,[x])\in V^*\times\mathbb{P}(V)|\theta(x)=0\}&\to V^* \nonumber\\
(\theta,[x])&\mapsto \theta, \nonumber \end{aligned}$$ where $V=H^0(\kappa_{\Sigma_t}-D)$, so that the construction of Lemma 5.4 of [@DS] lets us extend $J$ to the closure of some open neighborhood $U$ of $ AJ^{-1}(Im \,i)$. But then since $AJ$ is a $\mathbb{P}^{r-g}$-bundle over the complement of $ AJ^{-1}(Im
\,i)$, the problem of extending $J$ suitably to all of $X_r(f)$ amounts to the problem of extending the connection induced by $J$ from $\partial U$ to the entire bundle, which is possible because, again, our bundle is the projectivization of a vector bundle and connections on vector bundles can always be extended from closed subsets.
Our vanishing results are consequences of the following:
([@Smith],p.965)\[neg\] Assume that $b^+(X)>b_1(X)+1$. For any fixed compatible smooth almost complex structure $J_1$ on $X_{2g-2-r}(f)$ and for generic smooth compatible almost complex structures $J_2$ such that $J_2|_{Im\,i}=i_*J_1$, all $J_1$-holomorphic sections of $P_r(f)$ are contained in $i(X_{2g-2-r}(f))$.
This follows from the fact that, as Smith has shown, the index of the $\bar{\partial}$-operator on sections of $P_r(f)$ is $1+b_1-b^+$, which under our assumption is negative, and so since $J_2$ may be modified as we please away from $Im\,i$, standard arguments show that for generic $J_2$ as in the statement of the lemma all sections will be contained in $Im\,i$.
Proof of Theorem \[famvan\] {#vanproof}
===========================
\[bigvan2\] If $b^+(X)>b_1(X)+4n+1$, then $\mathcal{FDS}^{n}_{f}(\alpha-2\sum e_i)=0$ for all $\alpha \in
H^2(X;\mathbb{Z})$ such that $r=\langle \alpha,[\Phi]\rangle$ satisfies $r>\max\{g(\Phi)+3n,2g(\Phi)-2\}$.
Let $(J'_b)_{b\in X_n}$ be a smooth family of almost complex structures on the relative Picard scheme $P_r(f)$ such that
- For each $b$, the map $G{\colon\thinspace}P_r(f)\to
S^2$ is pseudoholomorphic with respect to $J'_b $, and for all critical values $t$ of $f$ $J$ agrees near $G^{-1}(t)$ with the standard complex structure on the relative Picard scheme induced by an integrable complex structure near $f^{-1}(t)$;
- For each $b=(p_1,\ldots,p_n)$, where $t_i=f\circ\pi^{(p_1,\ldots,p_{i-1})}(p_i)$, $J'_b$ also agrees near each $G^{-1}(t_i)$ with the standard complex structure induced by an integrable complex structure near $f^{-1}(t_i)$.
Thanks to the assumption that $b^+(X)>b_1(X)+4n+1$ and the fact that the index of the $\bar{\partial}$-operator on sections of $P_r(f)$ is $1+b_1-b^+$, for a generic such family $(J'_b)_{b\in X_n}$ there will be no $J'_b$ holomorphic sections of $P_r(f)$ for any $b$. Now, as in Section \[review\], since $r>2g-2$, so that $AJ{\colon\thinspace}X_r(f)\to P_r(f)$ is a projective-space bundle, we can construct a family $J_b$ of almost complex structures on $X_r(f)$ such that $AJ{\colon\thinspace}X_r(f)\to P_r(f)$ is $(J_b,J'_b)$-holomorphic for each $b$. By construction, for each $b$ $J_b$ agrees with the standard complex structure on the relative Hilbert scheme $F{\colon\thinspace}X_r(f)\to S^2$ near each singular fiber and also near each $F^{-1}(t_i)$ where the $t_i$ are as above. Since $X^b$ is formed from $X$ by performing blowups at points in $f^{-1}(t_i)$, for $b\in X'_n$ $J_b$ lifts to an almost complex structure $\tilde{J}_b$ on $X^{b}_{r}(f^b)$ such that the map $\Pi^b{\colon\thinspace}X^{b}_{r}(f^b)\to X_r(f)$ induced by blowup is $(\tilde{J}_b,J_b)$-holomorphic.
Let $J^m$ be almost complex structures on the $\mathcal{X}^{n}_{r}(f)$ from the Baire set in the definition of $\mathcal{FDS}$ which converge to an almost complex structure that agrees on each $X^{b}_{r}(f^b)$ with $\tilde{J}_b$. If the invariant were nonzero, we would obtain $J^m$-holomorphic sections $s^m$ of some $X^{b_m}_{r}(f^{b_m})$ ($b_m\in X'_n$); after passing to a subsequence we assume $b_m\to \bar{b}\in X_n$ (since $X_n$, though not $X'_n$, is compact). By the definition of our class of almost complex structures (see the text before Lemma \[cpct2\]) there are compatible almost complex structures $J^{m}_{b_m}$ on $X_{r}(f)$ such that $\Pi^{b_m}{\colon\thinspace}X^{b_m}_{r}(f^{b_m})\to X_r(f)$ is $(J^m,J^{m}_{b_m})$-holomorphic; further, we will have $J^{m}_{b_m}\to J_{\bar{b}}$. So the $\Pi^{b_m}\circ s^m$ are $J^{m}_{b_m}$-holomorphic sections of $X_{r}(f)$, whence after passing to a subsequence they converge modulo bubbling to a $J_{\bar{b}}$-holomorphic section $\bar{s}$. (As usual, even if bubbling occurs, the bubble tree will contain a component which is a $J_{\bar{b}}$-holomorphic section by virtue of the fact that all bubbles will be contained in the fibers.) But then $AJ\circ \bar{s}$ would be a $J'_{\bar{b}}$-holomorphic section, contradicting the fact that no $J'_b$-holomorphic sections exist for any $b\in X_n$.
The intermediate case where $\max\{g(\Phi)+3n+d(\alpha),(4g(\Phi)-11)/3\}<r\leq 2g(\Phi)-2$ takes slightly more work. In this case, as in Section \[review\] we use the fact that combining the Abel-Jacobi map with Serre duality gives a map $$i{\colon\thinspace}X_{2g-2-r}(f)\to P_{r}(f);$$ as before since $3r>4g-11$ generic choices of the complex structures on the fibers of $f$ result in this map being an embedding. Similarly to the proof of Lemma \[bigvan2\], consider families of almost complex structures $J''_b$ ($b\in
X_n$) on $X_{2g-2-r}(f)$ which make $X_{2g-2-r}(f)\to S^2$ holomorphic and are standard near the singular fibers and near the fibers containing the points which are blown up to form $X^b$. Form almost complex structures $J'_b$ on $P_r(f)$ restricting to $i(X_{2g-2-r}(f))$ as $i_*J''_b$ and which are also standard near the singular fibers and near the fibers containing the points which are blown up to form $X^b$. The fact that $b^+>b_1+1+4n$ implies that if the family $J'_b$ is chosen generically among almost complex structures with this property, then any $J'_b$ holomorphic sections of $P_r(f)$ for any $b$ must be contained in $i(X_{2g-2-r}(f))$.
We then form almost complex structures $J_b$ on $X_r(f)$ such that $AJ{\colon\thinspace}X_r(f)\to P_r(f)$ is $(J_b,J'_b)$-holomorphic. As in the proof of Lemma \[bigvan2\], a nonvanishing invariant $\mathcal{FDS}^{n}_{f}(\alpha-2\sum e_i)$ would give rise to a sequence of sections of $X_r(f)$ in the homotopy class $c_{\alpha}$ which converge modulo bubbling to a $J_{\bar{b}}$-holomorphic section $\bar{s}$ of $X_r(f)$. Since all fibers of $f$ are irreducible, any bubbles that arise will descend to a multiple covering of one of the fibers of $f$, and so for some $m\geq 0$ we will have $\bar{s}\in c_{\alpha-m PD[\Phi]}$ where as usual $[\Phi]$ is the class of the fiber.
$AJ\circ \bar{s}$ will then be a $J'_{\bar{b}}$-holomorphic section of $P_r(f)$, and so must be contained in $i(X_{2g-2-r}(f))$. By the construction of $i$, then, $i^{-1}\circ AJ\circ \bar{s}$ is a $J''_{\bar{b}}$-holomorphic section of $X_{2g-2-r}(f)$ in the homotopy class $c_{\kappa_X-\alpha+mPD[\Phi]}$.
Now one computes using the adjunction formula for the fiber $\Phi$ that $$\begin{aligned}
d(\kappa_{X^b}-\alpha+mPD[\Phi])
&=d(\kappa_X-\alpha)+d(m\Phi)+m\langle\kappa_X-\alpha,[\phi]\rangle
\\ &=d(\alpha)-\frac{m}{2}\langle \kappa_X,[\Phi]\rangle+m\langle
\kappa_X-\alpha,[\Phi]\rangle \\ &=d(\alpha)-m(r-g(\Phi)+1).\end{aligned}$$
Thus by choosing the $4n$-real-dimensional family $J''_b$ generically we ensure that $m=0$ thanks to the assumption that $r>d(\alpha)+g(\Phi)+3n$ in the statement of the theorem.
Now take a family of almost complex structures $j_b$ on $X$ which are standard near the singular fibers of the fibrations $f$ and also near the fibers containing the points blown up to form $X^b$; these induce tautological almost complex structures $\mathbb{J}_{j_b}$ on $X_{2g-2-r}(f)$. Let $J''^{m}_{b}$ be families of smooth almost complex structures on $X_{2g-2-r}(f)$ which are generic in the sense of the previous paragraph and which converge in Hölder norm to the $\mathbb{J}_{j_b}$. For each $m$ there is some $b_m$ such that $J''^{m}_{b_m}$ admits a holomorphic section in the class $c_{\kappa_{X}-\alpha}$, so Gromov compactness guarantees the existence of a $\mathbb{J}_{j_{b_0}}$-holomorphic section of some $X_{2g-2-r}(f)$ in $c_{\kappa_{X}-\alpha}$ for some $b_0$; this section then tautologcally corresponds to a $j_{b_0}$-holomorphic curve $C$ Poincaré dual to $\kappa_{X}-\alpha$; setting $j=j_{b_0}$, this is the curve that we desire.
To get the $j$-holomorphic curve Poincaré dual to $\alpha$, we simply consider the almost complex structures $j_b$ on the members $X^b$ of the family blowup induced in the almost complex category by $j$. Let $j_{b}^{m}$ be a sequence of almost complex structures $C^0$-approximating the $j_b$ which are integrable near the exceptional spheres, and apply Gromov compactness to a sequences of almost complex structures on $\mathcal{X}^{n}_{r}(f)$ whose restrictions to $X^{b}_{r}(f^b)$ Hölder-approximate the family $\mathbb{J}_{j_{b}^{m}}$; in this way our nonvanishing invariant guarantees the existence of a $\mathbb{J}_{j_{b_m}^{m}}$-holomorphic section of some $X^{b_m}_{r}(f^{b_m})$ in the class $c_{\alpha-2\sum e_i}$ and so of a $j_{b_m}^{m}$-holomorphic curve Poincaré dual to $\alpha-2\sum e_i$. Appealing to Gromov compactness for these curves then gives a $j_b$-holomorphic curve, and this latter is sent by the blowdown map to the $j$-holomorphic curve which we desire. Theorem \[famvan\] is thus proven.
If $X$ admits an integrable complex structure $j$ making the fibration holomorphic, then for our original family of almost complex structures $j_b$ we can take the constant family $j$, justifying a statement made near the end of the introduction. For arbitrary $j$, though, this argument does not work, because it was crucial in the construction of the curve Poincaré dual to $\kappa_{X}-\alpha$ that each of the $j_b$ was integrable near the fibers containing the points blown up in forming $X^b$.
Two technical matters
=====================
Blowing up a point in an almost complex manifold {#app2}
------------------------------------------------
In the proof of Theorem \[famsame\] we have used the fact that, if $\pi{\colon\thinspace}X'\to X$ is the blowup of a 4-manifold at a point and $J$ is an almost complex structure on $X$, then there is a Lipschitz almost complex structure $J'$ on $X'$ such that $\pi$ is $(J',J)$-holomorphic. Since we have not found a proof of this fact in the literature, we present one here. As the dimension of $X$ does not affect the argument, we prove the result for almost complex manifolds of arbitrary complex dimension $n$. The blowup, of course, has the effect of replacing the point $p$ being blown up with an exceptional divisor $E\cong
\mathbb{C}P^{n-1}$; we note that, as will be seen in the proof, $J'|_{TE}$ agrees with the standard complex structure on $\mathbb{C}P^{n-1}$. If $(X,\omega)$ is symplectic, recall from, *e.g.*, Chapter 7 of [@MS2] that $X'$ can be endowed with symplectic forms $\omega_{\operatorname{\epsilon}}$ for small $\operatorname{\epsilon}>0$, with the parameter $\operatorname{\epsilon}$ reflecting the size of the exceptional divisor $E$ in the symplectic manifold $(X',\omega_{\operatorname{\epsilon}})$. One can easily check that if the almost complex structure $J$ on $X$ is $\omega$-tame, then $J'$ will be $\omega_{\operatorname{\epsilon}}$-tame for small enough $\operatorname{\epsilon}$.
Our method only proves Lipschitz regularity for $J'$; it is unclear whether $J'$ is differentiable in directions normal to $E$. In principle, one would also like to be able to blow up almost complex submanifolds $V\subset (X,J)$ of arbitrary dimension in the almost complex category. Our method does not readily extend to show that the pullback of $J$ under the blowup extends even continuously over the exceptional divisor of the blowup when $\dim V>0$. Nonetheless, the case of blowing up a point suffices for our application.
We begin with the following lemma, which will later be used to construct coordinate charts on the blowup.
\[fix\] Let $J$ be an almost complex structure on $\mathbb{C}^n$ agreeing at the origin with the standard complex structure $J_0$. Given $\kappa_0\in \mathbb{C}P^{n-1}$ there exists a constant $\rho_0$ with the following property. Let $\rho<\rho_0$ and let $U_{\rho}$ be the ball of radius $\rho$ around $\kappa_0$ in $\mathbb{C}P^{n-1}$ and $D_{\rho}$ the disc of radius $\rho$ in $\mathbb{C}$. There is a smooth map $$\Theta{\colon\thinspace}D_{\rho}\times U_{\rho}\to \mathbb{C}^n$$ such that each $\Theta|_{D_{\rho}\times \{\kappa\}}$ ($\kappa\in
U_{\rho}\subset \mathbb{C}P^{n-1}$) is an embedding whose image is a $J$-holomorphic disc which is tangent at the origin to the line $l_{\kappa}\subset \mathbb{C}^n$ determined by $\kappa$.
The proof quite closely parallels some of the arguments in Section 5 of [@TSW]; we outline it for completeness. By a complex linear change of coordinates we may assume that $\kappa_0 =[1:0:\cdots :0]$. Where $c=(c_1,\ldots,c_{n-1})\in (D_{\rho})^{n-1}$ and $\kappa=[1:\kappa_1:\cdots :\kappa_{n-1}]$ is close to $[1:0:\cdots :0]$, we search for a $J$-holomorphic disc $$q_{c,\kappa}(z)=(z,c_1+\kappa_1
z+u_1(c,\kappa,z),\ldots,c_{n-1}+\kappa_{n-1}z+u_{n-1}(c,\kappa,z))$$ defined for $z\in D_{\rho}$. As in [@TSW], this is equivalent to a system of equations $$\frac{\partial
u_i}{\partial
\bar{z}}=Q_i\left(c,\kappa,u_1(c,\kappa,z),\ldots,u_{n-1}(c,\kappa,z)\right)$$ such that for certain constants $\gamma_k$ we have $$\label{qsmall} \|Q_i\|_{C^k}\leq \gamma_k\|J-J_0\|_{C^k(D_{2\rho}^{n})}.$$ Note that by decreasing $\rho$ and rescaling the coordinates we can make the right hand side of (\[qsmall\]) as small as we like.
Now introduce a cutoff function $\chi_{\rho}{\colon\thinspace}\mathbb{C}\to
[0,1]$ which equals $1$ for $|z|<\rho$ and $0$ for $|z|>3\rho/2$, and search for a solution to $$\frac{\partial
u_i}{\partial\bar{z}}=\chi_{\rho}Q_i \quad (i=1,\ldots,n-1)$$ by, on the class of $(n-1)$-tuples of $C^{2,1/2}$ functions $u_i$ restricting to the circle of radius $4\rho$ around zero in the span of $\{e^{ik\theta}|k<0\}$, searching for a tuple $(u_1,\ldots,u_{n-1})$ obeying $$\label{fixedpt}
\left(
u_i(z)=\frac{1}{\pi}\int\frac{\chi_{\rho}Q_i(c,\kappa,u_i(c,z))}{z-w}d^2w\right)_{i=1,\ldots,n-1}$$ Applying the contractive mapping theorem on this class of functions (viewed as a Banach space using the $(n-1)$-fold direct sum of the norm used on p. 886 of [@TSW]), thanks to the smallness of the $Q_i$ we can find a unique small solution of (\[fixedpt\]). Furthermore as in Lemma 5.5 of [@TSW] the solution varies smoothly in each of $z$, $c$, and $\kappa$, and satisfies bounds $$\left|\frac{\partial
u}{\partial c_i}\right|<C\rho,\quad \left|\frac{\partial
u}{\partial \kappa_i}\right|<C\rho^2,\quad
\|u\|_{C^0}<C(\rho^2+\rho(|c|+|\kappa|)),\quad
\|u\|_{C^1}<C(\rho+(|c|+|\kappa|)).$$
Letting $\sigma$ denote the map which assigns to $(c,\kappa)$ the pair consisting of $q_{c,\kappa}(0)$ and the tangent space to $Im\, q_{c,\kappa}$ at $q_{c,\kappa}(0)$, the implicit function theorem then allows us to solve the equation $\sigma(c,\tilde{\kappa})=((0,\ldots,0),\kappa)$ for $c$ and $\tilde{\kappa}$ in terms of $\kappa$. The desired map $\Theta$ is then $$\begin{aligned}
\Theta{\colon\thinspace}D_{\rho}\times U_{\rho}&\to \mathbb{C}^{n} \\
(z,\kappa)&\mapsto q_{c(\kappa),\tilde{\kappa}(\kappa)}(z).\end{aligned}$$
For any even-dimensional manifold $X$ with $p\in X$, we form the blowup $X'$ of $X$ at $p$ as a topological manifold by removing a ball $B^{2n}$ around $p$, embedding $B^{2n}$ in $\mathbb{C}^n$ in standard fashion, and replacing $B^{2n}$ in $X$ by $B'= \{(l,e)\in
\mathbb{C}P^{n-1}\times \mathbb{C}^n|e\in l\cap B^{2n}\}$. The blowdown map $\pi{\colon\thinspace}X'\to X$ is of course just the identity outside $B'$ and the map $(l,e)\to e$ inside $B'$. The exceptional divisor is $E=\{(l,e)\in B'|e=0\}\subset X'$.
If $\kappa_0=[1:0:\cdots :0]$ in Lemma \[fix\] and we write $\kappa$ near $\kappa_0$ as $[1:\kappa_1\cdots:\kappa_{n-1}]$, the map $\Theta$ has the form $$(z,\kappa)\mapsto (z,\kappa_1z+
\tilde{u}_1(\kappa,z),\ldots,\kappa_{n-1}z+\tilde{u}_{n-1}(\kappa,z))$$ where the $\tilde{u}_i$ are smooth functions satisfying $|\tilde{u}_i(\kappa,z)|<C|z|^2$ for an appropriate constant $C$. (In the notation of the proof of Lemma \[fix\], $\tilde{u}_i(\kappa,z)=u_i(c(\kappa),\tilde{\kappa}(\kappa),z)+(\tilde{\kappa}_i(\kappa)-\kappa_i)z$.)
We hence obtain a local homeomorphism $\tilde{\Theta}=\tilde{\Theta}_{\kappa_0}{\colon\thinspace}D^2\times D^2\to
\tilde{\mathbb{C}}^n$ such that, where $\pi{\colon\thinspace}\tilde{\mathbb{C}}^n\to\mathbb{C}^n$ is the blowdown, $\pi\circ
\tilde{\Theta}=\Theta$. We use the $\tilde{\Theta}_{\kappa_0}$ as $\kappa_0$ varies over $\mathbb{C}P^{n-1}$ as an atlas for $\tilde{\mathbb{C}}^n$ near the exceptional divisor $E$ (away from $E$ we of course just use charts pulled back by $\pi$ from charts on $\mathbb{C}^n$ not containing the origin). From the definition of the $\tilde{\Theta}_{\kappa_0}$ and the fact that tangencies of $J$-holomorphic curves in $\mathbb{C}^n$ are $C^1$-diffeomorphic to tangencies between $J_0$-holomorphic curves [@Sik], one can see that the transition functions have the form $$\tilde{\Theta}_{\kappa_0}^{-1}\circ\tilde{\Theta}_{\kappa'_0}(z,\kappa_1,\ldots,\kappa_{n-1})=\qquad
\qquad$$ $$\left(z,\kappa_1+z^{-1}(f_1(\kappa)z^2+O(|z|^3)),\ldots,\kappa_{n-1}+z^{-1}(f_{n-1}(\kappa)z^2+O(|z|^3))\right),$$ and in particular are $C^1$. We have thus provided an atlas for $\tilde{\mathbb{C}}^{n}$ as a $C^1$ manifold.
This atlas depends on the almost complex structure $J$, and it is worth noting that the charts corresponding to different $J$ might not be $C^1$-related. For example, for a particular $J$ $\Theta_{[1:0:\cdots:0]}$ could conceivably have the form $$\Theta_{[1:0:\cdots:0]}(z,\kappa_1,\ldots,\kappa_n)=(z,\kappa_1z+\bar{z}^2,\kappa_2z,\ldots,\kappa_{n-1}z).$$ In this case, in terms of the *standard* smooth coordinates on $\tilde{\mathbb{C}}^n$ (equivalently, those induced by the above construction using the standard complex structure $J_0$ ), $$\tilde{\Theta}_{[1:0:\cdots:0]}(z,\kappa_1,\ldots,\kappa_{n-1})=(z,\kappa_1+\bar{z}^2/z,\kappa_2,\ldots,\kappa_{n-1}),$$ which is Lipschitz but not $C^1$ along the exceptional divisor $\{z=0\}$. Of course, these resulting manifolds are still abstractly $C^1$-diffeomorphic; this is somewhat reminiscent of the fact that distinct complex structures on a Riemann surface $\Sigma$ induce smooth charts on the symmetric products $S^d\Sigma$ which are related by transition maps that are only Lipschitz, as noted for instance in Remark 4.4 of [@Sa].
Let $\pi{\colon\thinspace}\tilde{\mathbb{C}}^n\to \mathbb{C}^n$ denote the blowup of $\mathbb{C}^n$ at the origin, and let $J$ be an almost complex structure on $\mathbb{C}^n$ agreeing with the standard almost complex structure $J_0$ at the origin. Then there is a unique Lipschitz continuous almost complex structure $\tilde{J}$ on $\tilde{\mathbb{C}}^n$ such that $\pi$ is a $(\tilde{J},J)$ holomorphic map.
Let $E\cong\mathbb{C}P^{n-1}$ denote the exceptional divisor of the blowup $\pi$. Of course, $\pi$ restricts to a diffeomorphism $\tilde{\mathbb{C}^n}\setminus E\to
\mathbb{C}^n\setminus (0,\ldots,0)$, so our $\tilde{J}$ must agree away from $E$ with $\pi^{*}J=\pi_{*}^{-1}\circ J\circ\pi_*$ away from $E$ and uniqueness even of a continuous almost complex structure $\tilde{J}$ is clear from the fact that $\tilde{\mathbb{C}}^{n}\setminus E$ is dense in $\tilde{\mathbb{C}}^{n}$. We show now that $\pi^{*}J$ extends over $E$ in Lipschitz fashion by exhibiting a Lipschitz continuous basis of vector fields for its antiholomorphic tangent space $T^{0,1}\subset T\tilde{\mathbb{C}}^n\otimes \mathbb{C}$ near any given point $x\in E$.
Lemma \[fix\] and the remarks thereafter provide us with one element of this basis: the maps $\Theta_{\kappa_0}$ map each $D_{\rho}\times \{\kappa\}$ diffeomorphically to a $J$-holomorphic disc $\Delta_{\kappa}$ in $\mathbb{C}^n$ in a way that varies smoothly in $\kappa$. We then obtain a (complexified) vector field $\tilde{\alpha}_{\kappa}$ along each $D_{\rho}\times
\{\kappa\}$ defined by the property that $\alpha_{\kappa}=(\Theta_{\kappa_0})_{*}\tilde{\alpha}$ generates the $J$-antiholomorphic tangent space to $\Delta_{\kappa}$. Choosing the $\alpha_{\kappa}$ to depend smoothly on $\kappa$ causes the $\tilde{\alpha}_{\kappa}$ to do so as well, and so to give a vector field $\alpha$ on a neighborhood of our basepoint $x$ which is transverse to $E$ and which is antiholomorphic for the pulled back almost complex structure $\pi^{*}J$ where the latter is defined.
After a complex linear change of coordinates on $\mathbb{C}^n$ we may assume that $x=([1:0:\cdots:0],(0,\ldots,0))$ and $\pi(x)=(0,\ldots,0)$. In terms of the coordinate chart given by $\Theta_{[1:0\cdots:0]}$, the blowdown map $\pi$ has the form $$(s,t_1,\ldots,t_{n-1})\mapsto (s,st_1+u_1(s,t_1,\ldots,t_{n-1}),\ldots,st_{n-1}+u_{n-1}(s,t_1,\ldots,t_{n-1})),$$ where $|u_{i}(s,t_1,\ldots,t_{n-1})|<C|s|^2$. Away from the exceptional sphere $s=0$, this is a diffeomorphism whose complexified linearization with respect to the coordinates $(s,\bar{s},t_1,\bar{t}_1,\ldots,t_{n-1}\bar{t}_{n-1})$ has inverse of the form $$((\pi_*)^{-1})_{\pi(s,t_1,\ldots,t_{n-1})}=\left(\begin{matrix}
1&0&\cdot&\cdot&\cdot&\cdot&0\\
0&1&0&\cdot&\cdot&\cdot&0\\
-t_1/s&0&1/s&0&\cdot&\cdot&0\\
0&-\bar{t}_1/\bar{s}&0&1/\bar{s}&0&\cdots&0\\
\vdots&\vdots& \vdots& \ddots&\ddots&\ddots&\vdots\\
-t_{n-1}/s&0&\cdot&\cdots&0&1/s&0\\
0&-\bar{t}_{n-1}/s&0&\cdot&\cdots&0&
1/\bar{s}\end{matrix}\right)+B(s,t_1,\ldots,t_n),$$ where $B$ is smooth away from $s=0$ and bounded (but not necessarily continuous) as $s\to 0$.
Write the coordinates on $\mathbb{C}^n$ as $(w,z_1,\ldots,z_{n-1})$. Since $J$ agrees with $J_0$ at the origin, for $i=1,\ldots, n-1$ there are $J$-antiholomorphic vector fields $$\begin{aligned}
\beta_i=\partial_{\bar{z}_i}+\sum_j
a_{ij}(z_1,\ldots,z_n)\partial_{z_j}+\sum_{j\neq
i}b_{ij}(z_0,\ldots,z_n)\partial_{\bar{z}_j}+c_i(z_0,\ldots,z_n)\partial_{w},\end{aligned}$$ where $a_{ij}(0,\ldots,0)=b_{ij}(0,\ldots,0)=c_i(0,\ldots,0)=0$. Away from $E$ we then have $$\begin{aligned}
(\pi_{*}^{-1}\beta_i)_{(u,v_1,\ldots,v_{n-1})}&=\frac{1}{\bar{u}}\partial_{\bar{v}_i}+\sum_j
a_{ij}(\pi(u,v_1,\ldots,v_{n-1}))\left(\frac{1}{u}\partial_{v_j}\right)\\&+
\sum_{j\neq i}
b_{ij}(\pi(u,v_1,\ldots,v_{n-1}))\left(\frac{1}{\bar{u}}\partial_{\bar{v}_j}\right)\\&+c_i(\pi(u,v_1,\ldots,v_{n-1}))\left(\partial_u-\sum_j\frac{v_j}{u}\partial_{v_j}\right)+\tilde{\gamma_i}\end{aligned}$$ where $\tilde{\gamma_i}=B\beta_i$ has bounded coeffecients. So $$\begin{aligned}
\tilde{\beta}_i&:=\bar{u}\pi_{*}^{-1}\beta_i=\partial_{\bar{v}_i}+\sum\frac{\bar{u}}{u}\left(a_{ij}(u,uv_1,\ldots,uv_{n-1})-v_jc_{ij}(u,uv_1,\ldots,uv_{n-1})\right)\partial_{v_j}
\\&+\sum_{j\neq
i}b_{ij}(u,uv_1,\ldots,uv_{n-1})\partial_{\bar{v}_j}+\bar{u}c_i(u,uv_1,\ldots,uv_{n-1})\partial_w+\bar{u}\tilde{\gamma_i}\end{aligned}$$ is an antiholomorphic tangent vector for $\pi^{*}J$ away from $E=\{u=0\}$. Further, we note that since $a_{ij},b_{ij}$, and $c_i$ are differentiable and vanish at the origin while $\tilde{\gamma_i}$ is $L^{\infty}$, so that $|a_{ij}(u,uv_1,\ldots,uv_{n-1})|$, $|b_{ij}(u,uv_1,\ldots,uv_{n-1})|$, $|c_{i}(u,uv_1,\ldots,uv_{n-1})|$, and $\|\bar{u}\gamma_i\|$ are all bounded by a constant times $|u|$, $\tilde{\beta}_i$ extends over $E$ in Lipschitz fashion, agreeing with $\partial_{\bar{v}_i}$ at $E$.
Hence, defining $\tilde{J}$ near $x$ by $$T^{0,1}_{\tilde{J}}=\langle
\tilde{\alpha},\tilde{\beta_1},\ldots,\tilde{\beta}_{n-1}\rangle,$$ we see that $\tilde{J}$ is Lipschitz and agrees with $\pi^{*}J$ where the latter is defined. So since $\tilde{J}$ preserves $TE$ and since at each point of $E$ there is a $\tilde{J}$-holomorphic disc transverse to $E$ mapped holomorphically to a $J$-holomorphic disc by $\pi$, we conclude that $\pi{\colon\thinspace}\tilde{\mathbb{C}}^n\to
\mathbb{C}^n$ is $(J',J)$-holomorphic.
Let $(X,J)$ be an almost complex manifold with $p\in
X$, and let $X'$ denote the blowup of $X$ at $p$. Then there is a unique almost complex structure $J'$ on $X'$ which is Lipschitz continuous such that $\pi{\colon\thinspace}X'\to X$ is $(J',J)$-holomorphic. Further $J'$ restricts to $E$ as the standard complex structure on $\mathbb{C}P^{n-1}$.
Since $\pi$ is a diffeomorphism away from $E=\pi^{-1}(p)$ (which thus determines $J'$ on $X'\setminus E$ as the smooth almost complex structure $\pi^{*}J$), this follows from the proposition and its proof by choosing a chart around $p$ which sends $(p,J|_{T_p X})$ to $(0,J_{0}|_{T_0\mathbb{C}^n})$ in $\mathbb{C}^n$ (as may easily be done by modifying any chart around $p$ by an appropriate real linear map).
The diagonal in the relative Hilbert scheme {#app1}
-------------------------------------------
Let $F{\colon\thinspace}\mathcal{H}_r\to D^2$ denote the $r$-fold relative Hilbert scheme of the map $f{\colon\thinspace}(z,w)\mapsto zw$; the spaces $\mathcal{H}_r\times \mathbb{C}^{s-r}$ form the local models for the relative Hilbert scheme $X_s(g)$ of a Lefschetz fibration $g$ near points of $X_s(g)$ which correspond to divisors containing $r$ copies of a critical point of $g$. In this subsection we prove the fact, used in the proof of the compactness result underlying the construction of $\mathcal{FDS}$, that at a point in the diagonal $\Delta$ of the relative Hilbert scheme $\mathcal{H}_r$ corresponding to the divisor in the nodal fiber $f^{-1}(0)$ consisting of $r$ copies of $(0,0)$, the tangent cone to the diagonal is contained in the tangent cone to the fiber $F^{-1}(0)\subset \mathcal{H}_r$. (Note that since the natural map $F^{-1}(t)\to S^r f^{-1}(t)$ is an isomorphism if and only if $t$ is a regular value of $f$, there are many points in $F^{-1}(0)$ corresponding to $\{(0,0),\ldots,(0,0)\}$, as will be seen later on when we review the definition of $\mathcal{H}_r$.) Our proof of this fact uses the description of the relative Hilbert scheme in terms of linear algebra provided in Section 3 of [@Smith] based on work of Nakajima [@N], and boils down to a rather arcane fact about the discriminants of the characteristic polynomials of certain matrices. It would certainly not surprise us if there exists a more elegant way of proving this result via algebraic geometry, but the argument we give presently is the only one we have at the moment. As will be seen later on, the relevant characteristic polynomials have the form considered in the following lemma.
\[discrim\]There is a universal, nonzero polynomial $P(c_{k+1},\ldots ,c_{k+l+1})$ with $P(0,\ldots,0)=0$ such that, given a degree $r=k+l+1$ polynomial $$\label{poly}
f(x)=x^{r}+\sum_{a=1}^{k}\operatorname{\epsilon}(c_a+O(\operatorname{\epsilon}))x^{r-a}+\sum_{b=1}^{l+1}\operatorname{\epsilon}^b(c_{k+b}+O(\operatorname{\epsilon}))x^{l+1-b},$$ the discriminant $\delta (f)$ of $f$ has the form $$\label{discform} \delta (f)=P(c_{k+1},\ldots
,c_{k+l+1})\operatorname{\epsilon}^{r+l^2-1}+O(\operatorname{\epsilon}^{r+l^2}).$$
For $i=0,\ldots,r=k+l+1$, let $a_i$ be the coefficient of $x^{r-i}$ in $f$ (so in particular $a_0=1$). Recall that $\delta
(f)=(-1)^{r(r-1)/2}a_{0}^{-1} Res(f,f') $ (“$Res$” denoting the resultant; see, *e.g.*, Section V.10 of [@Lang]), so it suffices to prove the expansion (\[discform\]) for $Res(f,f')$. $Res(f,f')$ is given as the determinant $$\label{res} \left|
\begin{matrix}
a_0 & a_1 & a_2 &\cdots &\cdots & a_r & & & \\
& a_0 & a_1 & a_2 &\cdots & \cdots & a_r & & \\
& & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \\
& & & a_0 & a_1 & a_2 &\cdots & \cdot & a_r \\
ra_0 & (r-1)a_1 & (r-2)a_2 &\cdots & a_{r-1} & & & \\
&\cdot &\cdot & \cdot & \cdot & \cdot & & \\
& & \cdot & \cdot & \cdot & \cdot & \cdot & \\
& & & ra_0 & (r-1)a_1 & (r-2)a_2 &\cdots & a_{r-1} & \\
& & & & ra_0 & (r-1)a_1 & (r-2)a_2 &\cdots & a_{r-1}
\end{matrix}\right|$$
Each term in the expansion of this determinant will be a constant times $\prod_{j=0}^{r} a_{j}^{i_j}$ for some natural numbers $i_j$ satisfying $$\sum i_j = 2r-1$$ (since this is a $(2r-1)\times
(2r-1)$ matrix) and $$\sum ji_j =r(r-1)$$ (since if the roots of $f$ are $\alpha_1,\ldots,\alpha_r$, the discriminant $\prod_{a<b}(\alpha_a-\alpha_b)^2$ has degree $r(r-1)$ in the $\alpha_b$, while the coefficient $a_j$ has degree $j$ in the $\alpha_b$). Let $$e(i_0,\ldots,i_r)=\max\{e\in
\mathbb{N}|a_{0}^{i_0}\cdots a_{r}^{i_r}=O(\operatorname{\epsilon}^e)\}.$$ To prove the lemma we need to show that:
- For each $\prod a_{j}^{i_j}$ appearing in the expansion of the resultant (\[res\]), $e(i_0,\ldots,i_r)\geq r+l^2-1$, with equality implying that $i_1=\cdots =i_k=0$ (the latter condition being needed to show that our polynomial $P$ depends only on $c_{k+1},\ldots, c_r$ and vanishes when all of these $c_j$ are $0$); and
- There are particular values of the $c_j$ for which $Res(f,f')\neq O(\operatorname{\epsilon}^{r+l^2})$.
Point (ii) above is easy: in the statement of the lemma, let $$c_j=\begin{cases} 1 & i=k+1,n \\ 0 &\text{otherwise,}
\end{cases}$$ so that $$f(x)=x^r+(\operatorname{\epsilon}+O(\operatorname{\epsilon}^2))x^l+(\operatorname{\epsilon}^{l+1}+O(\operatorname{\epsilon}^{l+2})).$$ We then see that the unique lowest-order term in the expansion of the determinant \[res\] is obtained by choosing $a_0=1$ from the first $k+1$ columns, $(r-k-1)a_{k+1}=l\operatorname{\epsilon}+O(\operatorname{\epsilon}^2)$ from the next $r$ columns, and $a_n=\operatorname{\epsilon}^{l+1}+O(\operatorname{\epsilon}^{l+2})$ from the last $l-1$ columns, so that $$Res(f,f')=\pm(l\operatorname{\epsilon})^r(\operatorname{\epsilon}^{l+1})^{l-1}+\mbox{higher order
terms}=\pm l^n\operatorname{\epsilon}^{r+l^2-1}+O(\operatorname{\epsilon}^{r+l^2}).$$
We now set about the proof of point (i). Assume that $\prod
a_{j}^{i_j}$ is a term appearing in the expansion of the determinant (\[res\]). Let $q$ be the quotient and $p$ be the remainder when $\sum_{m=0}^{l}mi_{r-m}$ is divided by $l$, and set $s=\sum_{m=0}^{l}i_{r-m}-q$ (note that the above sums only go up to $l=r-k-1$). We then have $$\sum_{j=r-l}^{r}ji_j=\sum_{j=0}^{l}(r-l+j)i_{r-l+j}=(r-l)q+rs-p.$$
Now since $\sum_{j=0}^{r}i_j=\sum_{j=0}^{k}i_j+q+s=2r-1$ and since $2r-1=r+k+l$, we see $$\begin{aligned}
s&=2r-1-q-\sum_{j=0}^{k}(i_j-1)-(k+1) \nonumber \\
&=r+l-1-q-\sum_{j=0}^{k}(i_j-1). \nonumber \end{aligned}$$
Hence $$\begin{aligned}
r^2-r&=
\sum_{j=0}^{r}ji_j=\sum_{j=0}^{k}ji_j+q(r-l)+rs-p \nonumber \\
&=\sum_{j=0}^{k}ji_j+q(r-l)-p+r(r+l-1-q-\sum_{j=0}^{k}(i_j-1))
\nonumber \\ &=r^2-r+l(r-q)-p+\sum_{j=0}^{k}(ji_j-r(i_j-1)),
\nonumber\end{aligned}$$ *i.e.*, $$\label{lrq}
l(r-q)=p+\sum_{j=0}^{k}(r(i_j-1)-ji_j).$$
Meanwhile $$\begin{aligned}
e(i_0,\ldots,i_r)&=\sum_{j=1}^{k}i_j
+\sum_{j=0}^{l}(1+j)i_{r-l+j} \nonumber \\ &=\sum_{j=1}^{k}
i_j+q+s(l+1)-p \nonumber\\
&=\sum_{j=1}^{k}i_j+q+\left(r+l-1-q-\sum_{j=0}^{k}(i_j-1)\right)(l+1)-p
\nonumber
\\&=l(r-q)+r+l^2-1-(l+1)\sum_{j=0}^{k}(i_j-1)+\sum_{j=1}^{k}i_j-p
\nonumber \\
&=r+l^2-1+\sum_{j=0}^{k}\left(r(i_j-1)-ji_j\right)-(l+1)\sum_{j=0}^{k}(i_j-1)+\sum_{j=1}^{k}i_j
\nonumber \\ &=
r+l^2-1+k\sum_{j=0}^{k}(i_j-1)+\sum_{j=1}^{k}(1-j)i_j
\label{rearrange}, \end{aligned}$$ where in the penultimate equality we have used (\[lrq\]) and in the last we have used the fact that $r-(l+1)=k$.
In our term $\prod a_j^{i_j}$ in the expansion of the determinant (\[res\]), each of those $a_j$ which are chosen from the first $(k+1)$ columns necessarily has $j\leq k$. For each $j$ write $i_j=w_j+z_j=w_j+x_j+y_j$ where $w_j$ denotes the number of $a_j$’s chosen from the first $(k+1)$ columns and $x_j$ denotes the number of $a_j$’s chosen from columns $k+2$ through $2k+1$; evidently $w_j=0$ for $j>k$ while $\sum_{j=0}^{k} w_j=k+1$, *i.e.*, $$\label{k+1} \sum_{j=0}^{k}(w_j-1)=0.$$
Rearrange our term $\prod_{j=0}a_{j}^{i_j}$ as $$a_{p_1}\cdots
a_{p_{2r-1}},$$ where the entry $a_{p_n}$ is culled from the $nth$ column in the matrix in (\[res\]); label the row from which $a_{p_n}$ is taken as $m_n$. Denoting $$\bar{m}=\begin{cases} m & m\leq r-1 \\
m+1-r & m\geq r \end{cases},$$ we see from the form of the resultant matrix that $$\bar{m}_n= n-p_n.$$ Consider the quantity $$\sum_{n=1}^{2k+1}
\bar{m}_n.$$ Obviously, the way to minimize this quantity is by using rows $1,2,\ldots,k,r,r+1,\ldots,r+k$ (or, just as well, rows $1,\ldots,k+1,r,\ldots,r+k-1$) when we pick the $a_{p_1},\ldots,a_{p_{2k+1}}$; such a choice then yields $\{\bar{m}_n|n\leq 2k+1\}=\{1,1,\ldots,k,k,k+1\}$ and $$\sum_{n=1}^{2k+1}
\bar{m}_n=\frac{k(k+1)}{2}+\frac{(k+1)(k+2)}{2}=(k+1)^2.$$ If $x_0\neq 0$, we have some $n\in [k+2,2k+1]$ with $p_n=0$ and so $\bar{m}_n=n >k+1$; in this vein, one may easily check that $$\sum_{n=1}^{2k+1} \bar{m}_n\geq (k+1)^2+\frac{x_0(x_0+1)}{2};$$ in particular $$\sum_{n=1}^{2k+1} \bar{m}_n\geq (k+1)^2 +x_0,$$ with equality requiring that either $x_0=0$ and $\{\bar{m}_n|n\leq
2k+1\}=\{1,1,\ldots,k,k,k+1\}$ or $x_0=1$ and $\{\bar{m}_n|n\leq
2k+1\}=\{1,1,\ldots,k,k,k+2\}$.
Thus, $$\begin{aligned}
(k+1)^2+x_0&\leq
\sum_{n=1}^{2k+1}\bar{m}_n=\sum_{n=1}^{2k+1}(n-p_n) \nonumber \\
&= (k+1)(2k+1)-\sum_{n=1}^{k+1}p_n-\sum_{n=k+2}^{2k+1}p_n
\nonumber \\
&=(k+1)^2-\sum_{j=1}^{k}jw_j+\sum_{n=k+2}^{2k+1}(k+1-p_n)
\nonumber \\ &\leq (k+1)^2-\sum_{j=1}^{k}jw_j+\sum_{n=k+2,p_n\leq
k}^{2k+1}(k+1-p_n) \nonumber \\
&=(k+1)^2-\sum_{j=1}^{k}jw_j+\sum_{j=0}^{k}(k+1-j)x_j \end{aligned}$$
So $$kz_0+\sum_{j=1}^k(k+1-j)z_j\geq
kx_0+\sum_{j=1}^{k}(k+1-j)x_j\geq \sum_{j=1}^{k}jw_j\geq
\sum_{j=1}^{k}(j-1)w_j,\label{zx}$$ *i.e.*, $k\sum_{j=0}^{k}z_j+\sum_{j=1}^{k}(1-j)(w_j+z_j)\geq 0$, so that since $\sum_{j=0}^{k}(w_j-1)=0$ and $i_j=w_j+z_j$, we at last conclude that $$\label{geq}
k\sum_{j=0}^{k}(i_j-1)+\sum_{j=1}^{k}(1-j)i_j \geq
0.$$ In light of Equation \[rearrange\], this shows that $e(i_0,\ldots,i_r)\geq r+l^2-1$ with equality if and only if equality holds in (\[geq\]); equality in (\[geq\]) requires among other things that
- either $x_0=0$ and $\{\bar{m}_n|n\leq
2k+1\}=\{1,1,\ldots,k,k,k+1\}$ or $x_1=1$ and $\{\bar{m}_n|n\leq
2k+1\}=\{1,1,\ldots,k,k,k+2\}$; and
- due to (\[zx\]), $z_j=x_j$ for $j\leq k$ (so that for $j\leq k$ all of the $a_j$ in our term $\prod_{j=0}^{r}a_{j}^{i_j}$ come from the first $2k+1$ columns of the resultant matrix).
For $n=1,2,3,4$ let $M_n$ denote the $(2k+1)\times (2k+1)$ matrix constructed from the resultant matrix $(\ref{res})$ by taking columns $1$ through $2k+1$ and rows $1,\ldots,k,r,\ldots,r+k$ (for $n=1$), rows $1,\ldots,k+1,r,\ldots,r+k-1$ (for $n=2$), rows $1,\ldots,k,r,r+k-1,\ldots,r+k+1$ (for $n=3$), or rows $1,\ldots,k,k+2,r,\ldots,r+k-1$ (for $n=4$). Let $M'_n$ be the $(2r-2k-2)\times (2r-2k-2)$ constructed from the other rows and columns. Assume that our term $\prod_{j=0}^{r}a_{j}^{i_j}$ in the resultant gives rise to the lowest possible value of $e(i_0,\ldots,i_j)$. (i) above then ensures that $\prod_{j=0}^{r}a_{j}^{i_j}$ is constructed by multiplying a term in the determinant of one of the $M_n$ by a term in the determinant of the corresponding $M'_n$. In searching for the optimal such monomial, we may then vary the contributions from $M_n$ and $M'_n$ separately. But on examining the form of the $M_n$, one sees immediately that the term in $\det(M_n)$ giving rise to the *strictly* lowest possible power of $\operatorname{\epsilon}$ is obtained by a product of $k+1$ $a_0$’s (from columns 1 through $k+1$ for $n=1,2$ and columns $1,\ldots,k,k+2$ for $n=3,4$) and $k$ $a_{k+1}$’s (and in particular contains no $a_j$ for $1\leq
j\leq k$). By (ii), any optimal monomial from $M'_n$ can’t contain any $a_j$ with $j\leq k$. Thus any $\prod_{j=1}^{r}a_{j}^{i_j}$ with $(i_1,\ldots,i_k)\neq
(0,\ldots,0)$ must have $e(i_1,\ldots,i_r)$ strictly greater than the lowest possible value (which has been shown above to be $n+l^2-1$). This proves the lemma.
We now recall the linear algebra definition of the relative Hilbert scheme from [@Smith]. Let $$\label{hrmodel} \tilde{\mathcal{H}}_r=\{(A,B,t,v)\in
M_r(\mathbb{C})^2\times D^2\times\mathbb{C}^r| AB=BA=tId,
(*)\},$$ where the stability condition (\*) states that the matrices A and B share no proper invariant subspaces containing the vector $v$. The relative Hilbert scheme of the map $(z,w)\mapsto zw$ is then $$\mathcal{H}_r=\tilde{\mathcal{H}}_r/GL_r(\mathbb{C}),$$ where $GL_{r}(\mathbb{C})$ acts by $$g\cdot
(A,B,t,v)=(gAg^{-1},gBg^{-1},t,gv).$$ The projection map $F{\colon\thinspace}\mathcal{H}_r\to D^2$ is just $[A,B,t,v]\mapsto t$. To briefly motivate this, remark that a point of the $r$-fold relative Hilbert scheme of $f$ is naturally viewed from an algebro-geometric standpoint as an ideal $I\leq \mathbb{C}[z,w]$ with the property that $V=\mathbb{C}[z,w]/I$ is an $r$-dimensional vector space and, for some $t$, $I$ is supported on $f^{-1}(t)$ (*i.e.*, $\langle zw-t\rangle <I$). To go from such an ideal to an element of $\mathcal{H}_r$, let $v\in V$ be the image of $1\in \mathbb{C}[z,w]$ under the projection, and let $A$ and $B$ be the operators on $V$ defined by multiplication by the polynomials $z$ and $w$ respectively. For more details see [@N] and [@Smith].
Given $[A,B,t,v]\in \mathcal{H}_r$, the fact that $A$ and $B$ commute implies that they can be simultaneously conjugated to be upper triangular; assuming that this has been done, the natural map $\phi_t{\colon\thinspace}F^{-1}(t)\to Sym^r f^{-1}(t)$ takes $[A,B,t,v]$ to $\{(A_{11},B_{11}),\ldots,(A_{rr},B_{rr})\}$. For $t\neq 0$, according to (\[hrmodel\]), $A$ is invertible and $B=tA^{-1}$, so $\phi_t$ is an isomorphism; $\phi_0$, meanwhile, is a nontrivial partial resolution. On the diagonal $\Delta\subset
\mathcal{H}_r$, $A$ and $B$ will both have repeated eigenvalues, occurring in corresponding Jordan blocks.
The main result of this section is:
\[appmain\] Let $F{\colon\thinspace}\mathcal{H}_r\to D^2$ denote the $r$-fold relative Hilbert scheme of the map $(z,w)\mapsto zw$, $\phi_0$ the partial resolution map $F^{-1}(0)\to Sym^r\{zw=0\}$, and $\Delta\subset \mathcal{H}_r$ the diagonal stratum. At any point $p\in \Delta\cap F^{-1}(0)$ with $\phi_0(p)=\{(0,0),\ldots,(0,0)\}$, where $T_p\Delta$ is the tangent cone to $\Delta$ at $p$, we have $T_p\Delta\subset T_p
F^{-1}(0)$.
According to the above description, the points $p$ under concern are of the form $[A,B,0,v]$ with $A$ and $B$ both nilpotent matrices such that $AB=BA=0$ Further, letting $k$ be such that $A^kv\neq 0$ but $A^{k+1}v=0$, the stability condition $(*)$ in (\[hrmodel\]) ensures that, where $r=k+l+1$, $$\{A^kv,\ldots,Av,B^lv,\ldots,Bv,v\}$$ is a basis for $\mathbb{C}^r$. All operators on $V\cong \mathbb{C}^r$ appearing in the rest of the proof will be written as matrices in terms of this basis.
Since $AB=0$ we can write $$B^{l+1}v=aA^kv+\sum_{i=1}^{l}b_{l-i}B^iv.$$ With respect to our above basis, we have $$A=\left( \begin{array}{c c c c| c c c c
|c}0
& 1 & & & 0 & \cdot & \cdot & 0 & 0 \\ & \ddots & \ddots & & \cdot & \cdot & &\cdot & \vdots \\ & & 0 & 1 & \cdot & & \cdot &\cdot & 0 \\
& & & 0 & 0 &\cdot & \cdot & 0 & 1 \\ \hline 0 & & \cdots & 0&0 &
\cdots & & & 0\\\vdots & & & & & & & &\vdots \\ 0 & & \cdots & 0&0
& \cdots & & & 0 \end{array}\right),$$ $$B=\left( \begin{array}{c c c c|c| c c
c c}0&\cdot&\cdot&0&a&0&\cdot&\cdot&0\\
\cdot &\cdot & &\cdot & 0&\cdot & & & \cdot\\
\cdot& &\cdot&\cdot& \vdots&\cdot & &\ddots &\cdot \\
\cdot & & & \cdot & 0 & 0 & \cdot &\cdot & 0\\
\hline 0 &\cdot &\cdot & 0 & b_1 & 1 & 0 &\cdots & 0\\
\hline \cdot &\cdot & & \cdot & \vdots & 0 & 1 &\cdots & 0\\
\cdot & & & \cdot & b_{l-1} & \cdot & \ddots & \ddots & \vdots \\
\cdot& & \cdot & \cdot &b_l & \cdot & &\cdot & 1 \\
0 & \cdot & \cdot & 0 & 0 & 0 &\cdot & \cdot & 0
\end{array} \right),$$ and $v=e_r=(0,\ldots,0,1)$ (in both of the above matrices, the upper left block is of size $k\times k$). Let $$(C,D,\mu,w)\in T_{(A,B,0,e_r)}\tilde{\mathcal{H}_r}.$$ Letting $\pi{\colon\thinspace}\tilde{\mathcal{H}}_r\to \mathcal{H}_r$ be the projection, we have $\mu=F_*(\pi_*)_{(A,B,0,e_r)}(C,D,\mu,w)$, so our goal is to show that if $(C,D,\mu,w)$ is tangent to $\pi^{-1}\Delta$ then $\mu=0$. Linearizing the defining equations for $\tilde{H}_r$ gives $$CB+AD=BC+DA=\mu Id,$$ which implies, among other things, $$\mbox{For $i>k$, }\begin{cases} aC_{i1}+\sum b_m
C_{i,k+m}=\mu\delta_{i,j} \\ C_{i,j-1}=\mu\delta_{i,j} \mbox{ if
} j\geq k+2\end{cases}$$$$\mbox{For $j=1$ or $k+1\leq j\leq
r-1$, } \begin{cases} aC_{k+1,j}=\mu\delta_{1,j} \\
b_{i-k}C_{k+1,j}+C_{i+1,j}=\mu\delta_{i,j} \mbox{ if } k+1\leq
i\leq r-1\end{cases}$$
If $a=0$, we have $\mu=\mu\delta_{1,1}=aC_{k+1,1}=0$ and we are done.
If $a\neq 0$, we find from the above equations that$$C=\left(
\begin{array}{c c c c| c c c c |c}
* & * &* &* & *& * & * & * & * \\
*& *& *& * & * & * &* &*& * \\
* & *&* & * & * &* & *&*& *\\
* &* & *& * & * &*& *& * & * \\\hline
\mu/a & * & *& *&0 & 0 &\cdot &0 & *\\
\hline -b_1\mu/a & * & *& *&\mu &0 &\cdots & 0&* \\
-b_2\mu/a &* &* & *&0 & \mu& \cdot & 0 & * \\
\vdots & * & * & * & \cdot & \cdot & \ddots& \cdot & * \\
-b_l\mu/a & * & * & * & 0 & \cdot & \cdot &\mu & *
\end{array}\right),$$ where again the upper left block is size $k\times k$ and all asterisks denote undetermined entries.
We consider now the characteristic polynomials of the matrices $A+\operatorname{\epsilon}C$ for small $\operatorname{\epsilon}$. The matrix $A+\operatorname{\epsilon}C-\lambda Id$ is $$\left(
\begin{array}{c c c c| c c c c |c}
-\lambda+\operatorname{\epsilon}C_{11} & 1+\operatorname{\epsilon}C_{12} &* &* & *& * & * & * & * \\
*& -\lambda+\operatorname{\epsilon}C_{22}& \ddots& * & * & * &* &*& * \\
* & *&\ddots & 1+\operatorname{\epsilon}C_{k-1,k} & * &* & *&*& *\\
* &* & *& -\lambda+\operatorname{\epsilon}C_{kk} & * &*& *& * & 1+\operatorname{\epsilon}C_{rk} \\\hline
\operatorname{\epsilon}\mu/a & * & *& *&-\lambda & 0 &\cdot &0 & *\\
\hline -\operatorname{\epsilon}b_1\mu/a & * & *& *&\operatorname{\epsilon}\mu &-\lambda &\ddots & 0&* \\
-\operatorname{\epsilon}b_2\mu/a &* &* & *&0 & \operatorname{\epsilon}\mu& \cdot & 0 & * \\
\vdots & * & * & * & \vdots & \cdot & \ddots& -\lambda & * \\
-\operatorname{\epsilon}b_l\mu/a & * & * & * & 0 & \cdot & \cdot &\operatorname{\epsilon}\mu &
-\lambda+\operatorname{\epsilon}C_{rr} \end{array}\right)$$ where an asterisk in the $(i,j)$th entry signifies $\operatorname{\epsilon}C_{ij}$. When we expand the determinant of this matrix, among the terms that we obtain are $$(-\lambda)^r \mbox{ and } \ \pm (-\operatorname{\epsilon}b_m\mu/a)\cdot
1^{k-1}(-\lambda)^{m-1}(\operatorname{\epsilon}\mu)^{l-m+1}=\pm
\frac{\mu^{l-m+2}b_m}{a}\operatorname{\epsilon}^{l-m+2}\lambda^{m-1};$$ note that these latter have combined degree exactly $l+1$ in $\operatorname{\epsilon}$ and $\lambda$. Any other term in the expansion of the determinant will have degree at least 1 in $\operatorname{\epsilon}$ and at least $l+2$ in $\operatorname{\epsilon}$ and $\lambda$ combined, the reason being that each of the entries denoted with an asterisk above lies in either the same row or the same column as an entry of form $1+\operatorname{\epsilon}C_{ij}$, so a term in the determinant containing one of the asterisked entries can contain at most $k-1$ of the $k$ $(1+\operatorname{\epsilon}C_{ij})$’s and hence must contain at least $r-(k-1)=l+2$ other terms, each of which is of combined order at least 1 in $\operatorname{\epsilon}$ and $\lambda$. In other words, for constants $c_1,\ldots,c_{k+l+1}$ where $$c_{k+m}=\pm \frac{b_{l+1-m}\mu^m}{a} \mbox{ for $1\leq m\leq l$
and } c_{k+l+1}=\frac{\mu^{l+1}}{a},$$ the characteristic polynomial of $A+\operatorname{\epsilon}C$ has form $$p_{A+\operatorname{\epsilon}C}(x)=(-x)^r+\operatorname{\epsilon}\sum_{a=1}^{k}(c_a+O(\operatorname{\epsilon}))(-x)^{r-a}+\sum_{b=1}^{l+1}\operatorname{\epsilon}^b(c_{k+b}+O(\operatorname{\epsilon}))(-x)^{l+1-b},$$ which, since $r=k+l+1$, is precisely the sort of polynomial considered in Lemma \[discrim\]. By replacing $\operatorname{\epsilon}$ with $\nu\operatorname{\epsilon}$ in the statement of that lemma, we see that the polynomial $P(a_{r-l},\ldots,a_r)$ provided by its conclusion scales as $$P(\nu a_{r-l},\nu^2
a_{r-l+1},\ldots,\nu^{l+1}a_r)=\nu^{r+l^2-1}P(a_{r-l},\ldots,a_r),$$ so that $$\begin{aligned}
P(c_{k+1},\ldots,c_r)&=P\left(\pm\frac{b_l\mu}{a},\pm\frac{b_{l-1}\mu^2}{a},\ldots,\pm\frac{b_1\mu^l}{a},\frac{\mu^{l+1}}{a}\right)\\
&=\mu^{r+l^2-1}P(\pm b_l/a,\pm b_{l-1}/a,\ldots,\pm
b_1/a,1/a).\end{aligned}$$ So since $P$ is not the zero polynomial, at least for a generic initial choice of our base point $[A,B,0,e_r]$ (equivalently, for generic $a,b_1,\ldots,b_l$) we conclude that if $(C,D,\mu,w)\in
T_{(A,B,0,e_r)}\tilde{\mathcal{H}}_r$, we have $$\label{conc} \delta(p_{A+\operatorname{\epsilon}C})=\mu^{r+l^2-1}M\operatorname{\epsilon}^{n+l^2-1}+O(\operatorname{\epsilon}^{n+l^2}),$$ where $M$ is a nonzero constant depending only on $A$. Let $$\begin{aligned}
\tilde{\Delta}_1&=\{(A',B',t,v')\in
\tilde{\mathcal{H}}_r|A' \mbox{ has a repeated eigenvalue}\}\\
&=\{(A',B',t,v')\in
\tilde{\mathcal{H}}_r|\delta(p_A)=0\}\end{aligned}$$ Equation \[conc\] then shows that, for $(C,D,\mu,w)\in
T_{(A,B,0,e_r)}\tilde{\mathcal{H}}_r$, $$(C,D,\mu,w)\in
T_{(A,B,0,e_r)}\tilde{\Delta}_1\Leftrightarrow \mu=0$$ ($T_{(A,B,0,e_r)}\tilde{\Delta}_1$ denoting the tangent cone at $(A,B,0,e_r)$ if $\tilde{\Delta}_1$ is singular there). Where again $\pi{\colon\thinspace}\tilde{\mathcal{H}}_r\to\mathcal{H}_r$ is the projection, we have $T\Delta\subset \pi_*T\tilde{\Delta}_1$, so if $\alpha\in T_{[A,B,0,e_r]}\Delta$, writing $\alpha=\pi_*(C,D,\mu,w)$, we have that $F_*\alpha=\mu=0$. This conclusion initially only applies at those $[A,B,0,e_r]\in \Delta$ which are generic in the sense that $P(\pm b_l/a,\pm
b_{l-1}/a,\ldots,\pm b_1/a,1/a)\neq 0$, but then since the conclusion is a closed condition it in fact applies to all $[A,B,0,e_r]$ lying on the diagonal $\Delta$.
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abstract: 'Computing the determinant of a matrix with the univariate and multivariate polynomial entries arises frequently in the scientific computing and engineering fields. In this paper, an effective algorithm is presented for computing the determinant of a matrix with polynomial entries using hybrid symbolic and numerical computation. The algorithm relies on the Newton’s interpolation method with error control for solving Vandermonde systems. It is also based on a novel approach for estimating the degree of variables, and the degree homomorphism method for dimension reduction. Furthermore, the parallelization of the method arises naturally.'
address:
- 'Department of Mathematics, Sichuan University, Chengdu 610064, PR China'
- 'Chengdu Institute of Computer Applications, Chinese Academy of Sciences, Chengdu 610041, PR China'
author:
- Xiaolin Qin
- Zhi Sun
- Tuo Leng
- Yong Feng
bibliography:
- 'mybibfile.bib'
title: Computing the determinant of a matrix with polynomial entries by approximation
---
\[section\] \[section\] \[section\] \[section\] \[section\] \[section\] \[section\]
symbolic determinant ,approximate interpolation ,dimension reduction ,Vandermonde systems ,error controllable algorithm
Introduction
============
In the scientific computing and engineering fields, such as computing multipolynomial resultants [@Cox], computing the implicit equation of a rational plane algebraic curve given by its parametric equations [@Delvaux2009], and computing Jacobian determinant in multi-domain unified modeling [@Qin2013], computing the determinant of a matrix with polynomial entries (also called symbolic determinant) is inevitable. Therefore, computing symbolic determinants is an active area of research \[4–12\]. There are several techniques for calculating the determinants of matrices with polynomial entries, such as expansion by minors [@Gentleman1976], Gaussian elimination over the integers [@Sasaki1982; @Kaltofen1992], a procedure which computes the characteristic polynomial of the matrix [@Lipson1969], and a method based on evaluation and interpolation \[5–7\]. The first three algorithms belong to symbolic computations. As is well known, symbolic computations are principally exact and stable. However, they have the disadvantage of intermediate expression swell. The last one is the interpolation method, which as an efficient numerical method has been widely used to compute resultants and determinants, etc.. In fact, it is not approximate numerical computations but big number computations, which are also exact computations and only improve intermediate expression swell problem. Nevertheless, the main idea of black box approach takes an external view of a matrix, which is a linear operator on a vector space [@CEK2002]. Therefore, it is particularly suited to the handling of large sparse or structured matrices over finite fields. In this paper, we propose an efficient approximate interpolation approach to remedy these drawbacks.
Hybrid symbolic-numerical computation is a novel method for solving large scale problems, which applies both numerical and symbolic methods in its algorithms and provides a new perspective of them. The approximate interpolation methods are still used to get the approximate results \[12–15\]. In order to obtain exact results, one usually uses exact interpolation methods to meliorate intermediate expression swell problem arising from symbolic computations [@Marco2004; @Li2009; @Chen2013; @Kaltofen2007]. Although the underlying floating-point methods in principle allow for numerical approximations of arbitrary precision, the computed results will never be exact. Recently, the exact computation by intermediate of floating-point arithmetic has been an active area of solving the problem of intermediate expression swell in \[16–20\]. The nice feature of the work is as follows: The initial status and final results are accurate, whereas the intermediate of computation is approximate. The aim of this paper is to provide a rigorous and efficient algorithm to compute symbolic determinants by approximate interpolation. In this paper, we restrict our study to a non-singular square matrix with polynomial entries and the coefficients of polynomial over the integers.
The rest of this paper is organized as follows. Section 2 first constructs the degree matrix of symbolic determinant on variables and gives theoretical support to estimate the upper bounds degree of variables, and then analyzes the error controlling for solving Vandermonde systems of equations by Newton’s interpolation method, finally proposes a reducing dimension method based on degree homomorphism. Section 3 proposes a novel approach for estimating the upper bound on degree of variables in symbolic determinant, and then presents algorithms of dimension reduction and lifting variables and gives a detailed example. Section 4 gives some experimental results. The final section makes conclusions.
Preliminary results
===================
Throughout this paper, $\mathbb{Z}$ and $\mathbb{R}$ denote the set of the integers and reals, respectively. There are $v$ variables named $x_i$, for $i=1$ to $v$. Denote the highest degree of each $x_i$ by $d_i$. Denoted by ${\Phi}_{m,n} (\mathbb{F})$ the set of all $m$ by $n$ matrices over field $\mathbb{F} = \mathbb{R}$, and abbreviate ${\Phi}_{n,n}(\mathbb{F})$ to ${\Phi}_{n}(\mathbb{F})$.
Estimating degree of variables
------------------------------
In this subsection, a brief description to Chio’s expansion is proposed. We also give the Theorem \[lem:maxdeg\] for estimating the upper bound on degree of variables in symbolic determinant.
\[chio’s theorem\] ([@Howard1966]) Let $A=[a_{ij}]$ be an $n\times n$ matrix and suppose $a_{11}\neq 0$. Let $K$ denote the matrix obtained by replacing each element $a_{ij}$ in $A$ by $\begin{vmatrix}
a_{11}& a_{1j}\\
a_{i1}&a_{ij}
\end{vmatrix}$. Then $|A|=|K|/a_{11}^{n-2}$. That is, $$|A|=\frac{1}{a_{11}^{n-2}}
\begin{vmatrix}
\begin{vmatrix}
a_{11}&a_{12}\\
a_{21}&a_{22}
\end{vmatrix}
&\begin{vmatrix}
a_{11}&a_{13}\\
a_{21}&a_{23}
\end{vmatrix}&\cdots &
\begin{vmatrix}
a_{11}&a_{1n}\\
a_{21}&a_{2n}
\end{vmatrix}\\
\begin{vmatrix}
a_{11}&a_{12}\\
a_{31}&a_{32}
\end{vmatrix}
&\begin{vmatrix}
a_{11}&a_{13}\\
a_{31}&a_{33}
\end{vmatrix}&\cdots &
\begin{vmatrix}
a_{11}&a_{1n}\\
a_{31}&a_{3n}
\end{vmatrix}\\
\cdots &\cdots &\cdots &\cdots\\
\begin{vmatrix}
a_{11}&a_{12}\\
a_{n1}&a_{n2}
\end{vmatrix}
&\begin{vmatrix}
a_{11}&a_{13}\\
a_{n1}&a_{n3}
\end{vmatrix}&\cdots &
\begin{vmatrix}
a_{11}&a_{1n}\\
a_{n1}&a_{nn}
\end{vmatrix}
\end{vmatrix}.$$
\[remchio\] The proof of Lemma \[chio’s theorem\] is clear. Multiply each row of $A$ by $a_{11}$ except the first, and then perform the elementary row operations, denote $Op(2-a_{21}\cdot 1)$, $Op(3-a_{31}\cdot 1)$, $\cdots$, $Op(n-a_{n1}\cdot 1)$, where $'1', '2', \cdots, 'n'$ represents for the row index. We get $$a_{11}^{n-1}|A|=
\begin{vmatrix}
a_{11}& a_{12}&\cdots &a_{1n}\\
a_{11}a_{21}& a_{11}a_{22}&\cdots &a_{11}a_{2n}\\
\vdots&\vdots&\ddots&\vdots\\
a_{11}a_{n1}& a_{11}a_{n2}&\cdots &a_{11}a_{nn}
\end{vmatrix}=$$ $$\begin{vmatrix}
a_{11}& a_{12}&a_{13}&\cdots &a_{1n}\\
0& \begin{vmatrix}
a_{11}&a_{12}\\
a_{21}&a_{22}
\end{vmatrix}&\begin{vmatrix}
a_{11}&a_{13}\\
a_{21}&a_{23}
\end{vmatrix}&\cdots &\begin{vmatrix}
a_{11}&a_{1n}\\
a_{21}&a_{2n}
\end{vmatrix}\\
\vdots&\vdots&\vdots&\ddots&\vdots\\
0& \begin{vmatrix}
a_{11}&a_{12}\\
a_{n1}&a_{n2}
\end{vmatrix}&\begin{vmatrix}
a_{11}&a_{13}\\
a_{n1}&a_{n3}
\end{vmatrix}&\cdots &\begin{vmatrix}
a_{11}&a_{1n}\\
a_{n1}&a_{nn}
\end{vmatrix}
\end{vmatrix}
=a_{11}|K|.$$ We observe that $K$ is $(n-1)\times (n-1)$ matrix, then the above procedure can be repeated until the $K$ is $2 \times 2$ matrix. It is a simple and straightforward method for calculating the determinant of a numerical matrix.
\[lem:deg\] Given two polynomials $f(x_1)$ and $g(x_1)$, the degree of the product of two polynomials is the sum of their degrees, i.e., $$deg( f(x_1) \cdot g(x_1), x_1 ) = deg(f(x_1), x_1) + deg(g(x_1), x_1).$$ The degree of the sum (or difference) of two polynomials is equal to or less than the greater of their degrees, i.e., $$deg(f(x_1) \pm g(x_1), x_1) \leq max\{deg(f(x_1), x_1),
deg(g(x_1), x_1)\},$$ where $f(x_1)$ and $g(x_1)$ are the univariate polynomials over field $\mathbb{F}$, and $deg(f(x_1), x_1)$ represents the highest degree of $x_1$ in $f(x_1)$.
Let $M=[M_{ij}]$ be an $n\times n$ matrix and suppose $M_{ij}$ is a polynomial with integer coefficients consisting of variables $x_1, x_2, \cdots, x_v$, where the order of $M$ is $n\geq 2$. Without loss of generality, we call it the degree matrix $\Omega_1 = (\sigma_{ij})$ [^1] for $x_1$ defined as:
${\sigma_{ij}}=\begin{cases}
highest \ degree \ of \ x_1 \ appears\ in \ the \ element \ M_{ij}, i.e., deg(M_{ij}, x_1),\\
0, \;\;\; if \ x_1 \ does \ not \ occur \ in \ M_{ij}.
\end{cases}$
So, we can construct the degree matrix from $M$ for all variables, respectively.
\[lem:maxdeg\] $M$ is defined as above. Suppose the $2 \times 2$ degree matrix can be obtained from $M$ for $x_i (1\leq i \leq v)$, denotes $$\begin{aligned}
\Omega_i=
\left[\begin{array}{cc}
\sigma_{(n-1)(n-1)}^{(n-2)}&\ \ \ \ \sigma_{(n-1)n}^{(n-2)}\\
\sigma_{n(n-1)}^{(n-2)}&\ \ \sigma_{nn}^{(n-2)}\\
\end{array}\right],\end{aligned}$$ then $$maxdeg=\max\{\sigma_{(n-1)(n-1)}^{(n-2)}+\sigma_{nn}^{(n-2)}, \sigma_{(n-1)n}^{(n-2)}+\sigma_{n(n-1)}^{(n-2)}\}.$$ That is, the maximum degree of variable is no more than $$maxdeg-\sum_{i=3}^{n}(i-2)\sigma_{(n-i+1)(n-i+1)}^{(n-i)},$$ where $\sigma^{(n-2)}_{(n-1)(n-1)}=deg(M_{(n-1)(n-1)}^{(n-2)}, x_i).$[^2]
Considering the order $n$ of symbolic determinant $$|M| = \left| {\begin{array}{cccc}
M_{11}&\ M_{12}& \ \cdots &M_{1n}\\
M_{21}&\ M_{22}&\ \cdots &M_{2n}\\
\vdots&\ \vdots&\ddots&\ \vdots\\
M_{n1}&\ M_{n2}&\ \cdots &\ M_{nn}
\end{array} } \right|$$ by Chio’s expansion is from Remark \[remchio\], then $$|M|=\frac{1}{M_{11}^{n-2}}
\left| {\begin{array}{cccc}
M_{22}^{(1)}&\ M_{23}^{(1)}&\ \cdots&\ M_{2n}^{(1)}\\
M_{32}^{(1)}&\ M_{33}^{(1)}&\ \cdots&\ M_{3n}^{(1)}\\
\vdots&\ \vdots&\ \ddots&\ \vdots\\
M_{n2}^{(1)}& \ M_{n3}^{(1)}&\ \cdots&\ M_{nn}^{(1)}\\
\end{array} } \right|$$ $$=\frac{1}{M_{11}^{n-2}}\frac{1}{{M_{22}^{(1)}}^{n-3}}\cdots\frac{1}{M_{(n-2)(n-2)}^{(n-3)}}\begin{vmatrix}
M_{(n-1)(n-1)}^{(n-2)}&\ M_{(n-1)n}^{(n-2)}\\
M_{n(n-1)}^{(n-2)}&\ M_{nn}^{(n-2)}
\end{vmatrix},$$
where $$M_{22}^{(1)}=M_{11}M_{22}-M_{12}M_{21}, M_{32}^{(1)}=M_{11}M_{32}-M_{12}M_{31}, \cdots, M_{nn}^{(1)}=M_{11}M_{nn}-M_{1n}M_{n1}.$$
By Lemma \[lem:deg\], for $x_i$ we get $$deg(|M|, x_i)\leq
\max\{\sigma_{(n-1)(n-1)}^{(n-2)}+\sigma_{nn}^{(n-2)},
\sigma_{(n-1)n}^{(n-2)}+\sigma_{n(n-1)}^{(n-2)}\} -(n-2)\sigma_{11}
-(n-3)\sigma_{22}^{(1)}-\cdots-\sigma^{(n-3)}_{(n-2)(n-2)}$$ $$= maxdeg-\sum_{i=3}^{n}(i-2)\sigma^{(n-i)}_{(n-i+1)(n-i+1)},$$ where $$maxdeg=\max\{\sigma_{(n-1)(n-1)}^{(n-2)}+\sigma_{nn}^{(n-2)}, \sigma_{(n-1)n}^{(n-2)}+\sigma_{n(n-1)}^{(n-2)}\}.$$ The proof of Theorem \[lem:maxdeg\] is completed. It can be applied to all variables.
We present a direct method for estimating the upper bound on degrees of variables by computation of the degree matrices. Our method only needs the simple recursive arithmetic operations of addition and subtraction. Generally, we can obtain the exact degrees of all variables in symbolic determinant in practice.
Newton’s interpolation with error control
-----------------------------------------
Let $M$ be defined as above. Without loss of generality, we consider the determinant of a matrix with bivariate polynomial entries, and then generalize the results to the univariate or multivariate polynomial. A good introduction to the theory of interpolation can be seen in [@Boor1994].
The Kronecker product of ${A} = [a_{i,j}]\in
{\Phi}_{m,n}(\mathbb{F})$ and ${B} = [b_{ij}] \in
{\Phi}_{p,q}(\mathbb{F})$ is denoted by ${A}\otimes {B}$ and is defined to the block matrix $${A}\otimes{B}=\left(\begin{array}{cccc}a_{11}{B}&a_{12}{B}&\cdots
&a_{1n}{B}\\
a_{21}{B}&a_{22}{B}&\cdots
&a_{2n}{B}\\
\vdots&\vdots&\ddots&\vdots\\
a_{m1}{B}&a_{m2}{B}&\cdots&a_{mn}{B}
\end{array}\right)\in{M}_{mp,nq}(\mathbb{F}).$$ Notice that ${A}\otimes {B} \neq {B}\otimes {A} $ in general.
With each matrix ${A} = [a_{ij}] \in {\Phi}_{m, n}(\mathbb{F})$, we associate the vector ${\mathrm{vec}({A})}\in \mathbb{F}^{mn}$ defined by $${\mathrm{vec}({A})} \equiv [a_{11},\cdots
a_{m1},a_{12},\cdots,a_{m2},\cdots,a_{1n},\cdots,a_{mn}]^T,$$ where $^T$ denotes the transpose of matrix or vector.
Let the determinant of $M$ be $f(x_1,x_2)=\sum_{i,j}a_{ij}x_1^ix_2^j$ which is a polynomial with integer coefficients, and $d_1$, $d_2$ [^3]be the bounds on the highest degree of $f(x_1,x_2)$ in $x_1$, $x_2$, respectively. We choose the distinct scalars $(x_{1i}, x_{2j})$ ($i=
0, 1, \cdots, d_1$; $j = 0, 1, \cdots, d_2$), and obtain the values of $f(x_1, x_2)$, denoted by $f_{ij}\in\mathbb{R}$ ($i = 0, 1,
\cdots, d_1; j = 0, 1, \cdots, d_2$). The set of monomials is ordered as follows:
$$(1, x_1, x_1^2, \cdots, x_1^{d_1}) \times (1, x_2, x_2^2, \cdots, x_2^{d_2}),$$
and the distinct scalars in the corresponding order is as follows: $$(x_{10}, x_{11}, \cdots, x_{1d_1}) \times (x_{20}, x_{21}, \cdots,
x_{2d_2}).$$
Based on the bivariate interpolate polynomial technique, which is essential to solve the following linear system:
$$\label{equ:tem1}
({V}_{x_1}\otimes{V}_{x_2}){\mathrm{vec}({a})}={\mathrm{vec}({F})},$$
where the coefficients ${V}_{x_1}$ and ${V}_{x_2}$ are Vandermonde matrices: $${V}_{x_1}=\left(
\begin{array}{ccccc}
1&x_{10}&x_{10}^2&\cdots&x_{10}^{d_1}\\
1&x_{11}&x_{11}^2&\cdots&x_{11}^{d_1}\\
\vdots&\vdots&\vdots&\ddots&\vdots\\
1&x_{1d_1}&x_{1d_1}^2&\cdots&x_{d_1}^{1d_1}
\end{array}\right),\quad
{V}_{x_2}=\left(
\begin{array}{ccccc}
1&x_{20}&x_{20}^2&\cdots&x_{20}^{d_2}\\
1&x_{21}&x_{21}^2&\cdots&x_{21}^{d_2}\\
\vdots&\vdots&\vdots&\ddots&\vdots\\
1&x_{2d_2}&x_{2d_2}^2&\cdots&x_{2d_2}^{d_2}
\end{array}\right),$$ and $${a}=\left( \begin{array}{cccc}
a_{00}&a_{01}&\cdots& a_{0d_2}\\
a_{10}&a_{11}&\cdots& a_{1d_2}\\
\vdots&\vdots&\ddots&\vdots\\
a_{d_10}&a_{d_11}&\cdots&a_{d_1d_2}
\end{array}
\right),\quad {F}=\left( \begin{array}{cccc}
f_{00}&f_{01}&\cdots& f_{0d_2}\\
f_{10}&f_{11}&\cdots& f_{1d_2}\\
\vdots&\vdots&\ddots&\vdots\\
f_{d_10}&f_{d_11}&\cdots&f_{d_1d_2}
\end{array}
\right).$$
Marco et al. [@Marco2004] have proved in this way that the interpolation problem has a unique solution. This means that ${V}_{x_1}$ and ${V}_{x_2}$ are nonsingular and therefore ${V}=
{V}_{x_1}\otimes{V}_{x_2}$, then the coefficient matrix of the linear system (\[equ:tem1\]) is nonsingular. The following lemma shows us how to solve the system (\[equ:tem1\]).
\[theo:kronecker\_equation\] ([@Horn1991]) Let $\mathbb{F}$ denote a field. Matrices ${A}\in{\Phi}_{m,n}(\mathbb{F})$, ${B}\in{\Phi}_{q,p}(\mathbb{F})$, and ${C}\in{\Phi}_{m,q}(\mathbb{F})$ are given and assume ${X}\in{\Phi}_{n,p}(\mathbb{F})$ to be unknown. Then, the following equation: $$\label{equ:kronecker_equ}
({B}\otimes{A}){\mathrm{vec}({X})}={\mathrm{vec}({C})}$$ is equivalent to matrix equation: $$\label{equ:matrix_equ}
{AXB}^T={C}.$$ Obviously, equation (\[equ:matrix\_equ\]) is equivalent to the system of equations $$\left\{\begin{array}{l}
{AY}={C} \\
{BX}^T={Y}^T.
\end{array}\right.$$
Notice that the coefficients of system (\[equ:tem1\]) are Vandermonde matrices, the reference [@Bjorck1970] by the Newton’s interpolation method presented a progressive algorithm which is significantly more efficient than previous available methods in $O(d_1^2)$ arithmetic operations in Algorithm \[alg:dual\].
Input: a set of distinct scalars $(x_i, f_i) (0\leq i \leq d_1)$;\
Output: the solution of coefficients $a_0, a_1, \cdots, a_{d_1}$.
[Step :]{}
$c_i^{(0)} := f_i(i=0, 1, \cdots, d_1)$\
$c_{i}^{(k+1)} := \frac{c_i^{(k)}-c_{i-1}^{(k)}}{x_i-x_{i-k-1}}(i=d_1, d_1-1, \cdots,
k+1)$
$a_i^{(d_1)} := c_i^{(d_1)}(i=0, 1, \cdots, d_1)$\
$a_{i}^{(k)} := a_i^{(k+1)}-x_{k}a_{i+1}^{(k+1)}(i=k, k+1, \cdots,
d_1-1)$
Return $a_i := a_i^{(0)}(i=0, 1, \cdots, d_1)$.
In general, we can compute the equation (\[equ:tem1\]) after choosing $d_1+1$ distinct scalars $(x_{10},x_{11},\cdots,x_{1d_1})$ and $d_2+1$ distinct scalars $(x_{20}, x_{21}, \cdots, x_{2d_2})$, then obtain their corresponding exact values $(f_{00},f_{01},
\cdots,f_{0d_2},\cdots, $\
$f_{10},f_{11},\cdots,f_{1d_2},\cdots,f_{d_10},f_{d_11},\cdots, f_{d_1d_2})$. However, in order to improve intermediate expression swell problem arising from symbolic computations and avoid big integer computation, we can get the approximate values of $f(x_1, x_2)$, denoted by $(\tilde{f}_{00},
\tilde{f}_{01}, $ $ \cdots, \tilde{f}_{0d_2},\tilde{f}_{10},
\tilde{f}_{11}, \cdots, \tilde{f}_{1d_2}, \tilde{f}_{d_10},
\tilde{f}_{d_11},\cdots,$ $\tilde{f}_{d_1d_2})$.
Based on Algorithm \[alg:dual\], together with Lemma \[theo:kronecker\_equation\] we can obtain the approximate solution $\tilde{{a}}=[\tilde{a}_{ij}]$($i = 0, 1, \cdots, d_1; j = 0, 1,
\cdots, d_2$). So an approximate bivariate polynomial $\tilde{f}(x_1, x_2) = \sum_{i,j}\tilde{a}_{ij}x_1^ix_2^j$ is only produced. However, we usually need the exact results in practice. Next, our main task is to bound the error between approximate coefficients and exact values, and discuss the controlling error $\varepsilon$ in Algorithm \[alg:dual\]. The literature [@Feng2011] gave a preliminary study of this problem. Here, we present a necessary condition on error controlling $\varepsilon$ in floating-point arithmetic. In Step 1 of Algorithm \[alg:dual\], it is the standard method for evaluating divided differences($c_k^{(k)}=f[x_0, x_1, \cdots, x_k]$). We consider the relation on the $f_{ij}-\tilde{f}_{ij}$ with $a_{ij}-\tilde{a}_{ij}$ and the propagation of rounding errors in divided difference schemes. We have the following theorem to answer the above question.
\[lem:unierror\] $c_i$ and $f_{i}$ are defined as in Algorithm \[alg:dual\], $\tilde{c}_i$ and $\tilde{f}_{i}$ are their approximate values by approximate interpolation, $\lambda=\min\{|x_{2i}-x_{2j}|: i \neq
j\}(0<\lambda<1)$. Then $$|c_i- \tilde{c}_i|\leq
(\frac{2}{\lambda})^{d_2}\max\{|f_{i}-\tilde{f}_{i}|\}.$$
From Algorithm \[alg:dual\], we observe that Step 1 is recurrences for $c_i^{(k+1)}, (k=0, 1, \cdots, d_2-1, i=d_2, d_2-1, \cdots,
k+1)$, whose form is as follows: $$\begin{aligned}
c_i^{(d_2)}=\frac{1}{\lambda}(c_i^{(d_2-1)}-c_{i-1}^{(d_2-1)}).\end{aligned}$$ However, when we operate the floating-point arithmetic in Algorithm \[alg:dual\], which is recurrences for $\tilde{c}_i^{(k+1)}$, which form is as follows: $$\begin{aligned}
\tilde{c}_i^{(d_2)}=\frac{1}{\lambda}(\tilde{c}_i^{(d_2-1)}-\tilde{c}_{i-1}^{(d_2-1)}).\end{aligned}$$ Therefore, $$\begin{aligned}
|c_i^{(d_2)}-
\tilde{c}_i^{(d_2)}|=\frac{1}{\lambda}|c_i^{(d_2-1)}-\tilde{c}_i^{(d_2-1)}+\tilde{c}_{i-1}^{(d_2-1)}-c_{i-1}^{(d_2-1)}|
\leq\frac{1}{\lambda}(|c_i^{(d_2-1)}-\tilde{c}_i^{(d_2-1)}|+|c_{i-1}^{(d_2-1)}-\tilde{c}_{i-1}^{(d_2-1)}|).\end{aligned}$$ The bounds are defined by the following recurrences, $$\begin{aligned}
|c_i^{(d_2)}-
\tilde{c}_i^{(d_2)}|\leq\frac{2}{\lambda}|c_{i-1}^{(d_2-1)}-\tilde{c}_{i-1}^{(d_2-1)}|\leq\cdots\leq
(\frac{2}{\lambda})^{d_2}\max\{|f_{i}-\tilde{f}_{i}|\}.\end{aligned}$$ This completes the proof of the lemma.
\[theo:errocontrol\] Let $\varepsilon=\max\{|f_{ij}-\tilde{f}_{ij}|\}$, $\lambda=\min\{|x_{1i}-x_{1j}|, |x_{2i}-x_{2j}|: i \neq
j\}(0<\lambda<1)$. Then $$\max\{|a_{ij}-\tilde{a}_{ij}|\}\le(\frac{2}{\lambda})^{d_1}(\frac{2}{\lambda})^{d_2}\varepsilon.$$
From equation (\[equ:tem1\]), it holds that $${V}{\mathrm{vec}({\tilde{a}}-{a})}={\mathrm{vec}({\tilde{F}}-{F})},$$ where ${V} = {V}_{x_1} \otimes {V}_{x_2}$. By Lemma \[theo:kronecker\_equation\], the above equation is equivalent to the following equation: $${V}_{x_2}{({\tilde{a}}-{a})}{V}_{x_1}^T={\tilde{F}}-{F}.$$ Thus, it is equivalent to
$$\begin{aligned}
&&{V}_{x_2}{z}={\tilde{F}}-{F} \label{equ:a}\\
&&{V}_{x_1}({\tilde{a}}-{a})^T={z}^T \label{equ:b}\end{aligned}$$
where ${z}=[z_{ij}]$. Matrix equation (\[equ:a\]) is equivalent to $${V}_{x_2}{z}_{.i}={\tilde{F}}_{i.}-{F}_{i.}, \quad i=1, 2, \cdots d_2+1$$ where ${z}_{.i}$ stands for the $i$-th column of ${z}$ and ${F}_{i.}$ the $i$-th row of matrix ${F}$.
From Lemma \[lem:unierror\] and Algorithm \[alg:dual\], it holds that $$\max_{j=0}^{d_2}|z_{ji}|<(\frac{2}{\lambda})^{d_2}|f_{i\cdot}-\tilde{f}_{i\cdot
}|, \ for \ each \ i.$$ Hence, we conclude that $$\max_{i,j}|z_{ji}|<(\frac{2}{\lambda})^{d_2}|f_{i\cdot}-\tilde{f}_{i\cdot
}|.$$
Let $\delta =(\frac{2}{\lambda})^{d_2}|f_{i\cdot}-\tilde{f}_{i\cdot
}|$, argue equation (\[equ:b\]) in the same technique as do above, we deduce that $$\max_{i,j}|a_{ij}-\tilde{a}_{ij}|\leq (\frac{2}{\lambda})^{d_1}(\frac{2}{\lambda})^{d_2}\varepsilon.$$ The proof is finished.
In order to avoid the difficulty of computations, we restrict our study to the coefficients of polynomial over $\mathbb{Z}$. So we need to solve the Vandermonde system and take the nearest integer to each component of the solution. The less degree of bounds on variables we obtain, the less the amount of computation is for obtaining approximate multivariate polynomial. Once an upper bound $d_1$ and $d_2$ are gotten, we choose $(d_1+1)\cdot(d_2+1)$ interpolate nodes and calculate $$\label{equ:error_control}
\varepsilon=0.5{(\frac{\lambda}{2})}^{d_1+d_2}.$$ Then, compute the values $\tilde{f}_{ij}\approx f(x_{1i},x_{2j})$ for $i=0,1,\cdots,d_1,$ $ j=0,1,\cdots,d_2$ with an error less than $\varepsilon$. By interpolation method, we compute the approximate interpolation polynomial $\tilde{f}(x_1, x_2)$ with coefficient error less than 0.5.
As for the generalization of the algorithm to the case $v>2$, we can say that the situation is completely analogous to the bivariate case. It comes down to solving the following system: $$\label{equ:general}
\underbrace{({V}_{x_1}\otimes{V}_{x_2}\cdots\otimes{V}_{x_v})}_{v}{\mathrm{vec}({a})}
= {\mathrm{vec}({F})}.$$ Of course, we can reduce the multivariate polynomial entries to bivariate ones on symbolic determinant. For more details refer to Section 2.3.
We can analyze the computational complexity of the derivation of above algorithm. For the analysis of floating-point arithmetic operations, the result is similar with the exact interpolation situation [@Marco2004]. However, our method can enable the practical processing of symbolic computations in applications.
Our result is superior to the literature [@Feng2011]. Here we make full use of advantage of arbitrary precision of floating-point arithmetic operations on modern computer and symbolic computation platform, such as Maple. In general, it seems as if at least some problems connected with Vandermonde systems, which traditionally have been considered too ill-conditioned to be attached, actually can be solved with good precision.
Reducing dimension method
-------------------------
As the variables increased, the storage of computations expands severely when calculated high order on symbolic determinant. The literature [@Moenck1976] is to map the multivariate problem into a univariate one. For the general case, the validity of the method is established by the following lemma.
\[lem:reducedim\] ([@Moenck1976]) In the polynomial ring $R[x_1, x_2, \cdots,
x_v], v>2$. The mapping: $$\begin{aligned}
\phi: R[x_1, x_2, \cdots, x_v] \rightarrow R[x_1] \\
\phi: x_i \mapsto x_1^{n_i}, 1\leq i \leq v\end{aligned}$$ where $n_v> n_{v-1}>\cdots >n_1=1$ is a homomorphism of rings.
Let $d_i(f(x_1, x_2, \cdots, x_v))$ be the highest degree of the polynomial $f(x_1, x_2, \cdots, x_v)$ in variable $x_i$. The following lemma relates the ${n_i}$ of the mapping to $d_i$ and establishes the validity of the inverse mapping.
\[lem:liftingvar\] ([@Moenck1976]) Let $\psi$ be the homomorphism of free R-modules defined by:\
$\psi: R[x_1] \rightarrow R[x_1, x_2, \cdots, x_v]$\
$\psi: x_1^{k} \mapsto \makeatletter
\begin{cases}
1 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mbox{if~ $k=0,$} \\
\psi(x_1^r)\cdot x_i^q \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mbox{otherwise} \\
\end{cases}
$
where $n_{i+1}>k\geq n_i, k= q\cdot n_i+ r, 0\leq r <
n_i$ and $
n_v>\cdots >n_1 =1$.\
Then for all $f(x_1, x_2, \cdots, x_v)\in R[x_1, x_2, \cdots, x_v],
\psi(\phi(f)) = f$, and for all $i$ if and only if $$\label{equ:condition}
\sum_{j=1}^{i}d_j(f)n_j<n_{i+1}, 1 \leq i< v.$$
We apply the degree homomorphism method to reduce dimension for computing the determinant of a matrix with multivariate polynomial entries, which is distinguished from the practical fast polynomial multiplication [@Moenck1976]. We note that relation (\[equ:condition\]) satisfying is isomorphic to their univariate images. Thus any polynomial ring operation on entries of symbolic determinant, giving results in the determinant, will be preserved by the isomorphism. In this sense $\phi$ behaves like a ring isomorphism on the symbolic determinant of polynomials. Another way to view the mapping given in the theorems is: $$\phi: x_i \mapsto x_{i-1}^{n_i}, 2\leq i \leq v.$$
Derivation of the algorithm
===========================
The aim of this section is to describe a novel algorithm for estimating the degree of variables on symbolic determinant, and the degree homomorphism method for dimension reduction.
Description of algorithm
------------------------
Algorithm \[alg:maxdeg\] is to estimate the degree of variables on symbolic determinant by computation of the degree matrix, and Algorithm \[alg:reducedim\] and \[alg:liftingvar\] are used to reduce dimension and lift variables.
Input: given the order $n$ of symbolic determinant $M$, list of variables $vars$;\
Output: the exact or upper bounds on degree of variables.
[Step :]{}
Select variable from $vars$ respectively, and repeat the following steps
Obtain the degree matrix $\Omega=(\sigma_{ij}) (1 \leq i,j\leq n)$ from $M$; $maxdeg := \max\{\sigma_{11} + \sigma_{22}, \sigma_{12} + \sigma_{21}\}$ $temp := \sigma_{i1}+\sigma_{1j}$ $\sigma_{ij} := \max\{\sigma_{ij}+\sigma_{11}, temp\}$ $maxdeg := maxdeg - \sigma_{11}$
Return $maxdeg$
Algorithm \[alg:maxdeg\] works correctly as specified and its complexity is $O(n^2)$, where $n$ is the order of symbolic determinant.
Correctness of the algorithm follows from Theorem \[lem:maxdeg\].\
The number of arithmetic operations required to execute $(n-1)\times(n-1)$ additions and simultaneous comparisons, and remain $n-2$ substructions and one comparison by using degree matrix. Therefore, the total arithmetic operations are $n^2-n$, that is $O(n^2)$.
Input: given the order $n$ of symbolic determinant $M$, list of variables $vars$;\
Output: the order $n$ of symbolic determinant $M'$ with bivariate polynomial entries.
[Step :]{}
Call Algorithm \[alg:maxdeg\] to evaluate the bounds on degree of the variables in $M$, denoted by $d_i(1\leq i\leq v)$.
Reducing dimension
Divide the $vars$ into the partitions: $[x_1, x_2, \cdots, x_t], [x_{t+1}, x_{t+2}, \cdots,
x_v]$; $D_i := \prod_{j=i+1}^{t}(d_j+1)$, $x_i \leftarrow x_t^{D_i}$ $D_i := \prod_{j=i+1}^{v}(d_j+1)$, $x_i\leftarrow x_v^{D_i}$
Obtain the symbolic determinant $M'$ on variables $vars=[x_t, x_v$\];
Return $M'$.
The beauty of this method is in a substitution trick. In Algorithm \[alg:reducedim\], $t = ceil(\frac{n}{2})$, where $ceil(c)$ is a function which returns the smallest integer greater than or equal the number $c$. We note that the lexicographic order $x_v \succ
x_{v-1}\succ \cdots \succ x_1$ and divide the $vars$ into two parts. Then the symbolic determinant can be translated into the entries with bivariate polynomial. It can be highly parallel computation when the variables are more than three.
Input: given the set of monomial on $x_t, x_v$ in $L$;\
Output: the polynomial with $x_1, x_2, \cdots, x_v$.
[Step :]{}
Obtain the corresponding power set on $x_t, x_v$, respectively;
Lifting variables
Call Algorithm \[alg:reducedim\], extract the power $D_i(1\leq i \leq t-1, t+1 \leq i \leq v-1)$; $temp := deg(x_t)$ $d_i := iquo(temp, D_i), temp := irem(temp, D_i)$ $d_i := temp, temp := deg(x_v)$ $d_i := iquo(temp, D_i), temp := irem(temp, D_i)$ $d_i := temp$
Obtain the new set of monomial $L'$ on $x_1, x_2, \cdots,
x_v$;
Return $L'$.
To sum up, based on Algorithm \[alg:maxdeg\] to estimate bounds on degree of variables, Algorithm \[alg:reducedim\] to reduce dimension for multivariate case, Algorithm \[alg:dual\] to solve the Vandermonde coefficient matrix of linear equations with error controlling, and finally Algorithm \[alg:liftingvar\] to lift variables for recovering the multivariate polynomial.
In this paper, we consider the general symbolic determinant, which is not sparse. Applying the substitutions to the matrix entries as described above and assuming the monomial exists in the determinant then the bivariate form of unknown polynomial is a highest degree of $$D=\sum_{i=1}^{ceil(\frac{n}{
2})}(d_i\cdot\prod_{k=i+1}^{ceil(\frac{n}{ 2})}(d_k+1)).$$ While this upper bound on degree of variable is often much larger than needed, which is the worst case and thus is suitable to all cases.
A small example in detail
-------------------------
\[exam1\] For convenience and space-saving purposes, we choose the symbolic determinant is three variables and order 2 as follows. $$|M|=
\begin{vmatrix}
5x_1^2-3x_1x_2+2x_3^2&\ \ \ \ \ \ \ -9x_1-3x_2^2-x_3^2\\
-x_1+x_2+3x_2x_3 &x_3-4x_2^2
\end{vmatrix},$$ At first, based on Algorithm \[alg:maxdeg\] we estimate the degree on $x_1, x_2, x_3$. For the variable $x_1$, we get $$\begin{aligned}
\Omega_1=
\left[\begin{array}{cc}
2&\ \ 1\\
1&\ \ 0
\end{array}\right].\end{aligned}$$ Then $$\max\{2+0,1+1\} = 2.$$ Therefore, the maximum degree of the variable $x_1$ is $2$. As the same technique for $x_2, x_3$, we can get $3$ and $3$.
Call Algorithm \[alg:reducedim\], by substituting $x_1=x_2^4$, we get $$|M'|=
\begin{vmatrix}
5x_2^8-3x_2^5+2x_3^2&\ \ \ \ \ \ \ -9x_2^4-3x_2^2-x_3^2\\
-x_2^4+x_2+3x_2x_3&x_3-4x_2^2
\end{vmatrix}.$$ Then, based on Algorithm \[alg:maxdeg\] we again estimate the degree on $x_2, x_3$ for $[10, 3]$.
Based on the derivation of algorithm in Section 3.1 and Algorithm \[alg:dual\], computing exact polynomial $f(x_2, x_3)$ as follows: Choose the different floating-point interpolation nodes by using the distance between two points 0.5; $\lambda=0.5$, compute $\varepsilon=0.745\times10^{-8}$ from Theorem \[theo:errocontrol\]. Compute the approximate interpolate datum $\tilde{f}_{ij}$ such that $|f_{ij}-\tilde{f}_{ij}|<\varepsilon$. We get the following approximate bivariate polynomial: $$4.99995826234x_2^8x_3-20.0000018736x_2^{10}+24.0010598569x_2^5x_3+12.0025760656x_2^7+2.00000000000x_3^3$$ $$-8.00094828634x_2^2x_3^2
-9.00045331720x_2^8+9.01977448800x_2^5-3.00897542075x_2^6
+3.02270681750x_2^3$$ $$+9.00076124850x_2^3x_3-1.00207248277x_2^4x_3^2
+1.00018098282x_2x_3^2+2.99986559933x_2x_3^3.$$ Next, based on Algorithm \[alg:liftingvar\] we lift the variables to obtain the following multivariate polynomial: $$4.99995826234x_1^2x_3-20.0000018736x_2^2x_1^2+24.0010598569x_1x_2x_3
+12.0025760656x_2^3x_1+2.00000000000x_3^3$$ $$-8.00094828634x_2^2x_3^2
-9.00045331720x_1^2+9.01977448800x_1x_2-3.00897542075x_2^2x_1
+3.02270681750x_2^3$$ $$+9.00076124850x_2^3x_3-1.00207248277x_1x_3^2+1.00018098282x_2x_3^2+2.99986559933x_2x_3^3.$$ Finally, we easily recover the integer coefficients of above approximate polynomial to the nearest values as follows: $$5x_1^2x_3-20x_1^2x_2^2+24x_1x_2x_3+12x_1x_2^3+2x_3^3-8x_3^2x_2^2
-9x_1^2+9x_1x_2-3x_2^2x_1+3x_2^3+9x_2^3x_3-x_3^2x_1+x_3^2x_2+3x_3^3x_2.$$
Experimental results
====================
Our algorithms are implemented in *Maple*. The following examples run in the same platform of *Maple* under Windows and <span style="font-variant:small-caps;">amd</span> Athlon(tm) 2.70 Ghz, 2.00 GB of main memory(RAM). Figures 1 and 2 present the $Time$ and $RAM$ of computing for symbolic determinants to compare our method with symbolic method($det$, see *Maple*’s help), and exact interpolation method [@Marco2004; @Li2009; @Chen2013]. Figure 1 compared with time for computing, Figure 2 compared with memory consumption for computing, the $order$ of $x$-coordinate represents for the order of symbolic determinants.
From Figures 1 and 2, we have the observations as follows:
1. In general, the $Time$ and $RAM$ of algorithm $det$ are reasonable when the $order$ is less than nine, and two indicators increase very rapidly when the $order$ is to nine. However, two indicators of interpolation algorithm is steady growth.
2. Compared with the exact interpolation method, the approximate interpolation algorithm has the obvious advantages on the $Time$ and $RAM$ when the $order$ is more than eight.
All examples are randomly generated using the command of *Maple*. The symbolic method has the advantage of the low order or sparse symbolic determinants, such as expansion by minors, Gaussian elimination over the integers. However, a purely symbolic algorithm is powerless for many scientific computing problems, such as resultants computing, Jacobian determinants and some practical engineering always involving high-order symbolic determinants. Therefore, it is necessary to introduce numerical methods to improve intermediate expression swell problem arising from symbolic computations.
Conclusions
===========
In this paper, we propose a hybrid symbolic-numerical method to compute the symbolic determinants. Meanwhile, we also present a novel approach for estimating the bounds on degree of variables by the extended numerical determinant technique, and introduce the reducing dimension algorithm. Combined with these methods, our algorithm is more efficient than exact interpolation algorithm for computing the high order symbolic determinants. It can be applied in scientific computing and engineering fields, such as computing Jacobian determinants in particular. Thus we can take fully advantage of approximate methods to solve large scale symbolic computation problems.
References {#references .unnumbered}
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[^1]: $\Omega_1, \Omega_2, \cdots, \Omega_v$ denote the degree matrix of $x_1, x_2, \cdots, x_v$, respectively.
[^2]: $\sigma^{(\cdot)}_{ij}$ is defined by the same way for the rest of this paper.
[^3]: $d_1, d_2$ are defined by the same way for the rest of this paper.
|
[Compact Group Actions On Operator Algebras]{}
[ and Their Spectra]{}
[Costel Peligrad]{}
*Department of Mathematical Sciences, University of Cincinnati, 4508 French Hall West, Cincinnati, OH 45221-0025, United States. E-mail address: [email protected]*
[Abstract. ]{}We consider a class of dynamical systems with compact non abelian groups that include C\*-, W\*- and multiplier dynamical systems. We prove results that relate the algebraic properties such as simplicity or primeness of the fixed point algebras as defined in Section 3., to the spectral properties of the action, including the Connes and strong Connes spectra.
Keywords: dynamical system, compact group, simple C\*-algebra, prime C\*-algebra, von Neumann factor, Connes spectrum, strong Connes spectrum.
2010 Mathematics Subject Classifications: Primary 46L05, 46L10, 46L55, Secondary 46L40, 37B99.
Introduction
============
In \[2\], Connes introduced the invariant $\Gamma(U)$ known as the Connes spectrum of the action $U$ of a locally compact abelian group on a von Neumann algebra and used it in his seminal classification of type III von Neumann factors. Soon after, Olesen \[10\] defined the Connes spectrum of an action of a locally compact abelian group on a C\*-algebra. In \[11\], using the definition of the Connes spectrum in \[10\], it is proven an analog of a result of Connes and Takesaki \[3, Chapter III, Corollary 3.4.\] regarding the significance of the Connes spectrum of a locally compact abelian group action on a C\*-algebra for the ideal structure of the crossed product. In particular, in \[11\] is discussed a spectral characterization for the crossed product to be a prime C\*-algebra. This definition of the Connes spectrum in \[10\] cannot be used to prove similar results for the simplicity of the crossed product, unless the group is discrete \[11\]. Kishimoto \[8\] defined the strong Connes spectrum for C\*-dynamical systems with locally compact abelian groups that coincides with the Connes spectrum for the W\*-dynamical system and with the Connes spectrum defined by Olesen for discrete abelian group actions on C\*-algebras and he proved the Connes-Takesaki result for simple crossed products. In \[2\] Connes obtained results that relate the spectral properties of the von Neumann algebra with the algebraic properties of the fixed point algebra. These results were extended in \[12\] to C\*-algebras and compact abelian groups. In \[6\], \[14\] we considered the problems of simplicity and primeness of the crossed product by compact, non abelian group actions. In particular, in \[6\] we have defined the Connes and strong Connes spectra for such actions that coincide with Connes spectra \[2\], \[10\], respectively with the strong Connes spectra \[8\] for compact abelian groups. Further, in \[15\] we have considered the case of one-parameter $\mathcal{F}$-dynamical systems that include the C\*- the W\*- and the multiplier one-parameter dynamical systems. In particular, we have obtained extensions of some results in \[2\], \[12\] for $W^{\ast}-,$ respectively $C^{\ast}-$ dynamical systems to the case of$\ \mathcal{F}$-dynamical systems with compact abelian groups \[15, Theorems 3.2 and 3.4.\]. In this paper we will prove results for $\mathcal{F}$-dynamical systems with compact non abelian groups. Our results contain and extend to the case of compact non abelian groups the following: \[2 Proposition 2.2.2. b) and Theorem 2.4.1\], \[12, Theorem 2\], \[13, Theorem 8.10.4\] and \[15, Theorems 3.2. and 3.4.\]. In Section 2. we will set up the framework and state some results that will be used in the rest of the paper. In Section 3. we discuss the connection between the strong Connes spectrum, $\widetilde{\Gamma
}_{\mathcal{F}}(\alpha),$ of the action and the $\mathcal{F}$-simplicity of the fixed point algebras $(X\otimes B(H_{\pi}))^{\alpha\otimes ad\pi}$. In Section 4. we will get similar results about the connection between the $\mathcal{F}$-primeness of the fixed point algebras and the Connes spectrum, $\Gamma_{\mathcal{F}}(\alpha),$ of the action.
Notations and preliminary results
=================================
This section contains the definitions of the basic concepts used in the rest of the paper, the notations and some preliminary results.
**2.1. Definition. (\[1\], \[16 \])** *A dual pair of Banach spaces is, by definition, a pair* $(X,\mathcal{F})$ *of Banach spaces with the following properties:*
*a)* $\mathcal{F}$ *is a Banach subspace of the dual* $X^{\ast}$ *of* $X.$
*b)* $\left\Vert x\right\Vert =\sup\left\{ \left\vert \varphi
(x)\right\vert :\varphi\in\mathcal{F},\left\Vert \varphi\right\Vert
\leq1\right\} ,x\in X.$
*c)* $\left\Vert \varphi\right\Vert =\sup\left\{ \left\vert
\varphi(x)\right\vert :x\in X,\left\Vert x\right\Vert \leq1\right\}
,\varphi\in\mathcal{F}.$
*d)* *The convex hull of every relativel*y $\mathcal{F}$-*compact subset of* $\mathit{X}$* is relatively* $\mathcal{F}$-*compact*.
*e)* *The convex hull of every relatively* $X$*-compact subset of* $\mathcal{F}$ *is relatively* $X$-*compact*.
*In the rest of the paper* $X$* will be assumed to be a C\*-algebra with the additional property*
*f) The involution of* $X$* is* $\mathcal{F}$*-continuous and the multiplication in* $X$* is separately* $\mathcal{F}$*-continuous.*
The property d) implies the existence of the weak integrals of continuous functions defined on a locally compact measure space, ($S,\mu)$ with values in $X$ endowed with the $\mathcal{F}$-topology:
If $f$ is such a function, we will denote by$$\int f(s)d\mu$$ the unique element $y$ of $X$ such that $$\varphi(y)=\int_{S}\varphi(f(s))d\mu$$ for every $\varphi\in\mathcal{F}$ \[1, Proposition 1.2.\]$.$ The propery e) was used by Arveson \[1, Proposition 1.4.\] to prove the continuity in the $\mathcal{F}$-topology of some linear mappings on $X\ $(in particular the mappings $P_{\alpha}(\pi)$ and ($P_{\alpha})_{ij}(\pi)$ defined below).
**2.2**. **Examples.** *a) \[1\] If* $X$* is a C\*-algebra and and* $\mathcal{F}=X^{\ast},$ *conditions 1)-5) are satisfied.*
*b) \[1\] If* $X$* is a W\*-algebra and* $\mathcal{F}=X_{\ast}$ *is its predual then conditions 1)-5) are satisfied.*
*c) \[4\] If* $X=M(Y)$* is the multiplier algebra of* $Y$* and* $\mathcal{F}=Y^{\ast}$* then conditions 1)-5) are satisfied. In addition, in this case, the* $\mathcal{F}$*-topology on* $X$* is compatible with the strict toplogy on* $X=M(Y).$
Let $(X,\mathcal{F})$ be a dual pair of Banach spaces $G$ a compact group and $\alpha:G\rightarrow Aut(X)$ a homeomorphism of $G$ into the group of $\ast
-$automorphisms of $X$. We say that $(X,G,\alpha)$ is an $\mathcal{F}$-dynamical system if the mapping$$g\rightarrow\varphi(\alpha_{g}(x))$$ is continuous for every $x\in X$ and $\varphi\in\mathcal{F}.$
**2.3**. **Examples.** *a) If* $\mathcal{F}=X^{\ast},$ the dual of $X$ *then, by \[7 p. 306\] the above condition is equivalent to the continuity of* the mapping $g\rightarrow\alpha_{g}(x)$ from $G$* to* $X$* endowed with the norm topology for every* $x\in
X$*, so, in this case* $(X,G,\alpha)$ *is a C\*-dynamical system.*
*b) If* $X$* is a von Neumann algebra and* $\mathcal{F}=X_{\ast},$ the predual of $X$ *then* $(X,G,\alpha)$ *is a W\*-dynamical system.*
*c) If* $X=M(Y)$* is the multiplier algebra of* $Y$* and* $\mathcal{F}=Y^{\ast},$ *then* $(X,G,\alpha)$ *is said to be a multiplier dynamical system.*
Let $(X,G,\alpha)$ be an $\mathcal{F}$-dynamical system with $G$ compact. Denote by $\widehat{G}$ the set of unitary equivalence classes of irreducible representations of $G.$ For each $\pi\in\widehat{G}$ denote also by $\pi$ a fixed representative of that class. If $\chi_{\pi}(g)=d_{\pi}\sum
_{i=1}^{d_{\pi}}\pi_{ii}(g^{-1})=d_{\pi}\sum\overline{\pi_{ii}(g)}$ is the character of $\pi,$ denote by $$P_{\alpha}(\pi)(x)=\int_{G}\chi_{\pi}(g)\alpha_{g}(x)dg.$$ Then $P_{\alpha}(\pi)$ is a projection of $X$ onto the spectral subspace $$X_{1}(\pi)=\left\{ x\in X:P_{\alpha}(\pi)(x)\right\} .$$ where the integral is taken in the weak sense defined in (1) above. As in \[14\] one can also define for every $1\leq i,j\leq d_{\pi}$$$(P_{\alpha})_{ij}(\pi)(x)=\int_{G}\overline{\pi_{ji}(g)}\alpha_{g}(x)dg.$$ where $d_{\pi}$ is the dimension of the Hilbert space $H_{\pi}$ of $\pi
$ and show that $$(P_{\alpha})_{ij}(\pi)(X)\subset X_{1}(\pi).$$ Using \[1, Proposition 1.4.\] it follows that $P_{\alpha}(\pi),(P_{\alpha})_{ij}(\pi)$ are $\mathcal{F}$-continuous. If $\pi$ is the identity one dimensional representation $\iota$ of $X,$ we will denote $$P_{\alpha}(\iota)=P_{\alpha}.$$ and $$X_{1}(\iota)=X^{\alpha}.$$ is the fixed point algebra of the action.
**2.4. Remark.** $\overline{\sum_{\pi\in\widehat{G}}X_{1}(\pi)}^{\sigma
}=X,$ *where* $\overline{\sum_{\pi\in\widehat{G}}X_{1}(\pi)}^{\sigma}$* denotes the closure of* $\sum_{\pi\in\widehat{G}}X_{1}(\pi
)$ *in the* $\mathcal{F}$*-topology of* $X.$
Suppose that there exists $\varphi\in\mathcal{F}$ such that $\varphi
(X_{1}(\pi))=\left\{ 0\right\} $ for every $\pi\in\widehat{G}.$ Since, as noticed above, $(P_{\alpha})_{ij}(\pi)(X)\subset X_{1}(\pi),$ it follows that$$\int_{G}\overline{\pi_{ij}(g)}\varphi(\alpha_{g}(x))dg=0.$$ for every $x\in X$ and every $\pi\in\widehat{G}.$ Since $\left\{ \pi
_{ij}(g):\pi\in\widehat{G},1\leq i,j\leq d_{\pi}\right\} $ is an orthogonal basis of $L^{2}(G),$ and $\varphi(\alpha_{g}(x))$ is a continuous function of $g,$ for every $x\in X,$ it follows that $\varphi(x)=0$ for every $x\in
X$ so $\varphi=0$ and we are done.
In (\[9\], \[14\], \[6\]) it is pointed out that the spectral subspaces$$X_{2}(\pi)=\left\{ a\in X\otimes B(H_{\pi}):(\alpha_{g}\otimes\iota
)(a)=a(1\otimes\pi_{g})\right\} .$$ where $\iota$ is the identity automorphism of $B(H_{\pi})$ are, in some respects more useful$.$ In \[14\] it is shown that $X_{2}(\pi)$ consists of all matrices $$\left\{ a=\left[ (P_{\alpha})_{ij}(\pi)(x)\right] (=\left[ a_{ij}\right]
)\in X\otimes B(H_{\pi}):x\in X,1\leq i,j\leq d_{\pi}\right\} .$$ It is straightforward to prove that, if $a\in X_{2}(\pi)$ and $x=\sum
_{i}a_{ii}$, then $a_{ij}=(P_{\alpha})_{ij}(\pi)(x).$ In what follows, if $b\in X\otimes B(H_{\pi})$ we will denote$$tr(b)=\sum b_{ii}$$ which is an $\mathcal{F}$-continuous linear mapping from $X\otimes B(H_{\pi})$ to $X.$ The following lemma is proven for compact non abelian group actions on C\*-algebras in \[6, Lemma 2.3.\] and for compact abelian $\mathcal{F}$-dynamical systems in \[15\]. Since the proof is very similar with the proof of \[6, Lemma 2.3.\] we will state it without proof
**2.5. Lemma.** *Let (*$X,G,\alpha)$ *be an* $\mathcal{F}$*-dynamical system with* $G$* compact and* $J$* a two sided ideal of* $X^{\alpha}.$ *Then*$$(\overline{XJX}^{\sigma})^{\alpha}=\mathcal{F}\text{\textit{-closed linear
span of }}\left\{ tr(X_{2}(\pi)JX_{2}(\pi)^{\ast}):\pi\in\widehat{G}\right\}
.$$ *where, if* $a=\left[ a_{kl}\right] \in X\otimes B(H_{\pi})$ *and* $j\in X,$* by* $ja$* we mean the matrix* $\left[
ja_{kl}\right] $ *and the multiplications* $\overline{XJX}^{\sigma},$ ** $X_{2}(\pi)JX_{2}(\pi)^{\ast}\ $*are defined in 2.6. below.*
We will use the following notations
**2.6.** **Notation.** *Let (*$X,\mathcal{F})$ *be a dual pair of Banach spaces with* $X$* a C\*-algebra satisfying conditions 1)-6).* I*f* $Y,Z$* are subsets of* $X$* denote:*
*a)* $lin\left\{ Y\right\} $* is the linear span of* $Y$*.*
*b)* $Y^{\ast}=\left\{ y^{\ast}:y\in Y\right\} .$
*c)* $YZ=lin\left\{ yz:y\in Y,z\in Z\right\} .$
*d)* $\overline{Y}^{\sigma}=\mathcal{F}-$*closure of* $Y$ in $X.$
*e)* $\overline{Y}^{\left\Vert {}\right\Vert }=$ *norm closure of* $Y.$
*f)* $\overline{Y}^{w}=$ *the* $w^{\ast}$*-closure of* $Y$* in* $\mathcal{F}^{\ast}.$
*If (*$X,G,\alpha)$ *is an* $\mathcal{F}\mathit{-}$*dynamical system denote*
*g)* $\mathcal{H}_{\sigma}^{\alpha}(X)$ *the set of all non-zero globally* $\alpha-$*invariant* $\mathcal{F}\mathit{-}$*closed hereditary C\**$\mathit{-}$*subalgebras of* $X.$
Notice that if *(*$X,G,\alpha)$ is an $\mathcal{F}-$dynamical system and if $X_{2}(\pi)$ is the spectral subspace defined above, then $X_{2}(\pi)X_{2}(\pi)^{\ast}$ is a two sided ideal of $X^{\alpha}\otimes B(H_{\pi})$ and $X_{2}(\pi)^{\ast}X_{2}(\pi)\ $is a two sided ideal of $(X\otimes
B(H_{\pi}))^{\alpha\otimes ad\pi}$ where $\alpha\otimes ad\pi$ is the action$$(\alpha_{g}\otimes ad\pi_{g})(a)=(1\otimes\pi_{g})[\alpha_{g}(a_{ij})](1\otimes\pi_{g^{-1}}).$$ on $X\otimes B(H_{\pi}).$
**2.7. Definition.** *a)* $sp(\alpha)=\left\{ \pi\in\widehat
{G}:X_{1}(\pi)\neq\left\{ 0\right\} \right\} .$*b)* $sp_{\mathcal{F}}(\alpha)=\left\{ \pi\in\widehat{G}:\overline{X_{2}(\pi)^{\ast}X_{2}(\pi)}^{\sigma}\text{ \textit{is essential in} }(X\otimes
B(H_{\pi}))^{\alpha\otimes ad\pi}\right\} .$*c)* $\widetilde{sp}_{\mathcal{F}}(\alpha)=\left\{ \pi\in\widehat{G}:\overline{X_{2}(\pi)^{\ast}X_{2}(\pi)}^{\sigma}=(X\otimes B(H_{\pi}))^{\alpha\otimes ad\pi}\right\} .$*Corresponding to the above Arveson type spectra b) and c) we define two Connes type spectrad)* $\Gamma_{\mathcal{F}}(\alpha)=\cap\left\{ sp_{\mathcal{F}}(\alpha|_{Y}):Y\in\mathcal{H}_{\sigma}^{\alpha}(X)\right\} .$*e)* $\widetilde{\Gamma}_{\mathcal{F}}(\alpha)=\cap\left\{ \widetilde
{sp}_{\mathcal{F}}(\alpha|_{Y}):Y\in\mathcal{H}_{\sigma}^{\alpha}(X)\right\}
.$
Clearly, $\widetilde{sp}_{\mathcal{F}}(\alpha)\subset sp_{\mathcal{F}}(\alpha)\subset sp(\alpha),$ so $\widetilde{\Gamma}_{\mathcal{F}}(\alpha)\subset\Gamma_{\mathcal{F}}(\alpha).$ The definition of $\widetilde
{\Gamma}_{\mathcal{F}}(\alpha)$ is a direct generalization of the strong Connes spectrum of Kishimoto to compact non abelian groups. Our motivation for the definition of $\Gamma_{\mathcal{F}}(\alpha)$ above (and $\Gamma(\alpha)$ for C\*-dynamical systems in \[6\]) is the following observation
**2.8. Remark *a)*** *If (*$X,G,\alpha)$ *is* *an* $\mathcal{F}\mathit{-}$*dynamical system with* $G$* compact abelian, then* $$\cap\left\{ sp(\alpha|_{Y}):Y\in\mathcal{H}_{\sigma}^{\alpha}(X)\right\}
=\cap\left\{ sp_{\mathcal{F}}(\alpha|_{Y}):Y\in\mathcal{H}_{\sigma}^{\alpha
}(X)\right\} .$$ *and the left hand side of the above equality is the Connes spectrum for W\**$\mathit{-}$*as well as for C\*-dynamical systems.*
*b) If* $G$* is not abelian, the equality in part a) is not true.*
a\) We have to prove only one inclusion, the opposite one being obvious. Let $\gamma\in\cap\left\{ sp(\alpha|_{Y}):Y\in\mathcal{H}_{\sigma}^{\alpha
}(X)\right\} $ and $Y\in\mathcal{H}_{\sigma}^{\alpha}(X).$ Suppose that $aY_{\gamma}^{\ast}Y_{\gamma}=\left\{ 0\right\} $ for some $a\in Y^{\alpha
},a\neq0.$ Then $aY_{\gamma}^{\ast}=\left\{ 0\right\} .$ Therefore, if we denote $Z=\overline{aYa^{\ast}}^{\sigma}$, it follows that $Z\in
\mathcal{H}_{\sigma}^{\alpha}(X)$ and $Z_{\gamma}^{\ast}=\left\{ 0\right\} $ which is in contradiction with the hypothesis that $\gamma\in\gamma\in
\cap\left\{ sp(\alpha|_{Y}):Y\in\mathcal{H}_{\sigma}^{\alpha}(X)\right\}
\subset sp(\alpha|_{Z}).$
b\) In \[14, Example 3.9.\] we provided an example of an action of an action of $G=S_{3}$ the permutation group on three elements on the algebra $X$ of $2\times2$ matrices such that $sp(\alpha)=\widehat{G},H_{\sigma}^{\alpha
}=\left\{ X\right\} ,$ so $$\cap\left\{ sp(\alpha|_{Y}):Y\in\mathcal{H}_{\sigma}^{\alpha}(X)\right\}
=sp(\alpha)$$ and we have shown that there exists $\pi\in\widehat{G}$ such that ($X\otimes B(H_{\pi}))^{\alpha\otimes ad\pi}$ has nontrivial center and, therefore, it is not a prime C\*-algebra. By \[6, Thm. 2.2.\], it follows that $\cap\left\{ sp(\alpha|_{Y}):Y\in\mathcal{H}_{\sigma}^{\alpha}(X)\right\}
\neq\cap\left\{ sp_{\mathcal{F}}(\alpha|_{Y}):Y\in\mathcal{H}_{\sigma
}^{\alpha}(X)\right\} =\Gamma(\alpha).$
$\mathcal{F}-$ simple fixed point algebras
==========================================
Let $(X,G,\alpha)$ be an $\mathcal{F}$-dynamical system with $G$ compact. In the rest of this paper we will study how the $\mathcal{F}$-simplicity (respectively $\mathcal{F}$-primeness) as defined below, of the fixed point algebras $(X\otimes B(H_{\pi}))^{\alpha\otimes ad\pi}$ is reflected in the spectral properties of the action.
**3.1. Definition.** *Let* $(B,\mathcal{F)}$ *be a dual pair of Banach spaces with* $B$* a C\*-algebra.*
*a)* $B$* is called* $\mathcal{F}$*-simple if every non zero two sided ideal of* $B$* is* $\mathcal{F}$*-dense in* $B.$
*b)* $B$* is called* $\mathcal{F}$*-prime if the annihilator of every non zero two sided ideal of* $B$ *is trivial, or, equivalently, every non zero two sided ideal of* $B$* is an essential ideal (using Definition 2.1. f) it is easy to see that* $X$* is* $\mathcal{F}-$*prime if and only if* $X$* is prime as a C\*-algebra).*
*Let (*$X,G,\alpha)$ *be an* $\mathcal{F}$*-dynamical system.*
*c)* $X$* is called* $\mathcal{\alpha}\mathit{-}$*simple if every non zero* $\alpha-$*invariant two sided ideal of* $X$* is* $\mathcal{F}\mathit{-}$*dense in* $X.$
*d)* $X$* is called* $\mathcal{\alpha}\mathit{-}$*prime if every non zero* $\alpha-$*invariant two sided ideal of* $X$* is an essential ideal.*
In the particular case when $B$ is a C\*-algebra and $\mathcal{F}=B^{\ast}$ is its dual, then, clearly, the concepts of $\mathcal{F}-$simple, (respectively $\mathcal{F}-$prime) in the above Definition 3.1. a) (respectively b)) coincide with the usual concepts of simple (respectively prime) C\*-algebras. Similarly, if *(*$X,G,\alpha)$ is a C\*-dynamical system, that is if $X$ is a C\*-algebra and $\mathcal{F}=X^{\ast}$ is its dual, then the notions of $\mathcal{\alpha}\mathit{-}$*simple* and $\mathcal{\alpha}\mathit{-}$*prime* coincide with the usual ones for C\*-dynamical systems.
If $B$ is a von Neumann algebra and $\mathcal{F}=B_{\ast}$ is its predual, then, since the weak closure of every essential ideal equals $B,$ it follows that $B$ is $\mathcal{F}$-simple if and only if $B$ is $\mathcal{F}-$prime, so, if and only if $B$ is a factor. It is also obvious that if *(*$X,G,\alpha)\ $is a W\*-dynamical system, that is if $X$ is a von Neumann algebra and $\mathcal{F}=X_{\ast}$ is its predual, then $X$ is $\mathcal{\alpha}-$*simpl*e if and only if it is $\mathcal{\alpha}-$*prime*, and this holds if and only if $\alpha$ acts ergodically on the center of $X$ (i.e. every fixed element in the center of $X$ is a scalar).
The above observations and the next Remark show that for W\*-dynamical systems, $(X,G,\alpha)$ with $G$ compact, the results in the current Section 3 and Section 4 are equivalent.
**3.2. Remark.** *Let (*$X,G,\alpha)$ *be a W\*-dynamical system, that is, an* $\mathcal{F}$*-dynamical system with* $X$* a von Neumann algebra and* $\mathcal{F}=X_{\ast}$ *its predual.* *Then* $\widetilde{\Gamma}_{\mathcal{F}}(\alpha
)=\Gamma_{\mathcal{F}}(\alpha).$
This follows from the fact that if $X$ is a von Neumann algebra, $p\in
X^{\alpha}$ an $\alpha$-invariant projection and $\overline{pX_{2}(\pi)p^{\ast}X_{2}(\pi)p}^{\sigma}$ is essential in $(pXp\otimes B(H_{\pi
}))^{\alpha\otimes ad\pi}$, then $\overline{pX_{2}(\pi)p^{\ast}X_{2}(\pi
)p}^{\sigma}$=$(pXp\otimes B(H_{\pi}))^{\alpha\otimes ad\pi}.$
The next lemma will be used in the proofs of the main results of the current Section 3 and the next Section.
**3.3. Lemma.** *Let* $(B,G,\alpha)$* be an* $\mathcal{F}$*-dynamical system with* $G$* compact. Then*
*a) If* $\left\{ e_{\lambda}\right\} $* is an approximate identity of* $B^{\alpha}$* in the norm topology, then*$$(norm)\lim_{\lambda}e_{\lambda}x=(norm)\lim_{\lambda}xe_{\lambda}=(norm)\lim_{\lambda}e_{\lambda}xe_{\lambda}=x.$$ *for every* $x\in\overline{\sum_{\pi\in\widehat{G}}B_{1}(\pi
)}^{\left\Vert {}\right\Vert }.$* *
*b) If* $b\in B$* is such that* $B^{\alpha}bB^{\alpha}=\left\{
0\right\} $* then* $b=0.$
*c)* $\overline{B^{\alpha}BB^{\alpha}}^{\sigma}=\overline{B^{\alpha}B}^{\sigma}=\overline{BB^{\alpha}}^{\sigma}=B.$
*d)* $\overline{B^{\alpha}B_{1}(\pi)}^{\sigma}=\overline{B_{1}(\pi)B^{\alpha}}^{\sigma}=B_{1}(\pi),\pi\in\widehat{G}.$
a\) This follows from the proof of \[5, Lemma 2.7\] in the more general case of compact quantum group actions.
b\) If $\left\{ e_{\lambda}\right\} $ is an approximate identity of $B^{\alpha},$ then $e_{\lambda}be_{\lambda}=0$ implies$$e_{\lambda}P_{\alpha}(\pi_{ij})(b)e_{\lambda}=P_{\alpha}(e_{\lambda
}be_{\lambda})=0.$$ for every $\pi\in\widehat{G},1\leq i,j\leq d_{\pi},$ so, by a), $P_{\alpha
}(\pi_{ij})(b)=0$ Therefore, $$\varphi(P_{\alpha}(\pi_{ij})(b))=\int_{G}\pi_{ji}(g)\varphi(\alpha
_{g}(b))dg=0.$$ for every $\varphi\in\mathcal{F},\pi\in\widehat{G},1\leq i,j\leq d_{\pi}.$ Since $\left\{ \pi_{ij}(g):\pi\in\widehat{G},1\leq i,j\leq d_{\pi}\right\} $ form an orthogonal basis of $L^{2}(G),$ and $\varphi(\alpha_{g}(b))$ is continuous on $G,$ it follows that $\varphi(\alpha_{g}(b))=0$ for every $g\in
G,\varphi\in\mathcal{F},$ so $b=0.$
c\) We will prove only that ** $\overline{B^{\alpha}BB^{\alpha}}^{\sigma}=B,$ the proofs of the other equalities being similar. Let $\left\{
e_{\lambda}\right\} $ be an approximate identity of $B^{\alpha}.$ By a), $$(norm)\lim_{\lambda}e_{\lambda}xe_{\lambda}=xforeveryx\in\overline{\sum
_{\pi\in\widehat{G}}B_{1}(\pi)}^{\left\Vert {}\right\Vert }.$$ Therefore$$\overline{\sum_{\pi\in\widehat{G}}B_{1}(\pi)}^{\left\Vert {}\right\Vert
}\subset\overline{B^{\alpha}BB^{\alpha}}^{\left\Vert {}\right\Vert }\subset\overline{B^{\alpha}BB^{\alpha}}^{\sigma}.$$ Since, by Remark 2.4., the $\mathcal{F}$-closure of $\overline{\sum_{\pi
\in\widehat{G}}B_{1}(\pi)}^{\left\Vert {}\right\Vert }$ equals $B$ it follows that $$B=\overline{\sum_{\pi\in\widehat{G}}B_{1}(\pi)}^{\sigma}\subset\overline
{B^{\alpha}BB^{\alpha}}^{\sigma}.$$ so $\overline{B^{\alpha}BB^{\alpha}}^{\sigma}=B.$
d\) The proof is similar with the proof of part c).
Theorem 3.4. below is an extension of \[2, Proposition 2.2.2. b)\] to the case of $\mathcal{F}$-dynamical systems with compact groups, not neccessarily abelian, for the strong Connes spectrum, $\widetilde{\Gamma}_{\mathcal{F}}(\alpha).$
**3.4. Theorem.** *Let* $(X,G,\alpha)$ *be an* $F$*-dynamical system with* $G$* compact. Then*$$\widetilde{\Gamma}_{\mathcal{F}}(\alpha)=\cap\left\{ \widetilde
{sp}_{\mathcal{F}}(\alpha|_{\overline{JXJ}^{\sigma}}):J\subset X^{\alpha
},\text{ }\mathcal{F}\text{-\textit{closed two sided ideal}}\right\}$$
Clearly, since $\overline{JXJ}^{\sigma}\in\mathcal{H}_{\sigma}^{\alpha}(X),$$$\widetilde{\Gamma}_{\mathcal{F}}(\alpha)\subset\cap\left\{ \widetilde
{sp}_{\mathcal{F}}(\alpha|_{\overline{JXJ}^{\sigma}}):J\subset X^{\alpha
},\text{ }\mathcal{F}\text{-closed two sided ideal}\right\} .$$ Let $\pi\in\cap\left\{ \widetilde{sp}_{\mathcal{F}}(\alpha|_{\overline
{JXJ}^{\sigma}}):J\subset X^{\alpha},\text{ }\mathcal{F}\text{-closed two
sided ideal}\right\} $ and $Y\in\mathcal{H}_{\sigma}^{\alpha}(X),$ so $Y^{\alpha}\in\mathcal{H}_{\sigma}(X^{\alpha}).$ We will prove that $\overline{Y_{2}(\pi)^{\ast}Y_{2}(\pi)}^{\sigma}=(Y\otimes B(H_{\pi}))^{\alpha\otimes ad\pi}$ and thus $\pi\in\widetilde{sp}_{\mathcal{F}}(\alpha|_{Y}).$ Since $Y\in\mathcal{H}_{\sigma}^{\alpha}(X)$ is arbitrary, it will follow that $\pi\in\widetilde{\Gamma}_{\mathcal{F}}(\alpha).$ Denote by $J$ the following ideal of $X^{\alpha}$$$J=\overline{X^{\alpha}Y^{\alpha}X^{\alpha}}^{\sigma}.$$ It is clear that $J=\overline{JX^{\alpha}J}^{\sigma}$ (actually it is quite easy to show that this equality holds without the closure, but we do not need this fact). Also$$\begin{aligned}
\overline{Y^{\alpha}JY^{\alpha}}^{\sigma} & =\overline{Y^{\alpha}X^{\alpha
}Y^{\alpha}X^{\alpha}Y^{\alpha}}^{\sigma}=\overline{(Y^{\alpha}X^{\alpha
}Y^{\alpha})(Y^{\alpha}X^{\alpha}Y^{\alpha})}^{\sigma}=\label{1}\\
& =\overline{Y^{\alpha}Y^{\alpha}}^{\sigma}=Y^{\alpha}.\nonumber\end{aligned}$$ Denote $Z=\overline{JXJ}^{\sigma}.$ Notice that, since $Y\in\mathcal{H}_{\sigma}^{\alpha}(X),$ we have $\overline{Y^{\alpha}XY^{\alpha}}^{\sigma}=Y,$ so $$\begin{gathered}
Z=\overline{X^{\alpha}Y^{\alpha}X^{\alpha}}^{\sigma}X\overline{X^{\alpha
}Y^{\alpha}X^{\alpha}}^{\sigma}=\overline{X^{\alpha}Y^{\alpha}X^{\alpha
}XX^{\alpha}Y^{\alpha}X^{\alpha}}^{\sigma}=\label{2}\\
\overline{X^{\alpha}Y^{\alpha}(X^{\alpha}XX^{\alpha})Y^{\alpha}X^{\alpha}}^{\sigma}=\overline{X^{\alpha}Y^{\alpha}XY^{\alpha}X^{\alpha}}^{\sigma
}=\overline{X^{\alpha}YX^{\alpha}}^{\sigma}.\nonumber\end{gathered}$$ Since $\pi\in\cap\left\{ \widetilde{sp}_{\mathcal{F}}(\alpha|_{\overline
{JXJ}^{\sigma}}):J\subset X^{\alpha},\text{ }\mathcal{F}\text{-closed two
sided ideal}\right\} $, it follows that $\pi\in\widetilde{sp}_{\mathcal{F}}(\alpha|_{Z}),$ so $$\overline{Z_{2}(\pi)^{\ast}Z_{2}(\pi)}^{\sigma}=(Z\otimes B(H_{\pi}))^{\alpha\otimes ad\pi}. \label{3}$$ Using the equalities (\[2\]) above, the fact that $Y$ is a hereditary C\*-subalgebra of $X$, and the obvious equality$$P_{ij}(\pi)(xyz)=xP_{ij}(\pi)(y)z$$ for every $x,z\in X^{\alpha},y\in X,$ and $1\leq i,j\leq\dim H_{\pi},$ the relation (\[3\]) becomes$$\overline{X^{\alpha}Y_{2}(\pi)^{\ast}Y_{2}(\pi)X^{\alpha}}^{\sigma}=\overline{X^{\alpha}(Y\otimes B(H_{\pi}))^{\alpha\otimes ad\pi}X^{\alpha}}^{\sigma}. \label{4}$$ where, for $x\in X^{\alpha}$ and $a\in X\otimes B(H_{\pi}),a=\left[
a_{kl}\right] ,$ by $xa$ we mean the matrix whose $kl$ entry is $xa_{kl}.$ Therefore, by applying Lemma 3.3. d) to $B=Y,$ we get$$\overline{X^{\alpha}Y^{\alpha}Y_{2}(\pi)^{\ast}Y_{2}(\pi)Y^{\alpha}X^{\alpha}}^{\sigma}=\overline{X^{\alpha}Y^{\alpha}(Y\otimes B(H_{\pi}))^{\alpha\otimes
ad\pi}Y^{\alpha}X^{\alpha}}^{\sigma}. \label{5}$$ By multiplying (\[5\]) on the right and on the left by $Y^{\alpha}$ and taking into account that, by Lemma 3.3. c) $\overline{Y^{\alpha}YY^{\alpha}}^{\sigma}=Y$ and consequently, $\overline{Y^{\alpha}X^{\alpha}Y^{\alpha}}^{\sigma}=Y^{\alpha},$ it follows that $$\overline{Y_{2}(\pi)^{\ast}Y_{2}(\pi)}^{\sigma}=(Y\otimes B(H_{\pi}))^{\alpha\otimes ad\pi}.$$ Therefore, $\pi\in\widetilde{sp}_{\mathcal{F}}(\alpha|_{Y})$ and the proof is complete.
In the next Lemma and the rest of the paper, a subalgebra of $X\otimes
B(H_{\pi})$ will be called $\mathcal{F}-$simple (respectively $\mathcal{F}-$prime) if it is $\mathcal{F}\otimes B(H_{\pi})^{\ast}-$simple (respectively $\mathcal{F}\otimes B(H_{\pi})^{\ast}-$prime) where $B(H_{\pi})^{\ast}$ denotes the dual of $B(H_{\pi}).$ Clearly, a subalgebra of $X\otimes B(H_{\pi
})$ is $\mathcal{F}-$prime if and only if it is a prime C\*-algebra. The similar statement for the $\mathcal{F}-$simple case is not true.
**3.5. Lemma.** *Let* $(X,G,\alpha)$ *be an* $\mathcal{F}$*-dynamical system with* $G$* compact. Then, if* $X^{\alpha}$* is* $\mathcal{F}$*-simple, it follows that (*$X\otimes
B(H_{\pi}))^{\alpha\otimes ad\pi}$* is* $\mathcal{F}-$*simple.*
Let $\pi\in\widetilde{sp}_{\mathcal{F}}(\alpha)\subset sp(\alpha).$ Since $X^{\alpha}$ is $\mathcal{F}$-simple, so $X^{\alpha}\otimes B(H_{\pi})$ is also $\mathcal{F}$-simple and $X_{2}(\pi)X_{2}(\pi)^{\ast}$ is an ideal of $X^{\alpha}\otimes B(H_{\pi}),$ it follows that $\overline{X_{2}(\pi
)X_{2}(\pi)^{\ast}}^{\sigma}=X^{\alpha}\otimes B(H_{\pi}).$ To prove that ($X\otimes B(H_{\pi}))^{\alpha\otimes ad\pi}$ is simple, let $I\subset
(X\otimes B(H_{\pi}))^{\alpha\otimes ad\pi}$ be a non-zero ideal. Then it can be easily verified that$$\begin{aligned}
J & =\overline{lin}^{\sigma}\left\{ yy^{\ast}:y\in X_{2}(\pi)I\right\} =\\
& =\overline{X_{2}(\pi)IX_{2}(\pi)^{\ast}}^{\sigma}.\end{aligned}$$ is an ideal of $X^{\alpha}\otimes B(H_{\pi})$ and, since the latter algebra is $\mathcal{F}$-simple, it follows that $J=X^{\alpha}\otimes B(H_{\pi}).$ Therefore, since $\pi\in\widetilde{sp}_{\mathcal{F}}(\alpha),$ we have $\overline{X_{2}(\pi)^{\ast}X_{2}(\pi)}^{\sigma}=(X\otimes B(H_{\pi}))^{\alpha\otimes ad\pi}$ and consequently, since, by Lemma 3.3. d) $\overline{X^{\alpha}X_{2}(\pi)}^{\sigma}=X_{2}(\pi),$ we have $$(X\otimes B(H_{\pi}))^{\alpha\otimes ad\pi}=\overline{X_{2}(\pi)^{\ast}JX_{2}(\pi)}^{\sigma}\subset\overline{X_{2}(\pi)^{\ast}X_{2}(\pi)IX_{2}(\pi)^{\ast}X_{2}(\pi)}^{\sigma}\subset I.$$ Thus $I=(X\otimes B(H_{\pi}))^{\alpha\otimes ad\pi}$ and we are done.
The following result extends \[2, Théorème 2.4.1\], \[12, Theorem 2. i)$\Leftrightarrow ii)]$\] and \[15, Theorem 3.4.\] to the more general case of $\mathcal{F}$-dynamical systems and non abelian compact groups $G.$
**3.6. Theorem.** *Let* $(X,G,\alpha)$ *be an* $\mathcal{F}$*-dynamical system with* $G$* compact. The following conditions are equivalent:i) (*$X\otimes B(H_{\pi
}))^{\alpha\otimes ad\pi}$ *is* $\mathcal{F}$*-simple for all* $\pi\in sp(\alpha).$*ii)* $X$ *is* $\alpha
$*-simple and* $sp(\alpha)=\widetilde{\Gamma}_{\mathcal{F}}(\alpha).$
$i)\Rightarrow ii)$ Suppose that ($X\otimes B(H_{\pi}))^{\alpha\otimes ad\pi}$ is $\mathcal{F}$-simple for all $\pi\in sp(\alpha)$. Then, it follows immediately from the definitions that $sp(\alpha)=\widetilde{sp}_{\mathcal{F}}(\alpha).$ Let $\pi\in sp(\alpha)$ be arbitrary. Since, in particular, $X^{\alpha}$ is $\mathcal{F}$-simple, so it has no non-trivial $\mathcal{F}$-closed ideals, from Theorem 3.4. it follows that $\pi\in\widetilde{\Gamma
}_{\mathcal{F}}(\alpha),$ so $sp(\alpha)=\widetilde{\Gamma}_{\mathcal{F}}(\alpha).$ Let us prove that $X$ is $\alpha$-simple. If $I$ is an $\mathcal{F}$-closed $\alpha$-invariant ideal of $X,$ then $I^{\alpha}$ is an $\mathcal{F}$-closed ideal of $X^{\alpha}$, so $I^{\alpha}=X^{\alpha}.$ By Lemma 3.3. c) applied to $B=I,$ and to $B=X$ it follows that $\overline
{I^{\alpha}II^{\alpha}}^{\sigma}=I$ and $\overline{X^{\alpha}XX^{\alpha}}^{\sigma}=X$, so$$X=\overline{X^{\alpha}XX^{\alpha}}^{\sigma}=\overline{I^{\alpha}XI^{\alpha}}^{\sigma}\subset I.$$ Therefore, $\ I=X,$ hence $X$ is $\alpha$-simple.$ii)\Rightarrow i).$ Suppose that $X$ is $\alpha$-simple and $sp(\alpha)=\widetilde{\Gamma
}_{\mathcal{F}}(\alpha).$ We will prove first that $X^{\alpha}$ is $\mathcal{F}$-simple. Let $J\subset X^{\alpha}$ be a non zero ideal and $\pi\in\widetilde{\Gamma}_{\mathcal{F}}(\alpha).$ Since $\overline
{JXJ\ }^{\sigma}\in\mathcal{H}_{\sigma}^{\alpha}(X),$ and $\pi\in
\widetilde{\Gamma}_{\mathcal{F}}(\alpha),$ it follows that $$\overline{JX_{2}(\pi)^{\ast}JX_{2}(\pi)J}^{\sigma}=\overline{J(X\otimes
B(H_{\pi}))^{\alpha\otimes ad\pi}J}^{\sigma}. \label{6}$$ where, for $j\in J\subset X^{\alpha}$ and $a\in X\otimes B(H_{\pi}),a=\left[
a_{kl}\right] ,$ by $ja$ we mean the matrix whose $kl$ entry is $ja_{kl}.$ By multiplying the above relation on the left by $X_{2}(\pi)$ and on the right by $X_{2}(\pi)^{\ast},$ we get$$\overline{X_{2}(\pi)JX_{2}(\pi)^{\ast}JX_{2}(\pi)JX_{2}(\pi)^{\ast}}^{\sigma
}=\overline{X_{2}(\pi)J(X\otimes B(H_{\pi}))^{\alpha\otimes ad\pi}JX_{2}(\pi)^{\ast}}^{\sigma}. \label{7}$$ From the above relations (\[6\]) and (\[7\]) it follows that$$X_{2}(\pi)JX_{2}(\pi)^{\ast}=\overline{X_{2}(\pi)JX_{2}(\pi)^{\ast}}^{\sigma
}\subset\overline{X_{2}(\pi)J(X\otimes B(H_{\pi}))^{\alpha\otimes ad\pi}JX_{2}(\pi)^{\ast}}^{\sigma}=$$$$=\overline{X_{2}(\pi)JX_{2}(\pi)^{\ast}JX_{2}(\pi)JX_{2}(\pi)^{\ast}}^{\sigma
}\subset\overline{(X^{\alpha}\otimes B(H_{\pi}))J(X^{\alpha}\otimes B(H_{\pi
}))}^{\sigma}\subset$$$$J\otimes B(H_{\pi})$$ It follows that $tr(X_{2}(\pi)JX_{2}(\pi)^{\ast})\subset J.$ From Lemma 2.5. it follows that $(\overline{XJX}^{\sigma})^{\alpha}\subset J.$ Since $X$ is $\alpha$-simple, we have $\overline{XJX}^{\sigma}=X,$ so $J=X^{\alpha}$ and therefore, $X^{\alpha}$ is $\mathcal{F}$-simple. Applying lemma 3.5. it follows that ($X\otimes B(H_{\pi}))^{\alpha\otimes ad\pi}$ is $\mathcal{F}$-simple for all $\pi\in sp(\alpha)=\widetilde{\Gamma}_{\mathcal{F}}(\alpha).$
$\mathcal{F}$-prime fixed point algebras
========================================
This section is concerned with the relationship between the $\mathcal{F}$-primeness of the fixed point algebras and the spectral properties, involving the Connes spectrum $\Gamma_{\mathcal{F}}(\alpha)$ of the $\mathcal{F}$-dynamical system $(X,G,\alpha).$
Theorem 4.1. below is an extension of \[2, Proposition 2.2.2. b)\] to the case of $\mathcal{F}$-dynamical systems with compact groups, not neccessarily abelian, for the Connes spectrum, $\Gamma_{\mathcal{F}}(\alpha).$ By Remark 3.2. and the discussion preceding it, if $(X,G,\alpha)$ is a W\*-dynamical system (that is $X$ is a von Neumann algebra and $\mathcal{F}=X_{\ast}$ its predual), then the next Theorem 4.1. is equivalent with Theorem 3.4.
**4.1. Theorem**. *Let* $(X,G,\alpha)$ *be an* $\mathcal{F}$-*dynamical system*. *Then*$$\Gamma_{\mathcal{F}}(\alpha)=\cap\left\{ sp_{\mathcal{F}}(\alpha
|_{\overline{JXJ}^{\sigma}}):J\subset X^{\alpha},\text{ \textit{a non-zero}
}\mathcal{F}\text{-\textit{closed two sided ideal}}\right\} \text{.}$$
Since $\overline{JXJ}^{\sigma}\in\mathcal{H}_{\sigma}^{\alpha}(X)$ for every non-zero $\mathcal{F}$-closed two sided ideal $J\subset X^{\alpha}$, we have$,$$$\Gamma_{\mathcal{F}}(\alpha)\subset\cap\left\{ sp_{_{\mathcal{F}}}(\alpha|_{\overline{JXJ}^{\sigma}}):J\subset X^{\alpha},\text{ }\mathcal{F}\text{-closed two sided ideal}\right\} .$$ Now let $\pi\in\cap\left\{ sp_{_{\mathcal{F}}}(\alpha|_{\overline
{JXJ}^{\sigma}}):J\subset X^{\alpha},\text{ }\mathcal{F}\text{-closed two
sided ideal}\right\} $ and $Y\in\mathcal{H}_{\sigma}^{\alpha}(X),$ so $Y^{\alpha}\in\mathcal{H}_{\sigma}(X^{\alpha}).$ We will prove that $\overline{Y_{2}(\pi)^{\ast}Y_{2}(\pi)}^{\sigma}$ is essential in $(Y\otimes
B(H_{\pi}))^{\alpha\otimes ad\pi}.$ As in the proof of Theorem 3.4., let $J=\overline{X^{\alpha}Y^{\alpha}X^{\alpha}}^{\sigma}$ and $Z=\overline
{JXJ}^{\sigma}\in H_{\sigma}^{\alpha}(X).$ Since $\pi\in\cap\left\{
sp_{_{\mathcal{F}}}(\alpha|_{\overline{JXJ}^{\sigma}}):J\subset X^{\alpha
},\text{ }\mathcal{F}\text{-closed two sided ideal}\right\} ,$ we have that $\pi\in sp_{_{\mathcal{F}}}(\alpha|_{Z}).$ Therefore, $\overline{Z_{2}(\pi)^{\ast}Z_{2}(\pi)}^{\sigma}$ is essential in $(Z\otimes B(H_{\pi
}))^{\alpha\otimes ad\pi}.$ As noticed in the proof of Theorem 3.4., $$\overline{Z_{2}(\pi)^{\ast}Z_{2}(\pi)}^{\sigma}=\overline{X^{\alpha}Y_{2}(\pi)^{\ast}Y_{2}(\pi)X^{\alpha}}^{\sigma}.$$ and $$(Z\otimes B(H_{\pi}))^{\alpha\otimes ad\pi}=\overline{X^{\alpha}(Y\otimes
B(H_{\pi}))^{\alpha\otimes ad\pi}X^{\alpha}}^{\sigma}\text{.}$$ Let $a\in(Y\otimes B(H_{\pi}))^{\alpha\otimes ad\pi}$ be such that $$a\overline{Y_{2}(\pi)^{\ast}Y_{2}(\pi)}^{\sigma}=\left\{ 0\right\} .$$ Then, by Lemma 3.3. c) $Y=\overline{Y^{\alpha}Y}^{\sigma}=\overline{Y^{\alpha
}X^{\alpha}Y^{\alpha}Y}^{\sigma}$ and, by Lemma 3.3. d), $Y_{2}(\pi)^{\ast
}=\overline{Y^{\alpha}Y_{2}(\pi)^{\ast}}^{\sigma}.$ Then, it follows that$$a\overline{Y_{2}(\pi)^{\ast}Y_{2}(\pi)}^{\sigma}=\overline{aY^{\alpha}Y_{2}(\pi)^{\ast}Y_{2}(\pi)}^{\sigma}=\overline{aY^{\alpha}X^{\alpha}Y^{\alpha}Y_{2}(\pi)^{\ast}Y_{2}(\pi)}^{\sigma}=\left\{ 0\right\} .$$ Therefore$$\overline{Y^{\alpha}aY^{\alpha}X^{\alpha}Y^{\alpha}Y_{2}(\pi)^{\ast}Y_{2}(\pi)X^{\alpha}}^{\sigma}=\left\{ 0\right\} .$$ so, since $\overline{Z_{2}(\pi)^{\ast}Z_{2}(\pi)}^{\sigma}=\overline
{X^{\alpha}Y_{2}(\pi)^{\ast}Y_{2}(\pi)X^{\alpha}}^{\sigma}$ is essential in $\overline{X^{\alpha}(Y\otimes B(H_{\pi}))^{\alpha\otimes ad\pi}X^{\alpha}}^{\sigma},$ we have $Y^{\alpha}aY^{\alpha}=0$ and therefore, by Lemma 3.3. b) applied to $B=Y,$ it follows that $a=0.$
**4.2. Lemma.** *Let* $(X,G,\alpha)$ *be an* $\mathcal{F}$*-dynamical system with* $G$* compact. Then, if* $X^{\alpha}$* is* $\mathcal{F}$*-prime, it follows that (*$X\otimes
B(H_{\pi}))^{\alpha\otimes ad\pi}$* is* $\mathcal{F}-$*prime for every* $\pi\in sp_{\mathcal{F}}(\alpha)$*.*
Since $X^{\alpha}$ is $\mathcal{F}$-prime. it follows that $X^{\alpha}\otimes
B(H_{\pi})$ is $\mathcal{F}$-prime for every $\pi\in\widehat{G}$. Since $X_{2}(\pi)X_{2}(\pi)^{\ast}$ is a non zero ideal of $X^{\alpha}\otimes
B(H_{\pi}),$ it follows that $X_{2}(\pi)X_{2}(\pi)^{\ast}$is an essential ideal$.$ To prove that ($X\otimes B(H_{\pi}))^{\alpha\otimes ad\pi}$ is $\mathcal{F}$-prime, let $I\subset(X\otimes B(H_{\pi}))^{\alpha\otimes ad\pi}$ be a non-zero ideal. Then, as in the proof of Lemma 3.5., consider the following ideal of $X^{\alpha}\otimes B(H_{\pi})$ $$\begin{aligned}
J & =\overline{lin}^{\sigma}\left\{ yy^{\ast}:y\in X_{2}(\pi)I\right\} =\\
& =\overline{X_{2}(\pi)IX_{2}(\pi)^{\ast}}^{\sigma}.\end{aligned}$$ Since $X^{\alpha}\otimes B(H_{\pi})$ is $\mathcal{F}$-prime, it follows that $J$ is essential in $X^{\alpha}\otimes B(H_{\pi}).$ Therefore, if $a\in(X\otimes B(H_{\pi}))^{\alpha\otimes ad\pi}$ and $aI=\left\{ 0\right\}
.$ we have $$aX_{2}(\pi)^{\ast}JX_{2}(\pi)\subset\overline{aX_{2}(\pi)^{\ast}X_{2}(\pi)IX_{2}(\pi)^{\ast}X_{2}(\pi)}^{\sigma}\subset\overline{aI}^{\sigma
}=\left\{ 0\right\} .$$ so $$(X_{2}(\pi)aX_{2}(\pi)^{\ast}J)X_{2}(\pi)X_{2}(\pi)^{\ast}=\left\{ 0\right\}$$ Thus, since $X_{2}(\pi)X_{2}(\pi)^{\ast}$ is essential in $X^{\alpha}\otimes
B(H_{\pi}),\ $it follows that $X_{2}(\pi)aX_{2}(\pi)^{\ast}J=\left\{
0\right\} .$ Since $X_{2}(\pi)aX_{2}(\pi)^{\ast}\subset X^{\alpha}\otimes
B(H_{\pi}),$ $J$ is essential in $X^{\alpha}\otimes B(H_{\pi})$ and $\pi\in
sp_{\mathcal{F}}(\alpha),$ it follows that $a=0.$
The next result extends \[2, Théorème 2.4.1.\], and \[13, Theorem 8.10.4.\] to the case of $\mathcal{F}$-dynamical systems with compact non abelian groups.
**4.3. Theorem.** *Let* $(X,G,\alpha)$ *be an* $\mathcal{F}$*-dynamical system with* $G$* compact. The following conditions are equivalent:i) (*$X\otimes B(H_{\pi
}))^{\alpha\otimes ad\pi}$ *is* $\mathcal{F}$*-prime for all* $\pi\in sp(\alpha).$*ii)* $X$ *is* $\alpha
$*-prime and* $sp(\alpha)=\Gamma_{\mathcal{F}}(\alpha).$
$i)\Rightarrow ii)$ Suppose that ($X\otimes B(H_{\pi}))^{\alpha\otimes ad\pi}$ is $\mathcal{F}$-prime for all $\pi\in sp(\alpha)$. Then, it follows immediately from i) and the definitions that $sp(\alpha)=sp_{\mathcal{F}}(\alpha).$ Let $\pi\in sp(\alpha)$ be arbitrary. We will use Theorem 4.1. to show that $\pi\in\Gamma_{\mathcal{F}}(\alpha)$. Indeed, let $J$ be a non-trivial ideal of $X^{\alpha}$ and $Z=\overline{JXJ}^{\sigma}\in H_{\sigma
}^{\alpha}(X).$ We will show that $\pi\in sp(\alpha|_{Z})$, that is $Z_{2}(\pi)^{\ast}Z_{2}(\pi)\ $is.esential in ($Z\otimes B(H_{\pi}))^{\alpha\otimes ad\pi}.$ Notice that$$Z_{2}(\pi)=JX_{2}(\pi)J.$$ so$$Z_{2}(\pi)^{\ast}Z_{2}(\pi)=JX_{2}(\pi)^{\ast}JX_{2}(\pi)J.$$ Since, in particular, $X^{\alpha}$ is prime, and $J\ $is an essential ideal of $X^{\alpha},$ we have$\ Z_{2}(\pi)\neq\left\{ 0\right\} .$ Indeed as observed after 2.6.$,X_{2}(\pi)X_{2}(\pi)^{\ast}$ is an ideal of $X^{\alpha
}\otimes B(H_{\pi}),$ so, as $X^{\alpha}$ is $\mathcal{F}$-prime it follows that $X^{\alpha}\otimes B(H_{\pi})$ is a prime C\*-algebra$\ $ and therefore $JX_{2}(\pi)X_{2}(\pi)^{\ast}\neq\left\{ 0\right\} ,$ hence $JX_{2}(\pi)\neq\left\{ 0\right\} $ and $X_{2}(\pi)^{\ast}J\neq\left\{
0\right\} ,$ so, $X_{2}(\pi)^{\ast}JX_{2}(\pi)\neq\left\{ 0\right\}
.$ Using the hypothesis that *(*$X\otimes B(H_{\pi}))^{\alpha\otimes
ad\pi}$ is $\mathcal{F}-$prime and the fact that $J$ is a non-trivial ideal of $X^{\alpha},$ it follows that $X_{2}(\pi)^{\ast}JX_{2}(\pi)J\neq\left\{
0\right\} ,$ so, $Z_{2}(\pi)\neq\left\{ 0\right\} .$ As noticed above, $X_{2}(\pi)^{\ast}JX_{2}(\pi)$ is a non-trivial ideal of ($X\otimes B(H_{\pi
}))^{\alpha\otimes ad\pi}.$ If $a\in$($Z\otimes B(H_{\pi}))^{\alpha\otimes
ad\pi},a\geq0$ is such that $aZ_{2}(\pi)^{\ast}Z_{2}(\pi)=\left\{ 0\right\}
,$ then$$aJX_{2}(\pi)^{\ast}JX_{2}(\pi)J=\left\{ 0\right\} .$$ Hence$$JaJX_{2}(\pi)^{\ast}JX_{2}(\pi)J=\left\{ 0\right\} .$$ Since, as noticed above, $X_{2}(\pi)^{\ast}JX_{2}(\pi)$ is non-trivial and ($X\otimes B(H_{\pi}))^{\alpha\otimes ad\pi}$ is $\mathcal{F}$-prime it follows that $$JaJ=\left\{ 0\right\} .$$ so $Ja=\left\{ 0\right\} .$ Hence $Jtr(a)=\left\{ 0\right\} .$ Since $X^{\alpha}$ is $\mathcal{F}$-prime, we deduce that $tr(a)=0,$ so $a=0$ because $a$ was assumed to be non negative. Therefore, $\pi\in
\Gamma_{\mathcal{F}}(\alpha),$ so $sp(\alpha)=\Gamma_{\mathcal{F}}(\alpha).$ It remains to prove that $X$ is $\mathcal{F}$-prime. Let $I\subset X$ be an $\alpha$-invariant non-trivial ideal and $x\in X,x\geq0$ be such that $xI=\left\{ 0\right\} .$ Then, in particular, $xI^{\alpha}=\left\{
0\right\} ,$ so $P(x)I^{\alpha}=\left\{ 0\right\} .$ Since $X^{\alpha}$ is $\mathcal{F}$-prime and $I^{\alpha}$ is a non trivial ideal of $X^{\alpha}$ we have $P(x)=0$ so, since $P\ $is faithful, $x=0.$ii)$\Rightarrow i)$ Suppose that $X$ is $\alpha$-prime and $sp(\alpha
)=\Gamma_{\mathcal{F}}(\alpha).$ We will prove first that $X^{\alpha}$ is $\mathcal{F}$-prime. Let $J\subset X^{\alpha}$ be a non zero ideal and $a\in
X^{\alpha},a\geq0,a\neq0$ such that $Ja=\left\{ 0\right\} .$ Since $X$ is $\alpha-$prime, and $XJX$ is a non zero $\alpha$-invariant ideal of $X,$ it follows that $XJXa\neq\left\{ 0\right\} $ so $JXa\neq\left\{ 0\right\}
.$ Therefore, since by Remark 2.4., $$X=\overline{\sum_{\pi\in sp(\alpha)}X_{1}(\pi)}^{\sigma}.$$ there exists $\pi\in sp(\alpha)$ such that $JX_{1}(\pi)a\neq\left\{
0\right\} .$ Denote $Z=\overline{aXa}^{\sigma}\in H_{\sigma}^{\alpha}(X).$ Then, since $\pi\in sp(\alpha)=\Gamma_{\mathcal{F}}(\alpha),$ $Z_{2}(\pi)^{\ast}Z_{2}(\pi)$ is essential in ($Z\otimes B(H_{\pi}))^{\alpha\otimes
ad\pi}.$ But$$\overline{Z_{2}(\pi)^{\ast}Z_{2}(\pi)}^{\sigma}=\overline{aX_{2}(\pi)^{\ast
}a^{2}X_{2}(\pi)a}^{\sigma}.$$ and$$(Z\otimes B(H_{\pi}))^{\alpha\otimes ad\pi}=\overline{a((X\otimes B(H_{\pi
}))^{\alpha\otimes ad\pi})a}^{\sigma}.$$ Taking into account that $X_{2}(\pi)a^{2}X_{2}(\pi)^{\ast}\subset X^{\alpha
}\otimes B(H_{\pi})$ and $Ja=\left\{ 0\right\} $ we immediately get that $$JX_{2}(\pi)a^{2}X_{2}(\pi)^{\ast}\subset J\otimes B(H_{\pi}).$$ Hence $$\overline{(aX_{2}(\pi)^{\ast}J)X_{2}(\pi)a^{2}X_{2}(\pi)^{\ast}(a^{2}X_{2}(\pi)a)}^{\sigma}=\left\{ 0\right\} .$$ Therefore $$\overline{(aX_{2}(\pi)^{\ast}JX_{2}(\pi)a)(aX_{2}(\pi)^{\ast}a^{2}X_{2}(\pi)a)}^{\sigma}=\left\{ 0\right\} .$$ It follows that$$(aX_{2}(\pi)^{\ast}JX_{2}(\pi)a)(Z_{2}(\pi)^{\ast}Z_{2}(\pi))=\left\{
0\right\} .$$ Since $Z_{2}(\pi)^{\ast}Z_{2}(\pi)$ is essential in $(Z\otimes B(H_{\pi
}))^{\alpha\otimes ad\pi}$ and obviously $aX_{2}(\pi)^{\ast}JX_{2}(\pi
)^{\ast}a\subset(Z\otimes B(H_{\pi}))^{\alpha\otimes ad\pi}$ it follows that $JX_{2}(\pi)a=\left\{ 0\right\} $ and hence $JX_{1}(\pi)a=\left\{
0\right\} ,$ but this is in contradiction with our choice of $\pi$ in $sp(\alpha)$ so $X^{\alpha}$ is $\mathcal{F}$-prime. From Lemma 4.2. it follows that ($X\otimes B(H_{\pi}))^{\alpha\otimes ad\pi}$ is $\mathcal{F}$-prime for all $\pi\in\Gamma_{\mathcal{F}}(\alpha)=sp_{\mathcal{F}}(\alpha)=sp(\alpha)$ and we are done.
[References]{}
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\[3\] A. Connes, M. Takesaki, The flow of weights on factors of type III, Tôhoku Math. J. 29 (1977) 473-575.
\[4\] C. D’Antoni, L. Zsido, Groups of linear isometries on multiplier C\*-algebras, Pacific J. Math. 193 (2000) 279-306.
\[5\] R. Dumitru, C. Peligrad, Compact quantum group actions and invariant derivations, Proc. Amer. Math. Soc. 135 (2007) 3977-3984.
\[6\] E. C. Gootman, A. J. Lazar, C. Peligrad, Spectra for compact group actions, J. Operator Theory 31 (1994) 381-399.
\[7\] E. Hille, R. Phillips, Functional Analysis and Semi-Groups, AMS, 1957.
\[8\] A. Kishimoto, Simple crossed products of C\*-algebras by locally compact abelian groups, Yokohama Math. J. 28 (1980) 69-85.
\[9\] M. B. Landstad, Algebras of spherical functions associated with covariant systems over a compact group, Math. Scand. 47 (1980), 137-149.
\[10\] D. Olesen, Inner $^{\ast}$-automorphisms of simple C\*-algebras, Comm. Math. Phys..44 (1975) 175-190.
\[11\] D. Olesen, G.K. Pedersen, Applications of the Connes spectrum to C\*-dynamical systems, J. Funct. Anal. 30 (1978) 179-197.
\[12\] D. Olesen, G.K. Pedersen, E. Stormer, Compact abelian groups of automorphisms of simple C\*-algebras, Invent. Math. 39 (1977) 55-64.
\[13\] G.K. Pedersen, C\*-algebras and Their Automorphism Groups, Academic Press, 1979.
\[14\] C. Peligrad, Locally compact group actions and compact subgroups, J. Funct. Anal. 76 (1988) 126-139.
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\[16\] L. Zsido, Spectral and ergodic properties of the analytic generators, J. Approx. Theory 20 (1977) 77-138.
|
---
abstract: 'We study the radiation from a collision of black holes with equal and opposite linear momenta. Results are presented from a full numerical relativity treatment and are compared with the results from a “close-slow” approximation. The agreement is remarkable, and suggests several insights about the generation of gravitational radiation in black hole collisions.'
author:
- |
John Baker$^1$, Andrew Abrahams $^2$, Peter Anninos$^2$, Steve Brandt$^{2,3,4}$,\
Richard Price$^5$, Jorge Pullin$^1$ and Edward Seidel$^{2,3,4,6}$\
1. [*Center for Gravitational Physics and Geometry, Department of Physics\
The Pennsylvania State University, 104 Davey Lab, University Park, PA 16802*]{}\
2. [*National Center for Supercomputing Applications, 605 E. Springfield Ave., Champaign, IL 61820*]{}\
3. [*Max-Plank-Institute für Gravitationphysik, Albert-Einstein-Institute, 14473 Potsdam, Germany*]{}\
4. [*Department of Physics, University of Illinois at Urbana-Champaign, 61801*]{}\
5. [*Department of Physics, University of Utah, Salt Lake City, UT 84112*]{}\
6. [*Department of Astronomy, University of Illinois at Urbana-Champaign, 61801*]{}
title: The collision of boosted black holes
---
epsf
CGPG-96/8-3\
gr-qc/9608064\
Introduction
============
The collision of two black holes is now being studied extensively via the techniques of numerical relativity [@grandchallenge]. Collisions are of great importance as the most interesting source of gravitational waves that might be observable with interferometric detectors[@ligo]. The study is also of great inherent interest to relativity theory in that supercomputers allow us to investigate strong field gravity effects without symmetries which might preclude interesting or crucial phenomena. In dealing with such a daunting problem, useful checks, guidelines, and insights have been provided by analytical approximations, in particular by the close-limit approximation [@AbPr]. In principle, this method applies when the holes are initially very close together. In this case, the horizon is initially only slightly nonspherical and the spacetime that evolves outside the horizon can be treated as a perturbed single black hole. The highly nonspherical nature of the spacetime inside the horizon is causally disconnected from the exterior, and from the generation of outgoing gravitational waves. The exterior spacetime can be evolved forward in time from the initial data hypersurface with the linearized equations of perturbation theory.
This method turns out to be remarkably successful[@PrPu; @Anninosetal; @AbPr2]. The details of this success may give insights into the nature of collisions of holes. For holes that are initially momentarily stationary, the close-limit predictions of radiated energy and waveforms are quite good (i.e., in agreement with the results of numerical relativity) even when the initial horizon is highly distorted, violating the assumptions underlying the method. The close limit has been used by Abrahams and Cook[@ac94] for the head-on collision of holes with initial momenta towards each other. This momentum causes horizons to form when the holes are at larger separation and makes the exterior spacetime more spherical, so it is not surprising that the close limit should be successful for these cases. Puzzling results emerge, however, when close-limit calculations are combined with Newtonian trajectories to estimate the radiated energy for initially [*large*]{} separations of initially stationary holes. The success of these estimates suggests, among other things, that to a large extent the role of the early weak-field phase of the evolution is to only to determine what the momentum of the holes will be when they start to interact nonlinearly.
With that suggestion as one of our motivations, we consider here equal mass holes which are initially moving towards each other with equal and opposite momentum $P$. We analyze the problem with an approximation simple enough to allow insight, and we present, for comparison, the results of full numerical relativity for the same initial black hole configuration. In a certain sense, this study complements that of Ref. [@ac94]. The initial data sets being studied are representations of the same physical system; in Ref. [@ac94] the data were “exact” (up to numerical error) solutions to the initial value problem, however, in the current study we have more control over the approximations implicit in the perturbative analysis. In Sec. II we present the general formalism for the problem and briefly discuss the full numerical solution. In Sec. III we describe an approximation based on the close limit [*and*]{} on slow initial motion. Results of both methods are presented and discussed in section IV. Throughout the paper we use units in which $c=G=1$, and $M$ represents the total ADM mass on the initial hypersurface.
Initially moving holes
======================
The initial value equations for general relativity are [@BoYo], $$\begin{aligned}
\nabla^a (K_{ab} - g_{ab} K) &=& 0\\
{}^3R-K_{ab} K^{ab} + K^2 &=&0\end{aligned}$$ where $g_{ab}$ is the spatial metric, $K_{ab}$ is the extrinsic curvature and ${}^3R$ is the scalar curvature of the three metric. One proposes a three metric that is conformally flat $g_{ab} = \phi^4
\widehat{g}_{ab}$, with $\widehat{g}_{ab}$ a flat metric, and $\phi^4$ the conformal factor, and one uses a decomposition of the extrinsic curvature $K_{ab} = \phi^{-2} \widehat{K}_{ab}$. The constraints become, $$\begin{aligned}
\widehat{\nabla}^a \widehat{K}_{ab} &=& 0\label{momentum}\\
\widehat{\nabla}^2 \phi &=& -\frac{1}{8}
\phi^{-7} \widehat{K}_{ab} \widehat{K}^{ab}\ ,\label{hamil}\end{aligned}$$ where $\widehat{\nabla}$ is a flat-space covariant derivative.
In describing how (\[momentum\]) and (\[hamil\]) and the 3+1 evolution equations are solved numerically, it is useful to have at hand three different coordinate systems. Of greatest relevance to the numerical method are the Čadež coordinates, a system which is particularly well-suited for the collision of two black holes and which has been used extensively in numerical studies[@Smarr; @anninosPRL]. These coordinates are spherical near the throats of both holes and in the asymptotic wave zone, so they simplify the application of both inner and outer boundary conditions. It is useful also to refer to two coordinatizations of the flat conformal three space: cylindrical coordinates $\rho,z,\varphi$, and the bispherical-like Misner[@misner] coordinates $\mu,\eta,\varphi$. The fact that the problem is axisymmetric, of course, reduces the spatial computational grid to a two dimensional one. By choosing to consider only equal mass holes with equal and opposite momenta, we have a further symmetry which reduces the size of the computational grid to a quadrant, ($\varphi=0, z>0$). We characterize the separation of the holes with the Misner parameter $\mu_0$, and construct the coordinate grid independently for each choice of $\mu_0$. Details of the grid computation are given in Refs. [@Cadez71; @Anninos94a].
To solve the momentum constraint (\[momentum\]) we follow the prescription of York and coworkers[@Kulkarni83] and Cook [@Cook90; @Cook93]. This starts with a solution to (\[momentum\]) that represents the momentum of one hole, $$\label{onehole}
\hat{K}^{\rm one}_{ab} = {3 \over 2 r^2} \left[ 2 P_{(a} n_{b)} -(\delta_{ab}
-n_a n_b)P^c n_c\right]\ . \label{boyok}$$ Here the hole is associated with some point in the flat conformal space, $\vec{r}$ is the vector from that point, and $\vec{n}$ is the unit vector in the $\vec{r}$ direction. The next step is to modify (\[onehole\]) to represent holes centered at $z=\pm \coth\mu_0$, the centers of the circles $\mu=\pm\mu_0$ in the conformally flat metric. Since the momentum constraint (\[momentum\]) is linear, one can simply add two expressions of the form (\[onehole\]): $$\label{two}
\widehat{K}^{\rm two}_{ij} =
\widehat{K}^{\rm one}_{ij}\left(z \rightarrow z-\coth\mu_0 \right)+
\widehat{K}^{\rm one}_{ij}\left(z \rightarrow z+
\coth\mu_0,P \rightarrow -P\right)\ .$$
For convenience, the initial data is forced to obey an isometry condition,i.e., we operate on the momentum constraint solution with a reflection procedure equivalent to adding image charges in electrostatics. The result of this procedure is to create a solution which corresponds to two identical asymptotically flat universes connected by two Einstein-Rosen bridges. The nature of this symmetrization process, and the boundary condition it provides for (\[hamil\]), affects the mass of the holes being represented. Cook[@Cook93; @cook91] has also used this approach to develop codes to compute symmetric initial data solutions for axisymmetric and full 3D data.
The Hamiltonian constraint is solved by linearizing equation (\[hamil\]) around a solution $\phi_1$ so that $\phi = \phi_1 +
\delta \phi$, discretizing to second order the resulting linear elliptic equation, solving the matrix equation for $\delta \phi$ with a multigrid method, then iterating the procedure until a convergence tolerance of $\delta\phi/\phi_1 < 10^{-10}$ is achieved. It has been verified that for $\widehat{K}_{ab}=0$, the solution for $\phi$ converges quadratically with cell size to the time–symmetric Misner data[@misner].
The symmetrized initial data for $\phi$ and for $K_{ab}$ are now used as the starting point for numerical integration. The evolution employs maximal time slices and the shift is determined by an elliptic condition that forces the 3-metric (in Cadez coordinates) into diagonal form[@Anninosetal]. The numerical errors inherent in the method (to be described elsewhere) are similar to those in Ref. [@Anninosetal]. We have verified that the convergence rate for the total radiated energy scales quadratically with grid spacing and that differences in the dominant $\ell=2$ waveforms are on the order of a few percent at the grid resolutions used here. The errors are small on the scale of Fig. \[energy\], and do not affect any conclusions to be drawn from that figure. The methods used for the numerical evolution are described in detail in Ref.[@anninosPRL]; we modified only slightly the code described there for evolving the time symmetric Misner data.
Approximation method
====================
The close-limit approach can be applied to the Cook[@cook91] initial data, as has been done in Ref. [@ac94]. But the Cook initial solution is numerical. To facilitate insights we make a further approximation. We assume that the black holes are initially close, and that the initial momentum $P$ is small. Our solution for the extrinsic curvature $\widehat{K}_{ab}$ is $\widehat{K}^{\rm two}_{ab}$ from (\[two\]), the simple superposition (without symmetrization; this effect will be discussed later) of two one-hole solutions. We denote by $\vec{n}^+$ and $\vec{n}^-$ the normal vectors corresponding, respectively, to the one hole solutions at $z=+L/2$ and at $-L/2$, and we define $R$ to be the distance to a field point, in the flat conformal space, from the point midway between the holes. For large $R$, the normal vectors $\vec{n}^+$ and $\vec{n}^-$ almost cancel[@PuCa]. More specifically $\vec{n}^+=-\vec{n}^-+O(L/R)$. A consequence of this is that the total initial $\widehat{K}^{ab}$ is first order in $L/R$, and its ($R,\theta,\varphi$ coordinate basis) components can be written as $$\label{Kapprox}
\hat{K}_{ab} = {3 P L \over 2R^3} \left[
\begin{array}{ccc} -4 \cos^2 \theta&0&0\\0&R^2 (1+\cos^2 \theta)&0\\
0&0&R^2 \sin^2 \theta (3 \cos^2 \theta -1)
\end{array}\right]\ .$$
In addition to being first order in $L$, the solution for $\widehat{K}^{ab}$ is first-order in $P$ and therefore the source term on the right in the hamiltonian constraint (\[hamil\]) is quadratic in $P$. If we limit ourselves to a solution to first order in $P$ we can ignore this quadratic source term. (In Sec. IV, a more thorough discussion will be given for this step of ignoring the source term.) Without the source term the hamiltonian constraint reduces to the zero momentum case, the Laplace equation. The symmetric solution to this (i.e., the solution for two identical asymptotically flat universes) is the Misner solution[@misner], and this is the solution we take. The Misner geometry is characterized by a dimensionless parameter $\mu_0$ which describes the separation of the throats. We must, of course, choose $\mu_0$ appropriate to the parameters of the extrinsic curvature we are using. We choose therefore a Misner geometry characterized by the same value of $L$ as in (\[Kapprox\]). Since $L$ there represents not the physical distance, in any sense, between the holes, but the formal distance in the conformally flat space, we choose a Misner geometry with the same value of $L$ in the conformally flat part of the Misner metric. The relationship of $L$ to $\mu_0$ is (see, e.g., [@PrPu]) $$\label{Lvsmu}
L/M=\frac{\rm{\coth}\mu_0}{2\Sigma_1}\ \ \hspace*{30pt}
\ \ \ \Sigma_1\equiv\sum_{n=1}\frac{1}{\sinh n\mu_0}\ .$$ This completes the description of the initial data to first order in $L$ and to first order in $P$ (the close, slow approximation). We now view the spacetime exterior to the horizon as a perturbation of a single Schwarzschild hole described in standard Schwarzschild coordinates $t,r,\theta,\phi$. Even-parity perturbations are then described by a Zerilli function $\psi$. According to the general prescription given in reference [@AbPr] the value of $\psi$ on a $t=0$ initial hypersurface is found from the initial value of the three geometry. Our initial geometry, to first order in $P$, is exactly the same as the zero $P$ solution in reference[@PrPu], where the Zerilli function is denoted $\psi_{\rm pert}$, and is given in eq. (4.29), along with (4.10),(4.27) and (4.28). In that reference it is shown that in the close limit, the quadrupole contribution dominates, with contributions for $\ell>2$ higher order in the separation parameter. Here we shall consider only the $\ell=2$ contribution, and shall denote the Zerilli function, corresponding to this Misner (i.e., $P=0$)problem, as $\psi_{\rm Mis}(r,t)$.
The initial value of $\dot{\psi}$, the time derivative of the Zerilli function, follows from the extrinsic curvature as explained in [@AbPr]. The extrinsic curvature is given, in our approximation, by multiplying $\widehat{K}^{ab}$ in (\[Kapprox\]) by the squared reciprocal of the conformal factor for the Schwarzschild geometry, $\phi_{\rm Schw}=1+M/2R$. We must map the coordinates of the initial value solution to the coordinates for the Schwarzschild background. To do this, we use the same mapping used for the initial value of $\psi$ in [@PrPu]: we interpret the $R$ of (\[Kapprox\]) as the isotropic radial coordinate of a Schwarzschild spacetime, and we relate it to the usual Schwarzschild radial coordinate $r$ by $R=(\sqrt{r}+\sqrt{r-2M})^2/4$. From this we arrive at the following expression for the (Schwarzschild coordinate basis) components of the extrinsic curvature: $$\label{finKapp}
K_{ab} =
{3 P L \over 2r^3} \left[
\begin{array}{ccc} \frac{-4 \cos^2 \theta}{1-2M/r}&0&0\\
0& r^2 (1+\cos^2 \theta)&0\\
0&0& r^2 \sin^2 \theta (3 \cos^2 \theta -1)
\end{array}\right]\ .$$ Here we have used the fact that $$\phi^2\approx\phi_{\rm Mis}^2\approx
\phi_{\rm Schw}^2=r/R
=\frac{1}{\sqrt{1-2M/r}}\,\frac{dr}{dR}\ .$$ From (\[finKapp\]), which contains both monopole and quadrupole parts, we can project out the $\ell=2$ part and read off the initial value of the time derivative of the Zerilli function to be $$\label{initdot}
\dot{\psi}|_{t=0}=-{24 P L \sqrt{1-2M/r}\over r^2 (2+3 M/r)} \sqrt{4 \pi\over
5} (4 + {3 M\over r})\ .$$ Along with $\psi|_{t=0}=\psi_{\rm Mis}(r,t=0)$, this completes the specification of the Cauchy data for $\psi$.
Given this Cauchy data, the time evolution is obtained by evolving the Zerilli equation, $$\label{zerilli}
\partial^2 \psi / \partial t^2 -
\partial^2 \psi / \partial r_*^2 +V(r) \psi = 0\ ,$$ where $r_*=r+\log(r/2M-1)$ and the Zerilli function $\psi$ is a coordinate invariant combination of the perturbed metric coefficients; the $\ell=2$ “potential” V(r) can be seen in reference [@PrPu].
The evolved $\psi$ can be decomposed into two components $$\label{decomp}
\psi=\psi_{\rm Mis}+\psi_{\rm Mom}\ .$$ The first term is the solution of (\[zerilli\]) for cauchy data $\psi=\psi_{\rm Mis}(r,t=0)$ and $\dot{\psi}=0$ at $t=0$. The second term is the solution for $t=0$ cauchy data $\psi=0$, and with $\dot{\psi}$ given by (\[initdot\]). The two contributions are respectively zero order in $P$ and first order in $P$; the decomposition then represents a separation into parts of $\psi$ due to the masses, and to the momenta. The radiated energy is given by[@Anninosetal] $$E= {1 \over 384 \pi}\int_0^\infty
\dot{\psi}^2 dt \ ,$$ and can be written, in terms of the decomposition above, as: $$\begin{aligned}
\label{energyint}
E = {1 \over 384 \pi} \left(
\int_0^\infty\dot{\psi}_{\rm Mis}^2\ dt
+2\int_0^\infty\dot{\psi}_{\rm Mis}\dot{\psi}_{\rm Mom}\ dt
+\int_0^\infty\dot{\psi}_{\rm Mom}^2\ dt\right)\label{erad}\ .\end{aligned}$$ The first term gives the same result as in the momentarily stationary case; it is simply the radiation for the Misner initial geometry, as computed in reference[@PrPu]. The second term is linear in the momentum of each hole. The coefficient of it is given by the “correlation” of $\psi_{\rm Mis}$ and $\psi_{\rm Mom}$. As can be seen in Fig. \[anticorr\], this “correlation” integral is negative.
The anticorrelation is compatible with previous simulations done by Ref. [@ac94] using numerical initial data (see figures 3a,b in their paper). This means that for small values of $P$, the radiated energy [*decreases*]{} with increasing momentum. The effect is clearly visible in Fig. \[energy\] where we show the radiated energy as a function of the momentum,
Note that the first term is simply a function of the Misner parameter $\mu_0$. The second term depends on $\mu_0$, but also depends on $L$ and $P$. We can write $L$ in terms of $\mu_0$ with (\[Lvsmu\]) to express all dependencies in (\[energyint\]) only in terms of $\mu_0$ and $P/M$. With the correct numerical factors we get the final result of the close-slow approximation, a simple formula for the radiated energy simply and explicitly expressed in terms of the parameters of the collision: $$\label{clsleq}
\frac{E}{M}=2.51\times10^{-2}\kappa_2^2(\mu_0)
-2.06\times10^{-2}\frac{{\rm \coth}\,\mu_0\kappa_2(\mu_0)}{\Sigma_1}\left(
\frac{P}{M}
\right)
+5.37\times10^{-3}\left(
\frac{{\rm \coth}\,\mu_0}{\Sigma_1}
\right)^2
\left(
\frac{P}{M}
\right)^2\ ,$$ where $\kappa_2$, as defined in Ref. [@Anninosetal], is $$\label{kapdef}
\kappa_2(\mu_0)\equiv\frac{1}{\left(4\Sigma_1\right)^3}
\sum_{n=1}^\infty
\frac{(\coth{n\mu_0})^2}{\sinh{n\mu_0}}\ .$$
In Fig. \[energy\], we plot the radiated energy computed from (\[clsleq\]) for several values of initial separation $\mu_0$, and for a wide range of $P/M$ . On this plot, also, are presented the results for radiated energy from numerical results computations which make no approximations. The agreement between the numerical results and the results of the approximation is remarkably good, even at rather large values of $P/M$.
Results
=======
Two features of Fig. \[energy\] stand out. The first is “momentum dominance”: the radiated energy is dominated by the third integral in (\[energyint\]) unless the momentum is very small. The second obvious feature is that the approximation method works very well even for sizeable values of $P/M$.
To understand the implications of these features, let us start by reviewing the difference between the exact, nonlinear numerical computation, and the approximation scheme of Sec. III. In the exact method we start with an exact solution to the initial value equations described by two parameters, one a dimensionless measure of the separation of the holes, the other a dimensionless measure of the momentum. The process of generating the solution consists of four steps: (i) One starts with a very simple prescription for $\widehat{K}_{ab}$ constructed by superposing two solutions of form (\[onehole\]) corresponding to two coordinate positions in the conformally flat space. (ii) Equation (\[hamil\]) is then solved for the conformal factor and hence for the three geometry. (iii) The solution for the extrinsic curvature and the initial geometry is then “symmetrized” by an iterative process equivalent to adding image charges. (iv) This solution is numerically evolved off the initial hypersurface with the full nonlinear Einstein equations.
By contrast, the steps for the approximate solution are: (i) The (conformal) extrinsic curvature is taken to be the unsymmetrized superposition of two contributions with the form of (\[onehole\]). (ii) The conformal factor, and therefore the three geometry, is taken to be the symmetrized solution corresponding to throats located at the same points in the conformally flat space as the points in $\widehat{K}_{ab}$. (iii) This approximate initial data is then treated as initial data for the nonspherical perturbations of a Schwarzschild hole, and the perturbations are evolved with the linearized Einstein equations.
The difference in evolution off the initial hypersurface (full Einstein equations in one case, linearized equations in the other) is not a major source of error in the interesting cases, those with high momentum. As momentum increases, the location of the horizon in the initial geometry moves outward. The high momentum cases, therefore, correspond to throats which, on the initial hypersurface, are well inside an all-encompassing horizon. This is the situation in which the “close-limit” approximation method should work very well. It is also not surprising that no large error is introduced by the failure, in the approximation method, to symmetrize the extrinsic curvature. One way of understanding this is to note that $\psi_{\rm Mom}$ lacks the “image” contributions needed for symmetrization. These images only influence the form of $\widehat{K}_{ab}$ very close to the holes. As the separation between the holes gets smaller the horizon moves further from the throats and the effect of the images on $\widehat{K}_{ab}$ outside the horizon diminishes. We have checked numerically that the difference between the symmetrized and unsymmetrized $\widehat{K}_{ab}$, for all cases considered, is negligibly small outside the horizon.
These two aspects of the approximation method rely on the throats being “close” in some sense, an approximation that seems well justified. What remains to be explained is how the slow-limit approximation does such a good job of approximating the very “unslow” correct initial data. We must also justify the apparent inconsistency in how the approximation scheme deals with orders of $P$. In the computation of $\psi$ the scheme explicitly omits corrections of order $P^2$ in (\[hamil\]). Formally, then, we should only be able to keep terms of first order in $P$ in (\[energyint\]). But it is the apparently inconsistent $P^2$ terms, of course, which dominate at most points in Fig. \[energy\] (“momentum dominance” in generation of radiation). Not only do the $P^2$ terms agree with the results of numerical relativity, but the agreement remains good for rather high values of $P/M$. This raises the question: just what momentum contributions has our approximation really omitted?
The momentum enters into the construction of the initial data in only two direct ways. First, it is an overall scaling parameter for $\widehat{K}_{ab}$. The expression in (\[Kapprox\]) is an approximation for small $L$, but it is exact in $P$. The process of symmetrizing does not change this. Up to a conformal factor, then, the extrinsic curvature is exactly linear in $P$. Second, $P$ enters the determination of the conformal factor through (\[hamil\]). The success of the slow approximation must be directly ascribed to the relatively unimportant role played by the right hand side of (\[hamil\]).
Further work will be needed for a real understanding of this, but some reasonable speculations can already be made. Due to momentum dominance the details of the initial three geometry are not crucial, so any quadrupolar distortion induced by $\widehat{K}_{ab}$ at large $P$ will be insignificant compared to the radiation generated by the extrinsic curvature. The “slow” approximation, of course, is not perfect; at sufficiently high momentum it begins to fail. We speculate that the reason for this failure is not primarily due to $\widehat{K}_{ab}$ generating quadrupolar distortions of the initial three geometry. Rather, it is the effect of that source on the monopole part of the conformal factor, and hence on the ADM mass “$M$,” that is used to scale physical quantities. When we do a comparison in Fig. \[energy\] between the numerical relativity results and those of the approximation, we are comparing two cases for the same $\mu_0$ (i.e., the same coordinate separation in conformal space) and for the two cases we compare $E/M$ at a given value of $P/M$. We are therefore placing on an equal footing the true value of $M$ in the numerical relativity solution, and the $P\rightarrow0$ value of $M$ in the approximation. It should be possible, in principle, to correct for this and, in effect, reduce the approximation to one in which we have only ignored the quadrupolar part of the source in (\[hamil\]).
The present results greatly help us to understand the success of the results of Ref. [@ac94]. That
success seems to require two things about the generation of gravitational radiation in collisions from large distances: (i) There must be negligible radiation during the early motion, when the holes are in each other’s weak field region. (ii) The only important consequence of the early, weak-field, motion must be to give the holes momentum when each reaches the strong field region of the other. The first requirement is relatively easy to check. In Fig. \[time\] we plot radiated energy, computed by methods of numerical relativity, as a function of time, first for initial data representing two black holes falling from large separation. (The oscillations are due to the fact that almost all the energy comes off as “quasinormal ringing” of the final hole formed.) We also show the result of a second calculation. Cook[@cook91] initial data are taken corresponding to the separation and momentum that the black holes would have after falling to a fairly close separation. A comparison of the curves verifies that the early stage of motion does not produce a significant contribution to the total outgoing radiation.
Our present results, and in particular momentum dominance, strongly support the second requirement for the success of the ideas of Ref. [@ac94]. Since $\psi_{\rm Mom}$ is the source of essentially all the radiation, one can see that what is important about the early stages of the coalescence is only the development of extrinsic curvature. This does not, of course, explain why there seems to be insensitivity to the details of the extrinsic curvature. (Surely, the Bowen-York extrinsic curvature, symmetrized or not, is not actually the extrinsic curvature that evolves from earlier stationary conditions. Yet, it seems to be adequate to give good predictions.) A more satisfactory answer to this question means that we must understand the relationship between data on an initial hypersurface and how this evolves to data on subsequent hypersurfaces. We must also understand the importance of confining ourselves to conformally flat data on hypersurfaces. Progress on these questions will probably require comparable results from four distinct classes of initial data sets. These are (a) Misner data with large hole separation, (b) the non conformally-flat data with close holes that evolves from (a), (c) boosted conformally-flat data with close holes, and (d) boosted conformally-flat data in the close-slow approximation. In addition, one requires reasonable measures of physical separation and momentum so that correspondence can be drawn between disparate initial data sets.
There is strong motivation for carrying out such studies. The results so far achieved, both by numerical relativity and with the close and the slow approximation, are limited to head-on collisions. The situation of astrophysical interest, of course, is very different: the coalescence of orbiting holes. If the last few orbits in a coalescence are to be studied with numerical relativity, it will be crucial to understand what initial data are to be used to start the computation. Studies with the head-on collision provide a useful starting point to understanding the sensitivity of the radiation generation to the details of the initial data.
A rather different, and more speculative, motivation for a better understanding of these issues, is the hope that our approximation methods might be as successful with orbital problems as with head-on coalescence. These results might provide “easy” approximate answers over a reasonable range of orbital coalescences, and may therefore serve as a guide to the numerical studies.
This work was supported in part by grants NSF-PHY-9423950, NSF-PHY-9396246, NSF-PHY-9207225, NSF-PHY-9507719, NSF-PHY-9407882, research funds of the Pennsylvania State University, the University of Utah, the Eberly Family research fund at PSU and PSU’s Office for Minority Faculty development. JP acknowledges support of the Alfred P. Sloan foundation through a fellowship.
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---
abstract: |
We present the results of a new evaluation of the anomalous magnetic moment $a_\mu=(g_\mu-2)/2$ of the muon where the role of input data needed in the evaluation is lowered in the interval between $1.2$ and $3.0{{\rm\,GeV}}$ below charm threshold. This is achieved by decreasing the size of the weight function in the dispersion integral over the experimental ratio $R(s)$ by subtracting a polynomial from the weight function which mimics its energy dependence in that given energy interval. In order to compensate for this subtraction, the same polynomial weight integral is added again but is now evaluated on a circular contour in the complex plane using QCD and global duality. For the hadronic contribution to the shift in the anomalous magnetic moment of the muon we obtain $a_\mu^{\rm had}({\rm LO})=(701.3\pm 6.4)\times 10^{-10}$ at leading order in the electromagnetic coupling. In addition, using the same procedure, we recalculate the next-to-leading contribution $a_\mu^{\rm had}({\rm NLO})=(-10.3\pm0.2)\times 10^{-10}$. Including QED, electroweak, and light-by-light contribution, we obtain a value $a_\mu=(11\,659\,185.6\pm6.4_{\rm had}\pm3.5_{\rm LBL}\pm0.4_{\rm QED+EW})
\times 10^{-10}$.
---
MZ-TH/03-14\
hep-ph/0309226\
September 2003\
[**QCD improved determination**]{}\
[**of the hadronic contribution to the**]{}\
[**anomalous magnetic moment of the muon**]{}\
[S. Groote, J.G. Körner and J. Maul]{}\
Institut für Physik, Johannes-Gutenberg-Universität,\
Staudinger Weg 7, 55099 Mainz, Germany
Introduction
============
The anomalous magnetic moment $a_\mu$ of the muon is one of a few physical parameters which can be determined with high precision and therefore can serve as precision test for the Standard Model of elementary particle physics. In this paper we join the attempts to determine the hadronic contribution of the anomalous magnetic moment of the muon with a better accuracy (for an overview over the status of calculations of different collaborations two years ago see e.g. Ref. [@CzarneckiMarciano; @MarcianoRoberts]). While the dominant QED contribution $a_\mu^{\rm QED}=(11\,658\,470.6\pm 0.3)\times 10^{-10}$ and the weak contribution $a_\mu^{\rm weak}=(15.4\pm 0.1\pm 0.2)\times 10^{-10}$ are known with very good accuracy (see e.g.Refs. [@Hughes:fp; @Czarnecki:1998nd; @Czarnecki:2002nt] and references therein), the main uncertainty is given by the hadronic contribution. As an example for the actual calculations of the leading order hadronic contribution we cite the value $a_\mu^{\rm had}=(683.1\pm 5.9 \pm 2.0)\times 10^{-10}$ [@Hagiwara].
The calculation of the hadronic contribution to the anomalous magnetic moment of the muon mostly relies on experimental data in the $e^+e^-$ channel. In principle one can also make use of data from $\tau$-decay [@Aleph]. However, the inclusion of $\tau$-decay data introduces systematic uncertainties originating from isospin symmetry breaking effects which are difficult to estimate [@Hoecker]. We have therefore decided to only include $e^+e^-$ data in our analysis.
There are different concepts for using the $e^+e^-$ data sets, either to its full extent [@EidelmanJegerlehner] or in part by dividing the energy range into resonance regions close to the pair production thresholds where the experimental values are taken and those regions far from thresholds where perturbation theory is assumed to be valid. In this paper we present an approach which has been successfully applied for the determination of the running fine structure constant at the $Z^0$ boson resonance, $\alpha(M_Z^2)$ [@GKSN].
The subject of a precision determination of the anomalous magnetic moment became rather important again as a precision measurement for the positive muon could be accomplished at the Brookhaven National Laboratory (BNL) [@BNL:2001]. The value $$a_\mu^{\rm exp}=(11\,659\,202.3\pm 15.1)\times 10^{-10}$$ cited in Ref. [@BNL:2001] showed a deviation from the Standard Model prediction at that time, $$a_\mu^{\rm SM}=(11\,659\,159.7\pm 6.7)\times 10^{-10}$$ as weighted average over the calculation results of different collaborations [@MarcianoRoberts]. The deviation was $2.6$ standard deviations which seemed to have opened the window for new physics. Many such suggestions were published, including concepts of supersymmetry, leptoquarks, lepton number violating models, technicolor models, string theory concepts, extra dimensions and so on. The discrepancy between measurement and the Standard Model prediction became smaller, though, when a sign error was discovered in the theoretical calculation of the light-by-light contribution [@Hayakawa:2001bb]. Therefore, it is worth still to analyze the situation of the Standard Model prediction thoroughly, giving special emphasis on the error estimate, in order to compare it with the experimental world average $$a_\mu^{\rm exp}=(11\,659\,203\pm 8)\times 10^{-10}$$ which is dominated by the measurement of Ref. [@BNL:2002], as we will do in this paper.
Theoretical background
======================
The leading hadronic contribution to the anomalous magnetic moment of the muon $a_\mu^{\rm had}$ can be extracted from the $O(\alpha^2)$ vertex correction shown in Fig. \[fig1\].
The analytic expression for $a_\mu^{\rm had}$ from this diagram is a weighted dispersion integral over the imaginary part of the vector part of the hadronic vacuum polarisation, $$\label{disper}
a^{\rm had}_\mu=\frac{4\alpha^2}\pi\int_{4m_{\pi}^2}^\infty
{{\rm Im\,}}\Pi^{\rm had}(s)\frac{K(s)}sds$$ with the QED kernel $$\begin{aligned}
K(s)&=&\ln(1+x)\frac{(1+x^2)(1+x)^2}{x^2}
+\frac{x^2(1+x)}{1-x}\ln(x)\nonumber\\&&
+\frac{x^2}2(2-x^2)+\frac{(1+x^2)(1+x)^2}{x^2}\left(-x+\frac{x^2}2\right).\end{aligned}$$ As usual we make use of the kinematic variables $$x(s)=\frac{\beta(s)-1}{\beta(s)+1},\qquad\beta(s)=\sqrt{1-\frac{4m_\mu^2}s}.$$
${{\rm Im\,}}\Pi^{\rm had}(s)$ is calculated from the hadronic current-current correlator, $$\Pi^{\rm had}_{\mu\nu}(q^2)=i\int e^{iqx}
\langle0|T\;j_\mu(x)j_{\nu}(x')|0\rangle d^4x
=\Pi^{\rm had}(q^2)\left(q^2g_{\mu\nu}-q_\mu q_\nu\right)$$
Higher perturbative QCD corrections are well-known up to $O(\alpha_s^2)$ with $O(m_q^{12}/q^{12})$ quark mass corrections which are supplemented by massless terms up to $O(\alpha_s^4)$ [@Harlander]. This perturbative part is denoted by $\Pi^{\rm P}(q^2)$. For brevity we only cite the first few terms of this expansion in $\alpha_s$ and $m_q^2/q^2$, $$\Pi^{\rm P}(q^2)=\frac3{16\pi^2}\sum_{i=1}^{n_f}Q_i^2\Bigg[
\frac{20}9+\frac43L+C_F\left(\frac{55}{12}-4\zeta(3)+L\right)
\frac{\alpha_s}\pi+O(\alpha_s^2,m_q^2/q^2)\Bigg]$$ with $L=\ln(\mu^2/q^2)$. In our analysis we use the full $O(\alpha_s^2,m_q^{12}/q^{12})$ expression given in Ref. [@Harlander]. The number of active flavours is denoted by $n_f$ which changes according to the energy interval under consideration. For the zeroth order term in the $m_q^2/q^2$ expansion we have included the known higher order terms in $\alpha_s$, $$\frac3{16\pi^2}\sum_{i=1}^{n_f}Q_i^2\Bigg[\left(c_3+k_2L
+\frac12(k_0\beta_1+2k_1\beta_0)L^2+\frac13k_0\beta_0^2L^3\right)
\left(\frac{\alpha_s}\pi\right)^3+O(\alpha_s^4)\Bigg]$$ with $k_0=1$, $k_1=1.63982$ and $k_2=6.37101$. We have denoted the yet unknown constant term in the four-loop contribution by $c_3$ which, however, does not contribute to our calculations since it has no absorption part.
For the nonperturbative part the operator product expansion leads to [@Gorishnii] $$\begin{aligned}
\lefteqn{\Pi^{\rm NP}(q^2)\ =\ \frac1{18(q^2)^2}
\left(1+\frac{7\alpha_s}{6\pi}\right)
\langle\frac{\alpha_s}{\pi}G^2\rangle}\nonumber\\&&
+\frac8{9(q^2)^2}\left(1+\frac{\alpha_s}{4\pi}C_F+\ldots\ \right)
\langle m_u\bar uu\rangle
+\frac2{9(q^2)^2}\left(1+\frac{\alpha_s}{4\pi}C_F+\ldots\ \right)
\langle m_d\bar dd\rangle\nonumber\\&&
+\frac2{9(q^2)^2}\left(1+\frac{\alpha_s}{4\pi}C_F
+(5.8+0.92L)\frac{\alpha_s^2}{\pi^2}\right)
\langle m_s\bar ss\rangle\nonumber\\&&
+\frac{\alpha_s^2}{9\pi^2(q^2)^2}(0.6+0.333L)
\langle m_u\bar uu+m_d\bar dd\rangle\nonumber\\&&
-\frac{C_Am_s^4}{36\pi^2(q^2)^2}
\left(1+2L+(0.7+7.333L+4L^2)\frac{\alpha_s}{\pi}\right)
-\frac{448\pi}{243(q^2)^3}\alpha_s|\langle\bar qq\rangle|^2\nonumber\\&&
+\frac1{(q^2)^4}(0.48{{\rm\,GeV}})^8-\frac1{(q^2)^5}(0.3{{\rm\,GeV}})^{10}+O((q^2)^{-6})\end{aligned}$$ where $C_F=4/3$, $C_A=3$, $T_F=1/2$ are $SU(3)$ colour factors. We will use these results for the evaluation of the theoretical contributions.
Experimental contributions
==========================
The evaluation of the integral involves experimental data via the optical theorem which connects the imaginary part of the vacuum polarisation with the ratio $$R(s)=\frac{\sigma(e^+e^-\rightarrow{\rm hadrons})}{\sigma(e^+e^-\rightarrow
\mu^+\mu^-)}=12\pi{{\rm Im\,}}\Pi^{\rm had}(s).$$ Especially in the ${{\rm\,MeV}}$ and the low ${{\rm\,GeV}}$ energy range and in regions of resonances where perturbative QCD cannot be applied, data sets taken from experimental measurements are crucial. For our evaluation we have used the combined $e^+e^-$ data sets from Ref. [@EidelmanJegerlehner; @BES] and complement them in the dominant low energy range from $610{{\rm\,MeV}}$ to $961{{\rm\,MeV}}$ by recent two pion data [@Akhmetshin]. By comparing with the new data for three and four pion decays [@Achasov; @Novosib] one ensures that their influence for the sub ${{\rm\,GeV}}$-range where the two pion data were recorded is neglible.
All data sets are combined by weighting their relative contribution to the total experimental value with the corresponding statistic and systematic errors.
Introduction of the method
==========================
Global duality states that QCD can be used in weighted integrals over a spectral function if the spectral function is multiplied by polynomial functions. However, this does not work if the polynomial function is replaced by a singular function such as the weight $K(s)/s$ in the present case. Nevertheless, local duality is expected to hold for large values of $s$ far from resonances and threshold regions, i.e.${{\rm Im\,}}\Pi^{\rm had}(s)\simeq{{\rm Im\,}}\Pi^{\rm QCD}(s)$.
In our approach we attempt to reduce the influence of experimental $R(s)$ data in regions where the data has large uncertainties. Specifically, this holds for the interval between $1.2$ and $3.0{{\rm\,GeV}}$. The essence of our method is to diminish the magnitude of the weight function by subtracting a polynomial function which mimics the weight function at those energies. In order to compensate for this subtraction, the same polynomial function is added again, but now its contribution is evaluated by using global duality on a circular contour in the complex plane, according to $$\begin{aligned}
\lefteqn{\frac{\alpha^2}{3\pi^2}\int_{s_1}^{s_2}R(s)\frac{K(s)}{s}ds
\ =\ \frac{\alpha^2}{3\pi^2}\int_{s_1}^{s_2}R(s)
\left(\frac{K(s)}s-P_n(s)\right)ds}\nonumber\\&&
+\frac{4\alpha^2}\pi\Bigg[\frac1{2\pi i}\oint_{|s|=s_1}\Pi^{\rm had}(s)
P_n(s)ds-\frac1{2\pi i}\oint_{|s|=s_2}\Pi^{\rm had}(s)P_n(s)ds\Bigg].\end{aligned}$$
Application of the method to $a_\mu^{\rm had}$
==============================================
The usual procedure to evaluate Eq. (\[disper\]) is to use the experimental cross section up to some high momentum transfer and to calculate the remaining part from QCD using local duality.
Indeed, in the energy region up to $1.2{{\rm\,GeV}}$ we use the $e^+e^-$ cross section data set from Refs. [@EidelmanJegerlehner] and [@Akhmetshin]. The corresponding dispersion integral reads $$\label{BreitWigner}
(\Delta a_\mu^{\rm had})_1=\frac{\alpha^2}{3\pi^2}
\int_{4m_{\pi}^2}^{(1.2{{\rm\,GeV}})^2} \left[R^{e^+e^-}(s)+R^{\rm BW}(s)\right]
\frac{K(s)}sds.$$ Narrow resonances are added in explicit form. In Eq. (\[BreitWigner\]) the missing resonances are parametrized by a Breit-Wigner fit $R^{\rm BW}(s)$ and added to the remaining data sets.
In the subsequent interval from $1.2$ to $3.0{{\rm\,GeV}}$ the relevant $e^+e^-$ data from Refs. [@EidelmanJegerlehner], combined with data sets from the BES Collaboration [@BES], can be efficiently replaced by theoretical input from QCD. By subtracting a polynomial function from the weight function $K(s)/s$ we reduce the experimental input in this energy interval. Since the polynomial function is an analytic function in the whole complex plane, the remaining contribution can be evaluated off the real axis on a circular contour in the complex plane using perturbative and nonperturbative QCD results, $$\begin{aligned}
\label{split}
(a_\mu^{\rm had})_2&=&\frac{\alpha^2}{3\pi^2}
\int_{s_1=(1.2{{\rm\,GeV}})^2}^{s_2=(3.0{{\rm\,GeV}})^2}R^{e^+e^-}(s)
\left(\frac{K(s)}{s}-P_n(s)\right)ds\nonumber\\&&
-\frac{4\alpha^2}{\pi}\frac1{2\pi i}\oint_{|s_2|=(3.0{{\rm\,GeV}})^2}
\left[\Pi^{\rm P}(s)+\Pi^{\rm NP}(s)\right]P_n(s)ds\nonumber\\&&
+\frac{4\alpha^2}{\pi}\frac1{2\pi i}\oint_{|s_1|=(1.2{{\rm\,GeV}})^2}
\left[\Pi^{\rm P}(s)+\Pi^{\rm NP}(s)\right]P_n(s)ds\nonumber\\[3pt]
&=&(a_\mu^{\rm exp})_2+(a_\mu^{\rm the})_2\end{aligned}$$ where $n_f=3$ is taken for this interval.
At first glance the polynomial degree $n$ seems to be arbitrary. However, if $n$ is chosen too large, both the theoretical error from the strong coupling $\alpha_s(M_Z)$ and from unknown nonperturbative contributions arising from higher order condensates lead to larger uncertainties.
Moreover, we have to ensure that the calculated value is stable with respect of the variations of the lower polynomial degrees $n$ where the less known contributions from condensates are negligible. This point will help us find an optimal energy domain for our method.
An optimal choice for the polynomial degree should guarantee a good approximation of the polynomial function to the weight function and thus effectively replace the experimental data by theoretical input while reducing the total uncertainty from this region to a minimum.
Above $3.0{{\rm\,GeV}}$ the $e^+e^-$ data in Ref. [@EidelmanJegerlehner; @BES] show low uncertainties and can be integrated directly, $$(a_\mu^{\rm had})_3=\frac{\alpha^2}{3\pi^2}
\int_{s_2=(3.0{{\rm\,GeV}})^2}^{(40{{\rm\,GeV}})^2}\left[R^{e^+e^-}(s)+R^{\rm BW}(s)\right]
\frac{K(s)}{s}ds.$$ In the last step the high energy tail starting from $40{{\rm\,GeV}}$ can be calculated from theory using local duality. This is reasonable since there are no resonance contributions above $40{{\rm\,GeV}}$. In this region it is sufficient to use the lowest order contribution given by $${{\rm Im\,}}\Pi^{\rm had}(s)=3\!\!\!\sum_{{\it flavours}\ f}\!\!\!Q_f^2
\frac{\alpha}{12\pi}\sqrt{1-\frac{4m_f^2}s}\left(1+\frac{2m_f^2}s\right).$$
in units of $10^{-10}$ $n=1$ $n=2$ $n=3$ $n=4$
------------------------ ---------- --------- --------- ---------
$(a_\mu^{\rm the})_2$ $78.52$ $74.59$ $71.02$ $68.37$
$(a_\mu^{\rm exp})_2$ $-8.63$ $-3.06$ $-0.93$ $-0.51$
$(a_\mu^{\rm had})_2$ $71.60$ $71.49$ $69.21$ $67.08$
data contribution $12.1\%$ $4.3\%$ $1.3\%$ $0.8\%$
: \[tab1\]Contributions to $(a_\mu^{\rm had})_2$ for different fitting polynomial functions $P_n(s)$ with degree $n$. The purely experimental and theoretical contributions according to Eq. (\[split\]) are listed separately. Because the error estimate becomes worse for $n=3$ (cf. Table \[tab2\]), polynomial degrees $n=1$ and $n=2$ are used to obtain a mean value and a methodical error estimate, $(71.55\pm 0.13)\times 10^{-10}$.
Numerical results
=================
Using the recent $e^+e^-$ data from Refs. [@EidelmanJegerlehner] and [@Akhmetshin] we obtain for the dominant low energy component of $a_\mu^{\rm had}$, $$(a_\mu^{\rm had})_1=(518.1\pm 4.8)\times 10^{-10}.$$ We have added the statistical and point-to-point systematic errors in quadrature. In addition to this experimental value we have to consider contributions from the $\omega$ and $\phi$ resonances for which we obtain the values $(38.9\pm1.4)\times 10^{-10}$ and $(40.4\pm1.3)\times 10^{-10}$, respectively, from an integration over the Breit–Wigner distribution function where the values for the parameters are taken from Ref. [@PDG].
Applying our polynomial technique for the range between $1.2$ and $3.0{{\rm\,GeV}}$, we incorporate QCD-expressions for the hadronic current-current correlator with corresponding uncertainties for the parameters occuring in these expressions. The errors from the masses of the light quarks $u$, $d$, and $s$ can be neglected at energies above $1{{\rm\,GeV}}$. Thus the theoretical uncertainty is dominated by the strong coupling $\alpha_s(M_Z)$ whose error originates from the uncertainty in the QCD scale $\Lambda_{\overline{\rm MS}}=(380\pm 60){{\rm\,MeV}}$. For the condensate terms we assign generous errors of $100\%$. Thus we have $$\langle\frac{\alpha_s}\pi G^2\rangle=(0.04\pm 0.04){{\rm\,GeV}}^4,\qquad
\alpha_s|\langle\bar qq\rangle|^2=(4\pm 4)\times 10^{-4}{{\rm\,GeV}}^6.$$ With increasing polynomial degrees the condensate error contributions grow fast.
in units of $10^{-10}$ $n=1$ $n=2$ $n=3$ $n=4$
-------------------------------------------------- --------- --------- --------- ---------
uncert. due to $\alpha_s(M_Z)$ $2.5$ $5.1$ $7.7$ $10.5$
uncert. due to $\langle(\alpha_s/\pi)G^2\rangle$ $0.044$ $0.196$ $0.512$ $1.043$
uncertainty of $(a_\mu^{\rm the})_2$ $2.5$ $5.1$ $7.7$ $10.6$
: \[tab2\]Theoretical uncertainties of $(a_\mu^{\rm had})_2$ from QCD parameters and condensates. Because the uncertainty is worse for higher polynomial degrees, the degrees $n=1$ and $n=2$ are selected for the analysis. The error estimate is given by the average value $3.80\times 10^{-10}$. The uncertainty due to $\alpha_s|\langle\bar qq\rangle|^2$ is less than $10^{-14}$ in all cases and is therefore omitted in the table.
However, as can be clearly seen in Tab. \[tab1\] or in Fig. \[fig2\], good approximations to the kernel of the dispersion integral and consequently a very high reduction of the experimental data influence in the interval between $1.2$ and $3.0{{\rm\,GeV}}$ can be achieved even by low degrees $n=1$ and $n=2$. This can be understood by the fact that one is far away from the singularity at $s=0$ and thus the weight function can be well approximated by low degree polynomials.
We thus take the values $(a_\mu^{\rm had})_2$ for $n=1$ and $n=2$ and calculate the algebraic mean. Therefore, we have to take into account a small methodical error due to the variation of the polynomial degree, $$(a_\mu^{\rm meth})_2=\pm0.06\times 10^{-10}.$$ Summing up the error contributions in quadrature and the algebraic mean uncertainty (for $n=1$ and $n=2$) from Table \[tab2\], one obtains $$(a_\mu^{\rm had})_2=(71.55\pm3.83)\times 10^{-10}.$$
In the interval following $3.0{{\rm\,GeV}}$ up to the end of the $e^+e^-$ data set at $40{{\rm\,GeV}}$ as given in Ref. [@EidelmanJegerlehner] we obtain $$(a_\mu^{\rm had})_3=(23.2\pm0.2)\times 10^{-10}.$$ The contributions from narrow charmonium and bottonium resonances, $(8.8\pm0.6)\times 10^{-10}$ and $(0.11\pm0.01)\times 10^{-10}$, respectively, are taken by integrating over the narrow resonance distribution function where values for the parameters are again taken from Ref. [@PDG].
We finally obtain a tiny continuum contribution from the last interval, $$(a_\mu^{\rm had})_4=\frac{4\alpha^2}\pi\int_{(40{{\rm\,GeV}})^2}^\infty
{{\rm Im\,}}\Pi^{\rm had}(s)\frac{K(s)}sds=0.15\times 10^{-10}.$$
interval for $\sqrt s$ contributions to $a_\mu^{\rm had}$ comments
-------------------------------------- ------------------------------------ -----------------------------
$[0.28{{\rm\,GeV}},1.2{{\rm\,GeV}}]$ $(518.1\pm4.8)\times 10^{-10}$ $e^+e^-$ cross section data
$\omega$ resonance $(38.9\pm1.4)\times 10^{-10}$ Breit-Wigner
$\phi$ resonances $(40.4\pm1.3)\times 10^{-10}$ narrow resonances
$[1.2{{\rm\,GeV}},3.0{{\rm\,GeV}}]$ $(71.6\pm3.8)\times 10^{-10}$ polynomial method
$J/\psi$ resonances $(8.8\pm0.6)\times 10^{-10}$ narrow resonances
$[3.0{{\rm\,GeV}},40{{\rm\,GeV}}]$ $(23.2\pm0.2)\times 10^{-10}$ $e^+e^-$ annihilation data
$\Upsilon$ resonances $(0.11\pm0.01)\times 10^{-10}$ narrow resonances
$[40{{\rm\,GeV}},\infty]$ $0.15\times 10^{-10}$ theory
top quark contr. $<10^{-13}$ theory
hadronic contr. $(701.3\pm6.4)\times 10^{-10}$
: \[tab3\]The different contributions to the hadronic part of the anomalous magnetic moment $a_\mu^{\rm had}$ of the muon.
Summing up the contributions from the different energy intervals as shown in Table \[tab3\], we finally obtain $$a_\mu^{\rm had}=(701.3\pm6.4)\times 10^{-10}$$ for the hadronic contribution to $a_\mu$. We have thereby assumed that errors with different origin are uncorrelated and have to be added quadratically in order to obtain the total error.
Higher order correction
=======================
In the next-to-leading order (NLO) we have to consider three types of diagrams, those of type (2a) with an additional photon exchange, those of type (2b) with an electron loop inserted in one of the photon lines in Fig. \[fig1\], and finally the one of type (2c) with two correlator functions included in the photon line. For the former two the contribution reads $$\label{exp0vacNLO}
a_\mu^{\rm had}({\rm NLO})=\frac13{\left(\frac{\alpha}{\pi}\right)}^3\int_{4m_\pi^2}^\infty
\frac{ds}{s}K^{(2)}(s)R(s).$$ For numerical purposes it is convenient to represent the kernel functions $K^{(2a)}(s)$ and $K^{(2b)}(s)$ in terms of power series expansions in terms of $m^2/s$ [@Krause] ($m=m_\mu=105.6583568\pm0.0000052{{\rm\,MeV}}$ [@PDG] is the mass of the muon). One has $$\begin{aligned}
\lefteqn{K^{(2a)}(s)\ =\ 2\frac{m^2}{s}\Bigg\{
\left(\frac{223}{54}-\frac{\pi^2}{3}-\frac{23}{36}\ln{\left(\frac{s}{m^2}\right)}\right)}
\nonumber\\&&
+\frac{m^2}{s}\left(\frac{8785}{1152}-\frac{37\pi^2}{48}
-\frac{367}{216}\ln{\left(\frac{s}{m^2}\right)}+\frac{19}{144}\ln^2{\left(\frac{s}{m^2}\right)}\right)
\\&&
+\frac{m^4}{s^2}\left(\frac{13072841}{432000}-\frac{883\pi^2}{240}
-\frac{10079}{3600}\ln{\left(\frac{s}{m^2}\right)}+\frac{141}{80}\ln^2{\left(\frac{s}{m^2}\right)}\right)
+\ldots\Bigg\},\nonumber\\
\lefteqn{K^{(2b)}(s)\ =\ 2\frac{m^2}{s}\Bigg\{
\left(-\frac{1}{18}+\frac{1}{9}\ln{\left(\frac{s}{m_f^2}\right)}\right)}
\nonumber\\&&
+\frac{m^2}{s}\left(-\frac{55}{48}+\frac{\pi^2}{18}
+\frac{5}{9}\ln{\left(\frac{s}{m_f^2}\right)}+\frac{5}{36}\ln{\left(\frac{m^2}{m_f^2}\right)}
-\frac{1}{6}\ln^2{\left(\frac{s}{m_f^2}\right)}+\frac{1}{6}\ln^2{\left(\frac{m^2}{m_f^2}\right)}\right)
\\&&
+\frac{m^4}{s^2}\left(-\frac{11299}{1800}+\frac{\pi^2}{3}
+\frac{10}{3}\ln{\left(\frac{s}{m_f^2}\right)}-\frac{1}{10}\ln{\left(\frac{m^2}{m_f^2}\right)}
-\ln^2{\left(\frac{s}{m_f^2}\right)}+\ln^2{\left(\frac{m^2}{m_f^2}\right)}\right)
+\ldots\Bigg\}\nonumber\end{aligned}$$ where for $m_f$ we used the electron mass, $m_f=510.998902\pm0.000021{{\rm\,keV}}$ [@PDG], since the contributions from the muon is already included in the contribution (2a). The contribution of the $\tau$ is suppressed by $m_\tau^2/m_\mu^2$ and numerically negligible. The kernels of the different orders in the electromagnetic coupling, $K(s)$ on the one hand and $K^{(2a)}(s)$ and $K^{(2b)}(s)$ on the other hand, are compared in Fig. \[fig4\]. For illustration, we like to sum the results from the evaluation of Eq. (\[exp0vacNLO\]) which was performed in a completely analogous way as in the previous chapter. The final result $(-21.07\pm0.21)\times 10^{-10}$ in case of $K^{(2a)}(s)$ (cf.Table \[tab4\]) and $(10.78\pm0.08)\times 10^{-10}$ in case of $K^{(2b)}(s)$ (Table \[tab5\]) are in good agreement with the literature [@Alemany]. Finally, for the small contribution from diagrams of type (2c) with two correlator functions included into the photon line we take the value $(0.27\pm 0.01)\times 10^{-10}$ from Ref. [@Krause]. We have checked that to the required accuracy a recalculation using the method presented in this paper is not necessary. The total hadronic NLO contribution then sums up to a value of $(-10.3\pm0.2)\times 10^{-10}$, as compared to the value $(-10.1\pm0.6)\times 10^{-10}$ given in Ref. [@Krause]. Our central value is $3\%$ smaller than that given in Ref. [@Krause] and the error is reduced from $0.6\times 10^{-10}$ to $0.2\times 10^{-10}$. Finally, summing up leading and next-to-leading order contributions together with the afore-mentioned QED and weak contributions and the so-called light-by-light contribution $a_\mu^{\rm had}({\rm LBL})=(8.6\pm 3.5)\times 10^{-10}$ [@Knecht:2001qf; @Kinoshita; @Bijnens], we obtain $a_\mu=(11\,659\,185.6\pm
6.4_{\rm had}\pm3.5_{\rm LBL}\pm0.4_{\rm QED+EW})\times 10^{-10}$.
interval for $\sqrt s$ contributions to $a_\mu^{\rm had}$ comments
-------------------------------------- ------------------------------------ -----------------------------
$[0.28{{\rm\,GeV}},1.2{{\rm\,GeV}}]$ $(-14.04\pm0.12)\times 10^{-10}$ $e^+e^-$ cross section data
$\omega$ resonance $(-1.11\pm0.04)\times 10^{-10}$ Breit-Wigner
$\phi$ resonances $(-1.29\pm0.04)\times 10^{-10}$ narrow resonances
$[1.2{{\rm\,GeV}},3.0{{\rm\,GeV}}]$ $(-2.84\pm0.14)\times 10^{-10}$ polynomial method
$J/\psi$ resonances $(-0.44\pm0.03)\times 10^{-10}$ narrow resonances
$[3.0{{\rm\,GeV}},40{{\rm\,GeV}}]$ $(-1.35\pm0.09)\times 10^{-10}$ $e^+e^-$ annihilation data
$\Upsilon$ resonances $<10^{-13}$ narrow resonances
$[40{{\rm\,GeV}},\infty]$ $<10^{-12}$ theory
top quark contr. $<10^{-14}$ theory
hadronic contr. $(-21.07\pm0.21)\times 10^{-10}$
: \[tab4\]The different contributions to the hadronic part of the anomalous magnetic moment $a_\mu^{\rm had}$ of the muon for the kernel $K^{(2a)}$.
interval for $\sqrt s$ contributions to $a_\mu^{\rm had}$ comments
-------------------------------------- ------------------------------------ -----------------------------
$[0.28{{\rm\,GeV}},1.2{{\rm\,GeV}}]$ $(7.98\pm0.07)\times 10^{-10}$ $e^+e^-$ cross section data
$\omega$ resonance $(0.59\pm0.02)\times 10^{-10}$ Breit-Wigner
$\phi$ resonances $(0.62\pm0.02)\times 10^{-10}$ narrow resonances
$[1.2{{\rm\,GeV}},3.0{{\rm\,GeV}}]$ $(1.09\pm0.01)\times 10^{-10}$ polynomial method
$J/\psi$ resonances $(0.14\pm0.01)\times 10^{-10}$ narrow resonances
$[3.0{{\rm\,GeV}},40{{\rm\,GeV}}]$ $(0.364\pm0.003)\times 10^{-10}$ $e^+e^-$ annihilation data
$\Upsilon$ resonances $<10^{-13}$ narrow resonances
$[40{{\rm\,GeV}},\infty]$ $<10^{-12}$ theory
top quark contr. $<10^{-14}$ theory
hadronic contr. $(10.78\pm0.08)\times 10^{-10}$
: \[tab5\]The different contributions to the hadronic part of the anomalous magnetic moment $a_\mu^{\rm had}$ of the muon for the kernel $K^{(2b)}$.
Conclusion
==========
We have presented an alternative determination of the hadronic contribution to the anomalous magnetic moment of the muon where we have made use of theoretical QCD results to reduce the influence of the poor experimental data in the range between $1.2$ and $3.0{{\rm\,GeV}}$. Our analysis includes the leading and next-to-leading contribution.
Using the polynomial method in the range between $1.2$ and $3.0{{\rm\,GeV}}$, we could suppress the influence of experimental data effectively and thereby reduce the error on the determination of $a_\mu^{\rm had}$. Note that using this method we were able to use a value for the lower limit of this range lower than usually used for pure QCD methods.
Using only $e^+e^-$ data and QCD as input, we obtain a result which is $1.6\sigma$ away from the measured world average, stating that the deviation between theory and experiment may be smaller than commonly suggested. Other recent results to compare with are $(11\,659\,180.9\pm7.2_{\rm had}\pm
3.5_{\rm LBL}\pm0.4_{\rm QED+EW})\times10^{-10}$ [@Davier:2003pw] and $(11\,659\,166.9\pm7.4)\times10^{-10}$ [@Hagiwara].
Acknowledgements {#acknowledgements .unnumbered}
----------------
We would like to thank K. Schilcher for discussions. S.G. acknowledges a grant given by the DFG, Germany through the Graduiertenkolleg “Eichtheorien” at the University of Mainz.
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|
---
abstract: 'By averaging over an ensemble of field configurations, a classical field theory can display many of the characteristics of quantum field theory, including Lorentz invariance, a loop expansion, and renormalization effects. There is additional freedom in how the ensemble is chosen. When the field mode amplitudes have a Gaussian distribution, and the mode phases are randomly distributed, we review the known differences between the classical and quantum theories. When the mode amplitudes are fixed, or have a nongaussian distribution, the quartic and higher correlations among the free fields are modified, seemingly in a nonlocal way. We show how this in turn affects the perturbative expansion. We focus on $\lambda\phi^4$ theory in $1+1$ dimensions and use lattice simulations to augment our study. We give examples of how these nonlocal correlations induce behavior more similar to quantum field theory, at both weak and strong coupling.'
author:
- |
B. Holdom[^1]\
*Department of Physics, University of Toronto*\
*Toronto ON Canada M5S1A7*
title: |
**Approaching quantum behavior\
with classical fields**
---
Classical fields masquerading as quantum fields\[S2\]
=====================================================
We begin by considering free field configurations of a classical field theory having the same energy spectrum as the vacuum fluctuations of quantum field theory. From an ensemble of such configurations the $n$-point correlation functions can be obtained, and we will discuss some dependence that these functions have on the choice of ensemble. The relation to quantum field theory will become more clear in later sections, where we discuss how the free configurations are to be perturbatively corrected to account for interactions. The choice of ensemble that we discuss here will have a nontrivial effect on this perturbative expansion.
For a classical field theory with Lagrangian $${\cal L}=\frac{1}{2}\partial_\mu\phi\partial^\mu\phi-\frac{1}{2}m^2\phi^2-\frac{1}{4}\lambda\phi^4
,\label{e2}$$ we may consider a field configuration consisting of a sum over the free modes, $$\phi_0(x)=\sum_\mathbf{p}a_\mathbf{p}\cos(\omega_\mathbf{p} t+\mathbf{p}\cdot\mathbf{x}+\theta_{\mathbf{p}})
.$$ These are solutions for $\lambda=0$ and $\omega_\mathbf{p}=\sqrt{\mathbf{p}^{2}+m^{2}}$. We will typically work in a finite volume $V$ and so have a discrete set of modes. We constrain the value of the free Hamiltonian for these configurations to be $$H_0=\sum_\mathbf{p}\frac{1}{2}\hbar\omega_\mathbf{p}
\label{e7}$$ and thus fix the amplitudes as $$a_\mathbf{p}^2=\frac{\hbar}{V\omega_\mathbf{p}}
.\label{e10}$$
Now consider an ensemble of such configurations, differing only through the choice of random phases $\theta_{\mathbf{p}}$ which we take to be uniformly distributed. In the limit of a sufficiently large number of configurations we can obtain the expectation value of a product of fields by simply averaging over the phases. This gives [@F; @C] $$\langle\phi_{0}(x)\phi_{0}(y)\rangle\equiv D^0(x-y)=\sum_\mathbf{p}\frac{\hbar}{2\omega_\mathbf{p} V}\cos(p(x-y)) \mbox{, with $p=(\omega_\mathbf{p},\mathbf{p})$}
\label{e14}.$$ In the $V\rightarrow\infty$ limit we could write this as $$\begin{aligned}
\langle \phi_0(x)\phi_0(y)\rangle&=&\int \frac{d^4p}{(2\pi)^4} G_{\phi\phi}^0(p)e^{-ip(x-y)},\\
G_{\phi\phi}^0(p)&=&\hbar\pi\delta(p^2-m^2)
,\label{e9}\end{aligned}$$ thus showing that the $\frac{1}{2}\hbar\omega$ spectrum is Lorentz invariant in this limit.
A more conventional choice of ensemble would have the mode amplitudes also varying from one configuration to the next, in such a way that only the average value of the energy of each mode is $\frac{1}{2}\hbar\omega_\mathbf{p}$. Then the amplitudes can have a Gaussian probability distribution, $$\rho(a_\mathbf{p})=\frac{2 a_\mathbf{p}}{\sigma_\mathbf{p}^2}\exp(- \frac{a_\mathbf{p}^2}{\sigma_\mathbf{p}^2}),\quad\quad\sigma_\mathbf{p}^2=\hbar/(V\omega_\mathbf{p})
,\label{e8}$$ while treating the $a_\mathbf{p}$ as nonnegative polar variables. The result (\[e14\]) for the 2-point function again follows. This choice bears a resemblance to the state vectors of a quantum field theory when written as wave functionals $\Omega[\phi]$, since the solution to the free functional Schrodinger equation is of course a Gaussian.[^2]
The choice of ensemble affects the quartic and higher free $n$-point functions. They are obtained as before by taking the ensemble average of products of fields, which in turn are sums of modes. For example the free 4-point function in the Gaussian case is as expected, $$\begin{aligned}
&&\langle\phi_{0}(x_{1})\phi_{0}(x_2)\phi_{0}(x_3)\phi_{0}(x_{4})\rangle_\mathrm{G} =\nonumber\\&&\;\;\;\;D^0(x_{1}-x_{2})D^0(x_{3}-x_{4})+D^0(x_{1}-x_{3})D^0(x_{2}-x_{4})+D^0(x_{1}-x_{4})D^0(x_{2}-x_{3}).\label{e16}\end{aligned}$$ When the amplitudes do not vary between configurations, the fixed amplitude case, there are “unusual” contributions in the 4-point function due to terms where the product of modes involves only a single mode, so the four factors share a single phase and a single amplitude. The average is now over the phase only, and this changes the result to $$\begin{aligned}
&&\langle\phi_{0}(x_{1})\phi_{0}(x_2)\phi_{0}(x_3)\phi_{0}(x_{4})\rangle_\theta =\langle\phi_{0}(x_{1})\phi_{0}(x_2)\phi_{0}(x_3)\phi_{0}(x_{4})\rangle_\mathrm{G}\nonumber\\&&\;\;\;\;\;-\;\frac{1}{2}\Delta(x_1,x_2;x_3,x_4)-\frac{1}{2}\Delta(x_1,x_3;x_2,x_4)-\frac{1}{2}\Delta(x_1,x_4;x_2,x_3),\\
&&\Delta(x_1,x_2;x_3,x_4)=\sum_\mathbf{p}\frac{1}{(2\omega_\mathbf{p} V)^2}\cos(p(x_1-x_2))\cos(p(x_3-x_4)).\label{e12}\end{aligned}$$ Alternatively we may write this as $$\begin{aligned}
&&\langle\phi_{0}(x_{1})\phi_{0}(x_2)\phi_{0}(x_3)\phi_{0}(x_{4})\rangle_\theta =\nonumber\\&&\;\;\;\;[[D^0(x_{1}-x_{2})D^0(x_{3}-x_{4})+D^0(x_{1}-x_{3})D^0(x_{2}-x_{4})+D^0(x_{1}-x_{4})D^0(x_{2}-x_{3})]]_\frac{1}{2}.\label{e11}\end{aligned}$$ $[[..]]_\frac{1}{2}$ indicates that we must reduce the terms with a common $\mathbf{p}$, after inserting (\[e14\]) for $D^0$, by a factor of $1/2$.
In the case of the free 6-point function, contributions with two of the momenta the same are again reduced by $1/2$, while the terms with all three of the momentum the same are reduced by a factor of $1/6$. More generally, contributions to free correlation functions with $n$ momenta the same are reduced by a factor of $1/n!$.
Of course one could engineer a probability distribution $\rho(a_\mathbf{p})$ to produce quite arbitrary changes in the coincident momentum behavior of free correlation functions. In terms of the second moment of the distribution, $\sigma_\mathbf{p}^2$, if the $(2n)$th moment for $n>1$ is $C_n\sigma_\mathbf{p}^{2n}$, then the contributions with $n$ momenta coincident at $\mathbf{p}$ are changed by a factor of $C_n/n!$. The Gaussian and fixed amplitude cases have $C_n=n!$ and $C_n=1$ respectively, and it can be seen that the fixed amplitude case minimizes the $C_n$. A simple family of distributions is $$\rho(a_\mathbf{p})\propto \frac{a_\mathbf{p}^{d-1}}{\sigma_\mathbf{p}^d}\exp(- \frac{d a_\mathbf{p}^2}{2\sigma_\mathbf{p}^2}),\quad0<d<\infty$$ where Gaussian and fixed amplitude cases have $d=2$ and $d\rightarrow\infty$ respectively.
Thus we see that the freedom in the choice of ensemble modifies the free higher $n$-point functions, while holding the free $2$-point function fixed. We notice that $\Delta$ in (\[e12\]) does not factorize and it does not vanish for large spacelike separations of pairs of points. These modifications then violate cluster decomposition and thus lie outside the realm of correlations produced by a local quantum field theory. In this sense they could be considered nonlocal, even though we see that they can occur naturally in the free field configurations of a local classical field theory. We will refer to them as nonlocal for lack of a better description. As a measure of the nonlocality in the 4-point function we have $$\beta\equiv2-C_2=\frac{d-2}{d}
,\label{e19}$$ so that $[[..]]_\frac{1}{2}$ in (\[e11\]) is replaced by $[[..]]_{1-\beta/2}$.
In the classical field theory these free correlations form the basis of the perturbative expansion, and thus the first question is whether this expansion can be significantly affected by the nonlocal correlations. The perturbative expansion is a loop expansion involving momentum loop integrations, and it might appear that the nonlocal correlations would have vanishing effect in the infinite volume limit since the correction terms involve fewer momentum sums. On the other hand the loop integrands may display singular behavior in regions of momentum space where these corrections contribute. We will show, both through a direct sample calculation and by lattice simulation, that the nonlocal correlations do in fact alter the perturbative expansion in significant ways.
The Gaussian classical theory is known to deviate from the behavior of the corresponding interacting quantum field theory, and we will remind the reader of the differences at the perturbative level. In certain situations of high temperature and/or high occupation number, these differences may be controlled and the classical theory used as an approximation scheme [@G]. But at zero temperature the departures from the quantum theory that are evident for the Gaussian classical theory could be taken to define what truly quantum phenomena are. Thus it is of interest to compare these truly nonclassical effects of the quantum theory to the nonlocal effects of the nongaussian classical theory. We will find that the nonlocal correlations of the free fields can act to bring the perturbative expansion of the classical theory closer to that of the quantum theory. This helps to explain the findings of [@C; @D].
In the next section we will develop the Gaussian classical theory, highlighting both its resemblance to quantum field theory and its qualitative differences. Section 3 will pinpoint the perturbative difference, thus making clear what is missing relative to the quantum theory. In section 4 we consider a 2-loop example where we identify the missing piece, and we compare it to a similar contribution coming from the nonlocal correlations in the nongaussian classical theory. In sections 5 and 6 we use lattice simulations of the classical theory to investigate this more fully, and we find further evidence of quantum-like perturbative effects. In particular damping and thermalization effects are greatly reduced. The simulations can also be used at strong coupling, and in section 7 we will see quantum-like critical behavior emerging in the nongaussian classical theory. We end with some comments and open questions in the final section.
The Gaussian classical theory
=============================
It is possible to develop a perturbative expansion by writing the field equation as an integral equation and then iterating [@C]. Instead we shall make use of a path integral that does nothing but enforce the classical field equations, and then obtain the perturbative expansion from the path integral in the usual way. The integration over the fields in the path integral will produce the sum over the ensemble of classical field configurations.
The expectation values of products of fields in some ensemble of configurations can be expressed as follows, $$\langle \phi(x_1)\phi(x_2)...\rangle=\int d\phi d\chi\; \rho[\phi(x_i)]\phi(x_1)\phi(x_2)...e^{\frac{i}{\hbar}\int_{t_i}^{t_f}d^4x {\cal L}_\chi[\phi,\chi]}
\label{e5}$$ where $t_i<\{t_1,t_2,...\}<t_f$. (A similar construction appears for example in [@ms].) The Lagrangian ${\cal L}_\chi$ is linear in the auxiliary field $\chi$ $${\cal L}_\chi=\partial_\mu\phi\partial^\mu\chi-m^2\phi\chi-\lambda\phi^3\chi
,\label{e17}$$ and we may require that $\chi(x)$ vanishes at times $t_i$ and $t_f$. The functional integral over $\chi$ gives the constraint $$\prod_{\tau,\mathbf{x}}2\pi\hbar\delta\left[(\Box+m^2)\phi(\tau,\mathbf{x})+\lambda\phi^3(\tau,\mathbf{x})\right]$$ for $t_i<\tau<t_f$. In this way the field $\phi$ is constrained to satisfy the interacting classical field equation. Notice that a $\hbar^{-1}$ has been inserted in the action, even though it has no significance at this stage.
$\rho[\phi(x_i)]$, where $x_i$ is the coordinate on the initial time-slice $t=t_i$, selects the configurations in our ensemble. Since we are interested in a perturbation theory it is sufficient to describe this ensemble in terms of the free field modes, and in this section we are making the Gaussian choice. These configurations are acting as the initial conditions for evolution by the full field equations.
Now the point of this construction is to leave the $\chi$ field in the theory to develop the perturbation theory. The Lagrangian specifies the $\phi\chi$ and $\chi\phi$ propagators up to boundary conditions, and since we are studying a classical theory it is appropriate to choose the retarded and advanced Greens functions, $$G_{\phi\chi}^0=\frac{i\hbar}{p^2-m^2+i\varepsilon p_0}\;\;\;\;\;G_{\chi\phi}^0=\frac{i\hbar}{p^2-m^2-i\varepsilon p_0}
.$$ With this choice we see diagrammatically that the $\chi$ end of a line always appears at a later time than the $\phi$ end. But each vertex has only one $\chi$ and three $\phi$’s, so closed loops of $G_{\phi\chi}^0$ or $G_{\chi\phi}^0$ lines do not form. The resulting tree graph expansion is the expected result for a classical theory.
This would be all in the absence of background fields, but due to our ensemble of field configurations we also have $G_{\phi\phi}^0(p)=\hbar\pi\delta(p^2-m^2)$. The $\hbar$ appearing here, originating in the mode amplitudes, does have significance. Now the branches of the various trees can be interconnected with $G_{\phi\phi}^0$ lines, and a nontrivial loop expansion emerges. In fact the same relation between the number of loops in a diagram and the power of $\hbar$ occurs in this classical theory as it does in quantum field theory. The $\hbar^{-1}$ in the action was not necessary for this but it makes this result easier to see, since then factors of $\hbar$ are associated with propagators and vertices in the usual way.
The free 2-point functions are also such that $G_{\chi\chi}^0=0$ and $$G_{\phi\phi}^0=\frac{1}{2}\epsilon(p_0)(G_{\phi\chi}^0-G_{\chi\phi}^0)
.\label{e1}$$ We can make some statements about the full 2-point functions (denoted by $G$) and the 2PI self-energies (denoted by $\Pi$). Diagramatically it can be checked that $\Pi_{\phi\phi}=0$, and similarly $G_{\chi\chi}=0$ continues to hold in the interacting theory. Then the full $G_{\phi\chi}$ and $G_{\chi\phi}$ satisfy $$\begin{aligned}
G_{\phi\chi}&=&G^0_{\phi\chi}+G^0_{\phi\chi}\Pi_{\chi\phi}G_{\phi\chi} \\
G_{\chi\phi}&=&G^0_{\chi\phi}+G^0_{\chi\phi}\Pi_{\phi\chi}G_{\chi\phi} \end{aligned}$$ and thus $$\begin{aligned}
G_{\phi\chi} & = & \frac{i\hbar}{p^2-m^2-\Pi_r+i(\Pi_i+\varepsilon p_0)},\;\;\;\;\;\;i\hbar\Pi_{\chi\phi} \equiv \Pi_r-i\Pi_i \\
G_{\chi\phi} & = & \frac{i\hbar}{p^2-m^2-\Pi_r-i(\Pi_i+\varepsilon p_0)},\;\;\;\;\;\;i\hbar\Pi_{\phi\chi} \equiv \Pi_r+i\Pi_i \end{aligned}$$ These $G$’s have the same analytic structure as in the free theory, so $\Pi_i$ is an odd function of $p_0$ with $\Pi_i(p_0)>0$ for $p_0>0$.
Of more interest is the full 2-point function $G_{\phi\phi}$ for the original scalar field. The summation of the diagrams involves various classes of diagrams. $$\begin{aligned}
G_{\phi\phi}&=&G^0_{\phi\phi}+G^0_{\phi\phi}\Pi_{\phi\chi}G_{\chi\phi}+G_{\phi\chi}\Pi_{\chi\phi}G^0_{\phi\phi}+G_{\phi\chi}\Pi_{\chi\phi}G^0_{\phi\phi}\Pi_{\phi\chi}G_{\chi\phi}+G_{\phi\chi}\Pi_{\chi\chi}G_{\chi\phi}\\
&=&G_{\phi\chi}\Pi_{\chi\chi}G_{\chi\phi}\label{e3}\end{aligned}$$ The cancellations among the various classes follow from the expressions for $G^0_{\phi\phi}$, $G_{\phi\chi}$ and $G_{\chi\phi}$. We can compare this to the result of using (\[e1\]) as a relation among the full 2-point functions, which gives $$\begin{aligned}
G_{\phi\phi}&=&\frac{\epsilon(p_0)\Pi_i}{(p^2-m^2-\Pi_r)^2+\Pi_i^2}\label{e15}\\
&=&G_{\phi\chi}[\epsilon(p_0)\Pi_i] G_{\chi\phi}\label{e4}
\label{e6}\end{aligned}$$ Comparison of (\[e3\]) and (\[e4\]) would imply the relation $\Pi_{\chi\chi}=\epsilon(p_0)\Pi_i$. This is in fact true, as can be checked diagramatically.
It turns out that $\Pi_i$ (or $\Pi_{\chi\chi}$) is nonvanishing on mass shell, in which case we have the finite width (and shifted mass due to $\Pi_r$) form of a pseudoparticle correlator. The emergence of a pseudoparticle is typical in classical field theory applications, in thermal systems for example, and the pseudoparticle width is often referred to as plasmon or Landau damping. A sampling of references to this and related applications of classical field theories is given in [@therm].
This pseudoparticle distinguishes the classical theory from the particle description of a quantum field theory. Nevertheless there is a loop expansion characterized by powers of $\hbar$, and a renormalization of the parameters $m$ and $\lambda$ in ${\cal L}_\chi$ similar to quantum field theory. In addition to the $\chi\chi$ amplitude, the classical theory will perturbatively generate amputated amplitudes of the form $\phi^r\chi^s$ with $r+s$ even. We have seen that a $\phi\phi$ self-energy is not generated and more generally amplitudes are not generated unless $s\ge 1$. A real contribution to the $\phi\chi^3$ amplitude is expected, and in the next section we will find this amplitude to be of special interest.
\[Another implication is that ${\cal L}_\chi$ is not corrected by a constant term, a cosmological constant. Here it is useful again to emphasize the difference between the Lagrangian ${\cal L}_ \chi$ appearing in the path integral and the original classical Lagrangian ${\cal L}$ in (\[e2\]). It is the latter the defines the energy momentum tensor, whose the expectation value can be calculated perturbatively using the path integral involving ${\cal L}_\chi$. A contribution to the cosmological constant could be obtained in this way.\]
A small change and the true quantum theory
==========================================
Now let us consider a new theory by adding a $\phi\chi^3$ interaction to our previous theory defined by ${\cal L}_\chi$ in (\[e5\]-\[e17\]), where we continue to assume the Gaussian ensemble. Denoting the new Lagrangian by ${\cal L}_\alpha$, $${\cal L}_\alpha={\cal L}_\chi+\alpha\frac{\lambda}{4}\phi\chi^3
,$$ it is clear that the direct link to the classical evolution via integrating out the $\chi$ field has been lost. In fact this additional interaction term is all that separates the classical theory from the full quantum field theory. (This observation [@G] of the relation between classical and quantum field theories originates in studies of classical fields used to model quantum fields at high temperature or high occupation number.) This may be somewhat surprising, given that we have just observed that this amplitude is generated at some level in the classical theory.
The following redefinition of fields $$\chi=\sqrt{\frac{1}{\alpha}}(\phi_+-\phi_-)
,\;\;\;\;\;\;
\phi=\frac{1}{2}(\phi_++\phi_-)
,$$ yields $${\cal L}_\alpha=\sqrt{\frac{1}{\alpha}}\left[{\cal L}(\phi_+)-{\cal L}(\phi_-)\right]
,$$ where ${\cal L}$ is the original classical Lagrangian in (\[e2\]). Thus ${\cal L}$ is now appearing in the path integral itself. If $\alpha=1$ the overall factor in the action is the usual one for a quantum field theory, given the $\hbar^{-1}$ factor already in the action of (\[e5\]).
In this case of $\alpha=1$ the transformed versions of the original free propagators take the following form. $$\begin{aligned}
G_{++}^0&=&\frac{i\hbar}{p^2-m^2+i\epsilon}\\
G_{--}^0&=&\frac{i\hbar}{p^2-m^2-i\epsilon}\\
G_{-+}^0&=&2\hbar\pi\theta(p_0)\delta(p^2-m^2)\\
G_{+-}^0&=&2\hbar\pi\theta(-p_0)\delta(p^2-m^2)\end{aligned}$$ $G_{++}^0$ is the Feynman propagator. We have now arrived at the Schwinger-Keldysh formalism for the calculation of “in-in” matrix elements in quantum field theory [@rj]. This formalism is complementary to the more standard formalism of quantum field theory, concerned with the calculation of “in-out” matrix elements.
For example [@rj] consider the full $G_{++}$ 2-point function $$G_{++}(x_1,x_2)\propto\int d\phi_+ d\phi_-\; \rho[\phi_+(x_i)]\phi_+(x_1)\phi_+(x_2)e^{\frac{i}{\hbar}\int_{t_i}^{t_f}d^4x ({\cal L}(\phi_+)-{\cal L}(\phi_-))}$$ The original constraints on the $\chi$ field now read $\phi_-(x_i)=\phi_+(x_i)$ and $\phi_-(x_f)=\phi_+(x_f)$. Other than this the $\phi_+$ and $\phi_-$ path integrals have decoupled, and each is a path integral that can be identified with a “in-out” transition amplitude or correlation function of quantum field theory. Thus $$\begin{aligned}
G_{++}&=&\sum_s\langle \textrm{in, vac}|\textrm{out, }s\rangle
\langle \textrm{out, }s|T\phi(x_1)\phi(x_2)|\textrm{in, vac}\rangle \\
&=&\langle \textrm{in, vac}|T\phi(x_1)\phi(x_2)|\textrm{in, vac}\rangle\end{aligned}$$ $G_{--}$, $G_{+-}$ and $G_{-+}$ have analogous expressions. The sum over “out” states corresponds to the fact that no constraint has been placed on $\phi(x_f)$ in the path integral. And $\rho[\phi(x_i)]$ has given us an explicit representation of what we mean by the “in” vacuum state; for the Gaussian choice we have completed the equivalence to quantum field theory.
Thus the modification of the theory through the addition of the $\phi\chi^3$ term has replaced the classical correlation functions by their quantum counterparts. We see that the whole essence of quantum field theory (for a scalar field) lies in a particular value for the coefficient of the $\phi\chi^3$ term. The essence is not the $\hbar$, or the loop expansion, or renormalization, since all this appears in the classical theory. And as the coefficient $\alpha$ varies from 0 to 1, the theory extrapolates continuously from classical to quantum.
Quantum behavior at 2-loops\[S1\]
=================================
In this section we first isolate a truly quantum perturbative contribution to the self-energy, a contribution that is due to the $\phi\chi^3$ term. We will see explicitly that this contribution for $\alpha=1$ has the effect of cancelling the classical contribution to the imaginary self-energy $\Pi_i$ on mass shell. Then the width of the pseudoparticle goes to zero, and we end up with a stable particle as described by quantum field theory. But the main point here will be to continue our discussion of section 1, and obtain the nonlocal corrections to the same diagram that can occur in the classical theory.
A truly quantum contribution to the self-energy first appears at the 2-loop level. We return to the $(\phi,\chi)$ basis where a $\phi\chi^3$ term generates loops of $G^0_{\phi\chi}$ or $G^0_{\chi\phi}$ lines, thus allowing more than one $G^0_{\phi\chi}$ or $G^0_{\chi\phi}$ line to connect two vertices. We show the resulting classical and quantum pieces of the sunset diagram in Fig. (\[C\]). We have seen that the $\Pi_{\chi\chi}$ diagrams are just the imaginary parts of the $\Pi_{\phi\chi}$ ones, so we need only consider the top two diagrams.
We restrict ourselves to $1+1$ dimensions as we do in the lattice simulations to follow, and consider the classical 2-loop contribution to $\Pi_{\phi\chi}$, the first diagram in Fig. (\[C\]). Without Wick rotating and in finite volume $V=L$ this is $$\Pi_\textrm{classical}^\textrm{2-loop}(k)=\frac{6!^2\lambda^2}{2L^2}\sum_{p_1,q_1}\int\frac{dp_0}{2\pi}\frac{dq_0}{2\pi}\frac{\pi\delta(p_0^2-p_1^2-m^2)\pi\delta(q_0^2-q_1^2-m^2)}{(k_0-p_0-q_0+i\epsilon)^2-(k_1-p_1-q_1)^2-m^2}
.\label{e13}$$ After the temporal momentum integrals are done we are left with sums over the spatial momenta $p_1$ and $q_1$ whose discrete values are multiples of $2\pi/L$. These sums are infrared dominated. They are evaluated numerically while keeping the $\epsilon$ in the retarded propagator small but nonvanishing. Of interest is the on-shell self-energy, obtained by setting $(k_0,k_1)=(m,0)$. In a similar way we may consider the quantum contribution proportional to $\alpha$, the second diagram in Fig. (\[C\]).
Combining the results for the real and imaginary parts we have $$\begin{aligned}
\Pi_r^\textrm{2-loop}(m) & \propto & 1-\alpha/3, \\
\Pi_i^\textrm{2-loop}(m) & \propto & 1-\alpha.\end{aligned}$$ Thus in the quantum theory defined by $\alpha=1$ we find that $\Pi_r^\textrm{2-loop}(m)$ is 2/3 of its size in the classical theory. And $\Pi_i^\textrm{2-loop}(m)$ vanishes as expected. The classical and quantum pieces of $\Pi_r^\textrm{2-loop}(m)$ are each finite and quite insensitive to how the $V\rightarrow\infty$ and $\epsilon\rightarrow0$ limits are taken. For $\Pi_i^\textrm{2-loop}(m)$ on the other hand there is a weak divergence in each piece as $\epsilon\rightarrow0$, only canceling when $\alpha=1$. We note that this divergence is an artifact of the perturbative calculation, since the full $\Pi_i(m)$ is certainly finite in the classical theory. We will see this in the simulations. Although we will not pursue it here, one could consider replacing the free propagators by the full propagators $G_{\phi\phi}$ and $G_{\phi\chi}$ in the 2-loop calculation.
Now that we have isolated the “truly quantum” from the classical, we may consider the effect of making another choice for the ensemble in the classical theory. We return to the classical theory defined by $\alpha=0$, but now account for a modified perturbative expansion corresponding to the different choice of $\rho[\phi(x_i)]$ appearing in (\[e5\]). According to section 1, as we depart from the Gaussian prescription the free $n$-point functions are no longer simply expressed in terms of the free $2$-point function. We parametrized the nonlocality in the 4-point function by $\beta$, where $\beta=0$ and $\beta=1$ correspond to the Gaussian and fixed amplitude cases respectively. The question is how a nonvanishing $\beta$ affects the 2-loop calculation in (\[e13\]).
The Feynman rules for the two $G_{\phi\phi}^0$ lines in the diagram arose by writing the $D^0$ in (\[e14\]) as $$D^0(x-y)=\sum_{p_1}\frac{1}{V}\int \frac{dp_0}{2\pi} G_{\phi\phi}^0(p)e^{-ip(x-y)}
\label{e18}.$$ We notice that relative to (\[e14\]) this incorporates some inconsequential changes of sign of $p_1$ in the terms that have negative $p_0$. But to identify which contributions in (\[e13\]) need to be modified, we should undo these changes of sign by replacing $p_1$ inside the sum in (\[e18\]) by $\mathrm{sign}(p_0)p_1$, and the same for $p_1$ and $q_1$ in (\[e13\]). Then the rule from section 1 is to reduce the terms with $p_1=q_1$ in the discrete momentum sums by a factor of $(1-\beta/2)$. The net result is easier to describe in terms of the two 2-vectors $p_\mu$ and $q_\mu$, now with the sign factors included in their definition. The rule is to reduce by $(1-\beta/2)$ the contributions where the 2-momenta of the two $G_{\phi\phi}^0$ lines satisfy either $p_\mu=q_\mu$ or $p_\mu=-q_\mu$.[^3]
But this is not all, since what is directly measurable is the 2-point function rather than the self-energy. And one of the two external lines of the sunset diagram contribution to the 2-point function must be a $G_{\phi\phi}^0$ line (second diagram in Fig. (\[D\])). The same is true at 1-loop (first diagram in Fig. (\[D\])), which in this case implies that there are two $G_{\phi\phi}^0$ lines in that diagram. Thus there is also a nonlocal correction to the 1-loop contribution to the 2-point function. For an external line at vanishing spatial momentum, the correction involves the zero momentum contribution to the loop. This then is a factor of $(1-\beta/2)$ smaller than usual while all the other modes contribute in the loop as usual. It can easily be checked that this nonlocal correction at 1-loop will vanish in the $V\rightarrow\infty$ limit for fixed mass.
Returning to the sunset diagram, the external $G_{\phi\phi}^0$ line means that there are three $G_{\phi\phi}^0$ lines in this contribution to the 2-point function. Thus when either of the internal $G_{\phi\phi}^0$ lines has vanishing spatial momentum, these contributions should also be reduced by $(1-\beta/2)$. Unlike the 1-loop case, we find that these corrections, along with the previous $p_\mu=q_\mu$ or $p_\mu=-q_\mu$ corrections, do not vanish in the $V\rightarrow\infty$ limit.[^4] They produce a finite correction to $\Pi_r^\textrm{2-loop}(m)$ as long as the limit $\epsilon\rightarrow0$ is taken before the limit $V\rightarrow\infty$ (in the opposite order it vanishes). More precisely $\epsilon$ must be sufficiently small compared to lattice spacing $2\pi/L$ of the momentum space lattice used to evaluate the sums. The finite correction illustrates how the singularity structure of momentum loop integrals, arising from $\epsilon\rightarrow0$, can offset the naive volume suppression of the correction.
The actual result from numerical analysis is $$\begin{aligned}
\Pi_r^\textrm{2-loop}(m) & \propto & 1-\beta/2, \\
\Pi_i^\textrm{2-loop}(m) & \propto & 1-\beta/2.\label{e20}\end{aligned}$$ Each of the three ways of having pairwise coincident momentum in the three $G_{\phi\phi}^0$ lines contribute equally in $\Pi_r^\textrm{2-loop}(m)$, whereas only the case of coincident momentum on the two internal lines contributes in $\Pi_i^\textrm{2-loop}(m)$. But since each result is proportional to $1-\beta/2$, this shows that *only* the coincident momentum terms contribute; the contributions without coincident momenta vanish!
We see that $\Pi_r^\textrm{2-loop}(m)$ can be reduced from the Gaussian classical value as in the quantum theory. $\Pi_i^\textrm{2-loop}(m)$ is still plagued with the overall $\epsilon\rightarrow0$ divergence, so result (\[e20\]) is not particularly useful. To see how this quantity is actually behaving we turn to the lattice simulations of the classical theory ($\alpha=0$). We will be comparing the Gaussian $\beta=0$ and the fixed amplitude $\beta=1$ cases.
Lattice simulations
===================
We wish to obtain both the real and imaginary parts of the self-energy through a direct lattice study of interacting classical fields in $1+1$ dimensions. The details of the simulations are found in [@C; @D]. They are performed by numerically evolving each configuration according to the full field equations in real time. For lattice spacing $a$, the spatial size of the lattice is $Na$ where $N=256$ unless otherwise specified. The definition of the initial configurations requires a mass parameter, and this is chosen to match the physical mass as determined by the simulation. The expectation values of products of fields are directly obtained in coordinate space. An accurate determination of the mass corrections can be more easily determined from time-like rather than space-like correlators, since the latter falls exponentially. We will consider two time-like correlators, where the first one involves the zero mode only, $$\begin{aligned}
D_\mathit{zm}(t)&=&\int\frac{dp_0}{2\pi} \left. G_{\phi\phi}(p)\right|_{p_1=0}e^{-i p_0 t},\\
D_t(t)&=&\frac{1}{L}\sum_{p_1}\int\frac{dp_0}{2\pi}G_{\phi\phi}(p)e^{-i p_0 t}
.\end{aligned}$$ In the free case with $G_{\phi\phi}\rightarrow G_{\phi\phi}^0=\hbar\pi\delta(p^2-m^2)$ we have $$D_\mathit{zm}^0(t)=\frac{1}{2m}\cos(m t),\;\;\;\;\;D_t^0(t)=\frac{1}{L}\sum_{p_1}\frac{1}{2\omega}\cos(\omega t)
.$$ When we instead use $G_{\phi\phi}$ from (\[e15\]), and use a narrow width approximation where the self-energy functions are replaced by their on-shell values, then $D_\mathit{zm}(t)$ takes the form of an oscillation with decaying amplitude. This is the standard manifestation of plasmon damping [@therm], where the plasmon damping rate is $\gamma=\Pi_i(m)/2m$.
We thus determine the plasmon decay rate $\gamma$ by fitting the resulting decaying form of $D_\mathit{zm}(t)$ to what emerges from the simulation. We compare $\gamma$ for the Gaussian and fixed amplitude cases for a range of $ma$. For $(ma)^{-1} \lesssim5$ we find that $\gamma$’s are comparable. For larger $(ma)^{-1}$ the two $\gamma$’s rapidly depart from each other, and for $(ma)^{-1}=(15, 23, 30)$ we find that the $\gamma$ in the fixed amplitude case is $\approx (6, 28, 45)$ times *smaller* than for the Gaussian case.[^5] This is for small coupling where the leading 2-loop contribution should dominate.[^6] Thus we once again see that the choice of ensemble has a significant affect at the 2-loop level. Given that $\gamma$ should vanish in the quantum limit, we see that the fixed amplitude ensemble is causing the lattice simulation to move towards quantum-like behavior.
We should stress that “fixed amplitudes” refers only to how the amplitudes are chosen in the initial configurations; after the initial time the amplitude of each mode is free to evolve as the dynamics dictates. The large reduction in the plasmon decay rate indicates that these configurations are more stable; thermalization processes are much slower than for Gaussian classical simulations. When we next study the real part of the self-energy we are forced to use the fixed amplitude ensemble, since it is the slow thermalization that makes the investigation of this quantity possible.
$\Pi_r^\textrm{2-loop}(m)$ is more difficult to extract than $\Pi_i^\textrm{2-loop}(m)$, since there is also the real 1-loop contribution. The latter is not infrared dominated (it is log divergent in the continuum) and thus completely dominates the 2-loop contribution for small coupling. To isolate the 2-loop contribution, values of $\Pi_r(m)$ may be extracted for a range of the effective dimensionless coupling $\lambda/m^2$ so that both the 1- and 2-loop contributions can be fit simultaneously to the measured linear plus quadratic dependence. This procedure was used in [@D], where its relation to the 2-loop gap equation was described.
We shall adopt a complementary procedure here and will vary $\hbar$ rather than $\lambda/m^2$, since the 1-loop and 2-loop diagrams are proportional to $\hbar$ and $\hbar^2$ respectively. In the classical theory $\hbar$ originates in the normalization of the mode amplitudes, so we can just vary this overall normalization. It is a good check on the simulation to find that these two methods give similar results. As we vary $\hbar$ we also vary the bare mass $m_0$ in the field equation so as to hold the physical mass $m$ fairly constant. It is the difference between $m$ and $m_0$ that is fit to terms linear and quadratic in $\hbar$, and thus obtaining the 1- and 2-loop contributions to $\Pi_r(m)$.
For this analysis we use the 2-point function $D_t(t)$ rather than $D_\mathit{zm}(t)$. In $1+1$ dimensions the form of $D_t(t)$ reflects the contributions from both high and low momentum; it has both a rapidly oscillating component and an oscillation with $1/m$ time scale. Since $D_t(t)$ receives contributions from all the modes, it is expected to be a more reliable quantity than $D_\mathit{zm}(t)$.
We first consider the 1-loop mass correction, since the extracted value of this quantity displays a small deviation from the naively expected result, as was noted in [@D]. Indeed this discrepancy arises from the nonlocal correlations, as we have already described in the previous section as a finite volume effect. The effect increases for smaller $ma$, and amounts at most to about a 3% reduction in the 1-loop diagram. The observed and calculated values of this discrepancy agree, and this is a further indication that the simulation sees the nonlocal correlations. Returning to our fit of the linear and quadratic behavior, we now input our calculated linear behavior and use the data to extract the quadratic behavior only.[^7]
We compare the simulations to the explicit calculations of classical (Gaussian with $\alpha=0$) and quantum ($\alpha=1$) values of $\Pi_r^\textrm{2-loop}/m^2$, which at $\lambda/m^2=0.4$ are 0.022 and 0.015 respectively. In Fig. (\[A\]) we display the extracted values of $\Pi_r^\textrm{2-loop}$ divided by the classical value as a function of $(m a)^{-1}$. We see that as $ma$ decreases that $\Pi_r^\textrm{2-loop}$ gradually drops below the quantum value and becomes consistent with the calculated value of $1/2$ from the previous section.
In Fig. (\[B\]) we display the associated values of $\Pi_i^\textrm{2-loop}/m^2$, which was fit simultaneously to the same simulation data. In the context of the plasmon decay rate we have already noted how small this quantity is compared to its Gaussian classical value. In the latter case this quantity increases with increasing $(ma)^{-1}$, but with the fixed amplitude prescription it falls towards the vanishing quantum value.
Interpretation
==============
We have used lattice simulations at weak coupling to extract 2-loop effects, and have confirmed that the choice of ensemble can have a significant effect, by comparing the Gaussian and fixed amplitude prescriptions. We then focused on the latter and found interesting and almost quantum-like behavior in the real and imaginary 2-loop corrections for a range of $ma$. The question now is what is the meaning of this range of $ma$.
Some insight is provided by our direct evaluation of the 2-loop mass shift $\Pi_r^\textrm{2-loop}(m)$ in section \[S1\]. For the latter we found that we had to take the limit $\epsilon\rightarrow0$ before the $V\rightarrow\infty$ limit. If $V$ is measured with respect to $1/m$ then for the simulation with a fixed number of lattice points, increasing $V$ corresponds to increasing $m$. And $\epsilon$ corresponds loosely to the measured plasmon decay rate $\gamma$. Thus if the same order of limits is to apply in the simulation then there is an upper bound on $V$ or $m$, since $\gamma$, although small, is not exactly zero in the simulation. Thus we see why quantum-like behavior requires smaller $m$.
What is nontrivial is the fact that $\gamma$ is as small as it is, allowing nearly quantum-like behavior for a range of $ma$. Our direct evaluation in section \[S1\] did not anticipate this, suffering as it did with a $\epsilon\rightarrow0$ divergence. The real part of the self-energy was properly anticipated, but this basically *assumed* that $\gamma$ would turn out to be sufficiently small.
We note that the interesting behavior is occurring for values of $(ma)^{-1}$ that are smaller than where conventional finite volume effects are expected, which is when $(ma)^{-1}$ becomes closer to $N$. In fact finite volume effects do appear to show up in the simulation for $(ma)^{-1}\gtrsim 35$. There the mass extracted after long evolution times shows a tendency to drift downward. A drift down would result in spuriously large values of $\Pi_r^\textrm{2-loop}(m)$, and thus these finite volume effects behave oppositely to the observed departure of $\Pi_r^\textrm{2-loop}(m)$ from the Gaussian classical value.
To more directly test for conventional finite volume and discretization effects we performed a direct lattice calculation of the 2-loop graph from quantum field theory, after Wick rotation [@D]. The loop integrations become sums over the discrete Euclidean momenta on a $N\times N$ lattice. $$\begin{aligned}
\Pi^\mathrm{2\;loop\;latt}(k) & = & \frac{1}{N^{4}}\sum_{\{p_{1},p_2,q_1,q_2\}=-N/2}^{N/2-1}G(p_{1}+k,p_{2})G(q_{1},q_{2})G(p_{1}+q_{1},p_{2}+q_{2}),\\
G(p_{1},p_{2}) & = & \frac{1}{4\sin^2(\pi p_{1}/N)+4\sin^2(\pi p_{2}/N)+m^{2}}\end{aligned}$$ This yields the self-energy for space-like or vanishing momentum. Of interest is the $m$ dependence of the result, and we find for $N=256$ that $\Pi_r^\textrm{2-loop-latt}(0)/m^2$ varies by less than 3% for $10<(ma)^{-1}<48$. We may also compare this lattice result to a non-Wick-rotated calculation of the same diagram in the continuum limit, along the lines of section \[S1\] but evaluated at zero momentum. We find agreement to within a few percent in the same range of $ma$. Thus conventional discretization or finite volume effects are significantly smaller than the effects induced by the choice of ensemble.[^8]
We mention a few more details of the simulation. $D_t(t)$ is measured over a time interval equal to half the spatial size of the lattice $\Delta t=a N/2$. We evolved each configuration forward for a total time equal to $(n_\Delta+1)\Delta t$, and then measured $D_t(t)$ on each of the $\Delta t$ time intervals except the first.[^9]$^,$[^10] $n_\Delta=1$ for the largest value of $m$ and we scaled $n_\Delta$ as $1/m$ for smaller values of $m$. By averaging $D_t(t)$ over this many time intervals for each configuration we reduced the noise in the extracted masses. By using smaller $n_\Delta$ for larger $m$ we avoided the phenomena observed in [@C], namely a slight upward drift in the masses observed for increasing evolution time (if the evolution time was increased much beyond $(n_\Delta+1)\Delta t$). The drift is more pronounced for larger mass. In fact we suspect that the enhanced drift for larger mass is related to the observation that the system behaves more classically for larger mass; greater plasmon damping implies enhanced thermalization processes, meaning that our field configurations are less stable for larger mass.
Strong Coupling
===============
In this section we shall describe some evidence of quantum-like behavior occurring beyond the 2-loop level. At sufficiently strong coupling the quantum theory is known to have a broken phase where $\langle \phi \rangle\neq 0$, with a critical line in the $m_0$-$\lambda$ plane separating the two phases. This is usually defined as a line of fixed $g=\lambda/m_\mathrm{gap}^2$ where $m_\mathrm{gap}^2$ is the 1-loop gap mass. We may study this phenomena in the classical theory, where a priori there is no reason for the onset of $\langle \phi \rangle\neq 0$ to occur at a coupling $g$ similar to the quantum critical coupling.
A direct comparison of the quantum and classical lattice theories has the complication that the lattice regularizations are usually different. A square Euclidean lattice is often used for quantum Monte Carlo simulations, whereas the classical simulations are Lorentzian with the discretization in the time direction finer grained than the space direction. As described in [@D] there is a special line of constant $m_\mathrm{gap}^2$ in the $m_0$-$\lambda$ plane that is the same for the two regularizations; thus by comparing the location of the critical coupling on this line we may reduce the dependence on the lattice regularization. The result from the quantum lattice theory for the critical coupling is $g\approx10$ [@A], which on the special line corresponds to $\lambda\approx50/N^2a^2$.
In the classical simulation at strong coupling we no longer have the luxury of long simulation times, but there is still a window where quantum behavior can set in before thermalization effects dominate. We determine the value of $\langle \phi \rangle$ by averaging $\phi$ over space and over times between $Na/2$ and $3Na/2$, and then averaging over configurations. The mass parameter in the initial configuration is chosen to be the value that defines the special line, $m_\mathrm{gap}^2\approx 5/N^2a^2$. A constant is also added to the initial configuration to provide a nonzero $\langle \phi \rangle$. It is adjusted to match the most probable value that is measured at the later times, since this value should be close to the minimum of the associated effective potential. To find the most probable value at later times we find the peak of the histogram obtained by binning the values of the field. After iterating so that the input value of $\langle \phi \rangle$ matches this most probable value, we then determine $\langle \phi \rangle$ as the actual average of the field at the later times. In the absence of a nonzero most probable value, $\langle \phi \rangle$ is taken to vanish.
We start with the fixed amplitude prescription. In Fig. (\[E\]) we display three examples of histograms, with $g=(7,9,10)$, with a vanishing $\langle \phi \rangle$, a slightly nonvanishing $\langle \phi \rangle$, and a larger $\langle \phi \rangle$ respectively. From this we can see that the critical coupling is a little smaller than 9. We can test for the effect of lattice size by repeating this procedure for both a smaller $N=128$ and a larger $N=512$ lattice. Here we determine the couplings that give the same slightly nonvanishing $\langle \phi \rangle$ as the $g=9$, $N=256$ case; for $N=(128, 512)$ the corresponding couplings are $g\approx (6.5, 10)$. Thus the finite size effects are diminishing for the larger lattices. We see that these results are quite consistent with a critical coupling of $g\approx 10$ as found in the quantum theory. (A less detailed analysis of this effect was described in [@C]. In that reference another method to locate the critical line, involving the overlap of the time and space correlators, was perhaps less precise but gave similar results.)
Now we can consider the effect of using the Gaussian distribution prescription instead, keeping everything else the same. Here we find that the critical behavior shifts towards weaker coupling; the coupling that gives the same slightly nonvanishing $\langle \phi \rangle$ is now $g\approx3$ or 4. The clearly discernible difference between these two cases again shows the effect of the choice of ensemble, this time at strong coupling. And once again it is the fixed amplitude prescription that produces quantum-like behavior.
Comments
========
1\) In section 1 we noted that the amount of nonlocal correlations in ensembles of classical fields was controlled by the probability distribution for the amplitudes. For the 4-point function we parametrized the departure from locality by $\beta$, where a Gaussian distribution has $\beta=0$. In section 3 we described a class of Gaussian theories having a new interaction term with coefficient $\alpha$, for which $\alpha=1$ corresponded to quantum field theory. Thus the first class of theories is nonlocal and the second is nonclassical, and they could be thought of as different ways to deform the $\alpha=\beta=0$ theory which is both local and classical. All these theories have $\hbar$ and a loop expansion. In fact the two classes of theories belong to a larger class of theories in which both nonlocal and nonclassical features are introduced in varying amounts (both $\alpha$ and $\beta$ nonvanishing). But in this work we have focused on nonlocal effects in interacting classical theories and have shown that they can resemble the nonclassical effects of the corresponding local quantum field theories. These two classes of theories may have more in common than commonly thought.
2\) We observe that the existence of spatial dimensions are crucial for producing quantum-like behavior. In $0+1$ dimensions then there is just one site and one mode. Averaging over the phase and holding the amplitude fixed in this case gives results that are in no way quantum-like. In fact the 1-loop correction to the mass is $1/2$ its quantum-mechanical value, as can be deduced from our discussion of the correction to the 1-loop contribution in section \[S1\]. This can also be found through a direct perturbative solution of the $0+1$ dimensional field equation. Thus as the number of lattice points in the spatial direction becomes smaller, we would expect our results to move away from quantum-like behavior, and approach the non-quantum-mechanical behavior of the $0+1$ dimension limit. But in the other limit of fine-grained spatial dimension(s), a nonrelativistic limit of the theory could presumably be taken as done in quantum field theory, and in this way making an approach to something that has some resemblance to quantum mechanics [@M].
3\) The stability of the classical system and quantum-like behavior are both related to the vanishing of the imaginary self-energy on mass shell. We found some tendency for this vanishing in the lattice simulation even though there was a $\epsilon\rightarrow0$ divergence in this quantity in perturbation theory. Can the perturbative description be improved, possibly in some other classical theory?
4\) The 2-loop sunset diagram turned out to be maximally sensitive to the nonlocal correlations of classical fields. How general is that result?
5\) In the examples we have given, the quantum-like behavior has emerged in quantities that are infrared dominated. Is it essential that the theory itself be infrared dominated, and do asymptotically free gauge theories qualify?
Acknowledgments {#acknowledgments .unnumbered}
===============
I thank T. Hirayama for discussions and S. Jeon for patiently explaining the Schwinger-Keldysh formalism to me. This work was supported in part by the National Science and Engineering Research Council of Canada.
[1]{} T. H. Boyer, Phys. Rev. D11 (1975) 790 and 809. T. Hirayama and B. Holdom, Classical Simulation of Quantum Fields I, hep-th/0507126. G. Aarts and J. Smit, Phys. Lett. B393 (1997) 395; G. Aarts and J. Smit, Nucl. Phys. B511 (1998) 451. T. Hirayama, B. Holdom, R. Koniuk, T. Yavin, Classical Simulation of Quantum Fields II, hep-lat/0507014. A. H. Mueller and D .T. Son, Phys. Lett. B582 (2004) 279; R. D. Jordan, Phys. Rev. D33 (1986) 444. S. Yu. Khlebnikov, I. I. Tkachev, Phys. Rev. Lett. 77 (1996) 219; W. Buchmuller, A. Jakovac Phys. Lett. B407 (1997) 39; K. Blagoev, F. Cooper, J. Dawson, B. Mihaila, Phys. Rev. D64 (2001) 125003; A. Rajantie, P. M. Saffin and E. J. Copeland, Phys. Rev. D63 (2001) 123512; G. D. Moore, JHEP 0111 (2001) 021; G. Aarts, J. Berges, Phys. Rev. Lett. 88 (2002) 041603; S. Jeon, The Boltzmann equation in classical and quantum field theory, hep-ph/0412121. W. Loinaz and R. S. Willey, Phys. Rev. D58 (1998) 076003, hep-lat/9712008; T. Sugihara, JHEP 0405 (2004) 007, hep-lat/0403008. For some discussion on the violation of Bell’s inequalities in random field models see P. Morgan, J. Phys. A39 (2006) 7441, cond-mat/0403692.
[^1]: [email protected]
[^2]: Gaussian distributions also arise for classical fields in thermal equilibrium, but the ensembles we are considering are very far from thermal.
[^3]: We notice the Lorentz invariance of this rule.
[^4]: The contribution when all three spatial momenta vanish does vanish.
[^5]: We also checked that analogous behavior occurs on a larger lattice, $N=512$.
[^6]: These ratios are obtained by considering different couplings in the two cases that give comparable $\gamma$’s, and then accounting for the different couplings by multiplying the ratio of these $\gamma$’s by the inverse ratio of squared couplings.
[^7]: The fact that this was not done in [@D] accounts for the slightly different results there.
[^8]: Discretization could be affecting the results for $(ma)^{-1}\lesssim10$, but this is not the main region of interest in this work.
[^9]: The first time interval allows some time for the free initial configurations to evolve.
[^10]: In our use of $D_\mathit{zm}(t)$ to determine the plasmon decay rate, discussed above, we measured $D_\mathit{zm}(t)$ once over the whole time interval $n_\Delta\Delta t$. Unlike $D_\mathit{zm}(t)$, $D_t(t)$ drastically changes its behavior for time intervals larger than $\approx2\Delta t$.
|
---
abstract: 'Given an abelian group $\cG$ endowed with a $\T=\R/\Z$-pre-symplectic form, we assign to it a symplectic twisted group $*$-algebra $\cW_\cG$ and then we provide criteria for the uniqueness of states invariant under the ergodic action of the symplectic group of automorphism. As an application, we discuss the notion of natural states in quantum abelian Chern-Simons theory.'
---
[**ON THE UNIQUENESS OF INVARIANT STATES**]{}
[^1]
[**by**]{}
[ **Federico Bambozzi**]{}\
[*Mathematical Institute, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK*]{}\
email: [[email protected]]{}\
[ **Simone Murro**]{}\
[*Mathematisches Institut,* ]{}[*Universität Freiburg,* ]{} [*D-79104 Freiburg, Germany*]{}\
email: [[email protected]]{}\
\
#### Keywords: {#keywords .unnumbered}
twisted group algebra, invariant states, symplectic group.
#### MSC 2010: {#msc-2010 .unnumbered}
Primary: 46L30, 22D15; Secondary: 46L55, 47H25.\
Introduction
============
The study of invariant states for the action of a group of $*$-automorphisms on $*$-algebras is a well-established research topic. Indeed, the type of a factor $\mathfrak{R}$ can be inferred by analyzing which and how many states are invariant, see [@Kadison; @stormer1; @stormer67; @StormerJFA; @StormerACTA]. More precisely, if there exists a unique invariant state $\omega$, then $\mathfrak{R}$ is of type III or $\omega$ is a trace and $\mathfrak{R}$ is of type II$_1$ or I$_n$ with $n<\infty$, [@Hugenholtz]. When the group of $*$-automorphism $\mathcal{E}$ acts ergodically on $\mathfrak{R}$ (or more generally on a Von Neumann algebra), Kovács and Szücs proved in [@KovacsSzucs] that there exists a unique $\mathcal{E}$-invariant state. Ten years later, Størmer showed in [@Stormer] that, under the additional assumption that $\mathcal{E}$ is a locally compact abelian group, the unique $\mathcal{E}$-invariant state is a trace state. For several years it has been an open problem if the same results hold with weaker assumptions ([@OPK]) and a positive answer was given in [@HLS], where it was shown that if $\mathcal{E}$ is a compact ergodic group of automorphism acting on a unital $C^*$-algebra, the unique $\mathcal{E}$-invariant state is a trace. Let us underline that the existence and uniqueness of a $\mathcal{E}$-invariant state on a $C^*$-algebra (which is not necessary a Von Neumann algebra) is guaranteed by the fact that $\mathcal{E}$ is compact, see [@stormer1].\
In most of the models inspired by mathematical physics, the group of $*$-automorphisms is not compact nor Abelian, in abelian Chern-Simons theory, as studied in [@DMS], $\mathcal{E}$ coincides with the symplectic group of automorphism. Nonetheless, it would be desirable to classify all the invariant states inasmuch they are interesting from mathematical and physical point of views. Indeed, they provide a ‘noncommutative generalization’ of the invariant measures in ergodic theory and the representations of $C^*$-algebras are implemented by unitary representations of $\mathcal{E}$ acting on a Hilbert space. From a physical perspective instead, they represent equilibrium states in statistical mechanics, see [@Araki; @DKS; @HHW; @KR]. In [@BMP] the authors classified invariant states by the symplectic $*$-group of automorphisms on the group $*$-algebras that define irrational non-commutative tori using elementary, and mostly algebraic, methods. More precisely, it was shown that for irrational rotational algebras the only state invariant under the action of the symplectic group is the canonical trace state. These $*$-algebras are obtained as twisted group algebras for the abelian groups $\Z^{2n}$ equipped with a symplectic form and the action of the symplectic group $\Sp(2n,\Z)$ on $\Z^{2n}$ can be lifted to an ergodic group of $*$-automorphisms. In the present paper, we generalize this result for many other twisted group algebras. To this end, in Section \[Sec: SAB\] we introduce the abstract notion of a $\cR$-pre-symplectic abelian group, an abelian group equipped with a $\cR$-pre-symplectic form, being $\cR$ a fixed abelian group where the pre-symplectic form takes values. This notion is considered for encompassing in a single concept different examples of symplectic forms arising in different contexts. From Section \[sec:group algebra\], we will restrict our attention to the case $\cR = \R/\Z$ (that we denote $\T$). After recalling how to assign a twisted group $*$-algebra $\cW_\cG$ to a $\T$-pre-symplectic abelian group $(\cG, \sigma)$, where $\sigma$ denotes a fixed pre-symplectic form, in Section \[sec:invariant states\] we study $\Sp(\cG)$-invariant states on $\cW_\cG$ (where $\Sp(\cG)$ is the symplectic group of $\cG$) for some specific $\cG$. A summary of the main results obtained is the following (see also Theorem \[thm:sum\_up\]):
\[thm:introduction\] Let $(\cG, \sigma)$ be a $\T$-pre-symplectic abelian group, then
1. if $\sigma$ is degenerate, then $\cW_\cG$ admits plenty of $\Sp(\cG)$-invariant states;
2. if $(\cG, \sigma)$ is symplectic, irrational (in the sense of Definition \[defn:irrational\]), the symplectic form is diagonalizable (in the sense of Notation \[not: diagonalization\]) and $(\cG, \sigma)$ satisfies a technical assumption (specified in Theorem \[thm:NCtori\_no\_torsion\]), then $\cG$ is torsion-free and the canonical trace is the unique $\Sp(\cG)$-invariant state on $\cW_\cG$;
3. if $(\cG, \sigma) \cong \bigoplus_{i \in I} ((\Z/n\Z)^2, \sigma_2)$, where $I$ has infinite cardinality and $\sigma_2$ is the canonical symplectic form, then the canonical trace is the unique $\Sp(\cG)$-invariant state on $\cW_\cG$.
Notice that in point 2. of Theorem \[thm:introduction\] it is not assumed that $\cG$ is finitely generated (otherwise the theorem reduces to the result of [@BMP]) and neither that it is a free module over $\Z$. We conclude the paper with some conjectures and by giving an application of our results to abelian Chern-Simons theory.
[Acknowledgements]{} {#acknowledgements .unnumbered}
--------------------
We would like to thank Sebastiano Carpi, Claudio Dappiaggi, Nicoló Drago, Francesco Fidaleo, Kobi Kremnizer and Vincenzo Morinelli for helpful discussions related to the topic of this paper.
Notations and conventions {#notations-and-conventions .unnumbered}
-------------------------
- The set of prime numbers is denoted by ${{\mathbb P}}$.
- For any $p\in {{\mathbb P}}$, we denote with $\F_p$ the finite field $ \Z / p \Z$, with $\Q_p$ the field of $p$-adic.
- The symbol $\K$ denotes one of the elements of the set $\{\Z,\Q,\R,\C,\F_p,\Q_p\}$.
- With $\T$ we denote the torus $\R/\Z$.
- $\cG$ and $\cR$ denote abelian groups.
- $\sigma_{2 n}$ denotes the *canonical symplectic form* as defined in Example \[exa:prototypical\].
Symplectic abelian groups {#Sec: SAB}
=========================
A *$\cR$-symplectic form* on $\cG$ is a map of abelian groups $\sigma: \cG \times \cG \to \cR$ that satisfies the following properties:
1. *Bilinearity:* For any $x,y,x',y' \in \cG$, it holds that $$\sigma(x + y, x' + y') = \sigma(x, x') + \sigma(x, y') + \sigma(y, x') + \sigma(y, y').$$
2. *Skew-symmetricity:* For all $x \in \cG$, it holds that $\sigma(x,x) = 0$.
3. *Non-degeneracy*: If $\sigma(x,y) = 0$ for all $y \in \cG$, then $x = 0$.
If we drop the requirement that $\sigma$ is non-degenerate we say that the form is *$\cR$-pre-symplectic*. By denoting with $0_\cG$ the unit in $\cG$, it is easy to see that for all $x,y\in\cG$, a $\cR$-pre-symplectic form satisfy the relations $$\sigma(0_\cG, x) = 0, \qquad \sigma(x, y) = - \sigma(y, x)\,.$$
\[defn:symplectic\_abelian\_group\] A *$\cR$-(pre-)symplectic abelian group* is a pair consisting of an abelian group $\cG$ together with a $\cR$-(pre-)symplectic form $\sigma:\cG\times\cG\to\cR$.
\[rmk:quasi\_symplectic\_groups\] We remark that in the case when $\cG$ is a discrete abelian group and $\cR=\T$, our definition of $\cR$-symplectic abelian group coincides with the definition of *quasi-symplectic spaces* given in [@LP].
Clearly, Definition \[defn:symplectic\_abelian\_group\] is not the most general one. Namely, it is possible to generalize it by considering pairs consisting of a module ${\tt M}$ over a base ring ${\tt R}$ and a skew-symmetric, non-degenerate ${\tt R}$-bilinear form on ${\tt M}$ with values on a fixed ${\tt R}$-module. We are not interested in developing this version of the theory in this work, although it can be done by easily adapting our discussion. Before proving some properties of $\cR$-symplectic abelian groups, we give some examples which are present in the literature.
\[exa:prototypical\] The prototypical example of a $\cR$-symplectic abelian group is obtained by endowing $\K^{2 n}$ with the standard $\K$-symplectic form $\sigma: \K^{2 n}\times \K^{2 n} \to \K$ (in this case $\cR = \K$ as an abelian group). After fixing a base for $\K^{2n}$ and the canonical scalar product $<\cdot\,,\cdot>_{\K^{2n}}$, the $\K$-symplectic forms read $$\sigma(x,y)=<x,\sigma_{2n} \,y>_{\K^{2n}}
\qquad \text{with}\qquad
\sigma_{2n}=\begin{pmatrix}
0 & \Id_n \\
-\Id_n & 0
\end{pmatrix}$$ and $\Id_n$ the $n\times n$ identity matrix. The $\K$-symplectic abelian groups have been used in several topics, e.g. for $\K=\Z$ in noncommutative geometry [@BMP; @DegliEspositi] and in abelian Chern-Simons theory [@DMS], for $\K=\R,\C$ in quantum mechanics [@AS; @FV], in symplectic geometry and deformation quantization [@defquant], for $\K=\F_p$ in modal quantum theory [@SW1; @SW2] and for $\K=\Q_p$ in $p$-adic quantum mechanics [@Zel].
With a slight abuse of notation, we refer to $\sigma_{2n}$ as the *canonical symplectic form*.
Arithmetic geometry is another source of examples of symplectic abelian groups.
\[exa:weil\_pairing\] The Tate modules associated to an elliptic curve equipped with the *Weil pairing* are $\T$-symplectic abelian groups. Explicitly, let $E$ be an elliptic curve defined over $\C$ and let $l \in \N$. The $l$-torsion points of $E$ are denoted by $E[l]$ and it is well-known that $E[l] \cong (\Z/l\Z)^2$. The Weil pairing on $E[l]$ is defined by the map $$e_l: E[l] \times E[l] \to \T$$ given by $${ e_l((a,b),(c,d)) = e^{2 \pi {{\imath}}\frac{(ad - bc)}{l}}}.$$ It is easy to check that $e_l$ is bilinear, non-degenerate and skew-symmetric. Notice also that $e_{m l}$ is compatible with $e_l$, for any $m \in \N$, in the sense that $$e_{m l}((a,b),(c,d)) = e_l ((m a, m b),(c, d)) = e_l ((a, b),(m c, m d)).$$ Hence, fixing a prime $p$, one can pass to the limit and obtain a Weil paring $$e_p: \limpro_{n} E[p^n] \times \limpro_{n} E[p^n] \to \T$$ on the Tate modules $T_p(E) = \underset{n}\limpro E[p^n]$.
As we shall see in the next example, not every symplectic abelian group appearing in natural examples is finite-dimensional.
\[exa:QFT\] Consider a $\K^n$-vector bundle $E$ over an oriented manifold $M$ and denote the space of compactly supported sections with $\Gamma_c(E)$. Then $\big(\Gamma_c(E) \oplus \Gamma_c(E),\sigma\big)$ forms a $\K$-symplectic abelian group, where $\sigma$ is given by $$\label{eq:sigma_QFT}
\sigma(f,g):=\frac{\int_M (f ^T \sigma_{2n}\,\, g )(x) \, vol_M}{\int_M vol_M}$$ being $\sigma_{2n}$ the canonical symplectic form and $vol_M$ the volume form of $M$. These symplectic abelian groups have been intensively used to quantize (bosonic) field theory on Lorentzian manifolds – see e.g. [@BDH; @gerard; @FR16; @ThomasAlex] for reviews or textbooks.
By defining a *morphism* $\phi: (\cG_1, \sigma_{\cG_1}) \to (\cG_2, \sigma_{\cG_2})$ to be a group homomorphism of the underlying groups that preserves the values of the symplectic forms, the class of $\cR$-pre-symplectic abelian groups form a category. We make this concept more precise in the next definition.
\[defn:cateogry\_symplectic\_groups\] We denote by $\bSymp_\cR$ (resp. $\bP\bSymp_\cR$) the category whose objects are $\cR$-symplectic (resp. $\cR$-pre-symplectic) abelian groups $(\cG_i, \sigma_{\cG_i})$ and whose morphisms are group homomorphisms $\phi: \cG_1 \to \cG_2$ for which the following diagram commutes $$\begin{tikzpicture}
\matrix(m)[matrix of math nodes,
row sep=2.6em, column sep=2.8em,
text height=1.5ex, text depth=0.25ex]
{ \cG_1 \otimes_\Z \cG_1 & \cR \\
\cG_2 \otimes_\Z \cG_2 & \\};
\path[->,font=\scriptsize]
(m-1-1) edge node[left] {$\phi \otimes \phi$} (m-2-1);
\path[->,font=\scriptsize]
(m-1-1) edge node[auto] {$\sigma_{\cG_1}$} (m-1-2);
\path[->,font=\scriptsize]
(m-2-1) edge node[right] {$\sigma_{\cG_2}$} (m-1-2);
\end{tikzpicture}.$$
The category of abelian groups fully faithfully embeds in $\bP\bSymp_\cR$ via the functor that associates to an abelian group $\cG$ the $\cR$-pre-symplectic abelian group $(\cG, \sigma_0)$ where $\sigma_0$ is the trivial pre-symplectic form that has value identically $0$, nevertheless $\bP\bSymp_\cR$ is not additive because the sum of two morphisms does not preserve the values of the pre-symplectic form, in general.
In analogy with symplectic geometry, for any subset $A \subset \cG$ of a $\cR$-pre-symplectic abelian group we define the *orthogonal* subset as $$A^\perp = \{ x \in \cG | \sigma(x, a) = 0, \forall a \in A \}.$$
We will use the following simple construction for producing new $\cR$-pre-symplectic abelian groups from known ones.
\[prop:direct\_sum\] Let $\{ (\cG_i, \sigma_i) \}_{i \in I}$ be a (small) family of objects of $\bP\bSymp_\cR$. Then the direct sum (coproduct) of this family is the $\cR$-pre-symplectic abelian group given by $$\cG = \bigoplus_{i \in I}(\cG_i, \sigma_{\cG_i}) = \Big(\bigoplus_{i \in I} \cG_i, \sum_{i \in I} \sigma_{\cG_i} \Big).$$ If all $\{ (\cG_i, \sigma_{\cG_i}) \}_{i \in I}$ are $\cR$-symplectic abelian groups then also the direct sum is a $\cR$-symplectic abelian group.
Using the fact that the tensor product functor commutes with colimits, as it is a left adjoint functor, we can immediately conclude that $$\cG \otimes_\Z \cG \cong \bigoplus_{(i,j) \in I \times I} \cG_i \otimes_\Z \cG_j,$$ as plain abelian groups. Moreover, since any element of $\displaystyle{\cG = \bigoplus_{i \in I} \cG_i}$ is zero in all but finitely many components, then we can define $\displaystyle{\sum_{i \in I} \sigma_{\cG_i}}$ in the following way: For any $g = (g_i)_{i \in I}, h = (h_i)_{i \in I} \in \cG$ $$(\sum_{i \in I} \sigma_{\cG_i})(g,h) = \sum_{i \in I} \sigma_{\cG_i}(g_i, h_i).$$ The latter sum is always finite and therefore well-defined. By the universal property of $\cG$, there are canonical morphisms $\iota_i: \cG_i \to \cG$ which induce morphisms $\iota_i \otimes \iota_i: \cG_i \otimes \cG_i \to \cG \otimes \cG$ by functoriality. We need to check that $\iota_i$ are morphisms of $\cR$-pre-symplectic abelian groups. This amounts to show that the diagram $$\begin{tikzpicture}
\matrix(m)[matrix of math nodes,
row sep=2.6em, column sep=2.8em,
text height=1.5ex, text depth=0.25ex]
{ \cG_i \otimes \cG_i & \cR \\
\cG \otimes \cG & \\};
\path[->,font=\scriptsize]
(m-1-1) edge node[left] {$\iota_i \otimes \iota_i$} (m-2-1);
\path[->,font=\scriptsize]
(m-1-1) edge node[auto] {$\sigma_{\cG_i}$} (m-1-2);
\path[->,font=\scriptsize]
(m-2-1) edge node[right] {$\underset{i \in I}\sum \sigma_{\cG_i}$} (m-1-2);
\end{tikzpicture}$$ is commutative. But this follows immediately by the definition of $\displaystyle{\sum_{i \in I} \sigma_{\cG_i}}$ and by the fact that its restriction on the image of $\iota_i$ is equal to $\sigma_{\cG_i}$. Hence, it follows easily that $\cG$ has the universal property of the coproduct in $\bP\bSymp_\cR$. It is also immediate to check that if $\{ (\cG_i, \sigma_{\cG_i}) \}_{i \in I}$ are $\cR$-symplectic abelian groups then also $\cG$ is a $\cR$-symplectic abelian group.
Given a $\cR$-symplectic abelian group, it is important to study subgroups which are compatible with the symplectic structure, in the sense that non-degeneracy is preserved. This leads us to the following definition.
\[defn:sub\_symplectic\_abelian\_group\] Let $(\cG, \sigma)$ be a $\cR$-symplectic abelian group. A subgroup $\cH \subset \cG$ is called *$\cR$-symplectic abelian subgroup* if the symplectic form $\sigma$ restricts to a symplectic form on $\cH$.
Definition \[defn:sub\_symplectic\_abelian\_group\] is less obvious than one expects at first glance. Indeed, the restriction of a symplectic form to a subgroup of $\cG$ does not preserve, in general, the non-degeneracy property of the form.
\[prop:hyperbolic plane\] Let $(\cG, \sigma)$ be a $\cR$-symplectic abelian group and $x \in \cG$. Then, there exists a $y \in \cG$ such that the abelian subgroup of $\cG$ generated by $x$ and $y$ is a $\cR$-symplectic abelian subgroup of $\cG$.
We denote by $\lt x, y \gt \subset \cG$ the subgroup of $\cG$ generated by $x$ and $y$. Since $\sigma$ is non-degenerate, by definition for any $x \in {\cG}$ there exists a $y$ such that $\sigma(x, y) \ne 0$. Consider such a $y \in \cG$. Then, any element in $z \in \lt x, y \gt$ can be written as $$z = n_x x + n_y y, \ \ \ n_x, n_y \in \Z$$ and for any such, non-null, element we can find a $z' \in \lt x, y \gt$ such that $$\sigma(z, z') \ne 0$$ by writing $z' = m_x x + m_y y$ and using bilinearity to get $$\sigma(n_x x + n_y y, m_x x + m_y y) = n_x m_x \sigma(x,x) + n_x m_y \sigma(x,y) + n_y m_x \sigma(y,x) + n_y m_y \sigma(y,y).$$ As by hypothesis $\sigma(x,y) \ne 0$ we get that it is always possible to find a $z'$ such that this expression is not zero.
Using the jargon of symplectic geometry, Proposition \[prop:hyperbolic plane\] can be restated by saying that each $x \in \cG$ is contained in a hyperbolic plane. Therefore, we give the following definition.
\[defn;hyperbolic\_plane\] We call *hyperbolic plane $\cH$* of $\cG$ any $\cR$-symplectic subgroup of $\cG$ given as in Proposition \[prop:hyperbolic plane\].
\[rmk:rank\_hyperbolic\_plane\] Notice that any hyperbolic plane $\cH \subset \cG$ is a finitely generated abelian group of rank $2$. It can be easily shown that if $\cG$ is torsion-free it is isomorphic to $\Z^2_r = (\Z^2, r \sigma_2)$, where for any $r\in\cR$ we use the notation $$r \sigma_2 = \begin{pmatrix}
0 & r \\
-r & 0
\end{pmatrix}.$$
Our notation in Remark \[rmk:rank\_hyperbolic\_plane\] is well-defined due to the fact that for $x_1, x_2 \in \cG$ the products $$\begin{pmatrix}
0 & r \\
-r & 0
\end{pmatrix}
\begin{pmatrix}
x_1 \\
x_2
\end{pmatrix} =\begin{pmatrix}
-r x_2 \\
rx_1
\end{pmatrix}$$ are well-defined elements of $\cR$ (although we cannot multiply two such matrices). Instead, it does not make sense to compute the determinant of $r\sigma_2$ as $\cR$ is just an abelian group (written additively) and the product $r (- r)$ is not defined. With the next proposition we investigate further the hyperbolic planes.
Consider $\cR = \Z$, $\Z^2_1 = (\Z^2, \sigma_2)$ and $\Z^2_2 = (\Z^2, 2 \sigma_2)$. Then $\Z^2_1$ and $\Z^2_2$ are not isomorphic as $\cR$-symplectic groups.
In order to prove our claim, we show that it does not exists a group homomorphism $\phi: \Z^2_1\to\Z^2_2$ which is also a morphism of $\Z$-symplectic groups. This can be understood via the following argument: Since any group homomorphism is defined by its action on the generators, it is enough to notice that the elements $(1, 0), (0, 1) \in \Z^2_1$ cannot be mapped to any element of $\Z^2_2$ while preserving the values of the symplectic form. This because $2 \sigma_2$ never takes the value $1$ on $\Z^2_2$. Therefore, $\Hom(\Z^2_1, \Z^2_2) = \void$.
The same kind of reasoning apply also for $\Z^2_{r_1}$ and $\Z^2_{r_2}$ for any $r_1 \ne r_2$ in any abelian group $\cR$.
Having introduced the notation for hyperbolic planes $\Z^2_r$ we can see that, in favorable conditions, general symplectic abelian groups can be written in terms of them. Before stating our results, we need a preparatory definition.
\[defn:finite\_abelian group\] We say that an abelian group $\cR$ is of *rank $1$* if every finitely generated sub-group of $\cR$ is cyclic, it is generated by one element.
Abelian groups of rank 1 have been completely classified:
- A torsion-free abelian group of rank 1 is either $\Q$ or a sub-group of $\Q$.
- A torsion abelian group of rank $1$ is either $\Q/\Z$ or a sub-group of $\Q/\Z$.
\[thm:symplectic\_diagonalization\] Let $(\Z^{2n}, \sigma)$ be a $\cR$-symplectic abelian group and suppose $\cR$ to be of rank $1$. Then $$\label{eq: diagonalization}
(\Z^{2 n}, \sigma) \cong (\Z^2_{r_1} \oplus \cdots \oplus \Z^2_{r_n})$$ for some $r_1, \ldots, r_n \in \cR$.
First of all, we notice that assigning a symmetric bilinear form $\sigma: \Z^{2 n} \times \Z^{2 n} \to \cR$ is equivalent to specify a $2 n \times 2 n$ matrix with coefficients in $\cR$, in a similar fashion of what we explained so far for the case $n = 1$ in Example \[rmk:rank\_hyperbolic\_plane\]. Indeed, since a linear morphism $\Z \to \cR$ is uniquely determined by the value of $1$, we have $$\begin{aligned}
\Hom_\Z(\Z^{2 n} \otimes_\Z \Z^{2 n}, \cR) &\cong \Hom_\Z(\Z^{4 n^2}, \cR) \cong \Hom_\Z(\Z, \cR)^{4 n^2} \cong \cR^{4 n^2}\end{aligned}$$ where we used the fact that $\Hom_\Z(\Z, \cR) \cong \cR$. Now consider the sub-group $$\cS = \sigma(\Z^{2 n} \otimes_\Z \Z^{2 n}) \subset \cR.$$ Since $\cS$ is a finitely generated abelian sub-group of $\cR$ it is either (abstractly) isomorphic to $\Z$ or to $\Z/m\Z$ for some $m \in \N$. Notice that we can consider $\sigma$ to belong in $\Hom_\Z(\Z^{2 n} \otimes_\Z \Z^{2 n}, \cS)$ and hence we can suppose that $\sigma$ has values on a ring. Since $\Z$ and $\Z/m\Z$ are principal ideal rings we can apply Theorem IV.1 of [@NM] to conclude that $(\Z^{2 n}, \sigma)$ can be written in the claimed form.
\[cor:symplectic\_diagonalization\] Theorem \[thm:symplectic\_diagonalization\] remains true if we drop the hypothesis that $\sigma$ is non-degenerate.
Theorem IV.1 of [@NM] is true also for degenerated symplectic forms for which it gives $(\Z^{2 n}, \sigma) \cong (\Z^2_{r_1} \oplus \cdots \oplus \Z^2_{r_{m}} \oplus \cH)$, with $m \le n$ and $\cH$ a subspace where the symplectic form is identically null.
\[not: diagonalization\] With an abuse of language, we say that the isomorphism is a *‘diagonalization’* of $\sigma$.
For the next theorem we need to introduce some notation, based on Theorem \[thm:symplectic\_diagonalization\].
Let $\sigma: \Z^{2n} \otimes \Z^{2n} \to \cR$ be a symplectic form valued on a rank $1$ group. By identifying $\cR$ with a sub-group of $\Q/\Z$ (or with a sub-group of $\Q$ in the case $\cR$ is torsion-free), we can assume without loss of generality that the symplectic form $\sigma$ can be written with a matrix of the form $$\bM_{i,j}(\sigma) = q_\sigma (\bm_{i,j})$$ where $q_\sigma \in \Q/\Z$ (or $\Q$) and $\bm_{i,j} \in \Z$.
The last piece of notation we need is given by the *external direct sum* of a $\cR_1$-pre-symplectic abelian group with a $\cR_2$-pre-symplectic one. Namely, if $\cG$ is an abelian group and $\sigma_1$ is a $\cR_1$-pre-symplectic form on $\cG$ and $\sigma_2$ is a $\cR_1$-pre-symplectic form (always defined on $\cG$), then $\cG$ equipped with the pre-symplectic form $$(\sigma_1 \boxplus \sigma_2)(x) = (\sigma_1(x), \sigma_2(x))$$ is a $\cR_1 \oplus \cR_2$-pre-symplectic abelian group. We denote the group so obtained by $(\cG, \sigma_1)\boxplus (\cG, \sigma_2)$.
\[thm:symplectic\_diagonalization\_2\] Let $(\Z^{2n}, \sigma)$ be a $\cR$-symplectic abelian group and assume that $\cR \cong \cR_1 \oplus \ldots \oplus \cR_m$ with $\cR_i$ of rank $1$. Let us rewrite the symplectic form as $\sigma = \sigma_1 \boxplus \ldots \boxplus \sigma_m$ where $ \sigma_i: (\Z^{2n}, \sigma) \to \cR_i $ and assume that $$\bM(\sigma_i) \bM(\sigma_j) = \bM(\sigma_j) \bM(\sigma_i)$$ for all $i,j$, where $ \bM(\sigma_i)$ denotes the matrix associated to $\sigma_i$ as explained above. Then $$(\Z^{2 n}, \sigma) \cong (\Z^{2n}, \sigma_1) \boxplus \cdots \boxplus (\Z^{2n}, \sigma_m)$$ where each $(\Z^{2n}, \sigma_i)$ is diagonal as in Theorem \[thm:symplectic\_diagonalization\].
It is easy to see that the symplectic form $\sigma: \Z^{2n} \otimes \Z^{2n} \to \cR \cong \cR_1 \oplus \ldots \oplus \cR_m$ can always be written in the form $\sigma = \sigma_1 \boxplus \ldots \boxplus \sigma_m$. Notice that only $\sigma$ is supposed to be non-degenerate, and this does not imply that each $\sigma_i$ is non-degenerate. On account of Corollary \[cor:symplectic\_diagonalization\] we can assign to each $\sigma_i$ a matrix of the form $$\bM(\sigma_i) = q_{\sigma_i} (\bm_{j,l}).$$ Without any loss of generality, we can set $q_{\sigma_i} = \frac{1}{t_i}$ with $t_i \in \N$. In this way, if we write $t = \prod_{i = 1}^n t_i$ the matrices $${\widetilde}{\bM}(\sigma_i) = \frac{1}{t^{2n}} \left ( \left ( \frac{t}{t_i}\right) \bm_{j,l} \right )$$ are all integer valued (up to the factor $\frac{1}{t^{2n}}$), and they represent the same symplectic form associated to $\bM(\sigma_i)$, $\sigma_i$. But as all coefficients of ${\widetilde}{\bM}(\sigma_i)$ lie in $\Z[\, \frac{1}{t^{2n}}] \subset \Q$, we can simultaneously identify this subset of $\Q$ with $\Z$ by multiplication by $t^{2n}$ in all the copies of $\Q$ we have chosen. Once these identifications are done, we can assume that $\sigma$ is a symplectic form with values in $\Z^m$, whose diagonalization is equivalent to the existence of a base which diagonalizes all the matrices ${\widetilde}{\bM}(\sigma_i)$ at the same time, that is equivalent to $\bM(\sigma_i) \bM(\sigma_j) = \bM(\sigma_j) \bM(\sigma_i)$ for all $i,j$.
\[cor:symplectic\_diagonalization\_2\] Theorem \[thm:symplectic\_diagonalization\_2\] remains true if we drop the hypothesis that $\sigma$ is non-degenerate.
In the proof of Theorem \[thm:symplectic\_diagonalization\_2\] we used Corollary \[cor:symplectic\_diagonalization\] that does not require the symplectic forms to be non-degenerate.
\[ex:noncomm sympl form\] Theorem \[thm:symplectic\_diagonalization\_2\] cannot be generalized as it is easy to find skew-symmetric matrices which are not simultaneously diagonalizable by just considering two skew-symmetric matrices whose commutant is not zero. Such an example could be $$\sigma_1 = \begin{pmatrix}
0 & 1 & 1 & 0 \\
-1 & 0 & 0 & 0 \\
-1 & 0 & 0 & 1 \\
0 & 0 & -1 & 0 \\
\end{pmatrix}, \ \
\sigma_2 = \begin{pmatrix}
0 & 1 & 0 & 0 \\
-1 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 \\
0 & 0 & -1 & 0 \\
\end{pmatrix},$$ for which it holds $$\sigma_1\sigma_2-\sigma_2\sigma_1 = \begin{pmatrix}
-1 & 0 & 0 & 1 \\
0 & -1 & 0 & 0 \\
0 & -1 & -1 & 0 \\
0 & 0 & 0 & -1 \\
\end{pmatrix}- \begin{pmatrix}
-1 & 0 & 0 & 0 \\
0 & -1 & -1 & 0 \\
0 & 0 & -1 & 0 \\
1 & 0 & 0 & -1 \\
\end{pmatrix}= \begin{pmatrix}
0 & 0 & 0 & 1 \\
0 & 0 & 1 & 0 \\
0 & -1 & 0 & 0 \\
-1 & 0 & 0 & 0 \\
\end{pmatrix}.$$
For a generic abelian group $\cG$ there is a lot of freedom in defining symplectic forms on it, but a constrain must be taken into account, as we will see in the next lemma: If $\cG$ contains a torsion sub-group $\cG_\tors$, then the symplectic form cannot take values on a torsion-free group (the bilinear form must be degenerate).
\[lem:torsion\] Let $\cR_\free$ be a torsion-free abelian group. Then, for any abelian group $\cG$ such that $\cG_\tors \ne 0$, there does not exist a symplectic bilinear form $\sigma:\cG\times \cG \to \cR_\free$.
Suppose by contradiction that it is possible to define a symplectic form on $\cG$ with values in $\cR_\free$ $$\sigma: \cG \times \cG \to \cR_\free.$$ Then, given any $x \in \cG_\tors$, $y \in \cG$ there exists $n \in \Z - \{0\}$ such that $n x = 0$, hence by bilinearity $$n \sigma(x, y) = \sigma(n x, y) = \sigma(0, y) = 0 \then \sigma(x, y) = 0.\qedhere$$
The symplectic group and its orbits {#sec:orbit}
-----------------------------------
Studying the orbits of the action of $\Sp(\cG,\sigma)$ on $\cG$ will be important for the classification of $\Sp(\cG)$-invariant states, as we shall see in Section \[sec:invariant states\]. Let us begin by giving the definition of the symplectic group in our context.
\[defn:symplectic\_group\] Let $(\cG, \sigma)$ be a $\cR$-(pre-)symplectic abelian group. The *symplectic group* of $(\cG, \sigma)$ is defined as $$\Sp(\cG, \sigma) \doteq \Sp(\cG) \doteq \{ M \in \Aut(\cG) | \sigma \circ (M \otimes M) = \sigma \} \subset \Aut(\cG),$$ where $\Aut(\cG)$ is the group of automoprhisms of $\cG$ as an abelian group.
The definition just given is equivalent to say that $\Sp(\cG)$ is the group of automorphisms of $(\cG, \sigma)$ as an element of $\bP\bSymp_{\cR}$. Hence, it is obvious that $\Sp(\cG)$ is a sub-group of $\Aut(\cG)$.
The most classical example of symplectic group is the one associated to $\K^{2n}$ equipped with its canonical symplectic form, $\Sp(\K^{2n},\sigma_{2n})=\Sp(2n, \K)$. Notice that $\Sp(2n, \K)$ is a [sub-group]{} of the special linear group $\Sl(2n, \K)$, and for $n = 1$, it is precisely equal to $\Sl(2,\K)$. Moreover, for $\F_2$ we can notice that $\Sp(2n,\F_2)=\SO(2n,\F_2)$. More generally, these symplectic groups have been thoroughly studied for finite-dimensional vector spaces over any field $\K$, see e.g. [@A; @MT; @M; @T].
\[rmk:inversion\] Notice that the symplectic group is never trivial because the automorphism $\Inv: g \mapsto -g$ preserves symplectic form. Indeed, by the linearity of $\sigma$ one has that $$\sigma(-x, -y) = -(-\sigma(x,y))) = \sigma(x,y), \ \ \forall x,y \in \cG.$$
We are now ready for describing the orbits of $\Sp(\cG)$ for some important special cases. We introduce the following notion.
\[defn:primitive\_element\] An element in $x = (x_i) \in \Z^n$ is called *primitive* if $\gcd(x_i) = 1$.
Clearly any element $x = (x_i) \in \Z^n$ can be written as $\gcd(x_i) y$, with $y$ primitive.
\[lem:Orbite\_su\_Z\] Suppose that $r \in \cR$ is not a torsion element. Then, the classical symplectic group $\Sp(2 n,\Z)$ is the symplectic group of $(\Z^{2 n}, r \sigma_{2 n})$ for any $r \in \cR$ and any abelian group $\cR$. A set of representatives for the orbits of the action of $\Sp(2 n,\Z)$ on $\Z^{2 n}$ is given by $\cE = \{j y \, | \, j \in \N\}$ where $y$ is a fixed primitive element.
The first assertion is easy to check and a proof the fact that the action of $\Sp(2 n,\Z)$ is transitive on primitive elements can be found in Example 5.1 (ii) of [@MS]. The characterizations of the orbits directly follows form this fact because elements of $\Sp(2 n,\Z)$ have determinant $1$.
If $r \in \cR$ is a torsion element, then $(\Z^{2 n}, r \sigma_{2 n})$ can have more automorphisms than the ones coming from $\Sp(2 n,\Z)$, for $r=0$ (the trivial symplectic bilinear form).
\[lem:Orbite\_su\_Z\_razionali\] If $r \in \cR$ is a torsion element and $\cG = (\Z^{2 n}, r \sigma_{2 n})$, then $$\Sp(2 n,\Z) \subset \Sp(\cG) \subset \Sl(2 n,\Z).$$
The inclusion $\Sp(\cG) \subset \Sl(2 n,\Z)$ is obvious. The inclusion $\Sp(2 n,\Z) \subset \Sp(\cG)$ follows by standard computations.
For finite groups the situation is different and we show that for $\F_p^2$ there are only two orbits: one fixed point and its complement. Even if it is a standard result, we would like to recall it.
\[lem:Orbita\_su\_F\_p\] The classical symplectic group $\Sp(\F_q, 2)$ is the symplectic group of $(\F_q^2, r \sigma_2)$ for any $r \in \cR$ and any abelian group $\cR$. Moreover, for any elements $m,n \in \F^2_p$, different from $(0,0)$, there exists a $\Theta \in \Sp((\F^2_p, \sigma))$ such that $\Theta n = m$.
The first claim easily follows from Proposition \[lem:Orbite\_su\_Z\]. For the second claim let $m,n\in\F^2_p$ of the form $m=(m_1,m_2)$ and $n=(n_1,n_2)$ and consider a generic ${\Theta} \in \Sp(\F^2_p)=\Sl(\F^2_p)$ satisfying $$m=\Theta n = \begin{pmatrix} a n_1 + b n_2 \\ c n_1 + d n_2 \end{pmatrix}\,.$$ Since $\F_p$ is a field, and assuming that both $m$ and $n$ are not the null vector, we can always find $a,b,c,d \in \F_p$ which solve the equations $$\begin{aligned}
& a n_1 + b n_2= m_1 \qquad \qquad c n_1 + d n_2= m_2 \qquad \qquad a d - b c= 1 \,.\end{aligned}$$ Indeed, if $n_1 \neq 0 \neq m_1$, we can set $b=0$, $a=m_1 n_1^{-1}$, $d=n_1m_1^{-1}$ and $c=(m_2-n_2n_1m_1^{-1})n_1^{-1}$; if $n_1 \neq 0 \neq m_2$ we can set $d=0$, $c=m_2 n_1^{-1}$, $b=n_1m_2^{-1}$ and $a=(m_1-n_2n_1m_2^{-1})n_1^{-1}$; if $n_2 \neq 0 \neq m_1$ we can set $a=0$, $b=m_1 n_2^{-1}$, $c=n_2m_1^{-1}$ and if $d=(m_2-n_1n_2m_1^{-1})n_2^{-1}$; and finally for $n_2 \neq 0 \neq m_2$, we can use $c=0$, $d=m_2 n_2^{-1}$, $a=n_2m_2^{-1}$ and $b=(m_1-n_1n_2m_2^{-1})n_2^{-1}$. This concludes our proof.
Lemma \[lem:Orbita\_su\_F\_p\] can be readily generalized to $(\F_p^{2 n}, r \sigma_{2n})$ but we postpone the proof of our claim to Lemma \[lemma:infinite\_torsion\_orbits\] where we prove a more general result. Also, the description of the orbits of $((\Z/N\Z)^{2 n}, r \sigma_{2n})$ can be easily reduced to the study of congruences as the ones of Lemma \[lem:Orbita\_su\_F\_p\].
The description of symplectic group of a generic $\cR$-symplectic abelian group is much more complicated. A classification of all possible groups that arise in this way seems infeasible, therefore we just discuss some examples that go beyond the classical cases described so far. In order to do that, we introduce some notation. Let $$\cG_{n_1, n_2, r} = (\Z^{2 n_1}, r \sigma_{n_1}) \oplus (\Z^{2 n_2}, \sigma_0)$$ where $\sigma_0$ is the trivial pre-symplectic form. Notice that if $r \in \cR$ is not a torsion element then $$\Sp(\cG_{n_1, n_2, r}) \cong \Sp((\Z^{2 n_1}, r \sigma_{n_1})) \times \Sl(\Z^{2 n_2}).$$
\[prop:torsionfree\_symplectic\_group\] Let $\cG = (\Z^{2n}, \sigma)$ be a $\T$-symplectic abelian group. Then, $\Sp(\cG)$ can be written as an intersection inside $\GL(2 n, \Z)$ of a finite number of conjugates of the symplectic groups of the form $\Sp(\cG_{n_{i, 1}, n_{i, 2}, r_i})$ for some $n_{i, 1}, n_{i, 2} \in \N$ and $r_i \in \T$.
Since $\Z^{2n}$ has finite rank we can suppose that $\sigma$ has values on a finite rank sub-group of $\T$, that we can write as $\cR_1 \oplus \cdots \oplus \cR_m$. Therefore, we can write $\sigma$ as an external direct sum of $\sigma_1 \boxplus \cdots \boxplus \sigma_m$ where each of $\sigma_k$ has values on a rank one abelian group. By Theorem \[thm:symplectic\_diagonalization\] each $\sigma_k$ is diagonalizable (although not necessarily non-degenerate) and hence $\Sp(\cG_{n_{k, 1}, n_{k, 2}, r_k})$ for some $n_{i, 1}, n_{i, 2} \in \N$ and $r_i \in \T$.
Now, suppose that we have chosen a basis of $\Z^{2n}$ such that $\sigma_1$ is diagonalizable. By definition, we can identify $\Sp((\Z^{2n}, \sigma_1))$ with some $\Sp(\cG_{n_{1, 1}, n_{1, 2}, r_1})$. Then, in general $\sigma_2$ is not diagonalizable in the chosen base but it becomes by a suitable change of base, $\sigma_2=M^t \rho M$ for some diagonal $\rho$. Consider $A\in\Sp(\Z^{2n},\sigma_2)$. Then we have $${ M^t \rho M = \sigma_2 = A^t \sigma_2 A = A^t M^t \rho M A \,,}$$ which implies $M A M^{-1} \in \Sp(\Z^{2n}, \rho)$ and therefore isomorphisms $$\Sp(\Z^{2n},\sigma_2) \cong M^{} \Sp(\Z^{2n}, \rho) M^{-1} \cong \Sp(\cG_{n_{2, 1}, n_{2, 2}, r_2})\,.$$ Clearly an element of $\GL(2 n, \Z)$ belongs to $\Sp(\Z^{2n}, \sigma_1 \boxplus \sigma_2)$ if and only if it belongs to $\Sp(\Z^{2n}, \sigma_1) \cap M^{} \Sp(\Z^{2n}, \rho) M^{-1}$. Iterating this reasoning finitely many times we get the claimed description of $\Sp(\Z^{2n}, \sigma_1 \boxplus \cdots \boxplus \sigma_m) \cong \Sp(\cG)$.
\[ex:conjugate sympl group\]
1. Let $\cG=(\Z^4,\sigma_1 \boxplus \sigma_2)$ with $\sigma_1$ and $\sigma_2$ as in Example \[ex:noncomm sympl form\]. Then $\sigma_1=M^T \sigma_2 M$ with $M$ given by $$M= \begin{pmatrix}
1 & 0 & 0 & 0\\
0 & 1 & 0 & 0\\
0 & 0 & 1 & 0\\
1 & 0 & 0 & 1
\end{pmatrix} \,.$$ Then,by Proposition \[prop:torsionfree\_symplectic\_group\], we have $$\Sp(\cG) = \Sp(\Z^4,\sigma_1)\cap M \Sp(\Z^4,\sigma_1) M^{-1} = \{ A\in \Sp(4,\Z)\, | \, M A M^{-1} \in \Sp(4,\Z) \}\,.$$ We show that $\Sp(\cG)$ is a “big" group. For simplifying computations, and without changing the outcome, up to isomorphism, we suppose that $$\sigma_1 = \begin{pmatrix}
0 & 0 & 0 & 1\\
0 & 0 & 1 & 0\\
0 & -1 & 0 & 0\\
-1 & 0 & 0 & 0
\end{pmatrix} = \begin{pmatrix}
0 & \Id_2 \\
-\Id_2 & 0
\end{pmatrix}, \ \ \sigma_2 = M^t \sigma_1 M,$$ so that we can use nice blocks representations of matrices. We can also write $$M = \begin{pmatrix}
\Id_2 & 0 \\
A & \Id_2
\end{pmatrix}, \ \ M^t = \begin{pmatrix}
\Id_2 & A \\
0 & \Id_2
\end{pmatrix}, \ \ M^{-1} = \begin{pmatrix}
\Id_2 & 0 \\
-A & \Id_2
\end{pmatrix}$$ where $$A = \begin{pmatrix}
0 & 0 \\
1 & 0
\end{pmatrix}.$$ It is well-known (see [@Stanek]) that $\Sp(\Z^4, \sigma_1)$ is generated by block matrices of the form $$T_S = \begin{pmatrix}
\Id_2 & S \\
0 & \Id_2
\end{pmatrix}, \ \ R_U = \begin{pmatrix}
U & 0 \\
1 & (U^t)^{-1}
\end{pmatrix}, \ \ D_Q = \begin{pmatrix}
Q & \Id_2 - Q \\
Q - \Id_2 & Q
\end{pmatrix}$$ where $S = S^t$, $\det(U)\ne 0$ and $Q$ is a diagonal matrix with only $0$’s and $1$’s. This generating set is far from being minimal and indeed it is more interesting as a set of elements of $\Sp(\Z^4,\sigma_1)$ than as a generating set. For describing elements of $\Sp(\cG)$ we might check which of the above elements of $\Sp(\Z^4,\sigma_1)$ are also in $M \Sp(\Z^4,\sigma_1) M^{-1}$. By straighforward computations one gets that $$M T_S M^{-1} = \begin{pmatrix}
\Id_2 - S A & S \\
-A S A & A S + \Id_2
\end{pmatrix}$$ and that $$(M T_S M^{-1}) \sigma_1 (M T_S M^{-1})^t =$$ $$= \begin{pmatrix}
-S(A + A^t)S & (S A^t)^2 + S A^t + \Id_2 - S A S A^t + S A \\
-(AS)^2 + A S A^t S - A S + A^t S - \Id_2 & -(AS)^2 A^t - A S A + A(S A^t)^2 + A^t S A^t
\end{pmatrix}\,.$$ If we write $$S = \begin{pmatrix}
s_1 & s_2 \\
s_2 & s_3
\end{pmatrix},$$ then, in order to have that last expression is equal to $\sigma_1$ it is necessary that $$S(A + A^t)S = \begin{pmatrix}
s_1 s_2 + s_2^2 & s_1 s_3 + s_2 s_3 \\
s_2^2 + s_2 s_3 & s_2 s_3 + s_3^2
\end{pmatrix} = 0\,.$$ This implies $s_1 = s_2 = s_3$, together with $$-(AS)^2 A^t - A S A + A(S A^t)^2 + A^t S A^t = \begin{pmatrix}
0 & s_3 \\
s_2 & s_1 s_2 + s_1 s_3
\end{pmatrix} = 0\,.$$ In particular we have $s_2 = s_3 = 0$. Therefore, none of the matrices $T_S$ belong to $\Sp(\cG)$. Similarly $$M R_U M^{-1} = \begin{pmatrix}
U & 0 \\
A U & (U^t)^{-1}
\end{pmatrix}$$ and $$(M R_U M^{-1}) \sigma_1 (M R_U M^{-1})^t = \begin{pmatrix}
0 & \Id_2 \\
-\Id_2 & A - U^t A^t (U^t)^{-1}
\end{pmatrix}.$$ Therefore, for $R_U \in \Sp(\cG)$, we need to impose the equation $$A - U^t A^t (U^t)^{-1} = 0$$ and if we write $$U = \begin{pmatrix}
u_1 & u_2 \\
u_3 & u_4
\end{pmatrix}$$ we get $$A - U^t A^t (U^t)^{-1} = \frac{1}{\det U}\begin{pmatrix}
u_1 u_2 & -u_1^2 \\
\det U + u_2^2 & -u_1 u_2
\end{pmatrix}.$$ Hence, we get the conditions $$u_1 = 0, \ \ u_2^2 = - \det U = u_2 u_3 \then u_2 = u_3.$$ Therefore, we get a two dimensional family of elements of $\Sp(\cG)$ parametrized by the matrices of the form $U$ with $u_1 = 0$, $u_2 = u_3 \ne 0$. The same computations for the matrices $D_Q$ are lengthy and we omit them, we just mention that one can show that none of the matrices $D_Q$ belongs to $\Sp(\cG)$. We conclude this example by remarking that the family of elements we described is a two dimensional family that should not describe all the elements of $\Sp(\cG)$ that is expected to be of dimension $5$ as it is described as an intersection inside $\GL(4, \Z)$, that is of dimension $16$, of $\Sp(4,\Z)$ with its conjugate by the matrix $M$, that are of dimension $10$ (here the are considered as schemes over $\Z$). A full description of the group $\Sp(\cG)$ requires an analysis that is too long to be given here.
2. By the description of $\Sp(\Z^{2n}, \sigma)$ given in Proposition \[prop:torsionfree\_symplectic\_group\] one expects $\Sp(\Z^{2n}, \sigma)$ to be very small if the rank of the image of $\sigma$ is large enough. Indeed, this already happens if the rank is three because, as we described $\Sp(\Z^{2n}, \sigma)$ as an intersection of closed subvarieties of $\GL(\Z^{2n}, \sigma)$ of codimension $2 n^2 - n$, then one expects to have a minimal generic intersection already when we intersect three of them as $6 n^2 - 3 n > 4 n^2$ if $n > 1$. In such cases $\Sp(\Z^{2n}, \sigma) = \{ \pm \Id \}$. But still, even when the rank of the image of $\sigma$ is large, many interesting examples with non-trivial symplectic groups are possible for specific choices of matrices of change of bases.
3. Pre-symplectic abelian groups can have a big automorphisms group. For example, if we consider $(\cG, \sigma_0)$, where $\sigma_0$ is the trivial pre-symplectic form, we obtain that $\Sp(\cG)$ is isomorphic to the group of automorphism of the abelian group $\cG$.
4. $\cG = (\Q^2, \sigma_2)$ is an example of a non-finetely generated $\cR$-symplectic abelian group and $\Sp(\cG) = \Sp(\Q, 2)$.
We refrain to study here the symplectic groups that arise in the case when the underlying abelian group is not finitely generated. The complications involved in the study of such cases go beyond the scope of the present work.
$\T$-symplectic abelian groups
------------------------------
In this section we restrict our attention to the case of $\T = \R/\Z$-valued symplectic forms. This is motivated by Lemma \[lem:torsion\] and by the fact that in many (quantum) physical applications, the usual $\K$-valued symplectic form $\sigma$ is composed with a character of the form $\chi: \K \to S^1 \subset \C$, in order to implement the so-called *canonical commutation relations*, [@MSTV].
\[prop:direct\_sum\_torsion\] Let $(\cG, \sigma)$ be a $\T$-symplectic abelian group such that $\cG$ is a torsion group and $$\cG \cong \cG_1 \oplus \cG_2$$ with $\cG_1$ and $\cG_2$ of coprime torsion. Then $\sigma$ restricts to a symplectic form both on $\cG_1$ and on $\cG_2$; moreover $\cG_2 = \cG_1^\perp$ and $\cG_1 = \cG_2^\perp$.
It is enough to show that for any $x \in \cG_1$ and $y \in \cG_2$ then $\sigma(x, y) = 0$ (where we are identifying $\cG_1$ and $\cG_2$ with their image in $\cG$ via the canonical morphisms). Indeed, by the non-degeneracy of $\sigma$ there exists $x' \in \cG$ such that $\sigma(x,x')\ne 0$, therefore this must necessarily be in $\cG_1$. The same thing is true for $y$.
It remains to prove the claim. Suppose that there exist $x \in \cG_1$ and $y \in \cG_2$ such that $\sigma(x, y) \ne 0$. Then, consider $n,m \in \Z$, coprime such that $n x = 0, m y = 0$. Then, $$\sigma(x, n y) = n \sigma(x, y) = \sigma(n x, y) = \sigma(0, y) = 0$$ and on the other hand $$\sigma(m x, y) = m \sigma(x, y) = \sigma(x, m y) = \sigma(x, 0) = 0.$$ But the element $\sigma(x, y)$ cannot be both of $n$ and $m$ torsion because $m$ and $n$ are coprime. Therefore, $\sigma(x,y)=0$.
\[cor:direct\_sum\_torsion\] Let $(\cG, \sigma)$ be a torsion $\T$-symplectic abelian group, then $$(\cG, \sigma) \cong \bigoplus_{p \in {{\mathbb P}}} (\cG_p, \sigma_p)$$ where $\cG_p$ is the $p$-primary part of $\cG$ and $\sigma_p$ the restriction of $\sigma$ to $\cG_p$.
This is an immediate application of Proposition \[prop:direct\_sum\_torsion\] using the fact that any torsion abelian group $\cG$ can be written as $$\cG \cong \bigoplus_{p \in {{\mathbb P}}} \cG_p,$$ where $\cG_p$ is the $p$-primary part.
Symplectic twisted group algebras {#sec:group algebra}
=================================
Twisted group algebras arise in a wide variety of situations in mathematics, *e.g.* in the study of representations of nilpotent groups and connected Lie groups [@Moore; @Packer; @Hanna], noncommutative differential geometry [@Connes], the study of continuous $C^*$-trace [@ER; @RW], the algebraic approach to quantum statistical mechanics [@OA1] and quantum field theory [@AQFT1; @AQFT2], and in the realization of octonions, Clifford algebras and multiplicative invariant lattices in $\R^n$ [@Albu1; @Albu2; @Albu3]. In this paper, we shall focus our attention to twisted group algebras arising from symplectic abelian groups. Since there exists an extensive literature on this topic, see e.g. [@OA2; @OA3], we just introduce them briefly.\
Let $(\cG,\sigma)$ be a $\T$-pre-symplectic abelian group. From now on we use the multiplicative notation for the operation of $\cG$. We will denote the idendity of $\cG$ with $1_\cG$. It is also convenient to embed $\T$ into $\C^\times$ as the complex number of modulus $1$ and we denote with $\Omega: \cG \times \cG \to \C^\times$ the map defined by $$\Omega(x, y) \doteq e^{\imath \pi \sigma(x,y)} .$$
Let $(\cG, \sigma)$ be a $\T$-pre-symplectic abelian group. Then, for any $x,y\in\cG_\tors$, $\Omega(x,y)$ is a root of unity.
As shown in the proof of Lemma \[lem:torsion\], if $x,y\in\cG_\tors$ then $\Omega(x,y)$ must be a torsion element of $\C^{\times}$. These are precisely the roots of unity.
We observe that since $\sigma$ is bilinear and skew-symmetric $\Omega$ defines a group $2$-cocycle. Indeed, for any $1_\cG,g_1,g_2,g_2\in\cG$ we have $\Omega(1_\cG,g_1)=\Omega(g_1,1_\cG)=1$ together with $$\begin{aligned}
\Omega(g_1, g_2) \Omega(g_1 g_2, g_3) &= \Omega(g_1, g_2 ) \big(\Omega(g_1, g_3) \Omega(g_2, g_3) \big)= \\
&= \big( \Omega(g_1, g_2 ) \Omega(g_1, g_3) \big) \Omega(g_2, g_3) = \Omega(g_1, g_2 g_3) \Omega(g_2, g_3)\,.\end{aligned}$$ Next, we denote by *twisted group algebra $\C[\cG]^\Omega$* the set of all finite $\C$-linear combinations $$\aa=\sum_{g\in\cG} \a_g \,g \qquad \text{with $\a_g\in \C$} \,,$$ endowed with the twisted product defined by $$\label{eq: twisted prod}
\aa\bb= \sum_{g\in\cG} \left(\sum_{h\in\cG } \Omega(g,h) \a_h \beta_{h^{-1} g}\right) g.$$ It is straightforward to verify that twisting the product by a (non-trivial) group $2$-cocycle makes the algebra $\C[\cG]^\Omega$ non-commutative and associative, although $\cG$ itself is a commutative group. Moreover, the isomorphism class of the twisted algebra so obtained depends only on the group cohomology class of the $2$-cocycle and any cohomology class in $H^2(\cG, \T)$ can be represented by a pre-symplectic form ([@Kleppner Theorem 7.1]). We shall see in Theorem \[thm:CTori\] that this twisting reduces considerably the number of $\Sp(\cG)$-invariant states on $\C[\cG]^\Omega$ when the form is non-degenerate.\
In order to distinguish positive elements, we promote $\C[\cG]^\Omega$ to a twisted group $*$-algebra by endowing it with the involution $*:\C[\cG]^{\Omega}\to \C[\cG]^{\Omega}$ given by $$\label{eq: invol}
\aa^*:= \sum_{g\in\cG} \overline{\a}_g g^{-1} \,.$$ Indeed, any positive element $\bb$ in a complex $*$-algebra $\cA$ can be written as $\bb=\aa^*\aa$ for a suitable $\aa \in \cA$ – for more detalis see e.g. [@dixmier].
We are thus ready to summarize this short discussion with the following definition.
We denote by $\cW_{\cG, \Omega} \equiv \cW_\cG$ the symplectic twisted group $*$-algebra (*-algebra*) obtained endowing $\C[\cG]^\Omega$ with the product and with the involution .
[When completed by a canonical $C^*$-norm, the twisted group algebras $\cW_{\cG,\Omega}$ are also called Weyl $C^*$-algebras or exponential Weyl algebras in the literature, see e.g. [@Moretti; @Robinson1; @Robinson2]. The reader should not confuse these algebras with the rings of differential operators with polynomial coefficients which are also called Weyl algebras, see e.g. [@Dixmier1; @Dixmier2].]{}
The natural settings for -algebras are quantum mechanics and bosonic quantum field theory. Since there are many different bosonic QFT, let us consider for simplicity the case of a scalar field $\phi$ satisfying the wave equation and denote with $Sol$ the space of solutions. Then, we have $$\cW_{\Omega_{QM}}=\C[C^\infty(\R^{6})]^{\Omega_{QM}} \qquad \text{ and } \qquad \cW_{\Omega_{QFT}}=\C[\mathcal{E}(Sol)]^{\Omega_{QFT}}\,,$$ where $\mathcal{E}(\cdot)$ denotes the space of complex linear functionals, the group 2-cocycles are given by $\Omega_{i}(\cdot,\cdot)=e^{-\imath\hbar \sigma_i(\cdot,\cdot)}$, $\sigma_{QM}$ is obtained using the canonical symplectic form of $\R^6$ as in Example \[exa:QFT\], while $\sigma_{QFT}$ by .
Notice that have implicitly assumed that the spacetime is $\R^4$. For a generic globally hyperbolic spacetime, namely a $n$-dimensional oriented Lorentzian manifold diffeormophic to $\R\times \Sigma$, being $\Sigma$ a Cauchy surface, the -algebras are respectively defined as $$\cW_{\Omega_{QM}}=\C[C^\infty(T^*\Sigma)]^{\Omega_{QM}} \qquad \text{ and } \qquad \cW_{\Omega_{QFT}}=\C[\mathcal{E}(Sol)]^{\Omega_{QFT}}\,,$$ being $T^*\Sigma$ the cotangent bundle of $\Sigma$. For more details, we refer to [@FR16]. It is interesting to notice that, while the representations of $\cW_{QM}$ are all unitary equivalent, according to the Stone-Von Neumann’s Theorem [@SVNthm], there exist plenty of inequivalent representations of $\cW_{QFT}$ according to Haag’s Theorem [@Hthm]. Finally, we observe that to each positive element of $\cW$ corresponds a unique physical observable associated to the theory.
\[ex:NCT\] One of the most studied objects in noncommutative geometry is the so-called *noncommutative torus*. It is defined as the universal $C^*$-algebra $\fA_\theta$ generated by unitaries $U,V \in C(\T^2,\C)$ satisying the *canonical commutation relations* $$\label{eq:CCR NCT}
UV = e^{2\pi{{\imath}}\theta} VU \,.$$ It is easy to see that $\fA_\theta$ is isomorphic to the $C^*$-completion of the -algebra $\cW_{\Z^2,\Omega}$, where $\Omega$ is the group 2-cocycle on $\Z^2$ given by $$\Omega(n,m)=e^{2\imath\pi \theta \sigma_2(n,m)} \qquad \theta\in \R\,,$$ being $\sigma_2$ the canonical symplectic form on $\Z^2$. Indeed, this twisted group $C^*$-algebra is generated by the unitary operators $U=(1,0)$ and $V=(0,1)$ acting on $\ell^2(\Z^2)$ $$U f(n,m)=e^{-2\pi\imath m} f(n+1,m) \qquad \text{and} \qquad V f(n,m)=e^{-2\pi\imath n} f(n,m+1)\,.$$ By a straightforward computation it is easy to see that this generators satisfy the canonical commutation relation .
\[rmk:functor\] Notice that the association $(\cG, \sigma) \mapsto \cW_{\cG, \Omega}$ is a functor from the category $\bP \bSymp_{\T}$ of $\T$-pre-symplectic abelian groups to the category of $*$-algebras over $\C$. Therefore, the action of the symplectic group $\Sp(\cG)$ on $\cG$ can be lifted to an action on $\cW_\cG$ by algebra automorphism via the formula $$\Phi_\Theta(\aa)=\sum_{g\in\cG} \a_g \,\Theta(g) \quad\qquad \text{with $\,\a_g\in \C\,$ and $\,\Theta \in \Sp(\cG)$.}$$ If the only invariant subspace of $\cW$ under the action of $\Phi_\Theta$, for all $\Theta\in \Sp(\cG)$, is span$_\C\{1_\cG\}$ then the action of $\Sp(\cG)$ is said *ergodic*.
\[prop:ergodic\] For a symplectic abelian group $(\cG,\sigma)$ the following statements are equivalent:
- The only finite orbit of $\Sp(\cG)$ on $\cG$ is the fixed point $1_\cG$;
- The action of $\Sp(\cG)$ on $\cW_{\cG}$ is ergodic.
Assume that there exists a finite $\Sp(\cG)$-orbit $\cO \subset \cG$. Then any element of the form $\aa=\sum_{g\in\cO} g$ is a fixed point for the action of $\Sp(\cG)$ on $\cW_{\cG}$, namely for any $\Theta\in\Sp(\cG)$ we have $\Phi_\Theta(\aa)=\aa$. By logic transposition, this implies that if the action is ergodic then finite orbits for the action of $\Sp(\cG)$ on $\cG$ do not exist.\
Conversely, if the action is not ergodic, then there exists at least an element $\aa\neq \a 1_\cG$, with $\a\in\C$, such that, for all $\Theta\in\Sp(\cG)$, it satisfies $\Phi_\Theta (\aa)=\aa$. Now write $\aa$ as the finite sum $\aa=\sum_{g\in\cO} g$ (this is always possible because all the non-zero coefficients of $\aa$ must be equal), with $\cO$ a finite subset of $\cG$. By what we wrote above we get $$\Phi(\aa)=\sum_{\Theta g \in\cO} \Theta g =\sum_{g\in\cO} g = \aa \qquad \forall \Theta \in \Sp(\cG)\,.$$ This implies in particular that $\Sp(\cG)$ maps elements in $\cO$ to elements in $\cO$, therefore it is a finite orbit.
$\Sp(\cG)$-invariant states {#sec:invariant states}
===========================
In this section, we keep the notation of the last section and we denote the -algebra associated to a $\T$-pre-symplectic abelian group $\cG$ simply as $\cW_{\cG}$. Let now $\omega$ be a *state*, namely a linear functional from $\cW_{\cG} $ into $\C$ that is positive, $\omega(\aa^*\aa)\geq 0$ for any $\aa\in\cW_{\cG} $, and normalized, $\omega(1_\cG)=1$.
We say that a state $\omega$ is *$\Sp(\cG)$-invariant* if for any $*$-automorphism $\Phi_\Theta\in \Aut(\cW_{\cG})$, with $\Theta\in \Sp(\cG)$, it holds $\omega \circ \Phi_\Theta = \omega \,. $
As anticipated in Example \[ex:NCT\], the -algebra $\cW_{\cG}$ can be equipped with a $C^*$-norm in a canonical way, and hence it can be completed to a $C^*$-algebra $\overline{\cW}_{\cG}$. Since every positive functional on $\cW$ can be extended to a positive functional $\overline{\cW}_{\cG}$ ([@Naimark]) and every positive functional on $\overline{\cW}_{\cG}$ can be restricted to a positive functional on $\cW_{\cG}$, then in the rest of the paper we shall focus only on $\cW_{\cG}$.
In order to construct a ($\Sp(\cG)$-invariant) state $\omega$ on the -algebra $\cW_{\cG}$, it is enough to prescribe its values on each generator of $\cW_\cG$ and then to extend it by linearity to any element $\aa\in\cW_{\cG}$. So, the $\Sp(\cG)$-invariance condition for a state $\omega$ can be written $$\label{eq:state on W}
\omega(\Theta(g))=\omega(g)=\begin{cases} 1 & \text{ if } g=1_\cG \\ q^{(g)} \in \C & \text{ else} \end{cases}$$ for a sequence of values $q^{(g)}$. Following [@BMP Section 2], we can prove the following.
\[prop:real\_values\] Let $\cW_{\cG} $ be a -algebra and consider a $\Sp(\cG)$-invariant state $\omega$. Then, for any $g\in \cG$, it holds $$\omega(g) \in [-1,1] \,.$$
Since $\omega$ is a linear positive functional, then, for every $g\in\cG$, it holds $$\begin{aligned}
\omega\left( (g + 1_\cG)^*(g + 1_\cG)\right) = 2 +\omega\left(g^*\right) + \omega\left(g\right) \in [0,\infty) \,.\end{aligned}$$ According to Remark \[rmk:inversion\], there exists an element $\Inv\in\Sp(\cG)$ defined by $g\mapsto g^{-1}$ and it holds $$\ol{\omega(g)} = \omega(g^*) =\omega(g^{-1})=\omega(\Inv g) = \omega(g) \,,$$ where in the fourth equality we used the invariance of the state under the action of the symplectic group. Let now be $\aa_\pm=g \pm 1_\cG$. Since $\omega$ as a positive functional has to satisfy $$0\leq \frac{1}{2}\, \omega(\aa_\pm^*\aa_\pm)= 1 \pm \omega(g) \,,$$ which concludes our proof.
As anticipated in Section \[sec:group algebra\], the twisting of the product of $\C[\cG]$ by $\Omega$ plays an important role in the characterization of the $\Sp(\cG)$-invariant states. Indeed, if we simply consider the (untwisted) group $*$-algebra $\C[\cG]$ (that corresponds to considering the trivial pre-symplectic form on $\cG$), then there are uncountably many $\Sp(\cG)$-invariant states, as shown in the next proposition.
\[thm:CTori\] Let $\C[\cG]$ be the *(untwisted)* group $*$-algebra associated to the symplectic abelian group $(\cG, \sigma_0)$, where $\sigma_0$ is the trivial pre-symplectic form. Then, there exist infinitely many $\Sp(\cG)$-invariant states.
It is easy to notice that any constant functional given by $$\label{eq:state on W_CTori}
\omega(g)=\begin{cases} 1 & \text{ if } g = 1_\cG \\ q \in [0,1]\bigcap \R & \text{ else} \end{cases}$$ is $\Sp(\cG)$-invariant (in this case $\Sp(\cG)$ agrees with the group of automorphism of $\cG$). We need just to verify that it is positive. To this end, we notice that for every finite dimensional subspace of $\cV \subset \C[\cG]$, the map $\aa \mapsto \omega(\aa^*\aa)$ is a quadratic form, therefore it can be written as $$\omega(\aa^*\aa) = \overline{\a}^t \, \bH \, \a$$ where $\bH$ is a Hermitian matrix and $\a$ a vector with components $\a_j\in\C$. On the $(d+1)$-dimensional subspace spanned by the elements $\{ 1_\cG, h_j \}_{1 \le j \le d}$ the entries of the matrix $\bH$ can be described as $$\begin{aligned}
\label{eq:H-entries1}
(\bH)_{0,0}&=(\bH)_{j,j} = 1, \qquad & j \geq 1\\
\label{eq:H-entries2}
(\bH)_{i,j} &= (\bH)_{j,i} = q, \qquad & i\neq j
\end{aligned}$$ where holds because of the state normalization condition. By a straighforward computation, we have that the eigenvalues of $\bH$ are given by $$\lambda_1=dq+1 \qquad \text{and} \qquad \lambda_i=1-q \quad \text{ for $i=1,\dots,d$\,.}$$ Since analogous considerations hold for subspaces of $\C[\cG]$ not containing $1_\cG$ (and hence for any finite dimensional subspace), we can conclude that $\omega$ given by is indeed a positive and normalized $\Sp(\cG)$-invariant functional.
It is an easy corollary of Proposition \[thm:CTori\] that any abelian group equipped with a degenerate $\T$-pre-symplectic form always admit many invariant states (we will discuss a proof of this fact in Section \[sec:conclusion\]). In particular, it is immediate to check that the rational non-commutative torus admits many invariant states as it comes equipped with a degenerate $\T$-pre-symplectic form.
Now we start to study $\Sp(\cG)$-invariant states on $\cW_{\cG}$, with the hypothesis that the product is twisted in a non-trivial way and that the form is non-degenerate.
$\cG$ is torsion-free {#sec:torsfree}
---------------------
In this section, we analyze invariant states in the case of torsion-free groups. In particular, in Theorem \[thm:NCtori\_no\_torsion\] we provide a sufficient condition for the symplectic abelian group $(\cG,\sigma)$ to have a unique invariant state. The proof is obtained by reducing to the case solved in our previous work [@BMP]. The aim of this section is to deal with the case when $\cG$ is not finitely generated. To this end, we need preparatory lemmas and definitions. Recall that the rank of an abelian group $\cG$ is defined as the dimension of the $\Q$-vector space $\cG \otimes_\Z \Q$.
\[defn:completely\_decomposable\_group\] Let $\cG$ be a torsion-free abelian group, we say that $\cG$ is a *completely decomposable* if $\cG \cong \bigoplus_{i \in I} \cG_i$ where $\cG_i$ is a rank $1$ torsion-free group, a sub-group of $\Q$.
The class of completely decomposable abelian groups is an amenable class of non-finitely generated abelian groups , including the class of $\Q$-vector spaces. Nevertheless, groups appearing in applications might be not completely decomposable
The additive group $\Z_p$ of $p$-adic integers is not completely decomposable (it follows from [@Bae Example 7.4]).
Given a torsion-free $\T$-symplectic abelian group $(\cG, \sigma)$ we can always consider the $\Q$-vector space $\cG \otimes_\Z \Q$ generated by $\cG$ and extend the symplectic form to $\cG \otimes_\Z \Q$ by considering the $\T$-symplectic form $$(\sigma \otimes_\Z \Q)(q_1 g_1, q_2 g_2) = q_1 q_2 \sigma(g_1, g_2),$$ for any $g_1, g_2 \in \cG$ and $q_1, q_2 \in \Q$.
\[lemma:unique\_hyperbolic\_plane\] Let $(\cG, \sigma)$ be a $\T$-symplectic abelian group of rank $2$, then $\sigma = (r \sigma_2)|_{\cG}$ for a $r \in \T$.
By hypothesis $\cG \otimes_\Z \Q \cong \Q^2$. Let $x,y \in \cG$ be two generators of $\cG \otimes_\Z \Q$ as a $\Q$-vector space. Any element in $\cG \otimes_\Z \Q$ can be written as $q_x x + q_y y$ with $q_x, q_y \in \Q$. By writing $$q_x = \frac{n_x}{d_x}, \ \ q_y = \frac{n_y}{d_y}$$ we get by bilinearity that $$\sigma(q_x x, q_y y) = n_x n_y \sigma \left(\frac{x}{d_x}, \frac{y}{d_x} \right),$$ and the relations $$d_x \frac{x}{d_x} = x, \ \ d_y \frac{y}{d_y} = y$$ again by bilinearity imply $$\sigma(x, y) = d_x d_y \sigma \left(\frac{x}{d_x}, \frac{y}{d_x} \right) \then \sigma \left(\frac{x}{d_x}, \frac{y}{d_x} \right) = \frac{\sigma(x, y)}{d_x d_y}.$$ This implies that the knowledge of the value $\sigma(x, y)$ uniquely determines the simplectic form $\sigma$ as $\cG$ injectively embeds in $\cG \otimes_\Z \Q$.
\[defn:irrational\] We say that the symplectic form $\sigma: \cG \otimes \cG \to \T$ is *irrational* if $$\sigma(\cG \otimes \cG) \cap \Q/\Z = \{1\}\,.$$
We say that the state on the -algebra $\cW_\cG$ defined by the values $$\tau(g)=\begin{cases} 1 & \text{ if } g=1_\cG \\ 0 & \text{ otherwise } \end{cases}$$ and extended to $\cW_\cG$ by linearity is the *tracial state*. We also often use the following hypothesis.
\[ass:diagonalization\] On any finitely generated sub-group $\cF \subset \cG$ the restriction of $\sigma$ to $\cF$ satisfies the hypothesis of Theorem \[thm:symplectic\_diagonalization\_2\].
A first result about non-finitely generated torsion-free groups is the following.
\[lemma:NCtori\_no\_Q2\] Let $\cG$ be a $\T$-symplectic abelian group of rank $2$ such that $\sigma$ is irrational. Then, the only $\Sp(\cG)$-invariant state on $\cW_\cG$ is the tracial state.
By Lemma \[lemma:unique\_hyperbolic\_plane\] the claim follows by a direct application of Theorem 2.2 of [@BMP].
The next step is to generalize the previous lemma to all injective abelian groups.
\[lemma:NCtori\_no\_torsion\_injective\] Let $(\cG, \sigma)$ be an injective $\T$-symplectic abelian group such that $\sigma$ is irrational and that satisfies Assumption \[ass:diagonalization\]. Then, the only $\Sp(\cG)$-invariant state on $\cW_\cG$ is the tracial state.
The condition of $\cG$ being injective is equivalent to $\cG$ being a $\Q$-vector space. By Proposition \[prop:hyperbolic plane\] we know that there is a hyperbolic plane $\cH \subset \cG$ that contains $g$. Since $\cG$ is a $\Q$-vector space then we can consider the $\Q$-vector space generated by $\cH$ in $\cG$, that we denote $\cH_\Q$. The restriction of $\sigma$ to $\cH_\Q$ is a $\T$-symplectic form, hence $\cH_\Q$ is isomorphic to $(\cH_\Q, r \sigma_2)$, for some irrational $r$, as a consequence of Lemma \[lemma:unique\_hyperbolic\_plane\]. Since $\sigma$ is diagonalizable, it is easy to see that $$(\cG, \sigma) \cong (\cH_\Q, \sigma|_{\cH_\Q}) \oplus (\cH_\Q^\perp, \sigma|_{{\cH_\Q}^\perp})$$ as a $\T$-symplectic abelian group, because it is easy to check that $\sigma(h, h') = 0$ for $h \in \cH_\Q$, $h' \in \cH_\Q^\perp$. Therefore, we get an inclusion $\Sp(\cH_\Q) \subset \Sp(\cG)$. This implies that every $\Sp(\cG)$-invariant state $\omega$ on $\cW_\cG$ must restrict to a $\Sp(\cH_\Q)$-invariant state on $\cW_{\cH_\Q}$. This implies that $\omega$ is trivial on $\cH_\Q$ by Lemma \[lemma:NCtori\_no\_Q2\].
The next one is our first general result.
\[thm:NCtori\_no\_torsion\_CD\] Let $(\cG, \sigma)$ be completely decomposable $\T$-symplectic abelian group such that $\sigma$ is irrational and that satisfies Assumption \[ass:diagonalization\]. Then, the only $\Sp(\cG)$-invariant state on $\cW_\cG$ is the tracial state.
We can embed $\cG$ in its injective envelop $\cG \otimes_\Z \Q$, that comes equipped with a structure of $\T$-sympletic abelian group $\sigma \otimes_\Z \Q$ that canonically extends the one of $\cG$, and for any $g \in \cG$ we can consider a $\Q$-hyperbolic plane $\cH_\Q \subset \cG \otimes_\Z \Q$ containing $g$. Now, suppose that there exists a non-trivial $\Sp(\cG)$-invariant state on $\cW_\cG$, then this state would be such that $\omega(g) \ne 0$ for some $g \ne 1_\cG$. For such a $g$ we have that $(\cG, \sigma) \cong (\cH_\Q \cap \cG, \sigma) \oplus ((\cH_\Q \cap \cG)^\perp, \sigma)$, by the same reasoning used in Lemma \[lemma:NCtori\_no\_torsion\_injective\]. Therefore, $\Sp((\cH_\Q \cap \cG, \sigma))$ embeds in $\Sp(\cG)$, proving that any $\Sp(\cG)$-invariant state on $\cW_\cG$ must restrict to the trivial state on $(\cH_\Q \cap \cG, \sigma)$, contradicting to the hypothesis that $\omega(g) \ne 0$.
In the next theorem we axiomatize the proof of Theorem \[thm:NCtori\_no\_torsion\_CD\]. In order to do that we introduce the following concepts.
\[defn:split\_hyperbolic\_plane\] Let $(\cG, \sigma)$ be a $\T$-symplectic abelian group. Let $\cH \subset (\cG, \sigma)$ be a rank $2$ hyperbolic plane (by this we mean a sub-group of rank $2$ of $\cG$ where $\sigma$ restricts to a non-degenerate form). We say that $\cH$ is *split* if $$\cG \cong (\cH, \sigma_\cH) \oplus (\cH^\perp, \sigma_{\cH^\perp})$$ as a $\T$-symplectic abelian group.
Notice that for any split hyperbolic plane $\cH \subset (\cG, \sigma)$ we have an injective morphism $$\Sp(\cH) \to \Sp(\cG).$$
\[defn:plane\_automorphisms\] Let $(\cG, \sigma)$ be a $\T$-symplectic abelian group. Let $\{ \cH_i \}_{i \in I}$ be the family of split hyperbolic planes of $\cG$. We define the group of *plane automorphisms* of $\cG$ as $$\lt \{ \Sp(\cH_i) \}_{i \in I} \gt \subset \Sp(\cG),$$ and we denote it by $\Sp^H(\cG)$.
\[thm:NCtori\_no\_torsion\] Let $(\cG, \sigma)$ be a $\T$-symplectic abelian group such that $\sigma$ is irrational and Assumption \[ass:diagonalization\]. Assume also that $\Sp(\cG)\otimes_\Z \Q \supset \Sp^H(\cG \otimes_\Z \Q)$, where $\cG \otimes_\Z \Q$ is equipped with the $\T$-symplectic form $\sigma \otimes_\Z \Q$. Then, the only $\Sp(\cG)$-invariant state on $\cW_\cG$ is the tracial state.
Let $\cG \otimes_\Z \Q$ be the injective envelope of $\cG$. Notice that every automorphism of $\cG$ lifts to an automorphism of $\cG \otimes_\Z \Q$ because of the universal property of $\cG \otimes_\Z \Q$. This gives an injection of $\Sp(\cG)$ in $\Sp(\cG \otimes_\Z \Q)$ because $\cG \otimes_\Z \Q$ is equipped with the $\T$-symplectic form $\sigma \otimes_\Z \Q$. Since we assume that ${\Sp(\cG)\otimes_\Z \Q} \supset \Sp^H(\cG \otimes_\Z \Q)$, this implies that every $\Sp(\cG)$-invariant state induces an invariant state on $\cW_{\cG \cap \cH_i}$, for any split hyperbolic plane $\cH_i$ of $\cG \otimes_\Z \Q$. And this restriction must be trivial as a consequence of Lemma \[lemma:NCtori\_no\_Q2\]. Since each non-null element of $\cG \otimes_\Z \Q$ generates a split hyperbolic plane, we get that only the only $\Sp(\cG)$-invariant state on $\cW_\cG$ is the trivial state.
Without the hypothesis of diagonalization of Theorem \[thm:symplectic\_diagonalization\_2\], the symplectic group could be very small, as discussed in Example \[ex:conjugate sympl group\]. Therefore, its action could not be ergodic, allowing many invariant states in some specific cases.
$\cG$ is torsion and non-finitely generated {#inftors}
-------------------------------------------
In this section we investigate the uniqueness of $\Sp(\cG)$-invariant states for torsion $\T$-sympletic abelian groups $\cG$ of infinite rank. Since a complete classification of this class of groups is not known, we shall focus our attention on some important examples. First, consider an infinite direct sum $$\cG = \bigoplus_{i \in I} (\F_p^2, \sigma_2)$$ where $I$ is any set of infinite cardinality (as if $I$ is finite than the action of $\Sp(\cG)$ on $\cW_\cG$ is not ergodic, on account of Proposition \[prop:ergodic\], and it is not difficult to see that in this case $\cW_\cG$ has many invariant states). Without loss of generality, we assume $I = \N$, since the cases when $I$ has bigger cardinality can be dealt in the same way or reduced to this case.
\[lemma:infinite\_torsion\_orbits\] The action of $\Sp(\cG)$ on $\cG = \bigoplus_{i \in I} (\F_p^2, \sigma_2)$ has only one orbit besides the one of the identity of $\cG$.
Let $g, h \in \cG$ be two non null-elements. By definition, we can write $g = (g_i)$, $h = (h_i)$ with $g_i,h_i \in \F_p^2$ where only a finite number of $g_i$ and $h_i$ are not null (for simplicity in this lemma we are using the additive notation for the operation of $\cG$). Let us write $g_{i_1}, \ldots, g_{i_n}$ and $h_{j_1}, \ldots, h_{j_m}$ for the non-null components of $g$ and $h$. We can always find elements $\phi_g, \phi_h \in \Sp(\cG)$ such that $$\phi_g(g) = (\underbrace{(x, x), (x, x), \ldots, (x, x)}_{n \text{ times}}, (0,0), \ldots)$$ and $$\phi_h(h) = (\underbrace{(x, x), (x, x), \ldots, (x, x)}_{m \text{ times}}, (0,0), \ldots),$$ for an arbitrary fixed non null-element $x$ of $\F_p$. This is due to the fact that in a direct sum of the form $(\F_p^2, \sigma_2) \oplus (\F_p^2, \sigma_2)$ the map that sends an element $(x_1, x_2, x_3, x_4)$ to $(x_3, x_4, x_1, x_2)$ is easily seen to be in the symplectic group. The general result follows by induction.\
In this way, we can map $g$ and $h$ to vectors whose non-null components are in the first $n$ and $m$ coordinates respectively. Then, the symplectic group $\Sp((\F_p^2, \sigma_2))$ of each factor maps injectively in $\Sp(\cG)$ because they are split hyperbolic plane in the sense of Definition \[defn:split\_hyperbolic\_plane\]. Then, applying Lemma \[lem:Orbita\_su\_F\_p\] we get the desired form for $\phi_g(g)$ and $\phi_h(h)$.
It remains to check that we can always map the elements $\phi_g(g)$ and $\phi_h(h)$ to each other via an element of $\Sp(\cG)$. The general case can be reduced to the case $\phi_g(g) = ((x, x), (x, x))$ and $\phi_h(h) = ((x, x), (0,0))$ by induction, so we discuss only this case. Then, it is enough to show that there exists a symplectic automorphism $\psi: (\F_p^4, \sigma_4) \to (\F_p^4, \sigma_4)$ such that $\psi(\phi_g(g)) = \phi_h(h)$. Consider the following matrix $$\bM =
\begin{pmatrix}
1& 0 & 0 & 0 \\
0 & 1 & 1 & -1 \\
1 & 0 & 1 & 0 \\
1 & 0 & 0 & 1
\end{pmatrix}.$$ It is easy to check that the automorphism induced by $\bM$ on $(\F_p^4, \sigma_4)$ belongs to the symplectic group. Since we have $$\begin{pmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & 1 & -1 \\
1 & 0 & 1 & 0 \\
1 & 0 & 0 & 1
\end{pmatrix}
\begin{pmatrix}
x \\
x \\
0 \\
0
\end{pmatrix}
=
\begin{pmatrix}
x \\
x \\
x \\
x
\end{pmatrix}$$ we can conclude the proof.
As immediate consequence of Lemma \[lemma:infinite\_torsion\_orbits\] we obtain that a $\Sp(\cG)$-invariant state on $\cW_\cG$ must be constant on all generators. With the next theorem prove uniqueness of $\Sp(\cG)$-invariant states on $\cW_\cG$.
\[thm:infinite\_torsion\] Let $\cW_\cG$ be the symplectic twisted group algebra associated to $\cG = \bigoplus_{i \in I} (\F_p^2, \sigma_2)$ and consider the constant normalized functional $\omega: \cW_\cG \to \C$ given by $$\omega(g) = \begin{cases}
1 & \text{ if } g = 1_\cG \\
q \in [-1, 1], \text{ otherwise. }
\end{cases}$$ Then, $\omega$ is positive if and only if $q = 0$.
Before entering into the details of the proof, we define some useful notation.
\[eq:M\_notation\] With $\bM_n(p, q)$, we denote the $n \times n$ matrix given by $$\begin{aligned}
(\bM_n(p, q))_{j,j} &= 1 & j \geq 1\\
(\bM_n(p, q))_{1,j}&=(\bM_n(p, q))_{j,1} = p & j > 1\\
(\bM_n(p, q))_{i,j} &= (\bM_n(p, q))_{j,i} = q & \text{else}
\end{aligned}$$ for which $p, q \in \C$.
Consider any element in $g \in \cG$. We want to find, for any $n \in \N$, suitable sequences of elements $\aa_{1, k}, \ldots, \aa_{n p, k} \in \cW_\cG$, for $1 \le k \le {p}$ such that the matrix associated to the quadratic for $\omega(\aa_{i,k}^* \aa_{j,k})$, for $1 \le i {, j} \le n p$, is of the form $\bM_{n p}(q, q e^{2 \pi {{\imath}}\frac{k}{p}})$, using the notation of Notations . If $q$ is such that the functional $\omega$ is positive, then all matrices $\bM_{n p}(q, q e^{2 \pi {{\imath}}\frac{k}{p}})$ must be positive.
Suppose that such elements exist. Then, we can consider the matrix $$\bR_n = \frac{1}{p} \sum_{k = 1}^p \bM_{n p}(q, q e^{2 \pi {{\imath}}\frac{k}{p}}) = \bM_{n p}(q, 0).$$ By hypothesis this can be done for any $n$. Hence, for a fixed $q$ there exists a $n$ big enough such that the matrix $\bR_n$ is non-positive, as it follows immediately by computing the determinant of $\bR_n$. But this is in contradiction with the hypothesis that all the matrices $\bM_{pn}(q, q e^{2 \pi {{\imath}}\frac{k}{p}})$ are positive, as the convex sum of positive matrices must be a positive matrix.
It remains to show that it is always possible to find such elements $\aa_{i, k}$, for which the matrix associated to the quadratic for $\omega(\aa_{i,k}^* \aa_{j,k})$ is $\bM_{n p}(q, q e^{2 \pi {{\imath}}\frac{k}{p}})$. To this end, consider a generic element $e_1 \in \cG$. It is enough to find for any $n \in \N$ elements $e_2, \ldots, e_n \in \cG$ such that $$\label{eq:sigma_system}
\sigma(e_i, e_j) = e^{2 \pi {{\imath}}k/p }.$$ But since $e^{2 \pi {{\imath}}k/p }$ is a primitive $p$-th root of $1$, we can identify the elements $e^{2 \pi {{\imath}}k/p }$ with elements of $\F_p$ and the becomes a linear system of finitely many equations with $\F_p$ coefficients. Since $\F_p$ is a field and $\cG$ is an infinite dimensional vector space over it we can always find a suitable finite dimensional subspace of $\cG$ where the system admits solutions, concluding the proof.
We conclude this section, by showing how the proof of Theorem \[thm:infinite\_torsion\] can be generalized to the case when $\F_p$ is replaced with $\Z/n\Z$. Indeed, the case when $n = p^f$, and therefore $(\Z/n\Z)^2$ is equipped with is canonical $\T$-symplectic form determined by $\sigma_2((1,0), (0,1)) = e^{\frac{2 \pi {{\imath}}}{n}}$, can be dealt in the same way as done in Theorem \[thm:infinite\_torsion\], using the fact that in this case there are $f$ different orbits each of which with infinite element, besides the trivial orbit of the identity element. Therefore the following corollary.
\[cor:infinite\_torsion\] Theorem \[thm:infinite\_torsion\] remains true when for $\cG = \bigoplus_{i \in I} ((\Z/n\Z)^2, \sigma_2)$.
We already remarked that the case when $n = p^f$ is similar to the case of the theorem. The general case follows from Corollary \[cor:direct\_sum\_torsion\] and the observation that in this case $$\Sp(\bigoplus_{p \in {{\mathbb P}}} \cG_p) \cong \prod_{p \in {{\mathbb P}}} \Sp(\cG_p)$$ where $\cG_p$ is the $p$-primary part of $\cG$.
We conclude this section by remarking that it is easy to find examples of infinite torsion groups on which the action of any group of automorphism cannot give ergodic -algebras, as their $p$-primary parts are finite, $\bigoplus_{p \in {{\mathbb P}}} \F_p$.
Conclusions {#sec:conclusion}
===========
We conclude this paper with some conjectures and an application of our results to abelian Chern-Simons. To this end, let us remark that, in all cases we have considered so far, we proved the uniqueness of $\Sp(\cG)$-invariant states when the group $\cG$ was equipped with a non-degenerate pre-symplectic form. Indeed, it is easy to see that this is a necessary condition on $\cW_\cG$ for having a unique $\Sp(\cG)$-invariant state.
\[cor:degenerate sympl. form\] Let $(\cG,\sigma)$ be a $\T$-pre-symplectic abelian group. If $\sigma$ is degenerate, then $\cW_\cG$ admits plenty of $\Sp(\cG)$-invariant states.
If the pre-symplectic form on $\cG$ is degenerate, then $\cG^\perp \ne 0$ and $\Phi(\cG^\perp) \subset \cG^\perp$ for all $\Phi \in \Sp(\cG)$. Then, it is easy to check that any functional defined by $$\tau(g)=\begin{cases} 1 & \text{ if } g=1_\cG \\
0 & g \not\in \cG^\perp \\
q & g \in \cG^\perp \end{cases}$$ with $q \in (0, 1)$, is $\Sp(\cG)$-invariant and positive on account of the computations done in the proof of Proposition \[thm:CTori\].
The next theorem summarizes our main results on the uniqueness of $\Sp(\cG)$-invariant states, Corollary \[cor:degenerate sympl. form\], Theorem \[thm:NCtori\_no\_torsion\], Theorem \[thm:infinite\_torsion\] and Corollary \[cor:infinite\_torsion\].
\[thm:sum\_up\] Let $(\cG, \sigma)$ be a $\T$-pre-symplectic abelian group, then
1. if $\sigma$ is degenerate, then $\cW_\cG$ admits plenty of $\Sp(\cG)$-invariant states;
2. if $(\cG, \sigma)$ is symplectic, irrational (in the sense of Definition \[defn:irrational\]), the symplectic form is diagonalizable (in the sense of Notation \[not: diagonalization\]) and ${\Sp(\cG)\otimes_\Z \Q} \supset \Sp^H(\cG \otimes_\Z \Q)$, then $\cG$ is torsion-free and $\cW_\cG$ admits only one invariant state;
3. if $(\cG, \sigma) \cong \bigoplus_{i \in I} ((\Z/n\Z)^2, \sigma_2)$, where $I$ has infinite cardinality, then the associated -algebra admits a unique $\Sp(\cG)$-invariant state.
From these results, we conjecture the following to hold.
\[conj:symplectic\_ergodic\] If $(\cG, \sigma)$ is a $\T$-symplectic abelian group such that the action of $\Sp(\cG)$ on $\cW_\cG$ is ergodic, then there are no non-trivial $\Sp(\cG)$-invariant states on $\cW_\cG$.
A step in this direction is the following weak version of Conjecture \[conj:symplectic\_ergodic\].
\[conj:symplectic\_ergodic\_weak\] If $(\cG, \sigma)$ is a irrational $\T$-symplectic abelian group such that the action of $\Sp(\cG)$ on $\cW_\cG$ is ergodic and $\sigma$ is diagonalizable, then there are no non-trivial $\Sp(\cG)$-invariant states on $\cW_\cG$.
One possible approach to Conjecture \[conj:symplectic\_ergodic\_weak\] is to prove that the technical assumption ${\Sp(\cG) \otimes_\Z \Q} \supset \Sp^H(\cG \otimes_\Z \Q)$ in Theorem \[thm:NCtori\_no\_torsion\] is always satisfied.
We conclude this paper with an application of the results proven so far inspired by [@DMS]. In , the quantization of Abelian Chern-Simons theory is interpreted as a functor $$\mathfrak{A}: \Man_2 \to\stgAlg$$ which assigns symplectic twisted group $*$-algebras to 3-dimensional manifolds of the form $\R \times \Sigma$, where $\Sigma$ is a 2-dimensional oriented manifold. This assignment is obtained by composing the functor $$\mathfrak{S}:=( H_c^1(-,\Z),\sigma): \Man_2 \to \bP\bSymp_\T$$ which assign to every $\Sigma\in\Man_2$ its first (singular) homology group with compact support $H_c^1(\Sigma)$ endowed with a $\T$-valued pre-symplectic form $\sigma$, with the functor $$\mathfrak{CCR}: \bP\bSymp_\T \to \stgAlg$$ which assign to every $\Z$-pre-symplectic abelian group a symplectic twisted group algebra. We say that such a functor $\mathfrak{A}$ is a *Cherns-Simons functor*.
By functoriality, $\mathfrak{CCR}$ induces a representation of the group of orientation preserving diffeomorphisms Diff$^+ (\Sigma)$ of $\Sigma$, that is the group of automorphisms of $\Sigma$ in $\Man_2$, on $\mathfrak{A}(\Sigma)$ as a group of $*$-algebra automorphisms. This representation can be related to a representation of the mapping class group and, for compact $\Sigma$, to a representation of the discrete symplectic group $\Sp(2n,\Z)$.\
For a quantum physical interpretation of quantum Abelian Chern-Simons theory it is necessary to choose for each $\Sigma\in\Man_2$ a state $\omega_\Sigma: \mathfrak{A}(\Sigma) \to \C$ on the -algebra $\mathfrak{A}(\Sigma)$. Motivated by the functoriality of the association, it seems natural to demand that the family of states $\{\omega_\Sigma\}_{\Sigma\in\Man_2}$ is compatible with the functor $\mathfrak{A}: \Man_2 \to \stgAlg$ in the sense that $$\omega_{\Sigma'} \circ \mathfrak{A}(f) = \omega_\Sigma$$ for all $\Man_2$-morphisms $f : \Sigma \to \Sigma'$. Such compatible families of states are called *natural states* on $\mathfrak{A}: \Man_2 \to \stgAlg.$ Even though the idea of natural states is very beautiful and appealing, there are hard obstructions to the existence of natural states.
Using the results in Theorem \[thm:sum\_up\] we can now generalize the main result of [@DMS]. Before stating the theorem, let us given a definition.
We say that a Chern-Simons functor is *irrational and not degenerate* if each $\mathfrak{A}(\Sigma)$ is irrational in the sense of Definition \[defn:irrational\] and, for any $\Sigma$ such that $H^1_c(\Sigma;\Z)\simeq \Z^{2n}$, the symplectic form is *non-degenerate.*
There exists no natural state for an irrational and non-degenerate Chern-Simons functor.
Let us assume that there exists a natural state $\{\omega_{\Sigma}\}_{\Sigma\in\Man_2}$. Consider the $\Man_2$-diagram $$\bS^2 \stackrel{f_1}{\longleftarrow} \R\times \T\stackrel{f_2}{\longrightarrow} \T^2$$ describing an orientation preserving open embedding of the cylinder $\R\times\T$ into the $2$-sphere $\bS^2$ and the $2$-torus $\T^2$. The Chern-Simons functor assigns $*$-homomorphisms $$\mathfrak{A}(\bS^2) \stackrel{\mathfrak{A}(f_1)}{\longleftarrow} \mathfrak{A}(\R\times \T)
\stackrel{\mathfrak{A}(f_2)}{\longrightarrow} \mathfrak{A}(\T^2)$$ and the naturality of the state implies the condition $$\label{eqn:statecondition}
\omega_{\bS^2}^{}\circ \mathfrak{A}(f_1) = \omega_{\R\times\T}^{} = \omega_{\T^2}^{}\circ \mathfrak{A}(f_2)~.$$ Because of $H^1_{c}(\bS^2;\Z)=0$, it follows that $\mathfrak{A}(\bS^2) \simeq \C$ and hence $\omega_{\bS^2}^{} = \Id_{\C}$ has to be the unique state on $\C$. Using further that $H^1_c(\R\times \T) \simeq \Z$, it follows by the first equality in $$\label{eqn:condition1}
\omega_{\R\times\T}^{}\big(n\big)= 1~,$$ for all $n\in\Z$. By the non-degeneracy hypothesis on $\mathfrak{A}$ it follows that $(H^1_{c}(\T^2;\Z),\sigma_{\T^2})$ is isomorphic to the abelian group $\Z^2$ with an irrational $\T$-symplectic form, $\mathfrak{A}(\T^2)$ is an algebraic irrational non-commutative torus. We can choose $f_2$ such that the $\ast$-algebra homomorphism $\mathfrak{A}(f_2) : \mathfrak{A}(\R\times\T)\to \mathfrak{A}(\T^2)$ is given by $n \mapsto {(n,0)}$, for all $n\in \Z$. As a consequence of and , we obtain that $$\omega_{\T^2}^{}\big({(n,0)}\big) =1~,$$ for all $n\in\Z$, which is not positive by Theorem \[thm:NCtori\_no\_torsion\_CD\] and hence not a state.
[^1]: F. B. is supported the DFG research grant “Derived geometry and arithmetic” and he thanks the Mathematical Institute of the University of Freiburg for the kind hospitality during the preparation of this work as well as the DFG GRK 1821 “Cohomological Methods in Geometry” for supporting this stay. S. M. is supported by the research grant “Geometric boundary value problems for the Dirac operator” and partially supported by the DFG research training group GRK 1821 “Cohomological Methods in Geometry”.
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abstract: 'The resource theory of quantum coherence studies the off-diagonal elements of a density matrix in a distinguished basis, whereas the resource theory of purity studies all deviations from the maximally mixed state. We establish a direct connection between the two resource theories, by identifying purity as the maximal coherence which is achievable by unitary operations. The states that saturate this maximum identify a universal family of maximally coherent mixed states. These states are optimal resources under maximally incoherent operations, and thus independent of the way coherence is quantified. For all distance-based coherence quantifiers the maximal coherence can be evaluated exactly, and is shown to coincide with the corresponding distance-based purity quantifier. We further show that purity bounds the maximal amount of entanglement and discord that can be generated by unitary operations, thus demonstrating that purity is the most elementary resource for quantum information processing.'
author:
- Alexander Streltsov
- Hermann Kampermann
- Sabine Wölk
- Manuel Gessner
- Dagmar Bruß
bibliography:
- 'bibfile.bib'
title: Maximal Coherence and the Resource Theory of Purity
---
Introduction
============
A number of different quantum features are considered as important resources for applications of quantum information theory. Entanglement [@PhysRevLett.78.2275; @Bruss2002; @Plenio2007; @Horodecki2009], quantum discord [@Ollivier2002; @Henderson2001; @RevModPhys.84.1655; @Streltsov2014; @Adesso2016; @Adesso2016b], and quantum coherence [@Aberg2006; @Plenio2014; @Levi2014; @WinterResourceCoherence; @Marvian2016; @CoherenceResource] have been identified as necessary ingredients for the successful implementation of tasks, such as quantum cryptography [@RevModPhys.74.145], quantum algorithms [@NielsenChuang; @RevModPhys.74.145] and quantum metrology [@Giovannetti2011; @Demkowicz2012; @Varenna; @Toth2014; @PhysRevLett.112.210401]. Quantum resources can be formally classified in the framework of resource theories [@Horodecki2013; @SpekkensResourceTheory], where the state space is divided into free states and resource states. Moreover, a set of free operations, which cannot turn a free state into a resource state, is identified [^1]. The possibility of conversion between two resource states via free operations is a central issue within a resource theory, as it introduces a natural order of the resource states. A suitable measure for the resource must be non-increasing under free operations. Equipped with suitable measures, one is able to quantify the resource in any given quantum state.
States that maximize such measures are called extremal resource states [^2]. Every quantum state can then be characterized by the minimal rate of extremal resource states needed to create it (resource cost), or the maximal rate for creating an extremal resource state from it (distillable resource), using the free operations [^3]. A number of different resource theories have been developed in the context of quantum information theory [@Horodecki2013; @SpekkensResourceTheory], prominent examples being entanglement [@PhysRevLett.78.2275; @Bruss2002; @Plenio2007; @Horodecki2009] and coherence [@Aberg2006; @Plenio2014; @Levi2014; @WinterResourceCoherence; @Marvian2016; @CoherenceResource].
While the concept of coherence is basis-dependent by its very definition, both entanglement and quantum discord are locally basis-independent. However, entanglement and discord usually change if a global unitary is applied. It is clear, however, that the unitary activation of these resources must be limited in terms of some basis-independent quantity of the initial quantum state. As we will show in rigorous quantitative terms, this fundamental quantity is identified as purity. Specifically, we show how purity can be used to establish quantum coherence by a unitary operation. This further provides direct bounds on the amount of entanglement and discord that can be reached by unitary operations, since these quantities can be traced back to coherences in a specific many-body basis. These results hold for all suitable distance-based quantifiers.
A resource theory of purity was introduced in [@Horodecki2003] for the asymptotic limit of infinitely many copies of the quantum state. The finite-copy scenario was considered more recently [@GourResThOpPhysRep15]. Our results relate both of these approaches directly to the resource theory of coherence. In general, purity can be interpreted as the maximal coherence, maximized over all unitaries. Depending on the chosen coherence monotone, we recover either the asymptotic or the finite-copy resource theory of purity, by maximizing over unitary operations, or even more generally, over all unital operations. As one of our main results, we are able to identify the states that maximize any given coherence monotone for a fixed spectrum of the density matrix. These states define a universal set of maximally coherent mixed states. The coherence of these states can be evaluated exactly for any distance-based coherence monotone, and is shown to coincide with its distance-based purity.
Resource theory of quantum coherence
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In the following, we recall the resource theory of coherence [@Aberg2006; @Plenio2014; @Levi2014; @WinterResourceCoherence; @Marvian2016; @CoherenceResource] and then identify the family of maximally coherent mixed states. The free states of this resource theory are called *incoherent states*, these are states which are diagonal in a fixed basis $\{\ket{i}\}$, i.e., $$\sigma=\sum_{i}p_{i}\ket{i}\!\bra{i}.$$ The set of all incoherent states will be denoted by $\mathcal{I}$. The definition of free operations is not unique, and several approaches have been presented in the literature [@Marvian2016; @CoherenceResource].
The historically first and most general approach was suggested in [@Aberg2006], where the set of *maximally incoherent operations* (MIO) was considered. These are all operations which cannot create coherence, i.e., $$\Lambda_{\mathrm{MIO}}[\sigma]\in\mathcal{I}$$ for any incoherent state $\sigma\in\mathcal{I}$. Another important family is the set of *incoherent operations* (IO) [@Plenio2014]. These are operations which admit a Kraus decomposition $$\Lambda_{\mathrm{IO}}[\rho]=\sum_{i}K_{i}\rho K_{i}^{\dagger}$$ with incoherent Kraus operators $K_{i}$, i.e., $K_{i}\ket{m}\sim\ket{n}$, where the states $\ket{m}$ and $\ket{n}$ belong to the incoherent basis. We also note that IO is a strict subset of MIO [@WinterResourceCoherence; @Chitambar2016; @Chitambar2016b] $$\mathrm{IO}\subset\mathrm{MIO},\label{eq:IOMIO}$$ and the inclusion is strict even for single-qubit states [@ChitambarPhysRevA.95.019902]. While we will focus on the sets MIO and IO in this work, other relevant sets of operations have been discussed in recent literature, based on physical or mathematical considerations [@Levi2014; @WinterResourceCoherence; @Marvian2016; @Yadin2016; @Chitambar2016; @Chitambar2016b; @ChitambarPhysRevA.95.019902; @Marvian2016b; @GenuineCoherence]. An extension of quantum coherence to multipartite systems has also been presented [@Bromley2015; @Streltsov2015], which made it possible to investigate the resource theory of coherence in distributed scenarios [@Chitambar2016c; @Ma2016; @Streltsov2016; @Chitambar2016d; @Matera2016; @Streltsov2015b]. A review over alternative frameworks of coherence and their interpretation can be found in [@CoherenceResource].
The amount of coherence in a given state can be quantified via *coherence monotones*. These are nonnegative functions ${\mathcal{C}}$ which do not increase under the corresponding set of free operations, i.e., for a MIO monotone we have ${\mathcal{C}}(\Lambda_{\mathrm{MIO}}[\rho])\leq{\mathcal{C}}(\rho)$. Since MIO is the most general set of free operations for any resource theory of coherence, a MIO monotone is also a monotone in any other coherence theory. An important example are distance-based coherence monotones: $$\begin{aligned}
{\mathcal{C}}(\rho)=\inf_{\sigma\in\mathcal{I}}D(\rho,\sigma),\label{eq:C}\end{aligned}$$ where $D$ is a suitable distance on the space of quantum states. Such quantifiers were studied in [@Aberg2006; @Plenio2014], the most prominent example being the relative entropy of coherence $${\mathcal{C}}_{\mathrm{r}}(\rho)=\min_{\sigma\in\mathcal{I}}S(\rho||\sigma)$$ with the quantum relative entropy $S(\rho||\sigma)=\mathrm{Tr}[\rho\log_{2}\rho]-\mathrm{Tr}[\rho\log_{2}\sigma]$ [^4]. Remarkably, this quantity admits a closed expression [@Plenio2014] and coincides with the distillable coherence under MIO and IO and also with the coherence cost under MIO [@WinterResourceCoherence]: $${\mathcal{C}}_{\mathrm{r}}(\rho)=S(\Delta[\rho])-S(\rho).$$ Here, $S(\rho)=-\mathrm{Tr}[\rho\log_{2}\rho]$ is the von Neumann entropy and $\Delta[\rho]=\sum_{i}\braket{i|\rho|i}\ket{i}\!\bra{i}$ denotes dephasing in the incoherent basis.
For a general distance-based coherence quantifier as given in Eq. (\[eq:C\]) one usually considers nonnegative distances $D$ which are contractive under any quantum operation $\Lambda$: $$\begin{aligned}
D(\Lambda[\rho],\Lambda[\sigma])\leq D(\rho,\sigma).\label{eq:contractivity}\end{aligned}$$ Any such distance gives rise to a MIO monotone [@Plenio2014; @CoherenceResource]. Examples for such distances are the relative Rényi entropy $$D_{\alpha}(\rho||\sigma)=\frac{1}{\alpha-1}\log_{2}\mathrm{Tr}[\rho^{\alpha}\sigma^{1-\alpha}]$$ and the quantum relative Rényi entropy $$D_{\alpha}^{\mathrm{q}}(\rho||\sigma)=\frac{1}{\alpha-1}\log_{2}\mathrm{Tr}[(\sigma^{\frac{1-\alpha}{2\alpha}}\rho\sigma^{\frac{1-\alpha}{2\alpha}})^{\alpha}].$$ While $D_{\alpha}$ is contractive for $\alpha\in[0,2]$, the function $D_{\alpha}^{\mathrm{q}}$ is contractive in the range $\alpha\in[\frac{1}{2},\infty]$ [@QuantRenyEntrGeneralTomamichel; @Leditzky2016]. We can now define a family of coherence monotones in the following way: $${\mathcal{C}}_{\alpha}(\rho)=\begin{cases}
\inf_{\sigma\in\mathcal{I}}D_{\alpha}(\rho||\sigma) & \mathrm{for}\,\,0<\alpha<1,\\
\inf_{\sigma\in\mathcal{I}}D_{\alpha}^{\mathrm{q}}(\rho||\sigma) & \mathrm{for}\,\,\alpha>1.
\end{cases}\label{eq:Calpha}$$ This quantity is a MIO monotone in the range $\alpha\in[0,\infty]$. In the limit $\alpha\rightarrow1$ both functions $D_{\alpha}(\rho||\sigma)$ and $D_{\alpha}^{\mathrm{q}}(\rho||\sigma)$ coincide with the relative entropy $S(\rho||\sigma)$. Coherence quantifiers of this type were studied in [@Chitambar2016; @Chitambar2016b; @RenyiEntrCohMeas16]. A related approach based on Tsallis relative entropies has also been investigated [@Rastegin2016].
Several MIO monotones have additional desirable properties such as strong monotonicity under IO and convexity [@Plenio2014; @CoherenceResource]. This is in particular the case for the relative entropy of coherence [@Plenio2014]. While any MIO monotone is also an IO monotone, the other direction is less clear. In particular, the $l_{1}$-norm of coherence $${\mathcal{C}}_{l_{1}}(\rho)=\sum_{i\neq j}|\rho_{ij}|$$ is known to be an IO monotone [@Plenio2014], but violates monotonicity under MIO [@Bu2016]. Another IO monotone which is not a MIO monotone is the coherence of formation $${\mathcal{C}}_{\mathrm{f}}(\rho)=\min\sum_{i}p_{i}S(\Delta[\ket{\psi_{i}}\!\bra{\psi_{i}}]),$$ where the minimum is taken over all pure state decompositions $\{p_{i},\ket{\psi_{i}}\}$ of the state $\rho$ [@Yuan2015; @WinterResourceCoherence; @Hu2016].
We also note that coherence of formation is equal to coherence cost under IO [@WinterResourceCoherence], and $l_{1}$-norm of coherence is related to the path information in multi-path interferometer [@Bagan2016; @Bera2015].
Maximally coherent mixed states
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Since coherence is a basis-dependent concept, a unitary operation will in general change the amount of coherence in a given state. In the following, we will focus on the question: which unitary maximizes the coherence of a given state $\rho$? The corresponding figure of merit is given as follows: $$\begin{aligned}
{\mathcal{C}}_{\max}(\rho):=\sup_{U}{\mathcal{C}}(U\rho U^{\dagger}).\label{eq:Cmax}\end{aligned}$$ If the supremum in Eq. (\[eq:Cmax\]) is realized for the unitary $V$, the corresponding state $\rho_{\max}=V\rho V^{\dagger}$ will be called *maximally coherent mixed state*. This definition is in full analogy to maximally entangled mixed states investigated in [@PhysRevA.62.022310; @MaxEntMixed2QubitStatesPRA01; @PhysRevA.64.030302; @PhysRevA.67.022110]. Maximally coherent mixed states were first introduced for specific measures of coherence in [@Singh2015], and studied further more recently in [@Yao2016].
While the relative entropy of coherence admits a closed formula, the evaluation of general coherence monotones is considered as a hard problem [@CoherenceResource]. It is thus reasonable to believe that the maximization in Eq. (\[eq:Cmax\]) is out of reach. Quite surprisingly, we will now show that the supremum in Eq. (\[eq:Cmax\]) can be evaluated in a large number of relevant scenarios. In particular, we will see that there exists a *universal* maximally coherent mixed state, which does not depend on the particular choice of coherence monotone. These results will also lead us to a closed expression of ${\mathcal{C}}_{\max}$ for all distance-based coherence monotones.
\[thm:MCM\] Among all states $\rho$ with a fixed spectrum $\{p_{n}\}$, the state $$\begin{aligned}
\rho_{\max}=\sum_{n=1}^{d}p_{n}\ket{n_{+}}\!\bra{n_{+}},\label{eq:MCM}\end{aligned}$$ is a maximally coherent mixed state with respect to any MIO monotone. Here, $\{\ket{n_{+}}\}$ denotes a mutually unbiased basis with respect to the incoherent basis $\{\ket{i}\}$, i.e., $|\!\braket{i|n_{+}}\!|^{2}=\frac{1}{d}$, where $d$ is the dimension of the Hilbert space.
We will actually prove an even stronger statement. In particular, we will show that for any unitary $U$, the transformation $\rho_{\max}\rightarrow U\rho_{\max}U^{\dagger}$ can be achieved via MIO, i.e., $$\begin{aligned}
\Lambda_{\mathrm{MIO}}[\rho_{\max}]=U\rho_{\max}U^{\dagger}.\end{aligned}$$ The proof of the theorem then follows by using monotonicity of ${\mathcal{C}}$ under MIO: $${\mathcal{C}}(U\rho_{\max}U^{\dagger})={\mathcal{C}}(\Lambda_{\mathrm{MIO}}[\rho_{\max}])\leq{\mathcal{C}}(\rho_{\max}).$$ The operation $\Lambda_{\mathrm{MIO}}$ which achieves this transformation has Kraus operators $K_{n}=U\ket{n_{+}}\bra{n_{+}}$. Note that bases $\{\ket{n_{+}}\}$ and $\{\ket{i}\}$ are mutually unbiased, which implies that $\sum_{n}K_{n}\sigma K_{n}^{\dagger}=\openone/d$ for any incoherent state $\sigma$. This means that the operation $\Lambda_{\mathrm{MIO}}[\rho]=\sum_{n}K_{n}\rho K_{n}^{\dagger}$ is indeed maximally incoherent. In the final step, note that $\sum_{n}K_{n}\rho_{\max}K_{n}^{\dagger}=U\rho_{\max}U^{\dagger}$, and the proof is complete.
This theorem has several important implications. First, it implies that the state $\rho_{\max}$ is a resource with respect to all states with the same spectrum. Second, this theorem provides an alternative simple proof for the fact that $l_{1}$-norm of coherence can increase under MIO [@Bu2016]. This can be seen by combining Theorem \[thm:MCM\] with the fact that the state in Eq. (\[eq:MCM\]) is not a maximally coherent mixed state for the $l_{1}$-norm of coherence [@Yao2016]. Moreover, a unitary $V$ for an arbitrary state $\rho$ which achieves the supremum in Eq. (\[eq:Cmax\]) for any MIO monotone is given by $V=\sum_{n=1}^{d}\ket{n_{+}}\!\bra{\psi_{n}}$, where $\{\ket{\psi_{n}}\}$ are the eigenstates of $\rho$.
We will now go one step further and give an explicit expression for ${\mathcal{C}}_{\max}$ for any distance-based coherence monotone.
\[thm:Cmax\]For any distance-based coherence monotone as given in Eq. (\[eq:C\]) with a contractive distance $D$ the following equality holds: *$${\mathcal{C}}_{\max}(\rho)={\mathcal{C}}(\rho_{\max})=D\left(\rho,\openone/d\right).$$*
We refer to Appendix \[sec:Proof-2\] for the proof. Note that Theorem \[thm:Cmax\] also holds for all coherence quantifiers $${\mathcal{C}}_{p}=\min_{\sigma\in\mathcal{I}}||\rho-\sigma||_{p}$$ based on Schatten $p$-norms $||M||_{p}=(\mathrm{Tr}[(M^{\dagger}M)^{p/2}])^{1/p}$ for all $p\geq1$. Equipped with these results, we will show below in this paper that the resource theory of coherence is closely related to the resource theory of purity. Before we present these results, we review the main properties of the resource theory of purity in the following.
\[sec:ResourceTheories\]Resource theory of purity
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We will now review resource theories of purity based on different sets of free operations. The discussion summarizes results previously presented in [@GourResThOpPhysRep15]. There exists a hierarchy of quantum operations which generalize classical bistochastic (purity non-increasing) maps. We distinguish three types of quantum operations:
- Mixture of unitary operations: $$\begin{aligned}
\Lambda_{\mathrm{MU}}[\rho]=\sum_{i}p_{i}U_{i}\rho U_{i}^{\dag},\end{aligned}$$ with $p_{i}\geq0$, $\sum_{i}p_{i}=1$, and unitary operations $U_{i}$.
- Noisy operations: $$\begin{aligned}
\Lambda_{\mathrm{NO}}[\rho]={\text{Tr\,}}_{E}\left[U\left(\rho\otimes\openone_{E}/d\right)U^{\dag}\right],\end{aligned}$$ with a unitary operation $U$.
- Unital operations: $$\begin{aligned}
\Lambda_{\mathrm{U}}[\openone/d]=\openone/d,\end{aligned}$$ i.e., operations which preserve the maximally mixed state.
Note that in contrast to the discussion in [@GourResThOpPhysRep15], we only consider operations which preserve the dimension of the Hilbert space. It turns out that these operations form a subset hierarchy $$\begin{aligned}
\left\{ \Lambda_{\mathrm{MU}}\right\} \subset\left\{ \Lambda_{\mathrm{NO}}\right\} \subset\left\{ \Lambda_{\mathrm{U}}\right\} ,\end{aligned}$$ see, e.g., Lemma 5 in Ref. [@GourResThOpPhysRep15].
We call two resource theories equivalent if their respective sets of free states, as well as their sets of all states coincide, and additionally if for each $\Lambda_{1}$ with $\Lambda_{1}(\rho)=\sigma$ there exists a $\Lambda_{2}$, such that $\Lambda_{2}(\rho)=\sigma$, where $\Lambda_{1}$ $(\Lambda_{2})$ is a free operation of resource theory 1 (2). Due to Lemma 10 in [@GourResThOpPhysRep15] the state conversion abilities are equivalent for the three cases $\Lambda_{\mathrm{MU}}$, $\Lambda_{\mathrm{NO}}$, and $\Lambda_{\mathrm{U}}$. It follows that any resource theory which only deviates in the type of operations as defined above will be equivalent. If not stated otherwise, we will consider the resource theory of purity based on unital operations $\Lambda_{\mathrm{U}}$ in the following.
Within the resource theory of purity, the state conversion possibilities follow from the classical theory of bistochastic maps [@UhlmannStochBook; @GourResThOpPhysRep15], using the concept of majorization. A state $\rho$ majorizes another state $\sigma$, i.e., $\rho\succ\sigma$, if their spectra are in majorization order: $$\begin{aligned}
\sum\limits _{i=1}^{k}\lambda_{i}^{\downarrow}(\rho)\geq\sum\limits _{j=1}^{k}\lambda_{j}^{\downarrow}(\sigma)\end{aligned}$$ for all $k\geq1$. Here, $\lambda_{i}^{\downarrow}(\rho)$ denotes the eigenvalues of $\rho$ in non-increasing order. The aforementioned relation to the resource theory of purity is established via the following Lemma.
\[lem:unital\]Given two states $\rho$ and $\sigma$ of the same dimension, $\rho$ can be converted into $\sigma$ via some unital operation $\Lambda_{\mathrm{U}}$ if and only if $\rho$ majorizes $\sigma$: $$\begin{aligned}
\Lambda_{\mathrm{U}}[\rho]=\sigma\Leftrightarrow\rho\succ\sigma.\end{aligned}$$
For the proof of this Lemma we refer to Theorem 4.1.1 in [@NielsenLectureMajorization] (see also [@Uhlmann1970]). Due to the arguments mentioned above, it follows that the majorization relation is necessary and sufficient for state conversion via any set of operations presented above.
Fundamental questions in any resource theory address the number of extremal resource states that can be distilled from a state $\rho$. In the case of purity this poses the question, how many copies of a pure single-qubit state $\ket{\psi}_{2}$ can one extract via unital operations? We will call the corresponding figure of merit *single-shot distillable purity*. Its formal definition can be given as follows [^5]: $$\begin{aligned}
{\mathcal{P}}_{\mathrm{d}}^{1}(\rho) & =\max\!\left\{ m:\exists\:\Lambda_{\mathrm{U}}\text{, s.t. }\Lambda_{\mathrm{U}}\!\left[\rho\otimes\frac{\openone}{d_{2}}\right]=\psi_{2}^{\otimes m}\otimes\frac{\openone}{d_{1}}\right\} ,\label{eq:distillablepurity}\end{aligned}$$ where $\Lambda_{\mathrm{U}}$ is a unital operation, $\psi_{2}={{\ensuremath{| \psi \rangle \!\langle \psi |}}}_{2}$ is a pure single-qubit state, and $\openone/d_{i}$ is a maximally mixed state of dimension $d_{i}$. Correspondingly, we define the *single-shot purity cost* as the minimal number of pure single-qubit states which are required to create the state $\rho$ via unital operations: $$\begin{aligned}
{\mathcal{P}}_{\mathrm{c}}^{1}(\rho) & =\min\!\left\{ m:\exists\:\Lambda_{\mathrm{U}}\text{, s.t. }\Lambda_{\mathrm{U}}\!\left[\psi_{2}^{\otimes m}\otimes\frac{\openone}{d_{1}}\right]=\rho\otimes\frac{\openone}{d_{2}}\right\} .\label{eq:puritycost}\end{aligned}$$
Similar quantities were first studied in the asymptotic limit in [@Horodecki2003], allowing for infinitely many copies of a quantum state and a finite error margin that only vanishes in this limit. It was found that in the asymptotic case, the distillable purity and the purity cost coincide, and are both equal to the relative entropy of purity $${\mathcal{P}}_{\mathrm{r}}(\rho)=\log_{2}d-S(\rho).$$ The single-copy scenario was considered in [@GourResThOpPhysRep15], under the label of “nonuniformity”. There the Rényi $\alpha$-purities were identified as figures of merit using an approach based on Lorentz curves. We will discuss these results in more detail in the following, with particular focus on the resource theory of coherence.
Relation between the resource theories of purity and coherence
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Following established notions from the resource theories of entanglement [@PhysRevLett.78.2275; @Bruss2002; @Plenio2007; @Horodecki2009] and coherence [@Aberg2006; @Plenio2014; @Levi2014; @WinterResourceCoherence; @Marvian2016; @CoherenceResource], we will now introduce a framework for purity quantification. In particular, we distinguish between *purity monotones* and *purity measures*. Any purity monotone ${\mathcal{P}}$ should fulfill the following two requirements.
[00.00.0000]{}
*Nonnegativity*: ${\mathcal{P}}$ is nonnegative and vanishes for the state $\openone/d$.
*Monotonicity*: ${\mathcal{P}}$ does not increase under unital operations, i.e., ${\mathcal{P}}(\Lambda_{\mathrm{U}}[\rho])\leq{\mathcal{P}}(\rho)$ for any unital operation $\Lambda_{\mathrm{U}}$.
Similar as in the resource theories of entanglement and coherence, we regard these two properties as the most fundamental for any quantity which aims to capture the performance of some purity-based task. Purity measures will be monotones with the following additional properties.
[00.00.0000]{}
*Additivity*: ${\mathcal{P}}(\rho\otimes\sigma)={\mathcal{P}}(\rho)+{\mathcal{P}}(\sigma)$ for any two states $\rho$ and $\sigma$.
*Normalization*: ${\mathcal{P}}(\ket{\psi}_{d})=\log_{2}d$ for all pure states $\ket{\psi}_{d}$ of dimension $d$.
A purity monotone/measure ${\mathcal{P}}$ is further convex if it fulfills $\sum_{i}p_{i}{\mathcal{P}}(\rho_{i})\geq{\mathcal{P}}(\sum_{i}p_{i}\rho_{i})$. We note that purity monotones have also been previously studied in [@GourResThOpPhysRep15].
We can now introduce a family of coherence-based purity monotones as follows: $${\mathcal{P}}_{{\mathcal{C}}}(\rho):=\sup_{\Lambda_{\mathrm{U}}}{\mathcal{C}}(\Lambda_{\mathrm{U}}[\rho]),\label{eq:PC}$$ where the supremum is taken over all unital operations $\Lambda_{\mathrm{U}}$ and ${\mathcal{C}}$ is an arbitrary MIO monotone. Clearly, ${\mathcal{P}}_{{\mathcal{C}}}$ is nonnegative, vanishes for $\openone/d$, and does not increase under unital operations, i.e., it fulfills the requirements P1 and P2 for a purity monotone. Remarkably, as we show in Appendix \[sec:Proof-3\], for any MIO monotone ${\mathcal{C}}$ the corresponding purity monotone can be written as $${\mathcal{P}}_{{\mathcal{C}}}(\rho)={\mathcal{C}}(\rho_{\max})\label{eq:PC-1}$$ with the maximally coherent mixed state $\rho_{\max}$. If $\mathcal{C}$ is a distance-based coherence monotone with a contractive distance $D$, we can apply Theorem \[thm:Cmax\] to write the corresponding purity monotone explicitly as $${\mathcal{P}}_{D}(\rho)=D(\rho,\openone/d).\label{eq:PD}$$ Eq. (\[eq:PD\]) represents a general distance-based purity quantifier, in direct analogy to similar approaches for entanglement [@PhysRevLett.78.2275; @Bruss2002; @Plenio2007; @Horodecki2009], coherence [@Plenio2014; @CoherenceResource], and quantum discord [@PhysRevLett.104.080501; @RevModPhys.84.1655; @Streltsov2014; @Adesso2016; @Adesso2016b]. In contrast to these theories, a minimization over free states in Eq. (\[eq:PD\]) is not necessary due to the uniqueness of the free state in the resource theory of purity. For a single qubit the relation between coherence and purity can be visualized on the Bloch ball if coherence and purity are quantified via the trace norm, see Fig. \[fig:1\].
![\[fig:1\]Coherence and purity for a single qubit. The Bloch ball of all single-qubit states contains the incoherent axis (line connecting $\ket{0}\!\bra{0}$ and $\ket{1}\!\bra{1}$) and the maximally coherent (equatorial) plane. If for a state $\rho$ coherence and purity are quantified via the distance-based approach with the trace norm $||M||_{1}=\mathrm{Tr}\sqrt{M^{\dagger}M}$, the corresponding amount of coherence ${\mathcal{C}}$ (red dashed lines) and purity ${\mathcal{P}}$ (black dashed line) can be interpreted as the Euclidean distance to the incoherent axis and the center of the Bloch ball, respectively. The maximally coherent mixed state $\rho_{\max}$ can be obtained from $\rho$ via a rotation onto the maximally coherent plane.](BlochSphereR1a){width="0.58\columnwidth"}
Cases of particular interest can be derived from the coherence monotones introduced in Eq. (\[eq:Calpha\]). In these cases, Eq. (\[eq:PC\]) leads to the *Rényi $\alpha$-purity* $$\begin{aligned}
{\mathcal{P}}_{\alpha}(\rho) & =\log_{2}d-S_{\alpha}(\rho)\label{eq:Palpha}\end{aligned}$$ with the Rényi $\alpha$-entropy $S_{\alpha}(\rho)=\frac{1}{1-\alpha}\log_{2}({\text{Tr\,}}[\rho^{\alpha}])$. This quantity was studied in [@GourResThOpPhysRep15], and it admits an operational interpretation in the resource theories of purity. In particular, the single-shot distillable purity ${\mathcal{P}}_{\mathrm{d}}^{1}$, which was introduced in Eq. (\[eq:distillablepurity\]), can be expressed in terms of ${\mathcal{P}}_{\alpha}$ as follows: $${\mathcal{P}}_{\mathrm{d}}^{1}(\rho)=\left\lfloor \lim_{\alpha\rightarrow0}{\mathcal{P}}_{\alpha}(\rho)\right\rfloor =\left\lfloor \log_{2}(d/r)\right\rfloor ,\label{eq:distillablepurity-1}$$ where $r$ is the rank and $d$ is the dimension of $\rho$. Also the single-shot purity cost ${\mathcal{P}}_{\mathrm{c}}^{1}$, introduced in Eq. (\[eq:puritycost\]), can be written in terms of ${\mathcal{P}}_{\alpha}$ as follows: $${\mathcal{P}}_{\mathrm{c}}^{1}(\rho)=\left\lceil \lim_{\alpha\rightarrow\infty}{\mathcal{P}}_{\alpha}(\rho)\right\rceil =\left\lceil \log_{2}(d\lambda_{\max})\right\rceil ,\label{eq:puritycost-1}$$ with the maximum eigenvalue $\lambda_{\max}$ of $\rho$. These results for single-shot purity distillation and dilution were first found in [@GourResThOpPhysRep15]; we present alternative proofs in Appendix \[sec:Proof-4\] and \[sec:Proof-5\]. Finally, $${\mathcal{P}}_{\mathrm{r}}(\rho)=\lim_{\alpha\rightarrow1}{\mathcal{P}}_{\alpha}(\rho)=\log_{2}d-S(\rho)$$ is the **relative entropy of purity**. It coincides with both the distillable purity and the purity cost in the asymptotic limit, where the resource theory of purity becomes reversible [@Horodecki2003].
As is summarized in Appendix \[sec:RenyiPurity\], ${\mathcal{P}}_{\alpha}$ is a purity measure for all $\alpha\geq0$, i.e., it fulfills all requirements P1-P4, and it is convex for $0\leq\alpha\leq1$. We further note that ${\mathcal{P}}_{\alpha}(\rho)\geq{\mathcal{P}}_{\beta}(\rho)$ for $\alpha\geq\beta$, since the Rényi entropy $S_{\alpha}$ is nonincreasing in $\alpha$ [@Bengtsson2007]. Aside from the cases discussed before, another case of interest is the Rényi $2$-purity ${\mathcal{P}}_{2}(\rho)=\log_{2}(d{\text{Tr\,}}{[\rho^{2}]})$, a simple function of the linear purity ${\text{Tr\,}}[\rho^{2}]$ [^6], which can be directly measured by letting two copies of the state $\rho$ interfere with each other [@Ekert2002; @Pichler2013]. In this way, the purity of a composite system of ultracold bosonic atoms in an optical lattice, as well as the purity of its subsystems, have been determined experimentally [@Islam2015].
Relation to entanglement\
and quantum discord
=========================
Of particular interest for quantum information theory are non-classical properties of correlated quantum states in multipartite systems [@NielsenChuang; @Horodecki2009]. Our results about purity have immediate consequences for quantities such as entanglement and discord. Certain relations between entanglement and purity have already been reported. Bipartite entangled states, e.g., must have a linear purity above a threshold value of ${\text{Tr\,}}[\rho^{2}]=1/(d-1)$, with total dimension $d$, due to the existence of a finite-volume set of separable states around the maximally mixed state [@PhysRevA.58.883; @PhysRevA.66.062311; @Gurvits2003]. Furthermore, a bound for entanglement can be provided by comparing the purity of the composite system to the one of its subsystems [@Mintert2007; @Islam2015]. Similar investigations have also been performed for multipartite quantum systems [@GurvitsPhysRevA.72.032322; @HildebrandPhysRevA.75.062330].
In the following we focus on distance-based quantifiers for discord ${\mathcal{D}}$ and entanglement ${\mathcal{E}}$, in analogy to Eqs. (\[eq:C\]) and (\[eq:PD\]). In a multipartite system these can be defined as [@PhysRevLett.78.2275; @PhysRevLett.104.080501] $$\begin{aligned}
{\mathcal{D}}(\rho) & =\inf_{\sigma\in\mathcal{Z}}{D}(\rho,\sigma),\\
{\mathcal{E}}(\rho) & =\inf_{\sigma\in\mathcal{S}}{D}(\rho,\sigma),\end{aligned}$$ where $\mathcal{Z}$ and $\mathcal{S}$ denote the sets of zero-discord and separable states, respectively. The latter contains all convex combinations of arbitrary product states $\rho_{1}\otimes\cdots\otimes\rho_{N}$, whereas the set of zero-discord states can either be defined with respect to a particular subsystem, or symmetrically with respect to all subsystems. Here, we consider the symmetrical set $\mathcal{Z}$ [@RevModPhys.84.1655], encompassing all convex combinations of pure, locally orthonormal product states $|\varphi_{1}\rangle\langle\varphi_{1}|\otimes\cdots\otimes|\varphi_{N}\rangle\langle\varphi_{N}|$. With this choice, ${\mathcal{D}}(\rho)$ provides an upper bound for the non-symmetric definitions of discord.
Our general distance-based approach leads to the following natural ordering of resources: $${\mathcal{P}}(\rho)\geq{\mathcal{C}}_{N}(\rho)\geq{\mathcal{D}}(\rho)\geq{\mathcal{E}}(\rho).$$ Here, ${\mathcal{C}}_{N}(\rho)=\inf_{\sigma\in\mathcal{I}_{N}}D(\rho,\sigma)$ denotes a coherence monotone with respect to an $N$-partite incoherent product basis, where $\mathcal{I}_{N}$ is the set of $N$-partite incoherent states [@Bromley2015; @Streltsov2015]. This hierarchy holds true if purity, coherence, discord, and entanglement are defined via the same distance $D$. In this case, the statement follows directly by noting that $\openone/d\in\mathcal{I}_{N}\subset\mathcal{Z}\subset\mathcal{S}$, see also Fig. \[fig:2\]. Based on the same argument, this hierarchy can be easily extended beyond entanglement to include the concepts of steering and non-locality [@Wiseman2007; @Adesso2016]. It holds that ${\mathcal{D}}(\rho)=\inf_{U_{N}}{\mathcal{C}}_{N}(U_{N}\rho U_{N}^{\dag})$ with product unitary $U_{N}$; this was pointed out for the relative entropy in [@PhysRevA.92.022112], but holds for general distance-based quantifiers.
![\[fig:2\]Schematic representation of purity ${\mathcal{P}}$ (black dashed line), coherence ${\mathcal{C}}$ (red dashed lines), discord ${\mathcal{D}}$ (green dashed lines), and entanglement ${\mathcal{E}}$ (blue dashed lines) for distance-based quantifiers of the corresponding framework. Zero-discord states $\mathcal{Z}$ are a nonconvex measure-zero subset of separable states $\mathcal{S}$. Incoherent states $\mathcal{I}_{N}$ are a convex subset of $\mathcal{Z}$. All sets contain the maximally mixed state $\openone/d$. The maximally coherent mixed state $\rho_{\max}$ is obtained from $\rho$ via unitary rotation (dotted circle).](Distances){width="1\columnwidth"}
A change of the *global* basis, or equivalently, application of a collective unitary operation can generate entanglement and discord. As we will see below, the maximal achievable amount is again directly bounded by the purity. For this we introduce ${\mathcal{D}}_{\max}(\rho)=\sup_{U}{\mathcal{D}}(U\rho U^{\dagger})$ and similarly ${\mathcal{E}}_{\max}(\rho)=\sup_{U}{\mathcal{E}}(U\rho U^{\dagger})$, in analogy to Eq. (\[eq:Cmax\]). As we prove in Appendix \[sec:Proof-6\], these quantities obey the following relation $$\begin{aligned}
{\mathcal{P}}(\rho)={\mathcal{C}}_{\max}(\rho)\geq{\mathcal{D}}_{\max}(\rho)\geq{\mathcal{E}}_{\max}(\rho),\label{eq:maxhierarchy}\end{aligned}$$ which is visualized in Fig. \[fig:2\]. States related to ${\mathcal{D}}_{\max}$ and ${\mathcal{E}}_{\max}$ have been studied for the two-qubit case. For instance, the set of maximally entangled mixed states, i.e., states which maximize entanglement for a fixed spectrum, as well as states that maximize entanglement at a fixed value of purity, have been characterized for various quantifiers of entanglement and purity [@PhysRevA.62.022310; @MaxEntMixed2QubitStatesPRA01; @PhysRevA.64.030302; @PhysRevA.67.022110]. States satisfying ${\mathcal{E}}_{\max}(\rho)=0$ are also known as absolutely separable states, and have been studied in [@KusPhysRevA.63.032307; @Jivulescu2015276; @Filippov1367-2630-19-8-083010]. Similar studies were performed for discord [@MaxDiscMixed2Qubits; @PhysRevA.83.032101], based on the original definition [@Ollivier2002]. We also note that the relative entropy of purity ${\mathcal{P}}_{\mathrm{r}}$ coincides with the maximal mutual information $I_{\max}(\rho)=\max_{U}I(U\rho U^{\dagger})$, where $I(\rho)=S(\rho^{A})+S(\rho^{B})-S(\rho)$ is the mutual information and both subsystems $A$ and $B$ have the same dimension $\sqrt{d}$ [@Jevtic2012]. As a direct consequence of Theorem \[thm:Cmax\], purity further bounds the accessible entanglement under incoherent operations. This is discussed in more detail in Appendix \[sec:incoherentent\].
Experimental relevance
======================
In well-controllable quantum systems, quantum states with nearly maximal purity are usually easy to initialize but hard to maintain. Especially large quantum systems suffer immensely from purity losses due to noise. For example the linear purity ${\text{Tr\,}}[\rho^{2}]$ of the Greenberger-Horne-Zeilinger (GHZ) state decreases exponentially in time under global phase noise with a decay proportional to the number of particles squared [@Monz2011]. A principal challenge of single photon experiments is the creation of the temporal purity, which is necessary for coherent interaction between photons of two independent sources [@Qian2016].
In contrast to measures of entanglement, discord, or coherence, purity measures are rather easily accessible in experiments [@Ekert2002; @Pichler2013; @Islam2015]. For example the Rényi $\alpha$-purities (\[eq:Palpha\]) are essentially functions of the eigenvalue distribution $\{p_{i}\}$ of $\rho$. Any von Neumann measurement of $\rho$ immediately provides a lower bound for this distribution: if such a measurement is performed in a basis $\{|\varphi_{i}\rangle\}$, the measurement outcomes are distributed according to the probabilities $p'_{i}=\langle\varphi_{i}|\rho|\varphi_{i}\rangle$. Any purity monotone that is evaluated on the basis of the $\{p'_{i}\}$ is a lower bound for the true purity of $\rho$ [^7]. The most easily accessible basis in experiments is the energy eigenbasis. In this case the measured bound coincides with the purity for all thermal states. By virtue of Theorem \[thm:Cmax\], the measured purity also naturally provides an experimental bound on the amount of coherence, entanglement, and quantum discord.
Conclusions
============
As we have proven in this work, the resource theories of coherence and purity are closely connected. This connection was established by showing that any amount of purity can be converted into coherence by means of a suitable unitary operation. We further provided a closed expression for the optimal unitary operation, as well as the quantum states that achieve the maximal coherence. Remarkably, this set of maximally coherent mixed states is universal, i.e., these states maximize all coherence monotones for a fixed spectrum. For any distance-based coherence monotone the maximal coherence achievable via unitary operations can be evaluated exactly, and is shown to coincide with the corresponding distance-based purity monotone.
Based on these results, we defined a new family of coherence-based purity monotones which admit a closed expression and an operational interpretation in several relevant scenarios. We further proposed a general framework for quantifying purity, following related approaches for entanglement and coherence. This approach also provides quantitative bounds on the required amount of purity to achieve certain levels of entanglement and discord. Lower bounds for a large variety of purity measures are easily accessible in experiments.
We thank Gerardo Adesso, Kaifeng Bu, Dario Egloff, Xueyuan Hu, Martin Plenio, and Alexey Rastegin for discussions. We acknowledge financial support by the Alexander von Humboldt-Foundation, Bundesministerium für Bildung und Forschung, the National Science Center in Poland (POLONEZ UMO-2016/21/P/ST2/04054), and the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 665778.
\[sec:Proof-2\]Proof of Theorem \[thm:Cmax\]
============================================
Here we will prove that $${\mathcal{C}}_{\max}(\rho)=\sup_{U}{\mathcal{C}}(U\rho U^{\dagger})=\max_{U}{\mathcal{C}}(U\rho U^{\dagger})=D(\rho,\openone/d)$$ holds true for any distance-based coherence monotone ${\mathcal{C}}(\rho)=\inf_{\sigma\in\mathcal{I}}D(\rho,\sigma)$ with a contractive distance $D$, i.e., $$D(\Lambda[\rho],\Lambda[\sigma])\leq D(\rho,\sigma)\label{eq:contractivity-2}$$ for any quantum operation $\Lambda$.
Our proof will consist of two steps. In the first step, we will prove the inequality $$\begin{aligned}
{\mathcal{C}}_{\max}(\rho)\leq D\left(\rho,\openone/d\right).\label{eq:bound}\end{aligned}$$ This follows by noting that the maximally mixed state $\openone/d$ is incoherent, and thus gives an upper bound for any distance-based coherence monotone: ${\mathcal{C}}(\rho)\leq D(\rho,\openone/d)$. By contractivity (\[eq:contractivity-2\]) the distance $D$ must be invariant under unitaries, which implies that ${\mathcal{C}}(U\rho U^{\dagger})\leq D(\rho,\openone/d)$ for any unitary $U$. This completes the proof of Eq. (\[eq:bound\]).
To complete the proof of the theorem, we will now show the converse inequality $$\begin{aligned}
{\mathcal{C}}_{\max}(\rho)\geq D\left(\rho,\openone/d\right).\end{aligned}$$ For this, we introduce the unitary $V$ with the property that $\rho_{\max}=V\rho V^{\dagger}$, where $\rho_{\max}=\sum_{n}p_{n}\ket{n_{+}}\!\bra{n_{+}}$ is a maximally coherent mixed state. By definition of ${\mathcal{C}}_{\max}$, it must be that ${\mathcal{C}}_{\max}(\rho)\geq{\mathcal{C}}(V\rho V^{\dagger})$. We further define $\Delta_{+}$ as the dephasing operation in the maximally coherent basis: $$\begin{aligned}
\Delta_{+}[\rho]=\sum_{n}\braket{n_{+}|\rho|n_{+}}\ket{n_{+}}\!\bra{n_{+}}.\end{aligned}$$ It is important to note that the application of $\Delta_{+}$ to any incoherent state $\sigma\in\mathcal{I}$ leads to the maximally mixed state: $\Delta_{+}[\sigma]=\openone/d$. If we further define $\tau\in\mathcal{I}$ to be the closest incoherent state to $\rho_{\max}$, we arrive at the following result: $$\begin{aligned}
{\mathcal{C}}_{\max}(\rho) & \geq{\mathcal{C}}(\rho_{\max})=D(\rho_{\max},\tau)\\
& \geq D\left(\Delta_{+}[\rho_{\max}],\Delta_{+}[\tau]\right)\nonumber \\
& =D\left(\rho_{\max},\openone/d\right)=D\left(\rho,\openone/d\right).\nonumber \end{aligned}$$ In the second line we used contractivity (\[eq:contractivity-2\]) and in the last equality we used unitary invariance of the distance $D$. This completes the proof of the theorem.
We note that the same proof also applies for all coherence quantifiers ${\mathcal{C}}_{p}$ based on Schatten $p$-norms for $p\geq1$. This can be seen by using the same arguments as above, together with the fact that Schatten $p$-norms are contractive under unital operations for all $p\geq1$ [@Contractivity].
\[sec:Proof-3\]Proof of Eq. (\[eq:PC-1\])
=========================================
Here, we will show that any MIO monotone ${\mathcal{C}}$ fulfills the following inequality: $$\sup_{\Lambda_{\mathrm{U}}}{\mathcal{C}}(\Lambda_{\mathrm{U}}[\rho])=\max_{\Lambda_{\mathrm{U}}}{\mathcal{C}}(\Lambda_{\mathrm{U}}[\rho])={\mathcal{C}}(\rho_{\max}),\label{eq:max-unital}$$ where $\rho_{\max}=\sum_{n}p_{n}\ket{n_{+}}\!\bra{n_{+}}$ is a maximally coherent mixed state, $\{p_{n}\}$ is the spectrum of $\rho$, and the supremum is taken over all unital operations $\Lambda_{\mathrm{U}}$.
In the first step of the proof, we recall that unital operations are equivalent to mixtures of unitaries with respect to state transformations, see Lemma 10 in [@GourResThOpPhysRep15]. By using similar arguments as in Appendix \[sec:Proof-2\], we will now show that for any mixture of unitaries $\Lambda_{\mathrm{MU}}[\rho]=\sum_{i}q_{i}U_{i}\rho U_{i}^{\dagger}$ there exists a maximally incoherent operation $\Lambda_{\mathrm{MIO}}$ such that $$\Lambda_{\mathrm{MU}}[\rho_{\max}]=\Lambda_{\mathrm{MIO}}[\rho_{\max}].\label{eq:MU}$$ The desired maximally incoherent operation will be given by $\Lambda_{\mathrm{MIO}}[\rho]=\sum_{i,n}K_{i,n}\rho K_{i,n}^{\dagger}$ with Kraus operators $K_{i,n}=\sqrt{q_{i}}U_{i}\ket{n_{+}}\!\bra{n_{+}}$. It is straightforward to verify that $\sum_{i,n}K_{i,n}\sigma K_{i,n}^{\dagger}=\openone/d$ holds for any incoherent state $\sigma$, which means that the operation is indeed maximally incoherent. Moreover, it holds that $$\sum_{i,n}K_{i,n}\rho_{\max}K_{i,n}^{\dagger}=\sum_{i}q_{i}U_{i}\rho_{\max}U_{i}^{\dagger},$$ which completes the proof of Eq. (\[eq:MU\]).
Together with Lemma 10 in [@GourResThOpPhysRep15], this result implies that for any unital operation $\Lambda_{\mathrm{U}}$ there exists a maximally incoherent operation $\Lambda_{\mathrm{MIO}}$ such that $\Lambda_{\mathrm{U}}[\rho_{\max}]=\Lambda_{\mathrm{MIO}}[\rho_{\max}].$ To complete the proof of Eq. (\[eq:max-unital\]), recall that the states $\rho$ and $\rho_{\max}$ are related via a unitary, i.e., $\rho=U\rho_{\max}U^{\dagger}$. This immediately implies that ${\mathcal{C}}(\rho_{\max})\leq\sup_{\Lambda_{\mathrm{U}}}{\mathcal{C}}(\Lambda_{\mathrm{U}}[\rho])$. On the other hand, the results presented above imply the converse inequality: $$\begin{aligned}
{\mathcal{C}}(\rho_{\max}) & \geq\sup_{\Lambda_{\mathrm{U}}}{\mathcal{C}}(\Lambda_{\mathrm{U}}[\rho_{\max}])=\sup_{\Lambda_{\mathrm{U}}}{\mathcal{C}}(\Lambda_{\mathrm{U}}[\rho]),\end{aligned}$$ where the last equality follows from the fact that $\rho$ and $\rho_{\max}$ are related via a unitary. This completes the proof.
\[sec:Proof-4\]Proof of Eq. (\[eq:distillablepurity-1\])
========================================================
For proving the statement, let $m$ be an integer such that $$\begin{aligned}
\Lambda_{\mathrm{U}}\left[\rho\otimes\frac{\openone}{d_{2}}\right]=\psi_{2}^{\otimes m}\otimes\frac{\openone}{d_{1}}\label{eq:transformation-2}\end{aligned}$$ holds true for some unital operation $\Lambda_{\mathrm{U}}$ and some integers $d_{1}$ and $d_{2}$. Since we require that the unital operation does not change the dimension of the system, we have the additional constraint $$\begin{aligned}
\frac{d_{1}}{d_{2}}=\frac{d}{2^{m}}.\label{eq:condition-3}\end{aligned}$$ From Lemma \[lem:unital\], it follows that the rank of a state cannot decrease under unital operations. Thus, Eq. (\[eq:transformation-2\]) implies $$\begin{aligned}
\frac{d_{1}}{d_{2}}\geq r,\label{eq:condition-4}\end{aligned}$$ where $r$ is the rank of $\rho$. The inequality (\[eq:condition-4\]) implies the majorization relation $$\begin{aligned}
\rho\otimes\frac{\openone}{d_{2}}\succ\psi^{\otimes m}\otimes\frac{\openone}{d_{1}},\end{aligned}$$ as can be seen by recalling that the maximally mixed state is majorized by any other state of the same dimension. Thus, by Lemma \[lem:unital\], Eqs. (\[eq:condition-3\]) and (\[eq:condition-4\]) are necessary and sufficient conditions for the transformation in Eq. (\[eq:transformation-2\]).
Eqs. (\[eq:condition-3\]) and (\[eq:condition-4\]) further imply the inequality $$\begin{aligned}
m\leq\log_{2}\left(d/r\right),\end{aligned}$$ which proves that single-shot distillable purity is bounded above by $\left\lfloor \log_{2}\left(d/r\right)\right\rfloor $. Moreover, it is straightforward to check that Eqs. (\[eq:condition-3\]) and (\[eq:condition-4\]) hold true if we choose $m=\left\lfloor \log_{2}\left(d/r\right)\right\rfloor $, $d_{1}=d$, and $d_{2}=2^{m}$. This completes the proof.
\[sec:Proof-5\]Proof of Eq. (\[eq:puritycost-1\])
=================================================
In the first step of the proof, let $m$ be an integer such that $m$ copies of a pure single-qubit state $\psi_{2}$ can be transformed into the desired state $\rho$ via some unital operation $\Lambda_{\mathrm{U}}$, i.e., $$\begin{aligned}
\Lambda_{\mathrm{U}}\left[\psi_{2}^{\otimes m}\otimes\frac{\openone}{d_{1}}\right]=\rho\otimes\frac{\openone}{d_{2}}\label{eq:transformation}\end{aligned}$$ with some integers $d_{1}$ and $d_{2}$. Since we require that $\Lambda_{\mathrm{U}}$ preserves the dimension of the Hilbert space, it must be that $$\begin{aligned}
\frac{d_{1}}{d_{2}}=\frac{d}{2^{m}}.\label{eq:condition-1}\end{aligned}$$
A necessary requirement for the existence of the unital operation in Eq. (\[eq:transformation\]) is that due to Lemma \[lem:unital\] the maximal eigenvalue of $\rho\otimes\openone/d_{2}$ – which is $\lambda_{\max}/d_{2}$ – is smaller or equal than the maximal eigenvalue of the resource state $\psi_{2}^{\otimes m}\otimes\openone/d_{1}$, i.e. $$\begin{aligned}
\frac{\lambda_{\max}}{d_{2}}\leq\frac{1}{d_{1}}.\label{eq:condition-2}\end{aligned}$$ It is now crucial to note that due to the special form of the resource state, Eq. (\[eq:condition-2\]) directly implies the majorization relation $$\begin{aligned}
\rho\otimes\frac{\openone}{d_{2}}\prec\psi_{2}^{\otimes m}\otimes\frac{\openone}{d_{1}}.\end{aligned}$$ Thus, by Lemma \[lem:unital\], Eqs. (\[eq:condition-1\]) and (\[eq:condition-2\]) are necessary and sufficient for the transformation in Eq. (\[eq:transformation\]).
In the next step, we note that Eqs. (\[eq:condition-1\]) and (\[eq:condition-2\]) imply the following inequality: $$\begin{aligned}
m\geq\log_{2}(d\lambda_{\max}),\end{aligned}$$ which means that the single-shot purity cost is bounded below by $\left\lceil \log_{2}(d\lambda_{\max})\right\rceil $. In the last step, it is straightforward to check that Eqs. (\[eq:condition-1\]) and (\[eq:condition-2\]) hold true if we choose $m=\left\lceil \log_{2}(d\lambda_{\max})\right\rceil $, $d_{1}=d$, and $d_{2}=2^{m}$. This completes the proof.
\[sec:RenyiPurity\]Properties of Rényi $\alpha$-purities
=========================================================
Here we will prove that the Rényi $\alpha$-purity $$\begin{aligned}
{\mathcal{P}}_{\alpha}(\rho)=\log_{2}d-S_{\alpha}(\rho)\end{aligned}$$ is a purity measure, i.e., it fulfills the requirements P1-P4 stated in the main text. For this, we will use the fact that the Rényi entropy is Schur concave for all $\alpha\geq0$ [@Olkin2011]: $$\begin{aligned}
\rho\succ\sigma\Rightarrow S_{\alpha}(\rho)\leq S_{\alpha}(\sigma).\label{eq:SchurConcavity}\end{aligned}$$ We will now prove each of the conditions P1-P4.
- ${\mathcal{P}}_{\alpha}(\openone/d)=0$ follows immediately from $S_{\alpha}(\openone/d)=\log_{2}d$ for all $\alpha$. Furthermore Eq. (\[eq:SchurConcavity\]) and the fact that the maximally mixed state $\openone/d$ is majorized by any other state of the same dimension imply nonnegativity: $$\begin{aligned}
{\mathcal{P}}_{\alpha}(\rho)\geq{\mathcal{P}}_{\alpha}(\openone/d)=0.\end{aligned}$$
- Due to Lemma \[lem:unital\], we have $\rho\succ\Lambda_{\mathrm{U}}[\rho]$ for any unital operation $\Lambda_{\mathrm{U}}$. Eq. (\[eq:SchurConcavity\]) then implies that ${\mathcal{P}}_{\alpha}(\rho)\geq{\mathcal{P}}_{\alpha}(\Lambda_{\mathrm{U}}[\rho])$.
- The Rényi entropy is additive: $S_{\alpha}(\rho\otimes\sigma)=S_{\alpha}(\rho)+S_{\alpha}(\sigma)$. This directly implies additivity of ${\mathcal{P}}_{\alpha}$.
- ${\mathcal{P}}_{\alpha}(\ket{\psi}_{d})=\log_{2}d$, since $S_{\alpha}(\ket{\psi}_{d})=0$ for all $\alpha$.
The Rényi $\alpha$-purity is convex for $0\leq\alpha\le1$, since $S_{\alpha}$ is concave in this region [@Bengtsson2007]. For $\alpha>1$ the Rényi entropy $S_{\alpha}$ is neither concave nor convex [@RenyiEntrConcFiniteAlp78].
\[sec:Proof-6\]Proof of Eq. (\[eq:maxhierarchy\])
==================================================
Let us denote with $U_{{\mathcal{E}}}$ the unitary operation that provides the maximum for ${\mathcal{E}}_{\max}(\rho)$. We find $$\begin{aligned}
{\mathcal{E}}_{\max}(\rho) & ={\mathcal{E}}(U_{{\mathcal{E}}}\rho U_{{\mathcal{E}}}^{\dagger})\leq{\mathcal{D}}(U_{{\mathcal{E}}}\rho U_{{\mathcal{E}}}^{\dagger})\notag\\
& \leq\sup_{U}{\mathcal{D}}(U\rho U^{\dagger})={\mathcal{D}}_{\max}(\rho).\end{aligned}$$ Similarly, let $U_{{\mathcal{D}}}$ be the unitary that leads to ${\mathcal{D}}_{\max}(\rho)$. We obtain $$\begin{aligned}
{\mathcal{D}}_{\max}(\rho) & ={\mathcal{D}}(U_{{\mathcal{D}}}\rho U_{{\mathcal{D}}}^{\dagger})\leq{\mathcal{C}}_{N}(U_{{\mathcal{D}}}\rho U_{{\mathcal{D}}}^{\dagger})\leq\sup_{U}{\mathcal{C}}_{N}(U\rho U^{\dagger})\notag\\
& =\sup_{U}{\mathcal{C}}(U\rho U^{\dagger})={\mathcal{P}}(\rho),\end{aligned}$$ where we used Theorem \[thm:Cmax\] as well as the fact that any two bases can be mapped onto each other by a unitary operation.
\[sec:incoherentent\]Purity bounds on entanglement\
by incoherent operations
===================================================
The amount of entanglement which can be generated by an optimal incoherent operation is bounded by the coherence [@Streltsov2015]: $$\begin{aligned}
{\mathcal{C}}_{\mathrm{r}}(\rho^{A})= & \lim_{d_{B}\rightarrow\infty}\left\{ \sup_{\Lambda_{\mathrm{i}}}{\mathcal{E}}_{\mathrm{r}}^{A:B}\left(\Lambda_{\mathrm{i}}\left[\rho^{A}\otimes{{\ensuremath{| 0 \rangle \!\langle 0 |}}}^{B}\right]\right)\right\} ,\label{eq:activation}\end{aligned}$$ where the supremum is performed over all bipartite incoherent operations $\Lambda_{\mathrm{i}}$ [@Streltsov2015] and ${\mathcal{C}}_{\mathrm{r}}$ and ${\mathcal{E}}_{\mathrm{r}}$ are the relative entropy of coherence and entanglement respectively. Our results from Theorem \[thm:Cmax\] allow us to further connect these results to the relative entropy of purity: Using a unitary to rotate $\rho^{A}$ into a maximally coherent basis followed by the application of the optimal incoherent operation, the generated entanglement amounts to $$\begin{aligned}
{\mathcal{P}}_{\mathrm{r}}(\rho^{A})= & \sup_{U}\lim_{d_{B}\rightarrow\infty}\left\{ \sup_{\Lambda_{\mathrm{i}}}{\mathcal{E}}_{\mathrm{r}}^{A:B}\left(\Lambda_{\mathrm{i}}\left[U\rho^{A}U^{\dag}\otimes{{\ensuremath{| 0 \rangle \!\langle 0 |}}}^{B}\right]\right)\right\} \end{aligned}$$ with the relative entropy of purity ${\mathcal{P}}_{\mathrm{r}}$.
A similar result can be established for the geometric entanglement ${\mathcal{E}}_{\mathrm{g}}(\rho)=1-\max_{\sigma\in\mathcal{S}}F(\rho,\sigma)$ and the geometric coherence ${\mathcal{C}}_{\mathrm{g}}(\rho)=1-\max_{\sigma\in\mathcal{I}}F(\rho,\sigma)$, recalling that Eq. (\[eq:activation\]) also holds true for these quantities [@Streltsov2015]. If we introduce the geometric purity as ${\mathcal{P}}_{\mathrm{g}}(\rho)=1-F(\rho,\openone/d)=1-\frac{1}{d}({\text{Tr\,}}\sqrt{\rho})^{2}$, we immediately obtain the following result: $$\begin{aligned}
{\mathcal{P}}_{\mathrm{g}}(\rho^{A})= & \sup_{U}\lim_{d_{B}\rightarrow\infty}\left\{ \sup_{\Lambda_{\mathrm{i}}}{\mathcal{E}}_{\mathrm{g}}^{A:B}\left(\Lambda_{\mathrm{i}}\left[U\rho^{A}U^{\dag}\otimes{{\ensuremath{| 0 \rangle \!\langle 0 |}}}^{B}\right]\right)\right\} .\end{aligned}$$
In [@Orieux2015] a CNOT-gate ($U_{\mathrm{CNOT}}$) is used to create entanglement out of the two-qubit input state $$\begin{aligned}
\rho_{\text{in}}=\rho^{A}\otimes{{\ensuremath{| 0 \rangle \!\langle 0 |}}}^{B}\end{aligned}$$ with system $A$ being the control qubit system and $B$ being the target qubit, i.e. $\rho_{\text{out}}=U_{\mathrm{CNOT}}\rho_{\text{in}}U_{\mathrm{CNOT}}^{\dag}$. In this two-qubit scenario the entanglement of the state $\rho_{\text{out}}$ can be measured by the negativity $\mathcal{N}(\rho)=\sum_{j}|\lambda_{j}^{-}|$ where $\lambda_{j}^{-}$ are the negative eigenvalues of the partial transpose of $\rho$ [@Peres1996; @Horodecki1996; @PhysRevA.58.883; @Vidal2002]. The negativity of $\rho_{\mathrm{out}}$ is closely related to the $l_{1}$-norm of coherence of the state $\rho^{A}$ [@Nakano2013]: $$\begin{aligned}
\mathcal{N}(\rho_{\text{out}})={\left| \rho_{01}^{A}\right|}=\frac{{\mathcal{C}}_{\ell_{1}}(\rho^{A})}{2},\end{aligned}$$ with $\rho_{01}^{A}$ being the off-diagonal element of the chosen qubit basis.
For a single qubit there is a direct relation between ${\mathcal{C}}_{\ell_{1}}$ and the geometric coherence [@Streltsov2015]: $C_{\ell_{1}}=\sqrt{1-(1-2C_{\mathrm{g}})^{2}}$. Using Theorem \[thm:Cmax\] to bound the geometric coherence by the geometric purity, we obtain the following bound for the negativity $$\begin{aligned}
\mathcal{N}(\rho_{\text{out}})\leq\sqrt{1-(1-2{\mathcal{P}}_{\mathrm{g}})^{2}},\end{aligned}$$ where equality holds if the eigenstates of $\rho^{A}$ form a maximally coherent basis.
[^1]: Throughout this paper, a quantum operation (or just “operation”) is used as a synonym for a completely positive trace-preserving map.
[^2]: Here, the term “extremal” is not synonymous to extremal elements of convex sets.
[^3]: This has to be understood in the asymptotic setting, where “rate” means the asymptotic fraction of required (distilled) resource states per copy of the desired (given) quantum state $\rho$. Widely used examples for such asymptotic rates are entanglement cost and distillable entanglement, we refer to Ref. [@Plenio2007] for their formal definition.
[^4]: We note that the quantum relative entropy is not a distance in the mathematical sense, as it is not symmetric and does not fulfill the triangle inequality.
[^5]: Note that the dimensions $d_{1}$ and $d_{2}$ in Eq. (\[eq:distillablepurity\]) are arbitrary finite numbers, up to the requirement that $d\times d_{2}=2^{m}\times d_{1}$. This guarantees that the unital operation preserves the dimension of the Hilbert space. As we show in Appendix \[sec:Proof-4\], the optimal choice is $d_{1}=d$ and $d_{2}=2^{\left\lfloor \log_{2}(d/r)\right\rfloor }$, where $r$ is the rank of $\rho$. This only applies if $\log_{2}(d/r)\geq1$, as single-shot purity distillation does not work otherwise. By similar considerations, the optimal choice of dimensions in Eq. (\[eq:puritycost\]) is $d_{1}=d$ and d$_{2}=2^{\left\lceil \log_{2}(d\lambda_{\max})\right\rceil }$, where $\lambda_{\max}$ is the maximal eigenvalue of $\rho$, see Appendix \[sec:Proof-5\] for more details.
[^6]: Notice that $\mathrm{Tr}[\rho^{2}]-1/d$ corresponds to the purity monotone obtained from the squared Schatten 2-norm $||\rho-\openone/d||_{2}^{2}$.
[^7]: This is a direct consequence of the fact that nonselective von Neumann measurements are unital operations.
|
---
abstract: 'We develop a Feynman diagram approach to calculating correlations of the Cosmic Microwave Background (CMB) in the presence of distortions. As one application, we focus on CMB distortions due to gravitational lensing by Large Scale Structure (LSS). We study the Hu-Okamoto quadratic estimator for extracting lensing from the CMB and derive the noise of the estimator up to ${{\mathcal O}(\phi^4)}$ in the lensing potential $\phi$. By identifying the diagrams responsible for the previously noted large ${{\mathcal O}(\phi^4)}$ term, we conclude that the lensing expansion does not break down. The convergence can be significantly improved by a reorganization of the $\phi$ expansion. Our approach makes it simple to obtain expressions for quadratic estimators based on any CMB channel, including many previously unexplored cases. We briefly discuss other applications to cosmology of this diagrammatic approach, such as distortions of the CMB due to patchy reionization, or due to Faraday rotation from primordial axion fields.'
author:
- 'Elizabeth E. Jenkins'
- 'Aneesh V. Manohar'
- 'Wouter J. Waalewijn'
- 'Amit P. S. Yadav'
bibliography:
- 'lensing.bib'
title: 'Gravitational Lensing of the CMB: a Feynman Diagram Approach'
---
[^1]
[*Introduction–*]{} Primary anisotropies in the Cosmic Microwave Background (CMB) were generated around $375,000$ years after the “big bang," when the universe was still in the linear regime. The CMB field can be decomposed and studied in terms of its temperature $T$ and polarization modes $E$ and $B$. Primordial scalar perturbations create only $E$ modes of the CMB, while primordial tensor perturbations generate both parity-even $E$ modes and parity-odd $B$ polarization modes [@Seljak:1996gy; @Kamionkowski:1996ks; @Kamionkowski:1996zd]. The recent detection of primordial $B$ modes [@Ade:2014xna] constrains the ratio of tensor to scalar perturbations as well as the energy scale at which inflation happened [@Baumann:2008aq].
The primordial CMB generated at the surface of last scattering is statistically isotropic and Gaussian. However, during the photon’s journey to us, it encounters several distorting fields, which make the CMB non-Gaussian and statistically anisotropic. Examples of such distorting fields are (a) gravitational lensing which bends the light as photons travel though the LSS [@Seljak:1995ve; @Zaldarriaga:1998ar; @Seljak:1998nu; @Zaldarriaga:2000ud], (b) patchy reionization which modulates the CMB intensity because of scattering when Hydrogen reionizes [@Dvorkin:2008tf], and (c) cosmological rotation, due to parity-violating physics (e.g. axions), which rotates the plane of polarization of the CMB [@Carroll:1998zi; @Kamionkowski:2008fp; @Yadav:2009eb; @Gluscevic:2009mm]. By coupling different modes of the CMB, the distortion imprints its signature on the observed CMB by breaking statistical isotropy and introducing non-Gaussianities. All these distortions also produce $B$-modes that contaminate the primordial tensor $B$-mode signal.
One can utilize the statistical anisotropy of the observed CMB to reconstruct the distorting fields. Estimators based on the Hu-Okamoto quadratic estimator [@Hu:2001kj; @Okamoto:2003zw] are the most studied method for extracting these distortions. In this paper, we present a new diagrammatic way of studying distortions using such estimators and employ this method to investigate the noise properties of the estimator. We show that the previously unexplained large $N^{(2)}$ noise can be understood from the contributing diagrams, and reduced by reorganizing the expansion. Our approach automatically yields expressions for all possible channel combinations of the quadratic estimators, including cross channels like $TE\,EB$, some of which are particularly interesting due to their low noise [@Jenkins:2014hza].
[*Distortions in the CMB–*]{} The primordial CMB is statistically isotropic and Gaussian, so all information is contained in the power spectrum, $\langle x_{{\bm \ell}}\, y_{{\bm k}}\rangle_\text{CMB}=C^{xy}_{{\bm \ell}}(2\pi)^2 \delta^2({{\bm \ell}}+{{\bm k}})$, where the average is over CMB realizations and we work in the flat sky approximation. Here ${{\bm \ell}}$ and ${{\bm k}}$ denote the Fourier modes, and the power spectrum only depends on $\ell = |{{\bm \ell}}|$. The $x,y\in \{T,E,B\}$ are temperature and polarization components of the CMB, which can conveniently be combined into a column vector $X$ such that $C^{xy}_{{\bm \ell}}$ are components of a $3\times 3$ CMB power spectrum matrix $C_{{\bm \ell}}$, $$\begin{aligned}
\langle X_{{\bm \ell}}\, X^T_{{\bm k}}\rangle_\text{CMB} &= \langle \begin{pmatrix}T_{{\bm \ell}}\\ E_{{\bm \ell}}\\ B_{{\bm \ell}}\end{pmatrix} \begin{pmatrix}T_{{\bm k}}& E_{{\bm k}}&B_{{\bm k}}\end{pmatrix}\rangle_\text{CMB}
{\nonumber}\\ &
= C_{{\bm \ell}}(2\pi)^2 \delta^2({{\bm \ell}}+{{\bm k}})
\,.\end{aligned}$$ Secondary distortions, such as gravitational lensing, patchy reionization and Faraday rotation, will modify the components of $X$ [@Yadav:2009za; @Dvorkin:2008tf; @Yadav:2009eb; @Gluscevic:2009mm; @Kamionkowski:2008fp; @Yadav:2012uz]. The effect of these distortion fields on Fourier modes may generically be written as $$\label{eq:gen}
\widetilde X_{{\bm \ell}}= {\int\! \frac{d^2 {{\bm m}}}{(2\pi)^2}}\, D_{({{\bm \ell}},{{\bm m}})}\,X_{{\bm m}}\,,$$ where the matrix $D_{({{\bm \ell}},{{\bm m}})}$ can mix components of the CMB. $X_{{\bm m}}$ is the primordial spectrum, and $\widetilde X_{{\bm \ell}}$ is the distorted (observed) spectrum. We now focus on how to calculate the effect of gravitational lensing using Feynman diagrams, but we will comment on other distortions in the final discussion.
[*Gravitational lensing–*]{} Lensing deflects the path of CMB photons from the last scattering surface. This deflection results in a remapping of the CMB temperature/polarization pattern on the sky, ${\hat{\bm n}}\to {\hat{\bm n}}+ {{\bm d}}({\hat{\bm n}})$, and mixes the $E$ and $B$ polarization modes. The deflection is given by, ${{\bm d}}({\hat{\bm n}}) = \nabla\phi ({\hat{\bm n}})$, where the lensing potential $\phi({\hat{\bm n}})$ is obtained by integrating the gravitational potential along the line of the sight [@Seljak:1995ve]. There are higher-order corrections to the lensing potential [@Bernardeau:1996un; @Cooray:2002mj; @Hirata:2003ka; @Cooray:2005hm; @Namikawa:2011cs], which for simplicity we will ignore here. Treating $\phi$ as a Gaussian field with power spectrum $C^{\phi\phi}_\ell$ [@Jenkins:2014hza], $$\begin{aligned}
\label{eq:lensing}
D^\text{Lensing}_{({{\bm \ell}},{{\bm m}})} &= R_{({{\bm \ell}},{{\bm m}})}\,(2\pi)^2 \delta^2({{\bm \ell}}-{{\bm m}}-{{\mathcal P}})
{\nonumber}\\ & \quad \times
\exp\Big[ -{\int\! \frac{d^2 {{\bm k}}}{(2\pi)^2}}\,({{\bm k}}{\!\cdot\!}{{\bm m}})\, \phi_{{\bm k}}\Big]
\,,\end{aligned}$$ where ${{\mathcal P}}$ gives the total momentum of all the $\phi$ fields and $R_{({{\bm \ell}},{{\bm m}})}$ encodes the mixing of $E$ and $B$ polarizations, $$R_{({{\bm \ell}},{{\bm m}})} =
\begin{pmatrix}
1 & 0 & 0 \\
0 & \cos 2\varphi({{\bm \ell}},{{\bm m}}) & \sin 2\varphi({{\bm \ell}},{{\bm m}}) \\
0 & -\sin 2\varphi({{\bm \ell}},{{\bm m}}) & \cos 2\varphi({{\bm \ell}},{{\bm m}})
\end{pmatrix}
\,.$$ Here, $\varphi({{\bm \ell}},{{\bm m}})$ is the (oriented) angle between ${{\bm \ell}}$ and ${{\bm m}}$. The lowest order terms in [Eq. ]{} produce the familiar result [@Hu:2001kj] $$\begin{aligned}
D^\text{Lensing}_{({{\bm \ell}},{{\bm m}})} &= (2\pi)^2 \delta^2({{\bm \ell}}-{{\bm m}})
-R_{({{\bm \ell}},{{\bm m}})} \big[({{\bm \ell}}-{{\bm m}}) {\!\cdot\!}{{\bm m}}\big]\, \phi_{{{\bm \ell}}-{{\bm m}}} {\nonumber}\\ & \quad+ {{\mathcal O}(\phi^2)}
\,.\end{aligned}$$
----------- ----------------------------------------------------------------------------------------------------------
$C^{xy}_{{\bm m}}$
$C^{\phi\phi}_{{\bm k}}$
\[1cm\] $\begin{array}{c} R_{({{\bm \ell}},{{\bm m}})}^{xy} \prod_i (-{{\bm k}}_i {\!\cdot\!}{{\bm m}}) \\[10pt]
\text{with }{{\bm \ell}}= {{\bm m}}+ \sum_i {{\bm k}}_i \end{array}$
\[2.5cm\] $\dfrac{A_{{{\bm L}}}}{L^2} F^{xy}_{({{\bm \ell}},{{\bm L}}-{{\bm \ell}})}$
----------- ----------------------------------------------------------------------------------------------------------
[*Feynman diagrams for lensing–*]{} [Eq. ]{} yields a simple Feynman rule when calculating the average of several CMB modes $\langle \widetilde x({{\bm \ell}}) \widetilde y({{\bm k}}) \dots \rangle$ over CMB or LSS realizations (see [Fig. \[fig:rules\]]{}). In the calculation of such an average, each lensed field $\widetilde x({{\bm \ell}})$ is represented as a vertex with momentum ${{\bm \ell}}$ flowing in. It has one straight line (the unlensed field) and arbitrary many wiggly lines (the lensing field $\phi$) connected to it. When averaging over CMB realizations, each straight line must begin and end at a vertex. It corresponds to $C^{xy}_{{\bm m}}$ for the CMB components $x$ and $y$, where ${{\bm m}}$ is the momentum flowing through the line. Similarly, each wiggly line corresponds to $C^{\phi\phi}_{{\bm k}}$ when averaging over LSS realizations, where ${{\bm k}}$ is the wiggly line momentum. Momentum is conserved at the vertex and each unconstrained internal momentum ${{\bm k}}$ is integrated over with $d^2{{\bm k}}/(2\pi)^2$. In addition, there is a factor corresponding to total momentum conservation $(2\pi)^2 \delta^2({{\bm \ell}}+ {{\bm k}}+ \dots)$ which is typically pulled out front (see e.g. [Eq. ]{}). These rules are summarized in [Fig. \[fig:rules\]]{} and will be illustrated with explicit examples below.
As a simple example, we calculate the lensed CMB spectra $\widetilde C_{{\bm \ell}}$. The diagrams contributing to $\widetilde C_{{{\bm \ell}}}$ are shown in [Fig. \[fig:Cl\]]{}. Using the rules from [Fig. \[fig:rules\]]{} gives $$\begin{aligned}
\label{eq:C_lensed}
\langle \widetilde X_{{\bm \ell}}\widetilde X_{{{\bm \ell}}'}^T \rangle &= (2\pi)^2 \delta^2({{\bm \ell}}+ {{\bm \ell}}') \widetilde C_{{\bm \ell}}\\ &
= (2\pi)^2 \delta^2({{\bm \ell}}+ {{\bm \ell}}') \Big\{ C_{{\bm \ell}}+{\int\! \frac{d^2 {{\bm k}}}{(2\pi)^2}}\Big [ -C_{{{\bm \ell}}} C^{\phi \phi}_{{\bm k}}({{\bm k}}{\!\cdot\!}{{\bm \ell}})^2
{\nonumber}\\ & \quad
+ R_{({{\bm \ell}},{{\bm \ell}}-{{\bm k}})}C_{{{\bm \ell}}-{{\bm k}}}R^T_{({{\bm \ell}},{{\bm \ell}}-{{\bm k}})} C^{\phi \phi}_{{\bm k}}\big({{\bm k}}{\!\cdot\!}({{\bm \ell}}-{{\bm k}})\big)^2 \Big] \Big\}
\,,{\nonumber}\end{aligned}$$ where both $C$ and $\widetilde C$ are $3 \times 3$ matrices in $\{T,E,B\}$. Graph (a) is the unlensed spectrum $C_{{\bm \ell}}$ and graph (b) yields the third line. Graphs (c) and (d) give identical contributions, are multiplied by a symmetry factor of $1/2$, and simplify due to $R_{({{\bm \ell}}, {{\bm \ell}})} = 1$, giving the last term on the second line.
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --
![Diagrams contributing to lensed spectra (denoted by a double line). Graph (a) is the primordial contribution, and (b) - (d) describe corrections due to lensing.[]{data-label="fig:Cl"}](fig2a "fig:"){width="12.00000%"} = \[c\]\[c\][${{\bm \ell}}$]{} ![Diagrams contributing to lensed spectra (denoted by a double line). Graph (a) is the primordial contribution, and (b) - (d) describe corrections due to lensing.[]{data-label="fig:Cl"}](fig2b "fig:"){width="12.00000%"} + \[c\]\[c\] \[c\]\[c\][${{\bm \ell}}-{{\bm k}}$]{} ![Diagrams contributing to lensed spectra (denoted by a double line). Graph (a) is the primordial contribution, and (b) - (d) describe corrections due to lensing.[]{data-label="fig:Cl"}](fig2c "fig:"){width="12.00000%"}
$\widetilde C_{{\bm \ell}}$ (a) (b)
+\[c\]\[c\] \[c\]\[c\][${{\bm \ell}}$]{} ![Diagrams contributing to lensed spectra (denoted by a double line). Graph (a) is the primordial contribution, and (b) - (d) describe corrections due to lensing.[]{data-label="fig:Cl"}](fig2d "fig:"){width="14.00000%"} + \[c\]\[c\] \[c\]\[c\][${{\bm \ell}}$]{} ![Diagrams contributing to lensed spectra (denoted by a double line). Graph (a) is the primordial contribution, and (b) - (d) describe corrections due to lensing.[]{data-label="fig:Cl"}](fig2e "fig:"){width="14.00000%"}
(c) (d)
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --
[*Quadratic estimator and noise terms–*]{} Lensing breaks the statistical isotropy, correlating the CMB modes, $$\begin{aligned}
\langle \widetilde x_{{{\bm \ell}}} \widetilde y_{{{\bm L}}-{{\bm \ell}}}\rangle_\text{CMB}
&= (2\pi)^2 \delta^2({{\bm L}}) \widetilde C^{xy}_{{\bm L}}\! \nonumber \\
&+ \!\big[{f^{(\phi,0)xy}_{({{\bm \ell}},{{\bm L}}-{{\bm \ell}})}} \!+\! {f^{(\phi,1)xy}_{({{\bm \ell}},{{\bm L}}-{{\bm \ell}})}} \!+\! \dots \big] \phi_{{\bm L}}\nonumber\\ & \quad
+ {\int\! \frac{d^2 {{\bm m}}}{(2\pi)^2}}\, {f^{(\phi\phi,0)xy}_{({{\bm \ell}},{{\bm L}}-{{\bm \ell}},{{\bm m}})}} \phi_{{{\bm L}}-{{\bm m}}} \phi_{{\bm m}}+ \dots
\,, \label{eq:2point}\end{aligned}$$ which can be used to reconstruct the lensing field from the CMB. The superscript $0,1$ on $f$ denotes its order in powers of $C^{\phi\phi}$. A quadratic estimator for the lensing potential can be written as $$\begin{aligned}
\label{eq:estimator}
\hat \phi^{xy}_{{\bm L}}= \frac{A^{xy}_L}{L^2} {\int\! \frac{d^2 {{\bm \ell}}}{(2\pi)^2}}\, F^{xy}_{({{\bm \ell}},{{\bm L}}-{{\bm \ell}})}
\widetilde x_{{\bm \ell}}\widetilde y_{{{\bm L}}-{{\bm \ell}}}
\,,\end{aligned}$$ and its Feynman rule is given in [Fig. \[fig:rules\]]{}. Following [@Hu:2001kj], the normalization $A_L$ is chosen so that [Eq. ]{} yields an unbiased estimator, $\langle \hat \phi^{xy}_{{\bm L}}\rangle_\text{CMB} = \phi_{{\bm L}}$, and the filter $F^{xy}$ is determined by minimizing the variance $\langle \langle \hat \phi_{{\bm L}}^{xy} \hat \phi_{{{\bm L}}'}^{x'y'} \rangle_\text{CMB} - \langle \hat \phi_{{\bm L}}^{xy} \rangle_\text{CMB} \langle \hat \phi_{{{\bm L}}'}^{x'y'} \rangle_\text{CMB} \rangle_\text{LSS}$ at lowest order in the lensing expansion.
-------------- -----
\[-2ex\] (a) (b)
-------------- -----
$\!\!\!\!=$ $+$
-------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![Diagrams contributing at ${{\mathcal O}(\phi^2)}$. Diagram (a) produces $C^{\phi\phi}$ and (b) gives $N^{{(1)}}$. The cross graph for (b) (analogous to [Fig. \[fig:N0\]]{}(b)) is not shown.[]{data-label="fig:N1"}](fd31 "fig:"){width="17.50000%"}
\[-2ex\] (a) (b)
-------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Using the quadratic estimator to extract the lensing power spectrum introduces a bias $$\begin{aligned}
\label{eq:bias}
\langle \hat \phi_{{\bm L}}^{xy} \hat \phi_{{{\bm L}}'}^{x'y'} \rangle_\text{CMB,LSS}
&= (2\pi)^2 \delta^2({{\bm L}}\!+\! {{\bm L}}') \big[C^{\phi\phi}_{{\bm L}}\!+\! N^{xy,x'y' {{(0)}}}_{{\bm L}}\!
{\nonumber}\\ & \quad
+\!N^{xy,x'y' {{(1)}}}_{{\bm L}}+\!N^{xy,x'y' {{(2)}}}_{{\bm L}}\!+\! {{\mathcal O}(\phi^6)}\big]
\,,\end{aligned}$$ given by the noise terms $N^{xy,x'y' (n)}_{{\bm L}}$, which are ${{\mathcal O}(\phi^{2n})}$. The Gaussian noise $N^{{{(0)}}}_{{\bm L}}$ is expected to provide the dominant contribution to the variance. However, it has been recently noticed that the higher order noise term $N^{{{(2)}}}_{{\bm L}}$ can give a large contribution at small $L$ [@Hanson:2010rp; @Anderes:2013jw]. One of the main goals of our paper is to illustrate the power of Feynman diagrams in calculating the higher-order contributions $N^{{{(1)}}}_{{\bm L}}$ and $N^{{{(2)}}}_{{\bm L}}$, which also makes it easy to track down the origin of this large contribution.
The two diagrams contributing to the lowest order noise term are shown in [Fig. \[fig:N0\]]{} and lead to $$\begin{aligned}
\label{eq:N0}
\hspace{-2ex}N^{xy,x'y'(0)}_{{\bm L}}&= \frac{A^{xy}_{{\bm L}}A^{x'y'}_{{\bm L}}}{L^4} {\int\! \frac{d^2 {{\bm \ell}}}{(2\pi)^2}}\, F^{xy}_{({{\bm \ell}},{{\bm L}}-{{\bm \ell}})}
\nonumber \\
& \times \Big[ F^{x'y'}_{(-{{\bm \ell}},{{\bm \ell}}-{{\bm L}})}
{\overline C}^{xx'}_{{\bm \ell}}{\overline C}^{yy'}_{{{\bm L}}-{{\bm \ell}}} + F^{x'y'}_{({{\bm \ell}}-{{\bm L}},-{{\bm \ell}})} {\overline C}^{xy'}_{{\bm \ell}}{\overline C}^{x'y}_{{{\bm L}}-{{\bm \ell}}} \Big]
\,,\end{aligned}$$ in agreement with Ref. [@Hu:2001kj]. Here ${\overline C}^{xy}_{{\bm \ell}}=\widetilde C^{xy}_{{\bm \ell}}+ \Delta^2_{xy} \, e^{\ell (\ell+1)\sigma^2/8\ln2}$ is the observed spectra, $\sigma$ is the full-width-half-maximum of the experimental beam, and $\Delta_{xy}$ is experimental noise [@Knox:1995dq]. We will assume fully polarized detectors for which $\Delta_{EE}=\Delta_{BB}=\sqrt{2}\Delta_{TT}$, and $\Delta_{xx'}=0$ for $x \neq x^\prime$.
Although using lensed rather than unlensed spectra in $\overline C$ is formally beyond the order in $\phi$ of $N^{{(0)}}_{{\bm L}}$, it reduces the number of diagrams contributing to the higher-order noise. Specifically, corrections of the type shown in [Fig. \[fig:Cl\]]{}(b) - (d) are now already included. This approach is standard for the Gaussian noise $N^{{(0)}}_{{\bm L}}$, but we find that also using lensed spectra in the higher-order noise terms improves their convergence. We will compare using lensed vs. unlensed spectra when we present numerical results in [Fig. \[fig:Nl\]]{}. We also use lensed spectra everywhere in the filter $F$ of the estimator, which has been considered in Refs. [@Lewis:2011fk; @Anderes:2013jw].
The diagrams contributing at ${{\mathcal O}(\phi^2)}$ are shown in [Fig. \[fig:N1\]]{}, which we break into subgraphs involving the filter $f^{(\phi,0)}$ defined in [Eq. ]{}. This filter $f^{(\phi,0)}$ describes the distortion of the two-point function due to lensing, as shown in [Fig. \[fig:f0\]]{}. In these figures the “crossing out" of lines indicates that they do not produce a power spectrum in the corresponding expression. [Fig. \[fig:N1\]]{}(a) produces the lensing spectrum $C^{\phi\phi}_{{\bm L}}$ by construction. [Fig. \[fig:N1\]]{}(b) and the corresponding cross graph can be calculated using the Feynman rules in [Fig. \[fig:rules\]]{}, $$\begin{aligned}
\label{eq:N1}
{f^{(\phi,0)}_{({{\bm \ell}},{{\bm \ell}}')}}&=R_{({{\bm \ell}},{{\bm \ell}}')} C_{{{\bm \ell}}'} ({{\bm \ell}}+{{\bm \ell}}')\cdot {{\bm \ell}}'
\\ & \quad
+ C_{{\bm \ell}}R^T_{({{\bm \ell}}',{{\bm \ell}})} ({{\bm \ell}}+{{\bm \ell}}')\cdot {{\bm \ell}}\,,{\nonumber}\\
N^{xy,x'y'(1)}_{{\bm L}}&= \frac{A^{xy}_{{\bm L}}A^{x'y'}_{{\bm L}}}{L^4} {\int\! \frac{d^2 {{\bm \ell}}}{(2\pi)^2}}\frac{d^2 {{\bm k}}} {(2\pi)^2}\, C^{\phi \phi}_{{\bm k}}F^{xy}_{({{\bm \ell}},{{\bm L}}-{{\bm \ell}})}
{\nonumber}\\ & \quad \times
\Big[ F^{x'y'}_{({{\bm k}}-{{\bm \ell}},{{\bm \ell}}-{{\bm L}}-{{\bm k}})} {f^{(\phi,0)xx'}_{({{\bm \ell}},{{\bm k}}-{{\bm \ell}})}} {f^{(\phi,0)yy'}_{({{\bm L}}-{{\bm \ell}},{{\bm \ell}}-{{\bm L}}-{{\bm k}})}}
{\nonumber}\\ & \qquad
+ F^{x'y'}_{({{\bm \ell}}-{{\bm L}}-{{\bm k}},{{\bm k}}-{{\bm \ell}})} {f^{(\phi,0)xy'}_{({{\bm \ell}},{{\bm k}}-{{\bm \ell}})}} {f^{(\phi,0)yx'}_{({{\bm L}}-{{\bm \ell}},{{\bm \ell}}-{{\bm L}}-{{\bm k}})}} \Big]
\,.{\nonumber}\end{aligned}$$ This noise contribution was first determined by Kesden et al. [@Cooray:2002py; @Kesden:2003cc] (for $x=x'$ and $y=y'$). Note that corrections of the form shown in [Fig. \[fig:Cl\]]{}(b) through (d) were already part of the calculation of $N^{{(0)}}_{{\bm L}}$ by using lensed spectra there, and thus should not be included in $N^{{(1)}}_{{\bm L}}$. We will also consider using lensed spectra $\widetilde C_{{\bm \ell}}$ and $\widetilde C_{{{\bm \ell}}'}$ instead of $C_{{\bm \ell}}$ and $C_{{{\bm \ell}}'}$ in ${f^{(\phi,0)}_{}}$.
There are two classes of diagrams contributing to $N^{(2)}_{{\bm L}}$. The first class of diagrams is the same form as those in [Fig. \[fig:N1\]]{} and [Eq. ]{}, but with one of the $f^{(\phi,0)}$ vertices replaced by the higher order $f^{(\phi,1)}$. The second class of diagrams is shown in [Fig. \[fig:N2\]]{}, and involves the new filter $f^{(\phi\phi,0)}$ in [Fig. \[fig:g\]]{}. Expressions for $f^{(\phi,1)}$, $f^{(\phi\phi,0)}$ and $N^{(2)}$ will be given in Ref. [@Jenkins:2014hza]. We have identified the ${{\mathcal O}(\phi^4)}$ analogue of [Fig. \[fig:N1\]]{}(a) as the contribution that is responsible for the large size of $N^{{(2)}}$ that had been observed in Refs. [@Hanson:2010rp; @Anderes:2013jw]. To understand this, it is useful to first discuss the contributions at order $\phi^2$: [Fig. \[fig:N1\]]{}(a) and (b) yield $C^{\phi\phi}$ and $N^{(1)}$ and although they are formally of the same order in the lensing expansion, $C^{\phi\phi}$ is numerically larger. This is to be expected because [Fig. \[fig:N1\]]{}(a) has less loop integrals than (b). The same is true at ${{\mathcal O}(\phi^4)}$ and there is thus no breakdown of perturbation theory. Instead, the graph in [Fig. \[fig:N1\]]{}(a) and corresponding higher order contributions give a convergent expansion but one that is numerically larger than the diagrams in [Fig. \[fig:N1\]]{}(b), [Fig. \[fig:N2\]]{}, etc.
The numerical size of this contribution can be significantly reduced by organizing the expansion in terms of lensed spectra $\widetilde C^{xy}$, i.e. replacing $C^{xy} \to \widetilde C^{xy}$ in $f^{(\phi,0)}$ and compensating for this change in $f^{(\phi,1)}$ (with $f^{(\phi,1)}$ also written in terms of $\widetilde C^{xy}$). This essentially sums a class of higher order corrections, as we already discussed for $N^{{(0)}}$. The results are shown in [Fig. \[fig:Nl\]]{}, which compares using lensed to unlensed spectra in the computation of $N^{(2)}$ and will be discussed below. Since the estimator minimizes the leading order variance, and this reorganization changes what is called leading order, the estimator is modified as well.
------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------
![Diagrams contributing to the filter $f^{(\phi\phi,0)}$.[]{data-label="fig:g"}](fd32 "fig:"){width="17.50000%"} ![Diagrams contributing to the filter $f^{(\phi\phi,0)}$.[]{data-label="fig:g"}](fd33 "fig:"){width="17.50000%"}
(a) (b)
(c) (d)
------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------
$=$ $+$ $+$
[*Numerical Results–*]{} In [Fig. \[fig:Nl\]]{}, we show the noise $N^{TT}_L$ in estimating the lensing $C^{dd}_L= L^2 C^{\phi\phi}_L$ as a function of $L$ for a Planck-like experiment with experimental noise $\Delta_{EE}=56\, \mu \text{K}$-arcmin and beam size $\sigma=7$ arcmin. We consider noise calculated using two counting methods, the unlensed spectra (dotted curves) as well as lensed spectra (solid curves). Our results for the former agree with Ref. [@Hanson:2010rp], showing that at small $L$ the bias $N^{TT{{(2)}}}_L$ is large. As we explained, this originates from higher-order corrections to [Fig. \[fig:N1\]]{}(a), so it is not surprising that its shape is similar to $C^{dd}_L$. [Fig. \[fig:Nl\]]{} clearly illustrates that using lensed spectra greatly improves the convergence of the noise terms. The main difference with Ref. [@Anderes:2013jw] is that in addition to changing our estimator to use lensed spectra, we have also reorganized our noise in terms of lensed spectra. This use of lensed spectra modifies $N^{{(0)}}$ and $N^{{(1)}}$ and is responsible for the improved convergence we see, in contrast to the seemingly accidental cancellation between $N^{TT{{(1)}}}_L$ and $N^{TT{{(2)}}}_L$ found in Ref. [@Anderes:2013jw].
![Lensing signal $C^{dd}_L=L^2 C^{\phi\phi}_L$ and noise power spectra $N^{TT(n)}_L$ of the quadratic estimator for $C^{dd}_L$. Lensed spectra are used for the solid curves, as described in the text, improving the convergence. We have assumed experimental noise $\Delta_{EE}=56\,\mu \text{K}$-arcmin and beam $\sigma =7'$. The sign of the noise $N^{(2)}_L$ changes as a function of $L$; negative values are shown in red and positive values in magenta.[]{data-label="fig:Nl"}](NTT_planck){width="50.00000%"}
[*Discussion–*]{} We have shown how Feynman diagrams can be used to understand the CMB, illustrating their power in the context of gravitational lensing. This method allowed us to simultaneously obtain expressions for quadratic estimators based on any CMB channel and identify the origin of the (supposed) poor convergence of higher order noise terms. Additional details, as well as plots for the polarization channels, such as $TT\, EE$ and $EE\,EB$ are given in a subsequent publication [@Jenkins:2014hza].
Apart from lensing, there are other cosmological effects that can couple the modes of the CMB [@Yadav:2009za] such as screening from patchy reionization [@Dvorkin:2008tf], and rotation of the plane of polarization either due to primordial magnetic fields [@Kosowsky:1996yc; @Kosowsky:2004zh; @Yadav:2012uz] or parity-violating physics [@Kamionkowski:2008fp; @Yadav:2009eb; @Gluscevic:2009mm]. The formalism presented here can be used to study these effects as well (see [Eq. ]{}). Below we discuss cosmological rotation and patchy reionization and show how Feynman rules can be derived for them.\
[*Cosmological rotation and patchy reionization–*]{} Many theories predict parity-violating primordial fields such as axions, which have Chern-Simons couplings of the form $a F_{\mu \nu}\tilde F^{\mu \nu}$ [@Carroll:1998zi; @Pospelov:2008gg], that rotate the plane of polarization of light through an angle $d\alpha = 2 \it d\tau \dot a$ during propagation for a conformal time $d\tau$. The fluctuations in the axion field $a$ then will be imprinted in the rotation angle $\alpha$ of the polarization. The observed (rotated) and primordial CMB in terms of Stokes parameters are related by $(\widetilde Q\pm i \widetilde U)({\bf n})=e^{\pm 2i \alpha(\bf n)}( Q\pm i U)({\bf n})$, which we can write in terms of [Eq. ]{} as $$\begin{aligned}
\label{eq:daxion}
D^\text{Rotation}_{({{\bm \ell}},{{\bm m}})} &= (2\pi)^2 \delta^2({{\bm \ell}}-{{\bm m}}-{{\mathcal P}})\, R_{({{\bm \ell}},{{\bm m}})}
\\ & \quad \times
\exp\Big[ 2\lambda {\int\! \frac{d^2 {{\bm k}}}{(2\pi)^2}}\, \alpha_{{\bm k}}\Big]
{\nonumber}\\
& = (2\pi)^2 \delta^2({{\bm \ell}}-{{\bm m}})
+ 2 R_{({{\bm \ell}},{{\bm m}})} \lambda\, \alpha_{{{\bm \ell}}-{{\bm m}}}
+ {{\mathcal O}(\alpha^2)}
\,,{\nonumber}\end{aligned}$$ which is frequency independent, and mixes $E$ and $B$ through $\lambda$, $$\begin{aligned}
\lambda =
\begin{pmatrix}
0 & 0 & 0 \\
0 & 0 & 1 \\
0 & -1 & 0
\end{pmatrix}
\,.\end{aligned}$$ Reionization marks the time after decoupling when the vast majority of Hydrogen became ionized due to gravitational nonlinearities. When and how this process occurred is at present not well constrained. Inhomogeneous reionization produces several secondary anisotropies in the CMB. The patchy nature of reionization results in a Thomson scattering optical depth to recombination, $\tau(\bf{n})$, depending on direction $\bf{n}$. Such optical depth fluctuations act as a modulation effect on CMB fields by suppressing the primordial anisotropies with a factor of $e^{-\tau(\bf{n})}$, correlating different modes by $$\begin{aligned}
\label{eq:dpatchy}
D^\text{Reionization}_{({{\bm \ell}},{{\bm m}})} &= (2\pi)^2 \delta^2({{\bm \ell}}-{{\bm m}}-{{\mathcal P}})\, R_{({{\bm \ell}},{{\bm m}})}
\\ & \quad \times
\exp\Big[ - {\int\! \frac{d^2 {{\bm k}}}{(2\pi)^2}}\, \tau_{{\bm k}}\Big]
{\nonumber}\\
& = (2\pi)^2 \delta^2({{\bm \ell}}-{{\bm m}})
- R_{({{\bm \ell}},{{\bm m}})} \tau_{{{\bm \ell}}-{{\bm m}}}
+ {{\mathcal O}(\tau^2)}
\,.{\nonumber}\end{aligned}$$ From [Eqs. and ]{}, one can obtain the corresponding Feynman rules that allow one to calculate their effect of the correlation structure of the CMB and construct the appropriate estimators and noise terms. Assuming these effects are Gaussian and statistically isotropic, the only other ingredient is $C^{\alpha\alpha}$ and $C^{\tau\tau}$. In this case, the calculation is an expansion in $\alpha$ or $\tau$ instead of $\phi$.
[*Acknowledgements–*]{} The computational resources required for this work were accessed via the Glidein-WMS [@Sfiligoi2009] on the Open Science Grid [@Pordes2007]. Numerical integrations were carried out using the [Cuba]{} integration library [@Hahn:2004fe].
APSY would like to thank Matias Zaldarriaga for discussions at an early stage of this project. This work was supported in part by the U.S. Department of Energy through DOE grant DE-SC0009919. WJW is supported by Marie Curie Fellowship PIIF-GA-2012-328913.
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abstract: 'Sponsored search adopts generalized second price (GSP) auction mechanism which works on the concept of pay per click which is most commonly used for the allocation of slots in the searched page. Two main aspects associated with GSP are the bidding amount and the click through rate (CTR). The CTR learning algorithms currently being used works on the basic principle of ($\#\mbox{clicks}_i/ \#\mbox{impressions}_i$) under a fixed window of clicks or impressions or time. CTR are prone to fraudulent clicks, resulting in sudden increase of CTR. The current algorithms are unable to find the solutions to stop this, although with the use of machine learning algorithms it can be detected that fraudulent clicks are being generated. In our paper, we have used the concept of relative ranking which works on the basic principle of ($\#\mbox{clicks}_i /\#\mbox{clicks}_t$). In this algorithm, both the numerator and the denominator are linked. As $\#\mbox{clicks}_t$ is higher than previous algorithms and is linked to the $\#\mbox{clicks}_i$, the small change in the clicks which occurs in the normal scenario have a very small change in the result but in case of fraudulent clicks the number of clicks increases or decreases rapidly which will add up with the normal clicks to increase the denominator, thereby decreasing the CTR.'
author:
- 'Rahul Gupta[^1], Gitansh Khirbat [^2] and Sanjay Singh[^3]'
bibliography:
- 'myref.bib'
title: A Novel Method to Calculate Click Through Rate for Sponsored Search
---
Introduction
============
The sponsored search algorithms are the most vital aspect of any search engine as the sponsored ads are its main source of income. The free service that the search engines are providing is being possible because of the sponsored advertisements that appear on the web page whenever any commercial words are search. So each keyword is associated with a group of advertisers who are ready to bid for it. Ideally the slots available to display those ads are less than the number of bidders that are bidding for it. In 2011, Google ad-words resulted an income of \$37.9 billion which accounts for 96% of its total income [@abc]. The keyword ranges from Finance & Insurance, Retailers & Merchandise, Travel & Tourism, Jobs and Education Home & Garden, Computer & Consumer Electronics, Vehicles, Internet & Telecoms, Business & Industrial, Occasions & Gifts with companies some companies paying as high as \$50 million [@ghi] [@hij]. The same is the case of other search engine like yahoo too, out of \$2.02 billion revenue generated in 2011, display ads constitutes \$1.2 billion which is equivalent to 61% of its total income [@ijk]. So, display-ad is the most important part of any search engine. The sole motivation of taking this work is an attempt to learn how are the search engines able earn a very high profit and maintain top most rank in the fortune 100 companies year after year even though they are not charging from the users. Each year their earnings are growing exponentially, which accounts for the popularity these ads holds. Currently, there are more than 65,000 keywords for which real time auction goes on among 800,000 bidders [@jkl]. The most interesting feature is that this profit is unaffected by any political, social, economical aspects. People will be using the search engine for searching even though there is a hike in petrol price or a tax rebate. So the whole idea reflects that this is a business which will always be growing.
Related Work
============
This section briefs about the existing work in the area of sponsored search.
Pay Per Impression
------------------
In 1994, the first time ads appeared online. The first ad that appeared online was of AT&T appeared at Hotmail.com which works on the principle of pay per click. Cost per impression is used in online advertising and marketing related to web traffic. It refers to the cost of Internet marketing campaigns where advertisers pay for every time their ad is displayed, usually in the form of a banner ad on in Email advertising. An impression is the display of an ad to a user while viewing a web page. A single web page may contain multiple ads. In such cases, a single page-view would result in one impression for each ad displayed.
In order to count the impressions accurately and prevent fraud, an ad server may exclude certain non-qualifying activities such as page-refreshes or the user opening same pages from counting as impressions [@xyz].
Pay Per Acquisition
-------------------
In 1996, companies like Amazon, CDNow used the concept of pay per acquisition. According to which, the advertiser pays for each specified action (a purchase, a form submission, and so on) linked to the advertisement. Direct response advertisers consider CPA the optimal way to buy online advertising, as an advertiser only pays for the ad when the desired action has occurred. An action can be a product being purchased, a form being filled, etc. The desired action to be performed is determined by the advertiser. A merchant would team up with other merchants or partner websites to promote its products or services online. The merchant would only pay out when confirmed lead, sale, email or registration takes place. Ideally, an advertiser would always prefer a pricing model in which the advertiser pays only when a customer actually completes a transaction. The PPT (pay per transaction) models were born out of this contention. A prominent example of PPT models is Amazon.com’s Associates Program. Under this program, a website that sends customers to Amazon.com receives a percentage of customer’s purchases.
This mechanism is highly beneficial for the advertiser as they had to pay only when a particular task which they want is done by the user. But for the host website, this is not trustworthy. As when the user has left the site and is redirected to the other site, it is impossible to keep a track of activities that one performs on the site in which he was directed. So this method can only be adopted when the advertiser is highly trustworthy or he agrees to share all the data to its host site also in which its advertisement had appear.
Pay Per Click
-------------
This is the most popular mechanism which forms the basis for the sponsored ads. It was first time started by GoTo.com in 1998. In this method, the advertiser has to pay every time a user clicks on the ads held by the publisher website. It is a middle ware between PPI & PPA as it provides a well trusted way for both the advertiser and the host website.
Pay Per Click with Generalized First Price Auction
==================================================
Till now all the web page considered for advertisement have a static framework which means that there is already a well defined number of pages and the web page displayed will be amongst them. But when advertisement in search engine comes into picture, the count of web page is not static as it displays different result based on the queries. So here for each keyword an auction is being is conducted where auctioneers bid for slots in the search engine; the number of slots associated with a keyword is lesser than the no of companies bidding for it. In generalized first price auction, bidders place their bid in a sealed envelope and simultaneously hand them to the auctioneer. The envelopes are opened and the individual with the highest bid wins, paying a price equal to the exact amount that he or she bid.
This method was first adopted by Yahoo in 2001, where an online auction was conducted on a set of keywords [@bcd]. This method which became very popular in the beginning soon suffered from very high in congruence which is explained the next section.The number of ads that the search engine can show to a user is limited with different desirability for advertisers: an ad shown at the top of a page is more likely to be clicked than an ad shown at the bottom. In the search engine industry, there are three key players: the advertisers, the search engines and the potential customers. Search engines navigate potential customers to advertisers’ product web sites by displaying their web links when potential customers conduct keyword search requests. These advertisers’ links are called sponsored links. Sponsored links distinguish themselves from the organic (non-sponsored) web search results by whether or not a fee is paid to the search engine company. Every time a consumer clicked on a sponsored link, an advertiser’s account was automatically billed the amount of the advertiser’s most recent bid. The links to advertisers were arranged in descending order of bids, making highest bids the most prominent. The ease of use, the very low entry costs, and the transparency of the mechanism quickly led to the success of Overture’s paid search platform.However, the underlying auction mechanism itself was far from perfect. In particular, Overture and advertisers quickly learned that the mechanism was unstable due to the dynamic nature of the environment.
After just 6 months of the existence, GFP failed as it suffered from serious glitches. The auction in the search engine is dynamic which means that a user can any time change the amount of bid which he/she has bid. Due to the same, this system failed to work as the competition among them causes a instability. This could be well understood with the help of an example: Consider an example where we have 2 slots available for a particular keyword. There was a bidding being held for a particular keyword in which advertiser A with bid amount \$ 10 won the first slot and the advertiser B with the bidding amount of \$ 3 won the second slot. Now as we know that the companies can change their bidding amount at any instant. This prompts the advertiser A to change its bid to \$ 4 as with that bid too he can maintain its first position. Now the advertiser A holds the first spot with \$ 4 and the advertiser B holds the second spot with \$ 3. This attracts the advertiser B to increase its bid to \$ 5, so that it can acquire the first slot. This will again be followed by advertiser B increasing its bid to \$ 6 to regain its spot. This follows until a particular level is reached where advertiser B will stop the increase. And advertiser A will have the first position. Now at that instant advertiser B will suddenly lower its value which accounts for the least possible value to b in that slot. This will again result in the previous scenario making the advertiser A to lower its bid [@cde].
Generalized Second Price Auction Mechanism
==========================================
This is the auction mechanism currently being followed by most of the search engines which includes Google, Yahoo, Bing. In this concept, each bidder pays the amount which is given by the next bidder following him [@efg]. The highest bidder pays the amount of the second highest and the second highest pays the amount bidded by the third highest, this goes on till the last available slot is allotted [@lmn]. The slot $S_i$ if given to an advertiser A, it has to pay a price $P_i=b_{i-1}$.
Click Through Rate (CTR)
------------------------
Click Through Rate is the quantitative measure of the ads which defines the probability an ad will get the click if it is shown. Each ads has an individual CTR which is based on the prior experience that it holds in the particular search engine. It also denotes the popularity an ad has among the user. CTR is multiplied by the bid amount that the advertiser is ready to pay and the answer is the final bid amount for that advertisement. So higher CTR is considered better as it will increase the bid and as a result the advertiser will be able to get a better slot with less amount of payment.
Ideally if any of the advertisement wants its CTR to be increased, it should take care of the service it is providing. The better the service/user-interface, the user will be more prompt to visit that ad which will subsequently increase its CTR. But practically it is sometimes not being followed, fraudulent clicks are done to increase the CTR of their own ads or to increase the spending of other’s ad.
Clicks Fraud
============
Click Fraud is the act of clicking on the the ads with the sole intent of increasing or decreasing the spending of the advertiser. In a pay per click mechanism, the advertiser has to pay every times its ad is clicked. This makes the system prone to fraudulent clicks [@fgh].
Clicking on other’s ad- As every click resulted in the payment to be paid by the advertisement. If the rivalry company performs the fraudulent clicks to the other company’s advertisement, it will result in the unnecessary payment to be paid by the other company. Clicking on own’s ad- This is done in order to increase the CTR of own ads so that it can acquire the better slots. As it is general proven fact that the higher is the slot, higher will be the probability to click on that ad. Higher position in the slot not only gives more genuine clicks but also decrease the cost to be paid per click.
Generating Click Fraud
======================
Click fraud is done by many ways which are broadly classified in two ways:
Automatically- by scripting
----------------------------
Here, a script(code) is used to generate clicks on the particular advertisement. This leads to a sudden increase in the number of clicks thereby increasing the CTR. The implementation of this method is quite simple, as no external resources is being used. However this method can be detected by the search engines. This is done by a variety of steps that are developed by examining the patterns of clicks that are generated by scripted clicks. Like when a click is scripted each click will occur generally occur after a fixed interval of time generally in milliseconds. This is not possible in practical scenario, so whenever a similar patterns of clicks are encountered for a longer duration of time it is discarded.
By humans
---------
This is a way of generating fraudulent clicks where humans are employed to click on the particular ads. Here in this case an ads is attacked by the set people where they visit the ad shown and spend the time like a normal user do. As there is no fixed pattern and their likelihood to behave like the normal user, it is almost impossible to detect the fraud done by this method.
Existing CTR Learning Algorithm
===============================
There is in general 3 CTR learning algorithms currently being used which works on the basic principle of (\#clicks $_{i}$/\#impressions $_{i}$) under a fixed window of clicks or impressions or time. CTR are prone to fraudulent clicks, resulting in the sudden increase of CTR. The current algorithms are unable to find the solutions to stop this.
Averaging Over Fixed Time Window T
----------------------------------
Let X= No of clicks in last T hours and Y= No of impressions. $$CTR= X/Y$$
Averaging over fixed impression window Y
----------------------------------------
Let X= No of clicks received in last Y impression. $$CTR= X/Y$$
Averaging over fixed window X
-----------------------------
Let Y = impression since the Xth last click. $$CTR= X/Y$$
New CTR Learning Algorithm
==========================
Let A,B,C be three advertisers and $C_a$, $C_b$, $C_c$ be the clicks that the advertisers receive in time interval $T$. $$\begin{aligned}
CTR_{A}= C_{a}/(C_{a} + C_{b}+ C_{c})\\
CTR_{B}= C_{b}/(C_{a}+C_{b}+C_{c})\\
CTR_{C}= C_{c}/(C_{a}+C_{b}+C_{c})\end{aligned}$$
Results and Discussion
======================
Below is the table and the graph obtained by the old algorithms and the new algorithms when the fraudulent clicks are being generated. The graph clearly depicts that the new algorithm is better then the previous one.
Time Impressions Clicks CTR
------ ------------- -------- -------
1 16 2 0.111
2 28 6 0.176
3 40 12 0.230
4 52 18 0.257
5 64 24 0.272
6 76 30 0.283
7 88 42 0.290
8 100 48 0.295
9 112 54 3.0
10 124 60 0.303
11 136 66 0.306
12 148 72 0.308
13 160 78 0.310
14 172 84 0.312
15 184 90 0.313
16 196 96 0.314
17 208 102 0.315
18 220 108 0.316
19 232 114 0.317
20 244 120 0.318
: Data for the calculation of CTR using current algorithm[]{data-label="tab:t1"}
Time Impressions Clicks Total Clicks CTR
------ ------------- -------- -------------- -------
1 16 2 22 0.090
2 28 6 50 0.12
3 40 12 84 0.142
4 52 18 124 0.145
5 64 24 170 0.141
6 76 30 222 0.135
7 88 42 280 0.128
8 100 48 344 0.122
9 112 54 414 0.115
10 124 60 490 0.110
11 136 66 572 0.104
12 148 72 660 0.100
13 160 78 754 0.095
14 172 84 854 0.091
15 184 90 960 0.087
16 196 96 1072 0.083
17 208 102 1190 0.080
18 220 108 1314 0.077
19 232 114 0.317 0.074
20 244 120 1444 0.072
: Data for the calculation of CTR using new algorithm[]{data-label="tab:t2"}
[.5]{} ![CTR in case of fraudulent click with old and proposed algorithm[]{data-label="fig:test"}](old "fig:"){width="\linewidth"}
[.5]{} ![CTR in case of fraudulent click with old and proposed algorithm[]{data-label="fig:test"}](new "fig:"){width="\linewidth"}
Table \[tab:t1\] depicts the CTR which is calculated by the current algorithm in the case of fraudulent clicks. Figure \[fig:sub1\] shows the exponential variation of CTR with time in case of fraudulent clicks using old algorithm. This problem is overcome by our proposed algorithm, which uses the same data for clicks but it raises up to a certain level till the normal clicks take place but eventually goes down when there are too many clicks within a very short duration of time. Thus the problem of rise in CTR due to fraudulent clicks is solved which is illustrated in Fig. \[fig:sub2\].
Conclusion
==========
In this paper we have discussed the working of sponsored search and the business model of search engine based on sponsored search. Also we have discussed about the problem associated with the existing algorithm and shown the efficacy of the proposed algorithm over the existing one.
[^1]: This work was carried out at MIT, Manipal. Currently Rahul Gupta is working as Data Scientist at Fractal Analytics Inc, India, Email: [email protected]
[^2]: Gitansh Khirbat is at University of Melbourne, Australia, Email:[email protected]
[^3]: Sanjay Singh is with the Department of Information and Communication Technology, Manipal Institute of Technology, Manipal University, Manipal-576104, INDIA, E-mail: [email protected]
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bibliography:
- 'bigone.bib'
---
ITP–UH–22/16\
EMPG–16–18
1.0cm
[**Extended Riemannian Geometry I:\
Local Double Field Theory**]{} 1.cm [Andreas Deser$^{a}$ and Christian Sämann$^b$]{}
\
and\
[*Istituto Nationale di Fisica Nucleare\
Via P. Giuria 1\
10125 Torino, Italy* ]{}\
[*${}^b$ Department of Mathematics, Heriot-Watt University\
Colin Maclaurin Building, Riccarton, Edinburgh EH14 4AS, U.K.*]{}\
and\
[*Maxwell Institute for Mathematical Sciences, Edinburgh, U.K.*]{}\
and\
[*Higgs Centre for Theoretical Physics, Edinburgh, U.K.*]{}\
[Email: [[email protected] , [email protected]]{}]{}
1.0cm
[**Abstract**]{}
> We present an extended version of Riemannian geometry suitable for the description of current formulations of double field theory (DFT). This framework is based on graded manifolds and it yields extended notions of symmetries, dynamical data and constraints. In special cases, we recover general relativity with and without 1-, 2- and 3-form gauge potentials as well as DFT. We believe that our extended Riemannian geometry helps to clarify the role of various constructions in DFT. For example, it leads to a covariant form of the strong section condition. Furthermore, it should provide a useful step towards global and coordinate invariant descriptions of T- and U-duality invariant field theories.
Introduction and results
========================
One of the most prominent features differentiating string theory from field theories of point particles is the symmetry known as T-duality [@Giveon:1994fu]. This symmetry interchanges the momentum modes of a string with its winding modes along compact cycles and it is one of the main ingredients in the web of dualities connecting the various different string theories. Considering the success of low-energy space-time descriptions of string theory in terms of ordinary field theories of the massless string modes, it is tempting to ask if there is some manifestly T-dual completion of these descriptions. It is the aim of double field theory (DFT) to provide such a T-duality invariant version of supergravity. The idea of DFT goes back to the early 90ies [@Tseytlin:1990va; @Siegel:1993th; @Siegel:1993xq] and the development of double geometry [@Hull:2004in; @Hull:2009sg] led to the paper [@Hull:2009mi], which gave DFT its name and seems to have triggered most of the current interest in this area. A detailed review of DFT is found in the overview papers [@Zwiebach:2011rg; @Berman:2013eva; @Aldazabal:2013sca; @Hohm:2013bwa].
In this paper, we shall not concern ourselves with the issue of how reasonable the main objective of DFT is. Rather, we would like to explain some of the mathematical structures involved in currently available formulations. These formulations are local and linear in various senses, and we believe that the mathematical picture we provide points towards global descriptions. In particular, we give an appropriate extension of Hitchin’s generalized geometry [@Hitchin:2004ut; @Hitchin:2005in; @Gualtieri:2003dx].
There have been a number of previous papers clarifying the underlying geometrical structures from various perspectives [@Vaisman:2012ke; @Vaisman:2012px; @Hohm:2012mf; @Cederwall:2014kxa; @Deser:2014mxa; @Bakas:2016nxt]. We also want to draw attention to the phase space perspective described in [@Aschieri:2015roa].
In this paper, we choose to follow a different approach. The $B$-field of string theory is well-known to be part of the connective structure of an abelian gerbe [@Gawedzki:1987ak; @Freed:1999vc], which is a categorified ${\mathsf{U}}(1)$-principal bundle. In this picture, we have two kinds of symmetries: the diffeomorphisms on the base manifolds as well as the gauge transformations on the connective structure of the abelian gerbe. It is natural to expect that together, they form a categorified Lie group, or Lie 2-group for short. At the infinitesimal level, we have a Lie 2-algebra, which is most concisely described by a symplectic N$Q$-manifold. The latter objects are simply symplectic graded manifolds $({\mathcal{M}},\omega)$ endowed with a vector field $Q$ of degree 1 satisfying $Q^2=0$ and ${\mathcal{L}}_Q\omega=0$. They are familiar to physicists from BV-quantization and string field theory [@Lada:1992wc] and mathematicians best think of them as symplectic $L_\infty$-algebroids.
Our expectation is confirmed by the fact that the symplectic N$Q$-manifold known as Courant algebroid (a symplectic Lie 2-algebroid) already features prominently in generalized geometry and various other examples appear in mathematical discussions of $T$-duality. The Lie 2-algebra of infinitesimal symmetries can here be described by a derived bracket construction on the Courant algebroid [@Roytenberg:1998vn; @Roytenberg:1999aa] involving the Poisson structure induced by the symplectic form as well as the Hamiltonian of the homological vector field $Q$. It has long been known that also the Lie bracket of vector fields is a derived bracket $\iota_{[X,Y]}=[\iota_X,[{\mathrm{d}},\iota_Y]]$, cf. [@Kosmann-Schwarzbach:0312524]. It is therefore natural to ask if even the Lie 2-algebra underlying double field theory can be realized by a derived bracket construction. It was shown in [@Deser:2014mxa] that the C-bracket of double field theory, which is part of the Lie 2-algebra structure, is a derived bracket. Moreover by applying the easiest form of the strong section condition $\tilde \partial = 0$, it reduces to the Courant bracket of generalized geometry.
To develop the full Lie 2-algebra, however, one quickly realizes that ordinary symplectic N$Q$-manifolds $({\mathcal{M}},\omega,Q)$ are not sufficient. One rather needs to lift the condition $Q^2=0$ and subsequently restrict the algebra of functions ${\mathcal{C}}^\infty({\mathcal{M}})$ to a suitable subset. We will develop this more general picture of derived brackets in great detail, studying what we call pre-N$Q$-manifolds. These pre-N$Q$-manifolds still carry categorified Lie algebras induced by derived brackets on restricted sets of functions ${\mathcal{C}}^\infty({\mathcal{M}})$, and they are the right extension of Hitchin’s generalized geometry to capture the geometry of local double field theory.
We tried to make our presentation self-contained and in particular to provide a very detailed introduction to the language of N$Q$-manifolds, which might not be well-known by people studying double field theory. We therefore begin with reviews of generalized geometry, double field theory and N$Q$-manifolds in sections \[sec:review\] and \[sec:NQ\]. Note that unless stated otherwise, we always focus on local structures and spacetimes are contractible.
Our actual discussion then starts in section \[sec:extendedRG\] with the definition of pre-N$Q$-manifolds and the identification of an $L_\infty$-algebra structure arising from derived brackets. This yields the notion of extended vector fields generating infinitesimal extended symmetries as well as a restriction of the algebra of functions to a subset. We then introduce extended tensors and define an action of the $L_\infty$-algebra structure on those. In this picture, a natural notion of extended metric arises, together with candidate terms for an invariant action functional.
This rather abstract discussion is then filled with life by studying a number of examples in section \[sec:examples\]. We show that the pre-N$Q$-manifolds suitable for the description of the symmetries of Einstein-Hilbert-Deligne actions, by which we mean general relativity minimally coupled to $n$-form gauge potentials, are the Vinogradov algebroids $T^*[n]T[1]M$. The fact that the symmetries are recovered appropriately might not be too surprising to people familiar with N$Q$-manifolds; new in this context is the construction of DFT-like action functionals for Einstein-Hilbert-Deligne theories.
Our formalism develops its real strength when applied to double field theory. The relevant pre-N$Q$-manifold here, ${\mathcal{E}}_2(M)$, is obtained in a rather nice way as a half-dimensional submanifold from the Vinogradov algebroid $T^*[n]T[1]X$ for the doubled space $X=T^*M$. That is, double field theory can be regarded as a restriction of generalized geometry on a doubled space.
The action of extended symmetries on extended tensors is then indeed the generalized Lie derivative familiar from double field theory. Moreover, the restrictions imposed on extended functions, vectors and tensors are coordinate invariant versions of the strong section condition of double field theory. This is probably the most interesting aspect of our formalism presented in this paper: Since our restrictions arise classically as a condition for obtaining the right symmetry structures, they automatically yield well-defined, covariant and transparent conditions on all types of fields. Also, the usual examples of solutions to the strong section condition yield indeed correct restrictions on extended fields.
It is interesting to note that our restrictions are also slightly weaker for tensor fields than the usual strong section condition of double field theory. The latter is found to be too strong in various contexts [@Blumenhagen:2016vpb], and our slight weakening might provide a resolution to this issue.
Another point that becomes more transparent in our formalism is the problem of patching together the local descriptions of double field theory to a global framework and we will comment in more detail on this point in section \[ssec:global\]. In [@Berman:2014jba], the integrated action of the generalized Lie derivative on fields was used to patch local doubled fields to global ones. This, however, seems to be consistent only if the curvature of the $B$-field is globally exact [@Papadopoulos:2014mxa], which implies that the underlying gerbe is topologically trivial. From our perspective, this is rather natural as it is already known in generalized geometry that the Courant algebroid needs to be twisted by a closed 3-form $T$ to describe accurately the symmetries of a gerbe with Dixmier-Douady class $T$.
In our formalism, it is clear how the generalized Lie derivative of double field theory should be twisted and we give the relevant definitions and an initial study of such twists in section \[ssec:twisted\_extended\_symmetries\]. The resulting symmetries are the ones that should be used to patch together local descriptions of double field theory in the case involving topologically non-trivial gerbes.
Finally, we also comment on the definitions of covariant derivative, torsion and Riemann curvature tensors. The problem here is that the Lie bracket in the various definitions should be replaced by a Lie 2-algebra bracket, which in general breaks linearity of the tensors with respect to multiplication of extended vector fields by functions. After introducing an extended covariant derivative, we can write the Gualtieri torsion of generalized geometry [@Gualtieri:2007bq] in a nice form using our language. Recall that the Gualtieri torsion is also the appropriate notion of a torsion tensor in double field theory [@Hohm:2012mf]. Our expression for the torsion contains in particular the ordinary torsion tensor as a special case. The same holds for the Riemann tensor of double field theory [@Hohm:2012mf], see also [@Jeon:1011.1324; @Jeon:2011cn; @Hohm:2011si; @Jurco:2015xra]. It is noteworthy that in the discussion of the latter, our weakened and invariant strong section condition appears.
There are a number of evident open problems that we plan to attack in future work. First, an extension of our formalism suitable for exceptional field theory is in preparation [@Deser:2017aa]. Second, we would like to develop a better understanding of the extension of Riemannian geometry underlying the actions of double and exceptional field theory using our language. Third, one should study our twists of the generalized Lie derivative in more detail, which can be considered as a stepping stone to fourth: developing an understanding of an appropriate global picture for double field theory. Since our approach showed that the local description of classical double field theory is mathematically consistent and reasonable, we have no more reason to doubt the existence of a global description.
Lightning review of double field theory {#sec:review}
=======================================
Let us very briefly sum up the key elements of local generalized geometry and local double field theory, since we shall reproduce them in our formalism from a different starting point.
Generalized geometry {#ssec:generalised_geometry}
--------------------
Any target space description of classical strings has to include the massless excitations of the closed string. These consist of the spacetime metric $g$, the Kalb-Ramond $B$-field and the dilaton $\phi$. The former two can be elegantly described in Hitchin’s generalized geometry [@Hitchin:2004ut; @Hitchin:2005in; @Gualtieri:2003dx]. Underlying this description is a vector bundle $E_2\cong TM\oplus T^*M$ over some $D$-dimensional manifold $M$, which fits into the short exact sequence $$0 \longrightarrow T^*M {\ensuremath{\lhook\joinrel\relbar\joinrel\rightarrow}}E_2\xrightarrow{~\rho~} TM \longrightarrow 0~.$$ Sections $\sigma$ of the map $\rho$ are given by rank 2 tensors, which can be split into their symmetric and antisymmetric parts, $\sigma_{\mu\nu}=g_{\mu\nu}+B_{\mu\nu}$. Moreover, sections of $E_2$ describe the infinitesimal symmetries of the fields $g,B,\phi$ and they are encoded in a vector field $X$ capturing infinitesimal diffeomorphisms and a 1-form $\alpha$ describing the gauge symmetry. Note that $E_2$ comes with a natural inner product of signature $(D,D)$, which is given by $$(X_1+\alpha_1,X_2+\alpha_2)=\iota_{X_1}\alpha_2+\iota_{X_2}\alpha_1~.$$ To write this more concisely, we introduce generalized vectors $X^M=(X^\mu,\alpha_\mu)$ with index $M=1,\ldots,2D$. Then we have $$\label{eq:dft_eta}
(X_1,X_2)=X_1^MX_2^N\eta_{MN}{{\qquad\mbox{with}\qquad}}\eta_{MN}=\eta^{MN}=\left(\begin{array}{cc} 0 & {\mathbbm{1}}\\ {\mathbbm{1}}& 0 \end{array}\right)~.$$ Note that the metric $\eta$ reduces the structure group of the bundle $E_2$ to ${\mathsf{O}}(D,D)$. The Lie algebra ${\mathfrak{o}}(D,D)$ consists of matrices of the form $$\left(\begin{array}{cc} A & B \\ \beta & -A^T \end{array}\right)~,$$ where $A$ is an arbitrary matrix, generating ${\mathsf{GL}}(D)\subset {\mathsf{O}}(D,D)$, and $\beta\in \wedge^2 {\mathbbm{R}}^D$ and $B\in \wedge^2 ({\mathbbm{R}}^D)^*$ generate what are known as $\beta$-transformations and $B$-transformations in generalized geometry [@Gualtieri:2003dx; @Grana:2008yw].
The transformation rule for the metric and the $B$-field can also be written in a homogeneous way, by introducing the [*generalized metric*]{} [@Shapere:1988zv; @Giveon:1988tt; @Maharana:1992my; @Gualtieri:2003dx] $$\label{eq:metric_H_O(D,D)}
{\mathcal{H}}_{MN}=\left(\begin{array}{cc} g_{\mu\nu}-B_{\mu\kappa}g^{\kappa\lambda}B_{\lambda\nu} & B_{\mu\kappa}g^{\kappa\nu}\\ -g^{\mu\kappa}B_{\kappa\nu} &g^{\mu\nu} \end{array}\right)~.$$ Note that this metric is obtained from the ordinary metric $g$ via a finite $B$-field transformation with adjoint action $$\label{eq:B-field_trafo}
{\mathcal{H}}=\left(\begin{array}{cc} {\mathbbm{1}}& B \\ 0 & {\mathbbm{1}}\end{array}\right)\left(\begin{array}{cc} g & 0 \\ 0 & g^{-1} \end{array}\right)\left(\begin{array}{cc} {\mathbbm{1}}& B \\ 0 & {\mathbbm{1}}\end{array}\right)^T~.$$ This explains the terminology, a $B$-field transformation modifies linearly the value of the $B$-field. Moreover, we have the relation ${\mathcal{H}}^{-1}=\eta^{-1}{\mathcal{H}}\eta^{-1}$.
Under infinitesimal diffeomorphisms and gauge transformations encoded in the generalized vector $X$, ${\mathcal{H}}$ transforms according to $$\label{eq:action_generalized_Lie}
\delta_X {\mathcal{H}}_{MN}=X^P{\partial}_P{\mathcal{H}}_{MN}+({\partial}_M X^P-{\partial}^PX_M){\mathcal{H}}_{PN}+({\partial}_N X^P-{\partial}^PX_N){\mathcal{H}}_{MP}~,$$ where ${\partial}_M=({\partial}_\mu,{\partial}^\mu=0)$. Underlying this transformation is a [*generalized Lie derivative*]{} whose action, e.g. on tensors $T^M{}_N$, is defined as $$\label{eq:action_extended_Lie}
\hat {\mathcal{L}}_X T^M{}_N:=X^K{\partial}_K T^M{}_N+({\partial}^MX_K-{\partial}_KX^M)T^K{}_N+({\partial}_NX^K-{\partial}^KX_N)T^M{}_K~.$$ Note that $\hat{\mathcal{L}}_X$ is compatible with the Courant bracket $[-,-]_C$, which is the natural bracket between generalized vector fields: $$\hat{\mathcal{L}}_X\hat {\mathcal{L}}_Y-\hat {\mathcal{L}}_Y\hat {\mathcal{L}}_X=\hat{\mathcal{L}}_{[X,Y]_C}~,$$ where $$\label{eq:Courant_bracket}
\Bigl([X,Y]_C\Bigr)^M :=\,X^K\partial_K Y^M - Y^K\partial_K X^M -\tfrac{1}{2}\Bigl(X^K\partial^M Y_K - Y^K\partial^M X_K\Bigr)\;.$$ Formulas , and clearly hint at a completion of the picture where ${\partial}^\mu\neq 0$. Starting from ${\partial}_M=({\partial}_\mu,{\partial}^\mu=0)$ we can transform to a situation with ${\partial}^\mu\neq 0$ by applying a global ${\mathsf{O}}(D,D)$-transformations with non-trivial off-diagonal components $\beta$ or $B$. Certain subclasses of these transformations correspond to T-dualities, which leads us to formulas which are invariant under T-duality. This is the aim of double field theory.
Double field theory {#ssec:DFT}
-------------------
The moduli space of toroidal compactifications of closed string theory on the $D$-dimensional torus $T^D$ is the [*Narain moduli space*]{} [@Narain:1985jj] $$\mathfrak{M}={\mathsf{O}}(D,D,{\mathbbm{Z}})~\backslash~{\mathsf{O}}(D,D,{\mathbbm{R}})~/~{\mathsf{O}}(D,{\mathbbm{R}})\times {\mathsf{O}}(D,{\mathbbm{R}})~.$$ Here, ${\mathsf{O}}(D,D,{\mathbbm{R}})$ is the global symmetry group, ${\mathsf{O}}(D,D,{\mathbbm{Z}})$ are the T-dualities and${\mathsf{O}}(D,{\mathbbm{R}})\times {\mathsf{O}}(D,{\mathbbm{R}})$ are the remaining Lorentz transformations. The latter can be promoted to a local symmetry group. Beyond these three symmetry groups, we have the local diffeomorphisms together with the local gauge symmetries of the $B$-field. These are precisely the various symmetry groups appearing in the previous section.
To obtain an ${\mathsf{O}}(D,D,{\mathbbm{R}})$-invariant formulation, we complement the coordinates $x^\mu$ by an additional set, $x_\mu$, to the total $x^M=(x^\mu,x_\mu)$. Generalized vectors here are special sections of the bundle $E_2$ over the doubled space, and we write $$X=X^\mu\left({\frac{{\partial}}{{\partial}x^\mu}}+{\mathrm{d}}x_\mu\right)+X_\mu\left({\frac{{\partial}}{{\partial}x_\mu}}+{\mathrm{d}}x^\mu\right)~.$$
String theory now requires that states or fields satisfy the level matching condition $$(L_0-\bar L_0)f(x)=0~,$$ which, for the massless subsector $N=\bar N=1$, translates directly into $${\frac{{\partial}}{{\partial}x^\mu}}{\frac{{\partial}}{{\partial}x_\mu}}f(x)=0~.$$ This is the so-called [*weak*]{} or [*physical section condition*]{}. We expect that there is an algebra structure on the fields satisfying the weak section condition and in particular, multiplication should close. This requires that we also impose the [*strong section condition*]{}, $$\label{eq:strong_condition}
\eta^{MN}\left({\frac{{\partial}}{{\partial}x^M}}f(x)\right)\left({\frac{{\partial}}{{\partial}x^N}}g(x)\right)=\left({\frac{{\partial}}{{\partial}x^\mu}}f(x)\right)\left({\frac{{\partial}}{{\partial}x_\mu}}g(x)\right)+\left({\frac{{\partial}}{{\partial}x^\mu}}g(x)\right)\left({\frac{{\partial}}{{\partial}x_\mu}}f(x)\right)=0$$ on any pair of fields $f(x)$ and $g(x)$. A straightforward example of a solution to the strong section is simply to put ${\frac{{\partial}}{{\partial}x_\mu}}=0$ and for consistency, we should also put its dual, ${\mathrm{d}}x_\mu$ to zero. This leads back to generalized geometry.
Note that the meaning of the strong condition on tensor fields is rather opaque, as it is clearly only covariant on functions. Also, some constructions in double field theory suggest that this condition is too strong and problematic, as mentioned in the introduction.
The metric $g$ and the Kalb-Ramond field $B$ are again encoded in the generalized metric ${\mathcal{H}}$. Let us also consider the dilaton $d$ which is connected via the field redefinition ${\mathrm{e}}^{-2d}=\sqrt{|g|}{\mathrm{e}}^{-2\phi}$ to the usual dilaton field $\phi$. The transformation laws for ${\mathcal{H}}$ and $d$ read as $$\label{eq:rev:gauge transformations}
\begin{aligned}
\delta_X {\mathcal{H}}_{MN}&=X^P{\partial}_P{\mathcal{H}}_{MN}+({\partial}_M X^P-{\partial}^PX_M){\mathcal{H}}_{PN}+({\partial}_N X^P-{\partial}^PX_N){\mathcal{H}}_{MP}~,\\
\delta_X ({\mathrm{e}}^{-2d})&={\partial}_M (X^M{\mathrm{e}}^{-2d})~.
\end{aligned}$$ This transformation is given indeed by the generalized Lie derivative and the dilaton transforms as a scalar tensor density.
The biggest success of double field theory is probably the provision of an action[^1] [@Hohm:2010pp] $$S_{\rm DFT}=\int {\mathrm{d}}^{2D} x~{\mathrm{e}}^{-2d}~{\mathcal{R}}$$ based on the Ricci scalar $$\begin{aligned}
{\mathcal{R}}&=\tfrac18 {\mathcal{H}}_{MN}{\partial}^M{\mathcal{H}}_{KL}{\partial}^N {\mathcal{H}}^{KL}-\tfrac12 {\mathcal{H}}_{MN}{\partial}^M{\mathcal{H}}_{KL}{\partial}^L{\mathcal{H}}^{KN}\\
&~~~~~-2{\partial}^Md {\partial}^N {\mathcal{H}}_{MN}+4{\mathcal{H}}_{MN}{\partial}^M d{\partial}^N d~.
\end{aligned}$$ This action is invariant under the generalized diffeomorphisms . Moreover, upon imposing the strong section condition ${\frac{{\partial}}{{\partial}x_\mu}}=0$ and integrating by parts, it reduces to the usual action for the NS sector of supergravity: $$S_{\rm NS}=\int {\mathrm{d}}^D x~\sqrt{g}~{\mathrm{e}}^{-2\phi}\left(R+4({\partial}\phi)^2-\tfrac{1}{12} H^2\right)~.$$ A more detailed analysis shows that constructing consistent doubled versions of the torsion, Riemann and Ricci tensors is much more involved.
Let us close with a few remarks on exceptional field theory, in which the T-duality invariant target space formulation of the massless subsector of string theory is replaced by a U-duality invariant target space formulation of a subsector of M-theory. Instead of winding modes, one here has to account for the various wrapping modes of M2- and M5-branes around non-trivial cycles of the target space.
There is an obvious analogue of generalized geometry, based on truncations of the bundle $TM\oplus \wedge^2 T^*M\oplus \wedge^5 T^*M$. Sections of this bundle are capable of describing infinitesimal diffeomorphisms as well as gauge symmetries of 3- and 6-form potentials. The ${\mathsf{O}}(D,D,{\mathbbm{R}})$ symmetry is replaced by the exceptional group $E_n$, where $n$ is the dimension of the torus $T^n$ in the compactification. There are evident analogues of the generalized metric as well as the generalized Lie derivative and the Courant bracket.
Moreover, a U-duality invariant extension of these objects exists, known as [*exceptional field theory*]{}, again with a Ricci scalar and an action principle. An extension of our formalism suitable for U-duality will be presented in upcoming work [@Deser:2017aa].
NQ-manifolds {#sec:NQ}
============
In the following section, we set up our conventions for symplectic N$Q$-manifolds, which will be the mathematical structure underlying most of our considerations. We also explain the relation between the symplectic N$Q$-manifolds known as exact Courant algebroids and ${\mathsf{U}}(1)$-bundle gerbes.
NQ-manifolds and higher Lie algebras {#ssec:NQ-manifolds}
------------------------------------
Recall that an [*N-manifold*]{} ${\mathcal{M}}$ is an ${\mathbbm{N}}_0$-graded manifold. We usually denote the body, i.e. the degree 0 part, of ${\mathcal{M}}$ by ${\mathcal{M}}_0$. We shall be exclusively interested in N-manifolds arising from ${\mathbbm{N}}$-graded vector bundles over ${\mathcal{M}}_0$. Such N-manifolds were called [*split*]{} in [@Sheng:1103.5920], following the nomenclature for supermanifolds. Note that Batchelor’s theorem [@JSTOR:1998201] for supermanifolds can be extended to N-manifolds, and therefore any smooth N-manifold is diffeomorphic to a split N-manifold [@Bonavolonta:2012fh], see also [@Roytenberg:0203110] for special cases.
An [*N$Q$-manifold*]{} is an N-manifold endowed with a homological vector field $Q$. That is, $Q$ is of degree 1 and satisfies $Q^2=0$. If the N$Q$-manifold ${\mathcal{M}}$ is concentrated[^2] in degrees $0,\ldots, n$, then we call ${\mathcal{M}}$ a Lie $n$-algebroid. Lie $n$-algebroids ${\mathcal{M}}$ with trivial body ${\mathcal{M}}_0=*$ form Lie $n$-algebras or $n$-term $L_\infty$-algebras, given in terms of their Chevalley-Eilenberg description.
For example, consider an N$Q$-manifold concentrated in degree 1: ${\mathcal{M}}={\mathfrak{g}}[1]$, where in general $[p]$ indicates a shift of the degree of the fibers or relevant linear spaces by $p$. If we parametrize the vector space ${\mathfrak{g}}[1]$ by coordinates $\xi^\alpha$ of degree 1, the vector field $Q$ is necessarily of the form $Q=\frac12 c^\alpha_{\beta\gamma}\xi^\beta\xi^\gamma{\frac{{\partial}}{{\partial}\xi^\alpha}}$. The condition $Q^2=0$ is then equivalent to the Jacobi identity for the structure constants $c^\alpha_{\beta\gamma}$. Similarly, one obtains higher Lie $n$-algebras.
Since the case of Lie 2-algebras will be particularly important later on, let us present it in somewhat more detail. Here, the N$Q$-manifold is concentrated in degrees 1 and 2, and consists of two vector spaces $V[2]$ and $W[1]$ fibered over a point $*$: $${\mathcal{M}}= * \leftarrow W[1] \leftarrow V[2]\leftarrow * \leftarrow \ldots ~.$$ In terms of coordinates $v^a$ and $w^\alpha$ on ${\mathcal{M}}$ of degrees 2 and 1, respectively, the most general homological vector field reads as $$\label{eq:Q_for_Lie_2_algebras}
Q = - c^\alpha_a v^a {\frac{{\partial}}{{\partial}w^\alpha}} - \frac12 c^\alpha_{\beta\gamma} w^\beta w^\gamma {\frac{{\partial}}{{\partial}w^\alpha}}
- c^a_{\alpha b} w^\alpha v^b {\frac{{\partial}}{{\partial}v^a}} + \frac{1}{3!} c^a_{\alpha\beta\gamma} w^\alpha
w^\beta w^\gamma {\frac{{\partial}}{{\partial}v^a}}~.$$ Recall from above that in the Chevalley-Eilenberg description, ordinary Lie algebras ${\mathfrak{g}}$ appear as N$Q$-manifolds ${\mathfrak{g}}[1]$ concentrated in degree one. Similarly, the N$Q$-manifold concentrated in degrees 1 and 2 encodes the Lie 2-algebra ${\mathsf{L}}=V[1]\oplus W[0]$. In terms of a [*graded*]{} basis $t_a$ and $\tau_\alpha$ carrying degrees 1 and 0, respectively, the structure constants in encode brackets[^3] $$\begin{gathered}
\mu_1(t_a)=c^\alpha_a \tau_\alpha~,\\
\mu_2(\tau_\alpha,\tau_\beta)=c^\gamma_{\alpha\beta}\tau_\gamma~,~~~\mu_2(\tau_\alpha,t_b)=c^a_{\alpha b}t_a~,\\
\mu_3(\tau_\alpha,\tau_\beta,\tau_\gamma)=c^a_{\alpha\beta\gamma}t_a~.
\end{gathered}$$ The equation $Q^2=0$ now translates into the [*higher*]{} or [*homotopy Jacobi identities*]{}, which for a Lie 2-algebra read as
\[eq:homotopy\_relations\] $$\begin{aligned}
\mu_1(\mu_2(w,v))&=\mu_2(w,\mu_1(v))~,~~~\mu_2(\mu_1(v_1),v_2)=\mu_2(v_1,\mu_1(v_2))~,\\
\mu_1(\mu_3(w_1,w_2,w_3))&=\mu_2(w_1,\mu_2(w_2,w_3))+\mu_2(w_2,\mu_2(w_3,w_1))+\mu_2(w_3,\mu_2(w_1,w_2))~,\\
\mu_3(\mu_1(v),w_1,w_2)&=\mu_2(v,\mu_2(w_1,w_2))+\mu_2(w_2,\mu_2(v,w_1))+\mu_2(w_1,\mu_2(w_2,v))
\end{aligned}$$ and $$\label{eq:homotopy_relation4b}
\begin{aligned}
\mu_2(\mu_3(w_1,&w_2,w_3),w_4)-\mu_2(\mu_3(w_4,w_1,w_2),w_3)+\mu_2(\mu_3(w_3,w_4,w_1),w_2)\\
& -\mu_2(\mu_3(w_2,w_3,w_4),w_1)=\\
&\mu_3(\mu_2(w_1,w_2),w_3,w_4)-\mu_3(\mu_2(w_2,w_3),w_4,w_1)+\mu_3(\mu_2(w_3,w_4),w_1,w_2)\\
&-\mu_3(\mu_2(w_4,w_1),w_2,w_3)
-\mu_3(\mu_2(w_1,w_3),w_2,w_4)-\mu_3(\mu_2(w_2,w_4),w_1,w_3)~,
\end{aligned}$$
where $v,v_i\in V[1]$ and $w,w_i\in W[0]$. Finally, let us mention that this form of a Lie 2-algebra is usually called [*semistrict*]{} Lie 2-algebra. This is sufficiently general for all our purposes; the most general notion of a weak Lie 2-algebra was given in [@Roytenberg:0712.3461]. Further details on semistrict Lie 2-algebras can be found in [@Baez:2003aa] and references therein.
An important example of a Lie 2-algebra which will be very similar in form to the ones we will encounter later is the String Lie 2-algebra of a compact simple Lie group ${\mathsf{G}}$. Let ${\mathfrak{g}}:={\mathsf{Lie}}({\mathsf{G}})$ be the Lie algebra of ${\mathsf{G}}$, then the String Lie 2-algebra has underlying graded vector space ${\mathbbm{R}}[1]\oplus {\mathfrak{g}}$ endowed with higher products $$\mu_1(r)=0~,~~~\mu_2(X_1,X_2)=[X_1,X_2]~,~~~\mu_2(X,r)=0~,~~~\mu_3(X_1,X_2,X_3)=(X_1,[X_2,X_3])~,$$ where $X_i\in {\mathfrak{g}}$, $r\in {\mathbbm{R}}[1]$ and $(-,[-,-])$ is the generator of $H^3({\mathsf{G}},{\mathbbm{Z}})$ involving the Killing form on ${\mathsf{G}}$.
A [*morphism of $L_\infty$-algebras*]{} is now simply a morphism of N$Q$-manifolds. We shall be mostly interested in [*strict*]{} such morphisms $\varphi:{\mathsf{L}}\rightarrow {\mathsf{L}}'$, which consist of maps of graded vector spaces of degree 0 such that $$\mu'_i\circ \varphi^{\otimes i}=\varphi\circ \mu_i~.$$
Rarely discussed in the literature but still very useful for our analysis are the notions of action and semidirect product for $L_\infty$-algebras. A very detailed account[^4] is found in [@Mehta:2012ppa], where the rather evident generalizations of the notions action, extension and semidirect product of Lie algebras to $L_\infty$-algebras were explored.
First, recall that an action of a Lie algebra ${\mathfrak{g}}$ on a smooth manifold $M$ is a Lie algebra homomorphism from ${\mathfrak{g}}$ to the vector fields ${\mathfrak{X}}(M)$. Next, note that given a graded manifold ${\mathcal{M}}$, the vector fields ${\mathfrak{X}}({\mathcal{M}})$ form a graded Lie algebra, which is a particular case of an $L_\infty$-algebra. Thus, an [*action of an $L_\infty$-algebra*]{} ${\mathsf{L}}$ on a manifold ${\mathcal{M}}$ is a morphism of $L_\infty$-algebras from ${\mathsf{L}}$ to ${\mathfrak{X}}({\mathcal{M}})$. We shall be exclusively interested in actions corresponding to strict morphisms of $L_\infty$-algebras, given by a chain map $\delta$ of the form $$\label{eq:L_infity_action}
\xymatrixcolsep{4pc}
{\vcenter{\vbox{\xymatrix{
\ldots \ar@{->}[r]^{\mu_1} \ar@{->}[d]^{\delta} & {\mathsf{L}}_2 \ar@{->}[r]^{\mu_1}\ar@{->}[d]^{\delta}& {\mathsf{L}}_1 \ar@{->}[r]^{\mu_1}\ar@{->}[d]^{\delta}& {\mathsf{L}}_0 \ar@{->}[d]^{\delta}\\
\ldots \ar@{->}[r]^{{\mathrm{id}}} & {*} \ar@{->}[r]^{{\mathrm{id}}} & {*}\ar@{->}[r]^{0} & {\mathfrak{X}}({\mathcal{M}})\\
}}}}$$
[*Semidirect products of $L_\infty$-algebras*]{} can be similarly defined, using analogies with Lie algebras [@Mehta:2012ppa]. These have an underlying action $\rho$ of an $L_\infty$-algebra $({\mathsf{L}},\mu)$ on another $L_\infty$-algebra $({\mathsf{L}}',\mu')$ such that $\tilde {\mathsf{L}}={\mathsf{L}}\ltimes {\mathsf{L}}'$ forms an $L_\infty$-algebra with brackets $$\tilde \mu_2(X_1+w_1,X_2+w_2)=\mu_2(X_1,X_2)+\rho(X_1)w_2-\rho(X_2)w_1+\mu'_2(w_1,w_2)~,$$ where $X_{1,2}\in {\mathsf{L}}$ and $w_{1,2}\in {\mathsf{L}}'$.
Higher symplectic Lie algebroids and their associated Lie n-algebras
--------------------------------------------------------------------
An important but simple example of a Lie algebroid is the grade-shifted tangent space $T[1]M$. In terms of local coordinates $x^\mu$ and $\xi^\mu$ on the base and the fiber, we can define a vector field $Q=\xi^\mu{\frac{{\partial}}{{\partial}x^\mu}}$. The functions on $T[1]M$ can be identified with differential forms on $M$ with $Q$ being the de Rham differential.
A [*symplectic N$Q$-manifold of degree $n$*]{} is an N$Q$-manifold endowed with a symplectic form $\omega$ of ${\mathbbm{N}}_0$-degree $n$, such that it is compatible with $Q$: ${\mathcal{L}}_Q \omega=0$. We write $|\omega|=n$ for its degree. Given such a symplectic N$Q$-manifold $({\mathcal{M}},Q,\omega)$, we can introduce a corresponding Poisson structure $\{-,-\}$ as follows. The Hamiltonian vector field $X_f$ of a function is defined implicitly by[^5] $$\iota_{X_f}\omega={\mathrm{d}}f$$ and has grading $|f|-n$. The corresponding Poisson bracket reads as $$\{f,g\}:=X_f g=\iota_{X_f}{\mathrm{d}}g=\iota_{X_f}\iota_{X_g}\omega~.$$ As one readily verifies, this Poisson bracket is of grading $|\{f,g\}|=|f|+|g|-n$, it is graded antisymmetric,
\[eq:properties\_poisson\] $$\{f,g\}=-(-1)^{(|f|+n)(|g|+n)}\{g,f\}~,$$ and satisfies the graded Leibniz rule $$\label{eq:Leibniz-rule-n}
\{f,gh\}=\{f,g\}h+(-1)^{(n-|f|)|g|}g\{f,h\}$$ as well as the graded Jacobi identity $$\{f,\{g,h\}\}=\{\{f,g\},h\}+(-1)^{(|f|+n)(|g|+n)}\{g,\{f,h\}\}$$ or $$\{\{f,g\},h\}=\{f,\{g,h\}\}+(-1)^{(|h|+n)(|g|+n)}\{\{f,h\},g\}$$
for all $f,g,h\in {\mathcal{C}}^\infty({\mathcal{M}})$.
Because of ${\mathcal{L}}_Q \omega=0$, $Q$ is Hamiltonian, cf. [@Roytenberg:0203110 Lemma 2.2], and we denote its Hamiltonian function by ${\mathcal{Q}}$: $$Q f=X_{{\mathcal{Q}}}f=\{{\mathcal{Q}},f\}~.$$ Note that $$\begin{aligned}
Q\{f,g\}&=\{{\mathcal{Q}},\{f,g\}\}=\{\{{\mathcal{Q}},f\},g\}+(-1)^{(n+1+n)(|f|+n)}\{f,\{{\mathcal{Q}},g\}\}\\
&=\{Qf,g\}+(-1)^{|f|+n}\{f,Qg\}~.
\end{aligned}$$
A trivial example of a symplectic N$Q$-manifold is an ordinary symplectic manifold $(M,\omega)$ with $Q=0$. Thus, symplectic manifolds are symplectic Lie 0-algebroids. A more interesting example is the grade-shifted cotangent bundle $T^*[1]M$. Locally, $T^*[1]M$ is described by coordinates $x^\mu$ of degree 0 and coordinates $\xi_\mu$ of degree 1. A homological vector field $Q$ is necessarily of the form $Q=\pi^{\mu\nu}\xi_\mu {\frac{{\partial}}{{\partial}x^\nu}}$. Compatibility with the natural symplectic structure $\omega={\mathrm{d}}x^\mu\wedge {\mathrm{d}}\xi_\mu$ means that $\pi^{\mu\nu}{\frac{{\partial}}{{\partial}x^\mu}}\otimes {\frac{{\partial}}{{\partial}x^\nu}}\in \wedge^2 {\mathfrak{X}}({\mathcal{M}}_0)$ and $Q^2$ implies that the bivector $\pi$ yields a Poisson structure on $M$. In this sense, Poisson manifolds are symplectic Lie 1-algebroids.
An important feature of symplectic Lie $n$-algebroids $({\mathcal{M}},\{-,-\},{\mathcal{Q}})$ is that they come with an [*associated Lie $n$-algebra*]{} via a derived bracket construction, see [@Roytenberg:1998vn; @Fiorenza:0601312; @Getzler:1010.5859; @Ritter:2015ffa]. This Lie $n$-algebra has underlying ${\mathbbm{N}}$-graded vector space $$\label{eq:ordinary_L_infty_complex}
\begin{aligned}
{\mathsf{L}}({\mathcal{M}})~:=&&~{\mathcal{C}}^\infty_0({\mathcal{M}})&\rightarrow &{\mathcal{C}}^\infty_1({\mathcal{M}})&\rightarrow &\ldots &\rightarrow &{\mathcal{C}}^\infty_{n-2}({\mathcal{M}})&\rightarrow &{\mathcal{C}}^\infty_{n-1}({\mathcal{M}})\\
=&&~{\mathsf{L}}_{n-1}({\mathcal{M}})&\rightarrow &{\mathsf{L}}_{n-2}({\mathcal{M}})&\rightarrow &\ldots &\rightarrow &{\mathsf{L}}_1({\mathcal{M}})&\rightarrow &{\mathsf{L}}_0({\mathcal{M}})~~~,
\end{aligned}$$ where ${\mathcal{C}}^\infty({\mathcal{M}})={\mathcal{C}}^\infty_0({\mathcal{M}})\oplus {\mathcal{C}}^\infty_1({\mathcal{M}})\oplus {\mathcal{C}}^\infty_2({\mathcal{M}})\oplus \ldots $ is the decomposition of ${\mathcal{C}}^\infty({\mathcal{M}})$ into parts ${\mathcal{C}}^\infty_i({\mathcal{M}})$ of homogeneous grading $i$ with ${\mathcal{C}}^\infty_0({\mathcal{M}})={\mathcal{C}}^\infty({\mathcal{M}}_0)$. Note that the degree of elements of ${\mathcal{C}}^\infty_i({\mathcal{M}})$ in ${\mathsf{L}}({\mathcal{M}})$ is different from their ${\mathbbm{N}}$-degree $i$. We shall therefore make an explicit distinction, calling the former ${\mathsf{L}}$-degree and the latter ${\mathbbm{N}}$-degree. In particular elements of ${\mathcal{C}}^\infty_i({\mathcal{M}})$ have ${\mathsf{L}}$-degree $(n-1)-i$. In general, a subscript $i$ in the expressions ${\mathsf{L}}_i({\mathcal{M}})$ and ${\mathcal{C}}^\infty_i({\mathcal{M}})$ will always refer to the ${\mathsf{L}}$-degree and ${\mathbbm{N}}$-degree, respectively.
An explicit formula for the higher products for even $n$ can be found e.g. in [@Getzler:1010.5859]. The formulas for the lowest products $\mu_i$ for arbitrary $n$ are the following totally antisymmetrized derived brackets: $$\label{eq:L_infty_brackets}
\begin{aligned}
\mu_1(\ell)&=\left\{\begin{array}{ll}
0 & \ell\in {\mathcal{C}}^\infty_{n-1}({\mathcal{M}})={\mathsf{L}}_0({\mathcal{M}})~,\\
Q\ell & \mbox{else}~,\\
\end{array}\right.\\
\mu_2(\ell_1,\ell_2)&=\tfrac12\big(\{\delta\ell_1,\ell_2\}\pm\{\delta\ell_2,\ell_1\}\big)~,\\
\mu_3(\ell_1,\ell_2,\ell_3)&=-\tfrac{1}{12}\big(\{\{\delta\ell_1,\ell_2\},\ell_3\}\pm \ldots\big)~,
\end{aligned}$$ where $$\label{def:delta}
\delta(\ell)=\left\{\begin{array}{ll}
Q\ell & \ell\in {\mathcal{C}}^\infty_{n-1}({\mathcal{M}})={\mathsf{L}}_0({\mathcal{M}})~,\\
0 & \mbox{else}~,\\
\end{array}\right.\\$$ and the last sum runs over all ${\mathsf{L}}$-graded permutations. The signs in are the obvious ones to conform with the symmetries of the $\mu_k$.
As a final remark, note that if we had not antisymmetrized the arguments in the derived bracket construction for $n=2$, we would have obtained a [*hemistrict*]{} Lie $2$-algebra [@Baez:2008bu]. The connection to [*semistrict*]{} Lie $2$-algebras is as follows. In a categorification of a Lie algebra to a weak Lie 2-algebra [@Roytenberg:0712.3461], we lift two identities to isomorphisms, the alternator $\mathsf{Alt}:[x,y]\mapsto -[y,x]$ and the Jacobiator $\mathsf{Jac}:[x,[y,z]]\mapsto[[x,y],z]+[y,[x,z]]$. In the semistrict case, $\mathsf{Alt}$ is trivial while $\mathsf{Jac}$ is not and in the hemistrict case $\mathsf{Jac}$ is trivial while $\mathsf{Alt}$ is not. In the cases we are interested in, both hemistrict and semistrict Lie $2$-algebras turn out to be equivalent. An analogous statement should certainly hold also for higher $n$.
Vinogradov Lie n-algebroids {#ssec:Vinogradov}
---------------------------
A very important hierarchy of symplectic N$Q$-manifolds are the Vinogradov Lie $n$-algebroids[^6] $${\mathcal{V}}_n(M):=T^*[n]T[1]M~,$$ cf. [@Gruetzmann:2014ica; @Ritter:2015ffa]. In local coordinates $(x^\mu,\xi^\mu,\zeta_\mu,p_\mu)$ of ${\mathbbm{N}}$-degrees $0,1,n-1,n$, they form a symplectic N$Q$-manifold with $$\omega={\mathrm{d}}x^\mu\wedge{\mathrm{d}}p_\mu +{\mathrm{d}}\xi^\mu\wedge {\mathrm{d}}\zeta_\mu{{\qquad\mbox{and}\qquad}}{\mathcal{Q}}=\xi^\mu p_\mu~.$$ Note that ${\mathcal{Q}}$ is of ${\mathbbm{N}}$-degree $n+1$, which guarantees that its Hamiltonian vector field is of degree $1$. Explicitly, the Poisson bracket reads as $$\{f,g\}:=(-1)^{n^2}f\overleftarrow{{\frac{{\partial}}{{\partial}p_\mu}}}\overrightarrow{{\frac{{\partial}}{{\partial}x^\mu}}} g-f\overleftarrow{{\frac{{\partial}}{{\partial}x^\mu}}}\overrightarrow{{\frac{{\partial}}{{\partial}p_\mu}}} g+(-1)^{(n-1)^2}f\overleftarrow{{\frac{{\partial}}{{\partial}\zeta_\mu}}}\overrightarrow{{\frac{{\partial}}{{\partial}\xi^\mu}}} g-(-1)^{n}f\overleftarrow{{\frac{{\partial}}{{\partial}\xi^\mu}}}\overrightarrow{{\frac{{\partial}}{{\partial}\zeta_\mu}}} g$$ and the homological vector field $Q$ is given by $$Q=\xi^\mu{\frac{{\partial}}{{\partial}x^\mu}}+p_\mu{\frac{{\partial}}{{\partial}\zeta_\mu}}$$ and trivially satisfies $Q^2=\{{\mathcal{Q}},{\mathcal{Q}}\}=0$.
Functions $X\in {\mathcal{C}}^\infty_{n-1}({\mathcal{V}}_n(M))$ of ${\mathbbm{N}}$-degree $n-1$ are of the form $$X=X^\mu\zeta_\mu+\tfrac{1}{n!}X_{\mu_1\ldots \mu_{n-1}}\xi^{\mu_1}\cdots \xi^{\mu_{n-1}}$$ and correspond to global sections of the total space of the vector bundle $$E_n:=TM\oplus \wedge^{n-1} T^*M~.$$ In particular, the case $n=2$ reproduces all structures of the [*exact Courant algebroid*]{} [@Roytenberg:0203110], which underlies generalized geometry, cf. section \[ssec:generalised\_geometry\]. The case $n=3$ is relevant in exceptional field theory and sections of $E_3$ can accommodate the wrapping modes of M2-branes.
For $n>1$, functions of ${\mathbbm{N}}$-degree $n-1$ will be necessarily linear in $\zeta$. If we demand this also for $n=1$, then the resulting functions are sections of $TM\oplus {\mathbbm{R}}$ with derived bracket $$\label{eq:Courant-bracket-n=1}
\begin{aligned}
\mu_2(X+f,Y+g)&=\tfrac12\big(\{\{{\mathcal{Q}},X^\mu\zeta_\mu+f\},Y^\nu\zeta_\nu+g\}-\{\{{\mathcal{Q}},Y^\nu\zeta_\nu+g\},X^\mu\zeta_\mu+f\}\big)\\
&=[X,Y]+{\mathcal{L}}_Xg-{\mathcal{L}}_Yf~,
\end{aligned}$$ where $X,Y\in {\mathfrak{X}}(M)$ and $f,g\in {\mathcal{C}}^\infty(M)$. This Lie algebra describes the local gauge transformations of a metric and a connection one-form: The diffeomorphisms form a Lie algebra, which, together with the abelian Lie algebra of gauge transformations, forms a semidirect product of Lie algebras.
We shall be particularly interested in the case $n=2$, capturing the local gauge transformations for a metric and the Kalb-Ramond field $B$, which is part of the connective structure of an abelian bundle gerbe. Here, we find a Lie 2-algebra ${\mathcal{C}}^\infty(M)\xrightarrow{~\mu_1~}{\mathfrak{X}}(M)\oplus\Omega^1(M)$ with higher products $$\label{eq:ass_Courant_algebra}
\begin{aligned}
\mu_1(f)&={\mathrm{d}}f~,\\
\mu_2(X+\alpha,Y+\beta)&=\tfrac12\big(\{\{{\mathcal{Q}},X^\mu\zeta_\mu+\alpha_\mu\xi^\mu\},Y^\nu\zeta_\nu+\beta_\nu\xi^\nu\}- (X+\alpha)\leftrightarrow(Y+\beta)\big)\\
&=[X,Y]+{\mathcal{L}}_X\beta-{\mathcal{L}}_Y\alpha-\tfrac12{\mathrm{d}}\big(\iota_X\beta-\iota_Y\alpha)~,\\
\mu_2(X+\alpha,f)&=\tfrac12\{\{{\mathcal{Q}},X+\alpha\},f\}=\tfrac12\iota_X{\mathrm{d}}f~,\\
\mu_3(X+\alpha,Y+\beta,Z+\gamma)&=\tfrac{1}{3!}\big(\{\{\{{\mathcal{Q}},X+\alpha\},Y+\beta\},Z+\gamma\}+\ldots\big)\\
&=\tfrac{1}{3!}\big(\{X+\alpha,\mu_2(Y+\beta,Z+\gamma)\}+\mbox{cycl.}\big)\\
&=\tfrac{1}{3!}\big(\iota_X\iota_Y{\mathrm{d}}\gamma+\tfrac32\iota_X{\mathrm{d}}\iota_Y\gamma\pm\mbox{perm.}\big)
\end{aligned}$$ where $X,Y\in {\mathfrak{X}}(M)$ and $\alpha,\beta\in \Omega^1(M)$. The bilinear operation $\mu_2$ is also known as [*Courant bracket*]{}. It is the antisymmetrization of the [*Dorfman bracket*]{} $\nu_2$, which is recovered from the Courant bracket as $$\label{eq:Courant_to_Dorfman}
\nu_2(X+\alpha,Y+\beta)=\mu_2(X+\alpha,Y+\beta)+\tfrac12Q\{X+\alpha,Y+\beta\}~.$$ Courant and Dorfman bracket are the bilinear operations underlying semistrict and hemistrict Lie 2-algebras. These two Lie 2-algebras are equivalent in the sense of [@Roytenberg:0712.3461], which is suggested by equation .
Twisting Vinogradov algebroids {#ssec:twisted_Vinogradov}
------------------------------
Vinogradov Lie $n$-algebroids ${\mathcal{V}}_n(M)$ may be twisted by a closed $n+1$-form $T $ [@Severa:2001qm], which amounts to the shift $$\label{eq:twisted_Hamiltonian_Q}
{\mathcal{Q}}_T =\xi^\mu p_\mu+\tfrac{1}{(n+1)!}T _{\mu_1\ldots \mu_{n+1}}\xi^{\mu_1}\ldots \xi^{\mu_{n+1}}$$ or $$\begin{aligned}
Q_T =\xi^\mu{\frac{{\partial}}{{\partial}x^\mu}}+p_\mu{\frac{{\partial}}{{\partial}\zeta_\mu}}&-\frac{1}{(n+1)!}{\frac{{\partial}}{{\partial}x^\nu}}T _{\mu_1\ldots \mu_{n+1}}\xi^{\mu_1}\ldots \xi^{\mu_{n+1}}{\frac{{\partial}}{{\partial}p_\nu}}\\
&+\frac{1}{n!}T _{\nu\mu_1\ldots \mu_{n}}\xi^{\mu_1}\ldots \xi^{\mu_{n}}{\frac{{\partial}}{{\partial}\zeta_\nu}}~.
\end{aligned}$$ This $Q$ is automatically compatible with the symplectic structure, since ${\mathcal{L}}_{Q_T }\omega ={\mathrm{d}}\iota_{Q_{T }}\omega ={\mathrm{d}}^2{\mathcal{Q}}_T =0$. Note that $\{{\mathcal{Q}}_T ,{\mathcal{Q}}_T \}=Q^2_T =0$ is equivalent to ${\mathrm{d}}T =0$. The closed form $T $ is known as the [*Ševera class*]{} of the Vinogradov algebroid [@Severa:1998ab], see also [@Severa:2001qm] and [@Bressler:2002ur]. We will explain its relation to the characteristic class of a gerbe in the following section.
First, however, let us analyze the most general twist element, restricting ourselves to the case ${\mathcal{V}}_2(M)$. The most general Hamiltonian function of degree 3 reads as $$\begin{aligned}
{\mathcal{Q}}_{S,T}=\xi^\mu p_\mu&+S_\mu{}^\nu\xi^\mu p_\nu+S^{\mu\nu}\zeta_\mu p_\nu\\
&+\tfrac{1}{3!}T_{\mu\nu\kappa}\xi^\mu\xi^\nu\xi^\kappa+\tfrac{1}{2}T_{\mu\nu}{}^\kappa\xi^\mu\xi^\nu\zeta_\kappa+\tfrac{1}{2}T_{\mu}{}^{\nu\kappa}\xi^\mu\zeta_\nu\zeta_\kappa+\tfrac{1}{3!}T^{\mu\nu\kappa}\zeta_\mu\zeta_\nu\zeta_\kappa~,
\end{aligned}$$ where the coefficients $S_{\ldots}^{\ldots}$ and $T_{\ldots}^{\ldots}$ are functions on $M$. We readily compute $$\label{eq:Qsq_twist_GG}
\begin{aligned}
\{&{\mathcal{Q}}_{S,T},{\mathcal{Q}}_{S,T}\}=\\
&p_{\mu} p_{\nu} (2S_{\kappa }{}^{\mu} S^{\kappa \nu}+2 S^{\mu \nu})\\
&+p_{\mu} \xi^{\nu} \xi^{\kappa} \left(S^{\lambda \mu} T_{\nu \kappa \lambda}+S_{\lambda }{}^{\mu} T_{\nu \kappa }{}^{\lambda}+T_{\nu \kappa }{}^{\mu}+2 ({\partial}_{\nu}+S_{\nu }{}^{\lambda} {\partial}_{\lambda})S_{\kappa }{}^{\mu}\right)\\
&+p_{\mu} \zeta _{\nu} \xi^{\kappa} \left(2 S^{\lambda \mu} T_{\kappa \lambda }{}^{\nu}+2 T_{\kappa }{}^{\mu \nu}-2 S_{\lambda }{}^{\mu} T_{\kappa }{}^{\nu \lambda}-2 ({\partial}_{\kappa}+S_{\kappa }{}^{\lambda} {\partial}_{\lambda}) S^{\nu \mu}+2 S^{\nu \lambda} {\partial}_{\lambda} S_{\kappa }{}^{\mu}\right)\\
&+p_{\mu} \zeta _{\nu} \zeta _{\kappa} \left(S^{\lambda \mu} T_{\lambda }{}^{\nu \kappa}+T^{\mu \nu \kappa}+S_{\lambda }{}^{\mu} T^{\nu \kappa \lambda}+2S^{\nu \lambda} {\partial}_{\lambda} S^{\kappa \mu}\right)\\
&+\xi^{\mu} \xi^{\nu} \xi^{\kappa} \xi^{\lambda} \left(\tfrac{1}{2} T_{\mu \nu \rho} T_{\kappa \lambda }{}^{\rho}-\tfrac{1}{3} ({\partial}_{\lambda}+S_{\lambda }{}^{\rho} {\partial}_{\rho})T_{\mu \nu \kappa}\right)\\
&+\zeta _{\mu} \xi^{\nu} \xi^{\kappa} \xi^{\lambda} \left(T_{\lambda \rho }{}^{\mu} T_{\nu \kappa }{}^{\rho}-T_{\nu \kappa \rho} T_{\lambda }{}^{\mu \rho}-({\partial}_{\lambda}+S_{\lambda }{}^{\rho} {\partial}_{\rho}) T_{\nu \kappa }{}^{\mu}+\tfrac{1}{3} S^{\mu \rho} {\partial}_{\rho} T_{\nu \kappa \lambda}\right)\\
&+\zeta _{\mu} \zeta _{\nu} \xi^{\kappa} \xi^{\lambda} \left(2T_{\lambda \rho }{}^{\nu} T_{\kappa }{}^{\mu \rho}+\tfrac{1}{2} T_{\kappa \lambda }{}^{\rho} T_{\rho }{}^{\mu \nu}+\tfrac{1}{2} T_{\kappa \lambda \rho} T^{\mu \nu \rho}-({\partial}_{\lambda}+S_{\lambda }{}^{\rho} {\partial}_{\rho})T_{\kappa }{}^{\mu \nu}-S^{\nu \rho} {\partial}_{\rho} T_{\kappa \lambda }{}^{\mu}\right)\\
&+\zeta _{\mu} \zeta _{\nu} \zeta _{\kappa} \xi^{\lambda} \left(-T_{\lambda }{}^{\kappa \rho} T_{\rho }{}^{\mu \nu}+T_{\lambda \rho }{}^{\kappa} T^{\mu \nu \rho}-\tfrac{1}{3} ({\partial}_{\lambda}+S_{\lambda }{}^{\rho} {\partial}_{\rho}) T^{\mu \nu \kappa}+S^{\kappa \rho} {\partial}_{\rho} T_{\lambda }{}^{\mu \nu}\right)\\
&+\zeta _{\mu} \zeta _{\nu} \zeta _{\kappa} \zeta _{\lambda} \left(\tfrac{1}{2} T_{\rho }{}^{\mu \nu} T^{\kappa \lambda \rho}+\tfrac{1}{3} S^{\mu \rho} {\partial}_{\rho} T^{\nu \kappa \lambda}\right)
\end{aligned}$$ Note that the deformation $S_\mu{}^\nu=-\delta_\mu^\nu$ would remove the constant term in ${\mathcal{Q}}_{S,T}$, and therefore we will not consider it. In the remaining cases, we can verify the following statement by a short computation, cf. also [@Roytenberg:0112152] for a similar statement.
The antisymmetric part of the deformation parameter $S^{\mu\nu}$ can be set to zero by a symplectomorphism, which can be written as $$\label{eq:form_beta_trafo}
z\mapsto -\big\{z,\tau\}{{\qquad\mbox{with}\qquad}}\tau=\tfrac12 \tau^{\mu\nu}\zeta_\mu\zeta_\nu:=\tfrac12(\delta_\mu^\kappa-S_\mu{}^\kappa)^{-1} S^{\nu\kappa}\zeta_\mu\zeta_\kappa$$ for $z\in (x^\mu,\xi^\mu,\zeta_\mu,p_\mu)$. Explicitly, this transformation reads as $$\label{eq:coord_trafo_3.30}
\begin{aligned}
x^\mu\rightarrow x^\mu~,~~~\zeta_\mu\rightarrow \zeta_\mu~,~~~\xi^\mu\rightarrow \xi^\mu-\tau^{\mu\nu}\zeta_\nu{{\qquad\mbox{and}\qquad}}p_\mu\rightarrow p_\mu-\frac12 {\frac{{\partial}\tau^{\kappa\lambda}}{{\partial}x^\mu}}\zeta_\kappa\zeta_\lambda~.
\end{aligned}$$ If $S_\kappa{}^\mu\neq 0$, then $S^{\mu\nu}$ may have a symmetric part and still satisfy $\{{\mathcal{Q}}_{S,T},{\mathcal{Q}}_{S,T}\}=0$. Even this part can be gauged away in this way.[^7]
The coordinate transformation is precisely the $\beta$-transformation of generalized geometry, cf. section \[ssec:generalised\_geometry\].
Next, we can readily classify all admissible infinitesimal twist elements, see also [@Roytenberg:0203110] for a more abstract proof.
The non-trivial infinitesimal twist elements of ${\mathcal{Q}}$ consist of closed 3-forms $\tfrac{1}{3!}T_{\mu\nu\kappa}\xi^\mu\xi^\kappa\xi^\lambda$.
Consider the equation $\{{\mathcal{Q}}_{S,T},{\mathcal{Q}}_{S,T}\}=0$ for infinitesimal twist elements $S,T$. From the linearized version of , it follows that $$\begin{aligned}
S^{\mu\nu}+S^{\nu\mu}&=0~,~~~&T_{\nu\kappa}{}^\mu&=-2{\partial}_\nu S_{\kappa}{}^\mu~,~~~T_{\kappa}{}^{\mu\nu}={\partial}_\kappa S^{\nu\mu}~,\\
{\partial}_{[\mu}T_{\nu\kappa\lambda]}&=0~,~~~&T^{\mu\nu\kappa}&=0~,
\end{aligned}$$ and $T_{\nu\kappa\lambda}$ are thus the components of a closed 3-form. Note that the deformations $$d_1=S_\mu{}^\nu\xi^\mu p_\nu+\tfrac{1}{2}T_{\mu\nu}{}^\kappa\xi^\mu\xi^\nu\zeta_\kappa{{\qquad\mbox{and}\qquad}}d_2=S^{\mu\nu}\zeta_\mu p_\nu+\tfrac{1}{2}T_{\mu}{}^{\nu\kappa}\xi^\mu\zeta_\nu\zeta_\kappa$$ are then $Q$-exact with $$d_1=Q~S_\mu{}^\nu\xi^\mu\zeta_\nu{{\qquad\mbox{and}\qquad}}d_2=Q~S^{\mu\nu}\zeta_\mu\zeta_\nu$$ and therefore trivial, since $Q$-exact terms arise from symplectomorphisms.
Thus, we see that the Ševera class characterizes all infinitesimal deformations of the Vinogradov algebroid.
Exact Courant algebroids and gerbes {#ssec:Ex_Courant_and_Gerbes}
-----------------------------------
It is well known [@Gawedzki:1987ak; @Freed:1999vc] that the $B$-field of string theory should really be regarded as part of a Deligne 3-cocycle from a global perspective. Equivalently, it is the curving 2-form of a ${\mathsf{U}}(1)$-bundle gerbe with connective structure over the target space.
Recall that a principal ${\mathsf{U}}(1)$-bundle is topologically characterized by its first Chern class, whose image in de Rham cohomology is an integer curvature 2-form. Similarly, its higher analogue, a [*${\mathsf{U}}(1)$-gerbe*]{}, is topologically characterized by its Dixmier-Douady class, whose image in de Rham cohomology is an integer curvature 3-form $H={\mathrm{d}}B$.
In general, a gerbe is some geometric realization of an element in singular cohomology $H^3(M,{\mathbbm{Z}})$. The most accessible such realization is probably that in terms of Murray’s bundle gerbes [@Murray:9407015]. Let us very briefly review these in the following, a useful and detailed introduction is found in [@Murray:2007ps]. Assume that we have a suitable surjective map $\pi:Y\rightarrow M$ such that $Y$ covers our manifold $M$. This covering space $Y$ can be chosen to consist of local patches of an ordinary good covering, but $Y$ does not have to be locally diffeomorphic to $M$. Then we can form the fibered product $$Y^{[2]}=Y\times_M Y:=\{(y_1,y_2)\in Y\times Y~|~\pi(y_1)=\pi(y_2)\}~,$$ which generalizes double overlaps of patches. More generally, we define $$Y^{[n]}=Y\times_M \ldots \times_M Y:=\{(y_1,\ldots,y_n)\in Y\times \ldots\times Y~|~\pi(y_1)=\ldots=\pi(y_n)\}~,$$ which generalize double, triple and $n$-fold overlaps of local patches. A [*${\mathsf{U}}(1)$-bundle gerbe*]{} is a principal ${\mathsf{U}}(1)$-bundle $P$ over $Y^{[2]}$, together with an isomorphism of principal bundles $\mu_{123}:P_{12}\otimes P_{23}\rightarrow P_{13}$, called [*bundle gerbe multiplication*]{}, where $P_{ij}$ are the three possible pullbacks of $P$ along the three projections $Y^{[3]}\rightarrow Y^{[2]}$: $${\vcenter{\vbox{\xymatrix{
{\begin{array}{c}
\mu_{123}:P_{12}\otimes P_{23}\stackrel{\cong}{\rightarrow}P_{13}\\
P_{12,13,23}
\end{array}}\ar@{->}[d]& P \ar@{->}[d] & \\
Y^{[3]}\ar@<-3pt>[r] \ar@<0pt>[r]\ar@<3pt>[r]& Y^{[2]} \ar@<1.5pt>[r] \ar@<-1.5pt>[r] & Y \ar@{->}[d]^{\pi} \\
& & M
}}}}$$ This bundle gerbe multiplication, or rather its pullback to $Y^{[4]}$ has to satisfy $$\mu_{134}\circ \mu_{123}=\mu_{124}\circ \mu_{234}~.$$ Altogether, a ${\mathsf{U}}(1)$-bundle gerbe is given by the triple $(P,\pi,\mu)$.
As an example, consider the trivial bundle $P=M\times {\mathsf{U}}(1)$, which can be regarded as the trivial bundle gerbe for $Y=Y^{[2]}=Y^{[n]}=M$. Since $P\otimes P\cong P$ on $Y^{[3]}=M$, the bundle gerbe multiplication $\mu_{123}$ is simply the trivial isomorphism. This example is relevant for T-duality for trivial torus fibrations. Note that non-trivial principal ${\mathsf{U}}(1)$-bundles $P$ are [*not*]{} bundle gerbes because for them, there is no isomorphism between $P\otimes P$ and $P$ over $M$.
We can now endow the principal ${\mathsf{U}}(1)$-bundle $P$ over $Y$ in a bundle gerbe $(P,\pi,\mu)$ with a connection $\nabla$ compatible with the bundle gerbe multiplication. Its curvature $F_\nabla$ on $Y$ can be shown [@Murray:2007ps] to be necessarily the difference of the pullbacks of a 2-form $B$ on $Y$ along the two possible projections $Y^{[2]}\rightarrow Y$. Finally, the 3-form ${\mathrm{d}}B$ on $Y$ is necessarily the pullback of a global 3-form $H$ on $M$. This 3-form $H$ is integral, just as the two form representing the first Chern class of a principal ${\mathsf{U}}(1)$-bundle. That is, any integral over closed 3-dimensional manifolds in $M$ is an integer. We call the data $(\nabla,B)$ on a ${\mathsf{U}}(1)$-bundle gerbe a [*connective structure*]{}.
Inversely, consider an integral 3-form $H\in H^3_{\rm dR}(M)$ representing the Dixmier-Douady of a bundle gerbe. For simplicity, we restrict ourselves to an ordinary cover ${\mathfrak{U}}=(U_i)$. By the Poincaré lemma, there are potential 2-forms $B_i$ on local patches $U_i$ with $H|_{U_i}={\mathrm{d}}B_i$. These are glued together by gauge transformations on overlaps $U_i\cap U_j$ parameterized by 1-forms $\Lambda_{ij}$ according to $B_i-B_j={\mathrm{d}}\Lambda_{ij}$, which in turn give rise to functions $f_{ijk}$ with ${\mathrm{d}}f_{ijk}=\Lambda_{ij}-\Lambda_{ik}+\Lambda_{jk}$. The $f_{ijk}$ form the Čech cocycle describing the bundle gerbe multiplication $\mu_{ijk}={\mathrm{e}}^{{\mathrm{i}}f_{ijk}}$, analogously to transition functions of a principal ${\mathsf{U}}(1)$-bundle.
There are now two obvious local symmetries of a ${\mathsf{U}}(1)$-bundle gerbe: the gauge symmetry of the 2-form $B$ as well as diffeomorphisms acting on the base space $M$. At infinitesimal level, these are captured by a 1-form and a vector field. Since a gerbe is a categorified space, these infinitesimal symmetries do not form an ordinary Lie algebra but rather a Lie 2-algebra, and this Lie 2-algebra is precisely the Lie 2-algebra constructed from the twisted Hamiltonian ${\mathcal{Q}}_{S,T}$. Note that it is [*not*]{} a semidirect product of $L_\infty$-algebras, because if we write $$\mu_2(X,\beta)=\rho(X)\beta~,$$ then $\rho(X)$ is [*not*]{} an action, i.e. an element of ${\mathfrak{X}}(M)$, but contains second order derivatives, $${\mathrm{d}}\iota_X\beta=\xi^\mu {\frac{{\partial}}{{\partial}x^\mu}} \left(X^\nu {\frac{{\partial}}{{\partial}\xi^\nu}}\beta_\kappa \xi^\kappa\right)~.$$
In the Lie 2-algebra , elements $X+\alpha$ of ${\mathfrak{X}}(M)\oplus\Omega^1(M)$ parametrize infinitesimal diffeomorphisms and gauge transformations, while elements $f$ of ${\mathcal{C}}^\infty(M)$ yield gauge transformations between gauge transformations: $${\vcenter{\vbox{\xymatrix{
\bullet
\ar@/^4ex/[rr]^{X+\alpha}="g1"
\ar@/_4ex/[rr]_{X+\alpha+{\mathrm{d}}f}="g3"
\ar@{=>}^{f} "g1"+<0ex,-1.8ex>;"g3"+<0ex,1.8ex>
&& \bullet
}}}}$$
More generally, and as suggested by Ševera [@Severa:1998ab], the Courant algebroid with Ševera class $H\in H^3(M,{\mathbbm{Z}})$ is to be seen as the Atiyah algebroid of the gerbe with Dixmier-Douady class $H$, see also [@Rogers:2010sc]. The infinitesimal symmetries[^8] of a ${\mathsf{U}}(1)$-gerbe with characteristic class $H$ over $M$ are described locally over patches $U_i$ by the exact Courant algebroid ${\mathcal{V}}_2(U_i)$ with Ševera class $H$. The 1-forms $\Lambda_{ij}$ introduced above, which describe the connection on the gerbe can now be used to glue together the local descriptions $TU_i\oplus T^*U_i$ over overlaps of patches $U_i\cap U_j$ by mapping $$X+\alpha\mapsto X+\alpha+\iota_X{\mathrm{d}}\Lambda_{ij}~.$$ The result is the [*generalized tangent bundle*]{}, cf. [@Hitchin:2005in], and this is one of the first replacements one should make in globalizing Vinogradov algebroids in the presence of non-trivial $H$-flux. Note that in this paper, all our considerations will remain local.
These above observations for $n=2$ readily extend to higher $n$. Functions on ${\mathcal{V}}_n(M)$ of degree $n-1$ parametrize local gauge transformations of a metric and a connection $n$-form on a ${\mathsf{U}}(1)$-bundle $n-1$-gerbe. Such local gauge transformations form a Lie $n$-algebra, which is encoded in the $L_\infty$-algebra associated to the Vinogradov Lie $n$-algebroids ${\mathcal{V}}_n(M)$. A more comprehensive analysis of all this is found e.g. in [@Fiorenza:1304.6292].
In summary, we can say that the Vinogradov Lie $n$-algebroids ${\mathcal{V}}_n(M)$ twisted by a closed $n+1$ form $\omega$ are linearizations of $(n-1)$-gerbes with characteristic class $\omega$ in the sense that they capture their infinitesimal gauge and diffeomorphism symmetry Lie $n$-algebra.
Extended Riemannian geometry and pre-NQ-manifolds {#sec:extendedRG}
=================================================
Motivation
----------
To study differential geometry of Riemannian manifolds, we require a definition of tensors including a preferred symmetric tensor specifying the metric. Moreover, there is a Lie algebra of infinitesimal diffeomorphisms which is generated by vector fields and acts on tensors via the Lie derivative. This Lie algebra is simply the Lie algebra of vector fields. With it and the Lie derivative, we readily define the exterior derivative. Next, Cartan’s formula links the Lie derivative to a composition of exterior derivative and interior product. The interior product is naturally extended to a more general tensor contraction, which should be defined, too. Finally, we require some notion of curvature tensors, from which we can derive an action principle for geodesics, if desired.
Most of the above mentioned structures are readily described in terms of structures on N$Q$-manifolds. Consider, e.g., the Vinogradov Lie $2$-algebroid ${\mathcal{V}}_2(M)$ for some $D$-dimensional manifold $M$ with coordinates $(x^\mu,p_\mu,\xi^\mu,\zeta_\mu)$, $\mu=1,\ldots, D$, on the total space. Recall from section \[ssec:Vinogradov\] that ${\mathcal{V}}_2(M)$ is a symplectic N$Q$-manifold of ${\mathbbm{N}}$-degree $n=2$ and comes with a symplectic form $\omega$ and a homological vector field $Q$ with Hamiltonian ${\mathcal{Q}}$.
Forms and polyvector fields are now simply functions on ${\mathcal{V}}_2(M)$: $$\begin{aligned}
\frac{1}{k!}X^{\mu_1\ldots\mu_k}(x){\frac{{\partial}}{{\partial}x^{\mu_1}}}\wedge\ldots \wedge{\frac{{\partial}}{{\partial}x^{\mu_k}}} &\ \leftrightarrow\ \frac{1}{k!}X^{\mu_1\ldots\mu_k}(x)\zeta_{\mu_1}\ldots \zeta_{\mu_k}~,\\
\frac{1}{k!}\alpha_{\mu_1\ldots\mu_k}(x){\mathrm{d}}{x^{\mu_1}}\wedge\ldots \wedge{\mathrm{d}}{x^{\mu_k}} &\ \leftrightarrow\ \frac{1}{k!}\alpha_{\mu_1\ldots\mu_k}(x)\xi^{\mu_1}\ldots \xi^{\mu_k}~.
\end{aligned}$$ The Lie derivative is given by $${\mathcal{L}}_X \alpha=\{QX,\alpha\}~,$$ and the Lie bracket on vector fields is its antisymmetrization $$[X,Y]=\tfrac12\big(\{QX,Y\}-\{QY,X\}\big)=\{QX,Y\}~.$$ Note that plugging polyvector fields into this bracket, one recovers the Schouten bracket. The exterior derivative is then simply $${\mathrm{d}}\alpha=Q\alpha$$ and, compatible with Cartan’s formula, we have $$\iota_X \alpha=\{X,\alpha\}=(-1)^{p+1}\{\alpha,X\}~.$$
It is therefore natural to expect that one can formulate differential and Riemannian geometry in terms of expressions on N$Q$-manifolds, which then extends to more general situations. It will turn out that we can even go slightly beyond N$Q$-manifolds, and the resulting framework will reproduce the formulas of double field theory.
In this section, we axiomatize our definitions. In a subsequent section, we work out various examples of our framework, which we call [*extended Riemannian geometry*]{}.
Extended diffeomorphisms {#ssec:extended_diffeos}
------------------------
For our purposes, we will require a generalization of symplectic N$Q$-manifolds.
A is a symplectic N-manifold $({\mathcal{M}},\omega)$ of ${\mathbbm{N}}$-degree $n$ with a vector field $Q$ of ${\mathbbm{N}}$-degree 1 satisfying ${\mathcal{L}}_Q\omega=0$.
The fact that $Q$ generates symplectomorphisms on ${\mathcal{M}}$ implies again that it is Hamiltonian. In the following, let $\{-,-\}$ be the Poisson bracket induced by $\omega$ and let ${\mathcal{Q}}$ be the Hamiltonian of $Q$.
To describe symmetries in extended Riemannian geometry, we require a homogeneously graded set of extended vector fields ${\mathscr{X}}({\mathcal{M}})\subset {\mathcal{C}}^\infty({\mathcal{M}})$ (not to be confused with the vector fields on ${\mathcal{M}}$, ${\mathfrak{X}}({\mathcal{M}})$). These should form a semistrict Lie $n$-algebra with the higher brackets obtained as the derived brackets introduced in .
In particular, we have the binary bracket $$\begin{aligned}
&\mu_2: \wedge^2 {\mathscr{X}}({\mathcal{M}})\rightarrow {\mathscr{X}}({\mathcal{M}})~,\\
&\mu_2(X,Y):=\tfrac12\left(\{QX,Y\}-\{QY,X\}\right)~,
\end{aligned}$$ which requires the elements of ${\mathscr{X}}({\mathcal{M}})$ to be of ${\mathbbm{N}}$-degree $n-1$. We immediately have the following simple statement:
\[lem:PB\_X\_f\_trivial\] Since the Poisson bracket is of degree $-n$, we have $\{X,f\}=0$ for all $X\in{\mathscr{X}}({\mathcal{M}})$ and $f$ of ${\mathbbm{N}}$-degree 0.
Note that $\mu_2$ is related to the ordinary derived bracket $\nu_2$ as follows.
\[lem:Courant-Dorfman\] The map $\mu_2$ and the map $$\label{eq:def:nu2}
\begin{aligned}
&\nu_2: \otimes^2 {\mathscr{X}}({\mathcal{M}})\rightarrow {\mathscr{X}}({\mathcal{M}})~,\\
&\nu_2(X,Y):=\{QX,Y\}~,
\end{aligned}$$ are related to each other by the equations $$\mu_2(X,Y)=\tfrac12(\nu_2(X,Y)-\nu_2(Y,X)){{\qquad\mbox{and}\qquad}}\nu_2(X,Y)=\mu_2(X,Y)+\tfrac12 Q\{X,Y\}~.$$
As stated above, $\mu_2$ should be the binary bracket on the degree 0 part of a Lie $n$-algebra. One might additionally want $\nu_2$ to form a Leibniz algebra[^9]. We start with examining the latter constraint.
\[lem:nu\_2:Leibniz\] The map $\nu_2$ as defined in defines a Leibniz algebra (of ${\mathsf{L}}$-degree $0$) on ${\mathscr{X}}(M)$, i.e. $$\nu_2(X,\nu_2(Y,Z))=\nu_2(\nu_2(X,Y),Z)+\nu_2(Y,\nu_2(X,Z))~,$$ if and only if $$\{\{Q^2X,Y\},Z\}=0$$ for all $X,Y,Z\in {\mathscr{X}}(M)$.
We compute $$\begin{aligned}
&\nu_2(X,\nu_2(Y,Z))-\nu_2(\nu_2(X,Y),Z)-\nu_2(Y,\nu_2(X,Z))\\
&=\{QX,\{QY,Z\}\}-\{Q\{QX,Y\},Z\}-\{QY,\{QX,Z\}\}\\
&=\{\{QX,QY\},Z\}+\{QY,\{QX,Z\}\}-\{\{Q^2X,Y\},Z\}-\{\{QX,QY\},Z\}+\\
&\hspace{10.5cm}-\{QY,\{QX,Z\}\}\\
&=-\{\{Q^2X,Y\},Z\}~.
\end{aligned}$$
Note that this result is interesting in its own right. Derived brackets of this kind have been abstractly discussed in [@kosmann1996poisson], where they were called [*generalized Loday-Gerstenhaber brackets*]{}. We are not aware of more than a few examples of such brackets, and our construction of $\nu_2$ on pre-N$Q$-manifolds using a subset ${\mathscr{X}}(M)$ of the full set of functions provides a large class of such examples.
For our discussion, however, lemma \[lem:nu\_2:Leibniz\] merely serves as a stepping stone to the following statement.
\[thm:algebra\_structure\_of\_brackets\] The antisymmetrized derived bracket $\mu_2$ satisfies the homotopy Jacobi relations $$\mu_2(X,\mu_2(Y,Z))+\mu_2(Y,\mu_2(Z,X))+\mu_2(Z,\mu_2(X,Y ))=Q(\mu_3(X,Y,Z))~,$$ $$\mu_3(X,Y,Z)=3\{\{Q X,Y\},Z\}_{[X,Y,Z]}~,$$ if and only if $$\label{eq:Q2XYZ}
\{\{Q^2X,Y\},Z\}_{[X,Y,Z]}=0$$ holds for all $X,Y,Z\in {\mathscr{X}}({\mathcal{M}})$. Here, and in the following we use the notation $$\begin{aligned}
&F(X,Y,Z)_{[X,Y,Z]}=\\
&\hspace{1cm}\tfrac{1}{3!}\big(F(X,Y,Z)-F(Y,X,Z)+F(Y,Z,X)-F(Z,Y,X)+F(Z,X,Y)-F(X,Z,Y)\big)
\end{aligned}$$ for the weighted, totally antisymmetrized sum of some function $F$ with range in a vector space.
This statement follows from the proof of lemma \[lem:nu\_2:Leibniz\] together with the observation that $$\begin{aligned}
\mu_2(X,\mu_2(Y,Z))=\nu_2(X,\nu_2(Y,Z))-\{X,Q^2\{Y,Z\}\}+Q\{\{QY,Z\},X\}~,
\end{aligned}$$ which is a consequence of lemma \[lem:Courant-Dorfman\], and $\{Y,Z\}-\{Z,Y\}=0$.
Clearly, the condition for $\mu_2$ satisfying the homotopy Jacobi relation is weaker than that of $\nu_2$ forming a Leibniz algebra. Again, we are merely interested in the Lie $n$-algebra of symmetries, so we will constrain the set ${\mathscr{X}}({\mathcal{M}})$ by .
Let us now extend theorem \[thm:algebra\_structure\_of\_brackets\] to a full Lie $n$-algebra structure on a subset ${\mathsf{L}}({\mathcal{M}})$ of ${\mathcal{C}}^\infty({\mathcal{M}})$ with derived brackets . We start with the chain complex $${\mathsf{L}}({\mathcal{M}}):= {\mathsf{L}}_{n-1}({\mathcal{M}}) \xrightarrow{~Q~} {\mathsf{L}}_{n-2}({\mathcal{M}}) \xrightarrow{~Q~} \ldots \xrightarrow{~Q~} {\mathsf{L}}_{1}({\mathcal{M}}) \xrightarrow{~Q~} {\mathsf{L}}_{0}({\mathcal{M}}) \xrightarrow{~0~} 0~,$$ cf. . Note that the homotopy Jacobi relation $\mu_1^2:=Q^2=0$ is nontrivial only when applied to elements of ${\mathsf{L}}_k({\mathcal{M}})$ with $k>1$. We can therefore lift $Q^2=0$ on elements of ${\mathsf{L}}_1({\mathcal{M}})$ and ${\mathsf{L}}_0({\mathcal{M}})$, as done in theorem \[thm:algebra\_structure\_of\_brackets\]. Moreover, it is clear from the proof of theorem \[thm:algebra\_structure\_of\_brackets\] that in verifying the higher homotopy relations , $Q^2$ will never appear outside all Poisson brackets, but always in the form $\{\ldots\{Q^2-,-\},\ldots\}$. Therefore, the condition $Q^2=0$ is unnecessarily strict and can be relaxed to conditions like . As a consequence, however, we cannot expect an $L_\infty$-algebra structure on all of ${\mathcal{C}}^\infty({\mathcal{M}})$, as in the case of Lie $n$-algebroids. Instead, we have to choose a suitable subset.
Given a pre-N$Q$-manifold ${\mathcal{M}}$, an on ${\mathcal{M}}$ is a subset ${\mathsf{L}}({\mathcal{M}})$ of the functions ${\mathcal{C}}^\infty(M)$ such that the derived brackets close on ${\mathsf{L}}({\mathcal{M}})$ and form an $L_\infty$-algebra.
In special cases, this notion of $L_\infty$-algebra structure corresponds to a polarization or, as we will see later, to the strong section condition of double field theory up to a slight weakening.
We are particularly interested in the case of Lie 2-algebras for which we have the following theorem.
\[thm:Lie\_2\_subset\] Consider a subset ${\mathsf{L}}({\mathcal{M}})$ of ${\mathcal{C}}^\infty(M)$ concentrated in ${\mathsf{L}}$-degrees $0$ and $1$, i.e. ${\mathsf{L}}({\mathcal{M}})={\mathsf{L}}_1({\mathcal{M}})\oplus {\mathsf{L}}_0({\mathcal{M}})$, on which the derived brackets and the Poisson bracket close. Then ${\mathsf{L}}({\mathcal{M}})$ is an $L_\infty$-algebra if and only if $$\label{eq:conditions_thm_Lie2_subset}
\begin{aligned}
\{Q^2f,g\}+\{Q^2g,f\}&=0~,\\
\{Q^2X,f\}+\{Q^2f,X\}&=0~,\\
\{\{Q^2X,Y\},Z\}_{[X,Y,Z]}&=0
\end{aligned}$$ for all $f,g\in {\mathsf{L}}_1({\mathcal{M}})$ and $X,Y,Z\in {\mathsf{L}}_0({\mathcal{M}})$.
The first and second conditions are equivalent to the homotopy Jacobi identities$\mu_2(\mu_1(f),g)=\mu_2(f,\mu_1(g))$ and $\mu_1(\mu_2(X,f))=\mu_2(X,\mu_1(f))$, respectively. The third condition is equivalent to $$\mu_1(\mu_3(X,Y,Z))=\mu_2(X,\mu_2(Y,Z))+\mu_2(Y,\mu_2(Z,X))+\mu_2(Z,\mu_2(X,Y))$$ by theorem \[thm:algebra\_structure\_of\_brackets\]. The homotopy Jacobi identity $$\mu_3(\mu_1(f),X,Y)=\mu_2(f,\mu_2(X,Y))+\mu_2(Y,\mu_2(f,X))+\mu_2(X,\mu_2(Y,f))$$ yields the condition $$\{\{Q^2 f,X\},Y\}-\{\{Q^2f,Y\},X\}-\{\{Q^2 X,Y\},f\}+\{\{Q^2Y,X\},f\}=0~,$$ which is automatically satisfied, if the brackets close on ${\mathsf{L}}({\mathcal{M}})$ and the second condition holds. Here, we used that $\{X,f\}=0$ from lemma \[lem:PB\_X\_f\_trivial\] implies $$\{Q X,f\}-\{X,Qf\}=\{Qf,X\}+\{f,QX\}=0~.$$ Using the same relation, we find by direct computation that also the identity holds up to terms of the form $\{\{Q^2X,Y\},Z\}_{[X,Y,Z]}$.
Extended tensors {#ssec:extended_tensors}
----------------
The $L_\infty$-structure ${\mathsf{L}}({\mathcal{M}})$ on the pre-N$Q$-manifold ${\mathcal{M}}$ will play the role of extended infinitesimal diffeomorphisms and gauge transformations, which should act on extended tensors. Note that ${\mathcal{C}}^\infty({\mathcal{M}})$ already encodes totally antisymmetric extended tensors. In order to capture extended Riemannian geometry, and in particular the metric, we will have to allow for arbitrary extended tensor. This is done by replacing the graded symmetric tensor algebra ${\mathcal{C}}^\infty({\mathcal{M}})$ generated by the coordinate functions of positive grading on ${\mathcal{M}}$ by the free (associative) tensor algebra $T({\mathcal{M}})$ generated by the coordinate functions of positive grading. Functions of ${\mathbbm{N}}$-degree zero can be moved through the tensor product in $T({\mathcal{M}})$.
We would like our gauge $L_\infty$-algebra ${\mathsf{L}}({\mathcal{M}})$ to act on elements in $T({\mathcal{M}})$, and the natural candidate is a generalization of the derived bracket. For this, we extend the Poisson bracket via the Leibniz rule as follows.
We implicitly define an extension of the Poisson bracket on ${\mathcal{M}}$,$\{-,-\}:{\mathsf{L}}({\mathcal{M}})\times T({\mathcal{M}})\rightarrow T({\mathcal{M}})$, by $$\label{eq:ext_Poisson}
\{f,g\otimes h\}:=\{f,g\}\otimes h+(-1)^{(n-|f|)|g|}g\otimes \{f,h\}~.$$
Note that this equation is consistent and fixes the extension uniquely, as is readily seen by computing the expression $\{f,g_1\otimes g_2\otimes g_3\}$, $g_{1,2,3}\in {\mathcal{C}}^\infty({\mathcal{M}})$, in the two possible ways. We now have the following lemma:
The extended Poisson bracket satisfies the Jacobi identity $$\{f,\{g,t\}\}=\{\{f,g\},t\}+(-1)^{(|f|+n)(|g|+n)}\{g,\{f,t\}\}$$ for all $f,g\in {\mathcal{C}}^\infty({\mathcal{M}})$ and $t\in T({\mathcal{M}})$.
By direct verification of the identity for $t=t_1\otimes t_2$.
The extended Poisson bracket allows us now to define a natural action of the Lie $n$-algebra ${\mathsf{L}}({\mathcal{M}})$ on a subset of $T({\mathcal{M}})$ via $$\label{eq:L_infty-rep}
f{\vartriangleright}t:=\{\delta f,t\}~,$$ where $f\in {\mathsf{L}}({\mathcal{M}})$, $t\in {\mathsf{T}}({\mathcal{M}})\subset T({\mathcal{M}})$ and $\delta$ was defined in section \[ssec:NQ-manifolds\]. The subset ${\mathsf{T}}({\mathcal{M}})$ is then defined by the requirement that is indeed an action of an $L_\infty$-algebra as defined in section \[ssec:NQ-manifolds\].
\[thm:restrictions\_T\] The map defines an action of ${\mathsf{L}}({\mathcal{M}})$ on ${\mathsf{T}}({\mathcal{M}})$ if ${\mathsf{T}}({\mathcal{M}})$ is a subset of $T({\mathcal{M}})$ whose elements $t$ satisfy $$\label{eq:tensor-conditions}
\{\{Q^2X,Y\}-\{Q^2Y,X\},t\}=0{{\qquad\mbox{and}\qquad}}\{\{\{Q^2X,Y\}-\{Q^2Y,X\},QZ\},t\}=0$$ for all $X,Y,Z\in {\mathsf{L}}_0({\mathcal{M}})$. That is, ${\vartriangleright}$ defines a morphism of $L_\infty$-algebras from ${\mathsf{L}}({\mathcal{M}})$ to ${\mathfrak{X}}({\mathsf{T}}({\mathcal{M}}))$.
The first equation amounts to $$X{\vartriangleright}(Y{\vartriangleright}t)-Y{\vartriangleright}(X{\vartriangleright}t)=\mu_2(X,Y){\vartriangleright}t~,$$ for all $X,Y\in {\mathsf{L}}_0({\mathcal{M}})$, where $$\mu_2(X,Y)=\tfrac12(\{QX,Y\}-\{QY,X\})~.$$ The second equation guarantees that $X{\vartriangleright}$ is indeed an element of ${\mathfrak{X}}({\mathsf{T}}({\mathcal{M}}))$, that is $X{\vartriangleright}t\in {\mathsf{T}}({\mathcal{M}})$. Since $\delta$ vanishes on ${\mathsf{L}}_i({\mathcal{M}})$ for $i>0$, there is nothing else to check.
Note that for $t\in{\mathsf{L}}_0({\mathcal{M}})$, the first condition in is an antisymmetrization of that of lemma \[lem:nu\_2:Leibniz\], but stronger than that of theorem \[thm:algebra\_structure\_of\_brackets\]. Ideally, we would like elements of ${\mathsf{L}}({\mathcal{M}})$ to be simultaneously elements of ${\mathsf{T}}({\mathcal{M}})$, just as vectors are also tensors. Also, it is not clear to us whether the most general set ${\mathsf{T}}({\mathcal{M}})$ is really relevant or interesting. We therefore give the following definition.
An on a symplectic pre-N$Q$-manifold ${\mathcal{M}}$ of degree $n$ is an $L_\infty$-algebra structure ${\mathsf{L}}({\mathcal{M}})$, which is simultaneously carrying an action of ${\mathsf{L}}({\mathcal{M}})$. An is an element of ${\mathscr{X}}({\mathcal{M}})={\mathsf{L}}_0({\mathcal{M}})$ of ${\mathsf{L}}$-degree 0. It is a sections of the ${\mathscr{T}}{\mathcal{M}}$, which is a vector bundle over $M={\mathcal{M}}_0$.
An is now simply a map from ${\mathscr{X}}^*({\mathcal{M}})^{\otimes p}\otimes{\mathscr{X}}({\mathcal{M}})^{\otimes q}\rightarrow {\mathcal{C}}^\infty({\mathcal{M}}_0)$, which is multilinear over each point of ${\mathcal{M}}_0$.
An is an element of ${\mathcal{C}}^\infty({\mathcal{M}}_0)={\mathsf{L}}_{n-1}({\mathcal{M}})$ of ${\mathsf{L}}$-degree $n-1$.
Note that any $L_\infty$-algebra structure gives rise to an extended tangent bundle structure, as long as $Q^2=0$.
Extended tensor densities {#ssec:tensor_densitites}
-------------------------
For writing down actions, we shall also require an extended scalar tensor density and its transformations under extended symmetries. For example, the naive local top form ${\mathrm{d}}x^1\wedge \ldots \wedge {\mathrm{d}}x^D$ on a $D$-dimensional Lorentzian manifold is not invariant, but needs to be multiplied by a scalar tensor density, yielding $$\label{eq:vol_GR}
\sqrt{|g|}\,{\mathrm{d}}x^1\wedge \ldots \wedge {\mathrm{d}}x^D~~~\mbox{or}~~~\sqrt{|g|}\,{\mathrm{e}}^{-2\phi}\,{\mathrm{d}}x^1\wedge \ldots \wedge {\mathrm{d}}x^D~,$$ where $\phi$ is the ordinary dilaton field. We follow the convention of double field theory and combine both the tensor density and a potential exponential of a dilaton into an extended dilaton $d$ and always write $$\Omega:={\mathrm{e}}^{-2d}\,{\mathrm{d}}x^1\wedge \ldots \wedge {\mathrm{d}}x^D$$ for the volume form. Note that $d=-\tfrac12 \log \sqrt{|g|}$ in the case of ordinary geometry and trivial dilaton.
In a general extended geometry, the transformation law of ${\mathrm{e}}^{-2d}$ is derived from the fact that a scalar function $R$ together with $\Omega$ should give rise to the invariant action $$\label{eq:invariance_action}
S=\int \Omega R~,$$ such that $\delta S$ is an integral over a total derivative. That is, $$(X{\vartriangleright}{\mathrm{e}}^{-2d})\Omega R+{\mathrm{e}}^{-2d}(X{\vartriangleright}\Omega) R+{\mathrm{e}}^{-2d}\Omega (X{\vartriangleright}R)={\partial}_M(X^M{\mathrm{e}}^{-2d}\Omega R)~,$$ which reproduces in the case of ordinary differential geometry. We shall work out several further examples in sections \[sec:examples\] and \[sec:dft\].
Extended metric and action {#ssec:metric_and_action}
--------------------------
So far, we have described extended infinitesimal symmetries as Lie $n$-algebras and we have defined their action on extended tensor fields. It remains to provide a definition of an extended metric. For this, we generalize an idea found e.g. in [@Hull:2007zu].
Let ${\mathsf{G}}$ and $\varrho$ be the structure group and the relevant representation on the extended tangent bundle ${\mathscr{T}}({\mathcal{M}})$ on our pre-N$Q$-manifold ${\mathcal{M}}$ and let ${\mathsf{H}}$ be a maximal compact subgroup. An extended metric is simply a reduction of the bundle ${\mathscr{T}}({\mathcal{M}})$ to another one, $E$, with structure group ${\mathsf{H}}$ and restriction of the representation $\varrho$.
Explicitly, let ${\mathfrak{U}}=(U_i)$ be a cover of $M$ and let $g_{ij}:U_i\cap U_j\rightarrow \varrho({\mathsf{G}})$ be maps encoding a general cocycle with which defines ${\mathscr{T}}({\mathcal{M}})$ subordinate to ${\mathfrak{U}}$. Then there is a coboundary given by maps $\gamma_i:U_i\rightarrow \varrho({\mathsf{G}})$ between $g_{ij}$ and another cocycle $h_{ij}$ with values in $\varrho({\mathsf{H}})$, defining an isomorphic vector bundle: $$\label{eq:coboundary}
h_{ij}=\gamma_i g_{ij}\gamma_j^{-1}~.$$ This equation implies that $$\gamma_i=h_{ij}\gamma_j g_{ij}^{-1}~,$$ and therefore $\gamma_i$ encodes a bundle morphism from ${\mathscr{T}}({\mathcal{M}})$ to another bundle $E$ with structure group ${\mathsf{H}}$ and representation $\varrho|_{{\mathsf{H}}}$.
Note, however, that also a coboundary given by $h_i\gamma_i$ with $h_i:U_i\rightarrow \varrho|_{{\mathsf{H}}}$ defines such a bundle: $$\label{eq:remaining_equivalence}
h'_{ij}=h_i\gamma_i g_{ij} (h_j\gamma_j)^{-1}~.$$ Let us now assume that ${\mathscr{T}}({\mathcal{M}})$ carries an ${\mathsf{H}}$-invariant inner product and denote the corresponding adjoints by $-^*$. Then $h^*_i=h^{-1}_i$, but $\gamma_i^*\neq \gamma_i^{-1}$ in general and we have a simple way of factoring out the remaining equivalence by considering $${\mathcal{H}}:=\gamma^*\gamma~.$$ We will usually refer to ${\mathcal{H}}$ as the extended metric.
Action principles can be defined from the extended metric and its derivatives as functions which are invariant under the symmetry group ${\mathcal{H}}$ and the extended symmetries induced by the extended Lie derivative. Such action functionals have to be constructed individually for each extended geometry as a sum of ${\mathsf{H}}$-invariant terms which is invariant under extended diffeomorphisms. It turns out that in all the cases we shall discuss, the terms considered in the literature on double and exceptional field theory, cf. e.g. [@Berman:2011pe], are sufficient. That is, we will consider actions of the form $$\begin{aligned}
S=\int_M &{\mathrm{e}}^{-2d}\,{\mathrm{d}}x^1\wedge \ldots \wedge {\mathrm{d}}x^D \Big(c_0 {\mathcal{H}}_{MN}{\partial}^M{\mathcal{H}}_{KL}{\partial}^N {\mathcal{H}}^{KL}+c_1 {\mathcal{H}}_{MN}{\partial}^M{\mathcal{H}}_{KL}{\partial}^L{\mathcal{H}}^{KN}\\
&~~+c_2{\mathcal{H}}^{MN}({\mathcal{H}}^{KL}{\partial}_M{\mathcal{H}}_{KL})({\mathcal{H}}^{RS}{\partial}_N {\mathcal{H}}_{RS})+c_3{\mathcal{H}}^{MN}{\mathcal{H}}^{PQ}({\mathcal{H}}^{RS}{\partial}_P{\mathcal{H}}_{RS})({\partial}_M{\mathcal{H}}_{NQ})\\
&~~+c_4 {\partial}^Md {\partial}^N {\mathcal{H}}_{MN}+ c_5{\mathcal{H}}_{MN}{\partial}^M d{\partial}^N d\Big)
\end{aligned}$$ for some real constants $c_0,\ldots,c_5$, where $d$ is the extended dilaton introduced in section \[ssec:tensor\_densitites\].
It is an obvious question whether one can derive this action from an extended Riemann tensor. We will discuss the problems associated with extended Riemann and torsion tensors in detail in section \[ssec:Riemann\_tensor\].
Examples {#sec:examples}
========
Example: Riemannian geometry
----------------------------
To describe the symmetries of ordinary Riemannian geometry on a manifold $M$, we choose ${\mathcal{M}}={\mathcal{V}}_1(M)$ with coordinates $x^\mu,\zeta_\mu$ of ${\mathbbm{N}}$-degree 0 and $\xi^\mu,p_\mu$ of ${\mathbbm{N}}$-degree 1, cf. section \[ssec:Vinogradov\]. We trivially have $Q^2=0$, so we are free to choose any consistent subset of ${\mathcal{C}}^\infty(M)$ as our $L_\infty$-structure. The appropriate ${\mathsf{L}}({\mathcal{M}})$ here are the functions linear in $\zeta_\mu$ since these parametrize vector fields $X=X^\mu\zeta_\mu$, and their Lie $n$-algebra structure is simply the Lie algebra of vector fields: $$\mu_2(X,Y)=\tfrac12\big(\{QX,Y\}-\{QY,X\}\big)=X^\mu {\partial}_\mu Y^\nu\zeta_\nu-Y^\mu{\partial}_\mu X^\nu\zeta_\nu=[X,Y]~.$$ Since $Q^2=0$, the tensor fields ${\mathsf{T}}({\mathcal{M}})$ are all elements of $T({\mathcal{M}})$, and the action of ${\mathsf{L}}({\mathcal{M}})$ on ${\mathsf{T}}({\mathcal{M}})$ is just the usual transformation law of tensors under infinitesimal diffeomorphisms: $$\begin{aligned}
X{\vartriangleright}t^\mu{}_\nu \zeta_\mu\otimes \xi^\nu&:=\{\delta X,t^\mu{}_\nu \zeta_\mu\otimes \xi^\nu\}\\
&=\big(X^\mu({\partial}_\mu t^\nu{}_\kappa)-({\partial}_\mu X^\nu) t^\mu{}_\kappa+({\partial}_\kappa X^\mu) t^\nu{}_\mu\big)\zeta_\nu\otimes \xi^\kappa~.
\end{aligned}$$ The reduction from ${\mathsf{G}}$ to ${\mathsf{H}}$ defining the extended metric is degenerate in this case, and the coboundaries $\gamma_i$ are simply chosen to be ordinary vielbeins. This yields the ordinary metric as extended metric, and it transforms as expected: $$X{\vartriangleright}g_{\mu\nu}\xi^\mu\odot \xi^\nu=(X^\kappa{\partial}_\kappa g_{\mu\nu}+{\partial}_\mu X^\kappa g_{\kappa\nu}+{\partial}_\nu X^\kappa g_{\mu\kappa})\xi^\mu\odot \xi^\nu~.$$
By its definition in section \[ssec:tensor\_densitites\], the exponential of the extended dilaton ${\mathrm{e}}^{-2d}$ has to transform in such a way that the Lagrangian in transforms as a total derivative under the integral. As it is well known, this is the case if ${\mathrm{e}}^{-2d}$ transforms as an ordinary tensor tensity of weight 1, as e.g. $\sqrt{g}$.
The relevant action here, which is constructed from the terms listed in section \[ssec:metric\_and\_action\], reads as $$\begin{aligned}
S=\int_M {\mathrm{e}}^{-2d}\,{\mathrm{d}}x^1\wedge \ldots \wedge {\mathrm{d}}x^D \Big(& \tfrac14 {\mathcal{H}}^{MN}{\partial}_M{\mathcal{H}}^{KL}{\partial}_N {\mathcal{H}}_{KL}-\tfrac12 {\mathcal{H}}^{MN}{\partial}_M{\mathcal{H}}^{KL}{\partial}_L{\mathcal{H}}_{KN}\\
&~~-2 {\partial}_Md {\partial}_N {\mathcal{H}}^{MN}+ 4{\mathcal{H}}^{MN}{\partial}_M d{\partial}_N d\Big)~.
\end{aligned}$$ Instead of checking the invariance of this action under the usual diffeomorphisms, we directly verify that it reproduces the Einstein-Hilbert action coupled to a dilaton. To see this, we identify ${\mathrm{e}}^{-2d}=\sqrt{|g|}{\mathrm{e}}^{-2\phi}$, which implies $$d=\phi-\tfrac12 \log \sqrt{|g|}~.$$ We also recall that $${\partial}_\mu\log \sqrt{|g|}=\Gamma_\mu:=\Gamma^{\kappa}{}_{\mu\kappa}=\tfrac12 g^{\kappa\lambda}{\partial}_\mu g_{\kappa\lambda}$$ and note that $$\tfrac14 {\partial}^\mu g^{\kappa\lambda}{\partial}_\mu g_{\kappa\lambda}-\tfrac12 {\partial}_\mu g_{\kappa\lambda}{\partial}^\lambda g^{\kappa\mu}={\partial}_\nu g^{\nu\mu}\Gamma_\mu-{\partial}_\kappa g^{\mu\nu}\Gamma^\kappa{}_{\mu\nu}+g^{\mu\nu}(\Gamma^\kappa{}_{\mu\nu}\Gamma_\kappa-\Gamma^\lambda{}_{\mu\kappa}\Gamma^{\kappa}{}_{\nu\lambda})+\Gamma^\mu \Gamma_\mu~,$$ which is readily verified. Using the latter equation, we rewrite the action $$\begin{aligned}
S&=\int {\mathrm{d}}^D x~{\mathrm{e}}^{-2d}\Big(\tfrac14{\partial}^\mu g^{\kappa\lambda}{\partial}_\mu g_{\kappa\lambda}-\tfrac12{\partial}_\mu g_{\kappa\lambda}{\partial}^\lambda g^{\kappa\mu}+2{\partial}^\mu d {\partial}^\nu g_{\mu\nu}+4{\partial}_\mu d{\partial}^\mu d\Big)
\end{aligned}$$ as $$\begin{aligned}
S&=\int {\mathrm{d}}^D x~{\mathrm{e}}^{-2d}\Big(2{\partial}_\kappa d g^{\kappa\mu}{\partial}^\nu g_{\mu\nu}+{\partial}_\mu g^{\mu\nu} \Gamma_\nu-{\partial}_\kappa g^{\mu\nu}\Gamma^{\kappa}{}_{\mu\nu}\\
&\hspace{4cm}+g^{\mu\nu}(\Gamma^\kappa{}_{\mu\nu}\Gamma_\kappa-\Gamma^{\lambda}{}_{\mu\kappa}\Gamma^{\kappa}{}_{\nu\lambda})+g^{\mu\nu}\Gamma_\mu\Gamma_\nu+4{\partial}_\mu d{\partial}^\mu d\Big)~.
\end{aligned}$$ Since $g^{\kappa\mu}{\partial}^\nu g_{\mu\nu}=g^{\mu\nu}\Gamma^\kappa{}_{\mu\nu}+\Gamma^\kappa$, this equals $$\begin{aligned}
S&=\int {\mathrm{d}}^D x~{\mathrm{e}}^{-2d}\Big(2{\partial}_\kappa d (g^{\mu\nu}\Gamma^\kappa{}_{\mu\nu}-\Gamma^\kappa)+4{\partial}^\mu d\Gamma_\mu+{\partial}_\mu g^{\mu\nu} \Gamma_\nu-{\partial}_\kappa g^{\mu\nu}\Gamma^{\kappa}{}_{\mu\nu}\\
&\hspace{4cm}+g^{\mu\nu}(\Gamma^\kappa{}_{\mu\nu}\Gamma_\kappa-\Gamma^{\lambda}{}_{\mu\kappa}\Gamma^{\kappa}{}_{\nu\lambda})+g^{\mu\nu}\Gamma_\mu\Gamma_\nu+4{\partial}_\mu d{\partial}^\mu d\Big)~,
\end{aligned}$$ and partial integration of the first term finally yields $$\begin{aligned}
S&=\int {\mathrm{d}}^D x\sqrt{|g|}{\mathrm{e}}^{-2\phi}\Big(g^{\mu\nu}({\partial}_\kappa\Gamma^\kappa{}_{\mu\nu}-{\partial}_\nu\Gamma^\kappa{}_{\mu\kappa}+\Gamma^\kappa{}_{\mu\nu}\Gamma_\kappa-\Gamma^{\lambda}{}_{\mu\kappa}\Gamma^{\kappa}{}_{\nu\lambda})+\\
&\hspace{4cm}+g^{\mu\nu}\Gamma_\mu\Gamma_\nu+4\Gamma_\mu{\partial}^\mu d+4{\partial}_\mu d{\partial}^\mu d\Big)\\
&=\int {\mathrm{d}}^D x\sqrt{|g|}{\mathrm{e}}^{-2\phi}(R+4{\partial}_\mu \phi{\partial}^\mu\phi)~.
\end{aligned}$$
Example: Riemannian geometry with principal U(1)-bundle
-------------------------------------------------------
To describe an additional connection on a (trivial) principal ${\mathsf{U}}(1)$-bundle over $M$, we can again start from ${\mathcal{M}}={\mathcal{V}}_1(M)$, but with a more general $L_\infty$-structure. We choose ${\mathsf{L}}({\mathcal{M}})$ to be the constant and linear functions in the $\zeta^\mu$, which yields ${\mathfrak{u}}(1)$-valued functions together with the vector fields on $M$. Correspondingly, the extended tangent bundle is $TM\oplus {\mathbbm{R}}$. The resulting Lie $n$-algebra is an ordinary Lie algebra, namely the semidirect product of the diffeomorphisms with the gauge transformations: $$\begin{aligned}
\mu_2(f+X,g+Y)&=X^\mu {\partial}_\mu (g+Y^\nu\zeta_\nu)-Y^\mu{\partial}_\mu (f+X^\nu\zeta_\nu)\\
&=[X,Y]+{\mathcal{L}}_Xg-{\mathcal{L}}_Yf
\end{aligned}$$ for $f,g,X,Y\in {\mathsf{L}}({\mathcal{M}})\subset{\mathcal{C}}^\infty({\mathcal{M}}_0)$ with $f,g$ and $X,Y$ constant and linear in the $\zeta^\mu$, respectively, cf. .
Let us introduce indices $m=(\mu,\circ)$ on extended tangent vectors, where $\circ$ stands for the component in the new, additional direction. The action of extended symmetries on extended tensors is readily obtained from its definition $$X{\vartriangleright}t=\{\delta (X^\mu\zeta_\mu+f),t\}$$ for $X\in {\mathscr{X}}({\mathcal{V}}_1(M))$ and $t\in {\mathsf{T}}({\mathcal{V}}_1(M))$.
The extended metric then reads as $${\mathcal{H}}_{mn}=\left(\begin{array}{cc}
g_{\mu\nu} +A_\mu A_\nu& A_\nu \\ A_\mu & 1
\end{array}\right)$$ with inverse $${\mathcal{H}}^{mn}=\left(\begin{array}{cc}
g^{\mu\nu} & -g^{\mu\nu}A_\nu \\ -A_\mu g^{\mu\nu} & 1+g^{\mu\nu}A_\mu A_\nu
\end{array}\right)~.$$ These formulas already appeared in [@Maharana:1992my] in a different context. Note that ${\mathcal{H}}_{mn}$ has trivial kernel, because $${\mathcal{H}}=\left(\begin{array}{cc}
{\mathbbm{1}}_{TM} & A\\0 & 1
\end{array}\right)\left(\begin{array}{cc}
g & 0\\0 & 1
\end{array}\right)\left(\begin{array}{cc}
{\mathbbm{1}}_{TM} & A\\0 & 1
\end{array}\right)^T$$ and each of these matrices has trivial kernel. We could say that ${\mathcal{H}}$ arises from $A$-field transformations from the block-diagonal metric, analogous to equation .
The action of an extended vector $(f+X){\vartriangleright}{\mathcal{H}}$ induces the usual diffeomorphisms on the metric $g_{\mu\nu}$ as well as gauge transformations and diffeomorphisms on the gauge potential $A_\mu$.
The extended Lie derivative transforms the top form ${\mathrm{d}}x^1\wedge \ldots \wedge {\mathrm{d}}x^D$ as in the previous example, and so the extended dilaton transforms again identical to an ordinary tensor density of weight 1.
The invariant action here is the same as in the previous example: $$\begin{aligned}
S=\int_M {\mathrm{e}}^{-2d}\,{\mathrm{d}}x^1\wedge \ldots \wedge {\mathrm{d}}x^D \Big(& \tfrac14 {\mathcal{H}}^{MN}{\partial}_M{\mathcal{H}}^{KL}{\partial}_N {\mathcal{H}}_{KL}-\tfrac12 {\mathcal{H}}^{MN}{\partial}_M{\mathcal{H}}^{KL}{\partial}_L{\mathcal{H}}_{KN}\\
&~~-2 {\partial}_Md {\partial}_N {\mathcal{H}}^{MN}+ 4{\mathcal{H}}^{MN}{\partial}_M d{\partial}_N d\Big)~.
\end{aligned}$$ Identifying again ${\mathrm{e}}^{-2d}=\sqrt{|g|}{\mathrm{e}}^{-2\phi}$, we verify this action’s invariance by rewriting it in familiar form: $$\begin{aligned}
S=\int_M {\mathrm{e}}^{-2d}\,{\mathrm{d}}^D x~\Big(&\tfrac14 {\mathcal{H}}^{\mu \nu}{\partial}_\mu{\mathcal{H}}^{\kappa\lambda}{\partial}_\nu {\mathcal{H}}_{\kappa\lambda}+\tfrac12 {\mathcal{H}}^{\mu \nu}{\partial}_\mu{\mathcal{H}}^{\kappa\circ}{\partial}_\nu {\mathcal{H}}_{\kappa\circ}+\tfrac14 {\mathcal{H}}^{\mu \nu}{\partial}_\mu{\mathcal{H}}^{\circ\circ}{\partial}_\nu {\mathcal{H}}_{\circ\circ}\\
&~~-\tfrac12 {\mathcal{H}}^{\mu \nu}{\partial}_\mu{\mathcal{H}}^{\kappa\lambda}{\partial}_\lambda{\mathcal{H}}_{\kappa\nu}-\tfrac12 {\mathcal{H}}^{\mu \circ}{\partial}_\mu{\mathcal{H}}^{\kappa\lambda}{\partial}_\lambda{\mathcal{H}}_{\kappa\circ}-\tfrac12 {\mathcal{H}}^{\mu \nu}{\partial}_\mu{\mathcal{H}}^{\circ\lambda}{\partial}_\lambda{\mathcal{H}}_{\circ\nu}\\
&~~-\tfrac12 {\mathcal{H}}^{\mu \circ}{\partial}_\mu{\mathcal{H}}^{\circ\lambda}{\partial}_\lambda{\mathcal{H}}_{\circ\circ}-2{\partial}_\mu d {\partial}_\nu {\mathcal{H}}^{\mu\nu}+4{\mathcal{H}}^{\mu\nu}{\partial}_\mu d{\partial}_\nu d\Big)\\
=\int_M {\mathrm{e}}^{-2d}\,{\mathrm{d}}^D x~\Big(&\tfrac14 g^{\mu \nu}{\partial}_\mu g^{\kappa\lambda}{\partial}_\nu (g_{\kappa\lambda}+A_\kappa A_\lambda)-\tfrac12 g^{\mu \nu}{\partial}_\mu A^{\kappa}{\partial}_\nu A_\kappa\\
&~~-\tfrac12 g^{\mu \nu}{\partial}_\mu g^{\kappa\lambda}{\partial}_\lambda (g_{\kappa\nu}+A_\kappa A_\nu)+\tfrac12 A^\mu{\partial}_\mu g^{\kappa\lambda}{\partial}_\lambda A_\kappa+\tfrac12 g^{\mu \nu}{\partial}_\mu A^\lambda{\partial}_\lambda A_\nu\\
&~~-2{\partial}_\mu d {\partial}_\nu g^{\mu\nu}+4g^{\mu\nu}{\partial}_\mu d{\partial}_\nu d\Big)~.
\end{aligned}$$ Note that for $A_\mu=0$, this reduces to the action of the previous section. Thus, it remains to study the part $S_A$ of the action containing the gauge potential, which read as $$\begin{aligned}
S_A=\int_M {\mathrm{e}}^{-2d}\,{\mathrm{d}}^D x~\Big(&\tfrac14 g^{\mu \nu}{\partial}_\mu g^{\kappa\lambda}{\partial}_\nu (A_\kappa A_\lambda)-\tfrac12 g^{\mu \nu}{\partial}_\mu A^{\kappa}{\partial}_\nu A_\kappa\\
&~~-\tfrac12 g^{\mu \nu}{\partial}_\mu g^{\kappa\lambda}{\partial}_\lambda (A_\kappa A_\nu)+\tfrac12 A^\mu{\partial}_\mu g^{\kappa\lambda}{\partial}_\lambda A_\kappa+\tfrac12 g^{\mu \nu}{\partial}_\mu A^\lambda{\partial}_\lambda A_\nu\Big)~.
\end{aligned}$$ Since $$\begin{aligned}
-\tfrac14 F^2&=-\tfrac14 g^{\mu\kappa}g^{\nu\lambda}F_{\mu\nu}F_{\kappa\lambda}\\
&=-\tfrac12 {\partial}_\mu A^{\kappa}{\partial}^\nu A_\kappa+\tfrac12 A_\lambda {\partial}_\mu g^{\kappa\lambda}{\partial}^\mu A_\kappa+\tfrac12 {\partial}^\mu A^\lambda{\partial}_\lambda A_\mu-\tfrac12 A_\kappa{\partial}^\mu g^{\lambda\kappa}{\partial}_\lambda A_\mu~,
\end{aligned}$$ we obtain the desired expression: $$\begin{aligned}
S_A=\int_M {\mathrm{e}}^{-2d}\,{\mathrm{d}}^D x~\Big(&-\tfrac14 F^2\Big)~.
\end{aligned}$$
Example: Generalized geometry
-----------------------------
It is well known that the appropriate geometric structure underlying generalized geometry on a manifold $M$ is the symplectic Lie 2-algebroid ${\mathcal{M}}:={\mathcal{V}}_2(M)$ of ${\mathbbm{N}}$-degree 2. The antisymmetrized derived bracket is simply the Courant bracket, see equations . The resulting Lie 2-algebra is the semidirect product of the Lie algebra of diffeomorphism, regarded trivially as a Lie 2-algebra, and the abelian Lie 2-algebra of gauge transformations of the curving 2-form of a trivial abelian gerbe. The transformation law for tensors reads as $$\begin{aligned}
(\alpha+X){\vartriangleright}t^\mu{}_\nu~\zeta_\mu\otimes \xi^\nu:=~&\{\delta \alpha+\delta X,t^\mu{}_\nu~\zeta_\mu\otimes \xi^\nu\}\\
=~&\big(X^\mu({\partial}_\mu t^\nu{}_\kappa)-({\partial}_\mu X^\nu) t^\mu{}_\kappa+({\partial}_\kappa X^\mu) t^\nu{}_\mu\big)~\zeta_\nu\otimes \xi^\kappa+\\
&~+({\partial}_\mu \alpha_\nu -{\partial}_\nu \alpha_\mu)t^\nu{}_\lambda~\xi^\mu\otimes \xi^\lambda~.
\end{aligned}$$ The extended metric here and how it arises from a diagonal metric via $B$-field transformation was already discussed in \[ssec:generalised\_geometry\]. We have $$\label{eq:2metric_H_O(D,D)}
{\mathcal{H}}_{MN}=\left(\begin{array}{cc} g_{\mu\nu}-B_{\mu\kappa}g^{\kappa\lambda}B_{\lambda\nu} & B_{\mu\kappa}g^{\kappa\nu}\\ -g^{\mu\kappa}B_{\kappa\nu} &g^{\mu\nu} \end{array}\right)$$ with inverse $${\mathcal{H}}^{MN}=\left(\begin{array}{cc} g^{\mu\nu} & -g^{\mu\kappa}B_{\kappa\nu} \\ B_{\mu\kappa}g^{\kappa\nu}& g_{\mu\nu}-B_{\mu\kappa}g^{\kappa\lambda}B_{\lambda\nu} \end{array}\right)~.$$ The action of an extended vector $X{\vartriangleright}{\mathcal{H}}$ yields the action of diffeomorphisms on the metric as well as the combined action of diffeomorphisms and gauge transformations on the 2-form potential $B$. As in the two cases before, the extended Lie derivative transforms the top form ${\mathrm{d}}x^1\wedge \ldots \wedge {\mathrm{d}}x^D$ as usual, and therefore the extended dilaton transforms as an ordinary tensor density of weight 1.
The invariant action for generalized geometry is well known to be that of double field theory: $$\begin{aligned}
S=\int_M {\mathrm{e}}^{-2d}\,{\mathrm{d}}x^1\wedge \ldots \wedge {\mathrm{d}}x^D \Big(& \tfrac18 {\mathcal{H}}^{MN}{\partial}_M{\mathcal{H}}^{KL}{\partial}_N {\mathcal{H}}_{KL}-\tfrac12 {\mathcal{H}}^{MN}{\partial}_M{\mathcal{H}}^{KL}{\partial}_L{\mathcal{H}}_{KN}\\
&~~-2 {\partial}_Md {\partial}_N {\mathcal{H}}^{MN}+ 4{\mathcal{H}}^{MN}{\partial}_M d{\partial}_N d\Big)~.
\end{aligned}$$ Again, we bring this action into familiar form, focusing on the Lagrangian: $$\begin{aligned}
&\tfrac18 {\partial}_\mu {\mathcal{H}}^{KL}{\partial}^\mu {\mathcal{H}}_{KL}-\tfrac12 {\mathcal{H}}^{\mu N}{\partial}_\mu{\mathcal{H}}^{K\lambda}{\partial}_\lambda{\mathcal{H}}_{KN}-2 {\partial}_\mu d {\partial}_\nu g^{\mu\nu}+ 4{\partial}_\mu d{\partial}^\mu d\\
&=\tfrac18 {\partial}_\mu g^{\kappa\lambda}{\partial}^\mu (g_{\kappa\lambda}-B_{\kappa\rho}g^{\rho\sigma}B_{\sigma\lambda})-\tfrac18 {\partial}_\mu (g^{\kappa\rho}B_{\rho \lambda}){\partial}^\mu (B_{\kappa\sigma}g^{\sigma\lambda})-\tfrac18 {\partial}_\mu (B_{\kappa\rho}g^{\rho\lambda}){\partial}^\mu (g^{\kappa\sigma}B_{\sigma\lambda})\\
&~~+\tfrac18 {\partial}_\mu (g_{\kappa\lambda}-B_{\kappa\rho}g^{\rho\sigma}B_{\sigma\lambda}) {\partial}^\mu g^{\kappa\lambda}-\tfrac12 g^{\mu \nu}{\partial}_\mu g^{\kappa\lambda}{\partial}_\lambda(g_{\kappa\nu}-B_{\kappa\sigma}g^{\sigma\rho}B_{\sigma\nu})\\
&~~+\tfrac12 g^{\mu \nu}{\partial}_\mu(B_{\kappa\sigma}g^{\sigma\lambda}){\partial}_\lambda(g^{\kappa\rho}B_{\rho\nu})-\tfrac12 B_{\mu\nu}{\partial}^\mu g^{\kappa\lambda}{\partial}_\lambda(B_{\kappa\rho}g^{\rho\nu})+\tfrac12 B_{\mu\nu}{\partial}^\mu(B_{\kappa\sigma}g^{\sigma\lambda}){\partial}_\lambda g^{\kappa\nu}~.
\end{aligned}$$ The terms independent of $B$ combine to the familiar form $$\tfrac14 {\partial}^\mu g^{\kappa\lambda}{\partial}_\mu g_{\kappa\lambda}-\tfrac12 {\partial}_\mu g_{\kappa\lambda}{\partial}^\lambda g^{\kappa\mu}$$ of the Einstein-Hilbert Lagrangian plus the dilaton contributions. The terms dependent on $B$ reduce, as expected, to $H^2$ for $H={\mathrm{d}}B$ and we have $$S=\int_M {\mathrm{d}}^D x \sqrt{|g|}{\mathrm{e}}^{-2\phi}(R+4{\partial}_\mu\phi{\partial}^\mu \phi-\tfrac{1}{12}H^2)~,$$ see e.g. [@Hohm:2010jy] for the details.
Example: Higher generalized geometry {#ssec:3-form-GG}
------------------------------------
So far, we have discussed general relativity with 1- and 2-form gauge potentials. This sequence is readily continued to 3-form gauge potentials. The latter case is also physically interesting, as it corresponds to exceptional field theory after imposing a section condition. Therefore, and for completeness sake, let us also list the ingredients here.
From our above discussion, we are clearly led to consider the symplectic Lie 3-algebroid ${\mathcal{V}}_3(M)=T^*[3]T[1]M$ of ${\mathbbm{N}}$-degree 3. We use the usual coordinates $(x^\mu,\xi^\mu,\zeta_\mu,p_\mu)$ of degrees 0,1,2 and 3, respectively. Extended vector fields have degree 2 and are of the form $$X=X^\mu\zeta_\mu+\tfrac12 \alpha_{\mu\nu}\xi^\mu\xi^\nu~.$$ They thus contain a vector field $X^\mu{\frac{{\partial}}{{\partial}x^\mu}}$ as well as a 2-form $\tfrac12 \alpha_{\mu\nu}{\mathrm{d}}x^\mu\wedge {\mathrm{d}}x^\nu$. This is the right data for diffeomorphisms and gauge transformations of a 3-form potential $C$. Since $Q^2=0$, we get a canonical $L_\infty$-structure on ${\mathcal{V}}_3(M)$ with 2-bracket $$\mu_2(X+\alpha,Y+\beta)=[X,Y]+{\mathcal{L}}_X\beta-{\mathcal{L}}_Y\alpha-\tfrac12{\mathrm{d}}\big(\iota_X\beta-\iota_Y\alpha)~,$$ which has the same form as the Courant bracket . Note, however, that $\alpha$ and $\beta$ are here 2-forms. The tensor transformation law is again obtained from $$(\alpha+X){\vartriangleright}t:=\{\delta \alpha+\delta X,t\}~,$$ where $t$ carries indices $M=({}^\mu,{}_{\nu\kappa})$. One can, however, extend this action to a subset of more general functions on ${\mathcal{V}}_3(M)$.
As for all previous cases, the extended metric is obtained by a gauge field transformation acting on the diagonal metric. The relevant indices here are $M=(\mu,[\nu\kappa])$ such that ${\mathscr{T}}{\mathcal{M}}$ is of dimension $D+\tfrac12 D(D-1)$. Explicitly, we have $${\mathcal{H}}=\left(\begin{array}{cc}
{\mathbbm{1}}_{TM} & a C\\0 & {\mathbbm{1}}_{\wedge^2 T^*M}
\end{array}\right)\left(\begin{array}{cc}
g & 0\\0 & g^{-1}g^{-1}
\end{array}\right)\left(\begin{array}{cc}
{\mathbbm{1}}_{TM} & aC\\0 & {\mathbbm{1}}_{\wedge^2 T^*M}
\end{array}\right)^T~,$$ where we inserted a constant $a\in{\mathbbm{R}}$ allowing for a field rescaling of $C$, and therefore we have the component expressions $${\mathcal{H}}_{MN}={\mathcal{H}}_{(\mu,[\rho\sigma]),(\nu,[\kappa\lambda])}=\left(\begin{array}{cc}
g_{\mu\nu}+a^2C_{\mu\alpha\beta}g^{\alpha\gamma}g^{\beta\delta}C_{\gamma\delta\nu} & a C_{\mu\alpha\beta}g^{\alpha\kappa}g^{\beta\lambda}\\
a g^{\rho\alpha}g^{\sigma\beta}C_{\alpha\beta \nu} & g^{\rho\kappa} g^{\sigma\lambda}
\end{array}\right)$$ with inverse $${\mathcal{H}}^{MN}={\mathcal{H}}^{(\mu,[\rho\sigma]),(\nu,[\kappa\lambda])}=\left(\begin{array}{cc}
g^{\mu\nu} & -a g^{\mu\alpha}C_{\alpha\kappa\lambda} \\
-a C_{\rho\sigma\alpha}g^{\alpha\nu} & g_{\rho \kappa}g_{\sigma\lambda}+a^2C_{\rho\sigma\alpha}g^{\alpha\beta}C_{\beta\kappa\lambda}
\end{array}\right)~.$$
The relevant action here requires additional terms to those of the previous examples: $$\begin{aligned}
S=\int_M&{\mathrm{e}}^{-2d}\,{\mathrm{d}}x^1\wedge \ldots \wedge {\mathrm{d}}x^D \Big( \frac{1}{4(1+2D)}{\mathcal{H}}^{MN}{\partial}_M{\mathcal{H}}^{KL}{\partial}_N {\mathcal{H}}_{KL}-\tfrac12{\mathcal{H}}^{MN}{\partial}_M{\mathcal{H}}^{KL}{\partial}_L{\mathcal{H}}_{KN}\\
&\hspace{3cm}+\frac{1}{2(2D-1)^2(1+2D)}{\mathcal{H}}^{MN}({\mathcal{H}}^{KL}{\partial}_M{\mathcal{H}}_{KL})({\mathcal{H}}^{RS}{\partial}_N {\mathcal{H}}_{RS})\\
&\hspace{3cm}-2 {\partial}_Md {\partial}_N {\mathcal{H}}^{MN}+ 4{\mathcal{H}}^{MN}{\partial}_M d{\partial}_N d\Big)~.
\end{aligned}$$ Again, we verify its gauge invariance by reducing it to a familiar form. Since the computations get rather involved, we used a computer algebra program to complete this task. The result is $$S=\int {\mathrm{d}}^D x~{\mathrm{e}}^{-2d}\Big(\tfrac14{\partial}^\mu g^{\kappa\lambda}{\partial}_\mu g_{\kappa\lambda}-\tfrac12{\partial}_\mu g_{\kappa\lambda}{\partial}^\lambda g^{\kappa\mu}+2{\partial}^\mu d {\partial}^\nu g_{\mu\nu}+4{\partial}_\mu d{\partial}^\mu d+\tfrac{1}{48}G^2\Big)$$ with the choice $a=3\sqrt{2}$, which is the usual Einstein-Hilbert action of the first example coupled to the abelian 3-form potential $C$ with curvature $G={\mathrm{d}}C$.
Double field theory as extended Riemannian geometry {#sec:dft}
===================================================
So far, we only employed ordinary symplectic Lie $n$-algebroids in our examples, and there was no need to lift $Q^2=0$, generalizing to pre-N$Q$-manifolds, which are not N$Q$-manifolds. As we shall see now, the description of double field theory requires this lift.
Restriction of doubled generalized geometry {#ssec:E2M_from_restriction}
-------------------------------------------
For a description of double field theory, we have to double spacetime from $M$ to $M\times \hat M$. Since we have to restrict ourselves to a local description, the bundle $T^* M$ will have to do. On this bundle, we want to describe the gauge transformations of a trivial abelian gerbe, just as in generalized geometry. This suggest to use ${\mathcal{V}}_2(T^*M)=T^*[2]T[1](T^*M)$ with coordinates $(x^M,p_M,\xi^M,\zeta_M)=(x^\mu,x_\mu,\ldots,\zeta_\mu,\zeta^\mu)$ as our starting point. This Lie 2-algebroids comes with the usual symplectic structure and Hamiltonian for a homological vector field $Q$, $$\omega={\mathrm{d}}x^M\wedge {\mathrm{d}}p_M+{\mathrm{d}}\xi^M\wedge {\mathrm{d}}\zeta_M~,~~~{\mathcal{Q}}=\sqrt{2}\xi^M p_M~,$$ where we rescaled ${\mathcal{Q}}$ for convenience.
The underlying symmetry group is now ${\mathsf{GL}}(2D)$, which we have to restrict to the symmetry group ${\mathsf{O}}(D,D)$ of double field theory. This is done by introducing the metric $\eta_{MN}$ of split signature $(1,\ldots,1,-1,\ldots,-1)$, cf. . Using this metric, we can restrict the tangent space coordinates to the diagonal, using coordinates $$\theta^M=\frac{1}{\sqrt{2}}(\xi^M+\eta^{MN}\zeta_N){{\qquad\mbox{and}\qquad}}\beta^M=\frac{1}{\sqrt{2}}(\xi^M-\eta^{MN}\zeta_N)~.$$ Eliminating the dependence on $\beta^M$ of all our objects[^10] leaves us with the pre-N$Q$-manifold ${\mathcal{E}}_2(M):=(T^*[2]\oplus T[1])T^*M$ with coordinates $(x^M,\theta^M,p_M)$ of degrees $0$, $1$ and $2$, respectively. The reduction also leads to the following symplectic structure on ${\mathcal{E}}_2(M)$: $$\omega={\mathrm{d}}x^M\wedge {\mathrm{d}}p_M +\tfrac12\eta_{MN}{\mathrm{d}}\theta^M\wedge {\mathrm{d}}\theta^N~,~~~{\mathcal{Q}}=\theta^M p_M~.$$ The Poisson bracket reads as $$\{f,g\}:=f\overleftarrow{{\frac{{\partial}}{{\partial}p_M}}}\overrightarrow{{\frac{{\partial}}{{\partial}x^M}}} g-f\overleftarrow{{\frac{{\partial}}{{\partial}x^M}}}\overrightarrow{{\frac{{\partial}}{{\partial}p_M}}} g-f\overleftarrow{{\frac{{\partial}}{{\partial}\theta^M}}}\eta^{MN}\overrightarrow{{\frac{{\partial}}{{\partial}\theta^N}}} g$$\[eq:E2\_Poisson\] for $f,g\in{\mathcal{C}}^\infty({\mathcal{E}}_2(M))$ and the Hamiltonian vector field of ${\mathcal{Q}}$ is $$Q=\theta^M{\frac{{\partial}}{{\partial}x^M}}+p_M\eta^{MN}{\frac{{\partial}}{{\partial}\theta^N}}~.$$ Note that $Q^2=p_M\eta^{MN}{\frac{{\partial}}{{\partial}x^N}}\neq 0$, and ${\mathcal{E}}_2(M)$ is not an N$Q$-manifold. It is, however, a symplectic pre-N$Q$-manifold of ${\mathbbm{N}}$-degree $2$.
We thus need to choose a $L_\infty$-structure in the form of a subset ${\mathsf{L}}({\mathcal{E}}_2(M))\subset {\mathcal{C}}^\infty(M)$ satisfying the conditions of theorem \[thm:Lie\_2\_subset\]. A short computation yields the following.
For elements $f,g$ and $X=X_M\theta^M,Y=Y_M\theta^M,Z=Z_M\theta^M$ of ${\mathsf{L}}({\mathcal{E}}_2(M))$ of ${\mathsf{L}}$-degrees 1 and 0, respectively, we have the following relations. $$\label{eq:restrictions_LM}
\begin{gathered}
\{Q^2 f,g\}+\{Q^2 g,f\}=2\left({\frac{{\partial}}{{\partial}x^M}}f\right)\eta^{MN}\left({\frac{{\partial}}{{\partial}x^N}}g\right)=0~,\\
\{Q^2 X,f\}+\{Q^2 f,X\}=2\left({\frac{{\partial}}{{\partial}x^M}}X\right)\eta^{MN}\left({\frac{{\partial}}{{\partial}x^N}}f\right)=0~,\\
\{\{Q^2X,Y\},Z\}_{[X,Y,Z]}=2\theta^L\big(({\partial}^MX_L)({\partial}_MY^K)Z_K\big)_{[X,Y,Z]}=0~.
\end{gathered}$$
Note that the strong section condition of double field theory is sufficient, but not necessary to satisfy . Thus, choosing a specific $L_\infty$-algebra structure ${\mathsf{L}}({\mathcal{E}}_2(M))$ amounts to a choice of solution to and therefore corresponds essentially to “solving the strong section condition” in DFT parlance. Contrary to the strong section condition postulated in DFT, however, the left-hand expressions in equations are completely independent of a choice of coordinates.
Note also that the Poisson bracket yields the natural contraction of two extended vector fields $X,Y$: $$\{X,Y\}=X_M \eta^{MN} Y_N~.$$ We use again the shorthand notations ${\frac{{\partial}}{{\partial}x^M}}={\partial}_M$, $X^M:=\eta^{MN} X_N$ and $x_N=\eta_{MN} x^N$ introduced in section \[sec:review\].
As we shall see in section \[ssec:examples\_L\_infty\], an obvious $L_\infty$-structure on ${\mathcal{E}}_2(M)$ for $M={\mathbbm{R}}^n$ recovers the corresponding Vinogradov Lie 2-algebroid encoding generalized geometry. This implies that the C-bracket and all induced structures of double field theory reduce to the Courant bracket and the corresponding structures in generalized geometry.
Symmetries {#ssec:Symmetries_of_DFT}
----------
The extended vector fields ${\mathscr{X}}({\mathcal{E}}_2(M))={\mathsf{L}}_0({\mathcal{E}}_2(M))$ encode infinitesimal diffeomorphism and gauge transformations diffeomorphisms, while elements of ${\mathsf{L}}({\mathcal{E}}_2(M))_1\subset {\mathcal{C}}^\infty(M)$ describe morphisms between these, as discussed in section \[ssec:Vinogradov\]. Let us now give the explicit Lie 2-algebra structure.
An $L_\infty$-algebra structure on the pre-N$Q$-manifold ${\mathcal{E}}_2(M)$ is a Lie 2-algebra of symmetries given by a graded vector space $${\mathsf{L}}({\mathcal{E}}_2(M))={\mathsf{L}}_0({\mathcal{E}}_2(M))\oplus{\mathsf{L}}_1({\mathcal{E}}_2(M))\subset {\mathcal{C}}^\infty_1({\mathcal{E}}_2(M))\oplus {\mathcal{C}}^\infty_0({\mathcal{E}}_2(M))$$ together with higher products $$\begin{aligned}
\mu_1(f)&=Qf=\theta^M{\partial}_M f~,\\
\mu_2(X,Y)&=-\mu_2(Y,X)=\tfrac12\big(\{QX,Y\}-\{QY,X\}\big)\\
&=X^M{\partial}_M Y-Y^M{\partial}_M X+\tfrac12\theta^M(Y^K{\partial}_MX_K-X^K{\partial}_MY_K)~,\\
\mu_2(X,f)&=-\mu_2(f,X)=\tfrac12 X^M{\partial}_M f~,\\
\mu_3(X,Y,Z)&=\tfrac13\big(\{\mu_2(X,Y),Z\}+\{\mu_2(Y,Z),X\}+\{\mu_2(Z,X),Y\}\big)\\
&=X^MZ^N{\partial}_MY_N-Y^MZ^N{\partial}_MX_N+Y^MX^N{\partial}_MZ_N\\
&\hspace{1cm}-Z^MX^N{\partial}_MY_N+Z^MY^N{\partial}_MX_N-X^MY^N{\partial}_MZ_N~,
\end{aligned}$$ for $f\in {\mathsf{L}}_1({\mathcal{E}}_2(M))$ and $X,Y,Z\in {\mathsf{L}}_0({\mathcal{E}}_2(M))$, where we wrote $X=X_M\theta^M$, etc.
This Lie 2-algebra now has a clear relationship to the symmetry structures in double field theory, and we readily conclude the following statements.
Given an $L_\infty$-structure on ${\mathcal{E}}_2(M)$, the product $\mu_2(X,Y)$ is simply the so-called C-bracket of double field theory. The D-bracket is given by $$\nu_2(X,Y)=\mu_2(X,Y)+\tfrac12Q\{X,Y\}=\{QX,Y\}~.$$ Moreover, the action $X{\vartriangleright}t$ of vector fields $X\in{\mathsf{L}}_0({\mathcal{E}}_2(M))$ on tensors $t\in {\mathsf{T}}({\mathcal{E}}_2(M))$ is indeed the action of the extended Lie derivative on tensors as given in . In particular, for a rank 2-tensor $t_{MN}\theta^M\otimes \theta^N$, we have $$\begin{aligned}
&X{\vartriangleright}t_{MN} \theta^M\otimes \theta^N:=\{\delta X,t_{MN} \theta^M\otimes \theta^N\}\\
&\hspace{1cm}=X^N{\partial}_N t_{KL} \theta^K\otimes \theta^L+({\partial}_M X^N-{\partial}^N X_M)t_{NK} \theta^M\otimes \theta^K+\\
&\hspace{5cm}+({\partial}_M X^N-{\partial}^N X_M)t_{KN} \theta^K\otimes \theta^M~.
\end{aligned}$$
Note that since we are working with a pre-N$Q$-manifold, the restriction of theorem \[thm:restrictions\_T\] apply to extended tensors.
The transformation property of the dilaton ${\mathrm{e}}^{-2d}$ is fixed by the invariance of the action: $$S=\int {\mathrm{d}}x^1\wedge \ldots \wedge {\mathrm{d}}x^D\wedge {\mathrm{d}}x_1\wedge \ldots \wedge {\mathrm{d}}x_D~{\mathrm{e}}^{-2d}~{\mathcal{R}}~,$$ where ${\mathcal{R}}$ is an appropriate Ricci scalar. Note that contrary to ordinary differential geometry, the naïve top form is invariant: $$X{\vartriangleright}{\mathrm{d}}x^1\wedge \ldots \wedge {\mathrm{d}}x^D\wedge {\mathrm{d}}x_1\wedge \ldots \wedge {\mathrm{d}}x_D= X{\vartriangleright}\xi^1\ldots \xi^D\xi_1\ldots \xi_{D}=0~.$$ Since $X{\vartriangleright}({\mathrm{e}}^{-2d}{\mathcal{R}})=(X{\vartriangleright}{\mathrm{e}}^{-2d}){\mathcal{R}}+{\mathrm{e}}^{-2d}(X{\vartriangleright}{\mathcal{R}})$, we have to demand that $$X{\vartriangleright}{\mathrm{e}}^{-2d}={\partial}_\mu(X^\mu{\mathrm{e}}^{-2d})$$ in order to obtain a total derivative as a transformation of the action.
Examples of L-infinity-structures {#ssec:examples_L_infty}
---------------------------------
Recall that an $L_\infty$-structure on ${\mathcal{E}}_2(M)$ is a subset ${\mathsf{L}}({\mathcal{E}}_2(M))\subset {\mathcal{C}}^\infty({\mathcal{E}}_2(M))$ satisfying equations . Moreover, an $L_\infty$-structure is essentially a weaker replacement of a solution to the strong section condition in double field theory.
The most obvious $L_\infty$-structure is obtained after splitting coordinates $x^M=(x^\mu,x_\mu)$, $\theta^M=(\theta^\mu,\theta_\mu)$ and $p_M=(p_\mu,p^\mu)$ and restricting to functions independent of $x_\mu$. We can regard this as restricting ourselves to the subspace of ${\mathcal{E}}_2(M)$ given by $x_\mu=0$. Restricted to this subspace, the original symplectic structure $\omega$ is singular, unless we restrict further to $p^\mu=0$. The resulting subspace is simply the symplectic N$Q$-manifold ${\mathcal{V}}_2(M)$ underlying generalized geometry. In this way, double field theory reduces to generalized geometry.
Obviously, we can apply ${\mathsf{O}}(D,D)$-rotations to this $L_\infty$-structure to obtain new variants. In particular, the “dual” restriction to functions independent of $x^\mu$ and $p_\mu$ works similarly well, and produces an isomorphic solution. Also mixed versions exist: For example, putting $${\frac{{\partial}}{{\partial}x_\mu}}+\pi^{\mu\nu}(x){\frac{{\partial}}{{\partial}x^\nu}}=0$$ for any antisymmetric tensor field $\pi\in\Gamma(\wedge^2 TM)$ provides a solution to , as one easily verifies. All of these $L_\infty$-structures are also solutions to the strong section condition.
Clearly, it would be very interesting to study the existence of further, more general $L_\infty$-structures, in particular of those which do not satisfy the strong section condition.
Twisting extended symmetries {#ssec:twisted_extended_symmetries}
----------------------------
In section \[ssec:Ex\_Courant\_and\_Gerbes\] we explained how in generalized geometry, an exact Courant algebroid with Ševera class $H$ describes the infinitesimal symmetries of an ${\mathsf{U}}(1)$-bundle gerbe with 3-form curvature $H$. Because a solution to the section condition reduces double field theory to generalized geometry, it is clear that in the presence of non-trivial background fluxes, also the symmetries of double field theory require twisting. Moreover, such a twist should play an important role in developing a global description of double field theory, and we will return to this point in section \[ssec:global\].
The idea is to mimic the twist of a Courant algebroid and introduce a Hamiltonian $$\label{eq:E_2_twist}
{\mathcal{Q}}_{S,T}=\theta^Mp_M+S_{MN}p^M\theta^N+\tfrac{1}{3!}T_{MNK}\theta^M\theta^N\theta^K~,$$ where $T:=\tfrac{1}{3!}T_{MNK}\theta^M\theta^N\theta^K\in {\mathcal{C}}^\infty_3({\mathcal{E}}_2(M))$ is some extended 3-form on ${\mathcal{E}}_2(M)$ and $S:=S_{MN}p^M\theta^N\in {\mathcal{C}}^\infty_3({\mathcal{E}}_2(M))$ is another extended function of degree 3. This is in fact the most general deformation of ${\mathcal{Q}}$, as only elements of ${\mathbbm{N}}$-degree 3 are admissible. The corresponding Hamiltonian vector field with respect to the Poisson bracket reads as $$\begin{aligned}
Q_{S,T}&=\theta^M{\frac{{\partial}}{{\partial}x^M}}+p_M{\frac{{\partial}}{{\partial}\theta_M}}-\frac{1}{3!}{\frac{{\partial}}{{\partial}x^M}}T_{NKL}\theta^N\theta^K\theta^L{\frac{{\partial}}{{\partial}p_M}}+\tfrac12 T_{MNK}\theta^N\theta^K{\frac{{\partial}}{{\partial}\theta_M}}\\
&~~~~+S_{MN}\theta^N{\frac{{\partial}}{{\partial}x_M}}-{\frac{{\partial}}{{\partial}x^K}}S_{MN}p^M\theta^N{\frac{{\partial}}{{\partial}p_K}}+S_{MN}p^M{\frac{{\partial}}{{\partial}\theta_N}}~,
\end{aligned}$$ which is by construction a symplectomorphism on ${\mathcal{E}}_2(M)$. We now have $$\label{eq:E_2_Qsq}
\begin{aligned}
\{{\mathcal{Q}}_{S,T},{\mathcal{Q}}_{S,T}\}&=\theta^M\theta^N\theta^K\theta^L\left(\frac{1}{3}{\frac{{\partial}}{{\partial}x^M}}T_{NKL}+\frac13S_{PM}{\frac{{\partial}}{{\partial}x_P}}T_{NKL}+\frac14T_{MNP}T^P{}_{KL}\right)\\
&~~~~+\theta^M\theta^Np^K\left(2{\frac{{\partial}}{{\partial}x^M}}S_{KN}+S_{PM}{\frac{{\partial}}{{\partial}x_P}}S_{KN}+T_{KMN}+T_{MNL}S_K{}^{L}\right)\\
&~~~~+p^Mp^N\left(\eta_{MN}+2S_{MN}+S_{MK}S_N{}^K\right)~,
\end{aligned}$$ where $T^P{}_{KL}:=\eta^{PQ}T_{QKL}$ etc. Again, we should not require that $Q^2_{S,T}=0$, which is equivalent to $\{{\mathcal{Q}}_{S,T},{\mathcal{Q}}_{S,T}\}=0$, but merely demand that the conditions of theorem \[thm:Lie\_2\_subset\] are satisfied. This leads to a twisted $L_\infty$-algebra structure, containing twisted C- and D-brackets. Altogether, we make the following definition.
Given an $L_\infty$-algebra structure ${\mathsf{L}}({\mathcal{E}}_2(M))$ on ${\mathcal{E}}_2(M)$, we call ${\mathcal{Q}}_{S,T}$ in or $Q_{S,T}$ a if is satisfied for elements of ${\mathsf{L}}({\mathcal{E}}_2(M))$ if $Q$ is replaced by $Q_{S,T}$.
Given a twist of ${\mathsf{L}}({\mathcal{E}}_2(M))$, we define the by $$\begin{aligned}
\nu^{S,T}_2(X,Y)&:=\{Q_{S,T}X,Y\}~,\\
\mu_2^{S,T}(X,Y)&:=\tfrac12\big(\{Q_{S,T}X,Y\}-\{Q_{S,T}Y,X\}\big)~.
\end{aligned}$$
A detailed study of such twists is beyond the scope of this paper and left to future work. Let us merely present an discussion analogue to that of section \[ssec:twisted\_Vinogradov\] and look at infinitesimal twists.
We start by considering . The term quadratic in $p^Mp^N$ implies that $S_{MN}$ is antisymmetric if it is infinitesimal. By construction, the coordinate transformations $$z\mapsto -\{z,\tfrac12 \tau^{MN}\theta_M\theta_N\}$$ for $z=(x^M,\theta^M,p_M)$ are symplectomorphisms for infinitesimal $\tau^{MN}$. Explicitly, we have $$x^M\rightarrow x^M~,~~~\theta^M\rightarrow \theta^M-\tau^{MN}\theta_N~,~~~p_M=p_M-\tfrac12{\partial}_M\tau^{KL}\theta_K\theta_L~,$$ and we can use these transformations to put the contribution $S_{MN}p^M\theta^N$ to an infinitesimal twist to zero. Assuming that ${\partial}_{[M}T_{NKL]}=0$, we are then left with the conditions $$\label{eq:twist_condition_1}
\{\Xi f,g\}+\{\Xi g,f\}=0~,~~~\{\Xi X,f\}+\{\Xi f,X\}=0~,~~~\{\{\Xi X,Y\},Z\}_{[X,Y,Z]}=0$$ with $$\begin{aligned}
\Xi&=\{\theta^M\theta^Np^KT_{KMN},-\}\\
&=\theta^M\theta^NT_{KMN}{\frac{{\partial}}{{\partial}x_N}}-\theta^M\theta^Np^K\left({\frac{{\partial}}{{\partial}x^L}}T_{KMN}\right){\frac{{\partial}}{{\partial}p_L}}+2\theta^M T_{KMN}p^K{\frac{{\partial}}{{\partial}\theta_N}}~.
\end{aligned}$$ Equations then reduce to $$\begin{aligned}
&\theta^MX^NT_{KMN}{\frac{{\partial}}{{\partial}x_K}}f=0~,\\
&(Y^MX^NT_{KMN}{\frac{{\partial}}{{\partial}x_K}}Z+Z^MX^NT_{KMN}{\frac{{\partial}}{{\partial}x_K}}Y)_{[X,Y,Z]}=0~,
\end{aligned}$$ and we arrive at the following theorem.
Consider an $L_\infty$-structure ${\mathsf{L}}({\mathcal{E}}_2(M))$. Then ${\mathcal{Q}}=\theta^Mp_M+\tfrac{1}{3!}T_{MNK}\theta^M\theta^N\theta^K$ with $T_{MNK}$ infinitesimal is a twist of ${\mathsf{L}}({\mathcal{E}}_2(M))$ if $$T_{MNK}{\frac{{\partial}}{{\partial}x_K}} F=0$$ for any $F\in {\mathsf{L}}({\mathcal{E}}_2(M))$.
As a corollary, we directly obtain the usual twists of Courant algebroids in generalized geometry.
For the $L_\infty$-structure ${\mathsf{L}}({\mathcal{E}}_2(M))$ given by $${\mathsf{L}}({\mathcal{E}}_2(M))=\left\{F\in {\mathcal{C}}^\infty({\mathcal{E}}_2(M))~|~{\frac{{\partial}}{{\partial}x_\mu}}F=0\right\}~,$$ closed 3-forms are infinitesimal twists.
Outlook
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In this last section, we present some partial but interesting results on extended torsion and Riemann tensors as well as some comments on the global picture.
Extended torsion and Riemann tensors {#ssec:Riemann_tensor}
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The extension of torsion and Riemann curvature tensors on a general pre-N$Q$-manifold is an unsolved mathematical problem. It is not even clear, whether it is a good problem to pose. In the following, we present extensions for the case of the Vinogradov algebroids ${\mathcal{V}}_1(M)$, ${\mathcal{V}}_2(M)$ and the pre-N$Q$-manifold ${\mathcal{E}}_2(M)$, which are motivated by standard Riemannian geometry and DFT, respectively.
As a first step, we have to introduce a notion of covariant derivative on a symplectic pre-N$Q$-manifold ${\mathcal{M}}$. Let $n$ be the degree of ${\mathcal{M}}$. Just as all our symmetries were described in terms of Hamiltonians acting via the Poisson bracket, we ask that a covariant derivative in the direction $X$ is also a Hamiltonian function.
\[def:excovder\] An $\nabla$ on a pre-N$Q$-manifold ${\mathcal{M}}$ is a linear map from ${\mathscr{X}}({\mathcal{M}})$ to ${\mathcal{C}}^\infty({\mathcal{M}})$ such that the image $\nabla_X$ for $X\in {\mathscr{X}}({\mathcal{M}})$ gives rise to a map $$\{\nabla_X,-\}:{\mathscr{X}}({\mathcal{M}})\rightarrow {\mathscr{X}}({\mathcal{M}})~,$$ which readily generalizes to extended tensors and satisfies $$\{\nabla_{fX},Y\}=f\{\nabla_X,Y\}{{\qquad\mbox{and}\qquad}}\{\nabla_X,fY\}=\{Q X,f\}Y+f\{\nabla_X,Y \}$$ for all $f\in {\mathscr{C}}^\infty({\mathcal{M}})$ and extended tensors $Y$.
We now restrict to the three cases ${\mathcal{V}}_1(M)$, ${\mathcal{V}}_2(M)$ and ${\mathcal{E}}_2(M)$. To simplify out notation, we use $\hat{\mathscr{X}}({\mathcal{M}})$ to denote extended vectors and covectors. For the Vinogradov algebroids, $\hat {\mathscr{X}}({\mathcal{M}})$ are simply the functions linear in the coordinates $\zeta_\mu$ or $\xi^\mu$, cf. section \[ssec:Vinogradov\], and for ${\mathcal{E}}_2(M)$, $\hat {\mathscr{X}}({\mathcal{M}})$ are the functions linear in the coordinates $\theta^M$, cf. section \[ssec:E2M\_from\_restriction\]. That is, $\hat {\mathscr{X}}({\mathcal{M}})={\mathscr{X}}({\mathcal{M}})$ for ${\mathcal{V}}_2(M)$ and ${\mathcal{E}}_2(M)$, while ${\mathscr{X}}({\mathcal{M}})\subsetneq \hat {\mathscr{X}}({\mathcal{M}})$ for ${\mathcal{V}}_1(M)$. We also use repeatedly the algebra of functions ${\mathcal{C}}^\infty(M)={\mathcal{C}}^\infty({\mathcal{M}}_0)$.
We follow the historical development of generalized geometry and introduce the torsion tensor first.
\[def:ext\_torsion\] Let ${\mathcal{M}}$ be a pre-N$Q$-manifold. Given an extended connection $\nabla$, we define the ${\mathcal{T}}: \otimes^3 \hat {\mathscr{X}}({\mathcal{M}}) \rightarrow {\mathcal{C}}^\infty(M)$ for $X,Y,Z \in \hat {\mathscr{X}}({\mathcal{M}})$ by $$\label{extorsion}
\begin{aligned}
{\mathcal{T}}(X,Y,Z):=\, 3&\Bigl((-1)^{n|X|}\,\Bigl\{X,\{\nabla_{\pi(Y)},Z\}\Bigr\}\Bigr)_{[X,Y,Z]}\\
&+\frac{(-1)^{n(|Y|+1)}}{2}\left(\{X,\{QZ,Y\}\}-\{Z,\{QX,Y\}\}\right)~, \end{aligned}$$ where $|X|,|Y|$ denote the respective ${\mathbbm{N}}$-degrees and $\pi$ is the obvious projection $\hat {\mathscr{X}}({\mathcal{M}})\rightarrow {\mathscr{X}}({\mathcal{M}})$.
For ordinary differential geometry, which is captured by ${\mathcal{M}}={\mathcal{V}}_1(M)$, ${\mathcal{T}}(-,-,-)$ reduces to the ordinary torsion function $T(X,Y,Z) = \langle X,\nabla_Y Z -\nabla_Z Y - [Y,Z]\rangle$, where $X$ is a one-form, $Y,Z$ are vector fields and $\langle X,Y\rangle := \iota_Y X$. It vanishes for all other combinations of forms and vectors. In the cases of generalized geometry and double field theory, this extension of the torsion tensor boils down to the Gualtieri torsion [@Gualtieri:2007bq] and [@Hohm:2012mf], as we shall show later. In particular, it is a tensor and therefore ${\mathcal{C}}^\infty(M)$-linear in all its entries for the cases we consider.
In a similar way, we are able to provide a definition of an extended curvature operator which reduces to the curvature tensors of ordinary Riemannian geometry, generalized geometry and double field theory in the respective cases of ${\mathcal{V}}_1(M)$, ${\mathcal{V}}_2(M)$ and ${\mathcal{E}}_2(M)$. Again it uses the Poisson brackets for $\hat{\mathscr{X}}({\mathcal{M}})$.
\[def:ext\_Riemann\] Let ${\mathcal{M}}$ be a pre-N$Q$-manifold. Given an extended connection $\nabla$, the ${\mathcal{R}}: \otimes^4 \hat{\mathscr{X}}({\mathcal{M}}) \rightarrow {\mathcal{C}}^\infty(M)$ for $X,Y,Z,W \in \hat{\mathscr{X}}({\mathcal{M}})$ by $$\label{exCurvature}
\begin{aligned}
{\mathcal{R}}(X,Y,Z,W):=\; &\frac{1}{2}\Bigl(\Bigl\{\bigl\{\{\nabla_X,\nabla_Y\} -\nabla_{\mu_2(X,Y)},Z\bigr\},W\Bigr\} -(-1)^n( Z\leftrightarrow W) \\
& + \Bigl\{\bigl\{\{\nabla_Z,\nabla_W\}-\nabla_{\{\nabla_Z,W\} - \{\nabla_W,Z\}},X\bigr\},Y\Bigr\} - (-1)^n(X\leftrightarrow Y)\Bigr)\;.
\end{aligned}$$
In the following, we will discuss these two tensors for the three symplectic pre-N$Q$-manifolds of interest to us. In the case of ${\mathcal{E}}_2(M)$, we will make the remarkable observation that tensoriality of the curvature operator is established using the constraints , underlining the fact that these are the appropriate strong constraints for functions and vector fields.
Let us start with the case of Riemannian geometry, which is captured by the symplectic pre-N$Q$-manifold ${\mathcal{M}}={\mathcal{V}}_1(M)$. We need to show that definitions \[def:excovder\], \[def:ext\_torsion\] and \[def:ext\_Riemann\] reduce correctly to the expected objects of Riemannian geometry.
The following Hamiltonian for $X\in {\mathscr{X}}({\mathcal{M}})$ has the properties of definition \[def:excovder\] and reproduces the right expressions for the covariant derivative of forms and vector fields: $$\nabla_X =\,X^\mu p_\mu -X^\mu\Gamma^\rho{}_{\mu\nu}\zeta_\rho \xi^\nu\;.$$ The coefficients $\Gamma^\rho{}_{\mu\nu}$ are indeed the usual Christoffel symbols.
For ${\mathcal{M}}={\mathcal{V}}_1(M)$, let $X \in {\mathscr{X}}^*({\mathcal{M}})$ and $Y,Z \in {\mathscr{X}}({\mathcal{M}})$, then torsion as defined in reduces to the torsion operator $T(X,Y,Z)=\langle X,\nabla_Y Z - \nabla_Z Y - [Y,Z]\rangle$, where the bracket is the Lie bracket of vector fields. More generally, this is true whenever we take one element of ${\mathscr{X}}^*({\mathcal{M}})$ and the other two in ${\mathscr{X}}({\mathcal{M}})$. In all other cases the extended torsion vanishes.
The proof is done by simply writing out the Poisson brackets. Let $X\in{\mathscr{X}}^*({\mathcal{M}})$ and $Y,Z \in {\mathscr{X}}({\mathcal{M}})$. Then we get $$\begin{aligned}
3&\Bigl((-1)^{n|X|}\,\Bigl\{X,\{\nabla_Y,Z\}\Bigr\}\Bigr)_{[X,Y,Z]}\\=& \frac{1}{2}\Bigl(-\Bigl\lbrace X,\lbrace \nabla_Y Z\rbrace\Bigr\rbrace + \Bigr\lbrace X,\lbrace \nabla_Z,Y\rbrace\Bigr\rbrace +\Bigl\lbrace Y,\lbrace \nabla_Z,X\rbrace\Bigr\rbrace - \Bigl\lbrace Z,\lbrace \nabla_Y,X\rbrace\Bigr\rbrace \Bigr)\\
=&\frac{1}{2}\Bigl(-Y^\mu Z^\nu(\partial_\mu X_\nu - \partial_\nu X_\mu) + X_\rho Y^\mu Z^\nu(\Gamma^\rho{}_{\mu \nu} - \Gamma^\rho{}_{\nu \mu}) +\langle X,\nabla_Y Z - \nabla_Z Y\rangle \Bigr)\;.
\end{aligned}$$ Similarly, expanding out the second part in the definition of extended torsion, we get $$\frac{1}{2}\Bigl(\Bigl\lbrace X, \lbrace QZ,Y\rbrace \Bigr\rbrace - \Bigl\lbrace Z,\lbrace QX,Y\rbrace \Bigr\rbrace\Bigr)
=\frac{1}{2}\Bigl(-\langle X,[Y,Z]\rangle + Y^\mu Z^\nu(\partial_\mu X_\nu - \partial_\nu X_\mu)\Bigr)\;.$$ Because torsion components in Riemannian geometry are given by $\Gamma^\rho{}_{\mu \nu} - \Gamma^\rho{}_{\nu\mu}$, the first claim follows. The calculation is similar whenever there is one element in ${\mathscr{X}}^*({\mathcal{M}})$. If all elements are in ${\mathscr{X}}({\mathcal{M}})$, all terms are separately zero, as there are only Poisson brackets of vectors and vectors, which vanish. If there are two elements or all three elements in ${\mathscr{X}}^*({\mathcal{M}})$, one is left with brackets of two forms, which also vanish.
We also have the expected statement for the extended curvature operator:
For ${\mathcal{M}}={\mathcal{V}}_1(M)$, let $X,Y,Z \in {\mathscr{X}}({\mathcal{M}})$ and $W \in {\mathscr{X}}^*({\mathcal{M}})$. Then the extended curvature reduces to the standard curvature: $${\mathcal{R}}(X,Y,Z,W) = \langle W,\nabla_X\nabla_Y Z - \nabla_Y\nabla_X Z - \nabla_{\mu_2(X,Y)} Z \rangle\,.$$ Furthermore, if $X,Y,Z,W \in {\mathscr{X}}({\mathcal{M}})$ or if two, three or all of $X,Y,Z,W$ are in ${\mathscr{X}}^*({\mathcal{M}})$, we have ${\mathcal{R}}(X,Y,Z,W)=0$.
Similar to the previous proposition, the proof is done by checking the Poisson brackets. For the last statement, we recall that the binary product $\mu_2$ was defined as $$\mu_2(X,Y):= \,\tfrac{1}{2}(\lbrace QX,Y\rbrace - \lbrace QY,X\rbrace)\,.$$ If $X \in {\mathscr{X}}^*({\mathcal{M}})$ and $Y\in {\mathscr{X}}({\mathcal{M}})$, $\mu_2(X,Y)$ is in ${\mathscr{X}}^*({\mathcal{M}})$, so in this case $\nabla_{\mu_2(X,Y)}=0$. The statement follows by using the latter and checking the different cases.
So far, we have seen that for the right Vinogradov algebroid ${\mathcal{M}}={\mathcal{V}}_1(M)$, we recover the defining objects of ordinary Riemannian geometry. In this sense, we *extend* ordinary Riemannian geometry. In the following, we describe the remaining two cases, yielding essential aspects of torsion and curvature in generalized geometry and double field theory.
Since generalized geometry is readily obtained by reduction of double field theory, we focus now on the case ${\mathcal{M}}={\mathcal{E}}_2(M)$. Generalized vector fields are given by extended vector fields, which we denote by ${\mathscr{X}}({\mathcal{M}}) \ni X= X^M \theta_M$ and as noted above, we here have $\hat{\mathscr{X}}({\mathcal{M}})={\mathscr{X}}({\mathcal{M}})$. The extended covariant derivative with the properties of definition \[def:excovder\] now reads $$\label{Covder}
\nabla_X =\, X^Mp_M - \tfrac{1}{2}\,X^M\Gamma_{MNK}\theta^N\theta^K\;.$$ Already from the constancy of the bilinear form $\eta$, we see the antisymmetry of the extended connection coefficients: $$0=\,\partial_M\eta_{NK} = \,\Gamma_{MNK}+\Gamma_{MKN}\,.$$ As was shown in detail in [@Hohm:2012mf], the appropriate notion of torsion is the Gualtieri torsion, which in generalized geometry reads $$\textrm{GT}(X,Y,Z)=\,\langle \nabla_X Y - \nabla_Y X - \nu_2(X,Y),Z\rangle + \langle Y,\nabla_Z X\rangle\,,$$ where $\nu_2(-,-)$ denotes the Dorfman bracket of generalized geometry, and $X,Y,Z$ are generalized vectors. Furthermore, it was shown in [@Hohm:2012mf], that a Riemann curvature operator can be defined, see also [@Jurco:2015xra] for related results. We will now reformulate these objects in our language and investigate their ${\mathcal{C}}^\infty(M)$-linearity properties. We begin with the torsion tensors.
For extended vector fields $X,Y,Z \in {\mathscr{X}}({\mathcal{M}})$, the extended torsion tensor equals the Gualtieri-torsion: $${\mathcal{T}}(X,Y,Z) =\,\textrm{GT}(X,Y,Z)\;.$$
The proof is done by explicit calculation. Non-tensorial derivative terms cancel out and the remaining terms are $$\label{gualtieri2}
{\mathcal{T}}(X,Y,Z) =\,X^M Y^N Z^K(\Gamma_{MNK}-\Gamma_{NMK}+\Gamma_{KMN})\;,$$ coinciding with the original torsion found by Gualtieri and used in [@Hohm:2012mf] in the context of double field theory.
Now we turn to the extended curvature tensor in the case of ${\mathcal{M}}={\mathcal{E}}_2(M)$. We discuss the result of [@Hohm:2012mf] in terms of the language set up in this work. As a simplification, we restrict ourselves to the case of vanishing extended torsion . First, we observe that the Poisson-bracket of two extended covariant derivatives contains the tensorial part of the standard Riemann tensor, but on the doubled space[^11]: $$\label{standardRiem}
\begin{aligned}
\bigl\{\{\nabla_X,&\nabla_Y\},Z\bigr\} = \Bigl([X,Y]^M\partial_M Z^R +[X,Y]^K\,\Gamma_{KL}{}^R\,Z^L\Bigr)\theta_R \\
&+X^MY^NZ^K\bigl(\partial_M\Gamma_{NK}{}^R - \partial_N \Gamma_{MK}{}^R + \Gamma_M{}^{RQ}\Gamma_{NQK} - \Gamma_N{}^{RQ}\Gamma_{MQK}\bigr)\theta_R\;.
\end{aligned}$$ In standard Riemannian geometry, the Lie-derivative terms are canceled by the covariant derivative with respect to the Lie derivative of the corresponding vector fields. In double field theory, infinitesimal symmetries are determined by the C-bracket $\mu_2$, so we have to replace the Lie derivative by the latter. The resulting expression is not tensorial, and we have to add further terms to correct for this, as done in . We arrive at the following result, written in terms of Poisson brackets:
For ${\mathcal{M}}={\mathcal{E}}_2(M)$, the extended curvature operator ${\mathcal{R}}$, given by is tensorial up to terms that vanish after imposing the constraints .
First, note that for $n=2$ the sign factors in drop out. Denoting the standard Riemannian curvature combination by $R_{MNKR}$, i.e. $$R_{MNKR} = \partial_M\Gamma_{NKR} - \partial_N \Gamma_{MKR} + \Gamma_{MR}{}^{Q}\Gamma_{NQK} - \Gamma_{NR}{}^{Q}\Gamma_{MQK}\;,$$ and writing out the expression in terms of components, we get $$\begin{aligned}
{\cal R}(X,Y,&Z,W) =\; X^MY^NZ^KW^R(R_{MNKR}+R_{KRMN}) \\
&+\frac{1}{4}(X_M\partial^K Y^M - Y_M\partial^K X^M)(W_N\partial_K Z^N - Z_N\partial_K W^N) \\
&+\frac{1}{4}(X_M\partial^K Y^M - Y_M\partial^K X^M)\Gamma_{KPQ}(Z^PW^Q - W^P Z^Q) \\
&+\frac{1}{2}\Bigl( Z^N W^K(-\Gamma_{NK}{}^M + \Gamma_{KN}{}^M)(Y_P\partial_M X^P - X_P\partial_M Y^P)\Bigr) \\
&+\frac{1}{2}\Bigl(Z^N W^K(\Gamma_{NK}{}^M - \Gamma_{KN}{}^M)\Gamma_{MPQ}(-X^P Y^Q + Y^PX^Q)\Bigr)\;.
\end{aligned}$$ We now use the vanishing of the Gualtieri torsion to simplify the combinations of connection coefficients, which yields $$\begin{aligned}
{\cal R}(X,Y,Z,W)=\;&X^MY^NZ^KW^R(R_{MNKR}+R_{KRMN} + \Gamma^Q{}_{MN}\Gamma_{QKR})\\
&+\tfrac{1}{4}(X_M\partial^K Y^M - Y_M\partial^K X^M)(W_N\partial_K Z^N - Z_N\partial_K W^N)\;.
\end{aligned}$$ From the last expression, we see, that all terms in the first line are manifestly tensorial, whereas the term in the second line is not. However, the terms destroying tensoriality vanish due to the constraints . For example, the failure of ${\mathcal{C}}^\infty(M)$-linearity in $X$ reads as $${\cal R}(fX,Y,Z,W)-f{\cal R}(X,Y,Z,W) = -\tfrac{1}{4}\,X_MY^M\,\partial^Kf(W_N\partial_K Z^N - Z_N\partial_K W^N)\;,$$ and the terms of the form $\{\partial^kf\, \partial_K Z,W\}$ and $\{\partial^kf \,\partial_K W,Z\}$ vanish due to the strong constraint in the form of the first two lines of .
Finally we comment on the Ricci and scalar curvatures in the case of ${\mathcal{M}}={\mathcal{E}}_2(M)$. By definition, the Ricci scalar is the trace of an endomorphism of the doubled tangent bundle, more precisely, we have $$\textrm{Ric}(X,Y)=\; \textrm{tr}(Z\mapsto {\cal R}(X,Z)Y)\;.$$ In our case, this means that $\textrm{Ric}(X,Y) =\,{\cal R}(\theta^I,X,\theta_I,Y)$. Using the result , we see that the derivative terms which are only tensorial up to the constraints cancel. Furthermore, denoting the standard combination for the Ricci tensor using by $\textrm{Ric}_0$, we arrive at $$\textrm{Ric}(X,Y)=\; \textrm{Ric}_0(X,Y) + \textrm{Ric}_0(Y,X) +X^M Y^N\,\Gamma^{QK}{}_M\Gamma_{QKN}\;.$$ Finally, the Ricci-scalar is computed by contracting the Ricci-tensor. We note that this is done by the bilinear form $\eta$, in contrast to the generalized metric ${\mathcal{H}}$ (cf. e.g. [@Hohm:2011si]). Thus we obtain the Ricci scalar $$\label{scal}
{\cal R} = \eta^{MN}\,\textrm{Ric}(E_M,E_N) = 2R_0 + \Gamma^{MNK}\Gamma_{MNK}\;.$$ Note that here and in the following, we raise and lower indices with the constant form $\eta$. Contractions with the generalized metric ${\mathcal{H}}$ are written out explicitly to emphasize the appearance of the generalized metric as the dynamical field.
To write down actions, it is necessary to express the connection coefficients $\Gamma_{MNK}$ of in terms of the the generalized metric ${\mathcal{H}}$, which is the dynamical field of the theory. This is done in a similar way as in [@Hohm:2011si]. We review some details and state the result. In contrast to ordinary Riemannian geometry, where vanishing torsion and metricity uniquely determine the Levi-Civita connection via the Koszul formula, one has three different constraints on the connection coefficients in double field theory:
- $\textrm{GT}=0 \quad \leftrightarrow \quad \Gamma_{KMN}-\Gamma_{MKN} + \Gamma_{NKM} = 0\;,$
- $\nabla \eta = 0\quad \leftrightarrow \quad \Gamma_{KMN}+\Gamma_{KNM}=0\;,$
- $\nabla {\mathcal{H}}= 0 \quad \leftrightarrow \quad \partial_K {\mathcal{H}}_{MN} = \Gamma_{KMP} {\mathcal{H}}^P_N + \Gamma_{KNP}{\mathcal{H}}^P_M \;.$
In [@Hohm:2011si], it was shown that these conditions do not determine the connection coefficients uniquely. However, different choices of connection coefficients give rise to the same DFT actions, so we will use the most common solution to write down actions in our language. Before stating the solution according to [@Hohm:2011si], we remark that in case of a non-trivial dilaton there is a further constraint involving the trace of the connection coefficients:
- $\Gamma_{NMK}\eta^{NK} =\,-2\partial_M d\,$.
It was shown that the four constraints give a solution to the connection coefficients in terms of the generalized metric which is non-unique but different solutions lead to the same Ricci scalar:
[@Hohm:2011si]. In case of vanishing Gualtieri-torsion, ${\mathcal{H}}$-metricity, covariant constancy of the form $\eta$ and the dilaton constraint, a possible solution for the connection coefficients is given by $$\label{connectioncoef}
\begin{aligned}
\Gamma_{MNK} =&\,\tfrac{1}{2}{\mathcal{H}}_{KQ}\partial_M{\mathcal{H}}^Q{}_N + \tfrac{1}{2}(\delta_{[N}{}^P{\mathcal{H}}_{K]}{}^Q + {\mathcal{H}}_{[N}{}^P\delta_{K]}{}^Q)\partial_P{\mathcal{H}}_{QM} \\
&+\frac{2}{D-1}(\eta_{M[N}\delta_{K]}{}^Q + {\mathcal{H}}_{M[N}{\mathcal{H}}_{K]}{}^Q)(\partial_Q d + \tfrac{1}{4}{\mathcal{H}}^{PM}\partial_M{\mathcal{H}}_{PQ})\;,
\end{aligned}$$ where $D$ is the dimension of the manifold underlying the doubled manifold.
Using these connection coefficients, an action for double field theory was formulated in [@Hohm:2011si] using the extended Ricci scalar .
Comments on a global description {#ssec:global}
--------------------------------
There is now an obvious procedure for turning our local description of double field theory into a global one. The first step consists of integrating the infinitesimal symmetries described by the Lie 2-algebra structure on ${\mathcal{E}}_2(M)$ to finite symmetries. The second step is then to use the resulting finite symmetries to patch together local descriptions.
In principle, it is possible to integrate any Lie 2-algebra, using the various techniques in the literature [@Getzler:0404003; @Henriques:2006aa; @Severa:1506.04898]. These techniques, however, are very involved, and the outcome is often cumbersome in the sense that a categorical equivalence has to be applied to it to be useful.
Fortunately, we do not really require the finite symmetries themselves, but merely their action on extended tensors. This action is simply given by a Lie algebra, cf. section \[ssec:Symmetries\_of\_DFT\] and equation , which is readily integrated. Moreover, this has been done before in [@Hohm:2013bwa; @Hohm:2012gk].
A proposal of how to glue together local descriptions of double field theory with finite symmetries was made in [@Berman:2014jba]. In [@Papadopoulos:2014mxa], however, it was shown that this procedure works at most for gerbes with trivial or purely torsion characteristic class. That is, the 3-form curvature $H$ of the $B$-field is (globally) exact and thus does not exhibit any topological non-triviality. This is clearly too restrictive for a general application in string theory.
From our perspective, this is not very surprising, as non-trivial background fluxes require to work with the twisted symmetries introduced in section \[ssec:twisted\_extended\_symmetries\]. It is the Lie algebra of actions of the twisted infinitesimal symmetries that should be integrated and used in the patching of local descriptions of double field theories with non-trivial background fluxes. We hope to report on progress in this direction in future work.
One can, however, adopt a different standpoint, which also explains that a topologically non-trivial 3-form flux is precisely where things break down. As shown in [@Bouwknegt:2003vb], global T-duality with non-trivial $H$-flux induces a change in topology. In particular, the $H$-flux will encode the topology of the T-dual circle fibration. Even worse, performing a T-duality along the fibers of the torus bundle $T^2{{\hookrightarrow}}T^3\rightarrow S^1$ with $H$-flux the volume form, it was observed in [@Kachru:2002sk] that a circle disappears under T-duality. The T-dual was then interpreted in [@Mathai:2004qq] as a continuous field of stabilized noncommutative tori fibered over $S^1$. It is very hard to imagine that this situation can be captured by gluing together the current local description of double field theory. Inversely, however, there may be some room to turn pre-N$Q$-manifolds into ordinary noncommutative and/or non-associative N$Q$-manifolds.
Finally, let us note that in generalized geometry, one might be tempted to integrate the underlying Courant algebroid ${\mathcal{V}}_2(M)$, which is a symplectic Lie 2-algebroid, to a symplectic Lie 2-groupoid, using the methods of [@LiBland:2011aa; @Severa:1506.04898]. It would be natural to expect that the associated Lie $2$-algebra of infinitesimal symmetries survives this integration, but this is not clear at all.
Moreover, when trying to apply this integration method to the case of double field theory, we face the problem that the underlying geometric structure is a pre-N$Q$-manifold instead of an N$Q$-manifold.
Acknowledgements {#acknowledgements .unnumbered}
================
We would like to thank David Berman, Ralph Blumenhagen, André Coimbra, Dieter L[ü]{}st and Jim Stasheff for discussions. The work of CS was partially supported by the Consolidated Grant ST/L000334/1 from the UK Science and Technology Facilities Council. AD and CS want to thank the Department of Mathematics of Heriot-Watt University and the Institut für Theoretische Physik in Hannover for hospitality, respectively.
[^1]: A first action for double field theory was already suggested in [@Siegel:1993th].
[^2]: i.e. nontrivial
[^3]: In principle, we also complete these brackets by putting all other possible $\mu_i(...)$ taking $i$ basis elements as arguments to zero.
[^4]: with certain restrictions to finite dimensional $L_\infty$-algebras
[^5]: Recall the Koszul rule for interchanging two odd elements, e.g. $f {\mathrm{d}}g=(-1)^{|f|\,|g|}({\mathrm{d}}g)f$.
[^6]: The essential structure underlying this algebroid was first studied in [@MR1074539].
[^7]: To write also this transformation in the form , one should extend the Poisson brackets to the free tensor algebra of functions on ${\mathcal{V}}_2(M)$, along the lines of section \[ssec:extended\_tensors\].
[^8]: i.e. diffeomorphisms and gauge transformations of the connective structure
[^9]: A Leibniz algebra of degree $n$ is a graded vector space endowed with a bilinear operation satisfying the Jacobi identity of degree $n$, which differs from the Leibniz rule of degree $n$ given in , cf. [@Kosmann-Schwarzbach:0312524].
[^10]: Interestingly, the equation $\beta^M=0$ can be regarded as part of the equations of motion of a topological string as discussed in [@Bouwknegt:2011vn Section 7].
[^11]: Here we use the abbreviation $[X,Y]^K$ for the Lie-derivative part, i.e. $[X,Y]^K = \,X^M\partial_M Y^K - Y^M\partial_M X^K$.
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abstract: 'In [@bacher], Bacher and de la Harpe study conjugacy growth series of infinite permutation groups and their relationships with $p(n)$, the partition function, and $p(n)_{\textbf{e}}$, a generalized partition function. They prove identities for the conjugacy growth series of the finitary symmetric group and the finitary alternating group. The group theory in [@bacher] also motivates an investigation into congruence relationships between the finitary symmetric group and the finitary alternating group. Using the Ramanujan congruences for the partition function $p(n)$ and Atkin’s generalization to the $k$-colored partition function $p_{k}(n)$, we prove the existence of congruence relations between these two series modulo arbitrary powers of 5 and 7, which we systematically describe. Furthermore, we prove that such relationships exist modulo powers of all primes $\ell\geq 5$.'
address:
- '605 Asbury Circle, Box 122042, Atlanta, GA 30322'
- '605 Asbury Circle, Atlanta, GA 30322'
- '1192 Paresky Center, Williams College, Williamstown, MA 01267'
author:
- Tessa Cotron
- Robert Dicks
- Sarah Fleming
bibliography:
- 'references.bib'
title: Congruences between word length statistics for the finitary alternating and symmetric groups
---
Introduction
============
In a recent paper [@bacher], Bacher and de la Harpe study the conjugacy growth series of infinite permutation groups that are locally finite. Given $g \in G$, where $G$ is a group generated by some set $S$, define the *word length* $\ell_{G,S}(g)$ as the minimal non-negative integer $n$ where $g=s_1s_2\cdots s_n$ and $s_1,s_2,\ldots,s_n\in S\cup S^{-1}$. They define the *conjugacy length* $\kappa_{G,S}(g)$ as the minimal integer $n$ such that there exists an $h$ in the conjugacy class of $g$ for which $\ell_{G,S}(h)=n$. The number of conjugacy classes in $G$ consisting of elements $g$ where $\kappa_{G,S}(g)=n$ for $n\in{\mathbb{N}}$ is denoted $\gamma_{G,S}(n)\in{\mathbb{N}}\cup\{0\}\cup\{\infty\}$. If $\gamma_{G,S}(n)$ is finite for all $n\in{\mathbb{N}}$ for a given pair $(G, S)$, then the *conjugacy growth series* is defined to be $$C_{G,S}(q):=\sum_{n=0}^{\infty}\gamma_{G,S}(n)q^n.$$
Given a permutation $g$ of a non-empty set $X$, denote the *support* of $g$ as $\sup(g):=\{x \in X : g(x)\neq x\}$. The group of permutations of $X$ with finite support is the *finitary symmetric group* ${\operatorname{Sym}}(X)$. The subgroup of ${\operatorname{Sym}}(X)$ with permutations of even signature is the *finitary alternating group* ${\operatorname{Alt}}(X)$. Let $S\subset{\operatorname{Sym}}({\mathbb{N}})$ be a generating set of ${\operatorname{Sym}}({\mathbb{N}})$ such that $S_{{\mathbb{N}}}^{\text{Cox}}\subset S\subset T_{{\mathbb{N}}}$, where $$\label{cox}
S_{{\mathbb{N}}}^{\text{Cox}}:=\{(i,i+1):i\in{\mathbb{N}}\}$$ is such that $({\operatorname{Sym}}({\mathbb{N}}), S_{{\mathbb{N}}}^{\text{Cox}})$ is a Coxeter system, and $$\label{TN}
T_{{\mathbb{N}}}:=\{(x,y)\in{\operatorname{Sym}}({\mathbb{N}}): x,y\in{\mathbb{N}}\text{ are distinct}\}$$ is the conjugacy class of all transpositions in ${\operatorname{Sym}}({\mathbb{N}})$. Throughout this paper, we define $S$ to be the set described above. By Proposition 1 in [@bacher], the conjugacy growth series for the pair $({\operatorname{Sym}}({\mathbb{N}}), S)$ is given by $$\label{csym}
C_{{\operatorname{Sym}}({\mathbb{N}}),S}(q)=\sum_{n=0}^{\infty}p(n)q^n=\prod_{n=1}^{\infty}\frac{1}{1-q^n},$$ where $p(n)$ denotes the usual integer partition function. Let $S'\subset{\operatorname{Alt}}({\mathbb{N}})$ be a generating set of ${\operatorname{Alt}}({\mathbb{N}})$ such that $S_{{\mathbb{N}}}^{A}\subset S'\subset T_{{\mathbb{N}}}^{A}$, where we define $$\label{SNA}
S_{{\mathbb{N}}}^A:=\{(i, i+1, i+2)\in{\operatorname{Alt}}({\mathbb{N}}) : i\in N\}$$ and $$\label{TNA}
T_{{\mathbb{N}}}^A:=\cup_{g\in{\operatorname{Alt}}({\mathbb{N}})}g S_{{\mathbb{N}}}^A g^{-1}.$$ Throughout this paper, we define $S'$ to be the set described above. By Proposition 11 in [@bacher], the conjugacy growth series for the pair (${\operatorname{Alt}}({\mathbb{N}}), S')$ is given by $$\begin{aligned}
\label{calt1}
C_{{\operatorname{Alt}}({\mathbb{N}}),S'}(q)&=\frac{1}{2}\sum_{n=0}^{\infty}p\left(\frac{n}{2}\right)q^n+\frac{1}{2}\sum_{n=0}^{\infty}p_{2}(n)q^n\\
\nonumber&=\frac{1}{2}\prod_{n=1}^{\infty}\frac{1}{1-q^{2n}}+\frac{1}{2}\prod_{n=1}^{\infty}\frac{1}{(1-q^n)^2},\end{aligned}$$ where $p\left(\frac{n}{2}\right)=0$ for all odd $n$ and $p_2(n)$ denotes the number of 2-colored partitions of $n$. Combining (\[csym\]) and (\[calt1\]), we obtain $$\begin{aligned}
\label{calt2}
2\gamma_{{\operatorname{Alt}}({\mathbb{N}}),S'}(2n)&=p(n)+p_2(2n)\\
\nonumber&=\gamma_{{\operatorname{Sym}}({\mathbb{N}}),S}(n)+p_2(2n).\end{aligned}$$
The finite symmetric and alternating groups, $S_n$ and $A_n$, have the property that $A_n$ is a normal subgroup of $S_n$ and $[S_n:A_n]=2$, so one would naively expect a similar relationship between the finitary symmetric and alternating groups to hold. By (\[calt2\]), we see that $\gamma_{{\operatorname{Alt}}({\mathbb{N}}),S'}(2n)$ is one half of $\gamma_{{\operatorname{Sym}}({\mathbb{N}}),S}(n)$ together with a “discrepancy function,” $p_2(2n)$. In terms of size, we prove in [@cotron] that $\gamma_{{\operatorname{Alt}}({\mathbb{N}}),S'}(n)$ and $\gamma_{{\operatorname{Sym}}({\mathbb{N}}),S}(n)$ behave differently asymptotically and do not follow the pattern of their finite counterparts. More precisely, in [@cotron], the authors prove that as $n\rightarrow\infty$, we have that $$\gamma_{{\operatorname{Sym}}({\mathbb{N}}),S}(n)\sim\frac{e^{\pi{\sqrt}{2n/3}}}{4n{\sqrt}{3}}$$ and $$\gamma_{{\operatorname{Alt}}({\mathbb{N}}),S'}(n)\sim \frac{e^{2\pi{\sqrt}{n/3}}}{3^{\frac{3}{4}}\cdot 8n^{\frac{5}{4}}}.$$
It is natural to consider the question of congruence relations between the coefficients of the conjugacy growth series of each of these groups. By (\[calt2\]), there will exist congruences modulo powers of primes between $2\gamma_{{\operatorname{Alt}}({\mathbb{N}}),S'}(2n)$ and $\gamma_{{\operatorname{Sym}}({\mathbb{N}}),S}(n)$ whenever the “discrepancy function," $p_2(2n)$, is equivalent to 0. In [@cotron], we establish a method of proving congruences for generalized partition functions modulo a prime, including the following examples.
For all $n\equiv 2, 3, 4\pmod{5}$, we have that $$2\gamma_{{\operatorname{Alt}}({\mathbb{N}}),S'}(2n)\equiv\gamma_{{\operatorname{Sym}}({\mathbb{N}}),S}(n)\pmod{5}.$$
For all $n\equiv 17, 31, 38, 45\pmod{49}$, we have that $$2\gamma_{{\operatorname{Alt}}({\mathbb{N}}),S'}(2n)\equiv\gamma_{{\operatorname{Sym}}({\mathbb{N}}),S}(n)\pmod{7}.$$
It is natural to ask if these examples are part of a more general phenomenon and if there exists a method to determine congruences. Ramanujan stated congruences for the partition function $p(n)$ modulo powers of 5, 7, and 11, proved by Watson in [@watson]. In addition, Atkin proved the existence of congruences for the function $p_2(n)$ modulo powers of the primes 5, 7, and 13 in [@atkin]. Building off of these results, we obtain congruences between the coefficients of the conjugacy growth series for these groups modulo powers of 5 and 7.
Throughout, we let $S\subset{\operatorname{Sym}}({\mathbb{N}})$ be a generating set of ${\operatorname{Sym}}({\mathbb{N}})$ such that $S_{{\mathbb{N}}}^{\text{Cox}}\subset S\subset T_{{\mathbb{N}}}$, where $S_{{\mathbb{N}}}^{\text{Cox}}$ and $T_{{\mathbb{N}}}$ are defined by (\[cox\]) and (\[TN\]), respectively. In addition, we let $S'\subset{\operatorname{Alt}}({\mathbb{N}})$ be a generating set for ${\operatorname{Alt}}({\mathbb{N}})$ such that $S_{{\mathbb{N}}}^{A}\subset S'\subset T_{{\mathbb{N}}}^{A}$, where $S_{{\mathbb{N}}}^{A}$ and $T_{{\mathbb{N}}}^{A}$ are defined by (\[SNA\]) and (\[TNA\]), respectively. Using this notation, we arrive at the following theorem:
\[cong1\] Assume the notation above. Let $\ell=5$ or $7$ and let $j\geq 1$. Then for all $24n\equiv 1\pmod{\ell^j}$, we have that $$\gamma_{{\operatorname{Alt}}({\mathbb{N}}),S'}(2n)\equiv\gamma_{{\operatorname{Sym}}({\mathbb{N}}),S}(n)\equiv 0\pmod{\ell^{\lfloor{j/2-1}\rfloor}}.$$
For example, modulo $5$, $25$, and $125$, we obtain for all $n\geq0$ that $$\begin{aligned}
\gamma_{{\operatorname{Alt}}({\mathbb{N}}),S'}(2\cdot 5^4n+1198)&\equiv
0\pmod{5}\\
\gamma_{{\operatorname{Alt}}({\mathbb{N}}),S'}(2\cdot 5^6n+29948)&\equiv
0\pmod{25}\\
\gamma_{{\operatorname{Alt}}({\mathbb{N}}),S'}(2\cdot 5^8n+748698)&\equiv
0\pmod{125}.\end{aligned}$$ Likewise, modulo $7$, $49$, and $343$, we obtain for all $n\geq 0$ that $$\begin{aligned}
\gamma_{{\operatorname{Alt}}({\mathbb{N}}),S'}(2\cdot 7^4n+4602)&\equiv
0\pmod{7}\\
\gamma_{{\operatorname{Alt}}({\mathbb{N}}),S'}(2\cdot 7^6n+225494)&\equiv
0\pmod{49}\\
\gamma_{{\operatorname{Alt}}({\mathbb{N}}),S'}(2\cdot 7^8n+11049202)&\equiv
0\pmod{343}.\end{aligned}$$
It is a natural question to ask what holds for general primes $\ell\not\in\{5,7\}$. Following the work of Treneer [@treneer], we prove the existence of congruences between the coefficients of the conjugacy growth series for $({\operatorname{Alt}}({\mathbb{N}}),S')$ and $({\operatorname{Sym}}({\mathbb{N}}),S)$ modulo arbitrary powers of primes $\ell\geq 5$. Treneer’s work gives general congruences for coefficients of various types of modular forms. Here, we follow her method and make it explicit.
Let $\ell\geq 5$ be prime. We then define $$\label{m_l}
m_{\ell}:=
\begin{cases}
2 & 5\leq\ell\leq 23\\
1 & \ell\geq 29,
\end{cases}$$ $$\label{delta_l}
\delta_{\ell}:=
\frac{Q\ell^{m_{\ell}}\beta_{\ell}+1}{24},$$ and $$\label{beta_l}
\beta_{\ell}:=
\frac{23}{Q\ell^{m_\ell}}\pmod{24}.$$
Using this notation, we arrive at the following theorem:
\[sym\] Assume the above notation. Let $\ell\geq 5$ be prime and let $j\geq 1$. Then for a positive proportion of primes $Q\equiv -1\pmod{144\ell^j}$, we have that $$2\gamma_{{\operatorname{Alt}}({\mathbb{N}}),S'}(2Q\ell^{m_{\ell}}n+2\delta_\ell)\equiv\gamma_{{\operatorname{Sym}}({\mathbb{N}}),S}(Q\ell^{m_{\ell}}n+\delta_\ell)\pmod{\ell^j}$$ for all $24n+\beta_{\ell}$ coprime to $Q\ell$.
In this paper, we make effective the effort of [@treneer] by focusing on the specific case of the function $p_2(n)$, which arises from the conjugacy growth series for $({\operatorname{Alt}}({\mathbb{N}}),S')$. We use properties of modular forms to prove the existence of congruences between the coefficients of these two series.
Section \[modularity\] covers the basics of modular forms and cusps, and Section \[atkinsthm\] focuses on congruences for the partition function $p(n)$ and the $k$-colored partition function $p_k(n)$ modulo prime powers. In Section \[serresthm\] we state a theorem of Serre. Section \[pf\] provides a proof of Theorem \[cong1\] using the Ramanujan and Atkin congruences. In Section \[proof of thm\], we focus on and make effective a special case of Theorem 1.2 in [@treneer], which we require in order to prove the existence of congruences between $2\gamma_{{\operatorname{Alt}}({\mathbb{N}}),S'}(2n)$ and $\gamma_{{\operatorname{Sym}}({\mathbb{N}}),S}(n)$. In Section \[section4\], we use this result to prove Theorem \[sym\].
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors would like to thank Ken Ono and Olivia Beckwith for advising this project and for their many helpful conversations and suggestions. In addition, the authors would like to thank Emory University and the NSF for their support via grant DMS-1250467.
Preliminaries {#prelim}
=============
Here we state standard properties of modular forms that can be found in many texts such as [@apostol] and [@ono].
Modularity, Cusps, and Operators {#modularity}
--------------------------------
The proof of Theorem \[sym\] uses properties of modular forms, which requires the understanding of cusps. A *cusp* of $\Gamma\subseteq{\operatorname{SL}}_2({\mathbb{Z}})$ is an equivalence class in $\mathbb{P}^1(\mathbb{Q})=\mathbb{Q}\cup\{\infty\}$ under the action of $\Gamma$ [@ono p. 2]. We use the following definition of modular forms from [@apostol p. 114]:
A function $f$ is said to be an *entire modular form of weight k* on a subgroup $\Gamma\subseteq{\operatorname{SL}}_{2}({\mathbb{Z}})$ if it satisfies the following conditions:
1. $f$ is analytic in the upper-half $\mathbb{H}$ of the complex plane,
2. $f$ satisfies the equation $$f\left(\frac{az+b}{cz+d}\right)=(cz+d)^kf(z)$$ whenever $\begin{pmatrix}
a& b\\
c& d
\end{pmatrix}\in\Gamma$ and $z\in\mathbb{H}$, and
3. the Fourier expansion of $f$ has the form $$f(z)= \sum_{n=0}^{\infty} c(n)e^{2\pi i n z}$$ at the cusp $i\infty$, and $f$ has analogous Fourier expansions at all other cusps.
In particular, we study *Dedekind’s eta-function*, a weight 1/2 modular form defined as the infinite product $$\eta(z):=q^{1/24}{\displaystyle}\prod_{n=1}^{\infty}(1-q^n),$$ where $q:=e^{2\pi iz}$ and $z\in\mathbb{H}$. An *eta-quotient* is a function $f(z)$ of the form $$f(z):={\displaystyle}\prod_{\delta\mid N}\eta(\delta z)^{r_{\delta}},$$ where $N\geq 1$ and each $r_{\delta}$ is an integer. If each $r_{\delta}\geq 0$, then $f(z)$ is known as an *eta-product*.
If $N$ is a positive integer, then we define $\Gamma_0(N)$ as the congruence subgroup $$\Gamma_0(N):=\left\{\begin{pmatrix}
a& b\\
c& d
\end{pmatrix}\in SL_2({\mathbb{Z}}):c\equiv 0\pmod N\right\}.$$ We recall Theorem 1.64 from [@ono p. 18] regarding the modularity of eta-quotients:
\[mod\] If $f(z)=\prod_{\delta\mid N}\eta(\delta z)^{r_{\delta}}$ has integer weight $k=\frac{1}{2}\sum_{\delta\mid N}r_{\delta}$, with the additional properties that $$\sum_{\delta\mid N}\delta r_{\delta}\equiv 0\pmod{24}$$ and $$\sum_{\delta\mid N}\frac{N}{\delta}r_{\delta}\equiv 0\pmod{24},$$ then $f(z)$ satisfies $$\label{func}
f\left(\frac{az+b}{cz+d}\right)=\chi(d)(cz+d)^kf(z)$$ for every $\begin{pmatrix}
a& b\\
c& d
\end{pmatrix}\in\Gamma_0(N)$ where the character $\chi$ is defined by $\chi(d):=\left(\frac{(-1)^ks}{d}\right)$, where $s:=\prod_{\delta\mid N}\delta^{r_\delta}$.
Any modular form that is holomorphic (resp. vanishes) at all cusps of $\Gamma_0(N)$ and satisfies (\[func\]) is said to have Nebentypus character $\chi$, and the space of such forms is denoted $M_k(\Gamma_0(N),\chi)$ (resp. $S_k(\Gamma_0(N),\chi)$). In particular, if $k$ is a positive integer and $f(z)$ satisfies the conditions of Proposition \[mod\] and is holomorphic (resp. vanishes) at all of the cusps of $\Gamma_0(N)$, then $f(z)\in M_k(\Gamma_0(N),\chi)$ (resp. $S_k(\Gamma_0(N),\chi)$). If $f(z)$ satisfies the conditions of Proposition \[mod\] but has poles at the cusps of $\Gamma_0(N)$, then we say $f(z)$ is a *weakly holomorphic modular form*, and the space of such forms is denoted $M_k^{!}(\Gamma_0(N),\chi)$.
If $f(z)$ is a modular form, then we can act on it with various types of operators. If $\gamma=\begin{pmatrix}
a & b\\
c & d
\end{pmatrix}
\in{\operatorname{SL}}_2({\mathbb{Z}})$, then the action of the *slash operator* $\mid_{k}$ on $f(z)$ is defined by $$(f\mid_{k}\gamma)(z):=(cz+d)^{-k}f(\gamma z),$$ where $$\gamma z:=\frac{az+b}{cz+d}.$$
Next, we define $$\label{sigma}
\sigma_{v,t}:=\begin{pmatrix}
1 & v\\
0 & t
\end{pmatrix}.$$ If $f(z)=\sum_{n=n_0}^{\infty}a(n)q^n$ is a weight $k$ modular form, then the action of the *U-operator* $U(d)$ on $f(z)$ is defined by $$\label{U}
f(z)\mid U(d):=t^{\frac{k}{2}-1}\sum_{v=0}^{t-1}f(z)\mid_{k}\sigma_{v,t}=\sum_{n=n_0}^{\infty}a(dn)q^n.$$ Likewise, the action of the *V-operator* $V(d)$ is defined by $$f(z)\mid V(d):=t^{-\frac{k}{2}}f(z)\mid_{k}\begin{pmatrix}
t & 0\\
0 & 1
\end{pmatrix}
=
\sum_{n=n_0}^{\infty}a(n)q^{dn}.$$ Now, if $f(z)=\sum_{n=0}^{\infty}a(n)q^n\in M_k(\Gamma_0(N),\chi)$, then the action of the *Hecke operator* $T_{p,k,\chi}$ on $f(z)$ is defined by $$f(z)\mid T_{p,k,\chi}:={\displaystyle}\sum_{n=0}^{\infty}(a(pn)+\chi(p)p^{k-1}a(n/p))q^n,$$ where $a(n/p)=0$ if $p\nmid n$. We recall the following result from [@ono p. 21,28]:
\[hecke\] Suppose that $f(z)\in M_k(\Gamma_0(N),\chi)$.
1. If $d\mid N$, then $$f(z)\mid U(d)\in M_k(\Gamma_0(N),\chi).$$
2. If $d$ is a positive integer, then $$f(z)\mid V(d)\in M_k(\Gamma_0(Nd),\chi).$$
3. If $p\geq 2$, then $$f(z)\mid T_{p,k,\chi}\in M_k(\Gamma_0(N),\chi).$$
Ramanujan’s Congruences and Atkin’s Generalizations {#atkinsthm}
---------------------------------------------------
Ramanujan conjectured the following congruences for the partition function modulo powers of the primes 5 and 7, which Watson proved in [@watson].
\[ram\] Let $\ell=5$, $7$, or $11$ and let $j\geq 1$. Then if $24n\equiv 1\pmod{\ell^j}$, we have that $$\begin{cases}
p(n)\equiv 0\pmod{\ell^j}& \ell=5,11\\
p(n)\equiv 0\pmod{\ell^{\lfloor{j/2}\rfloor+1}}& \ell=7.
\end{cases}$$
In [@atkin], Atkin generalized the Ramanujan congruences modulo powers of 5, 7, and 11 to the function $p_k(n)$, which counts the number of $k$-colored partitions of $n$.
\[atkin\] Let $k>0$, $\ell=2,3,5,7$ or $13$, and $j\geq 1$. Then if $24n\equiv k\pmod{\ell^j}$, we have that $$p_k(n)\equiv 0\pmod{\ell^{\lfloor{\alpha j/2+\epsilon}\rfloor}},$$ where $\epsilon:=\epsilon(k)=O(\log k)$ and $\alpha=\alpha(k,\ell)$ depending on $\ell$ and the residue of $k$ modulo 24.
Atkin computes the value of $\alpha(k,\ell)$ in a table in [@atkin]. We note the following values of $\alpha$: $$\begin{aligned}
\alpha(2,5)&=\alpha(2,7)=1.\end{aligned}$$ In addition, following Atkin’s method to calculate $\epsilon$ exactly, we observe $$\epsilon=\begin{cases}
1-\lfloor{\log(48)/\log(\ell)}\rfloor=-1 & \ell=5\\
-\lfloor{\log(48)/\log(\ell)}\rfloor=-1 & \ell=7.
\end{cases}$$ Therefore, for the case where $k=2$ and $\ell=5$ or 7, we have that for all $24n\equiv k\pmod{\ell^j}$, $$\label{57}
p_2(n)\equiv 0\pmod{\ell^{\lfloor{j/2-1}\rfloor}}.$$
Serre’s Theorem {#serresthm}
---------------
We make use of the following theorem of Serre’s regarding congruences for certain types of modular forms, stated in [@treneer]:
\[serre\] Suppose that $f(z)=\sum_{n=1}^{\infty}a(n)q^n\in S_k(\Gamma_0(N),\chi)$ has coefficients in the ring of integers $\mathcal{O}_K$ of a number field $K$ and $M$ is a positive integer. Furthermore, suppose that $k\geq1$. Then a positive proportion of the primes $p\equiv -1\pmod{MN}$ have the property that $$f(z)\mid T_{p,k,\chi}\equiv 0\pmod{M}.$$
Serre’s theorem guarantees the existence of congruences for cusp forms with coefficients in the ring of integers of a number field, which we will use to prove properties of the coefficients of the conjugacy growth series for $({\operatorname{Alt}}({\mathbb{N}}),S')$ and $({\operatorname{Sym}}({\mathbb{N}}),S)$.
Proof of Theorem \[cong1\] {#pf}
==========================
We first prove congruences for arbitrary powers of $\ell=5$ or 7. Let $j\geq 1$ and suppose that $24n\equiv 1\pmod{\ell^j}$. Then by Theorem \[ram\], we have that $$\label{P1}
\gamma_{{\operatorname{Sym}}({\mathbb{N}}),S}(n)=p(n)\equiv0\pmod{\ell^{\lfloor{j/2}\rfloor+1}}.$$ Additionally, we have that $24(2n)\equiv 2\pmod{\ell^j}$. Using the case of Theorem \[atkin\] where $k=2$ and $\ell=5$ or 7, as in (\[57\]), we have that $$\label{P2}
p_2(2n)\equiv0\pmod{\ell^{\lfloor{j/2-1}\rfloor}}.$$ Therefore, for all $24n\equiv 1\pmod{\ell^j}$, we obtain from (\[calt2\]), (\[P1\]), and (\[P2\]) that $$\gamma_{{\operatorname{Alt}}({\mathbb{N}}),S'}(2n)\equiv\gamma_{{\operatorname{Sym}}({\mathbb{N}}),S}(n)\equiv0\pmod{\ell^{\lfloor{j/2-1}\rfloor}},$$ as desired.
Congruences for $p_{2}(n)$ {#proof of thm}
==========================
Because of the relationship between the conjugacy growth series for $({\operatorname{Alt}}({\mathbb{N}}),S')$ and the function $p_{2}(n)$, here, we focus on congruences for $p_{2}(n)$. By the definition of the 2-colored partition function $p_{2}(n)$, we have that $$\begin{aligned}
\label{p2}
\sum_{n=0}^{\infty}p_{2}(n)q^n&=\prod_{n=1}^{\infty}\frac{1}{(1-q^n)^2}=\frac{q^{\frac{1}{12}}}{\eta^2(z)}.\end{aligned}$$ Throughout, we let $$\label{f}
f(z):=\frac{1}{\eta(12z)^2}=\sum_{n=-1}^{\infty} a(n)q^n.$$ Then we have that $p_{2}\left(\frac{n+1}{12}\right)=a(n)$. In order to prove congruences between the coefficients of the conjugacy growth series for $({\operatorname{Alt}}({\mathbb{N}}),S')$ and $({\operatorname{Sym}}({\mathbb{N}}),S)$, we first prove a theorem concerning the coefficients of $f(z)$. This makes effective the following result of Treneer [@treneer] by determining the exact value of $m$ that is sufficiently large.
Suppose that $\ell$ is an odd prime and that $k$ and $m$ are integers. Let $N$ be a positive integer with $(N,p)=1$, and let $\chi$ be a Dirichlet character modulo $N$. Let $K$ be an algebraic number field with ring of integers $\mathcal{O}_K$, and suppose $f(z)=a(n)q^n\in M^{!}_{k}(\Gamma_0(N),\chi)\cap\mathcal{O}_K((q))$. If $m$ is sufficiently large, then for each positive integer $j$, a positive proportion of the primes $Q\equiv -1\pmod{N\ell^j}$ have the property that $$a(Q\ell^m n)\equiv 0\pmod{\ell^j}$$ for all $n$ coprime to $Q\ell$.
This section closely follows Section 3 in [@treneer]. Throughout this section, let $f(z)$ be defined by (\[f\]).
\[cong\] Let $\ell\geq 5$ be prime and $j\in{\mathbb{N}}$. Then for a positive proportion of primes $Q\equiv -1\pmod{144\ell^j}$, we have that $$a(Q\ell^{m_{\ell}} n)\equiv 0\pmod{\ell^j}$$ for all $n$ coprime to $Q\ell$.
The proof of Theorem \[cong\] requires the construction of a cusp form that preserves congruence properties of the function $f(z)$.
\[cusp\] For every positive integer $j$, there exists an integer $\beta\geq j-1$ and a cusp form $$g_{\ell,j}(z)\in S_{\kappa}(\Gamma_0(144\ell^2),\chi)\cap{\mathbb{Z}}((q)),$$ where $\kappa:=-1+\frac{\ell^{\beta}(\ell^2-1)}{2}$, with the property that $$g_{\ell,j}(z)\equiv\sum_{n=1}^{\infty}a(\ell^{m_\ell}n)q^n\pmod{\ell^j}.$$
We first require the following proposition concerning the Fourier expansion of $f(z)$ at a given cusp after being acted on by the $U(\ell^m)$ operator for $m\geq 1$.
\[expans\] Let $\gamma:=\begin{pmatrix}
a & b\\
c\ell^2 & d
\end{pmatrix}
\in{\operatorname{SL}}_2({\mathbb{Z}})$ where $c\in{\mathbb{Z}}$ and $ac>0$. Then there exists an integer $n_0\geq -24$ and a sequence $\{a_0(n)\}_{n\geq n_0}$ such that for each $m\geq 1$, we have that $$(f(z)\mid U_{\ell^m})\mid_{-1}\gamma=\sum_{\substack{n=n_0\\n\equiv 0\pmod{\ell^m}}}^{\infty}a_0(n)q_{24\ell^m}^{n},$$ where $q_{24\ell^m}:=e^{\frac{2\pi iz}{24\ell^m}}$.
The proof of this proposition makes use of the following lemma, which relies on the proof of Theorem 1 in [@honda].
\[honda\] Given any matrix $A\in{\operatorname{SL}}_2({\mathbb{Z}})$, we have that $$f(z)\mid_{-1}A=\sum_{n=n_0}^{\infty}a_0(n)q_{24}^{n}$$ where $a_0(n)\in{\mathbb{Z}}$ and $n_0\geq -24$.
Let $A:=
\begin{pmatrix}a & b\\ c& d\end{pmatrix}\in{\operatorname{SL}}_2({\mathbb{Z}})$. Then, as in the proof of Theorem 1 in [@honda], we can write $$\begin{pmatrix}
12 & 0\\
0 & 1
\end{pmatrix}
\begin{pmatrix}
a & b\\
c& d
\end{pmatrix}
=
\begin{pmatrix}
a' & b'\\
c' & d'
\end{pmatrix}
\begin{pmatrix}
\alpha & \beta\\
0 & \delta
\end{pmatrix}$$ where $\begin{pmatrix}
a' & b'\\
c' & d'
\end{pmatrix}\in{\operatorname{SL}}_2({\mathbb{Z}})$, $\alpha,\beta,\delta\in{\mathbb{Z}}$, and $\alpha,\delta>0$. Then we have that $12a=a'\alpha$ and $c=c'\alpha$, so $\alpha=(a'\alpha, c'\alpha)=(12a,c)=(12,c)\leq 12$. Again, by Theorem 1 in [@honda], we obtain $$f(z)\mid_{-1}A=\sum_{n=n_0}^{\infty}a_0(n)q_{24}^{n}$$ where $n_0:=\frac{-2\alpha}{\delta}>-2\alpha\geq -24$.
As in [@treneer], for each $0\leq v\leq \ell^m-1$, choose an integer $s_v$ such that $$s_vN\equiv (a+vc\ell^2)^{-1}(b+vd)\pmod{\ell^m}$$ and define $w_v:=s_vN$. We let $$\alpha_0:=\begin{pmatrix}
a & 0\\
c\ell^{m+2} & d-w_0c\ell^2
\end{pmatrix}.$$ By (\[U\]), we have that $$\label{eq4}
(f(z)\mid U_{\ell^m})\mid_{-1}\gamma=(\ell^m)^{-\frac{3}{2}}\sum_{v=0}^{\ell^m-1}f(z)\mid_{-1}\sigma_{v,\ell^m}\gamma.$$ We observe that $\sigma_{v,\ell^m}\gamma=\alpha_0\sigma_{w_v,\ell^m}$, so we have that $$(f(z)\mid U_{\ell^m})\mid_{-1}\gamma=(\ell^m)^{-\frac{3}{2}}\sum_{v=0}^{\ell^m-1}f(z)\mid_{-1}\alpha_0\sigma_{w_v,\ell^m}.$$ By Lemma \[honda\], we have that $$f(z)\mid_{-1}\alpha_0=\sum_{n=n_0}^{\infty}a_0(n)q_{24}^{n},$$ so we obtain $$\begin{aligned}
\label{eq3}
\sum_{v=0}^{\ell^m-1}f(z)\mid_{-1}\alpha_0\sigma_{w_v,\ell^m}&=\sum_{v=0}^{\ell^m-1}\ell^{\frac{m}{2}}\sum_{n=n_0}^{\infty}a_0(n)e^{\frac{2\pi in(z+w_v)}{24\ell^m}}\\
\nonumber&=\ell^{\frac{m}{2}}\sum_{n=n_0}^{\infty}a_0(n)q_{24\ell^m}^n\sum_{v=0}^{\ell^m-1}e^{\frac{2\pi inw_v}{24\ell^m}}.\end{aligned}$$ By Lemma 3.3 in [@treneer], the numbers $\frac{w_v}{24}$ run through the residue classes modulo $\ell^m$ as $v$ does. Therefore, we have that $$\label{eq2}
\sum_{v=0}^{\ell^m-1}e^{\frac{2\pi inw_v}{24\ell^m}}=\sum_{v=0}^{\ell^m-1}e^{\frac{2\pi inv}{\ell^m}}=\begin{cases}
\ell^m & n\equiv 0\pmod{\ell^m}\\
0 &\text{else.}
\end{cases}$$ Combining (\[eq3\]) and (\[eq2\]), we have that $$\label{eq1}
\sum_{v=0}^{\ell^m-1}f(z)\mid_{-1}\alpha_0\sigma_{w_v,\ell^m}=\ell^{\frac{3}{2}}\sum_{\substack{n=n_0\\n\equiv 0\pmod{\ell^m}}}^{\infty}a_0(n)q_{24\ell^m}^{n}.$$ Using (\[eq4\]) and (\[eq1\]), we obtain $$(f(z)\mid U_{\ell^m})\mid_{-1}\gamma=\sum_{\substack{n=n_0\\n\equiv 0\pmod{\ell^m}}}^{\infty}a_0(n)q_{24\ell^m}^{n},$$ the Fourier expansion of $f(z)\mid U_{\ell^m}$ at the cusp $\frac{a}{c\ell^2}$.
We now construct a weakly holomorphic modular form which vanishes at certain cusps of $\Gamma_0(144\ell^2)$.
\[vanish\] For each nonnegative integer $m$, define $$f_m(z):=f(z)\mid U_{\ell^m}-f(z)\mid U_{\ell^m+1}\mid V_{\ell}\in M_{-1}^{!}(\Gamma_0(144\ell^2),\chi).$$ Then $f_{m_\ell}$ vanishes at each cusp $\frac{a}{c\ell^2}$ of $\Gamma_0(144\ell^2)$ with $ac>0$.
By Proposition \[expans\], we have that $$(f(z)\mid U_{\ell^{m_\ell}})\mid\gamma=\sum_{\substack{n= n_0\\n\equiv 0\pmod{\ell^{m_\ell}}}}^{\infty}a_0(n)q_{24\ell^{m_\ell}}^{n}$$ where $n_0\geq-24$. We now consider two cases. If $5\leq\ell\leq 23$, we have that $$-\ell^{m_\ell}\leq -25<-24\leq n_0,$$ and if $\ell\geq 29$, we have that $$-\ell^{m_\ell}\leq -29<-24\leq n_0.$$ Suppose $a_0(n)\neq 0$. Then $n\geq n_0>-\ell^{m_{\ell}}$, but $n\equiv 0\pmod{\ell^{m_\ell}}$, so $n\geq 0$. Therefore, we obtain $$(f(z)\mid U_{\ell^{m_\ell}})\mid\gamma=\sum_{\substack{n=0\\n\equiv 0\pmod{\ell^{m_{\ell}}}}}^{\infty}a_0(n)q_{24\ell^{m_\ell}}^{n}$$ so $f(z)\mid U_{\ell^{m_{\ell}}}$ is holomorphic at the cusp $\frac{a}{c\ell^2}$.
Now, by Proposition 3.5 in [@treneer], we have that $$f_m(z)\mid_{-1}\gamma=\sum_{\substack{n=0\\n\equiv 0\pmod{\ell^m}}}^{\infty}a_0(n)q_{24\ell^m}^n-\sum_{\substack{n=0\\n\equiv 0\pmod{\ell^{m+1}}}}^{\infty}a_0(n)q_{24\ell^m}^n,$$ so the constant term in each expansion is $a_0(0)$, and they cancel. Therefore, $f_{m_{\ell}}$ vanishes at the cusp $\frac{a}{c\ell^2}$.
We are now ready to prove Proposition \[cusp\].
As in [@treneer], we define the eta-quotient $$F_{\ell}(z):=\frac{\eta^{\ell^2}(z)}{\eta(\ell^2 z)}\in M_{\frac{\ell^2-1}{2}}(\Gamma_0(\ell^2)).$$ By Theorem 1.65 in [@ono], we see that $F_{\ell}$ vanishes at every cusp $\frac{a}{c}$ of $\Gamma_0(144\ell^2)$ with $\ell^2\nmid c$. We also have that $F_{\ell}(z)^{\ell^{s-1}}\equiv 1\pmod{\ell^s}$ for any integer $s\geq 1$.
Now, define $$g_{\ell,j}(z):=f_{m_\ell}(z)\cdot F_{\ell}(z)^{\ell^\beta}$$ where $\beta\geq j-1$ is sufficiently large such that $g_{\ell,j}(z)$ vanishes at all cusps $\frac{a}{c}$ of $\Gamma_0(144\ell^2)$ where $\ell^2\nmid c$. By Theorem 1.65 in [@ono], it is possible to choose such a $\beta$ such that the order of vanishing of $g_{\ell,j}(z)$ is at least one at all such cusps. Then $g_{\ell,j}\in{\mathbb{Z}}((q))$ and $$g_{\ell,j}(z)\equiv f_{m_\ell}(z)\pmod{\ell^j}.$$ By our choice of $\beta$, $g_{\ell,j}(z)$ vanishes at all cusps $\frac{a}{c}$ of $\Gamma_0(144\ell^2)$ where $\ell^2\nmid c$. Furthermore, by Proposition \[vanish\], $g_{\ell,j}(z)$ vanishes at all cusps $\frac{a}{c}$ where $\ell^2\mid c$. Define $\kappa:=-1+\frac{\ell^{\beta}(\ell^2-1)}{2}$. Then we have that $$g_{\ell,j}(z)\in S_{\kappa}(\Gamma_0(144\ell^2),\chi).$$ By definition of $f_{m_{\ell}}$, we obtain $$g_{\ell,j}(z)\equiv \sum_{n=1}^{\infty}a(\ell^{m_\ell}n)q^n-\sum_{n=1}^{\infty}a(\ell^{m_\ell+1}n)q^n\equiv\sum_{\substack{n=1\\\ell\nmid n}}^{\infty}a(\ell^{m_\ell}n)q^n\pmod{\ell^j}.$$ Thus $g_{\ell,j}$ satisfies the conditions of Proposition \[cusp\].
Now that we have constructed the necessary cusp form, we arrive at the proof of Theorem \[cong\].
By Proposition \[cusp\], we can construct a cusp form $g_{\ell,j}\in S_{\kappa}(\Gamma_0(144\ell^2),\chi)\in{\mathbb{Z}}((q))$ such that $$g_{\ell,j}(z)\equiv\sum_{\substack{n=1\\{\ell\nmid n}}}^{\infty}a(\ell^{m_\ell}n)q^n\pmod{\ell^j}.$$ By Theorem \[serre\], for a positive proportion of the primes $Q\equiv -1\pmod{144\ell^{j+2}}$, we have that $$g_{\ell,j}(z)\mid T_{Q,\kappa,\chi}(Q)\equiv 0\pmod{\ell^{j}}.$$ We can then write $g_{\ell,j}(z)=\sum_{n=1}^{\infty}b(n)q^n$ to obtain $$\label{gheck}
g_{\ell,j}(z)\mid T_{Q,\kappa,\chi}=\sum_{n=1}^{\infty}\left(b(Qn)+\chi(Q)Q^{\kappa-1}b(n/Q)\right)q^n\equiv0\pmod{\ell^j}.$$ If $(Q,n)=1$, then the coefficient of $q^n$ in (\[gheck\]) is $b(Qn)$, so $$a(Q\ell^{m_{\ell}}n)\equiv b(Qn)\equiv 0\pmod{\ell^j}$$ for all $n$ coprime to $Q\ell$.
Proof of Theorem \[sym\] {#section4}
========================
We now make use of Theorem \[cong\] to prove congruences between the coefficients of the conjugacy growth series for $({\operatorname{Alt}}({\mathbb{N}}),S')$ and $({\operatorname{Sym}}({\mathbb{N}}),S)$.
By (\[calt2\]), it is enough to show that $p_{2}(2Q\ell^{m_{\ell}}n+2\delta_\ell)\equiv 0\pmod{\ell^j}$. By (\[p2\]) and (\[f\]), we observe $p_{2}\left(\frac{n+1}{12}\right)=a(n)$, so it suffices to prove the existence of congruences for $a(n)$.
By Theorem \[cong\], for a positive proportion of primes $Q\equiv -1\pmod{144\ell^j}$, we have that $$\label{pa}
p_{2}\left(\frac{Q\ell^{m_{\ell}}n+1}{12}\right)=a(Q\ell^{m_{\ell}}n)\equiv 0\pmod{\ell^j}$$ for all $n$ coprime to $Q\ell$. Defining $\delta_{\ell}$ and $\beta_{\ell}$ by (\[delta\_l\]) and \[beta\_l\]), respectively, we can rewrite the left-hand side of equation (\[pa\]) as $$p_{2}\left(2Q\ell^{m_{\ell}}n+2\delta_\ell\right)$$ for all $24n+\beta_{\ell}$ coprime to $Q\ell$. Therefore, for a positive proportion of primes $Q\equiv -1\pmod{144\ell^j}$, we have that $$p_{2}\left(2Q\ell^{m_{\ell}}n+2\delta_\ell\right)\equiv 0\pmod{\ell^j},$$ so we obtain $$2\gamma_{{\operatorname{Alt}}({\mathbb{N}}),S'}(2Q\ell^{m_{\ell}}n+2\delta_\ell)\equiv\gamma_{{\operatorname{Sym}}({\mathbb{N}}),S}(Q\ell^{m_{\ell}}n+\delta_\ell)\pmod{\ell^j},$$ as desired.
|
---
abstract: 'We discuss some constraints on the $x$ and $t$-dependence of $E(x,0,t)$ that arise from positivity bounds in the impact parameter representation. In addition, we show that $E(x,0,0)$ for the nucleon vanishes for $x\rightarrow 1$ at least as rapidly as $(1-x)^4$. Finally we provide an inequality that limits the contribution from $E$ to the angular momentum sum rule.'
author:
- Matthias Burkardt
bibliography:
- 'ineq.bib'
title: 'Some Inequalities for the Generalized Parton Distribution $E(x,0,t)$'
---
Introduction
============
Generalized parton distributions (GPDs) [@m; @ji; @r] are hybrid quantities that have features in common both with form factors and with the usual parton distribution functions (PDFs). They are defined as non-forward ($p\neq p^\prime$) matrix elements of the same operator $\hat{O} \equiv \int \frac{dx^-}{2\pi}e^{ix^-\bar{p}^+x}
\bar{q}\left(-\frac{x^-}{2}\right)
\gamma^+ q\left(\frac{x^-}{2}\right)$ whose forward matrix elements (i.e. expectation value) yield the usual parton distributions \[eq:defHE\] p\^| |p&=&H(x,,\^2)|[u]{}(p\^)\^+ u(p)\
&+& E(x,,\^2)|[u]{}(p\^) u(p) .Over the last few years there has been a strong interest in GPDs and meanwhile many observables have been identified that can be linked to them (for a recent review see Ref. [@d]). One of the most interesting observables that can be linked to GPDs is a quantity that has identified been identified with the total (spin plus orbital) angular momentum carried by the quarks in the nucleon [@ji] \[j\] J\_q= \_0\^1 dx x , where the subscript $q$ indicates that Eq. (\[j\]) holds for each quark flavor separately. Of course, $H_q(x,0,0)=q(x)$ is well known for the relevant values of $x$, but little is known about $E(x,0,0)$.
Although GPDs can be probed in Compton scattering experiments, they usually enter experimentally measurable cross sections only in terms of some integrals and therefore there may be some difficulties in unambiguously extracting GPDs from Compton scattering data. It is therefore desirable to use as many model-independent theoretical constraints as possible to help pin down the data on GPDs.
One class of such constraints are positivity constraints, where one uses the fact that any state in a Hilbert space has a non-negative norm. By using carefully constructed states one can thus derive inequalities relating physical observables.
Positivity Constraints in Impact Parameter Space
================================================
In the case of GPDs, the impact parameter space representation [@r] turns out to be very useful, since GPDs (for $\xi=0$) become diagonal in that basis [@s; @me:1st; @jr; @diehl; @ijmpa]. Parton distributions in impact parameter space are related to GPDs via a simple Fourier transform (throughout this paper, we use a notation where parton distributions in impact parameter space are denoted by script letters) (x,) &=& H(x,0,-)e\^[i]{}\
(x,) &=& (x,0,-)e\^[i]{}\
[E]{}(x,) &=& E(x,0,-)e\^[i]{}.In Ref. [@aussie], it was observed that the probabilistic interpretation of parton distributions in impact parameter space implies the positivity bound | [****]{}\_ [E]{}(x,) | (x,). \[ineq1\] Here $x>0$; for $x<0$ a similar inequality with ${\cal H}\rightarrow
-{\cal H}$ holds. In Ref. [@poby], an even stronger bound | [****]{}\_ [E]{}(x,) |\^2 |[H]{}(x,)|\^2 -|(x,)|\^2 \[ineq2\] was derived. Although Eqs. (\[ineq1\]) and (\[ineq2\]) are rigorous, their practical use has been rather limited so far since they are relations between several unknown quantities. In this paper, we will manipulate these positivity bounds into a form, where they should be more directly applicable to phenomenology. For this purpose we first take Eq. (\[ineq2\]) and simply integrate over impact parameter. Since the inequality is preserved under this operation, and since the norm is invariant under Fourier transformation (e.g. $\int d^2\bT \left|{\cal H}(x,\bT)\right|^2
= \int \frac{d^2\DT}{(2\pi)^2}\left| H(x,0,-\Ds)\right|^2$), one immediately finds \[ineqf\] & & \_[-]{}\^0 dt |[E]{}(x,0,t) |\^2\
& &\_[-]{}\^0 dt{ |[H]{}(x,0,t)|\^2 -|(x,0,t)|\^2}. Similar expressions can be derived by repeating this procedure with additional powers of $\left|\bT\right|$ in the integrand. While this result immediately deals with the GPDs rather than parton distributions in impact parameter space, its usefulness is still limited by the fact that it involves 3 unknown functions and therefore leaves too much room for model dependence. It would be much more useful if we had constraints relating $E(x,0,t)$ to some known functions, such as the forward PDFs $q(x)$ and $\Delta q(x)$. Deriving such relations will be the main goal in the rest of this paper.
For this purpose, we first introduce impact parameter dependent parton distributions for quarks with spins parallel \[${\cal H}_+(x,\bT)$\] and anti-parallel \[${\cal H}_-(x,\bT)$\] to the nucleon spin (longitudinally polarized target) \_(x,) && . In terms of ${\cal H}_\pm $, Eq. (\[ineq2\]) can be expressed in the form | [****]{}\_ [E]{}(x,) | 2. \[ineq3\] Integrating the l.h.s. of Eq. (\[ineq3\]) over the transverse plane yields \[rotational invariance implies ${\cal E}(x,\bT) ={\cal E}(x,b) $\] && d\^2| [****]{}\_ [E]{}(x,) | = \_0\^db b | \_b [E]{}(x,b)|\
&&|\_0\^db [E]{}(x,b)| = | d\^2|\
&&= |d\^2 |\
&&= | \_[-]{}\^0 dt |, \[ineq4\]where we used $\int d^2\bT \frac{e^{-i\bT\cdot\DT}}{
\left|\bT\right|} = \frac{2\pi}{\left|\DT\right|}$.
When integrating the r.h.s. of Eq. (\[ineq3\]), we use the Schwarz inequality to obtain \[ineq5\] & &2d\^2\
& &2\
& &= 2 ,where $q_\pm(x) \equiv \frac{1}{2}\left( q(x)\pm \Delta q(x)\right)$ are the parton distribution for quarks with spin parallel (anti-parallel) to the nucleon spin. Combining Eqs. (\[ineq4\]) and (\[ineq5\]) yields |\_[-]{}\^0 dt | < ,\[ineq6\] which is one of the results of this paper. Like Eqs. (\[ineq1\]) and (\[ineq2\]), this result holds for each quark flavor.
While Eq. (\[ineq6\]) is weaker than our starting point (\[ineq2\]), it may still be of more use at this point because it contains only one unknown quantity \[$E(x,0,t)$\] and relates it to $q_\pm(x)$, which are much better known from parton phenomenology
Although the r.h.s. of Eq. (\[ineq6\]) involves only known quantities, the l.h.s. still involves an integral. For practical applications it may be more useful to have an inequality that contains the unintegrated GPD $E$. For this purpose we now multiply Eq. (\[ineq3\]) by $\left|\bT\right|$ and integrate. For the l.h.s. we find && d\^2|| | [****]{}\_ [E]{}(x,) | = \_0\^db b\^2 | \_b [E]{}(x,b)|\
&&|\_0\^db b[E]{}(x,b)| = | d\^2(x,b)|\
&&= |E(x,0,0)|, \[ineq7\]which involves $E(x,0,0)$, i.e. the quantity entering the angular momentum sum rule [@ji], directly.
On the r.h.s. one can invoke the Schwarz inequality in different ways, and we choose to apply it in the form \[ineq8\] & &2d\^2\
& &2\
& &= 2 , where $H_\pm\equiv \frac{1}{2}\left(H\pm \tilde{H}\right)$. Combining Eqs. (\[ineq3\]),(\[ineq7\]), and (\[ineq8\]) we thus obtain |E(x,0,0)| . \[ineq9\] While we do not know the slope of $H_+(x,0,t)$, we know some general features. In particular, one expects that the transverse width of GPDs vanishes as $x\rightarrow 1$ [@ijmpa] \~(1-x)\^2 x1. \[width\] The reason for the vanishing of the transverse width for $x\rightarrow 1$ is that the variable conjugate to $\DT$ is the impact parameter $\bT$, which is measured w.r.t. the $\perp$ center of momentum. The latter is related to the distance from the active quark to the center of momentum of the spectators (which we denote by ${\bf B_\perp}$ via the relation $\bT = (1-x) {\bf B_\perp}$. The distance $\left|{\bf B_\perp}\right|$ between the active quark and the spectators should be roughly equal to the size of the nucleon or less. Being rescaled by a factor $(1-x)$, the typical scale for $\bT$ is therefore only $
(1-x)$ times that size, which leads to Eq. (\[width\]).
Making use of Eq. (\[width\]) in Eq, (\[ineq9\]) thus yields . H\_+(x,0,t)|\_[t=0]{}\~(1-x)\^[2+n\_+]{} x1 where $n_\pm$ characterizes the behavior of $q_\pm(x)$ for $x\rightarrow 1$ q\_(x) \~(1-x)\^[n\_]{} x1 . For example, if $n_+=3$ and $n_-=5$ (based on hadron helicity conservation [@BBS]), then E(x,0,0) \~(1-x)\^[1+]{}=(1-x)\^5 for $x\rightarrow 1$. Even if there is a small contribution to the negative helicity distribution $q_-(x)$ that vanishes with the same power as the positive helicity distribution $q_+(x)$, i.e. if $n_+=n_-=3$, then $E(x,0,0)$ would still behave like $(1-x)^4$ and therefore vanish faster than $H(x,0,0)$ as $x\rightarrow 1$. In either case we find \_[x1]{} =0.
For applications to the angular momentum sum rule (\[j\]), we can also try to convert Eq. (\[ineq9\]) into a statement about the $2^{nd}$ moment of $E$. Upon multiplying Eq. (\[ineq9\]) by $|x|$ and integrating from $-1$ to $1$ (antiquarks correspond to $x<0$), one finds \[ineq10\] &&|dx E(x,0,0) x|\
&& dx |E(x,0,0)| |x|\
&&dx\
&& , which contains only one unknown on the r.h.s., namely the slope of the second moment of $H_+$. To illustrate that this inequality may provide some useful bounds, let us insert some rough figures: not distinguishing between different flavors (i.e. implicitly adding all quark flavors) we approximate: $\int dx xq_-(x)\approx \int dx xq_+(x) \approx \frac{1}{4}$ and $\int dx^\prime \left.
\frac{d}{dt}x^\prime H_+(x^\prime ,0,t)\right|_{t=0}
= \int dx xq_+(x)\frac{R_+^2}{6}\approx \frac{1}{4}\frac{R_+^2}{6}$, where $R_+^2$ is the rms-radius corresponding to $\int dx x H_+$. We do not know the value of $R_+$ but it should be on the order of the rms radius of the nucleon. In fact, $R_+$ should be smaller than that since the slope of the form factor should decrease for increasing $x$-moments (see the discussion following Eq. (\[width\]) and also Ref. [@n]) , i.e. we approximate $R_+\approx 0.5 fm$. Inserting these rough figures, we find $\left|\int dx E(x,0,0)x\right| \leq \frac{R_+M}{\sqrt{6}}\approx 1$. Although this is not a very strong constraint, a better estimate may be available once the slope of the second moment of $H$ is known for different quark flavors.
Summary
=======
We started from positivity constraints for parton distributions in impact parameter space (\[ineq2\]) and derived several new positivity constraints on GPDs (\[ineqf\]),(\[ineq6\]), (\[ineq9\]), and (\[ineq10\]). Although the new constraints are weaker than the staring inequality (\[ineq2\]), the new inequalities may be more useful since they can be applied directly in momentum space, where the data is obtained. One of the new inequalities (\[ineq6\]) relates $\int \frac{dt}{\sqrt{|t|}} E(x,0,t)$ directly to the (forward) parton distributions $q(x)$ and $\Delta q(x)$ and therefore provides a direct constraint on the shape of $E$.
The third inequality that we derived (\[ineq9\]) is a bound on $E(x,0,0)$. Unfortunately, it involves the slope of $H(x,0,t)$ for $t=0$, which is currently not known.
If hadron helicity conservation (HHC) holds and $q_-(x)\sim (1-x)^5$ for quarks with spin antiparallel to the nucleon spin then $E(x,0,0)\sim (1-x)^5$ as $x\rightarrow 1$ for valence quarks in the nucleon. Even if HHC is violated, and $q_-(x)$ has a small component that vanishes only like $(1-x)^3$ then $E$ still vanishes faster than the leading valence distributions, i.e. $E(x,0,0)\sim (1-x)^4$. This is a consequence of the fact that the positivity constraints in impact parameter representation (\[ineq1\]) and (\[ineq2\]) involve ${\bf \nabla_{\bT}}$ together with the fact that the $\perp$ width of GPDs goes to zero as $x\rightarrow 1$. Knowing that $E(x,0,0)$ vanishes faster than $q(x)$ near be useful in estimating the contribution from $E$ to the angular momentum sum rule.
Finally we derived an inequality that can be used to constrain the second moment of $E(x,0,0)$. The only unknown in this inequality is the rms-radius for the second moment of $H$.
[**Acknowledgements:**]{} This work was supported by the DOE under grant number DE-FG03-95ER40965.
|
---
abstract: |
J.M. Howie, the influential St Andrews semigroupist, claimed that we value an area of pure mathematics to the extent that (a) it gives rise to arguments that are deep and elegant, and (b) it has interesting interconnections with other parts of pure mathematics.
This paper surveys some recent results on the transformation semigroup generated by a permutation group $G$ and a single non-permutation $a$. Our particular concern is the influence that properties of $G$ (related to homogeneity, transitivity and primitivity) have on the structure of the semigroup. In the first part of the paper, we consider properties of $S=\langle G,a\rangle$ such as regularity and idempotent generation. The second is a brief report on the synchronization project, which aims to decide in what circumstances $S$ contains an element of rank $1$. The paper closes with a list of open problems on permutation groups and linear groups, and some comments about the impact on semigroups are provided.
These two research directions outlined above lead to very interesting and challenging problems on primitive permutation groups whose solutions require combining results from several different areas of mathematics, certainly fulfilling both of Howie’s elegance and value tests in a new and fascinating way.
---
[PERMUTATION GROUPS AND TRANSFORMATION SEMIGROUPS: RESULTS AND PROBLEMS]{}[JOÃO ARAÚJO$^{\ast}$ and PETER J. CAMERON$^{\dagger}$]{}[$^{\ast}$Universidade Aberta and Centro de Algebra, Universidade de Lisboa, Av.Gama Pinto 2, 1649-003 Lisboa, PortugalEmail: [email protected]\
$^{\dagger}$Mathematical Institute, University of St Andrews, North Haugh, St Andrews, Fife, KY16 9SS, U.K.Email: [email protected]]{}
Regularity and generation
=========================
Introduction
------------
How can group theory help the study of semigroups?
If a semigroup has a large group of units, we can apply group theory to it. But there may not be any units at all! According to a widespread belief, almost all finite semigroups have only one idempotent, which is a zero, not an identity (see [@KRS] and [@JMS]). This conjecture, however, should not deter us from the general goal of investigating how the group of units shapes the structure of the semigroup. Infinitely many families of finite semigroups, and the most interesting, are composed by semigroups with group of units. Some of those families are interesting enough to keep many mathematicians busy their entire lives; in fact a unique family of finite semigroups, the endomorphism semigroups of vector spaces over finite fields, has been keeping experts in linear algebra busy for more than a century.
Regarding the general question of how the group of units can shape the structure of the semigroup, an especially promising area is the theory of *transformation semigroups*, that is, semigroups of mappings $\Omega\to\Omega$ (subsemigroups of the *full transformation semigroup* $T(\Omega)$, where $\Omega:=\{1,\ldots ,n\}$). This area is especially promising for two reasons. First, in a transformation semigroup $S$, the units are the permutations; if there are any, they form a *permutation group* $G$ and we can take advantage of the very deep recent results on them, chiefly the classification of finite simple groups (CFSG). Secondly, even if there are no units, we still have a group to play with, the *normaliser* of $S$ in ${\mathop{\mathrm{Sym}}}(\Omega)$, the set of all permutations $g$ such that $g^{-1}Sg=S$.
The following result of Levi and McFadden [@lm] is the prototype for results of this kind. Let $S_n$ and $T_n$ denote the symmetric group and full transformation semigroup on $\Omega:=\{1,2,\ldots,n\}$.
Let $a\in T_n\setminus S_n$, and let $S$ be the semigroup generated by the conjugates $g^{-1}ag$ for $g\in S_n$. Then
1. $S$ is idempotent-generated;
2. $S$ is regular;
3. $S=\langle a,S_n\rangle\setminus S_n$.
In other words, semigroups of this form, with normaliser $S_n$, have *very nice* properties!
Inspired by this result, we could formulate a general problem:
\[problem1\]
1. Given a semigroup property P, for which pairs $(a,G)$, with $a\in T_n\setminus S_n$ and $G\le S_n$, does the semigroup $\langle g^{-1}ag:g\in G\rangle$ have property P?
2. Given a semigroup property P, for which pairs $(a,G)$ as above does the semigroup $\langle a,G\rangle\setminus G$ have property P?
3. For which pairs $(a,G)$ are the semigroups of the preceding parts equal?
The following portmanteau theorem lists some previously known results on this problem. The first part is due to Levi [@levi96], the other two to Araújo, Mitchell and Schneider [@ArMiSc].
1. For any $a\in T_n\setminus S_n$ the semigroups $\langle g^{-1}ag:g\in S_n\rangle$ and $\langle g^{-1}ag:g\in A_n\rangle$ are equal.
2. $\langle g^{-1}ag:g\in G\rangle$ is idempotent-generated for all $a\in T_n\setminus S_n$ if and only if $G=S_n$ or $G=A_n$ or $G$ is one of three specific groups of low degrees.
3. $\langle g^{-1}ag:g\in G\rangle$ is regular for all $a\in T_n\setminus S_n$ if and only if $G=S_n$ or $G=A_n$ or $G$ is one of eight specific groups of low degrees.
Recently, we have obtained several extensions of these results. The first theorem is proved in [@ArCa].
\[arca\] Given $k$ with $1\le k\le n/2$, the following are equivalent for a subgroup $G$ of $S_n$:
1. for all rank $k$ transformations $a$, $a$ is regular in $\langle a,G\rangle$;
2. for all rank $k$ transformations $a$, $\langle a,G\rangle$ is regular;
3. for all rank $k$ transformations $a$, $a$ is regular in $\langle g^{-1}ag:g\in G\rangle$;
4. for all rank $k$ transformations $a$, $\langle g^{-1}ag:g\in G\rangle$ is regular.
Moreover, we have a complete list of the possible groups $G$ with these properties for $k\ge5$, and partial results for smaller values.
It is worth pointing out that in the previous theorem the equivalence between (a) and (c) is not new (it appears in [@lmm]). Really surprising, and a great result that semigroups owe to the classification of finite simple groups, are the equivalences between (a) and (b), and between (c) and (d).
The four equivalent properties above translate into a transitivity property of $G$ which we call the *$k$-universal transversal property*, which we will describe in the Subsection \[kut\].
In the framework of Problem \[problem1\], let P be the following property: the pair $(a,G)$, with $a\in T_n\setminus S_n$ and $G\le S_n$, satisfies $\langle a,G\rangle \setminus G = \langle a,S_n\rangle \setminus S_n$.
The classification of the pairs $(a,G)$ with this property poses a very interesting group theoretical problem. Recall that the rank of a map $a\in T_n$ is $|\Omega a|$ and the kernel of $a$ is $\ker(a):=\{(x,y)\in \Omega^2 \mid xa=ya\}$; by the usual correspondence between equivalences and partitions, we can identify $\ker(a)$ with a partition $\{A_1,\ldots , A_k\}$. Suppose $|\Omega|>2$ and we have a rank $2$ map $a\in T_n$. It is clear that $g^{-1}a\in \langle a,S_n\rangle$, for all $g\in S_n$. In addition, if $\ker(a)=\{A_1,A_2\}$, then $\ker(g^{-1}a)=\{A_1g,A_2g\}$. Therefore, in order to classify the groups with property P above we need to find the groups $G$ such that $$\begin{aligned}
\label{lambda}
\{\{A_1,A_2\}g\mid g\in G\}&=&\{\{A_1,A_2\}g\mid g\in S_n\}.\end{aligned}$$ If $|A_1|<|A_2|$, this is just $|A_1|$-homogeneity; but if these two sets have the same size, the property is a little more subtle.
Extending this analysis to partitions with more than two parts, we see that the group-theoretic properties we need to investigate are transitivity on ordered partitions of given shape (this notion was introduced by Martin and Sagan [@MS] under the name *partition-transitivity*) and the weaker notion of transitivity on unordered partitions of given shape. This is done in Section \[parthomog\], where we indicate the proof of the following theorem from [@AnArCa].
We have a complete list (in terms of the rank and kernel type of $a$) for pairs $(a,G)$ for which $\langle a,G\rangle\setminus G=
\langle a,S_n\rangle\setminus S_n$.
As we saw, the semigroups $\langle a,S_n\rangle\setminus S_n$ have very nice properties. In particular, the questions of calculating their automorphisms and congruences, checking for regularity, idempotent generation, etc., are all settled. Therefore the same happens for the groups $G$ and maps $a\in T_n\setminus S_n$ such that $\langle a,G\rangle\setminus G=
\langle a,S_n\rangle\setminus S_n$, and all these pairs $(a,G)$ have been classified.
Another long-standing open question was settled by the following theorem, from [@ArCaMiNe].
\[third\] The semigroups $\langle a,G\rangle\setminus G$ and $\langle g^{-1}ag:g\in G\rangle$ are equal for all $a\in T_n\setminus S_n$ if and only if $G=S_n$, or $G=A_n$, or $G$ is the trivial group, or $G$ is one of five specific groups.
It would be good to have a more refined version of this where the hypothesis refers only to all maps of rank $k$, or just a single map $a$.
Homogeneity and related properties {#homogeneity}
----------------------------------
A permutation group $G$ on $\Omega$ is *$k$-homogeneous* if it acts transitively on the set of $k$-element subsets of $\Omega$, and is *$k$-transitive* if it acts transitively on the set of $k$-tuples of distinct elements of $\Omega$.
It is clear that $k$-homogeneity is equivalent to $(n-k)$-homogeneity, where $|\Omega|=n$; so we may assume that $k\le n/2$. It is also clear that $k$-transitivity implies $k$-homogeneity.
We say that $G$ is *set-transitive* if it is $k$-homogeneous for all $k$ with $0\le k\le n$. The problem of determining the set-transitive groups was posed by von Neumann and Morgenstern [@vNM] in the first edition of their influential book on game theory. In the second edition, they refer to an unpublished solution by Chevalley, but the first published solution was by Beaumont and Peterson [@BP]. The set-transitive groups are the symmetric and alternating groups, and four small exceptions with degrees $5,6,9,9$.
In an elegant paper in 1965, Livingstone and Wagner [@lw] showed:
\[lwthm\] Let $G$ be $k$-homogeneous, where $2\le k\le n/2$. Then
1. $G$ is $(k-1)$-homogeneous;
2. $G$ is $(k-1)$-transitive;
3. if $k\ge5$, then $G$ is $k$-transitive.
In particular, part (a) of this theorem is proved by a short argument using character theory of the symmetric group. This can be translated into combinatorics, and generalised to linear and affine groups: see Kantor [@kantor:inc].
The $k$-homogeneous but not $k$-transitive groups for $k=2,3,4$ were determined by Kantor [@kantor:4homog; @kantor:2homog]. All this was pre-CFSG.
The $k$-transitive groups for $k>1$ are known, but the classification uses CFSG. Lists can be found in various references such as [@cam; @dixon].
The $k$-universal transversal property {#kut}
--------------------------------------
Let $G\le S_n$, and $k$ an integer smaller than $n$.
The group $G$ has the *$k$-universal transversal property*, or *$k$-ut* for short, if for every $k$-element subset $S$ of $\{1,\ldots,n\}$ and every $k$-part partition $P$ of $\{1,\ldots,n\}$, there exists $g\in G$ such that $Sg$ is a *transversal* or *section* for $P$: that is, each part of $P$ intersects $Sg$ in a single point.
For $k\le n/2$, the following are equivalent for a permutation group $G\le S_n$:
1. for all $a\in T_n\setminus S_n$ with rank $k$, $a$ is regular in $\langle a,G\rangle$;
2. $G$ has the $k$-universal transversal property.
In order to get the surprising equivalence (noted after Theorem \[arca\]) of “$a$ is regular in $\langle a,G\rangle$” and $``\langle a,G\rangle$ is regular”, we need to know that, for $k\le n/2$, a group with the $k$-ut property also has the $(k-1)$-ut property. This fact, the analogue of Theorem \[lwthm\](a), is not at all obvious.
We go by way of a related property: $G$ is *$(k-1,k)$-homogeneous* if, given any two subsets $A$ and $B$ of $\{1,\ldots,n\}$ with $|A|=k-1$ and $|B|=k$, there exists $g\in G$ with $Ag\subseteq B$.
Now the $k$-ut property implies $(k-1,k)$-homogeneity. (Take a partition with $k$ parts, the singletons contained in $A$ and all the rest. If $Bg$ is a transversal for this partition, then $Bg\supseteq A$, so $Ag^{-1}\subseteq B$.)
The bulk of the argument involves these groups. We show that, if $3\le k\le(n-1)/2$ and $G$ is $(k-1,k)$-homogeneous, then either $G$ is $(k-1)$-homogeneous, or $G$ is one of four small exceptions (with $k=3,4,5$ and $n=2k-1$).
It is not too hard to show that such a group $G$ must be transitive, and then primitive. Now careful consideration of the orbital graphs shows that $G$ must be $2$-homogeneous, at which point we invoke the classification of $2$-homogeneous groups (a consequence of CFSG).
One simple observation: if $G$ is $(k-1,k)$-homogeneous but not $(k-1)$-homogeneous of degree $n$, then colour one $G$-orbit of $(k-1)$-sets red and the others blue; by assumption, there is no monochromatic $k$-set, so $n$ is bounded by the Ramsey number $R(k-1,k,2)$. The values $R(2,3,2)=6$ and $R(3,4,2)=13$ are useful here; $R(4,5,2)$ is unknown, and in any case too large for our purposes.
Now we return to considering the $k$-ut property.
First, we note that the $2$-ut property says that every orbit on pairs contains a pair crossing between parts of every $2$-partition; that is, every orbital graph is connected. By Higman’s Theorem, this is equivalent to primitivity.
For $2<k<n/2$, we know that the $k$-ut property lies between $(k-1)$-homogeneity and $k$-homogeneity, with a few small exceptions. In fact $k$-ut is equivalent to $k$-homogeneous for $k\ge6$; we classify all the exceptions for $k=5$, but for $k=3$ and $k=4$ there are some groups we are unable to resolve (affine, projective and Suzuki groups), which pose interesting problems (see Problems \[p:ut1\] and \[p:ut2\]).
For large $k$ we have:
\[1.10\] For $n/2<k<n$, the following are equivalent:
1. $G$ has the $k$-universal transversal property;
2. $G$ is $(k-1,k)$-homogeneous;
3. $G$ is $k$-homogeneous.
In the spirit of Livingstone and Wagner, we could ask:
Without using CFSG, show any or all of the following implications:
1. $k$-ut implies $(k-1)$-ut for $k\le n/2$;
2. $(k-1,k)$-homogeneous implies $(k-2,k-1)$-homogeneous for $k\le n/2$;
3. $k$-ut (or $(k-1,k)$-homogeneous) implies $(k-1)$-homogeneous for $k\le n/2$.
Partition transitivity and homogeneity {#parthomog}
--------------------------------------
Let $\lambda$ be a partition of $n$ (a non-increasing sequence of positive integers with sum $n$). A partition of $\{1,\ldots,n\}$ is said to have *shape* $\lambda$ if the size of the $i$th part is the $i$th part of $\lambda$.
The group $G$ is *$\lambda$-transitive* if, given any two (ordered) partitions of shape $\lambda$, there is an element of $G$ mapping each part of the first to the corresponding part of the second. (This notion is due to Martin and Sagan [@MS].) Moreover, $G$ is *$\lambda$-homogeneous* if there is an element of $G$ mapping the first partition to the second (but not necessarily respecting the order of the parts).
Of course $\lambda$-transitivity implies $\lambda$-homogeneity, and the converse is true if all parts of $\lambda$ are distinct. If $\lambda=(n-t,1,\ldots,1)$, then $\lambda$-transitivity and $\lambda$-homogeneity are equivalent to $t$-transitivity and $t$-homogeneity.
The connection with semigroups is given by the next result, from [@AnArCa]. Let $G$ be a permutation group, and $a\in T_n\setminus S_n$, where $r$ is the rank of $a$, and $\lambda$ the shape of the kernel partition.
For $G\le S_n$ and $a\in T_n\setminus S_n$, the following are equivalent:
1. $\langle a,G\rangle\setminus G=\langle a,S_n\rangle\setminus S_n$;
2. $G$ is $r$-homogeneous and $\lambda$-homogeneous.
So we need to know the $\lambda$-homogeneous groups. First, we consider $\lambda$-transitive groups.
If the largest part of $\lambda$ is greater than $n/2$ (say $n-t$, where $t<n/2$), then $G$ is $\lambda$-transitive if and only if it is $t$-homogeneous and the group $H$ induced on a $t$-set by its setwise stabiliser is $\lambda'$-transitive, where $\lambda'$ is $\lambda$ with the part $n-t$ removed.
So if $G$ is $t$-transitive, then it is $\lambda$-transitive for all such $\lambda$.
If $G$ is $t$-homogeneous but not $t$-transitive, then $t\le 4$, and examination of the groups in Kantor’s list gives the possible $\lambda'$ in each case.
So what remains is to show that, if $G$ is $\lambda$-transitive but not $S_n$ or $A_n$, then $\lambda$ must have a part greater than $n/2$.
If $\lambda\ne(n),(n-1,1)$, then $G$ is primitive.
If $n\ge8$, then by *Bertrand’s Postulate*, there is a prime $p$ with $n/2< p\le n-3$. If there is no part of $\lambda$ which is at least $p$, then the number of partitions of shape $\lambda$ (and hence the order of $G$) is divisible by $p$. A theorem of Jordan (see Wielandt [@wie], Theorem 13.9) now shows that $G$ is symmetric or alternating.
The classification of $\lambda$-homogeneous but not $\lambda$-transitive groups is a bit harder. We have to use
1. a little character theory to show that either $G$ fixes a point and is transitive on the rest, or $G$ is transitive;
2. the argument using Bertrand’s postulate and Jordan’s theorem as before;
3. CFSG (to show that $G$ cannot be more than $5$-homogeneous if it is not $S_n$ or $A_n$).
The outcome is a complete list of such groups.
Normalising groups
------------------
We define a permutation group $G$ to be *normalising* if $\langle g^{-1}ag:g\in G\rangle=\langle a,G\rangle\setminus G$ for all $a\in T_n\setminus S_n$.
The classification of normalising groups given by Theorem \[third\] is a little different; although permutation group techniques are essential in the proof, we didn’t find a simple combinatorial condition on $G$ which is equivalent to this property. We will not discuss it further here.
Synchronization
===============
Introduction
------------
In this section, we give a brief report on synchronization.
A (finite deterministic) *automaton* consists of a finite set $\Omega$ of *states* and a finite set of maps from $\Omega$ to $\Omega$ called *transitions*, which may be composed freely.
In other words, it is a transformation semigroup with a distinguished set of generators.
An automaton is *synchronizing* if there is a map of rank $1$ (image of size $1$) in the semigroup. A word in the generators expressing such a map is called a *reset word*.
We will also call a transformation semigroup *synchronizing* if it contains an element of rank 1.
This example has four (numbered) states, and two transitions $A$ and $B$, shown as double and single lines respectively.
(50,40) (25,10)(0,30)[2]{} (10,25)(30,0)[2]{} (24,35)[$1$]{} (13,24)[$2$]{} (24,13)[$3$]{} (35,24)[$4$]{} (24.8,10.2)(-15,15)[2]{}[(1,1)[15]{}]{} (25.2,9.8)(-15,15)[2]{}[(1,1)[15]{}]{} (24.8,9.8)(15,15)[2]{}[(-1,1)[15]{}]{} (25.2,10.2)(15,15)[2]{}[(-1,1)[15]{}]{} (18,32)[$\swarrow$]{} (16,18)[$\searrow$]{} (30,18)[$\nearrow$]{} (28,32)[$\nwarrow$]{} (10,25,16,34,25,40) (14,36)[$\swarrow$]{} (10,25,5,23.5,3.5,25,5,26.5,10,25) (25,10,23.5,5,25,3.5,26.5,5,25,10) (40,25,45,23.5,46.5,25,45,26.5,40,25)
The reader can check easily that, irrespective of the starting state, following the path $BAAABAAAB$ always ends in state $2$, and hence this is a reset word of length $9$. In fact, this is the shortest reset word.
The *Černý Conjecture* asserts that if an $n$-state automaton is synchronizing, then it has a reset word of length at most $(n-1)^2$. The above example, with the square replaced by an $n$-gon, shows that this would be best possible. The problem has been open for about 45 years. The best known bound is cubic.
It is known that testing whether an automaton is synchronizing is in , but finding the length of the shortest reset word is -hard.
Graph homomorphisms and transformation semigroups
-------------------------------------------------
All graphs here are undirected simple graphs (no loops or multiple edges).
A *homomorphism* from a graph $X$ to a graph $Y$ is a map $f$ from the vertex set of $X$ to the vertex set of $Y$ which carries edges to edges. (We don’t specify what happens to a non-edge; it may map to a non-edge, or to an edge, or collapse to a vertex.) An *endomorphism* of a graph $X$ is a homomorphism from $X$ to itself.
Let $K_r$ be the complete graph with $r$ vertices. The *clique number* $\omega(X)$ of $X$ is the size of the largest complete subgraph, and the *chromatic number* $\chi(X)$ is the least number of colours required for a proper colouring of the vertices (adjacent vertices getting different colours).
1. There is a homomorphism from $K_r$ to $X$ if and only if $\omega(X)\ge r$.
2. There is a homomorphism from $X$ to $K_r$ if and only if $\chi(X)\le r$.
There are correspondences in both directions between graphs and transformation semigroups (not quite functorial, or a Galois correspondence, sadly!)
First, any graph $X$ has an *endomorphism semigroup* ${\mathop{\mathrm{End}}}(X)$.
In the other direction, given a transformation semigroup $S$ on $\Omega$, its *graph* ${\mathop{\mathrm{Gr}}}(S)$ has $\Omega$ as vertex set, two vertices $v$ and $w$ being joined if and only if there is no element of $S$ which maps $v$ and $w$ to the same place.
1. ${\mathop{\mathrm{Gr}}}(S)$ is complete if and only if $S\le S_n$;
2. ${\mathop{\mathrm{Gr}}}(S)$ is null if and only if $S$ is synchronizing;
3. $S\le{\mathop{\mathrm{End}}}({\mathop{\mathrm{Gr}}}(S))$ for any $S\le T_n$;
4. $\omega({\mathop{\mathrm{Gr}}}(S))=\chi({\mathop{\mathrm{Gr}}}(S))$; this is equal to the minimum rank of an element of $S$.
Now the main theorem of this section describes the unique obstruction to synchronization for a transformation semigroup.
A transformation semigroup $S$ on $\Omega$ is non-synchronizing if and only if there is a non-null graph $X$ on the vertex set $\Omega$ with $\omega(X)=\chi(X)$ and $S\le{\mathop{\mathrm{End}}}(X)$.
In the reverse direction, the endomorphism semigroup of a non-null graph cannot be synchronizing, since edges can’t be collapsed. In the forward direction, take $X={\mathop{\mathrm{Gr}}}(S)$; there is some straightforward verification to do. (For details see [@ArCa2].)
Maps synchronized by groups
---------------------------
Let $G\le S_n$ and $a\in T_n\setminus S_n$. We say that $G$ *synchronizes* $a$ if $\langle a,G\rangle$ is synchronizing.
By abuse of language, we say that $G$ is *synchronizing* if it synchronizes every element of $T_n\setminus S_n$.
Our main problem is to determine the synchronizing groups. From the theorem, we see that $G$ is non-synchronizing if and only if there is a $G$-invariant graph whose clique number and chromatic number are equal.
Rystsov [@rystsov] showed the following result, which implies that synchronizing groups are necessarily primitive.
A permutation group $G$ of degree $n$ is primitive if and only if it synchronizes every map of rank $n-1$.
We give a brief sketch of the proof, to illustrate the graph endomorphism technique. The backward implication is trivial; so suppose, for a contradiction, that $G$ is primitive but fails to synchronize the map $a$ of rank $n-1$. Then there are two points $x,y$ with $xa=ya$, and $a$ is bijective on the remaining points. Choose a graph $X$ with $\langle G,a\rangle\le{\mathop{\mathrm{End}}}(X)$. Note that $X$ is a regular graph. Since $a$ is an endomorphism, $x$ and $y$ are non-adjacent; so $a$ maps the neighbours of $x$ bijectively to the neighbours of $xa$, and similarly the neighbours of $y$ to those of $ya$. Since $xa=ya$, we see that $x$ and $y$ have the same neighbour set. Now “same neighbour set” is an equivalence relation preserved by $G$, contradicting primitivity.
So a synchronizing group must be primitive.
We have recently improved this: a primitive group synchronizes every map of rank $n-2$. The key tool in the proof is graph endomorphisms. Also, a primitive group synchronizes every map of kernel type $(k,1,\ldots,1)$. For both results, and further information, see [@ArCa2].
Also, $G$ is synchronizing if and only if there is no $G$-invariant graph, not complete or null, with clique number equal to chromatic number. For more on this see [@ArnoldSteinberg; @CK; @neumann; @JEP; @rystsov; @Tr; @Tr07]. Thus, a $2$-homogeneous group is synchronizing, and a synchronizing group is primitive. For if $G$ is $2$-transitive, the only $G$-invariant graphs are complete or null; and if $G$ is imprimitive, then it preserves a complete multipartite graph.
Furthermore, a synchronizing group is *basic* in the O’Nan–Scott classification, that is, not contained in a wreath product with the product action. (For non-basic primitive groups preserve Hamming graphs, which have clique number equal to chromatic number.) By the O’Nan–Scott Theorem, such a group is affine, diagonal or almost simple.
None of the above implications reverses. Indeed, there are non-synchronizing basic groups of all three O’Nan–Scott types.
We are a long way from a classification of synchronizing groups. The attempts to classify them lead to some interesting and difficult problems in extremal combinatorics, finite geometry, computation, etc. But that is another survey paper! We content ourselves here with a single result about an important class of primitive groups, namely the classical symplectic, orthogonal and unitary groups, acting on their associated polar spaces. The implicit geometric problem has not been completely solved, despite decades of work by finite geometers. We refer to Thas [@jat] for a survey.
A classical group, acting on the points of its associated polar space, is non-synchronizing if and only if the polar space possesses either an ovoid and a spread, or a partition into ovoids.
A conjecture
------------
We regard the following as the biggest open problem in the area. A map $a\in T_n$ is *non-uniform* if its kernel classes are not all of the same size.
A primitive permutation group synchronizes every non-uniform map.
We have some partial results about this (see [@ABC; @ArCa2]) but are far from a proof!
Problems {#spro}
========
One of the goals of this paper is to provide a list of problems that might help the interested reader involve himself in this fascinating topic. In addition to the problems included above, we collect here a number of problems on the general interplay between properties of the group of units and properties of the semigroup containing it.
We start by proposing a problem to experts in number theory. If this problem can be solved, the results on ${\mathop{\mathrm{AGL}}}(1,p)$, in [@ArCa], will be dramatically sharpened.
\[p:ut1\] Classify the prime numbers $p$ congruent to $11$ (mod $12$) such that for some $c\in {\mathop{\mathrm{GF}}}(p)^*$ we have $|\langle -1,c,c-1\rangle|<p-1$.
The primes less than $500$ with this property are $131$, $191$, $239$, $251$, $311$, $419$, $431$, and $491$.
\[p:ut2\] Do the Suzuki groups ${\mathop{\mathrm{Sz}}}(q)$ have the $3$-universal transversal property?
Classify the groups $G$ that have the $4$-ut property, when ${\mathop{\mathrm{PSL}}}(2,q) \le G \le{\mathop{\mathrm{P\Gamma L}}}(2,q)$, with either $q$ prime (except ${\mathop{\mathrm{PSL}}}(2,q)$ for $q\equiv1$ (mod $4$), which is not $3$-homogeneous), or $q=2^p$ for $p$ prime.
A group $G\leq S_n$ has the $(n-1)$-universal transversal property if and only if it is transitive. And $\langle a,G\rangle$ (for a rank $n-1$ map $a$) contains all the rank $n-1$ maps of $T_n$ if and only if $G$ is $2$-homogeneous. In this last case $\langle a,G\rangle$ is regular for all $a\in T_n$, because $\langle a,G\rangle=\{b\in T_n\mid |\Omega b|\leq n-1\}\cup G$, and this semigroup is well known to be regular.
Classify the groups $G\leq S_n$ such that $G$ together with any rank $n-k$ map, where $k\leq 5$, generate a regular semigroup. We already know that such $G$ must be $k$-homogeneous; so we know which groups to look at (see Theorem \[1.10\]).
The difficulty here (when rank $k>\lfloor \frac{n+1}{2}\rfloor$) is that a $k$-homogenous group is not necessarily $(k-1)$-homogenous. Therefore a rank $k$ map $a\in T_n$ might be regular in $\langle a, G\rangle$, but we are not sure that there exists $g\in G$ such that ${\mathop{\mathrm{rank}}}(bgb)={\mathop{\mathrm{rank}}}(b)$, for $b\in \langle a, G\rangle$ such that ${\mathop{\mathrm{rank}}}(b)<{\mathop{\mathrm{rank}}}(a)$.
It is clear that if $\langle a ,G\rangle\setminus G$ is idempotent generated, for all rank $k$ transformation $a\in T_n\setminus S_n$, then $G$ has the $k$-ut property (see [@ArMiSc]).
Classify the groups $G\leq S_n$ such that $\langle a ,G\rangle\setminus G$ is idempotent generated, for all rank $k$ maps, where $k\leq n/2$. Even if the classification of the groups with the $k$-ut property is only almost finished (Problem \[p:ut2\] is the missing part), it might be possible to settle the idempotent generation problem.
\[top\] The most general problem that has to be handled is the classification of pairs $(a,G)$, where $a\in T_n$ and $G\leq S_n$, such that $\langle a,G\rangle$ is a regular semigroup.
When investigating $(k-1)$-homogenous groups without the $k$-universal transversal property ($k$-ut property), it was common that some of the orbits on the $k$-sets have transversals for all the partitions. Therefore the following definition is natural.
A group $G\leq S_n$ is said to have the weak $k$-ut property if there exists a $k$-set $S\subseteq \Omega$ such that the orbit of $S$ under $G$ contains a section for all $k$-partitions. Such a set is called a $G$-universal transversal set. A solution to the following problem would have important consequences in semigroup theory.
\[five\] Classify the groups with the weak $k$-ut property; in addition, for each of them, classify their $G$-universal transversal sets.
In McAlister’s celebrated paper [@mcalister] it is proved that, if $e^2=e\in T_n$ is a rank $n-1$ idempotent, then $\langle G,e\rangle$ is regular for all groups $G\leq S_n$. In addition, assuming that $\{\alpha,\beta\}$ is the non-singleton kernel class of $e$ and $\alpha e=\beta$, if $\alpha$ and $\beta$ are not in the same orbit under $G$, then $\langle e,G\rangle$ is an orthodox semigroup (that is, the idempotents form a subsemigroup); and $\langle e,G\rangle$ is inverse if and only if $\alpha$ and $\beta$ are not in the same orbit under $G$ and the stabilizer of $\alpha$ is contained in the stabilizer of $\beta$.
Classify the groups $G\leq S_n$ that together with any idempotent \[rank $k$ idempotent\] generate a regular \[orthodox, inverse\] semigroup.
Classify the pairs $(G,a)$, with $a\in T_n$ and $G\leq S_n$, such that $\langle e,G\rangle$ is inverse \[orthodox\].
The theorems and problems in this paper admit linear versions that are interesting for experts in groups and semigroups, but also to experts in linear algebra and matrix theory.
Prove (or disprove) that if $G\leq {\mathop{\mathrm{GL}}}(n,q)$ such that for all singular matrix $a$ there exists $g\in G$ with ${\mathop{\mathrm{rank}}}(a)={\mathop{\mathrm{rank}}}(aga)$, then $G$ contains the special linear group.
For $n=2$ and for $n=3$, this condition is equivalent to irreducibility of $G$. But we conjecture that, for sufficiently large $n$, it implies that $G$ contains the special linear group.
Classify the groups $G\leq {\mathop{\mathrm{GL}}}(n,q)$ such that for all rank $k$ (for a given $k$) singular matrix $a$ we have that $a$ is regular in $\langle G,a\rangle$ \[the semigroup $\langle G,a\rangle$ is regular\].
To handle this problem it is useful to keep in mind the following results. Kantor [@kantor:inc] proved that if a subgroup of ${\mathop{\mathrm{P\Gamma L}}}(d,q)$ acts transitively on $k$-dimensional subspaces, then it acts transitively on $l$-dimensional subspaces for all $l\le k$ such that $k+l\le n$; in [@kantor:line], he showed that subgroups transitive on $2$-dimensional subspaces are $2$-transitive on the $1$-dimensional subspaces with the single exception of a subgroup of ${\mathop{\mathrm{PGL}}}(5,2)$ of order $31\cdot5$; and, with the second author [@cameron-kantor], he showed that such groups must contain ${\mathop{\mathrm{PSL}}}(d,q)$ with the single exception of the alternating group $A_7$ inside ${\mathop{\mathrm{PGL}}}(4,2)\cong A_8$. Also Hering [@He74; @He85] and Liebeck [@Li86], using CFSG, classified the subgroups of ${\mathop{\mathrm{PGL}}}(d,p)$ which are transitive on $1$-spaces.
Regarding synchronization, the most important question (in our opinion) is the following conjecture, stated earlier.
Is it true that every primitive group of permutations of a finite set $\Omega$ synchronizes every non-uniform transformation on $\Omega$?
Assuming the previous question has an affirmative answer (as we believe), an intermediate step in order to prove it would be to solve the following set of connected problems:
1. Prove that every map of rank $n-3$, with non-uniform kernel, is synchronized by a primitive group. This is known for idempotent maps (see [@ArCa2]).
2. Prove that a primitive group synchronizes every non-uniform map of rank $5$.
3. Prove that if in $S=\langle f,G\rangle$ there is a map of minimal rank $r>1$, there can be no map in $S$ with rank $r+2$.
The next class of groups lies strictly between primitive and synchronizing.
Is it possible to classify the primitive groups which synchronize every rank $3$ map?
Note that there are primitive groups that do not synchronize a rank $3$ map (see [@neumann]). And there are non-synchronizing groups which synchronize every rank $3$ map. Take for example ${\mathop{\mathrm{PGL}}}(2,7)$ of degree $28$; this group is non-synchronizing, but synchronizes every rank $3$ map, since $28$ is not divisible by $3$.
There are very fast polynomial-time algorithms to decide if a given set of permutations generates a primitive group, or a $2$-transitive group.
Find an efficient algorithm to decide if a given set of permutations generates a synchronizing group.
It would be quite remarkable if such an algorithm exists; as we saw, it would in particular resolve questions about ovoids and spreads in certain polar spaces (among other things).
There are a number of natural problems related to $\lambda$-homogeneity.
Let $H\leq {S_n}$ be a $2$-transitive group. Classify the pairs $(a,G)$, where $a\in {S_n}$ and $G\leq {S_n}$, such that $\langle a,G\rangle = H$.
Let $G\leq {S_n}$ be a $2$-transitive group. (The list of those groups is available in [@cam; @dixon].) For every $a\in {T_n}$ describe the structure of $\langle G,a\rangle\setminus G$. In particular (where $G$ is a $2$-transitive group and $a\in {T_n}$):
1. classify all the pairs $(a,G)$ such that $\langle a,G\rangle $ is a regular semigroup (that is, for all $x\in \langle a,G\rangle$ there exists $y \in \langle a,G\rangle$ such that $x=xyx$);
2. classify all the pairs $(a,G)$ such that $\langle a,G\rangle\setminus G$ is generated by its idempotents;
3. classify all the pairs $(a,G)$ such that $\langle a,G\rangle\setminus G =\langle g^{-1}ag\mid g\in G\rangle $;
4. describe the automorphisms, congruences, principal right, left and two-sided ideals of the semigroups $\langle a, G\rangle$ (when $G$ is a $2$-transitive group).
For each $2$-transitive group $G$ classify the $G$-pairs, that is, the pairs $(a,H)$ such that $H\leq {S_n}$, $a\in{T_n}$ and $\langle a,G\rangle\setminus G=\langle a,H\rangle\setminus H$.
Let $V$ be a finite dimension vector space. A pair $(a,G)$, where $a$ is a singular endomorphism of $V$ and $G\leq{\mathop{\mathrm{Aut}}}(V)$, is said to be an ${\mathop{\mathrm{Aut}}}(V)$-pair if $$\langle a,G\rangle\setminus G=\langle a,{\mathop{\mathrm{Aut}}}(V)\rangle\setminus{\mathop{\mathrm{Aut}}}(V).$$
Classify the ${\mathop{\mathrm{Aut}}}(V)$-pairs.
\[11\] Formulate and prove analogues of the results in this paper, but for semigroups of linear maps on a vector space.
Solve the analogue of Problem \[11\] for independence algebras (for definitions and fundamental results see [@ArEdGi; @arfo; @cameronSz; @F1; @gould]).
**Acknowledgment** The first author was partially supported by Pest-OE/MAT/UI0143/2011 of Centro de Algebra da Universidade de Lisboa, and by FCT and PIDDAC through the project PTDC/MAT/101993/2008.
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|
---
author:
- Manjul Bhargava
title: The density of discriminants of quintic rings and fields
---
Introduction
============
Let $N_n(X)$ denote the number of isomorphism classes of number fields of degree $n$ having absolute discriminant at most $X$. Then it is an old folk conjecture that the limit $$\label{basiclimit}
c_n = \lim_{X\rightarrow\infty} \frac{N_n(X)}{X} \vspace{.035in}$$ exists and is positive for $n>1$. The conjecture is trivial for $n\leq 2$, while for $n=3$ and $n=4$ it is a theorem of Davenport and Heilbronn [@DH] and of the author [@Bhargava5], respectively. In degrees $n\geq 5$, where number fields tend to be predominantly nonsolvable, the conjecture has not previously been known to be true for any value of $n$. The primary purpose of this article is to prove the above conjecture for $n=5$. In particular, we are able to determine the constant $c_5$ explicitly. More precisely, we prove:
Let $N_5^{(i)}(\xi,\eta)$ denote the number of quintic fields $K$, up to isomorphism, having $5-2i$ real embeddings and satisfying $\xi<\Disc(K)<\eta$. Then $$\begin{array}{rlcl}\label{dodqf}
\rm{(a)}& \displaystyle{\lim_{X\rightarrow\infty} \frac{N_5^{(0)}(0,X)}{X}}
&=&\!\! \displaystyle{\frac{1}{240}\prod_p (1+p^{-2}-p^{-4}-p^{-5})}; \\
\rm{(b)}& \displaystyle{\lim_{X\rightarrow\infty}
\frac{N_5^{(1)}(-X,0)}{X}} &=&\!\!\!\;\;\displaystyle{\frac{1}{24}\,\prod_p
(1+p^{-2}-p^{-4}-p^{-5})}; \\
\rm{(c)}&\displaystyle{\lim_{X\rightarrow\infty} \frac{N_5^{(2)}(0,X)}{X}}
&=& \!\!\!\;\;\displaystyle{\frac{1}{16}\,\prod_p (1+p^{-2}-p^{-4}-p^{-5})}.
\end{array}$$
The constants appearing in Theorem 1 (and thus their sum, $c_5
=\frac{13}{120}\,\prod_p (1+p^{-2}-p^{-4}-p^{-5})$) turn out to have very natural interpretations. Indeed, the constant $c_5$ takes the form of an Euler product, where the Euler factor at a place $\nu$ “counts” the total number of local étale quintic extensions of $\Q_\nu$, where each isomorphism class of local extension $K_\nu$ is counted with a certain natural weight to reflect the probability that a quintic number field $K$ has localization $K\otimes \Q_\nu$ isomorphic to $K_\nu$ at $\nu$. More precisely, let $$\label{binfdef}
\beta_\infty = \frac{1}{2}\sum_{[K_\infty:\R]=5 \mbox{\small \,\,\'etale}}
\frac{1}{|\Aut_\R(K_\infty)|},$$ where the sum is over all isomorphism classes $K_\infty$ of étale extensions of $\R$ of degree 5. Since $\Aut_\R(\R^5)=120$, $\Aut_\R(\R^3\oplus\C)=12$, and $\Aut_\R(\R\oplus\C^2)=8$, we have $\beta_\infty = \frac{1}{240}+\frac1{24}+\frac1{16}=\frac{13}{120}$. Similarly, for each prime $p$, let $$\label{bpdef}
\beta_p = \frac{p-1}{p}\sum_{[K_p:\Q_p]=5 \mbox{\small \,\,\'etale}}
\frac{1}{|\Aut_{\Q_p}(K_p)|}\cdot\frac{1}{\Disc_p(K_p)},$$ where the sum is over all isomorphism classes $K_p$ of étale extensions of $\Q_p$ of degree 5, and $\Disc_p(K_p)$ denotes the discriminant of $K_p$ viewed as a power of $p$. Then $$c_5 = \beta_\infty\cdot\prod_p \beta_p,$$ since we will show that $$\label{bpformula}
\beta_p=1+p^{-2}-p^{-4}-p^{-5}.$$ Thus we obtain a natural interpretation of $c_5$ as a product of counts of local field extensions. For more details on the evaluation of local sums of the form (\[bpdef\]), and for global heuristics on the expected values of the asymptotic constants associated to general degree $n$ $S_n$-number fields, see [@Bhargava6]. We obtain several additional results as by-products. First, our methods enable us to analogously count all [*orders*]{} in quintic fields:
Let $M_5^{(i)}(\xi,\eta)$ denote the number of isomorphism classes of orders $\O$ in quintic fields having $5-2i$ real embeddings and satisfying $\xi<\Disc(\O)<\eta$. Then there exists a positive constant $\alpha$ such that $$\begin{array}{rlcc}\label{dodqr}
\rm{(a)}& \displaystyle{\lim_{X\rightarrow\infty} \frac{M_5^{(0)}(0,X)}{X}}
&=&\!\! \displaystyle{\frac{\alpha}{240}}; \\
\rm{(b)}& \displaystyle{\lim_{X\rightarrow\infty}
\frac{M_5^{(1)}(-X,0)}{X}} &=&\!\!
\displaystyle{\frac{\alpha}{24}}; \\
\rm{(c)}&\displaystyle{\lim_{X\rightarrow\infty} \frac{M_5^{(2)}(0,X)}{X}}
&=& \!\!\displaystyle{\frac{\alpha}{16}}.
\end{array}$$
The constant $\alpha$ in Theorem 2 has an analogous interpretation. Let $\alpha_p$ denote the analogue of the sum (\[bpdef\]) for orders, i.e., $$\label{gpdef}
\alpha_p = \frac{p-1}{p}\sum_{[R_p:\Z_p]=5}
\frac{1}{|\Aut_{\Z_p}(R_p)|}\cdot\frac{1}{\Disc_p(R_p)},$$ where the sum is over all isomorphism classes of $\Z_p$-algebras $R_p$ of rank 5 over $\Z_p$ with nonzero discriminant. Then we will show that the constant $\alpha$ appearing in Theorem 2 is given by $$\label{rdef}
\alpha=\prod_p\alpha_p,$$ thus expressing $\alpha$ as a product of counts of local ring extensions. It is an interesting combinatorial problem to explicitly evaluate $\alpha_p$ in “closed form”, analogous to the formula (\[bpformula\]) that we obtain for $\beta_p$; see [@Bhargava7] for some further discussion on the evaluation of such sums.
Second, we note that the proof of Theorem 1 contains a determination of the densities of the various splitting types of primes in $S_5$-quintic fields. If $K$ is an $S_5$-quintic field and $K_{120}$ denotes the Galois closure of $K$, then the Artin symbol $(K_{120}/p)$ is defined as a conjugacy class in $S_5$, its values being $\langle e \rangle$, $\langle (12) \rangle$, $\langle (123) \rangle$, $\langle (1234)
\rangle$, $\langle (12345) \rangle$, $\langle (12)(34) \rangle$, or $\langle (12)(345)\rangle$, where $\langle x\rangle$ denotes the conjugacy class of $x$ in $S_5$. It follows from the Cebotarev density theorem that for fixed $K$ and varying $p$ (unramified in $K$), the values $\langle e \rangle$, $\langle (12) \rangle$, $\langle
(123) \rangle$, $\langle (1234) \rangle$, $\langle (12345) \rangle$, $\langle (12)(34) \rangle$, or $\langle (12)(345)\rangle$ occur with relative frequency 1:10:20:30:24:15:20 (i.e., proportional to the size of the respective conjugacy class). We prove the following complement to Cebatorev density:
Let $p$ be a fixed prime, and let $K$ run through all $S_5$-quintic fields in which $p$ does not ramify, the fields being ordered by the size of the discriminants. Then the Artin symbol $(K_{120}/p)$ takes the values $\langle e \rangle$, $\langle (12) \rangle$, $\langle (123) \rangle$, $\langle (1234)
\rangle$, $\langle (12345) \rangle$, $\langle (12)(34) \rangle$, or $\langle (12)(345)\rangle$ with relative frequency $1\!:\!10\!:\!20\!:\!30\!:\!24\!:\!15\!:\!20$.
Actually, we do a little more: we determine for each prime $p$ the density of $S_5$-quintic fields $K$ in which $p$ has the various possible ramification types. For example, it follows from our methods that a proportion of precisely $\frac{(p+1)(p^2+p+1)}{p^4+p^3+2p^2+2p+1}$ of $S_5$-quintic fields are ramified at $p$.
Lastly, our proof of Theorem 1 implies that nearly all—i.e., a density of 100% of—quintic fields have full Galois group $S_5$. This is in stark contrast to the quartic case [@Bhargava5 Theorem 3], where we showed that only about 91% of quartic fields have associated Galois group $S_4$:
When ordered by absolute discriminant, a density of $100\%$ of quintic fields have associated Galois group $S_5$.
In particular, it follows that 100% of quintic fields are nonsolvable. Note that, rather than counting quintic fields and orders up to isomorphism, we could instead count these objects within a fixed algebraic closure of $\Q$. This would simply multiply all constants appearing in Theorems 1 and 2 by five. Meanwhile, Theorems 3 and 4 of course remain true regardless of whether one counts quintic extensions up to isomorphism or within an algebraic closure of $\Q$.
The key ingredient that allows us to prove the above results for quintic (and thus predominantly nonsolvable) fields is a parametrization of isomorphism classes of quintic orders by means of four integral alternating bilinear forms in five variables, up to the action of $\GL_4(\Z)\times\SL_5(\Z)$, which we established in [@Bhargava4]. The proofs of Theorems 1–4 can then be reduced to counting appropriate integer points in certain fundamental regions, as in [@Bhargava5]. However, the current case is considerably more involved than the quartic case, since the relevant space is now 40-dimensional rather than 12-dimensional! The primary difficulty lies in counting points in the rather complicated cusps of these 40-dimensional fundamental regions (see Lemmas \[lem1\]–\[hard\]). The necessary point-counting is accomplished in Section 2, by carefully dissecting the “irreducible” portions of the fundamental regions into 152 pieces, and then applying a new adaptation of the averaging methods of [@Bhargava5] to each piece (see Lemma 11). The resulting counting theorem (see Theorem 6), in conjunction with the results of [@Bhargava4], then yields the asymptotic density of discriminants of pairs $(R,R')$, where $R$ is an order in a quintic field and $R'$ is a [*sextic resolvent ring*]{} of $R$. Obtaining Theorems 1–4 from this general density result then requires a sieve, which in turn uses certain counting results on resolvent rings and subrings obtained in [@Bhargava4] and in the recent work of Brakenhoff [@Jos], respectively. This sieve is carried out in the final Section 3. We note that the space of binary cubic forms that was used in the work of Davenport-Heilbronn to count cubic fields, the space of pairs of ternary quadratic forms that we used in [@Bhargava5] to count quartic fields, and the space of quadruples of alternating 2-forms in five variables that we use in this article, are all examples of what are known as prehomogeneous vector spaces. A [*prehomogeneous vector space*]{} is a pair $(G,V)$, where $G$ is a reductive group and $V$ is a linear representation of $G$ such that $G_\C$ has a Zariski open orbit on $V_\C$. The concept was introduced by Sato in the 1960’s and a classification of all irreducible prehomogeneous vector spaces was given in the work of Sato-Kimura [@SatoKimura], while Sato-Shintani [@SatoShintani] and Shintani [@Shintani] developed a theory of zeta functions associated to these spaces.
The connection between prehomogeneous vector spaces and field extensions was first studied systematically in the beautiful 1992 paper of Wright-Yukie [@WY]. In this work, Wright and Yukie determined the rational orbits and stabilizers in a number of prehomogeneous vector spaces, and showed that these orbits correspond to field extensions of degree 2, 3, 4, or 5. In their paper, they laid out a program to determine the density of discriminants of number fields of degree up to five, by considering adelic versions of Sato-Shintani’s zeta functions as developed by Datskovsky and Wright [@DW] in their extensive work on cubic extensions.
However, despite looking very promising, the program via adelic Shintani zeta functions encountered some difficulties and has not succeeded to date beyond the cubic case. The primary difficulties have been: (a) establishing cancellations among various divergent zeta integrals, in order to establish a “principal part formula” for the associated adelic Shintani zeta function; and (b) “filtering” out the correct count of extensions from the overcount of extensions that is inherent in the definition of the zeta function. In the quartic case, difficulty (a) was overcome in the impressive 1995 treatise of Yukie [@Yukie], while (b) remained an obstacle. In the quintic case, both (a) and (b) have remained impediments to obtaining a correct count of quintic field extensions by discriminant. (For more on the Shintani adelic zeta function approach and these related difficulties, see [@Bhargava5 §1] and [@Yukie].)
In [@Bhargava5] and in the current article, we overcome the problems (a) and (b) above, for quartic and quintic fields respectively, by introducing a different counting method that relies more on geometry-of-numbers arguments. Thus, although our methods are different, this article may be viewed as completing the program first laid out by Wright and Yukie [@WY] to count field extensions in degrees up to 5 via the use of appropriate prehomogeneous vector spaces. We now describe in more detail the methods of this paper, and give a comparison with previous methods. At least initially, our approach to counting quintic extensions using the prehomogeneous vector space $\C^4\otimes\wedge^2\C^5$ is quite similar in spirit to Davenport-Heilbronn’s original method in the cubic case [@DH] and its refinements developed in the quartic case [@Bhargava5]. Namely, we begin by giving an algebraic interpretation of the [*integer*]{} orbits on the associated prehomogeneous vector space which, in the quintic case, are the orbits of the group $G_\Z=\GL_4(\Z)\times\SL_5(\Z)$ on the 40-dimensional lattice $V_\Z=\Z^4\otimes\wedge^2\Z^5$. As we showed in [@Bhargava4], these integer orbits have an extremely rich algebraic interpret-ation and structure (see Theorem 5 for a precise statement), enabling us to consider not only quintic fields, but also more refined data such as all [*orders*]{} in quintic fields, the local behaviors of these orders, and their sextic resolvent rings. This interpretation of the integer orbits then allows us to reduce our problem of counting orders and fields to that of enumerating appropriate lattice points in a fundamental domain for the action of the discrete group $G_\Z$ on the real vector space $V_\R=V_\Z\otimes\R$.
Just as in [@DH] and [@Bhargava5], the main difficulty in counting lattice points in such a fundamental region is that this region is [not]{} compact, but instead has cusps (or “tentacles”) going off to infinity. To make matters even more interesting, unlike the case of binary cubic forms in Davenport-Heilbronn’s work—where there is one relatively simple cusp defined by small degree inequalities in four variables—in the case of quadruples of quinary alternating 2-forms, the cusps are numerous in number and are defined by polynomial inequalities of extremely high degree in 40 variables! These difficulties are further exacerbated by the fact that—contrary to the cubic case—in the quartic and quintic cases the number of nondegenerate lattice points in the cuspidal regions is of strictly [*greater*]{} order than the number of points in the noncuspidal part (“main body”) of the corresponding fundamental domains. The latter issue is indeed what lies behind the problems (a) and (b) above in the adelic zeta function method.
Following our work in the quartic case [@Bhargava5], we overcome these problems that arise from the cuspidal regions by counting lattice points not in a single fundamental domain, but over a [continuous, compact set of fundamental domains]{}. This allows one to “thicken” the cusps, thereby gaining a good deal of control on the integer points in these cuspidal regions. A basic version of this “averaging” method was introduced and used in [@Bhargava5] in the quartic case to handle points in these cusps, and thus enumerate quartic extensions by discriminant (see [@Bhargava5 §1] for more details). However, since the number, complexity and dimensions of the cuspidal regions are so much greater in the quintic case than in the quartic case, a number of new ideas and modifications are needed to successfully carry out the same averaging method in the quintic case.
The primary technical contribution of this article is the introduction of a method that allows one to systematically and canonically dissect the cuspidal regions into certain “nice” subregions on which a slightly refined averaging technique (see Sections 2.1–2.2) can then be applied in a uniform manner. Using this method, we divide up the fundamental region into 159 pieces. The first piece is the main body of the region, where we show using geometry-of-numbers arguments that the number of lattice points in the region is essentially its volume. For each of the remaining 158 cuspidal pieces, we show, by a uniform argument, that either the number of lattice points in that region is negligible (see Table 1, Lemma \[hard\]), [*or*]{} that the lattice points in that cuspidal piece are all [*reducible*]{}, i.e., they correspond to quintic rings that are not integral domains (see Lemma \[red\]). An asymptotic formula for the number of [irreducible]{} integer points in the entire fundamental domain is then attained. The interesting interaction between the algebraic properties of the lattice points (via the correspondence in [@Bhargava4]) and their geometric locations within the fundamental domain is therefore what allows us to overcome the problems (a) and (b) arising in the adelic Shintani zeta function method. As explained earlier, a sieving method can then be used to prove Theorems 1–4.
Our counting method in this article is quite robust and systematic, and should be applicable in many other situations. First, it can be used to reprove the density of discriminants of cubic and quartic fields, with much stronger error terms than have previously been known (in fact, in the cubic case it can be used, in conjunction with a sieve, to obtain an exact second order term; see [@simpledhc]). Second, the method can be suitably adapted to count cubic, quartic, and quintic field extensions of any base number field (see [@dode]). Third, the method can be used on prehomogeneous vector spaces having [*infinite*]{} stabilizer groups, which would also have a number of interesting applications (see, e.g., [@Bhargava8]). Finally, we expect that the methods should also be adaptable to representations of algebraic groups that are not necessarily prehomogeneous. We hope that these directions will be pursued further in future work.
On the class numbers of quadruples of $5\times 5$ skew-symmetric matrices
=========================================================================
Let $V=V_\R$ denote the space of quadruples of $5\times 5$ skew-symmetric matrices over the real numbers. We write an element of $V_\R$ as an ordered quadruple $(A,B,C,D)$, where the $5\times 5$ matrices $A$, $B$, $C$, $D$ have entries $a_{ij}$, $b_{ij}$, $c_{ij}$, $d_{ij}$ respectively. Such a quadruple $(A,B,C,D)$ is said to be [*integral*]{} if all entries of the matrices $A$, $B$, $C$, $D$ are integral.
The group $G_\Z=\GL_4(\Z)\times\SL_5(\Z)$ acts naturally on the space $V_\R$. Namely, an element $g_4\in\GL_4(\Z)$ acts by changing the basis of the $\Z$-module of matrices spanned by $A,B,C,D$; in terms of matrix multiplication, we have $(A\;B\;C\;D)^t\mapsto g_4\,
(A\;B\;C\;D)^t$. Similarly, an element $g_5\in \SL_5(\Z)$ changes the basis of the five-dimensional space on which the skew-symmetric forms $A,B,C,D$ take values, i.e., $g_5\cdot(A,B,C,D)=
(g_5Ag_5^t, g_5Bg_5^t, g_5 C g_5^t, g_5Dg_5^t)$. It is clear that the actions of $g_4$ and $g_5$ commute, and that this action of $G_\Z$ preserves the lattice $V_\Z$ consisting of the integral elements of $V_\R$.
The action of $G_\Z$ on $V_\R$ (or $V_\Z$) has a unique polynomial invariant, which we call the [*discriminant*]{}. It is a degree 40 polynomial in 40 variables, and is much too large to write down. An easy method to compute it for any given element in $V$ was described in [@Bhargava4].
The integer orbits of $G_\Z$ on $V_\Z$ have an important arithmetic significance. Recall that a [*quintic ring*]{} is any ring with unit that is isomorphic to $\Z^5$ as a $\Z$-module; for example, an order in a quintic number field is a quintic ring. In [@Bhargava4] we showed how quintic rings may be parametrized in terms of the $G_\Z$-orbits on $V_\Z$:
\[main\] There is a canonical bijection between the set of $G_\Z$-equivalence classes of elements $(A,B,C,D)\in V_\Z$, and the set of isomorphism classes of pairs $(R,R')$, where $R$ is a quintic ring and $R'$ is a sextic resolvent ring of $R$. Under this bijection, we have $\Disc(A,B,C,D)=\Disc(R)=\frac1{16}\cdot\Disc(R')^{1/3}$.
A [*sextic resolvent*]{} of a quintic ring $R$ is a sextic ring $R'$ equipped with a certain [*resolvent mapping*]{} $R\to\wedge^2R'$ whose precise definition will not be needed here (see [@Bhargava4] for details). In view of Theorem \[main\], we wish to try and understand the number of $G_\Z$-orbits on $V_\Z$ having absolute discriminant at most $X$, as $X\rightarrow\infty$. The number of integral orbits on $V_\Z$ having a fixed discriminant $\Delta$ is called a “class number”, and we wish to understand the behavior of this class number on average.
From the point of view of Theorem \[main\], we would like to restrict the elements of $V_\Z$ under consideration to those that are “irreducible” in an appropriate sense. More precisely, we call an element $(A,B,C,D)\in V_\Z$ [*irreducible*]{} if, in the corresponding pair of rings $(R,R')$ in Theorem \[main\], the ring $R$ is an integral domain. The quotient field of $R$ is thus a quintic field in that case. We say $(A,B,C,D)$ is [*reducible*]{} otherwise. One may also describe reducibility and irreducibility in more geometric terms. If $(A,B,C,D)\in V_\Z$, then one may consider the $4\times 4$ sub-Pfaffians $Q_1(t_1,t_2,t_3,t_4),\ldots,Q_5(t_1,t_2,t_3,t_4)$ of the single $5\times 5$ skew-symmetric matrix $At_1+Bt_2+Ct_3+Dt_4$ whose entries are linear forms in $t_1,t_2,t_3,t_4$. In other words, $Q_i=Q_i(w,x,y,z)$ is defined as a canonical squareroot of the determinant of the $4\times 4$ matrix obtained from $t_1A+t_2B+t_3C+t_4D$ by removing its $i$th row and column. Thus these $4\times 4$ Pfaffians $Q_1,\ldots,Q_5$ are quaternary quadratic forms and so define five quadrics in $\P^3$. If the element $(A,B,C,D)\in V_\Z$ has nonzero discriminant, then it is known that these five quadrics intersect in exactly five points in $\P^3$ (counting multiplicities); see e.g., [@WY], [@Bhargava4]. We refer to these five points as the [*zeroes of*]{} $(A,B,C,D)$ in $\P^3$. In [@Bhargava4] we showed that if $(A,B,C,D)$ corresponds to $(R,R')$, where $R$ is isomorphic to an order in a quintic field $K$, then there exists a zero of $(A,B,C,D)$ in $\P^3$ whose field of definition is $K$. (The other zeroes of $(A,B,C,D)\in V_\Z$ are thus defined over the conjugates of $K$.) Therefore, geometrically, we may say that $(A,B,C,D)$ is irreducible if and only if it possesses a zero in $\P^3$ having field of definition $K$, where $K$ is a quintic field extension of $\Q$. On the other hand, $(A,B,C,D)$ is reducible if and only if $(A,B,C,D)$ possesses a zero in $\P^3$ defined over a number field of degree smaller than five. The main result of this section is the following theorem:
\[cna\] Let $N(V^{(i)}_\Z;X)$ denote the number of $G_\Z$-equivalence classes of irreducible elements $(A,B,C,D)\in V_\Z$ having $5-2i$ real zeroes in $\P^3$ and satisfying $|\Disc(A,B,C,D)|<X$. Then $$\begin{array}{rlcl}\label{dodpotqf}
\rm{(a)}& \displaystyle{\lim_{X\rightarrow\infty} \frac{N(V^{(0)}_\Z;X)}{X}}
&=&\!\! \displaystyle{\frac{\zeta(2)^2\zeta(3)^2\zeta(4)^2\zeta(5)}{240}}; \\[.1in]
\rm{(b)}& \displaystyle{\lim_{X\rightarrow\infty} \frac{N(V^{(1)}_\Z;X)}{X}}
&=&\!\! \displaystyle{\frac{\zeta(2)^2\zeta(3)^2\zeta(4)^2\zeta(5)}{24}}; \\[.1in]
\rm{(c)}&\displaystyle{\lim_{X\rightarrow\infty} \frac{N(V^{(2)}_\Z;X)}{X}}
&=&\!\! \displaystyle{\frac{\zeta(2)^2\zeta(3)^2\zeta(4)^2\zeta(5)}{16}}.
\end{array}$$
Theorem \[cna\] is proven in several steps. In Subsection 2.1, we outline the necessary reduction theory needed to establish some particularly useful fundamental domains for the action of $G_\Z$ on $V_\R$. In Subsections 2.2 and 2.3, we describe a refinement of the “averaging” method from [@Bhargava5] that allows us to efficiently count integer points in various components of these fundamental domains in terms of their volumes. In Subsections 2.4 and 2.5, we investigate the distribution of reducible and irreducible integral points within these fundamental domains. The volumes of the resulting “irreducible” components of these fundamental domains are then computed in Subsection 2.6, proving Theorem \[cna\]. A version of Theorem \[cna\] for elements in $V_\Z$ satisfying any specified set of congruence conditions is then obtained in Subsection 2.7. In Section 3, we will show how these counting methods—together with a sieving argument—can be used to prove Theorems 1–4.
Reduction theory
----------------
The action of $G_\R=\GL_4(\R)\times \SL_5(\R)$ on $V_\R$ has three nondegenerate orbits $V_\R^{(0)}, V_\R^{(1)}, V_\R^{(2)}$, where $V^{(i)}_\R$ consists of those elements $(A,B,C,D)$ in $V_\R$ having nonzero discriminant and $5-2i$ real zeroes in $\P^3$. We wish to understand the number of irreducible $G_\Z$-orbits on $V^{(i)}_\Z=V^{(i)}_\R\cap V_\Z$ having absolute discriminant at most $X$ ($i=0,1,2$). We accomplish this by counting the number of integer points of absolute discriminant at most $X$ in suitable fundamental domains for the action of $G_\Z$ on $V_\R$.
These fundamental regions are constructed as follows. First, let $\FF$ denote a fundamental domain in $G_\R$ for $G_\Z\backslash G_\R$. We may assume that $\FF$ is contained in a standard Siegel set, i.e., we may assume $\FF$ is of the form $\FF=
\{nak\lambda:n\in N'(a),a\in A',k\in K,\lambda\in\Lambda\}$, where $$\begin{aligned}
\label{siegel}
K\,&=&\{\mbox{special orthogonal transformations in $G_\R$}\};\\
A'&=&\{a(s_1,s_2,\ldots,s_7):s_1,s_2,\ldots,s_7\geq c\},\;\mbox{where}\,\\
{}&{}&a({\boldmath{s}})={\footnotesize
\left(\left(\begin{array}{cccc} \!\!\!s_1^{-3}s_2^{-1}s_3^{-1}
\!\!\!\!\!\!\!\!\!\!\! &
{} & {} & {} \\[.04in] {} & \!\!\!\!\!\!\!\!\!s_1 s_2^{-1} s_3^{-1} \!\!\!\!\!\!\!\!
& {} &{} \\[.04in] {}&{}&\!\!\!\!\!\!\!\!s_1 s_2 s_3^{-1}\!\!\!\!\!\!\!\!
&{}\\[.04in] {}&{}&{}&
\!\!\!\!\!\!\!\!s_1 s_2 s_3^{3}\!\! \end{array}\right),
\left(\begin{array}{ccccc} \!\!s_4^{-4}s_5^{-3}s_6^{-2}s_7^{-1} \!\!\!\!\!\!\!\!
& {} & {} &{} &{}\\[.04in]
{}& \!\!\!\!\!\!\!\!\!\!\!\!{}\!\!\!\!s_4s_5^{-3}s_6^{-2}s_7^{-1}\!\!\!\!\!\!\!\!&{} &{}&{}\\[.04in]{}&{}&
\!\!\!\!\!\!\!\!\!\!\!{}\!\!s_4 s_5^{2}s_6^{-2}s_7^{-1}\!\!\!\!\!\!\!\!&{}&
{}\\[.04in]
{}&{}&{}& \!\!\!\!\!\!\!\!\!s_4s_5^{2}s_6^{3}s_7^{-1}\!\!\!\!\!\!\!\!
&{}\\[.04in] {}&{}&{}&{}&\!\!\!\!\!\!\!\!s_4s_5^{2}s_6^{3}s_7^4\!\! \end{array} \right)\right)};\,\,\\
\bar N'\,&=&\{n(u_1,u_2,\ldots,u_{16}):u=(u_1,u_2,\ldots,u_{16})\in\nu(a) \},\;\mbox{where}\,\\
{}&{}&
n({\boldmath{u}})={\footnotesize
\left(\left(\begin{array}{cccc} 1 & {} & {} & {} \\ {u_1} & 1
& {} &{} \\ {u_2}&{u_3}&1&{}\\{u_4}&{u_5}&{u_6}&1 \end{array}\right),
\left(\begin{array}{ccccc} 1 & {} & {} &{} &{}\\
{u_7}& 1 &{} &{}&{}\\{u_8}&{u_9}&
1&{}&{}\\{u_{10}}&{u_{11}}&{u_{12}}&1&{}\\
{u_{13}}&{u_{14}}&{u_{15}}&{u_{16}}&1\end{array} \right)\right)};\,\, \\
\Lambda\,&=&\{\{\lambda:\lambda>0\},\;\mbox{where}\,\\
{}&{}& \lambda \mbox{ acts by }{\footnotesize
\left(\left(\begin{array}{cccc} \lambda & {} & {} & {} \\ {} & \lambda
& {} &{} \\ {}&{}&\lambda&{}\\{}&{}&{}&\lambda \end{array}\right),
\left(\begin{array}{ccccc} 1 & {} & {} &{} &{}\\
{}& 1 &{} &{}&{}\\{}&{}&
1&{}&{}\\{}&{}&{}&1&{}\\
{}&{}&{}&{}&1\end{array} \right)\right)};\end{aligned}$$ here $c>0$ is an absolute constant and $\nu(a)$ is an absolutely bounded measurable subset of $\R^{16}$ dependent only on the value of $a\in A'$.
For $i=0,1,2$, let $n_i$ denote the cardinality of the stabilizer in $G_\R$ of any element $v\in V^{(i)}_\R$ (it follows from Proposition \[covering\] below that $n_1=120$, $n_2=12$, and $n_3=8$). Then for any $v\in V^{(i)}_\R$, $\FF v$ will be the union of $n_i$ fundamental domains for the action of $G_\Z$ on $V^{(i)}_\R$. Since this union is not necessarily disjoint, $\FF v$ is best viewed as a multiset, where the multiplicity of a point $x$ in $\FF v$ is given by the cardinality of the set $\{g\in\FF\,\,|\,\,gv=x\}$. Evidently, this multiplicity is a number between 1 and $n_i$. Even though the multiset $\FF v$ is the union of $n_i$ fundamental domains for the action of $G_\Z$ on $V^{(i)}_\R$, not all elements in $G_\Z\backslash V_\Z$ will be represented in $\FF v$ exactly $n_i$ times. In general, the number of times the $G_\Z$-equivalence class of an element $x\in V_\Z$ will occur in $\FF v$ is given by $n_i/m(x)$, where $m(x)$ denotes the size of the stabilizer of $x$ in $G_\Z$. We define $N(V_\Z^{(i)};X)$ to be the (weighted) number of irreducible $G_\Z$-orbits on $V_\Z^{(i)}$ having absolute discriminant at most $X$, where each orbit is counted by a weight of $1/m(x)$ for any point $x$ in that orbit. Thus $n_i\cdot N(V_\Z^{(i)};X)$ is the (weighted) number of points in $\FF v$ having absolute discriminant at most $X$, where each point $x$ in the multiset $\FF v$ is counted with a weight of $1/m(x)$.
We note that the $G_\Z$-orbits in $V_\Z$ corresponding to orders in non-Galois quintic fields will then each be counted simply with a weight of 1, since such orders can have no automorphisms. We will show (see Lemma \[3reducible2\]) that orbits having weight $< 1$ are negligible in number in comparison to those having weight $1$, and so points of weight $<1$ will not be important as they will not affect the main term of the asymptotics of $N(V_\Z^{(i)}; X)$ as $X\to\infty$.
Now the number of integer points can be difficult to count in a single fundamental region $\FF v$. The main technical obstacle is that the fundamental region $\FF v$ is not compact, but rather has a system of cusps going off to infinity which in fact contains infinitely many points, including many irreducible points. We simplify the counting of such points by “thickening” the cusp; more precisely, we compute the number of points in the fundamental region $\FF v$ by averaging over lots of such fundamental domains, i.e., by averaging over a continuous range of points $v$ lying in a certain special compact subset $H$ of $V$.
Averaging over fundamental domains
----------------------------------
Let $H=H(J)=\{w\in V : \|w\|\leq J,\;|\Disc(w)|\geq 1\}$, where $\|w\|$ denotes a Euclidean norm on $V$ fixed under the action of $K$, and $J$ is sufficiently large so that $H$ is nonempty and of nonzero volume. We write $V^{(i)}:=V^{(i)}_\R$. Then we have $$N(V^{(i)}_\Z;X) = \frac{\int_{v\in H\cap V^{(i)}}
\#\{x\in \FF v\cap V_\Z^\irr: |\Disc(x)|<X\}\;
|\Disc(v)|^{-1} dv}
{n_i\cdot\int_{v\in H\cap V^{(i)}} \:|\Disc(v)|^{-1} dv},$$ where $V_\Z^\irr\subset V_\Z$ denotes the subset of irreducible points in $V_\Z$. The denominator of the latter expression is, by construction, a finite absolute constant $M_i=M_i(J)$ greater than zero. We have chosen the measure $|\Disc(v)|^{-1}\,dv$ because it is a $G_\R$-invariant measure.
More generally, for any $G_\Z$-invariant subset $S\subset
V_\Z^{(i)}$, let $N(S;X)$ denote the number of irreducible $G_\Z$-orbits on $S$ having discriminant less than $X$. Then $N(S;X)$ can be expressed as $$\label{nsx}
N(S;X) = \frac{\int_{v\in H\cap V^{(i)}}
\#\{x\in \FF v\cap S^\irr: |\Disc(x)|<X\}\;
|\Disc(v)|^{-1} dv}
{n_i\cdot\int_{v\in H\cap V^{(i)}} \:|\Disc(v)|^{-1} dv},$$ where $S^\irr\subset S$ denotes the subset of irreducible points in $S$. We shall use this definition of $N(S;X)$ for any $S\subset V_\Z$, even if $S$ is not $G_\Z$-invariant. Note that for disjoint $S_1,S_2\subset V_\Z$, we have $N(S_1\cup S_2)=N(S_1)+N(S_2)$.
Now since $|\Disc(v)|^{-1}\,dv$ is a $G_\R$-invariant measure, we have for any $f\in C_0(V^{(i)})$, with $v,x\in V_\R^{(i)}$ and $g\in G_\R$ satisfying $v=gx$, that $f(v)|\Disc(v)|^{-1} dv= r_i\, f(gx)\,dg$ for some constant $r_i$ dependent only on whether $i=0$, $1$ or $2$; here $dg$ denotes a left-invariant Haar measure on $G_\R$. We may thus express the above formula for $N(S;X)$ as an integral over $\FF\subset
G_\R$: $$\begin{aligned}
N(S;X)&\!\!=\!\!& \frac{r_i}{M_i}\int_{g\in\FF}
\#\{x\in S^\irr\cap gH:|\Disc(x)|<X\}\,dg\\[.075in]
&\!\!=\!\!& \frac{r_i}{M_i}\int_{g\in N'(a)A'\Lambda K}
\#\{x\in S\cap \bar n(u)a(s) \lambda k H:|\Disc(x)|<X\}\,dg\,.
$$ Let us write $H(u,s,\lambda,X) =
\bar n(u)a(s)\lambda H\cap\{v\in V^{(i)}:|\Disc(v)|<X\}$. Noting that $KH=H$, $\int_K dk = 1$ (by convention), and $dg = s_1^{-12}s_2^{-8}s_3^{-12}s_4^{-20}s_5^{-30}s_6^{-30}s_7^{-20}
du\,d^\times s\, d^\times\lambda\, dk$ (up to scaling), we have $$\label{avg}
N(S;X) = \frac{r_i}{M_i}\int_{g\in N'(a)A'\Lambda}
\#\{x\in S^\irr\cap H(u,s,\lambda,X)\}
\,s_1^{-12}s_2^{-8}s_3^{-12}s_4^{-20}s_5^{-30}s_6^{-30}s_7^{-20} \,
du\, d^\times t\,d^\times \lambda\,.$$
We note that the same counting method may be used even if we are interested in counting both reducible and irreducible orbits in $V_\Z$. For any set $S\subset V_\Z^{(i)}$, let $N^*(S;X)$ be defined by (\[nsx\]), but where the superscript “irr” is removed. Thus for a $G_\Z$-invariant set $S\subset V_\Z^{(i)}$, $n_i\cdot N^*(S;X)$ counts the total (weighted) number of $G_\Z$-orbits in $S$ having absolute discriminant nonzero and less than $X$ (not just the irreducible ones). By the same reasoning, we have $$\label{avgS}
N^*(S;X) = \frac{r_i}{M_i}\int_{g\in N'(a)A'\Lambda}
\#\{x\in S\cap H(u,s,\lambda,X)\}
\, s_1^{-12}s_2^{-8}s_3^{-12}s_4^{-20}s_5^{-30}s_6^{-30}s_7^{-20} \,
du\, d^\times t\,d^\times \lambda\,.$$ The expression (\[avg\]) for $N(S;X)$, and its analogue (\[avgS\]) for $N^*(S,X)$, will be useful in the sections that follow.
A lemma from geometry of numbers
--------------------------------
To estimate the number of lattice points in $H(u,s,\lambda,X)$, we have the following elementary proposition from the geometry-of-numbers, which is essentially due to Davenport [@Davenport1]. To state the proposition, we require the following simple definitions. A multiset $\mathcal R\subset\R^n$ is said to be [*measurable*]{} if $\mathcal R_k$ is measurable for all $k$, where $\mathcal R_k$ denotes the set of those points in $\mathcal R$ having a fixed multiplicity $k$. Given a measurable multiset $\mathcal R \subset\R^n$, we define its volume in the natural way, that is, $\Vol(\mathcal R)=\sum_k
k\cdot\Vol(\mathcal R_k)$, where $\Vol(\mathcal R_k)$ denotes the usual Euclidean volume of $\mathcal R_k$.
\[genbound\] Let $\mathcal R$ be a bounded, semi-algebraic multiset in $\R^n$ having maximum multiplicity $m$, and which is defined by at most $k$ polynomial inequalities each having degree at most $\ell$. Let $\RR'$ denote the image of $\RR$ under any $($upper or lower$)$ triangular, unipotent transformation of $\R^n$. Then the number of integer lattice points $($counted with multiplicity$)$ contained in the region $\mathcal R'$ is $$\Vol(\mathcal R)+ O(\max\{\Vol(\bar{\mathcal R}),1\}),$$ where $\Vol(\bar{\mathcal R})$ denotes the greatest $d$-dimensional volume of any projection of $\mathcal R$ onto a coordinate subspace obtained by equating $n-d$ coordinates to zero, where $d$ takes all values from $1$ to $n-1$. The implied constant in the second summand depends only on $n$, $m$, $k$, and $\ell$.
Although Davenport states the above lemma only for compact semi-algebraic sets $\mathcal R\subset\R^n$, his proof adapts without essential change to the more general case of a bounded semi-algebraic multiset $\RR\subset\R^n$, with the same estimate applying also to any image $\mathcal R'$ of $\mathcal R$ under a unipotent triangular transformation.
Estimates on reducible quadruples $(A,B,C,D)$
---------------------------------------------
In this section we describe the relative frequencies with which reducible and irreducible elements sit inside various parts of the fundamental domain $\FF v$, as $v$ varies over the compact region $H$.
We begin by describing some sufficient conditions that guarantee that a point in $V_\Z$ is reducible.
\[lem1\] Let $(A,B,C,D)\in V_\Z$ be an element such that some non-trivial $\Q$-linear combination of $A,B,C,D$ has rank $\leq 2$. Then $(A,B,C,D)$ is reducible.
Suppose $E=rA+sB+tC+uD$, where $r,s,t,u\in\Q$ are not all zero. Let $Q_1,\ldots,Q_5$ denote the five $4\times 4$ sub-Pfaffians of $(A,B,C,D)$. Then we have proven in [@Bhargava4] that if $(A,B,C,D)\in V_\Z$ is irreducible, then the quadrics $Q_1=0,\ldots,Q_5=0$ intersect in five points in $\P^3(\bar\Q)$, and moreover, these five points are defined over conjugate quintic extensions of $\Q$. However, if rank$(E)\leq 2$, then $[r,s,t,u]\in\P^3(\Q)$ is a common zero of $Q_1,\ldots,Q_5$ and it is defined over $\Q$, contradicting the irreducibility of $(A,B,C,D)$.
\[lem2\] Let $(A,B,C,D)\in V_\Z$ be an element such that some non-trivial $\Q$-linear combination of $Q_1,\ldots,Q_5$ factors over $\Q$ into two linear factors, where $Q_1,\ldots,Q_5$ denote the five $4\times 4$ sub-Pfaffians of $(A,B,C,D)$. Then $(A,B,C,D)$ is reducible.
As noted in the proof of Lemma \[lem1\], the five associated quadratic forms $Q_1,\ldots,Q_5$ of an irreducible element $(A,B,C,D)\in V_\Z$ possess five common zeroes that are defined over conjugate quintic fields, and these zeroes are conjugate to each other over $\Q$. It follows that each of the ${5\choose 3}=10$ planes, going through subsets of three of those five points, cannot be defined over $\Q$, as these planes will each be part of a $\Gal(\bar\Q/\Q)$-orbit of size at least 5. However, if some rational quaternary quadratic form $Q$ factors over $\Q$ into linear factors, then (by the pigeonhole principle) at least one of these two rational factors must vanish at three of the five common points of intersection, a contradiction.
\[red\] Let $(A,B,C,D)\in V_\Z$ be an element such that all the variables in at least one of the following sets vanish:
$\{a_{12},a_{13},a_{14},a_{15},a_{23},a_{24},a_{25}\}$
$\{a_{12},a_{13},a_{14},a_{23},a_{24},a_{34}\}$
$\{a_{12},a_{13},a_{14},a_{15}\}\cup\{b_{12},b_{13},b_{14},b_{15}\} $
$\{a_{12},a_{13},a_{14},a_{23},a_{24}\}\cup\{b_{12},
b_{13},b_{14},b_{23},b_{24}\} $
$\{a_{12},a_{13},a_{14}\}\cup\{b_{12},b_{13},b_{14}\}\cup\{c_{12},
c_{13},c_{14}\}$
$\{a_{12},a_{13},a_{23}\}\cup\{b_{12},b_{13},b_{23}\}\cup\{c_{12},
c_{13},c_{23}\}$
$\{a_{12},a_{13}\}\cup\{b_{12},b_{13}\}\cup\{c_{12},c_{13}\}\cup\{d_{12},
d_{13}\}$
Then $(A,B,C,D)$ is reducible.
In cases (i) and (ii), one sees that $A$ has rank $\leq 2$, and thus $(A,B,C,D)$ is reducible by Lemma \[lem1\]. In the remaining cases (iii)–(vii), one finds that $Q_5$ factors into rational linear factors, and thus the result in these cases follows from Lemma \[lem2\].
We are now ready to give an estimate on the number of irreducible elements in $\FF v$, on average, satisfying $a_{12}=0$:
\[hard\] Let $v$ take a random value in $H$ uniformly with respect to the measure $|\Disc(v)|^{-1}\,dv$. Then the expected number of irreducible elements $(A,B,C,D)\in\FF v$ such that $|\Disc(A,B,C,D)|< X$ and $a_{12}=0$ is $O(X^{39/40})$.
As in [@Bhargava5], we divide the set of all $(A,B,C,D)\in V_\Z$ into a number of cases depending on which initial coordinates are zero and which are nonzero. These cases are described in the second column of Table 1. The vanishing conditions in the various subcases of Case $n+1$ are obtained by setting equal to 0—one at a time—each variable that was assumed to be nonzero in Case $n$. If such a resulting subcase satisfies the reducibility conditions of Lemma \[red\], it is not listed. In this way, it becomes clear that any irreducible element in $V_\Z$ must satisfy precisely one of the conditions enumerated in the second column of Table 1. In particular, there is no Case 14, because assuming any nonzero variables in Case 13 to be zero immediately results in reducibility by Lemma \[red\]. Let $T$ denote the set of all forty variables $a_{ij},b_{ij},c_{ij},d_{ij}$. For a subcase $\CC$ of Table 1, we use $T_0=T_0(\CC)$ to denote the set of variables in $T$ assumed to be 0 in Subcase $\CC$, and $T_1$ to denote the set of variables in $T$ assumed to be nonzero.
Each variable $t\in T$ has a [*weight*]{}, defined as follows. The action of $a(s_1,s_2,\ldots,s_7)\cdot\lambda$ on $(A,B,C,D)\in
V$ causes each variable $t$ to multiply by a certain weight which we denote by $w(t)$. These weights $w(t)$ are evidently rational functions in $\lambda,s_1,\ldots,s_7$.
Let $V(\CC)$ denote the set of $(A,B,C,D)\in V_\R$ such that $(A,B,C,D)$ satisfies the vanishing and nonvanishing conditions of Subcase $\CC$. For example, in Subcase 2a we have $T_0(\mbox{2a})=\{a_{12},a_{13}\}$ and $T_1(\mbox{2a})=\{a_{14},a_{23},b_{12}\}$; thus $V(\mbox{2a})$ denotes the set of all $(A,B,C,D)\in V_\Z$ such that $a_{12}=a_{13}=0$ but $a_{14},a_{23},b_{12}\neq 0$.
For each subcase $\CC$ of Case $n$ ($n>0$), we wish to show that $N(V(\CC);X)$, as defined by (\[nsx\]), is $O(X^{39/40})$. Since $N'(a)$ is absolutely bounded, the equality (\[avgS\]) implies that $$\label{estv0s}
N^*(V(\CC);X)\ll
\int_{\lambda=c'}^{X^{1/40}} \int_{s_1,s_2,\ldots,s_7=c}^\infty
\sigma(V(\CC))
\, s_1^{-12}s_2^{-8}s_3^{-12}s_4^{-20}s_5^{-30}s_6^{-30}s_7^{-20} \,
d^\times\! s \,d^\times\!\lambda,$$ where $\sigma(V(\CC))$ denotes the number of integer points in the region $H(u,s,\lambda,X)$ that also satisfy the conditions $$\label{cond}
\mbox{$t=0$ for $t\in T_0$ and $|t|\geq 1$ for $t\in T_1$}.$$
Now for an element $(A,B,C,D)\in H(u,s,\lambda,X)$, we evidently have $$\label{condt}
|t|\leq J{w(t)}$$ and therefore the number of integer points in $H(u,s,\lambda,X)$ satisfying (\[cond\]) will be nonzero only if we have $$\label{condt1}
J{w(t)}\geq 1$$ for all weights $w(t)$ such that $t\in T_1$. Now the sets $T_1$ in each subcase of Table 1 have been chosen to be precisely the set of variables having the minimal weights $w(t)$ among the variables $t\in
T\setminus T_0$ (by “minimal weight” in $T\setminus T_0$, we mean there is no other variable $t\in T\setminus T_0$ with weight having smaller exponents for all parameters $\lambda, s_1,s_2,\ldots,s_7$). Thus if the condition (\[condt1\]) holds for all weights $w(t)$ corresponding to $t\in T_1$, then—by the very choice of $T_1$—we will also have $Jw(t)\gg 1$ for all weights $w(t)$ such that $t\in
T\setminus T_0$.
Therefore, if the region $\H=\{(A,B,C,D)\in
H(u,s,\lambda,X):t=0\;\;\forall t\in T_0;\;\; |t|\geq 1\;\; \forall
t\in T_1\}$ contains an integer point, then (\[condt1\]) and Lemma \[genbound\] together imply that the number of integer points in $\H$ is $O(\Vol(\H))$, since the volumes of all the projections of $u^{-1}\H$ will in that case also be $O(\Vol(\H))$. Now clearly $$\Vol(\H)=O\Bigl(J^{40-|T_0|}\prod_{t\in T\setminus T_0} w(t)\Bigr),$$ so we obtain $$\label{estv1s}
N(V(\CC);X)\ll
\int_{\lambda=c'}^{X^{1/40}} \int_{s_1,s_2,\ldots,s_7=c}^\infty
\prod_{t\in T\setminus T_0} w(t)
\,\, s_1^{-12}s_2^{-8}s_3^{-12}s_4^{-20}s_5^{-30}s_6^{-30}s_7^{-20} \,\,\,
d^\times\! s \,d^\times\!\lambda.$$
The latter integral can be explicitly carried out for each of the subcases in Table 1. It will suffice, however, to have a simple estimate of the form $O(X^r)$, with $r<1$, for the integral corresponding to each subcase. For example, if the total exponent of $s_i$ in (\[estv1s\]) is negative for all $i$ in $\{1,\ldots,7\}$, then it is clear that the resulting integral will be at most $O(X^{(40-|T_0|)/40})$ in value. This condition holds for many of the subcases in Table 1 (indicated in the fourth column by “-”), immediately yielding the estimates given in the third column.
For cases where this negative exponent condition does not hold, the estimate given in the third column can be obtained as follows. The factor $\pi$ given in the fourth column is a product of variables in $T_1$, and so it is at least one in absolute value. The integrand in (\[estv1s\]) may thus be multiplied by $\pi$ without harm, and the estimate (\[estv1s\]) will remain true; we may then apply the inequalities (\[condt\]) to each of the variables in $\pi$, yielding $$\label{estv2s}
N(V(\CC);X)\ll
\int_{\lambda=c'}^{X^{1/40}} \int_{s_1,s_2,\ldots,s_7=c}^\infty
\prod_{t\in T\setminus T_0} w(t)\;w(\pi)
\,\, s_1^{-12}s_2^{-8}s_3^{-12}s_4^{-20}s_5^{-30}s_6^{-30}s_7^{-20} \,\,\,
d^\times\! s \,d^\times\!\lambda.$$ where we extend the notation $w$ multiplicatively, i.e., $w(ab)=w(a)w(b)$. In each subcase of Table 1, we have chosen the factor $\pi$ so that the total exponent of each $s_i$ in (\[estv2s\]) is negative. Thus we obtain from (\[estv2s\]) that $N(V(\CC);X)=O(X^{(40-\#T_0(\CC)+\#\pi)/40}$, where $\#\pi$ denotes the total number of variables of $T$ appearing in $\pi$ (counted with multiplicity), and this is precisely the estimate given in the third column of Table 1. In every subcase, aside from Case 0, we see that $40-\#T_0+\#\pi<40$, as desired.
Case The set $S\subset V_\Z$ defined by $N(S;X)\ll$ Use factor
------ ------------------------------------------------- ------------- --------------
0. ${a_{12}}\neq0\,$ $X^{40/40}$ -
1. ${a_{12}}=0\,;$ $X^{39/40}$ -
${a_{13}, b_{12}}\neq 0$
2a. ${a_{12}, a_{13}}=0\,;$ $X^{38/40}$ -
${a_{14}, a_{23}, b_{12}}\neq 0$
2b. ${a_{12}, b_{12}}=0\,;$ $X^{38/40}$ -
${a_{13}, c_{12}}\neq 0$
3a. ${a_{12}, a_{13}, a_{14}}=0\,;$ $X^{37/40}$ -
${a_{15}, a_{23}, b_{12}}\neq 0$
3b. ${a_{12}, a_{13}, a_{23}}=0\,;$ $X^{37/40}$ -
${a_{14}, b_{12}}\neq 0$
3c. ${a_{12}, a_{13}, b_{12}}=0\,;$ $X^{37/40}$ -
${a_{14}, a_{23}, b_{13}, c_{12}}\neq 0$
3d. ${a_{12}, b_{12}, c_{12}}=0\,;$ $X^{37/40}$ -
${a_{13}, d_{12}}\neq 0$
4a. ${a_{12}, a_{13}, a_{14}, a_{15}}=0\,;$ $X^{37/40}$ $ a_{23}$
${a_{23}, b_{12}}\neq 0$
4b. ${a_{12}, a_{13}, a_{14}, a_{23}}=0\,;$ $X^{37/40}$ $ a_{24}$
${a_{15}, a_{24}, b_{12}}\neq 0$
4c. ${a_{12}, a_{13}, a_{14}, b_{12}}=0\,;$ $X^{36/40}$ -
${a_{15}, a_{23}, b_{13}, c_{12}}\neq 0$
4d. ${a_{12}, a_{13}, a_{23}, b_{12}}=0\,;$ $X^{36/40}$ -
${a_{14}, b_{13}, c_{12}}\neq 0$
4e. ${a_{12}, a_{13}, b_{12}, b_{13}}=0\,;$ $X^{36/40}$ -
${a_{14}, a_{23}, c_{12}}\neq 0$
4f. ${a_{12}, a_{13}, b_{12}, c_{12}}=0\,;$ $X^{36/40}$ -
${a_{14}, a_{23}, b_{13}, d_{12}}\neq 0$
4g. ${a_{12}, b_{12}, c_{12}, d_{12}}=0\,;$ $X^{36/40}$ -
${a_{13}}\neq 0$
5a. ${a_{12}, a_{13}, a_{14}, a_{15}, a_{23}}=0\,;$ $X^{37/40}$ $ a_{24}^2 $
${a_{24}, b_{12}}\neq 0$
5b. ${a_{12}, a_{13}, a_{14}, a_{15}, b_{12}}=0\,;$ $X^{35/40}$ -
${a_{23}, b_{13}, c_{12}}\neq 0$
5c. ${a_{12}, a_{13}, a_{14}, a_{23}, a_{24}}=0\,;$ $X^{37/40}$ $ a_{34}^2 $
${a_{15}, a_{34}, b_{12}}\neq 0$
5d. ${a_{12}, a_{13}, a_{14}, a_{23}, b_{12}}=0\,;$ $X^{35/40}$ -
${a_{15}, a_{24}, b_{13}, c_{12}}\neq 0$
[**Table 1.**]{} Subcases 0–5d.
Case The set $S\subset V_\Z$ defined by $N(S;X)\ll$ Use factor
------ --------------------------------------------------------- ------------- --------------
5e. ${a_{12}, a_{13}, a_{14}, b_{12}, b_{13}}=0\,;$ $X^{35/40}$ -
${a_{15}, a_{23}, b_{14}, c_{12}}\neq 0$
5f. ${a_{12}, a_{13}, a_{14}, b_{12}, c_{12}}=0\,;$ $X^{35/40}$ -
${a_{15}, a_{23}, b_{13}, d_{12}}\neq 0$
5g. ${a_{12}, a_{13}, a_{23}, b_{12}, b_{13}}=0\,;$ $X^{35/40}$ -
${a_{14}, b_{23}, c_{12}}\neq 0$
5h. ${a_{12}, a_{13}, a_{23}, b_{12}, c_{12}}=0\,;$ $X^{35/40}$ -
${a_{14}, b_{13}, d_{12}}\neq 0$
5i. ${a_{12}, a_{13}, b_{12}, b_{13}, c_{12}}=0\,;$ $X^{35/40}$ -
${a_{14}, a_{23}, c_{13}, d_{12}}\neq 0$
5j. ${a_{12}, a_{13}, b_{12}, c_{12}, d_{12}}=0\,;$ $X^{35/40}$ -
${a_{14}, a_{23}, b_{13}}\neq 0$
6a. ${a_{12}, a_{13}, a_{14}, a_{15}, a_{23}, a_{24}}=0\,;$ $X^{37/40}$ $ a_{34}^3 $
${a_{25}, a_{34}, b_{12}}\neq 0$
6b. ${a_{12}, a_{13}, a_{14}, a_{15}, a_{23}, b_{12}}=0\,;$ $X^{35/40}$ $ a_{24}$
${a_{24}, b_{13}, c_{12}}\neq 0$
6c. ${a_{12}, a_{13}, a_{14}, a_{15}, b_{12}, b_{13}}=0\,;$ $X^{34/40}$ -
${a_{23}, b_{14}, c_{12}}\neq 0$
6d. ${a_{12}, a_{13}, a_{14}, a_{15}, b_{12}, c_{12}}=0\,;$ $X^{34/40}$ -
${a_{23}, b_{13}, d_{12}}\neq 0$
6e. ${a_{12}, a_{13}, a_{14}, a_{23}, a_{24}, b_{12}}=0\,;$ $X^{35/40}$ $ a_{34}$
${a_{15}, a_{34}, b_{13}, c_{12}}\neq 0$
6f. ${a_{12}, a_{13}, a_{14}, a_{23}, b_{12}, b_{13}}=0\,;$ $X^{34/40}$ -
${a_{15}, a_{24}, b_{14}, b_{23}, c_{12}}\neq 0$
6g. ${a_{12}, a_{13}, a_{14}, a_{23}, b_{12}, c_{12}}=0\,;$ $X^{34/40}$ -
${a_{15}, a_{24}, b_{13}, d_{12}}\neq 0$
6h. ${a_{12}, a_{13}, a_{14}, b_{12}, b_{13}, b_{14}}=0\,;$ $X^{34/40}$ -
${a_{15}, a_{23}, c_{12}}\neq 0$
6i. ${a_{12}, a_{13}, a_{14}, b_{12}, b_{13}, c_{12}}=0\,;$ $X^{34/40}$ -
${a_{15}, a_{23}, b_{14}, c_{13}, d_{12}}\neq 0$
6j. ${a_{12}, a_{13}, a_{14}, b_{12}, c_{12}, d_{12}}=0\,;$ $X^{34/40}$ -
${a_{15}, a_{23}, b_{13}}\neq 0$
6k. ${a_{12}, a_{13}, a_{23}, b_{12}, b_{13}, b_{23}}=0\,;$ $X^{34/40}$ -
${a_{14}, c_{12}}\neq 0$
6l. ${a_{12}, a_{13}, a_{23}, b_{12}, b_{13}, c_{12}}=0\,;$ $X^{34/40}$ -
${a_{14}, b_{23}, c_{13}, d_{12}}\neq 0$
6m. ${a_{12}, a_{13}, a_{23}, b_{12}, c_{12}, d_{12}}=0\,;$ $X^{34/40}$ -
${a_{14}, b_{13}}\neq 0$
[**Table 1.**]{} Subcases 5e–6m.
-----------------------------------------------------------------------------------------------------
Case The set $S\subset V_\Z$ defined by $N(S;X)\ll$ Use factor
------ ----------------------------------------------------------------- ------------- --------------
6n. ${a_{12}, a_{13}, b_{12}, b_{13}, c_{12}, c_{13}}=0\,;$ $X^{34/40}$ -
${a_{14}, a_{23}, d_{12}}\neq 0$
6o. ${a_{12}, a_{13}, b_{12}, b_{13}, c_{12}, d_{12}}=0\,;$ $X^{34/40}$ -
${a_{14}, a_{23}, c_{13}}\neq 0$
7a. ${a_{12}, a_{13}, a_{14}, a_{15}, a_{23}, a_{24}, b_{12}}=0\,;$ $X^{35 $ a_{34}^2 $
/40}$
${a_{25}, a_{34}, b_{13}, c_{12}}\neq 0$
7b. ${a_{12}, a_{13}, a_{14}, a_{15}, a_{23}, b_{12}, b_{13}}=0\,;$ $X^{34 $ a_{24}$
/40}$
${a_{24}, b_{14}, b_{23}, c_{12}}\neq 0$
7c. ${a_{12}, a_{13}, a_{14}, a_{15}, a_{23}, b_{12}, c_{12}}=0\,;$ $X^{34 $ a_{24}$
/40}$
${a_{24}, b_{13}, d_{12}}\neq 0$
7d. ${a_{12}, a_{13}, a_{14}, a_{15}, b_{12}, b_{13}, b_{14}}=0\,;$ $X^{34 $ b_{15}$
/40}$
${a_{23}, b_{15}, c_{12}}\neq 0$
7e. ${a_{12}, a_{13}, a_{14}, a_{15}, b_{12}, b_{13}, c_{12}}=0\,;$ $X^{34 $ d_{12}$
/40}$
${a_{23}, b_{14}, c_{13}, d_{12}}\neq 0$
7f. ${a_{12}, a_{13}, a_{14}, a_{15}, b_{12}, c_{12}, d_{12}}=0\,;$ $X^{34 $ b_{13}$
/40}$
${a_{23}, b_{13}}\neq 0$
7g. ${a_{12}, a_{13}, a_{14}, a_{23}, a_{24}, b_{12}, b_{13}}=0\,;$ $X^{34 $ a_{34}$
/40}$
${a_{15}, a_{34}, b_{14}, b_{23}, c_{12}}\neq 0$
7h. ${a_{12}, a_{13}, a_{14}, a_{23}, a_{24}, b_{12}, c_{12}}=0\,;$ $X^{34 $ a_{34}$
/40}$
${a_{15}, a_{34}, b_{13}, d_{12}}\neq 0$
7i. ${a_{12}, a_{13}, a_{14}, a_{23}, b_{12}, b_{13}, b_{14}}=0\,;$ $X^{33 -
/40}$
${a_{15}, a_{24}, b_{23}, c_{12}}\neq 0$
7j. ${a_{12}, a_{13}, a_{14}, a_{23}, b_{12}, b_{13}, b_{23}}=0\,;$ $X^{33 -
/40}$
${a_{15}, a_{24}, b_{14}, c_{12}}\neq 0$
7k. ${a_{12}, a_{13}, a_{14}, a_{23}, b_{12}, b_{13}, c_{12}}=0\,;$ $X^{33 -
/40}$
${a_{15}, a_{24}, b_{14}, b_{23}, c_{13}, d_{12}}\neq 0$
7l. ${a_{12}, a_{13}, a_{14}, a_{23}, b_{12}, c_{12}, d_{12}}=0\,;$ $X^{33 -
/40}$
${a_{15}, a_{24}, b_{13}}\neq 0$
7m. ${a_{12}, a_{13}, a_{14}, b_{12}, b_{13}, b_{14}, c_{12}}=0\,;$ $X^{34 $ d_{12}$
/40}$
${a_{15}, a_{23}, c_{13}, d_{12}}\neq 0$
7n. ${a_{12}, a_{13}, a_{14}, b_{12}, b_{13}, c_{12}, c_{13}}=0\,;$ $X^{34 $ d_{12}$
/40}$
${a_{15}, a_{23}, b_{14}, d_{12}}\neq 0$
7o. ${a_{12}, a_{13}, a_{14}, b_{12}, b_{13}, c_{12}, d_{12}}=0\,;$ $X^{34 $ c_{13}$
/40}$
${a_{15}, a_{23}, b_{14}, c_{13}}\neq 0$
7p. ${a_{12}, a_{13}, a_{23}, b_{12}, b_{13}, b_{23}, c_{12}}=0\,;$ $X^{33 -
/40}$
${a_{14}, c_{13}, d_{12}}\neq 0$
7q. ${a_{12}, a_{13}, a_{23}, b_{12}, b_{13}, c_{12}, c_{13}}=0\,;$ $X^{33 -
/40}$
${a_{14}, b_{23}, d_{12}}\neq 0$
-----------------------------------------------------------------------------------------------------
[**Table 1.**]{} Subcases 6n–7q.
----------------------------------------------------------------------------------------------------------
Case The set $S\subset V_\Z$ defined by $N(S;X)\ll$ Use factor
------ ------------------------------------------------------------------- ------------- -----------------
7r. ${a_{12}, a_{13}, a_{23}, b_{12}, b_{13}, c_{12}, d_{12}}=0\,;$ $X^{33 -
/40}$
${a_{14}, b_{23}, c_{13}}\neq 0$
7s. ${a_{12}, a_{13}, b_{12}, b_{13}, c_{12}, c_{13}, d_{12}}=0\,;$ $X^{34 $ d_{13}$
/40}$
${a_{14}, a_{23}, d_{13}}\neq 0$
8a. ${a_{12}, a_{13}, a_{14}, a_{15}, a_{23}, a_{24}, b_{12}, b_{13}} $X^{34/40}$ $ a_{34}^2 $
=0\,;$
${a_{25}, a_{34}, b_{14}, b_{23}, c_{12}}\neq 0$
8b. ${a_{12}, a_{13}, a_{14}, a_{15}, a_{23}, a_{24}, b_{12}, c_{12}} $X^{34/40}$ $ a_{25}a_{34}$
=0\,;$
${a_{25}, a_{34}, b_{13}, d_{12}}\neq 0$
8c. ${a_{12}, a_{13}, a_{14}, a_{15}, a_{23}, b_{12}, b_{13}, b_{14}} $X^{34/40}$ $ a_{24}b_{15}$
=0\,;$
${a_{24}, b_{15}, b_{23}, c_{12}}\neq 0$
8d. ${a_{12}, a_{13}, a_{14}, a_{15}, a_{23}, b_{12}, b_{13}, b_{23}} $X^{33/40}$ $ a_{24}$
=0\,;$
${a_{24}, b_{14}, c_{12}}\neq 0$
8e. ${a_{12}, a_{13}, a_{14}, a_{15}, a_{23}, b_{12}, b_{13}, c_{12}} $X^{34/40}$ $ a_{24}d_{12}$
=0\,;$
${a_{24}, b_{14}, b_{23}, c_{13}, d_{12}}\neq 0$
8f. ${a_{12}, a_{13}, a_{14}, a_{15}, a_{23}, b_{12}, c_{12}, d_{12}} $X^{34/40}$ $ a_{24}b_{13}$
=0\,;$
${a_{24}, b_{13}}\neq 0$
8g. ${a_{12}, a_{13}, a_{14}, a_{15}, b_{12}, b_{13}, b_{14}, c_{12}} $X^{34/40}$ $ b_{15}d_{12}$
=0\,;$
${a_{23}, b_{15}, c_{13}, d_{12}}\neq 0$
8h. ${a_{12}, a_{13}, a_{14}, a_{15}, b_{12}, b_{13}, c_{12}, c_{13}} $X^{34/40}$ $ b_{14}d_{12}$
=0\,;$
${a_{23}, b_{14}, d_{12}}\neq 0$
8i. ${a_{12}, a_{13}, a_{14}, a_{15}, b_{12}, b_{13}, c_{12}, d_{12}} $X^{34/40}$ $ c_{13}^2 $
=0\,;$
${a_{23}, b_{14}, c_{13}}\neq 0$
8j. ${a_{12}, a_{13}, a_{14}, a_{23}, a_{24}, b_{12}, b_{13}, b_{14}} $X^{33/40}$ $ a_{34}$
=0\,;$
${a_{15}, a_{34}, b_{23}, c_{12}}\neq 0$
8k. ${a_{12}, a_{13}, a_{14}, a_{23}, a_{24}, b_{12}, b_{13}, b_{23}} $X^{33/40}$ $ a_{34}$
=0\,;$
${a_{15}, a_{34}, b_{14}, c_{12}}\neq 0$
8l. ${a_{12}, a_{13}, a_{14}, a_{23}, a_{24}, b_{12}, b_{13}, c_{12}} $X^{33/40}$ $ a_{34}$
=0\,;$
${a_{15}, a_{34}, b_{14}, b_{23}, c_{13}, d_{12}}\neq 0$
8m. ${a_{12}, a_{13}, a_{14}, a_{23}, a_{24}, b_{12}, c_{12}, d_{12}} $X^{33/40}$ $ a_{15}$
=0\,;$
${a_{15}, a_{34}, b_{13}}\neq 0$
8n. ${a_{12}, a_{13}, a_{14}, a_{23}, b_{12}, b_{13}, b_{14}, b_{23}} $X^{33/40}$ $ a_{24}$
=0\,;$
${a_{15}, a_{24}, c_{12}}\neq 0$
8o. ${a_{12}, a_{13}, a_{14}, a_{23}, b_{12}, b_{13}, b_{14}, c_{12}} $X^{32/40}$ -
=0\,;$
${a_{15}, a_{24}, b_{23}, c_{13}, d_{12}}\neq 0$
8p. ${a_{12}, a_{13}, a_{14}, a_{23}, b_{12}, b_{13}, b_{23}, c_{12}} $X^{32/40}$ -
=0\,;$
${a_{15}, a_{24}, b_{14}, c_{13}, d_{12}}\neq 0$
8q. ${a_{12}, a_{13}, a_{14}, a_{23}, b_{12}, b_{13}, c_{12}, c_{13}} $X^{32/40}$ -
=0\,;$
${a_{15}, a_{24}, b_{14}, b_{23}, d_{12}}\neq 0$
----------------------------------------------------------------------------------------------------------
[**Table 1.**]{} Subcases 7r–8q.
----------------------------------------------------------------------------------------------------------------
Case The set $S\subset V_\Z$ defined by $N(S;X)\ll$ Use factor
------ ------------------------------------------------------------------- ------------- -----------------------
8r. ${a_{12}, a_{13}, a_{14}, a_{23}, b_{12}, b_{13}, c_{12}, d_{12}} $X^{32/40}$ -
=0\,;$
${a_{15}, a_{24}, b_{14}, b_{23}, c_{13}}\neq 0$
8s. ${a_{12}, a_{13}, a_{14}, b_{12}, b_{13}, b_{14}, c_{12}, c_{13}} $X^{34/40}$ $ c_{14}d_{12}$
=0\,;$
${a_{15}, a_{23}, c_{14}, d_{12}}\neq 0$
8t. ${a_{12}, a_{13}, a_{14}, b_{12}, b_{13}, b_{14}, c_{12}, d_{12}} $X^{34/40}$ $ c_{13}^2 $
=0\,;$
${a_{15}, a_{23}, c_{13}}\neq 0$
8u. ${a_{12}, a_{13}, a_{14}, b_{12}, b_{13}, c_{12}, c_{13}, d_{12}} $X^{34/40}$ $ d_{13}^2 $
=0\,;$
${a_{15}, a_{23}, b_{14}, d_{13}}\neq 0$
8v. ${a_{12}, a_{13}, a_{23}, b_{12}, b_{13}, b_{23}, c_{12}, c_{13}} $X^{33/40}$ $ d_{12}$
=0\,;$
${a_{14}, c_{23}, d_{12}}\neq 0$
8w. ${a_{12}, a_{13}, a_{23}, b_{12}, b_{13}, b_{23}, c_{12}, d_{12}} $X^{33/40}$ $ c_{13}$
=0\,;$
${a_{14}, c_{13}}\neq 0$
8x. ${a_{12}, a_{13}, a_{23}, b_{12}, b_{13}, c_{12}, c_{13}, d_{12}} $X^{33/40}$ $ d_{13}$
=0\,;$
${a_{14}, b_{23}, d_{13}}\neq 0$
9a. ${a_{12}, a_{13}, a_{14}, a_{15}, a_{23}, a_{24}, b_{12}, b_{13}, $X^{34/40}$ $ a_{34}^2 b_{15}$
b_{14}}=0\,;$
${a_{25}, a_{34}, b_{15}, b_{23}, c_{12}}\neq 0$
9b. ${a_{12}, a_{13}, a_{14}, a_{15}, a_{23}, a_{24}, b_{12}, b_{13}, $X^{33/40}$ $ a_{34}^2 $
b_{23}}=0\,;$
${a_{25}, a_{34}, b_{14}, c_{12}}\neq 0$
9c. ${a_{12}, a_{13}, a_{14}, a_{15}, a_{23}, a_{24}, b_{12}, b_{13}, $X^{34/40}$ $ a_{34}^2 d_{12}$
c_{12}}=0\,;$
${a_{25}, a_{34}, b_{14}, b_{23}, c_{13}, d_{12}}\neq 0$
9d. ${a_{12}, a_{13}, a_{14}, a_{15}, a_{23}, a_{24}, b_{12}, c_{12}, $X^{34/40}$ $ a_{25}^2 b_{13}$
d_{12}}=0\,;$
${a_{25}, a_{34}, b_{13}}\neq 0$
9e. ${a_{12}, a_{13}, a_{14}, a_{15}, a_{23}, b_{12}, b_{13}, b_{14}, $X^{33/40}$ $ a_{24}b_{15}$
b_{23}}=0\,;$
${a_{24}, b_{15}, c_{12}}\neq 0$
9f. ${a_{12}, a_{13}, a_{14}, a_{15}, a_{23}, b_{12}, b_{13}, b_{14}, $X^{34/40}$ $ a_{24}b_{15}d_{12}$
c_{12}}=0\,;$
${a_{24}, b_{15}, b_{23}, c_{13}, d_{12}}\neq 0$
9g. ${a_{12}, a_{13}, a_{14}, a_{15}, a_{23}, b_{12}, b_{13}, b_{23}, $X^{31/40}$ -
c_{12}}=0\,;$
${a_{24}, b_{14}, c_{13}, d_{12}}\neq 0$
9h. ${a_{12}, a_{13}, a_{14}, a_{15}, a_{23}, b_{12}, b_{13}, c_{12}, $X^{34/40}$ $ a_{24}b_{14}d_{12}$
c_{13}}=0\,;$
${a_{24}, b_{14}, b_{23}, d_{12}}\neq 0$
9i. ${a_{12}, a_{13}, a_{14}, a_{15}, a_{23}, b_{12}, b_{13}, c_{12}, $X^{34/40}$ $ a_{24}c_{13}^2 $
d_{12}}=0\,;$
${a_{24}, b_{14}, b_{23}, c_{13}}\neq 0$
9j. ${a_{12}, a_{13}, a_{14}, a_{15}, b_{12}, b_{13}, b_{14}, c_{12}, $X^{34/40}$ $ b_{15}c_{14}d_{12}$
c_{13}}=0\,;$
${a_{23}, b_{15}, c_{14}, d_{12}}\neq 0$
9k. ${a_{12}, a_{13}, a_{14}, a_{15}, b_{12}, b_{13}, b_{14}, c_{12}, $X^{34/40}$ $ b_{15}c_{13}^2 $
d_{12}}=0\,;$
${a_{23}, b_{15}, c_{13}}\neq 0$
9l. ${a_{12}, a_{13}, a_{14}, a_{15}, b_{12}, b_{13}, c_{12}, c_{13}, $X^{34/40}$ $ b_{14}d_{13}^2 $
d_{12}}=0\,;$
${a_{23}, b_{14}, d_{13}}\neq 0$
----------------------------------------------------------------------------------------------------------------
[**Table 1.**]{} Subcases 8r–9l.
-------------------------------------------------------------------------------------------------------------------
Case The set $S\subset V_\Z$ defined by $N(S;X)\ll$ Use factor
------ ------------------------------------------------------------------- ------------- --------------------------
9m. ${a_{12}, a_{13}, a_{14}, a_{23}, a_{24}, b_{12}, b_{13}, b_{14}, $X^{33/40}$ $ a_{34}b_{24}$
b_{23}}=0\,;$
${a_{15}, a_{34}, b_{24}, c_{12}}\neq 0$
9n. ${a_{12}, a_{13}, a_{14}, a_{23}, a_{24}, b_{12}, b_{13}, b_{14}, $X^{31/40}$ -
c_{12}}=0\,;$
${a_{15}, a_{34}, b_{23}, c_{13}, d_{12}}\neq 0$
9o. ${a_{12}, a_{13}, a_{14}, a_{23}, a_{24}, b_{12}, b_{13}, b_{23}, $X^{31/40}$ -
c_{12}}=0\,;$
${a_{15}, a_{34}, b_{14}, c_{13}, d_{12}}\neq 0$
9p. ${a_{12}, a_{13}, a_{14}, a_{23}, a_{24}, b_{12}, b_{13}, c_{12}, $X^{31/40}$ -
c_{13}}=0\,;$
${a_{15}, a_{34}, b_{14}, b_{23}, d_{12}}\neq 0$
9q. ${a_{12}, a_{13}, a_{14}, a_{23}, a_{24}, b_{12}, b_{13}, c_{12}, $X^{31/40}$ -
d_{12}}=0\,;$
${a_{15}, a_{34}, b_{14}, b_{23}, c_{13}}\neq 0$
9r. ${a_{12}, a_{13}, a_{14}, a_{23}, b_{12}, b_{13}, b_{14}, b_{23}, $X^{31/40}$ -
c_{12}}=0\,;$
${a_{15}, a_{24}, c_{13}, d_{12}}\neq 0$
9s. ${a_{12}, a_{13}, a_{14}, a_{23}, b_{12}, b_{13}, b_{14}, c_{12}, $X^{32/40}$ $ c_{14}$
c_{13}}=0\,;$
${a_{15}, a_{24}, b_{23}, c_{14}, d_{12}}\neq 0$
9t. ${a_{12}, a_{13}, a_{14}, a_{23}, b_{12}, b_{13}, b_{14}, c_{12}, $X^{32/40}$ $ c_{13}$
d_{12}}=0\,;$
${a_{15}, a_{24}, b_{23}, c_{13}}\neq 0$
9u. ${a_{12}, a_{13}, a_{14}, a_{23}, b_{12}, b_{13}, b_{23}, c_{12}, $X^{32/40}$ $ c_{23}$
c_{13}}=0\,;$
${a_{15}, a_{24}, b_{14}, c_{23}, d_{12}}\neq 0$
9v. ${a_{12}, a_{13}, a_{14}, a_{23}, b_{12}, b_{13}, b_{23}, c_{12}, $X^{32/40}$ $ c_{13}$
d_{12}}=0\,;$
${a_{15}, a_{24}, b_{14}, c_{13}}\neq 0$
9w. ${a_{12}, a_{13}, a_{14}, a_{23}, b_{12}, b_{13}, c_{12}, c_{13}, $X^{32/40}$ $ d_{13}$
d_{12}}=0\,;$
${a_{15}, a_{24}, b_{14}, b_{23}, d_{13}}\neq 0$
9x. ${a_{12}, a_{13}, a_{14}, b_{12}, b_{13}, b_{14}, c_{12}, c_{13}, $X^{34/40}$ $ c_{14}d_{13}^2 $
d_{12}}=0\,;$
${a_{15}, a_{23}, c_{14}, d_{13}}\neq 0$
9y. ${a_{12}, a_{13}, a_{23}, b_{12}, b_{13}, b_{23}, c_{12}, c_{13}, $X^{33/40}$ $ d_{13}^2 $
d_{12}}=0\,;$
${a_{14}, c_{23}, d_{13}}\neq 0$
10a. ${a_{12}, a_{13}, a_{14}, a_{15}, a_{23}, a_{24}, b_{12}, b_{13}, $X^{33/40}$ $ a_{34}^2 b_{15}$
b_{14}, b_{23}}=0\,;$
${a_{25}, a_{34}, b_{15}, b_{24}, c_{12}}\neq 0$
10b. ${a_{12}, a_{13}, a_{14}, a_{15}, a_{23}, a_{24}, b_{12}, b_{13}, $X^{34/40}$ $ a_{34}^2 b_{15}d_{12}$
b_{14}, c_{12}}=0\,;$
${a_{25}, a_{34}, b_{15}, b_{23}, c_{13}, d_{12}}\neq 0$
10c. ${a_{12}, a_{13}, a_{14}, a_{15}, a_{23}, a_{24}, b_{12}, b_{13}, $X^{31/40}$ $ a_{34}$
b_{23}, c_{12}}=0\,;$
${a_{25}, a_{34}, b_{14}, c_{13}, d_{12}}\neq 0$
10d. ${a_{12}, a_{13}, a_{14}, a_{15}, a_{23}, a_{24}, b_{12}, b_{13}, $X^{34/40}$ $ a_{34}^2 b_{14}d_{12}$
c_{12}, c_{13}}=0\,;$
${a_{25}, a_{34}, b_{14}, b_{23}, d_{12}}\neq 0$
10e. ${a_{12}, a_{13}, a_{14}, a_{15}, a_{23}, a_{24}, b_{12}, b_{13}, $X^{34/40}$ $ a_{25}^2 c_{13}^2 $
c_{12}, d_{12}}=0\,;$
${a_{25}, a_{34}, b_{14}, b_{23}, c_{13}}\neq 0$
10f. ${a_{12}, a_{13}, a_{14}, a_{15}, a_{23}, b_{12}, b_{13}, b_{14}, $X^{31/40}$ $ b_{15}$
b_{23}, c_{12}}=0\,;$
${a_{24}, b_{15}, c_{13}, d_{12}}\neq 0$
-------------------------------------------------------------------------------------------------------------------
[**Table 1.**]{} Subcases 9m–10f.
----------------------------------------------------------------------------------------------------------------------------
Case The set $S\subset V_\Z$ defined by $N(S;X)\ll$ Use factor
------ ------------------------------------------------------------------- ------------- -----------------------------------
10g. ${a_{12}, a_{13}, a_{14}, a_{15}, a_{23}, b_{12}, b_{13}, b_{14}, $X^{34/40}$ $ a_{24}b_{15}c_{14}d_{12}$
c_{12}, c_{13}}=0\,;$
${a_{24}, b_{15}, b_{23}, c_{14}, d_{12}}\neq 0$
10h. ${a_{12}, a_{13}, a_{14}, a_{15}, a_{23}, b_{12}, b_{13}, b_{14}, $X^{34/40}$ $ a_{24}b_{15}c_{13}^2 $
c_{12}, d_{12}}=0\,;$
${a_{24}, b_{15}, b_{23}, c_{13}}\neq 0$
10i. ${a_{12}, a_{13}, a_{14}, a_{15}, a_{23}, b_{12}, b_{13}, b_{23}, $X^{33/40}$ $ a_{24}b_{14}d_{12}$
c_{12}, c_{13}}=0\,;$
${a_{24}, b_{14}, c_{23}, d_{12}}\neq 0$
10j. ${a_{12}, a_{13}, a_{14}, a_{15}, a_{23}, b_{12}, b_{13}, b_{23}, $X^{31/40}$ $ c_{13}$
c_{12}, d_{12}}=0\,;$
${a_{24}, b_{14}, c_{13}}\neq 0$
10k. ${a_{12}, a_{13}, a_{14}, a_{15}, a_{23}, b_{12}, b_{13}, c_{12}, $X^{34/40}$ $ a_{24}b_{14}d_{13}^2 $
c_{13}, d_{12}}=0\,;$
${a_{24}, b_{14}, b_{23}, d_{13}}\neq 0$
10l. ${a_{12}, a_{13}, a_{14}, a_{15}, b_{12}, b_{13}, b_{14}, c_{12}, $X^{34/40}$ $ b_{15}c_{14}d_{13}^2 $
c_{13}, d_{12}}=0\,;$
${a_{23}, b_{15}, c_{14}, d_{13}}\neq 0$
10m. ${a_{12}, a_{13}, a_{14}, a_{23}, a_{24}, b_{12}, b_{13}, b_{14}, $X^{31/40}$ $ b_{24}$
b_{23}, c_{12}}=0\,;$
${a_{15}, a_{34}, b_{24}, c_{13}, d_{12}}\neq 0$
10n. ${a_{12}, a_{13}, a_{14}, a_{23}, a_{24}, b_{12}, b_{13}, b_{14}, $X^{31/40}$ $ c_{14}$
c_{12}, c_{13}}=0\,;$
${a_{15}, a_{34}, b_{23}, c_{14}, d_{12}}\neq 0$
10o. ${a_{12}, a_{13}, a_{14}, a_{23}, a_{24}, b_{12}, b_{13}, b_{14}, $X^{31/40}$ $ c_{13}$
c_{12}, d_{12}}=0\,;$
${a_{15}, a_{34}, b_{23}, c_{13}}\neq 0$
10p. ${a_{12}, a_{13}, a_{14}, a_{23}, a_{24}, b_{12}, b_{13}, b_{23}, $X^{31/40}$ $ c_{23}$
c_{12}, c_{13}}=0\,;$
${a_{15}, a_{34}, b_{14}, c_{23}, d_{12}}\neq 0$
10q. ${a_{12}, a_{13}, a_{14}, a_{23}, a_{24}, b_{12}, b_{13}, b_{23}, $X^{31/40}$ $ c_{13}$
c_{12}, d_{12}}=0\,;$
${a_{15}, a_{34}, b_{14}, c_{13}}\neq 0$
10r. ${a_{12}, a_{13}, a_{14}, a_{23}, a_{24}, b_{12}, b_{13}, c_{12}, $X^{31/40}$ $ d_{13}$
c_{13}, d_{12}}=0\,;$
${a_{15}, a_{34}, b_{14}, b_{23}, d_{13}}\neq 0$
10s. ${a_{12}, a_{13}, a_{14}, a_{23}, b_{12}, b_{13}, b_{14}, b_{23}, $X^{33/40}$ $ a_{24}c_{14}d_{12}$
c_{12}, c_{13}}=0\,;$
${a_{15}, a_{24}, c_{14}, c_{23}, d_{12}}\neq 0$
10t. ${a_{12}, a_{13}, a_{14}, a_{23}, b_{12}, b_{13}, b_{14}, b_{23}, $X^{31/40}$ $ c_{13}$
c_{12}, d_{12}}=0\,;$
${a_{15}, a_{24}, c_{13}}\neq 0$
10u. ${a_{12}, a_{13}, a_{14}, a_{23}, b_{12}, b_{13}, b_{14}, c_{12}, $X^{32/40}$ $ c_{14}d_{13}$
c_{13}, d_{12}}=0\,;$
${a_{15}, a_{24}, b_{23}, c_{14}, d_{13}}\neq 0$
10v. ${a_{12}, a_{13}, a_{14}, a_{23}, b_{12}, b_{13}, b_{23}, c_{12}, $X^{32/40}$ $ c_{23}d_{13}$
c_{13}, d_{12}}=0\,;$
${a_{15}, a_{24}, b_{14}, c_{23}, d_{13}}\neq 0$
11a. ${a_{12}, a_{13}, a_{14}, a_{15}, a_{23}, a_{24}, b_{12}, b_{13}, $X^{31/40}$ $ a_{34}b_{15}$
b_{14}, b_{23}, c_{12}}=0\,;$
${a_{25}, a_{34}, b_{15}, b_{24}, c_{13}, d_{12}}\neq 0$
11b. ${a_{12}, a_{13}, a_{14}, a_{15}, a_{23}, a_{24}, b_{12}, b_{13}, $X^{34/40}$ $ a_{34}^2 b_{15}c_{14}d_{12}$
b_{14}, c_{12}, c_{13}}=0\,;$
${a_{25}, a_{34}, b_{15}, b_{23}, c_{14}, d_{12}}\neq 0$
11c. ${a_{12}, a_{13}, a_{14}, a_{15}, a_{23}, a_{24}, b_{12}, b_{13}, $X^{36/40}$ $ a_{25}^2 a_{34}b_{15}c_{13}^3 $
b_{14}, c_{12}, d_{12}}=0\,;$
${a_{25}, a_{34}, b_{15}, b_{23}, c_{13}}\neq 0$
----------------------------------------------------------------------------------------------------------------------------
[**Table 1.**]{} Subcases 10g–11c.
----------------------------------------------------------------------------------------------------------------------------------------
Case The set $S\subset V_\Z$ defined by $N(S;X)\ll$ Use factor
------ ------------------------------------------------------------------- ------------- -----------------------------------------------
11d. ${a_{12}, a_{13}, a_{14}, a_{15}, a_{23}, a_{24}, b_{12}, b_{13}, $X^{33/40}$ $ a_{34}^2 b_{14}d_{12}$
b_{23}, c_{12}, c_{13}}=0\,;$
${a_{25}, a_{34}, b_{14}, c_{23}, d_{12}}\neq 0$
11e. ${a_{12}, a_{13}, a_{14}, a_{15}, a_{23}, a_{24}, b_{12}, b_{13}, $X^{31/40}$ $ a_{25}c_{13}$
b_{23}, c_{12}, d_{12}}=0\,;$
${a_{25}, a_{34}, b_{14}, c_{13}}\neq 0$
11f. ${a_{12}, a_{13}, a_{14}, a_{15}, a_{23}, a_{24}, b_{12}, b_{13}, $X^{34/40}$ $ a_{25}^2 b_{14}d_{13}^2 $
c_{12}, c_{13}, d_{12}}=0\,;$
${a_{25}, a_{34}, b_{14}, b_{23}, d_{13}}\neq 0$
11g. ${a_{12}, a_{13}, a_{14}, a_{15}, a_{23}, b_{12}, b_{13}, b_{14}, $X^{33/40}$ $ a_{24}b_{15}c_{14}d_{12}$
b_{23}, c_{12}, c_{13}}=0\,;$
${a_{24}, b_{15}, c_{14}, c_{23}, d_{12}}\neq 0$
11h. ${a_{12}, a_{13}, a_{14}, a_{15}, a_{23}, b_{12}, b_{13}, b_{14}, $X^{31/40}$ $ b_{15}c_{13}$
b_{23}, c_{12}, d_{12}}=0\,;$
${a_{24}, b_{15}, c_{13}}\neq 0$
11i. ${a_{12}, a_{13}, a_{14}, a_{15}, a_{23}, b_{12}, b_{13}, b_{14}, $X^{34/40}$ $ a_{24}b_{15}c_{14}d_{13}^2 $
c_{12}, c_{13}, d_{12}}=0\,;$
${a_{24}, b_{15}, b_{23}, c_{14}, d_{13}}\neq 0$
11j. ${a_{12}, a_{13}, a_{14}, a_{15}, a_{23}, b_{12}, b_{13}, b_{23}, $X^{33/40}$ $ a_{24}b_{14}d_{13}^2 $
c_{12}, c_{13}, d_{12}}=0\,;$
${a_{24}, b_{14}, c_{23}, d_{13}}\neq 0$
11k. ${a_{12}, a_{13}, a_{14}, a_{23}, a_{24}, b_{12}, b_{13}, b_{14}, $X^{33/40}$ $ a_{34}b_{24}c_{14}d_{12}$
b_{23}, c_{12}, c_{13}}=0\,;$
${a_{15}, a_{34}, b_{24}, c_{14}, c_{23}, d_{12}}\neq 0$
11l. ${a_{12}, a_{13}, a_{14}, a_{23}, a_{24}, b_{12}, b_{13}, b_{14}, $X^{31/40}$ $ b_{24}c_{13}$
b_{23}, c_{12}, d_{12}}=0\,;$
${a_{15}, a_{34}, b_{24}, c_{13}}\neq 0$
11m. ${a_{12}, a_{13}, a_{14}, a_{23}, a_{24}, b_{12}, b_{13}, b_{14}, $X^{31/40}$ $ c_{14}d_{13}$
c_{12}, c_{13}, d_{12}}=0\,;$
${a_{15}, a_{34}, b_{23}, c_{14}, d_{13}}\neq 0$
11n. ${a_{12}, a_{13}, a_{14}, a_{23}, a_{24}, b_{12}, b_{13}, b_{23}, $X^{31/40}$ $ c_{23}d_{13}$
c_{12}, c_{13}, d_{12}}=0\,;$
${a_{15}, a_{34}, b_{14}, c_{23}, d_{13}}\neq 0$
11o. ${a_{12}, a_{13}, a_{14}, a_{23}, b_{12}, b_{13}, b_{14}, b_{23}, $X^{33/40}$ $ a_{24}c_{14}d_{13}^2 $
c_{12}, c_{13}, d_{12}}=0\,;$
${a_{15}, a_{24}, c_{14}, c_{23}, d_{13}}\neq 0$
12a. ${a_{12}, a_{13}, a_{14}, a_{15}, a_{23}, a_{24}, b_{12}, b_{13}, $X^{33/40}$ $ a_{34}^2 b_{15}c_{14}d_{12}$
b_{14}, b_{23}, c_{12}, c_{13}}=0\,;$
${a_{25}, a_{34}, b_{15}, b_{24}, c_{14}, c_{23}, d_{12}}
\neq 0$
12b. ${a_{12}, a_{13}, a_{14}, a_{15}, a_{23}, a_{24}, b_{12}, b_{13}, $X^{36/40}$ $ a_{25}^2 a_{34}b_{15}b_{24}c_{13}^3 $
b_{14}, b_{23}, c_{12}, d_{12}}=0\,;$
${a_{25}, a_{34}, b_{15}, b_{24}, c_{13}}\neq 0$
12c. ${a_{12}, a_{13}, a_{14}, a_{15}, a_{23}, a_{24}, b_{12}, b_{13}, $X^{36/40}$ $ a_{25}^2 a_{34}b_{15}c_{14}^2 d_{13}^2 $
b_{14}, c_{12}, c_{13}, d_{12}}=0\,;$
${a_{25}, a_{34}, b_{15}, b_{23}, c_{14}, d_{13}}\neq 0$
12d. ${a_{12}, a_{13}, a_{14}, a_{15}, a_{23}, a_{24}, b_{12}, b_{13}, $X^{33/40}$ $ a_{25}^2 b_{14}d_{13}^2 $
b_{23}, c_{12}, c_{13}, d_{12}}=0\,;$
${a_{25}, a_{34}, b_{14}, c_{23}, d_{13}}\neq 0$
12e. ${a_{12}, a_{13}, a_{14}, a_{15}, a_{23}, b_{12}, b_{13}, b_{14}, $X^{33/40}$ $ a_{24}b_{15}c_{14}d_{13}^2 $
b_{23}, c_{12}, c_{13}, d_{12}}=0\,;$
${a_{24}, b_{15}, c_{14}, c_{23}, d_{13}}\neq 0$
12f. ${a_{12}, a_{13}, a_{14}, a_{23}, a_{24}, b_{12}, b_{13}, b_{14}, $X^{33/40}$ $ a_{15}b_{24}c_{23}d_{13}^2 $
b_{23}, c_{12}, c_{13}, d_{12}}=0\,;$
${a_{15}, a_{34}, b_{24}, c_{14}, c_{23}, d_{13}}\neq 0$
13. ${a_{12}, a_{13}, a_{14}, a_{15}, a_{23}, a_{24}, b_{12}, b_{13}, $X^{37/40}$ $ a_{25}^2 a_{34}b_{24}^2 c_{14}^2 d_{13}^3 $
b_{14}, b_{23}, c_{12}, c_{13}, d_{12}}=0\,;$
${a_{25}, a_{34}, b_{15}, b_{24}, c_{14}, c_{23}, d_{13}}
\neq 0$
----------------------------------------------------------------------------------------------------------------------------------------
[**Table 1.**]{} Subcases 11d–13.
Therefore, for the purposes of proving Theorem \[cna\], we may assume that $a_{12}\neq 0$.
The main term {#mainterm}
-------------
Let $\RR_X(v)$ denote the multiset $\{x\in
\FF v:|\Disc(x)|<X\}$. Then we have the following result counting the number of integral points in $\RR_X(v)$, on average, satisfying $a_{12}\neq 0$:
\[nonzeroa12\] Let $v$ take a random value in $H\cap V^{(i)}$ uniformly with respect to the measure $|\Disc(v)|^{-1}\,dv$. Then the expected number of integral elements $(A,B,C,D)\in\FF v$ such that $|\Disc(A,B,C,D)|< X$ and $a_{12}\neq0$ is $\Vol(\RR_X(v_i))
+ O(X^{39/40})$, where $v_i$ is any vector in $V^{(i)}$.
Following the proof of Lemma \[hard\], let $V^{(i)}(0)$ denote the subset of $V_\R$ such that $a_{12}\neq 0$. We wish to show that $$\label{toprove2}
N^*(V^{(i)}(0);X)=\frac{1}{n_i}\cdot\Vol(\RR_X(v_i)) + O(X^{39/40}).$$ We have $$\label{translate2}
N^*(V^{(i)}(0);X)=\frac{r_i}{M_i}\int_{\lambda=c'}^{X^{1/40}} \!\!
\int_{s_1,s_2,\ldots,s_7=c}^\infty
\int_{u\in N'(a(s))}
\sigma(V(0))
\, s_1^{-12}s_2^{-8}s_3^{-12}s_4^{-20}s_5^{-30}s_6^{-30}s_7^{-20} \,du\,
d^\times\! s \,d^\times\!\lambda,$$ where $\sigma(V(0))$ denotes the number of integer points in the region $H(u,s,\lambda,X)$ satisfying $|a_{12}|\geq 1$. Evidently, the number of integer points in $H(u,s,\lambda,X)$ with $|a_{12}|\geq 1$ can be nonzero only if we have $$\label{condt2}
J{w(a_{12})}=J\cdot\frac{\lambda}{s_1^3s_2s_3s_4^3s_5^6s_6^4s_7^2}\geq 1.$$ Therefore, if the region $\H=\{(A,B,C,D)\in
H(u,s,\lambda,X):|a_{12}|\geq 1\}$ contains an integer point, then (\[condt2\]) and Lemma \[genbound\] imply that the number of integer points in $\H$ is $\Vol(\H)+O(J^{-1}\Vol(\H)/w(a_{12}))$, since all smaller-dimensional projections of $u^{-1}\H$ are clearly bounded by a constant times the projection of $\H$ onto the hyperplane $a_{12}=0$ (since $a_{12}$ has minimal weight).
Therefore, since $\H=H(u,s,\lambda,X)-\bigl(H(u,s,\lambda,X)-\H\bigr)$, we may write $$\begin{aligned}
\label{bigint}
N^\ast(V^{(i)}(0);X) &\!\!\!=\!\!& \!\frac{r_i}{M_i}
\int_{\lambda=c'}^{X^{1/40}}
\int_{s_1,\ldots,s_7=c}^{\infty} \int_{u\in N'(a(s))}
\Bigl(\Vol\bigl(H(u,s,\lambda,X)\bigr)-\Vol\bigl(H(u,s,\lambda,X)-\H\bigr)
+ \\[.085in]\nonumber & & \,\,\,\,\,\,\,\,\,\,
\,\,\,\,\,\,\,O(\max\{{J^{39}\lambda^{39}s_1^3s_2s_3s_4^3s_5^6s_6^4s_7^2},1\})
\Bigr)
\, s_1^{-12}s_2^{-8}s_3^{-12}s_4^{-20}s_5^{-30}s_6^{-30}s_7^{-20} \,du\,
d^\times s\, d^\times \lambda.\end{aligned}$$ The integral of the first term in (\[bigint\]) is $(1/r_i)\cdot\int_{v\in H\cap V^{(i)}}
\Vol(\RR_X(v))|\Disc(v)|^{-1}dv$. Since $\Vol(\RR_X(v))$ does not depend on the choice of $v\in V^{(i)}$ (see Section \[volcomp\]), the latter integral is simply $[M_i/(n_i\,r_i)]\cdot \Vol(\RR_X(v))$. To estimate the integral of the second term in (\[bigint\]), let $\H'=
H(u,s,t,X)-\H$, and for each $|a_{12}|\leq 1$, let $\H'(a_{12})$ be the subset of all elements $(A,B,C,D)\in\H'$ with the given value of $a_{12}$. Then the 39-dimensional volume of $\H'(a_{12})$ is at most $O\Bigl(J^{39}\prod_{t\in T\setminus\{a_{12}\}}w(t)\Bigr)$, and so we have the estimate $$\Vol(\H') \ll \int_{-1}^1 J^{39}\prod_{t\in
T\setminus\{a_{12}\}}w(t)
\,\,da_{12} = O\Bigl(J^{39}\prod_{t\in T\setminus\{a_{12}\}}w(t)\Bigr).$$ The second term of the integrand in (\[bigint\]) can thus be absorbed into the third term.
Finally, one easily computes the integral of the third term in (\[bigint\]) to be $O(J^{39}X^{39/40})$. We thus obtain, for any $v\in V^{(i)}$, that $$\label{obtain}
N^\ast(V^{(i)};X) =
\frac1{n_i}\cdot\Vol(\RR_X(v))
+ O(J^{39}X^{39/40}/M_i(J)).$$
Note that the above proposition counts all integer points in $\RR_X(v)$ satisfying $a_{12}\neq 0$, not just the irreducible ones. However, in this regard we have the following lemma:
\[3reducible\] Let $v\in H\cap V^{(i)}$. Then the number of $(A,B,C,D)\in\FF v$ such that $a_{12}\neq 0$, $|\Disc(A,B,C,D)|<X$, and $(A,B,C,D)$ is not irreducible is $o(X)$.
Lemma \[3reducible\] will in fact follow from a stronger lemma. We say that an element $(A,B,C,D)\in V_\Z$ is [*absolutely irreducible*]{} if it is irreducible and the fraction field of its associated quintic ring is an $S_5$-quintic field (equivalently, if the fields of definition of its common zeroes in $\P^3$ are $S_5$-quintic fields). Then we have the following lemma, whose proof is postponed to Section 3:
\[3reducible2\] Let $v\in H\cap V^{(i)}$. Then the number of $(A,B,C,D)\in\FF v$ such that $a_{12}\neq 0$, $|\Disc(A,B,C,D)|<X$, and $(A,B,C,D)$ is not absolutely irreducible is $o(X)$.
Therefore, to prove Theorem \[cna\], it remains only to compute the fundamental volume $\Vol(\RR_X(v))$ for $v\in V^{(i)}$. This is handled in the next subsection.
Computation of the fundamental volume {#volcomp}
-------------------------------------
In this subsection, we compute $\Vol(\RR_X(v))$, where $\RR_X(v)$ is defined as in Section \[mainterm\]. We will see that this volume depends only on whether $v$ lies in $V^{(0)}$, $V^{(1)}$, or $V^{(2)}$; here $V^{(i)}$ again denotes the $G_\R$-orbit in $V_\R$ consisting of those elements $(A,B,C,D)$ having nonzero discriminant and possessing $5-2i$ real zeros in $\P^3$.
Before performing this computation, we first state two propositions regarding the group $G=\GL_4\times \SL_5$ and its 40-dimensional representation $V$.
\[covering\] The group $G_\R$ acts transitively on $V^{(i)}$, and the isotropy groups for $v\in V^{(i)}$ are given as follows:
$($[*i*]{}$)$ $S_5$, if $v\in V^{(0)}$;
$($[*ii*]{}$)$ $S_3\times C_2$, if $v\in V^{(1)}$; and
$($[*iii*]{}$)$ $D_4$, if $v\in V^{(2)}$.
In view of Proposition \[covering\], it will be convenient to use the notation $n_i$ to denote the order of the stabilizer of any vector $v\in V^{(i)}$. Proposition \[covering\] implies that we have $n_0=120$, $n_1=12$, and $n_2=8$.
Now define the usual subgroups $N$, $\bar N$, $A$, and $\Lambda$ of $G_\R$ as follows: $$\begin{aligned}
\label{subgroups}
N\,&=&\{n(x_1,x_2,\ldots,x_{16}):x_i\in\R \},\;\mbox{where}\,\\
{}&{}&
n({\boldmath{x}})={\footnotesize
\left(\left(\begin{array}{cccc} 1 & {x_1} & {x_2} & {x_3} \\
{} & 1
& {x_4} &{x_5} \\ {}&{}&1&{x_6}\\{}&{}&{}&1 \end{array}\right),
\left(\begin{array}{ccccc} 1 & {x_7} & {x_8} &{x_9} &{x_{10}}\\
{}& 1 &{x_{11}} &{x_{12}}&{x_{13}}\\{}&{}&
1&{x_{14}}&{x_{15}}\\{}&{}&{}&1&{x_{16}}\\
{}&{}&{}&{}&1\end{array} \right)\right)};\,\, \\
\bar N\,&=&\{\bar n(u_1,u_2,\ldots,u_{16}):u_i\in\R \},\;\mbox{where}\,\\
{}&{}&
\bar n({\boldmath{u}})={\footnotesize
\left(\left(\begin{array}{cccc} 1 & {} & {} & {} \\ {u_1} & 1
& {} &{} \\ {u_2}&{u_3}&1&{}\\{u_4}&{u_5}&{u_6}&1 \end{array}\right),
\left(\begin{array}{ccccc} 1 & {} & {} &{} &{}\\
{u_7}& 1 &{} &{}&{}\\{u_8}&{u_9}&
1&{}&{}\\{u_{10}}&{u_{11}}&{u_{12}}&1&{}\\
{u_{13}}&{u_{14}}&{u_{15}}&{u_{16}}&1\end{array} \right)\right)};\,\, \\
A&=&\{a(t_1,t_2,\ldots,t_7):t_1,t_2,\ldots,t_7\in\R_+\},\;\mbox{where}\,\\
{}&{}&a(\lambda,{\boldmath{t}})={\footnotesize
\left(\left(\begin{array}{cccc} t_1
&
{} & {} & {} \\[.04in] {} & t_2/t_1
& {} &{} \\[.04in] {}&{}& t_3/t_2
&{}\\[.04in] {}&{}&{}&
1/t_3 \end{array}\right),
\left(\begin{array}{ccccc} t_4
& {} & {} &{} &{}\\[.04in]
{}& t_5/t_4&{} &{}&{}\\[.04in]{}&{}&
t_6/t_5&{}&
{}\\[.04in]
{}&{}&{}& t_7/t_6
&{}\\[.04in] {}&{}&{}&{}& 1/t_7 \end{array} \right)\right)};\,\,\\
\Lambda\,&=&\{\{\lambda:\lambda>0\},\;\mbox{where}\,\\
{}&{}& \lambda \mbox{ acts by }{\footnotesize
\left(\left(\begin{array}{cccc} \lambda & {} & {} & {} \\ {} & \lambda
& {} &{} \\ {}&{}&\lambda&{}\\{}&{}&{}&\lambda \end{array}\right),
\left(\begin{array}{ccccc} 1 & {} & {} &{} &{}\\
{}& 1 &{} &{}&{}\\{}&{}&
1&{}&{}\\{}&{}&{}&1&{}\\
{}&{}&{}&{}&1\end{array} \right)\right)}.\end{aligned}$$
We define an invariant measure $dg$ on $G_\R$ by $$\int_G f(g) dg =
\int_{\R_+^\times}\int_{\R_+^{\times 7}}
\int_{\R^4}\int_{\R^4}
f(n(x)\bar n(u)a(t)\lambda)
\,dx\, du\, d^\times t\, d^\times\lambda.$$ With this choice of Haar measure on $G_\R$, it is known that $$\int_{G_\Z \backslash G^{\pm1}_\R} dg =
[\zeta(2)\zeta(3)\zeta(4)]\cdot[\zeta(2)\zeta(3)\zeta(4)\zeta(5)],$$ where $G^{\pm1}_\R\subset G_\R$ denotes the subgroup $\{(g_4,g_5)\in
G_\R:\det(g_4)=\pm1\}$ (see, e.g., [@Langlands]).
Now let $dy=dy_1\, dy_2 \cdots dy_{40}$ be the standard Euclidean measure on $V_\R$. Then we have:
\[volumes2\] For $i=0$[,]{} $1$[,]{} or $2$[,]{} let $f\in C_0(V^{(i)})$[,]{} and let $y$ denote any element of $V^{(i)}$. Then $$\label{twenty}
\int_{g\in G_\R} f(g\cdot y)dg
\,\,=\,\, \frac{\,n_i}{20}\cdot
\int_{v\in V^{(i)}}|\Disc(v)|^{-1}f(v)\,dv .$$
Put $$(z_1,\ldots,z_{40})=n(x) \bar n(u) a(t)\cdot y.$$ Then the form $\Disc(z)^{-1} dz_1\wedge\cdots\wedge dz_{12}$ is a $G_\R$-invariant measure, and so we must have $$\Disc(z)^{-1} dz_1\wedge\cdots\wedge dz_{40}=c\,\,dx\wedge du\wedge d^\times t\wedge
d^\times\lambda$$ for some constant factor $c$. An explicit Jacobian calculation shows that $c=-20$. (To make easier the calculation, we note that it suffices to check this on any fixed representative $y$ in $V^{(0)}$, $V^{(1)}$, or $V^{(2)}$.) By Proposition \[covering\], the group $G_\R$ is an $n_i$-fold covering of $V^{(i)}$ via the map $g\rightarrow
g\cdot y$. Hence $$\int_{G_\R} f(g\cdot y)dg = \frac{n_i}{20}\cdot\int_{V^{(i)}}|\Disc(v)|^{-1}f(v)dv.$$ as desired.
Finally, for any vector $y\in V^{(i)}$ of absolute discriminant 1, we obtain using Proposition \[volumes2\] that $$\frac1{n_i}\cdot\Vol(\RR_X(y)) = \frac{20}{n_i} \int_{1}^{X^{1/40}}
\lambda^{40}d^\times \lambda
\int_{G_\Z\backslash G^{\pm1}_\R}dg =
\frac{\zeta(2)^2\zeta(3)^2\zeta(4)^2\zeta(5)}{2 n_i}X,$$ proving Theorem \[cna\].
Congruence conditions
---------------------
We may prove a version of Theorem \[cna\] for a set in $V^{(i)}$ defined by a finite number of congruence conditions:
\[cong\] Suppose $S$ is a subset of $V^{(i)}_\Z$ defined by finitely many congruence conditions. Then we have $$\label{ramanujan}
\lim_{X\rightarrow\infty}\frac{N(S\cap V^{(i)};X)}{X}
= \frac{\zeta(2)^2\zeta(3)^2\zeta(4)^2\zeta(5)}{2n_i}
\prod_{p} \mu_p(S),$$ where $\mu_p(S)$ denotes the $p$-adic density of $S$ in $V_\Z$, and $n_i=120$, $12$, or $8$ for $i=0$, $1$, or $2$ respectively.
To obtain Theorem \[cong\], suppose $S$ is defined by congruence conditions modulo some integer $m$. Then $S$ may be viewed as the union of (say) $k$ translates $L_1,\ldots,L_k$ of the lattice $m\cdot
V_\Z$. For each such lattice translate $L_j$, we may use formula (\[avg\]) and the discussion following that formula to compute $N(S;X)$, but where each $d$-dimensional volume is scaled by a factor of $1/m^d$ to reflect the fact that our new lattice has been scaled by a factor of $m$. For a fixed value of $m$, we thus obtain $$\label{sestimate}
N(L_j;X) = m^{-40}\,\Vol(\RR_X(v)) + O(m^{-39}J^{39}X^{39/40}/M_i(J))$$ for $v\in V^{(i)}$, where the implied constant is also independent of $m$ provided $m=O(X^{1/40})$. Summing (\[sestimate\]) over $j$, and noting that $km^{-40}=\prod_p\mu_p(S)$, yields (\[ramanujan\]).
Quadruples of $5\times5$ skew-symmetric matrices and Theorems 1–4
=================================================================
Theorems \[main\] and \[cna\] of the previous section now immediately imply the following.
\[ringwithres\] Let $M_{5}^{*(i)}(\xi,\eta)$ denote the number of isomorphism classes of pairs $(R,R')$ such that $R$ is an order in an $S_5$-quintic field with $5-2i$ real embeddings, $R'$ is a sextic resolvent ring of $R$, and $\xi<\Disc(R)<\eta$. Then $$\begin{array}{rlcl}\label{dodqrr}
\rm{(a)}& \displaystyle{\lim_{X\rightarrow\infty} \frac{M_5^{*(0)}(0,X)}{X}}
&=&\! \displaystyle{\frac{\zeta(2)^2\zeta(3)^2\zeta(4)^2\zeta(5)}{240}}; \\[.1in]
\rm{(b)}& \displaystyle{\lim_{X\rightarrow\infty}
\frac{M_5^{*(1)}(-X,0)}{X}} &=&\!
\displaystyle{\frac{\zeta(2)^2\zeta(3)^2\zeta(4)^2\zeta(5)}{24}}; \\[.1in]
\rm{(c)}&\displaystyle{\lim_{X\rightarrow\infty} \frac{M_5^{*(2)}(0,X)}{X}}
&=& \!\displaystyle{\frac{\zeta(2)^2\zeta(3)^2\zeta(4)^2\zeta(5)}{16}}.
\end{array}$$
To obtain finer asymptotic information on the distribution of quintic rings (in particular, without the weighting by the number of sextic resolvents), we need to be able to count irreducible equivalence classes in $V_\Z$ lying in certain subsets $S\subset V_\Z$. If $S$ is defined, say, by [*finitely many*]{} congruence conditions, then Theorem \[cong\] applies in that case.
However, the set $S$ of elements $(A,B,C,D)\in V_\Z$ corresponding to maximal quintic orders is defined by infinitely many congruence conditions (see [@Bhargava4 §12]). To prove that (\[ramanujan\]) still holds for such a set, we require a uniform estimate on the error term when only finitely many factors are taken in (\[ramanujan\]). This estimate is provided in Section 3.1. In Section 3.2, we prove Lemma \[3reducible2\]. Finally, in Section 3.3, we complete the proofs of Theorems 1–4.
A uniformity estimate
---------------------
As in [@Bhargava4], for a prime number $p$ let us denote by $\mathcal U_p$ the set of all $(A,B,C,D)\in V_\Z$ corresponding to quintic orders $R$ that are maximal at $p$. Let $\mathcal W_p=V_\Z-\mathcal U_p$. In order to apply a sieve to obtain Theorems 1–4, we require the following proposition, analogous to Proposition 1 in [@DH] and Proposition 23 in [@Bhargava5].
\[errorestimate\] $N(\mathcal W_p;X) = O(X/p^2)$, where the implied constant is independent of $p$.
We begin with the following lemma.
\[upestimate\] The number of maximal orders in quintic fields, up to isomorphism, having absolute discriminant less than $X$ is $O(X)$.
Lemma \[upestimate\] follows immediately from Theorem \[ringwithres\], since we have shown that every quintic ring has a sextic resolvent ring ([@Bhargava4 Corollary 4]).
To estimate $N(\mathcal W_p;X)$ using Lemma \[upestimate\], we only need to know that (a) the number of subrings of index $p^k$ ($k\geq 1$) in a maximal quintic ring $R$ does not grow too rapidly with $k$; and (b) the number of sextic resolvents that such a subring possesses is also not too large relative to $p^{k}$. For (a), an even stronger result than we need here has recently been proven in the Ph.D. thesis [@Jos] of Jos Brakenhoff, who shows that the number of orders having index $p^k$ in a maximal quintic ring $R$ is at most $O(p^{\min\{2k-2,\frac{20}{11}k\}})$ for $k\geq1$, where the implied constant is independent of $p$, $k$, and $R$. Any such order will of course have discriminant $p^{2k}\:\!\Disc(R)$. As for (b), it follows from [@Bhargava4 Proof of Corollary 4] that the number of sextic resolvents of a quintic ring having content $n$ is $O(n^6)$; moreover, the number of sextic resolvents of a maximal quintic ring is 1. (Recall that the [*content*]{} of a quintic ring $R$ is the largest integer $n$ such that $R=\Z+nR'$ for some quintic ring $R'$.) Since every content $n$ quintic ring $R$ arises as $\Z+nR'$ for a unique content 1 quintic ring $R'$, and $\Disc(R)=n^8\;\!\Disc(R')$, we have $$N(\mathcal W_p;X)
=\sum_{n=1}^\infty \frac{O(n^6)}{n^8}\sum_{k=1}^\infty
\frac{O(p^{\min\{2k-2,\frac{20}{11}k\}})}
{p^{2k}}O(X)
= O(X/p^2),$$ as desired.
Proof of Lemma \[3reducible2\] {#lemmaproof}
------------------------------
We say a quintic ring is an [*$S_5$-quintic ring*]{} if it is an order in an $S_5$-quintic field. To prove Lemma \[3reducible2\], we wish to show that the expected number of integral elements $(A,B,C,D)\in\FF v$ ($v\in V^{(i)}$) that correspond to quintic rings that are not $S_5$-quintic rings, and such that $|\Disc(A,B,C,D)|< X$ and $a_{12}\neq 0$, is $o(X)$.
Now if a quintic ring $R=R(A,B,C,D)$ is not an $S_5$-quintic ring, then we claim that either the splitting type $(1112)$ or $(5)$ does not occur in $R$. Indeed, if both of these splitting types occur in $R$, then $R$ is clearly a domain (since $R/pR\cong \F_{p^5}$ for some prime $p$) and the Galois group associated with the quotient field of $R$ then must contain a 5-cycle and a transposition, implying that the Galois group is in fact $S_5$.
Therefore, to obtain an upper bound on the expected number of integral elements $(A,B,C,D)\in\FF v$ such that $R(A,B,C,D)$ is not an $S_5$-quintic ring, $|\Disc(A,B,C,D)|< X$, and $a_{12}\neq 0$, we may simply count those quintic rings in which $p$ does not split as $(1112)$ in $R$ for any prime $p<N$ and those quintic rings for which $p$ does not have splitting type $(5)$ for any prime $p<N$ (for some sufficiently large $N$). Now the $p$-adic density $\mu_p(T_p(1112))$ in $V_\Z$ of the set of those $(A,B,C,D)\in T_p(1112)$ approaches $1/12$ as $p\to\infty$ while the $p$-adic density $\mu_p(T_p(5))$ of those $(A,B,C,D)\in T_p(5)$ approaches $1/5$ as $p\to\infty$ (by [@Bhargava4 Lemma 20]). We conclude from (\[ramanujan\]) that the total number of such $(A,B,C,D)\in \FF v$ that do not lie in $T_p(1112)$ for any $p<N$ or do not lie in $T_p(5)$ for any $p<N$, and satisfy $|\Disc(A,B,C,D)|<X$ for sufficiently large $X=X(N)$, is at most $$\frac{\zeta(2)^2\zeta(3)^2\zeta(4)^2\zeta(5)}{2n_i}
\Bigl(\prod_{p<N} \bigl(1-\mu_p(T_p(1112))\bigr)
+ \prod_{p<N} \bigl(1-\mu_p(T_p(5))\bigr)\Bigr)X+o(X).$$ Letting $N\to\infty$, we see that asymptotically the above count of $(A,B,C,D)$ is less than $cX$ for any fixed positive constant $c$, and this completes the proof.
Proofs of Theorems 1–4
----------------------
[**Proof of Theorem 1:**]{} Again, let $\mathcal U_p$ denote the set of all $(A,B,C,D)\in V_\Z$ that correspond to pairs $(R,R')$ where $R$ is maximal at $p$, and let $\mathcal U=\cap_p \mathcal U_p$. Then $\mathcal U$ is the set of $(A,B,C,D)\in V_\Z$ corresponding to maximal quintic rings $R$. In [@Bhargava4 Theorem 21], we determined the $p$-adic density $\mu(\mathcal U_p)$ of $\mathcal U_p$: $$\label{totaludensity}
\mu(\mathcal U_p)=(p-1)^8p^{12}(p+1)^4(p^2+1)^2(p^2+p+1)^2
(p^4+p^3+p^2+p+1)(p^4+p^3+2p^2+2p+1)\,/\,p^{40}\,.$$ Suppose $Y$ is any positive integer. It follows from (\[ramanujan\]) and (\[totaludensity\]) that $$\begin{array}{rcl}
& &\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\displaystyle{\lim_{X\rightarrow\infty} \frac{N(\cap_{p<Y} \mathcal U_p\cap
V^{(i)};X)}{X}} \\[.175in]
& = & \displaystyle{\frac{\zeta(2)^2\zeta(3)^2\zeta(4)^2\zeta(5)}{2n_i}
\prod_{p<Y}[p^{-28}\:(p^2-1)^2(p^3-1)^2(p^4-1)^2(p^5-1)(p^5+p^3-p-1)].}
\end{array}$$ Letting $Y$ tend to $\infty$, we obtain immediately that $$\begin{array}{rcl}
& & \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\displaystyle{\limsup_{X\rightarrow\infty}
\frac{N(\mathcal U\cap V^{(i)};X)}{X}} \\[.175in]
& \leq &\displaystyle{\frac{\zeta(2)^2\zeta(3)^2\zeta(4)^2\zeta(5)}{2n_i}
\prod_{p}[p^{-28}(p^2-1)^2(p^3-1)^2(p^4-1)^2(p^5-1)(p^5+p^3-p-1)]}
\\[.065in]
& =& \displaystyle{\frac{\zeta(2)^2\zeta(3)^2\zeta(4)^2\zeta(5)}{2n_i}
\prod_p [(1-p^{-2})^2(1-p^{-3})^2(1-p^{-4})^2(1-p^{-5})
(1+p^{-2}-p^{-4}-p^{-5})]}.\\[.065in]
& =& \displaystyle{\frac{1}{2n_i} \prod_p (1+p^{-2}-p^{-4}-p^{-5})}.
\end{array}$$ To obtain a lower bound for $N(\mathcal U\cap V^{(i)};X)$, we note that $$\bigcap_{p<Y} \mathcal U_p \subset
(\mathcal U \cup \bigcup_{p\geq Y}\mathcal W_p).$$ Hence by Proposition \[errorestimate\], $$\begin{array}{rcl}
& & \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\displaystyle{\lim_{X\rightarrow\infty}
\frac{N(\mathcal U\cap V^{(i)};X)}{X}} \\[.175in]
&\!\!\!\!\!\!\!\!\!\!\geq&
\!\!\!\!\!\!\displaystyle{\frac{\zeta(2)^2\zeta(3)^2\zeta(4)^2\zeta(5)}
{2n_i}
\prod_{p<Y}[p^{-28}(p^2-1)^2(p^3-1)^2(p^4-1)^2(p^5-1)(p^5+p^3-p-1)]
- O(\sum_{p\geq Y} p^{-2}).}
\end{array}$$ Letting $Y$ tend to infinity completes the proof of Theorem 1. [$\Box$ ]{}
[**Proof of Theorem 2:**]{} For each (isomorphism class of) quintic ring $R$, we make a choice of sextic resolvent ring $R'$, and let $S\subset V_\Z$ denote the set of all elements in $V_\Z$ that yield the pair $(R,R')$ (under the bijection of Theorem \[main\]) for some $R$. Then we wish to determine $N(S\cap V^{(i)};X)$ for $i=0,1,2$; by equation (\[ramanujan\]), this amounts to determining the $p$-adic density $\mu_p(S)$ of $S$ for each prime $p$ for our choice of $S$. In this regard we have the following formula, which follows easily from the arguments in [@Bhargava4 Proof of Lemma 20]: $$\label{orderdensity}
\mu_p(S)\,=\,
\frac{|G(\F_p)|}{\Disc_p(R)\cdot|\Aut_{\Z_p}(R)|}.$$ Combining (\[ramanujan\]) and (\[orderdensity\]) together with the fact that $$|G(\F_p)|=(p-1)^8\; p^{16}\;(p+1)^4\;(p^2+1)^2\;(p^2+p+1)^2\;
(p^4+p^3+p^2+p+1),$$ and proceeding as in Theorem 1, now yields Theorem 2. [$\Box$ ]{}
[**Proof of Theorem 3:**]{} Let $K_5$ be an $S_5$-quintic field, and $K_{120}$ its Galois closure. It is known that the Artin symbol $(K_{120}/p)$ equals $\langle e \rangle$, $\langle (12) \rangle$, $\langle (123) \rangle$, $\langle (1234) \rangle$, $\langle (12345)
\rangle$, $\langle (12)(34) \rangle$, or $\langle (12)(345)\rangle$ precisely when the splitting type of $p$ in $R$ is $(11111)$, $(1112)$, $(113)$, $(14)$, $(5)$, $(122)$, or $(23)$ respectively, where $R$ denotes the ring of integers in $K_5$. As in [@Bhargava4], let $U_p(\sigma)$ denote the set of all $(A,B,C,D)\in
V_\Z$ that correspond to maximal quintic rings $R$ having a specified splitting type $\sigma$ at $p$. Then by the same argument as in the proof of Theorem 1, we have $$\lim_{X\rightarrow\infty}\frac{N(U_p(\sigma)\cap V^{(i)};X)}{X}
= \frac{\zeta(2)^2\zeta(3)^2\zeta(4)^2\zeta(5)}{2n_i}
\mu_p(U_p(\sigma))\prod_{q\neq p} \mu_q(\mathcal U_q) .$$ On the other hand, Lemma 20 of [@Bhargava4] gives the $p$-adic densities of $U_p(\sigma)$ for all splitting and ramification types $\sigma$; in particular, the values of $\mu_p(U_p(\sigma))$ for $\sigma = (11111)$, $(1112)$, $(113)$, $(14)$, $(5)$, $(122)$, or $(23)$ are seen to occur in the ratio $1\!:\!10\!:\!20\!:\!30\!:\!24\!:\!15\!:\!20$ for any value of $p$; this is the desired result. [$\Box$ ]{}
[**Proof of Theorem 4:**]{} This follows immediately from Theorem 1, Lemma \[hard\], and Lemma \[3reducible2\].
Acknowledgments {#acknowledgments .unnumbered}
===============
I am very grateful to B. Gross, H. W. Lenstra, P. Sarnak, A.Shankar, A. Wiles, and M. Wood for many helpful discussions during this work. I am also very thankful to the Packard Foundation for their kind support of this project.
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|
---
abstract: 'We calculate thermodynamical properties of the Hofstadter model using a recently developed quantum transfer matrix method. We find intrinsic oscillation features in specific heat that manifest the fractal structure of the Hofstadter butterfly. We also propose experimental approaches which use specific heat as an access to detect the Hofstadter butterfly.'
author:
- 'L. P. Yang$^{1}$'
- 'W. H. Xu$^{2}$'
- 'M. P. Qin$^{3}$'
- 'T. Xiang$^{3,1}$'
bibliography:
- 'references.bib'
title: Fractality of Hofstadter Butterfly in Specific Heat Oscillation
---
[^1]
The interplay between crystalline potential and magnetic field on a two-dimensional electronic gas remains a nontrivial problem for decades[@harper]. This issue provides a stage on which purely mathematical concepts, i.e.,the irrationality of a real number, interrupts our intuition of physical reality. Hofstadter[@PhysRevB.14.2239] studied the energy spectrum of the tight-binding limit of this problem, namely, the Hofstadter model. He proposed a fractal topology for the spectrum(Hofstadter butterfly), which reconciled the paradox raised by the irrationality. The experimental verification of the Hofstadter butterfly is challenging but some hints of the fractal structure have been observed in microwave measurements[@PhysRevLett.80.3232; @PhysRevB.43.5192], Hall conductivity[@PhysRevLett.86.147] and magnetic transport measurements[@PhysRevLett.62.1173] in analogous systems.
In this paper, we adopt a recently developed quantum transfer matrix method[@Yang] to study thermodynamic properties of the Hofstadter model. We focus on the behavior of internal energy and specific heat as functions of magnetic field. As far as we know, this is the first report that by theoretical method, the fractal structure in the Hofstadter butterfly can be studied by computing the specific heat in a magnetic field of a generic value. We also briefly discuss the feasibility of experimental observations of these features.
In a previous publication[@wenhu], we had used the quantum transfer matrix method to study the magnetic properties of Hofstadter model. The advantage of this method lies in that it directly computes the partition function of the model for arbitrary $\phi$, where $\phi$ is the magnetic flux through a unit cell, then the thermodynamic properties can be studied steadily. Conventional theoretical methods, such as Bethe ansatz[@PhysRevLett.72.1890; @PhysRevLett.73.1134] and exact diagonalization[@PhysRevLett.63.907; @PhysRevLett.63.1657], are mostly applied to $\phi=p/q$ cases, where $p$ and $q$ are mutually prime numbers, and $q$ is relatively small. Although detailed information of energy spectrum and wavefunction can be obtained with these methods, only limited cases of $\phi$ can be studied and most discussion was focused on ground state properties. Besides, at ground states, due to the fractality of the Hofstadter butterfly, the smoothness of physical quantities as functions of magnetic field, such as total energy, static magnetic susceptibility are significantly diminished. However, within the quantum transfer matrix formulation, the effect of finite temperature is embodied in the partition function at the beginning, and the singularities due to the fine fractality will be smeared out and the smoothness of physical quantities can be recovered, which makes the comparison to experimental results more straightforward.
Hofstadter model describes the dynamics of two-dimensional tight binding electrons in a uniform magnetic field[@PhysRevB.14.2239]. By applying Landau gauge, i.e., $A=H(0,x,0)$, the Hamiltonian is explicitly translationally invariant along the $y$-direction. Fourier transformation along the $y$-axis will then decouple the two-dimensional model $H$ into a series summation of one-dimensional Hamiltonian $H_k$: $$\begin{aligned}
\label{eqn:ham}
H&=&\sum_{k} H_{k},\\
H_{k} & = & \sum_x \Big[ tc^{\dag}_{k,x+1}c_{k,x} +
tc^{\dag}_{k,x}c_{k,x+1} \nonumber \\
&& +2t\cos\left(2\pi x\phi-k\right) c^{\dag}_{k,x}c_{k,x} \Big] ,\end{aligned}$$ where $k=2\pi n/N_y \, (n=0,1,...,N_y-1)$ are the quasimomenta and $N_y$ is the lattice dimension along $y$ direction. $x$ is the lattice coordinate of electrons along the $x$-axis. $\phi$ is the magnetic flux through each plaquette, with magnetic flux quanta $hc/e$ as unit.
$H_k$ does not generally have translational invariance along the $x$-axis. But for rational $\phi=p/q$, periodicity can be recovered by combining every $q$ cells to form a superlattice, and then the problem can be solved by diagonalizing a $q\times q$ matrix for each quasimomentum of the superlattice. Thus the full energy spectrum and thermodynamic properties can be steadily obtained, yet apparently, only up to relatively small $q$. To study cases with a generic $\phi$, the quantum transfer matrix method starts from the partition function $Z=\mathrm{Tr}[\exp(-\beta H)]$, which can be viewed as an trace of the evolution operator along the imaginary time. Since the trace naturally imposes a periodical boundary condition, a Fourier transformation can be well defined along the imaginary time(the inverse temperature), which is the key point leading us to the transfer matrix representation and to significantly simplify the calculation in Ref.\[\]. Given $k$, the partition function of $H_k$ is defined by $$Z_{k}=\mathrm{Tr} \exp(-\beta H_{k}),$$ where $\beta = 1/ k_BT$. The partition function of the whole system is simply a product of all $Z_k$s. By making use of the translational invariance along the imaginary time, $Z_k$ can be expressed as a product of $N_x$ $2\times2$ matrices. After multiplying different $k$ components, we can obtain the partition function of the system, from which one can calculate the free energy by $F=-(1/\beta)\ln Z$, and other thermodynamic quantities such as magnetic susceptibility and specific heat.
In Ref.\[\], the authors have discussed the effect of lattice size on the numerical results. Accordingly, we choose here $N_x=50000$ and $N_y=100$ to ensure the numerical accuracy as well as computational efficiency for the temperature range in this paper. For simplicity, we only consider the half-filling case, which corresponds to a particle-hole symmetry and automatically sets chemical potential $\mu$ to $0$.
First, we calculate the average internal energy as a function of $\phi$ at $T=0.01$.
![(Color online)\[fig:averE\] The internal energy as a function of $\phi$ for the Hofstadter model at half filling. $T=0.01$. Some values of $\phi$(the red number above the top axis) and the integers $(M, N)$ corresponding to local minima are marked. ](fig_1.eps){width="50.00000%"}
As shown in Fig. \[fig:averE\], at the local minima of the internal energy, the electron count $\nu$($=0.5$ for half-filling) and $\phi$ satisfy the relation in (\[eqn:EvsPhi\]), which was given Ref.[@PhysRevLett.63.907]. These minima are [*cusp*]{}-like. $$\label{eqn:EvsPhi}
\nu=M + N\phi,~~~~~~M, N \in Z.$$ The global minimum in Fig. \[fig:averE\] is consistent with the conclusion that there is an global minimum of the average energy[@PhysRevLett.63.1657; @PhysRevB.41.9174] when $\phi=\nu=1/2$, that is, each electron carries one flux quanta. We have marked the values of $\phi$(the red number on the top axis) located at distinguishable minima and the corresponding integers $M$ and $ N$ in Fig. \[fig:averE\]. At zero temperature, the average energy will not be smooth almost everywhere because there are infinite number of rational $\phi$s that satisfy (\[eqn:EvsPhi\]). But here the temperature will erase minor singularities and only keep the significant ones.
![(Color online)\[fig:C-T\] Temperature dependence of the specific heat $C$ at half filling for $\phi=1/2,1/3,1/4$. ](fig_2.eps){width="50.00000%"}
Then we compute the specific heat from the first order derivative of the internal energy with respect to the temperature. Fig. \[fig:C-T\] shows the specific heat $C$ as a function of temperature $T$ for some special $\phi$s. The chosen three $\phi$s belong to the pure cases in Hofstadter’s proposal[@PhysRevB.14.2239], i.e.,$\phi=1/N$, or $1-1/N$ when $N\geq2$. Under magnetic field of these values, the single Bloch band in zero magnetic field is split into $N$ subbands. If $N$ is odd, the central subband has a van-Hove singularity at the center point of the energy spectrum($E=0$). If $N$ is even, the density of states(DOS) goes to zero at $E=0$[@PhysRevLett.63.907]. When the temperature is so high that the thermal fluctuations are comparable to the energy difference between the lowest and the highest subband, the subbands will not be able to manifest their internal fine structures from specific heat. This can be observed from the high temperature tail in Fig. \[fig:C-T\].
The difference in the specific heat for various values of $\phi$ will emerge with the decreasing temperature. First, at low temperature, the behavior of specific heat can tell the singularity of DOS at the energy spectrum center point(the Fermi surface(FS) in our half-filling case). In the regime near zero temperature, the $\phi=1/2$ and $\phi=1/4$ curves are decreasing faster than that of $\phi=1/3$. A closer observation indicates that $\phi=1/2$ and $\phi=1/4$ decrease exponentially-like, while $\phi=1/3$ is linear-like. This is because of the different behavior of DOS at the spectrum center point[@PhysRevLett.63.907]. For $\phi=1/2$ and $\phi=1/4$, the original single band in zero field splits up into $2$ and $4$ bands. But the centermost two bands are not completely separated by a gap, rather they “kiss” at the center point, where DOS of both bands goes to zero. Therefore, we will expect a gap-like behavior at low temperature, which shows up as an exponential-like decrease in specific heat. For $\phi=1/3$, the Bloch band splits into $3$ bands, and DOS of the center band is singular at the center point. Thus we expect a much slower decrease near zero temperature, which is also observed in Fig. \[fig:C-T\].
In the intermediate temperature regime, the specific heat tells the information about gaps and redistribution of DOS along the energy spectrum. In Fig. \[fig:C-T\], the curves of $\phi=1/3,1/4$ show some similar minor hump structures, which is different from the case of $\phi=1/2$. For $\phi=1/2$, two subbands touch at FS, where DOS is zero, and there is no finite gap in the energy spectrum. Thus there is only one major hump in specific heat curve. For both $\phi=1/3$ and $\phi=1/4$, there is a finite gap lying above FS[@PhysRevLett.63.907], which separates the subband on($\phi=1/3$) or near($\phi=1/4$) FS from the higher band, and gives the extra minor hump in the specific heat curve.
![(Color online)(a) The specific heat coefficient $C/T$ as a function of $1/\phi$. Two different temperatures $T=0.01,0.1$ are compared; (b) Magnetizationat at $T=0.01$ as a function of $1/\phi$.[]{data-label="fig:C-phi"}](fig_3.eps){width="50.00000%"}
Fig. \[fig:C-phi\]-(a) shows the specific heat coefficient($C/T$) as a function of magnetic field at different temperatures, $T=0.1$ and $T=0.01$. The horizontal axis is chosen as $1/\phi$, so that the conventional de Haas-van Alphen(dHvA) -like oscillations are shown distinctly in the figure. The period $\triangle ( 1 / \phi )$ of this oscillation is about $2$ in both cases, which is consistent with that obtained from textbook formula[@solidstatephysics], $\triangle ( 1 / \phi )= 4\pi^{2}/S_{F}$, where $S_F$ is the Fermi volume. At half filling, $S_F = 2\pi^2$, thus $\triangle ( 1 / \phi
)=2$ in Fig. \[fig:C-phi\]-(a).
However, a more important observation is that subtle oscillations emerge within the dHvA-type period with decreasing temperature and strong field. The specific heat at $T=0.1$ displays a clean periodic oscillation on the weaker field side(large $1/\phi$), while this periodicity is disturbed in the stronger field regime(small $1/\phi$). This becomes more explicit with lower temperature $T=0.01$. In the first three periods, very sharp peaks and dips show up, and they make a peculiar type of oscillations within the period. Even for weaker field regime, some sharp structures are still observable.
For the purpose of comparison, Fig. \[fig:C-phi\]-(b) shows the magnetization oscillation with respect to the magnetic field. Similar to the specific heat, the main envelope of oscillation is the conventional dHvA oscillation. Besides, subtle structures emerge within the dHvA period[@wenhu]. By comparing the results for magnetization and specific heat, we find that the specific heat oscillations are more distinct and drastic.
![(Color online) Specific heat coefficient(black line) and magnetic susceptibility(red line) for the Hofstadter model at half-filling. The numerical values of specific heat are 6000 times larger than the original ones for comparison. The values of $\phi$s marked corresponds to some local maxima and minima in both specific heat and magnetic susceptibility. []{data-label="fig:C-phi_zoomin"}](fig_4.eps){width="50.00000%"}
To explore the information about the fractal structure of Hofstadter butterfly from specific heat, we zoom in the first period in Fig.\[fig:C-phi\] and then have Fig.\[fig:C-phi\_zoomin\] for low temperature specific heat $C$ and magnetic susceptibility $\chi$. The numerical values of the specific heat are enlarged to 6000 times of the original values for a clear comparison. Here $\phi$ is chosen to be the horizontal axis. The consistency between $C$ and $\chi$ is obvious if we compare the positions of local maxima and minima of both quantities. In Fig. \[fig:C-phi\] some fractional values of $\phi$s are marked where they are close to the local maxima and minima. Applying Hofstadter’s proposal of constructing the butterfly[@PhysRevB.14.2239], we can extract the structure of energy spectrum at thess $\phi$s and understand why the extrema of $C$ and $\chi$ are close to them. With Hofstadter’s proposal, each fractional $\phi$ can be decomposed to a set of more “fundamental ” fractions, or, “local variable” as in Ref.[@PhysRevB.14.2239], which then directly displays the splitting of subbands in the energy spectrum. For example, for $\phi=4/13$, the center local variable is $4/5$, which means there is a cluster of $5$ subbands centered at FS. Consequently, a van-Hove singularity of DOS shows up at FS and causes the local maximum in $C$ and the strong paramagnetism(local maximum in $\chi$)[@wenhu], while for $\phi=3/8$, the center local variable is $1/2$, thus there are two subbands lying above and below FS with a zero DOS at FS and consequently in Fig.\[fig:C-phi\_zoomin\], $C$ shows a small value(close to 0) around $\phi=3/8$ and $\chi$ is strongly diamagnetic around $\phi=3/8$.
Therefore, by decreasing the temperature, fractal structures of the Hofstadter butterfly manifest themselves by producing peculiar oscillatory features within conventional dHvA period. This emergence along with decreasing temperature is due to the fact that temperature provides the only energy scale that sets up the resolution of the spectrum. Temperature erases minor bands and gaps that are smaller than the scale of temperature and restores the smoothness of physical quantities. But fractal structures with an energy scale larger than the temperature survive, and are able to manifest themselves by displaying smoothened singularities in thermodynamic quantities. Thus the subtler fractal structures of Hofstadter butterfly can be probed by the measurement of the specific heat at lower temperatures.
We propose to adopt the superconducting thin films(for example the element Nb) with periodic arrays of pinning sites[@Harada] to realize this temperature-dependent emergence of fractal structures in specific heat. The artificial pinning centers hold great potential. Just below the onset temperature of superconducting transition, the electrons possess long mean free path. When the interval between adjacent sites comes to the order of 100nm, the experimentally accessible steady fields can enter the interesting regime of $\phi$. The resulting effective lattice subjected to perpendicular magnetic field is probably able to show the fractal properties of the Hofstadter model. In addition, the purity requirement of the sample is relaxed when considering the specific heat measurement.
In summary, adopting the quantum transfer matrix method, we compute the internal energy and specific heat of the Hofstadter model, and for the first time we study the oscillation of the specific heat with varying magnetic field as a signature of fractal structure of the Hofstadter butterfly. In low field regime, the oscillation period of specific heat is consistent with the conventional dHvA oscillation. When the temperature is decreased, sharp peaks and dips emerge in addition to the dHvA-type background. These peculiar oscillatory behaviors are direct indications of DOS in the fractal energy spectrum. We also suggest the possibility of making use of superconducting films to detect this fractal structure by measuring the specific heat.
Acknowledgement: We thank Shao-Jing Qin for helpful discussions. One of us(Li-Ping Yang) is pleased to acknowledge generous hospitality of Xiao Hu during her visit in Japan. This work was supported by the NSF-China and the National Program for Basic Research of MOST, China.
[^1]: Present address: Institute for Theoretical Solid State Physics, IFW Dresden, 01069 Dresden, Germany
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abstract: 'We present an algorithm for computation of cell adjacencies for well-based cylindrical algebraic decomposition. Cell adjacency information can be used to compute topological operations e.g. closure, boundary, connected components, and topological properties e.g. homology groups. Other applications include visualization and path planning. Our algorithm determines cell adjacency information using validated numerical methods similar to those used in CAD construction, thus computing CAD with adjacency information in time comparable to that of computing CAD without adjacency information. We report on implementation of the algorithm and present empirical data.'
address: 'Wolfram Research Inc., 100 Trade Centre Drive, Champaign, IL 61820, U.S.A. '
author:
- Adam Strzeboński
date: 'April 21, 2017'
title: CAD Adjacency Computation Using Validated Numerics
---
Introduction
============
A semialgebraic set is a subset of $\mathbb{R}^{n}$ which is a solution set of a system of polynomial equations and inequalities. Computation with semialgebraic sets is one of the core subjects in computer algebra and real algebraic geometry. A variety of algorithms have been developed for real system solving, satisfiability checking, quantifier elimination, optimization and other basic problems concerning semialgebraic sets [@C; @BPR; @CJ; @DSW; @GV; @HS; @LW; @R; @T; @W1]. Every semialgebraic set can be represented as a finite union of disjoint cells bounded by graphs of semialgebraic functions. The Cylindrical Algebraic Decomposition (CAD) algorithm [@C; @CJ; @S7] can be used to compute a cell decomposition of any semialgebraic set presented by a quantified system of polynomial equations and inequalities. Alternative methods of computing cell decompositions are given in [@CMXY; @S11; @S12]. For solving certain problems, for instance computing topological properties or visualization, it is not sufficient to know a cell decomposition of the set, but it is also necessary to know how the cells are connected together.
The CAD algorithm applied to the equation $y^{2}=x(x^{4}-1)$ gives four one-dimensional cells and three zero-dimensional cells shown in Figure \[fig\]. To find the connected components of the solution set it is sufficient to know which one-dimensional cells are adjacent to which zero-dimensional cells.
Several algorithms for computing cell adjacencies have been developed. The algorithm given in [@SS] computes cell adjacencies for CAD that are well-based. A CAD is well-based if none of the polynomials whose roots appear in cell description vanishes identically at any point. This is a somewhat stronger condition than well-orientedness required for the McCallum projection [@MC2], nevertheless a large portion of examples that appear in practice satisfies the condition. In a well-based CAD all cell adjacencies can be determined from adjacencies between cells whose dimensions differ by one. In $\mathbb{R}^{2}$ all CADs are well-based. Algorithm for computing cell adjacencies in arbitrary CADs in $\mathbb{R}^{3}$ has been given in [@ACM2]. For determining cell adjacencies [@SS] proposes methods based on fractional power series representations of polynomial roots. Another method, given in [@MCC], computes adjacencies between a zero-dimensional cell and one-dimensional cells by analyzing intersections of the one dimensional cells with sides of a suitable box around the zero-dimensional cell. For an alternative method of computing connectivity properties of semialgebraic sets see [@CA; @BPR1; @BPR; @SSC].
In this paper we present a new algorithm which computes cell adjacencies for well-based CAD. The algorithm uses validated numerical methods similar to those used in [@S7] for construction of CAD cell sample points. The method is based on computation of approximations of polynomial roots and increasing the precision of computations until validation criteria are satisfied. Unlike the previously known algorithms, it does not require polynomial computations over algebraic number fields or computation with fractional power series representations of polynomial roots. Also, unlike the CAD construction algorithm given in [@S7], the algorithm never needs to revert to exact algebraic number computations. We have implemented the algorithm as an extension to the CAD implementation in *Mathematica*. Empirical results show that computation of CAD with cell adjacency data takes time comparable to computation of CAD without cell adjacency data.
The general idea of the algorithm is as follows. It starts, similarly as the CAD algorithm, with computing a sample point in each cell in $\mathbb{R}^{k}$ for all $k\leq n$. The sample point of a cell in $\mathbb{R}^{k+1}$ extends the sample point of the projection of the cell on $\mathbb{R}^{k}$. Then for each pair of adjacent CAD cells $C$ and $C^{\prime}$ in $\mathbb{R}^{k}$ with $\dim C^{\prime}=\dim C-1$ the algorithm constructs a point $p\in C$ that is “sufficiently close” to the sample point $p^{\prime}$ of $C^{\prime}$. Here “sufficiently close” means that computing approximations of roots of projection polynomials at $p$ and $p^{\prime}$ is sufficient to identify which roots over $C$ tend to which roots over $C^{\prime}$ and to continue the construction to pairs of adjacent CAD cells in $\mathbb{R}^{k+1}$. The construction gives all pairs of adjacent cells in $\mathbb{R}^{n}$ whose dimensions differ by one. For well-based CAD this is sufficient to determine all cell adjacencies.
Preliminaries
=============
A *system of polynomial equations and inequalities* in variables $x_{1},\ldots,x_{n}$ is a formula$$S(x_{1},\ldots,x_{n})=\bigvee_{1\leq i\leq l}\bigwedge_{1\leq j\leq m_{i}}f_{i,j}(x_{1},\ldots,x_{n})\rho_{i,j}0$$ where $f_{i,j}\in\mathbb{R}[x_{1},\ldots,x_{n}]$, and each $\rho_{i,j}$ is one of $<,\leq,\geq,>,=,$ or $\neq$.
A subset of $\mathbb{R}^{n}$ is *semialgebraic* if it is a solution set of a system of polynomial equations and inequalities.
A *quantified system of real polynomial equations and inequalities* in free variables $x_{1},\ldots,x_{m}$ and quantified variables $x_{m+1},\ldots,x_{n}$ is a formula$$Q_{1}x_{m+1}\ldots Q_{n-m}x_{n}S(x_{1},\ldots,x_{n})\label{quantsyst}$$ Where $Q_{i}$ is $\exists$ or $\forall$, and $S$ is a system of real polynomial equations and inequalities in $x_{1},\ldots,x_{n}$.
By Tarski’s theorem (see [@T]), solution sets of quantified systems of real polynomial equations and inequalities are semialgebraic.
For $k\geq1$, let $\overline{a}$ denote a $k$-tuple $(a_{1},\ldots,a_{k})$ of real numbers and let $\overline{x}$ denote a $k$-tuple $(x_{1},\ldots,x_{k})$ of variables.
Every semialgebraic set can be represented as a finite union of disjoint *cells* (see [@L]), defined recursively as follows.
1. A cell in $\mathbb{R}$ is a point or an open interval.
2. A cell in $\mathbb{R}^{k+1}$ has one of the two forms$$\begin{aligned}
& \{(\overline{a},a_{k+1}):\overline{a}\in C_{k}\wedge a_{k+1}=r(\overline{a})\}\\
& \{(\overline{a},a_{k+1}):\overline{a}\in C_{k}\wedge r_{1}(\overline{a})<a_{k+1}<r_{2}(\overline{a})\}\end{aligned}$$ where $C_{k}$ is a cell in $\mathbb{R}^{k}$, $r$ is a continuous semialgebraic function, and $r_{1}$ and $r_{2}$ are continuous semialgebraic functions, $-\infty$, or $\infty$, and $r_{1}<r_{2}$ on $C_{k}$.
For a cell $C\subseteq\mathbb{R}^{n}$ let $\Pi_{k}(C)\subseteq\mathbb{R}^{k}$, for $k\leq n$, denote the projection of $C$ on $\mathbb{R}^{k}$. A finite collection $D$ of cells in $\mathbb{R}^{n}$ is *cylindrically arranged* if for any $C_{1},C_{2}\in D$ and any $k\leq n$ $\Pi_{k}(C_{1})$ and $\Pi_{k}(C_{2})$ are either disjoint or identical.
A *cylindrical algebraic decomposition (CAD) of $\mathbb{R}^{n}$* is a finite collection $D$ of pairwise disjoint cylindrically arranged cells in $\mathbb{R}^{n}$ such that $\bigcup_{C\in D}C=\mathbb{R}^{n}$.
Let $P\subset\mathbb{R}[x_{1},\ldots,x_{n}]$ be a finite set of polynomials. A CAD $D$ of *$\mathbb{R}^{n}$* is *P-invariant* if each element of $P$ has a constant sign on each cell of $D$.
Let *$A\subseteq\mathbb{R}^{n}$* be a semialgebraic set. A CAD $D$ of *$\mathbb{R}^{n}$* is *consistent with* $A$ if $A=\bigcup_{C\in D_{A}}C$ for some $D_{A}\subseteq D$.
Let $C_{1},C_{2}\in D$. $C_{1}$ and $C_{2}$ are *adjacent* if $C_{1}\neq C_{2}$ and $C_{1}\cup C_{2}$ is connected.
For a semialgebraic set $A$ presented by a quantified system of polynomial equations and inequalities (\[quantsyst\]), the CAD algorithm can be used to find a CAD $D$ of $\mathbb{R}^{n}$ consistent with $A$. The CAD $D$ is represented by a cylindrical algebraic formula (CAF). A CAF describes each cell by giving explicit semialgebraic function bounds and the Boolean structure of a CAF reflects the cylindrical arrangement of cells. Before we give a formal definition of a CAF, let us first introduce some terminology.
Let $k\geq1$ and let $f=c_{d}y^{d}+\ldots+c_{0}$, where $c_{0},\ldots,c_{d}\in\mathbb{\mathbb{Z}}[\overline{x}]$. A *semialgebraic function* given by the *defining polynomial* $f$ and a *root number* $\lambda\in\mathbb{N}_{+}$ is the function$$Root_{y,\lambda}f:\mathbb{R}^{k}\ni\overline{a}\longrightarrow Root_{y,\lambda}f(\overline{a})\in\mathbb{R}\label{rootfun}$$ where $Root_{y,\lambda}f(\overline{a})$ is the $\lambda$-th real root of $f(\overline{a},y)\in\mathbb{R}[y]$. The function is defined for those values of $\overline{a}$ for which $f(\overline{a},y)$ has at least $\lambda$ real roots. The real roots are ordered by the increasing value and counted with multiplicities. A real algebraic number $Root_{y,\lambda}f\in\mathbb{R}$ given by a *defining polynomial* $f\in\mathbb{Z}[y]$ and a *root number* $\lambda$ is the $\lambda$-th real root of $f$.
Let $C$ be a connected subset of $\mathbb{R}^{k}$. The function $Root_{y,\lambda}f$ is *regular* on *$C$* if it is continuous on $C$, $c_{d}(\overline{a})\neq0$ for all $\overline{a}\in C$, and there exists ** $m\in\mathbb{\mathbb{N}}_{+}$ such that for any $\overline{a}\in C$ $Root_{y,\lambda}f(\overline{a})$ is a root of $f(\overline{a},y)$ of multiplicity $m$.
The polynomial $f$ is *degree-invariant* on $C$ if there exists ** $e\in\mathbb{\mathbb{N}}$ such that $c_{d}(\overline{a})=\ldots=c_{e+1}(\overline{a})=0\wedge c_{e}(\overline{a})\neq0$ for all $\overline{a}\in C$.
A set $W=\{f_{1},\ldots,f_{m}\}$ of polynomials is *delineable* on $C$ if all elements of $W$ are degree-invariant on $C$ and for $1\leq i\leq m$$$f_{i}^{-1}(0)\cap(C\times\mathbb{R})=\{r_{i,1},\ldots,r_{i,l_{i}}\}$$ where $r_{i,1},\ldots,r_{i,l_{i}}$ are disjoint regular semialgebraic functions and for $i_{1}\neq i_{2}$ $r_{i_{1},j_{1}}$ and $r_{i_{2},j_{2}}$ are either disjoint or equal. Functions $r_{i,j}$ are *root functions of $f_{i}$ over $C$*.
Let $W$ be delineable on $C$, let $r_{1}<\ldots<r_{l}$ be all root functions of elements of $W$ over $C$, and let $r_{0}=-\infty$ and $r_{l+1}=\infty$. For $1\leq i\leq l$, the $i$-th *$W$-section over $C$* is the set$$\{(\overline{a},a_{k+1}):\overline{a}\in C\wedge a_{k+1}=r_{i}(\overline{a})\}$$ For $1\leq i\leq l+1$, the $i$-th *$W$-sector over $C$* is the set$$\{(\overline{a},a_{k+1}):\overline{a}\in C\wedge r_{i-1}(\overline{a})<a_{k+1}<r_{i}(\overline{a})\}$$ *$W$-stack over $C$* is the set of all $W$-sections and $W$-sectors over $C$.
A formula $F$ is an *algebraic constraint* with *bounds* $BDS(F)$ if it is a level-$k$ equational or inequality constraint with $1\leq k\leq n$ defined as follows. **
1. *A level*-$1$ *equational constraint* has the form $x_{1}=r$, where $r$ is a real algebraic number, and $BDS(F)=\{r\}$.
2. *A level*-$1$ *inequality constraint* has the form $r_{1}<x_{1}<r_{2}$, where $r_{1}$ and $r_{2}$ are real algebraic numbers, $-\infty$, or $\infty$, and $BDS(F)=\{r_{1},r_{2}\}\setminus\{-\infty,\infty\}$.
3. *A level*-$k+1$ *equational constraint* has the form $x_{k+1}=r(\overline{x})$, where $r$ is a semialgebraic function, and $BDS(F)=\{r\}$.
4. *A level*-$k+1$ *inequality constraint* has the form $r_{1}(\overline{x})<x_{k+1}<r_{2}(\overline{x})$, where $r_{1}$ and $r_{2}$ are semialgebraic functions, $-\infty$, or $\infty$, and $BDS(F)=\{r_{1},r_{2}\}\setminus\{-\infty,\infty\}$.
A level-$k+1$ algebraic constraint $F$ is *regular* on a connected set $C\subseteq\mathbb{R}^{k}$ if all elements of $BDS(F)$ are regular on $C$ and, if $F$ is an inequality constraint, $r_{1}<r_{2}$ on $C$.
An *atomic cylindrical algebraic formula (CAF)* $F$ in $(x_{1},\ldots,x_{n})$ has the form $F_{1}\wedge\ldots\wedge F_{n}$, where $F_{k}$ is a level-$k$ algebraic constraint for $1\leq k\leq n$ and $F_{k+1}$ is regular on the solution set of $F_{1}\wedge\ldots\wedge F_{k}$ for $1\leq k<n$.
*Level-$k$ cylindrical formulas* in $(x_{1},\ldots,x_{n})$ are defined recursively as follows
1. A level-$n$ cylindrical formula is $false$ or a disjunction of level-$n$ algebraic constraints.
2. A level-$k$ cylindrical formula, with $1\leq k<n$, is $false$ or has the form$$(F_{1}\wedge G_{1})\vee\ldots\vee(F_{m}\wedge G_{m})$$ where $F_{i}$ are level-$k$ algebraic constraints and $G_{i}$ are level-$k+1$ cylindrical formulas.
A *cylindrical algebraic formula (CAF)* is a level-$1$ cylindrical formula $F$ such that distributing conjunction over disjunction in $F$ gives $$DNF(F)=F_{1}\vee\ldots\vee F_{l}$$ where each $F_{i}$ is an atomic CAF. Let $C(F_{i})$ denote the solution set of $F_{i}$ and let $D(F)=\{C(F_{1}),\ldots,C(F_{l})\}$. The *bound polynomials* of $F$ is a finite set $BP(F)\subset\mathbb{R}[x_{1},\ldots,x_{n}]$ which consists of all polynomials $f$ such that $Root_{x_{k},\lambda}f\in BDS(G)$ for some $1\leq k\leq n$ and a level-$k$ algebraic constraint $G$ that appears in $F$.
Note that $C(F_{i})$ is a cell and $D(F)$ is a finite collection of pairwise disjoint cylindrically arranged cells.
For a CAF $F$ in $(x_{1},\ldots,x_{n})$, let $\Pi_{k}(F)$ denote the CAF in $(x_{1},\ldots,x_{k})$ obtained from $F$ by removing all level-$k+1$ subformulas. Then $$D(\Pi_{k}(F))=\{\Pi_{k}(C)\::\: C\in D(F)\}$$
Following the terminology of [@SS], we define a well-based CAF as follows.
A CAF ** $F$ is *well-based* if *$D(F)$* is a $BP(F)$-invariant CAD of $\mathbb{R}^{n}$ and for any $f\in BP(F)$ if $f\in\mathbb{R}[x_{1},\ldots,x_{k+1}]\setminus\mathbb{R}[x_{1},\ldots,x_{k}]$ then for any $\overline{a}\in\mathbb{R}^{k}$ $f(\overline{a},x_{k+1})$ is not identically zero.
In a CAD corresponding to a well-based CAF a closure of a cell is a union of cells and the only cells from other stacks adjacent to a given section are sections defined by the same polynomial. Moreover, any two adjacent cells have different dimensions and are connected through a chain of adjacent cells with dimensions increasing by one, and hence to determine all cell adjacencies it is sufficient to find all pairs of adjacent cells whose dimensions differ by one. These properties, stated precisely in Proposition \[WellBasedProp\], are essential for our algorithm.
\[WellBasedProp\]Let $F$ be a well-based CAF.
1. If $C\in D(F)$ then there exits cells $C_{1},\ldots,C_{m}\in D(F)$ such that $\overline{C}=C\cup C_{1}\cup\ldots\cup C_{m}$.
2. Let $C\in D(F)$ be a section$$C=\{(\overline{a},a_{n}):\overline{a}\in\Pi_{n-1}(C)\wedge a_{n}=Root_{x_{n},\lambda}f(\overline{a})\}$$ and let $C^{\prime}\in D(\Pi_{n-1}(F))$ be a cell adjacent to $\Pi_{n-1}(C)$ with $\dim C^{\prime}<\dim\Pi_{n-1}(C)$. Then either $\overline{C}\cap(C^{\prime}\times\mathbb{R})$ is equal to a section$$\{(\overline{a},a_{n}):\overline{a}\in C^{\prime}\wedge a_{n}=Root_{x_{n},\lambda^{\prime}}f(\overline{a})\}$$ for some $1\leq\lambda^{\prime}\leq\deg_{x_{n}}(f)$, or for any $\overline{a}\in C^{\prime}$ $$\lim_{\overline{b}\in\Pi_{n-1}(C),\overline{b}\rightarrow\overline{a}}Root_{x_{n},\lambda}f(\overline{a})=-\infty$$ or for any $\overline{a}\in C^{\prime}$ $$\lim_{\overline{b}\in\Pi_{n-1}(C),\overline{b}\rightarrow\overline{a}}Root_{x_{n},\lambda}f(\overline{a})=\infty$$
3. Let $C_{k},C_{l}\in D(F)$ be adjacent cells such that $\dim(C_{k})=k$ and $\dim(C_{l})=l$. Then $k\neq l$ and if $k<l$ then there exist cells $C_{k+1},\ldots,C_{l-1}\in D(F)$ such that $\dim(C_{j})=j$ and $C_{j}\subseteq\overline{C_{j+1}}$ for $k\leq j<l$.
Part $(1)$ is Lemma 1 of [@SS]. To prove $(2)$ first note that, by $(1)$, $C^{\prime}\subset\overline{\Pi_{n-1}(C)}$. By Lemma 2 of [@SS], there exists a unique continuous function $r:\overline{\Pi_{n-1}(C)}\rightarrow\mathbb{R}\cup\{-\infty,\infty\}$ extending $Root_{x_{n},\lambda}f(\overline{a})$. Moreover, $r$ is either infinite or a root of $f$. Since $f$ is delineable on $C^{\prime}$ and $C^{\prime}$ is connected, either $r$ is a root of $f$ on $C^{\prime}$, or $r\equiv-\infty$ on $C^{\prime}$, or $r\equiv\infty$ on $C^{\prime}$.
We will prove $(3)$ by induction on $n$. Note that by $(1)$ it is sufficient to find cells $C_{j}$ such that $\dim(C_{j})=j$ and $C_{j}$ and $C_{j+1}$ are adjacent for $k\leq j<l$. If $n=1$ then dimensions of any pair of adjacent cells differ by one, hence $(3)$ is true. To prove $(3)$ for $n>1$ we will use induction on $l-k$. If $l-k=1$ then $(3)$ is true. Suppose $l-k>1$. Let $C_{k^{\prime}}^{\prime}=\Pi_{n-1}(C_{k})$ and $C_{l^{\prime}}^{\prime}=\Pi_{n-1}(C_{l})$, where $\dim(C_{k^{\prime}}^{\prime})=k^{\prime}$ and $\dim(C_{l^{\prime}}^{\prime})=l^{\prime}$. If $C_{l}$ is a section, then, by Lemma 2 of [@SS], there exists a continuous function $r:\overline{C_{l^{\prime}}^{\prime}}\rightarrow\mathbb{R}\cup\{-\infty,\infty\}$ such that $C_{l}=\{(x,r(x))\::\: x\in C_{l^{\prime}}^{\prime}\}$ and $r$ is infinite or a root of an element of $BP(F)$. In this case set $s=r$. Similarly, if $C_{l}$ is a sector, then, by Lemma 2 of [@SS], there exists continuous functions $r,s:\overline{C_{l^{\prime}}^{\prime}}\rightarrow\mathbb{R}\cup\{-\infty,\infty\}$ such that $C_{l}=\{(x,y)\::\: x\in C_{l^{\prime}}^{\prime}\wedge r(x)<y<s(x)\}$ and $r$ and $s$ are infinite or roots of elements of $BP(F)$. Since $l-k>1$, $k^{\prime}<l^{\prime}$. Suppose first that $l^{\prime}-k^{\prime}=1$. Since $l-k>1$, $C_{k}$ is a section and $C_{l}$ is a sector. If $C_{k}=\{(x,t(x))\::\: x\in C_{k^{\prime}}^{\prime}\}$, where $t=r$ or $t=s$, then $t$ is finite on $C_{l^{\prime}}^{\prime}$, $C_{k}$ is adjacent to $C_{k+1}:=\{(x,t(x))\::\: x\in C_{l^{\prime}}^{\prime}\}$, and $C_{k+1}$ is adjacent to $C_{l}$. Since $l=k+2$, $(3)$ is true. Otherwise, let $C_{k+1}$ be the sector directly below $C_{k}$. Then $C_{k+1}\subset\{(x,y)\::\: x\in C_{k^{\prime}}^{\prime}\wedge r(x)<y<s(x)\}$, and hence $C_{k+1}$ is adjacent to $C_{l}$. Again, since $l=k+2$, $(3)$ is true. Now suppose that $l^{\prime}-k^{\prime}>1$. $\Pi_{n-1}(F)$ is well-based, hence, by the inductive hypothesis on $n$, there exist cells $C_{k^{\prime}+1}^{\prime},\ldots,C_{l^{\prime}-1}^{\prime}\in D(\Pi_{n-1}(F))$ such that $\dim(C_{j}^{\prime})=j$ and $C_{j}^{\prime}\subseteq\overline{C_{j+1}^{\prime}}$ for $k^{\prime}\leq j<l^{\prime}$. Let $x_{0}\in C_{k^{\prime}}^{\prime}$ and $(x_{0},y_{0})\in C_{k}$. Then $r(x_{0})\leq y_{0}\leq s(x_{0})$. Since $C_{k^{\prime}}^{\prime}$ is adjacent to $C_{l^{\prime}-1}^{\prime}$, there exist a sequence $\{x_{n}\}_{n\geq1}$ such that $x_{n}\in C_{l^{\prime}-1}^{\prime}$ and $\lim_{n\rightarrow\infty}x_{n}=x_{0}$. Put $y_{n}=\max(r(x_{n}),\min(s(x_{n}),y_{0}))$. Then $\lim_{n\rightarrow\infty}(x_{n},y_{n})=(x_{0},y_{0})$. The set $$S=\{(x,y)\::\: x\in C_{l^{\prime}-1}^{\prime}\wedge y\in\mathbb{R}\wedge r(x)\leq y\leq s(x)\}$$ is a union of a finite number of cells, $S\subset\overline{C_{l}}$, and $(x_{n},y_{n})\in S$. Hence, there exists a cell $C\subseteq S$ such that $C$ contains infinitely many elements of the sequence $\{(x_{n},y_{n})\}_{n\geq1}$. Therefore, $C$ is adjacent to both $C_{k}$ and $C_{l}$. Since $\dim C-k<l-k$ and $l-\dim C<l-k$, $(3)$ is true by the inductive hypothesis on $l-k$.
For a given semialgebraic set $A$ a well-based CAF $F$ such that $D(F)$ is consistent with $A$ may not exist in a given system of coordinates. However, as shown in [@SS], there always exists a linear change of variables after which a well-based CAF $F$ such that $D(F)$ is consistent with $A$ does exist.
If $A$ is the real solution set of $xy+xz+yz=0$ then a well-based CAF $F$ such that $D(F)$ is consistent with $A$ does not exist for any order of variables. A CAD computed using McCallum’s projection operator [@MC1] includes cells $$\begin{aligned}
C_{1} & = & \{(x,y,z)\::\: x>0\wedge y>-x\wedge z=-\frac{xy}{x+y}\}\\
C_{2} & = & \{(x,y,z)\::\: x=0\wedge y=0\}\end{aligned}$$ $\overline{C_{1}}$ is not a union of cells, since $\overline{C_{1}}\cap C_{2}=\{(x,y,z)\::\: x=0\wedge y=0\wedge z\geq0\}$, and section $C_{1}$ is adjacent to a sector $C_{2}$ from a different stack. After the linear change of variables $(x,y,z)\rightarrow(x,y+z,z)$ $A$ is transformed to the solution set of $z^{2}+z(y+2x)+xy=0$. The following CAF $F$ is well-based and $D(F)$ is consistent with the transformed $A$.$$\begin{aligned}
F & = & (x<0\wedge G_{1})\vee(x=0\wedge((y<Root_{y,1}g\wedge G_{1})\vee\\
& & (y=Root_{y,1}g\wedge G_{2})\vee(y>Root_{y,1}g\wedge G_{1})))\vee(x>0\wedge G_{1})\end{aligned}$$ where$$\begin{aligned}
f & = & z^{2}+z(y+2x)+xy\\
g & = & y^{2}+4x^{2}\\
G_{1} & = & z<Root_{z,1}f\vee z=Root_{z,1}f\vee Root_{z,1}f<z<Root_{z,2}f\vee\\
& & z=Root_{z,2}f\vee z>Root_{z,2}f\\
G_{2} & = & z<Root_{z,1}f\vee z=Root_{z,1}f\vee z>Root_{z,1}f\end{aligned}$$
Root isolation algorithms
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In this section we describe root isolation algorithms we will use in the algorithm computing cell adjacencies. Let us first introduce some notations and subalgorithms.
Let $\Delta(c,r)=\{z\in\mathbb{C}\::\:\mid z-c\mid\leq r\}$ denote a disk in the complex plane, let $\mathbb{Q}_{2}=\mathbb{Z}[\frac{1}{2}]=\{a2^{b}\::\: a,b\in\mathbb{Z}\}$ denote the set of dyadic rational numbers, and let $I_{2}(\mathbb{C})=\{\Delta(c,r)\::\: c\in\mathbb{Q}_{2}[\imath]\wedge r\in\mathbb{Q}_{2}\wedge r>0\}$ denote the set of discs in the complex plane with dyadic Gaussian rational centers and dyadic rational radii. For a disc $Z=\Delta(c,r)\in I_{2}(\mathbb{C})$, let $\gamma(Z):=c$ and $\rho(Z):=r$ denote the center and the radius of $Z$, let $\underline{Z}:=\max(0,\mid c\mid-r)$ and $\overline{Z}:=\mid c\mid+r$ denote the minimum and maximum of absolute values of elements of $Z$, let $conj(Z)$ denote the disc that consists of complex conjugates of elements of $Z$, and let $dbl(Z)=\Delta(c,2r)$ and $quad(Z)=\Delta(c,4r)$. When we refer to interval arithmetic operations we mean circular complex interval (disc) arithmetic (see e.g. [@PP]).
\[ApproxRootsProp\]There exists an algorithm (ApproximateRoots) that takes as input a polynomial$$g=b_{N}x^{N}+\ldots+b_{0}\in\mathbb{\mathbb{Q}}_{2}[\imath][x]$$ and $p\in\mathbb{\mathbb{N}}$ and outputs $(s_{1},\ldots,s_{N})\in\mathbb{\mathbb{Q}}_{2}[\imath]^{N}$ such that for any polynomial $$f=a_{N}x^{N}+\ldots+a_{0}=a_{N}(x-\sigma_{1})\cdots(x-\sigma_{N})\in\mathbb{C}[x]$$ and any $\epsilon>0$ there exits $p_{0}\in\mathbb{N}$ such that if $p\geq p_{0}$ and, for all $0\leq i\leq N$, $$\mid b_{i}-a_{i}\mid\leq2^{-p}\max_{0\leq i\leq N}|a_{i}|$$ then, after a suitable reordering of roots, for all $1\leq j\leq N$ $\mid s_{j}-\sigma_{j}\mid\leq\epsilon$.
The algorithm described in [@P1] satisfies Proposition \[ApproxRootsProp\].
Let us now describe a subalgorithm computing roots of polynomials with complex disc coefficients. The algorithm is based on the following proposition ([@S7], Proposition 4.1).
\[RootMultProp\]Let $f\in\mathbb{C}[z]$ be a polynomial of degree $N$, $z_{0}\in\mathbb{C}$, $r>0$, and let $c_{i}:=\mid\frac{f^{(i)}(z_{0})}{i!}\mid$. Suppose that $$\max{}_{0\leq i<k}(\frac{Nc_{i}}{c_{k}})^{\frac{1}{k-i}}<r<\min{}_{k<i\leq N}(\frac{c_{k}}{Nc_{i}})^{\frac{1}{i-k}}$$ Then $f$ has exactly $k$ roots, multiplicities counted, in the disc $\Delta(z_{0},r)$.
The following is an extended version of Algorithm 4.2 from [@S7]. The key difference is that this version does not assume that the leading coefficient does not contain zero and provides a lower bound on the absolute value of roots that tend to infinity when the leading coefficients that contain zero vanish.
\[IntervalRoots\](IntervalRoots)\
Input: $Z_{0},\ldots,Z_{N}\in I_{2}(\mathbb{C})$.\
Output: $D_{1},\ldots,D_{m}\in I_{2}(\mathbb{C})$, positive integers $k_{1},\ldots,k_{m}$, and a positive radius $R$, such that for any $a_{0}\in Z_{0},\ldots,a_{N}\in Z_{N}$ and any $1\leq i\leq m$ the polynomial $f=a_{N}z^{N}+\ldots+a_{0}$ has exactly $k_{i}$ roots in the disc $D_{i}$, multiplicities counted, and $f$ has no roots in $\Delta(0,R)\setminus\bigcup_{i=1}^{m}D_{i}$. Moreover, for any $1\leq i<j\leq m$, $D_{i}\cap D_{j}=\emptyset$, and $D_{i}\subseteq\Delta(0,R)$. The other possible output is $Failed$.
1. If $0\in Z_{i}$ for all $0\leq i\leq N$ return $Failed$. Otherwise let $d$ be the maximal $i$ such that $0\notin Z_{i}$.
2. Put $$R:=\min{}_{d<i\leq N}(\frac{\underline{Z_{d}}}{N\overline{Z_{i}}})^{\frac{1}{i-d}}$$ If $$R\leq\max{}_{0\leq i<d}(\frac{N\overline{Z_{i}}}{\underline{Z_{d}}})^{\frac{1}{d-i}}$$ return $Failed$.
3. Set $f_{c}=b_{d}z^{d}+\ldots+b_{0}$, where $b_{i}=\gamma(Z_{i})$ for $0\leq i\leq d$, and set $$p=-\max_{0\leq i\leq d}\log\rho(Z_{i})$$
4. Compute $(s_{1},\ldots,s_{d})=ApproximateRoots(f_{c},p)$.
5. Let $F=Z_{N}z^{N}+\ldots+Z_{0}$. For each $1\leq j\leq d$ and $0\leq i\leq N$ use complex disc arithmetic to compute $W_{i,j}:=\frac{F^{(i)}(s_{j})}{i!}$.
6. For each $1\leq j\leq d$ let $\tilde{k_{j}}$ be the smallest $k>0$ such that $\underline{W_{k,j}}\neq0$ and$$r_{j}:=\max{}_{0\leq i<k}(\frac{N\overline{W_{i,j}}}{\underline{W_{k,j}}})^{\frac{1}{k-i}}<\min{}_{k<i\leq N}(\frac{\underline{W_{k,j}}}{N\overline{W_{i,j}}})^{\frac{1}{i-k}}$$ If there is no $k$ satisfying the condition return $Failed$.
7. Find the connected components of the union of $\triangle(s_{j},r_{j})$. Let $J_{1},\ldots,J_{m}$ be the sets of indices $j$ corresponding to the connected components.
8. For each $1\leq l\leq m$, let $J_{l}=\{j_{l,1},\ldots j_{l,k_{l}}\}$. If for some $j\in J_{l}$ $\tilde{k_{j}}\neq k_{l}$ return $Failed$. Otherwise pick $j\in J_{l}$ with the minimal value of $r_{j}$, and set $D_{l}:=\Delta(s_{j},r_{j})$.
9. If $D_{l}\nsubseteq\Delta(0,R)$ for some $1\leq l\leq m$ return $Failed$.
10. Return $(D_{1},\ldots,D_{m})$, $(k_{1},\ldots,k_{m})$, and $R$.
To show correctness of Algorithm \[IntervalRoots\], suppose that the algorithm returned $(D_{1},\ldots,D_{m})$, $(k_{1},\ldots,k_{m})$, and $R$. Let $a_{0}\in Z_{0},\ldots,a_{N}\in Z_{N}$ and $f=a_{N}z^{N}+\ldots+a_{0}$. By Proposition \[RootMultProp\], and because the algorithm did not fail in step $(2)$, $f$ has exactly $d$ roots in $\Delta(0,R)$. The condition in step $(6)$ and Proposition \[RootMultProp\] imply that $f$ has exactly $k_{i}$ roots in the disc $D_{i}$. Since the algorithm did not fail in step $(8)$, $k_{1}+\ldots+k_{m}=d$, and hence $f$ has no roots in $\Delta(0,R)\setminus\bigcup_{i=1}^{m}D_{i}$. Step $(7)$ guarantees that for any $1\leq i<j\leq m$, $D_{i}\cap D_{j}=\emptyset$. Step $(9)$ ensures that $D_{i}\subseteq\Delta(0,R/2)$.
Computation of sample points in CAD cells requires a representation of vectors with algebraic number coordinates. The following gives a recursive definition of root isolation data and of representation of real algebraic vectors. Note that root isolation data provides information about roots of $f_{k}$ not only at $u$, but also in a neighbourhood of $u$ (point $(5)$ of the definition). This property is crucial for computing cell adjacencies.
\[RAV\]*$\Theta_{k}=((D_{1},\ldots,D_{m}),(k_{1},\ldots,k_{m}),R)$* is *root isolation data* for $f_{k}\in\mathbb{Q}[x_{1},\ldots,x_{k}]$ at $u=(\alpha_{1},\ldots,\alpha_{k-1})\in\mathbb{R}^{k-1}$ if
1. $D_{1},\ldots,D_{m}\in I_{2}(\mathbb{C})$, $k_{1},\ldots,k_{m}\in\mathbb{N}_{+}$, and $R>0$,
2. $g_{k}=f_{k}(\alpha_{1},\ldots,\alpha_{k-1},x_{k})$ has a root of multiplicity $k_{j}$ in $D_{j}$, for $1\leq j\leq m$, and has no other roots,
3. for any $1\leq j\leq m$ $dbl(D_{j})\subseteq\Delta(0,R/2)$,
4. for any $j_{1}\neq j_{2}$ $dbl(D_{j_{1}})\cap dbl(D_{j_{2}})=\emptyset$ and if $dbl(D_{j_{1}})\cap\mathbb{R}\neq\emptyset$ then $conj(dbl(D_{j_{1}}))\cap dbl(D_{j_{2}})=\emptyset$,
5. if $W_{i}$ is the isolating disc of $\alpha_{i}$ for $1\leq i<k$, and $\beta_{i}\in quad(W_{i})$ then $f_{k}(\beta_{1},\ldots,\beta_{k-1},x_{k})$ has exactly $k_{j}$ roots in $D_{j}$, multiplicities counted, and has no roots in $\Delta(0,R)\setminus\bigcup_{j=1}^{m}D_{j}$.
A *real algebraic vector* $v=RAV(\Omega_{k})=(\alpha_{1},\ldots,\alpha_{k})\in\mathbb{R}^{k}$ is represented by $$\Omega_{k}=(\Omega_{k-1},f_{k},W_{k},\Theta_{k})$$ where
- $\Omega_{k-1}$ represents the real algebraic vector $\Pi(v)=RAV(\Omega_{k-1})=(\alpha_{1},\ldots,\alpha_{k-1})\in\mathbb{R}^{k-1}$,
- $f_{k}\in\mathbb{Q}[x_{1},\ldots,x_{k}]$ is a *defining polynomial* of $\alpha_{k}$,
- $W_{k}\in I_{2}(\mathbb{C})$ is an *isolating disc* of $\alpha_{k}$,
- *$\Theta_{k}=((D_{1},\ldots,D_{m}),(k_{1},\ldots,k_{m}),R)$* is ** root isolation data ** for $f_{k}$ *at $\Pi(v)$*, $W_{k}=D_{j_{0}}$ for some $1\leq j_{0}\leq m$, and $W_{k}\cap\mathbb{R}\neq\emptyset$.
Define $\rho(\Omega_{k})=\max(\rho(\Omega_{k-1}),\max_{1\leq j\leq m}\rho(D_{j}))$.
To complete the recursive definition let $\Omega_{0}=()$ be the representation of the only element of $\mathbb{R}^{0}$.
We will say that $\Omega_{k}^{\prime}=(\Omega_{k-1}^{\prime},f_{k},W_{k}^{\prime},\Theta_{k}^{\prime})$ is a *refinement* of $\Omega_{k}$ if $RAV(\Omega_{k}^{\prime})=RAV(\Omega_{k})$, *$\Theta_{k}^{\prime}=((D_{1}^{\prime},\ldots,D_{m}^{\prime}),(k_{1},\ldots,k_{m}),R^{\prime})$,* *$\rho(D_{j}^{\prime})<\rho(D_{j})$* and $D_{j}^{\prime}\subseteq dbl(D_{j})$ for $1\leq j\leq m$, and $\Omega_{k-1}^{\prime}$ is a refinement of $\Omega_{k-1}$.
For any $v=(a_{1},\ldots,a_{k})\in\mathbb{R}^{k}$ and $a\in\mathbb{R}$, we will use notation $\Lambda(v)=a_{k}$ and $v\times a=(a_{1},\ldots,a_{k},a)\in\mathbb{R}^{k+1}$.
\[RefRem\]A refinement $\Omega_{k}^{\prime\prime}$ of a refinement $\Omega_{k}^{\prime}$ of $\Omega_{k}$ is a refinement of $\Omega_{k}$.
With notations from Definition \[RAV\], let $$\Omega_{k}^{\prime\prime}=(\Omega_{k-1}^{\prime\prime},f_{k},W_{k}^{\prime\prime},\Theta_{k}^{\prime\prime})$$ and ** $$\Theta_{k}^{\prime\prime}=((D_{1}^{\prime\prime},\ldots,D_{m}^{\prime\prime}),(k_{1},\ldots,k_{m}),R^{\prime\prime})$$ By induction, it suffices to show *$\rho(D_{j}^{\prime\prime})<\rho(D_{j})$* and $D_{j}^{\prime\prime}\subseteq dbl(D_{j})$ for $1\leq j\leq m$. $\rho(D_{j}^{\prime\prime})<\rho(D_{j})$ follows from *$\rho(D_{j}^{\prime\prime})<\rho(D_{j}^{\prime})$ and* $\rho(D_{j}^{\prime})<\rho(D_{j})$. ** $D_{j}^{\prime\prime}\subseteq dbl(D_{j}^{\prime})$ and $D_{j}^{\prime}\subseteq dbl(D_{j})$ implies that $D_{j}$, $D_{j}^{\prime}$ and $D_{j}^{\prime\prime}$ contain the same root of $f_{k}(\alpha_{1},\ldots,\alpha_{k-1},x_{k})$. Then *$\rho(D_{j}^{\prime\prime})<\rho(D_{j})$ implies* $D_{j}^{\prime\prime}\subseteq dbl(D_{j})$.
The algorithms we introduce next take a *working precision* argument. A working precision $p$ is a positive integer. One can think of it as the number of bits in floating-point numbers used in a numeric approximation algorithm. However, we will not attach any specific meaning to the working precision argument. Instead our algorithms will satisfy certain properties as $p$ tends to infinity. For instance, if we say that a certain quantity $\omega$ in the output of an algorithm tends to zero as $p$ tends to infinity, it means that for any $\epsilon>0$ there exists $N>0$ such that if the working precision $p>N$ then the algorithm will produce an output with $\omega<\epsilon$.
Let $v=(\alpha_{1},\ldots,\alpha_{k})$ be a real algebraic vector and let $$f\in\mathbb{Q}[x_{1},\ldots,x_{k},x_{k+1}]$$ be such that $f(\alpha_{1},\ldots,\alpha_{k},x_{k+1})$ does not vanish identically. We will now describe an algorithm $AlgRoots_{k}$, with $k\geq0$, which finds the root isolation data of $f$ at $v$ and the real roots of $f(\alpha_{1},\ldots,\alpha_{k},x_{k+1})$. The algorithm uses two subalgorithms $Refine_{k}$ and $ZeroTest_{k}$ that will be defined recursively in terms of $AlgRoots_{k-1}$. Given $$(\alpha_{1},\ldots,\alpha_{k})=RAV(\Omega_{k})$$ and a working precision $p>0$ $Refine_{k}$ computes a refinement $\Omega_{k}^{\prime}$ of $\Omega_{k}$ such that as $p$ tends to infinity $\rho(\Omega_{k}^{\prime})$ tends to zero. $ZeroTest_{k}$ decides whether $h(\alpha_{1},\ldots,\alpha_{k})$ is zero for a given $h\in\mathbb{Q}[x_{1},\ldots,x_{k}]$.
\[AlgRoots\]($AlgRoots_{k}$)\
Input: Real algebraic vector $v=(\alpha_{1},\ldots,\alpha_{k})=RAV(\Omega_{k})$, where $k\geq0$, $f\in\mathbb{Q}[x_{1},\ldots,x_{k},x_{k+1}]$, such that $f(\alpha_{1},\ldots,\alpha_{k},x_{k+1})$ does not vanish identically, and a working precision $p>0$.\
Output: Root isolation data $\Theta$ of $f$ at $v$, a refinement $\Omega_{k}^{\prime}$ of $\Omega_{k}$, and real algebraic vectors $v_{1}=RAV(\Omega_{k+1,1}),\ldots,v_{r}=RAV(\Omega_{k+1,r})\in\mathbb{R}^{k+1}$ such that $\Pi(v_{j})=RAV(\Omega_{k}^{\prime})$, for $1\leq j\leq r$, $\Lambda(v_{1}),\ldots,\Lambda(v_{r})$ are all the real roots of $f(\alpha_{1},\ldots,\alpha_{k},x_{k+1})$, and as $p$ tends to infinity $\rho(\Omega_{k+1,j})$ tends to zero.
1. Let $f=a_{N}x_{k+1}^{N}+\ldots+a_{0}$. Find $d$ such that $a_{d}(\alpha_{1},\ldots,\alpha_{k})\neq0$ and $$a_{d+1}(\alpha_{1},\ldots,\alpha_{k})=\ldots=a_{N}(\alpha_{1},\ldots,\alpha_{k})=0$$ using $ZeroTest_{k}$ if $k>0$. Set $g=a_{d}z^{d}+\ldots+a_{0}$.
2. Compute the principal subresultant coefficients $psc_{0},\ldots,psc_{n-1}$ of $g$ and $\frac{\partial g}{\partial z}$ with respect to $z$.
3. Find the largest integer $\mu\geq0$ such that $$psc_{0}(\alpha_{1},\ldots,\alpha_{k})=\ldots=psc_{\mu-1}(\alpha_{1},\ldots,\alpha_{k})=0$$ using $ZeroTest_{k}$ if $k>0$.
4. Set $p^{\prime}=p$ and $\Omega_{k}^{\prime}=\Omega_{k}$.
5. If $k>0$ compute $\Omega_{k}^{\prime}=Refine_{k}(\Omega_{k}^{\prime},p^{\prime})$ and let $W_{1},\ldots,W_{k}$ be the isolating discs of $\alpha_{1},\ldots,\alpha_{k}$ in $\Omega_{k}^{\prime}$.
6. For $0\leq i\leq N$
1. if $a_{i}\in\mathbb{Q}$ compute $Z_{i}\in I_{2}(\mathbb{C})$ such that $a_{i}\in Z_{i}$ and $\rho(a_{i})\leq2^{-p^{\prime}}$,
2. else compute $Z_{i}=a_{i}(quad(W_{1}),\ldots,quad(W_{k}))$ using interval arithmetic.
7. Compute $\Theta:=IntervalRoots(Z_{0},\ldots,Z_{N})$.
8. If $\Theta=Failed$ double $p^{\prime}$ and go to step $(5)$.
9. Let $\Theta=((D_{1},\ldots,D_{m}),(k_{1},\ldots,k_{m}),R)$. If $d-m>\mu$ or the conditions $(3)$ and $(4)$ of Definition \[RAV\] are not satisfied, double $p^{\prime}$ and go to step $(5)$.
10. Let $(j_{1},\ldots,j_{r})$ be the set of indices for which $D_{j_{l}}\cap\mathbb{R}\neq\emptyset$. For $1\leq l\leq r$ let $\Omega_{k+1,l}=(\Omega_{k}^{\prime},f,D_{j_{l}},\Theta)$ and $v_{l}=RAV(\Omega_{k+1,l})$.
11. Return $\Theta$, $\Omega_{k}^{\prime}$, and $v_{1},\ldots,v_{r}$.
To prove termination of the Algorithm \[AlgRoots\] we need to show that for sufficiently large $p^{\prime}$ the call to *IntervalRoots* in step $(7)$ succeeds and gives a result with $d-m=\mu$. The specification of the algorithm $Refine_{k}$ implies that as $p^{\prime}$ tends to infinity $\max_{1\leq i\leq k}\rho(W_{i})$ tends to zero. Hence also $\max_{1\leq i\leq N}\rho(Z_{i})$ tends to zero. Therefore, as $p^{\prime}$ tends to infinity, $\gamma(Z_{i})$ tends to $a_{i}(\alpha_{1},\ldots,\alpha_{k})$ and $\underline{Z_{i}}$ and $\overline{Z_{i}}$ tend to $|a_{i}(\alpha_{1},\ldots,\alpha_{k})|$ for all $0\leq i\leq N$. In particular, for sufficiently large $p^{\prime}$, $0\notin Z_{i}$ iff $a_{i}(\alpha_{1},\ldots,\alpha_{k})\neq0$, and hence $d$ in step $(1)$ of *IntervalRoots* is the same as $d$ computed in step $(1)$ of Algorithm \[AlgRoots\]. Since$$R:=\min{}_{d<i\leq N}(\frac{\underline{Z_{d}}}{N\overline{Z_{i}}})^{\frac{1}{i-d}}$$ tends to infinity and $$\max{}_{0\leq i<d}(\frac{N\overline{Z_{i}}}{\underline{Z_{d}}})^{\frac{1}{d-i}}$$ tends to a finite constant, the call to *IntervalRoots* does not fail in in step $(2)$ for sufficiently large $p^{\prime}$. Let $\sigma_{1},\ldots\sigma_{d}$ be the roots of $h(z)=g(\alpha_{1},\ldots,\alpha_{k},z)$, each repeated as many times as its multiplicity. Let $s_{1},\ldots,s_{d}$ be the roots computed in step $(4)$ of *IntervalRoots*. The specification of *ApproximateRoots* implies that, as $p^{\prime}$ tends to infinity, after a suitable reordering of roots, $s_{j}$ tends to $\sigma_{j}$ for each $1\leq j\leq d$. Hence for $W_{i,j}:=\frac{F^{(i)}(s_{j})}{i!}$ computed in step $(5)$ of *IntervalRoots* $\gamma(W_{i,j})$ tends to $\frac{h^{(i)}(\sigma_{j})}{i!}$ and $\rho(W_{i,j})$ tends to zero. Therefore, $\underline{W_{i,j}}$ and $\overline{W_{i,j}}$ tend to $|\frac{h^{(i)}(\sigma_{j})}{i!}|$. Hence, if $k_{j}$ is the multiplicity of $\sigma_{j}$, $$r_{j}:=\max{}_{0\leq i<k_{j}}(\frac{N\overline{W_{i,j}}}{\underline{W_{k_{j},j}}})^{\frac{1}{k_{j}-i}}$$ tends to zero and$$\min{}_{k_{j}<i\leq N}(\frac{\underline{W_{k_{j},j}}}{N\overline{W_{i,j}}})^{\frac{1}{i-k_{j}}}$$ is bounded away from zero for sufficiently large $p^{\prime}$. Therefore, for sufficiently large $p^{\prime}$, the condition in step $(6)$ of *IntervalRoots* is satisfied by $k_{j}$. Note that if $s_{j}$ is closer to $\sigma_{j}$ than to other roots of $h$ then the condition cannot be satisfied by any $k<k_{j}$. Otherwise, by Proposition \[RootMultProp\], $h$ would have exactly $k$ roots in $\Delta(s_{j},r_{j})$ which is impossible since if $\sigma_{j}\notin\Delta(s_{j},r_{j})$ then $\Delta(s_{j},r_{j})$ contains no roots of $h$ and else $\Delta(s_{j},r_{j})$ contains at least $k_{j}$ roots of $h$. Hence, for sufficiently large $p^{\prime}$, step $(6)$ does not fail and $\tilde{k_{j}}=k_{j}$ for each$1\leq j\leq d$. Since $r_{j}$ tends to zero for $1\leq j\leq d$, for sufficiently large $p^{\prime}$, $\Delta(s_{j_{1}},r_{j_{1}})$ and $\Delta(s_{j_{2}},r_{j_{2}})$ intersect iff $\sigma_{j_{1}}=\sigma_{j_{2}}$, and hence step $(8)$ of *IntervalRoots* does not fail. Since $R$ tends to infinity, for sufficiently large $p^{\prime}$, step $(9)$ of *IntervalRoots* does not fail and the whole algorithm succeeds. Since, for sufficiently large $p^{\prime}$, the discs returned by *IntervalRoots* correspond to distinct roots of $h$, $d-m=\mu$ in step $(9)$ of Algorithm \[AlgRoots\]. Since $\rho(D_{i})$ tends to zero for $1\leq i\leq m$ and $R$ tends to infinity, for sufficiently large $p^{\prime}$, we have $dbl(D_{i})\subseteq\Delta(0,R/2)$ for any $1\leq i\leq m$ and
$$\rho(D_{i})<\frac{\min_{\sigma_{j_{1}}\neq\sigma_{j_{2}}}|\sigma_{j_{1}}-\sigma_{j_{2}}|}{16}$$ Therefore for any $i_{1}\neq i_{2}$ $dbl(D_{i_{1}})\cap dbl(D_{i_{2}})=\emptyset$ and if $\sigma$ is the root of $h$ in $D_{i_{1}}$ then either $\sigma\notin\mathbb{R}$ and $dbl(D_{i_{1}})\cap\mathbb{R}=\emptyset$ or $\sigma\in\mathbb{R}$ and $conj(dbl(D_{i_{1}}))\cap dbl(D_{i_{2}})=\emptyset$. Hence for sufficiently large $p^{\prime}$, the conditions $(3)$ and $(4)$ of Definition \[RAV\] are satisfied and the algorithm terminates.
Proposition 4.3 of [@S7] and correctness of Algorithm \[IntervalRoots\] imply that $\Omega_{k+1,l}$ satisfy the conditions $(1)$, $(2)$, and $(5)$ of Definition \[RAV\], and the conditions $(3)$ and $(4)$ are ensured by step $(9)$ of Algorithm \[AlgRoots\]. Step $(10)$ selects all isolating discs that intersect the real line, hence $\Lambda(v_{1}),\ldots,\Lambda(v_{r})$ are all the real roots of $f(\alpha_{1},\ldots,\alpha_{k},x_{k+1})$.
To show that $\rho(\Omega_{k+1,j})$ tends to zero as $p$ tends to infinity, note that $p^{\prime}\geq p$, we have already shown that $\rho(D_{i})$ tends to zero for $1\leq i\leq m$ as $p^{\prime}$ tends to infinity, and since $\Omega_{k}^{\prime}$ is computed by $Refine_{k}$ with working precision $p^{\prime}$, $\rho(\Omega_{k}^{\prime})$ tends to zero as $p^{\prime}$ tends to infinity.
If $m=d$ in step $(10)$ of Algorithm \[AlgRoots\] then $$g(\alpha_{1},\ldots,\alpha_{k},z)$$ does not have multiple roots and computing the principal subresultant coefficients is not necessary. Hence instead of computing the principal subresultant coefficients in step $(2)$ it is sufficient to compute them only when the algorithm reaches step $(9)$ for the first time and $m<d$.
To complete the description of Algorithm \[AlgRoots\] let us now define the subalgorithms $Refine_{k}$ and $ZeroTest_{k}$.
\[Refine\]($Refine_{k}$)\
Input: Real algebraic vector $(\alpha_{1},\ldots,\alpha_{k})=RAV(\Omega_{k})$, where $k\geq1$, and a working precision $p>0$.\
Output: A refinement $\Omega_{k}^{\prime}$ of $\Omega_{k}$ such that as $p$ tends to infinity tends to zero.
1. Let $\Omega_{k}=(\Omega_{k-1},f_{k},W_{k},\Theta_{k})$, $$\Theta_{k}=((D_{1},\ldots,D_{m}),(k_{1},\ldots,k_{m}),R)$$ and $W_{k}=D_{j_{0}}$. Set $v=RAV(\Omega_{k-1})$ and $p^{\prime}=p$.
2. Compute $$(\Theta;\Omega_{k-1}^{\prime};v_{1},\ldots,v_{r})=AlgRoots_{k-1}(v,f_{k},p^{\prime})$$ where *$\Theta=((D_{1}^{\prime},\ldots,D_{m}^{\prime}),(k_{1},\ldots,k_{m}),R^{\prime})$.*
3. *If no reordering of indices in $\Theta$ yields* *$\rho(D_{j}^{\prime})<\rho(D_{j})$* and $D_{j}^{\prime}\subseteq dbl(D_{j})$ for $1\leq j\leq m$, double $p^{\prime}$ and go to $(2)$.
4. Let $v_{j}=RAV(\Omega_{k}^{\prime})$ be such that the isolating disk of $\Lambda(v_{j})$ is $D_{j_{0}}^{\prime}$. Return $\Omega_{k}^{\prime}$.
Since $\rho(D_{j}^{\prime})$ tends to zero as $p^{\prime}$ tends to infinity, for sufficiently large $p^{\prime}$ the condition in step $(3)$ is satisfied for pairs $D_{j}$ and $D_{j}^{\prime}$ containing the same root of $f_{k}(\alpha_{1},\ldots,\alpha_{k-1},x_{k})$, and hence the algorithm terminates. Correctness of Algorithm \[AlgRoots\] and the condition in step $(3)$ guarantee that $\Omega_{k}^{\prime}$ is a refinement of $\Omega_{k}$ and as $p$ tends to infinity $\rho(\Omega_{k}^{\prime})$ tends to zero.
\[ZeroTest\]($ZeroTest_{k}$)\
Input: Real algebraic vector $(\alpha_{1},\ldots,\alpha_{k})=RAV(\Omega_{k})$, where $k\geq1$, and $h\in\mathbb{Q}[x_{1},\ldots,x_{k}]$.\
Output: $true$ if $h(\alpha_{1},\ldots,\alpha_{k})=0$ and $false$ otherwise.
1. Let $\Omega_{k}=(\Omega_{k-1},f_{k},W_{k},\Theta_{k})$, $$\Theta_{k}=((D_{1},\ldots,D_{m}),(k_{1},\ldots,k_{m}),R)$$ and $W_{k}=D_{j_{0}}$. Set $\mu=k_{j_{0}}$, $\Omega_{k}^{\prime}=\Omega_{k}$, and set an initial value $p$ of working precision (e.g. to precision that was used to compute $\Omega_{k}$).
2. Compute $\Omega_{k}^{\prime}=Refine_{k}(\Omega_{k}^{\prime},p)$. Let $\Omega_{k}^{\prime}=(\Omega_{k-1}^{\prime},f_{k},W_{k}^{\prime},\Theta_{k}^{\prime})$. Set $v=RAV(\Omega_{k-1}^{\prime})$.
3. Compute $(\Theta;\Omega_{k-1}^{\prime};v_{1},\ldots,v_{r})=AlgRoots_{k-1}(v,f_{k}h,p)$.
4. If $W_{k}^{\prime}$ intersects the isolating disc of $\Lambda(v_{j})$ for more than one $j$, double $p$ and go to step $(2)$.
5. Let $j$ be the only index for which $W_{k}^{\prime}$ intersects the isolating disc $W$ of $\Lambda(v_{j})$. Let *$\Theta=((\tilde{D_{1}},\ldots,\tilde{D_{m}}),(\tilde{k_{1}},\ldots,\tilde{k_{m}}),\tilde{R})$*, and $W=\tilde{D_{\tilde{j_{0}}}}$.
6. If $\tilde{k_{\tilde{j_{0}}}}>\mu$ return $true$ otherwise return $false$.
Since as $p^{\prime}$ tends to infinity, $\rho(W_{k}^{\prime})$ and $\max_{1\leq j\leq r}\rho(\Lambda(v_{j}))$ tend to zero, for sufficiently large $p^{\prime}$, $W_{k}^{\prime}$ intersects only the isolating disc of $\Lambda(v_{j})=\alpha_{k}$, which proves termination and correctness of the algorithm.
Since $AlgRoots_{k}$ is defined for $k\geq0$ and $Refine_{k}$ and $ZeroTest_{k}$ are defined for $k\geq1$, the recursive definition of the algorithms is complete.
$ZeroTest_{k}$ is defined here in terms of $AlgRoots_{k-1}$ for simplicity of description. In practice, to decide whether $$h(\alpha_{1},\ldots,\alpha_{k})=0$$ we can first evaluate $h$ at the isolating discs of $\alpha_{1}\ldots,\alpha_{k}$ using interval arithmetic. If the result does not contain zero then $$h(\alpha_{1},\ldots,\alpha_{k})\neq0$$ Otherwise, we isolate roots of $$h_{\alpha}=h(\alpha_{1},\ldots,\alpha_{k-1},z)$$ and refine isolating discs of roots of $$g_{\alpha}=g(\alpha_{1},\ldots,\alpha_{k-1},z)$$ and roots of $h_{\alpha}$ until either the isolating disc of $\alpha_{k}$ does not intersect any isolating discs of roots of $h_{\alpha}$ or the number of intersecting isolating discs of roots of $g_{\alpha}$ and $h_{\alpha}$ agrees with the number of common roots of $g_{\alpha}$ and $h_{\alpha}$ computed by finding signs of principal subresultant coefficients of $g_{\alpha}$ and $h_{\alpha}$ (see Proposition 4.4 of [@S7]). When the algorithm is used in CAD construction we also use information about polynomials that are zero at the current point that was collected during the construction (see [@S7], Section 4.1).
Finding cell adjacencies
========================
Let $F$ be a well-based CAF in $x_{1},\ldots,x_{n}$. For simplicity let us assume that $BP(F)=\{f_{1},\ldots,f_{n}\}$, where $$f_{k}\in\mathbb{\mathbb{Q}}[x_{1},\ldots,x_{k}]\setminus\mathbb{\mathbb{Q}}[x_{1},\ldots,x_{k-1}]$$ This can be always achieved by multiplying all elements of $$BP(F)\cap(\mathbb{\mathbb{Q}}[x_{1},\ldots,x_{k}]\setminus\mathbb{\mathbb{Q}}[x_{1},\ldots,x_{k-1}])$$ We can also assume that $f_{1},\ldots,f_{n}$ are square-free.
The main algorithm *CADAdjacency* (Algorithm \[CADAdjacency\]) finds all pairs of adjacent cells $(C,C^{\prime})\in D(F)^{2}$ such that $\dim C-\dim C^{\prime}=1$. By Proposition \[WellBasedProp\], to determine all cell adjacencies for a well-based CAF it is sufficient to find all pairs of adjacent cells whose dimensions differ by one, hence Algorithm \[CADAdjacency\] is sufficient to fully solve the cell adjacency problem for well-based CAF.
The algorithm first calls *SamplePoints* (Algorithm \[SPT\]), which constructs a sample point $SPT(C)=(a_{1},\ldots,a_{k})\in\mathbb{R}^{k}$ in each cell $C\in D(\Pi_{k}(F))$, for $1\leq k\leq n$, and computes root isolation data $RTS(C)$ for each cell $C\in D(\Pi_{k}(F))$, for $1\leq k<n$. Let us describe the representation of sample points and give the specification of root isolation data. Let $I=(i_{1},\ldots,i_{l})$ be the set of indices $1\leq i\leq k$ such that $\Pi_{i}(C)$ is a section. For $i\notin I$, $a_{i}$ is a rational number and for $i\in I$, $a_{i}$ is an algebraic number. To represent sample points we will use combinations of rational vectors and algebraic vectors defined as follows. Let $1\leq k\leq n$, let $0\leq l\leq k$, let $I=\{i_{1},\ldots,i_{l}\}$, where $1\leq i_{1}<\ldots<i_{l}\leq k$, and let $J=\{1,\ldots,k\}\setminus I=\{j_{1},\ldots,j_{k-l}\}$, where $1\leq j_{1}<\ldots<j_{k-l}\leq k$. Let $v=(\alpha_{1},\ldots,\alpha_{l})=RAV(\Omega_{l})$ be a real algebraic vector and let $w=(q_{1},\ldots,q_{k-l})\in\mathbb{Q}^{k-l}$ be a rational vector. By $PT(v,w,I,J)$ we denote the point $(a_{1},\ldots,a_{k})\in\mathbb{R}^{k}$ such that $a_{i_{s}}=\alpha_{s}$ for $1\leq s\leq l$ and $a_{j_{t}}=q_{t}$ for $1\leq t\leq k-l$. Suppose $1\leq k<n$ and $SPT(C)=PT(v,w,I,J)$. Let $f_{k+1}^{C}\in\mathbb{Q}[x_{i_{1}},\ldots,x_{i_{l}},x_{k+1}]$ denote $f_{k+1}$ with $x_{j_{t}}$ replaced by $a_{j_{t}}$ for $1\leq t\leq k-l$. Then $RTS(C)$ computed by *SamplePoints* is root isolation data of $f_{k+1}^{C}$ at $v$.
Next *CADAdjacency* calls *AdjacencyPoints* (Algorithm \[ADP\]) which, for $1\leq k\leq n$, and for each pair of adjacent cells $(C,C^{\prime})$ of $D(\Pi_{k}(F))$ with $\dim C^{\prime}=\dim C-1$, constructs a point $ADP(C,C^{\prime})\in C$ which satisfies the following condition.
\[ADPCond\]If $SPT(C^{\prime})=(a_{1},\ldots,a_{k})$ and $ADP(C,C^{\prime})=(b_{1},\ldots,b_{k})$ then
- for each $1\leq i\leq k$ if $a_{i}$ is a root of $f_{i}(a_{1},\ldots,a_{i-1},x_{i})$ with isolating disc $W_{i}$ then $b_{i}\in dbl(W_{i})$,
- if $a_{i}$ is a rational number between roots of $f_{i}(a_{1},\ldots,a_{i-1},x_{i})$ then $b_{i}=a_{i}$.
Finally, *CADAdjacency* returns the pairs of cells $(C,C^{\prime})\in D(F)^{2}$ for which $ADP(C,C^{\prime})$ is defined.
\[CADAdjacency\](CADAdjacency)\
Input: A well-based CAF $F$ in $x_{1},\ldots,x_{n}$ with $BP(F)=\{f_{1},\ldots,f_{n}\}$, where $f_{k}\in\mathbb{\mathbb{Q}}[x_{1},\ldots,x_{k}]\setminus\mathbb{\mathbb{Q}}[x_{1},\ldots,x_{k-1}]$.\
Output: The set $A$ of all pairs of adjacent cells $(C,C^{\prime})\in D(F)^{2}$ such that $\dim C-\dim C^{\prime}=1$.
1. Compute $(SPT,RTS)=SamplePoints(F)$.
2. Compute $ADP=AdjacencyPoints(F,SPT,RTS)$.
3. Return the set of all pairs of cells $(C,C^{\prime})\in D(F)^{2}$ such that $ADP(C,C^{\prime})$ is defined.
\[SPT\](SamplePoints)\
Input: A well-based CAF $F$ in $x_{1},\ldots,x_{n}$ with $BP(F)=\{f_{1},\ldots,f_{n}\}$, where $f_{k}\in\mathbb{\mathbb{Q}}[x_{1},\ldots,x_{k}]\setminus\mathbb{\mathbb{Q}}[x_{1},\ldots,x_{k-1}]$.\
Output: $SPT$ and $RTS$ such that
- for $1\leq k\leq n$ and for each cell $C$ of $D(\Pi_{k}(F))$, $SPT(C)$ is a sample point in $C$,
- for $1\leq k<n$ and for each cell $C$ of $D(\Pi_{k}(F))$, $RTS(C)$ is root isolation data for $C$.
1. Set an initial value $p$ of working precision.
2. Compute $(\Theta;();v_{1},\ldots,v_{r})=AlgRoots_{0}((),f_{1},p)$. We have $$\Theta=((D_{1},\ldots,D_{m}),(k_{1},\ldots,k_{m}),R)$$
3. Pick rational numbers $-R<q_{1}<\Lambda(v_{1})<q_{2}<\ldots<q_{r}<\Lambda(v_{r})<q_{r+1}<R$ such that $q_{i}\notin\bigcup_{j=1}^{m}dbl(D_{j})$.
4. For $1\leq i\leq r$, let $C$ be the $i$-th $\{f_{1}\}$-section. Set $SPT(C)=PT(v_{i},(),\{1\},\{\})$.
5. For $1\leq i\leq r+1$, let $C$ be the $i$-th $\{f_{1}\}$-sector. Set $SPT(C)=PT((),(q_{i}),\{\},\{1\})$.
6. For $1\leq k<n$ and for each cell $C$ of $D(\Pi_{k}(F))$:
1. Let $(a_{1},\ldots,a_{k})=PT(v,w,I,J)=SPT(C)$.
2. Let $I=\{i_{1},\ldots,i_{l}\}$, $J=\{j_{1},\ldots,j_{k-l}\}$, and let $f_{k+1}^{C}$ be $f_{k+1}$ with $x_{j_{t}}$ replaced by $a_{j_{t}}$ for $1\leq t\leq k-l$.
3. Compute $(\Theta;\Omega_{l}^{\prime};v_{1},\ldots,v_{r})=AlgRoots_{l}(v,f_{k+1}^{C},p)$. We have $$\Theta=((D_{1},\ldots,D_{m}),(k_{1},\ldots,k_{m}),R)$$
4. Set $RTS(C)=\Theta$ and replace representations of algebraic vectors in $SPT(\Pi_{i}(C))$ for $i\leq k$ with their refinements that appear in $\Omega_{l}^{\prime}$.
5. Pick rational numbers $-R<q_{1}<\Lambda(v_{1})<q_{2}<\ldots<q_{r}<\Lambda(v_{r})<q_{r+1}<R$ such that $q_{i}\notin\bigcup_{j=1}^{m}dbl(D_{j})$, and let $w_{i}=w\times q_{i}$ for $1\leq i\leq r+1$.
6. For $1\leq i\leq r$, let $S$ be the $i$-th $\{f_{k+1}\}$-section over $C$. Set $$SPT(S)=PT(v_{i},w,I\cup\{k+1\},J)$$
7. For $1\leq i\leq r+1$, let $S$ be the $i$-th $\{f_{k+1}\}$-sector over $C$. Set $$SPT(S)=PT(v,w_{i},I,J\cup\{k+1\})$$
7. Return $SPT$ and $RTS$.
\[ADP\](AdjacencyPoints)\
Input: A well-based CAF $F$ in $x_{1},\ldots,x_{n}$ with $BP(F)=\{f_{1},\ldots,f_{n}\}$, where $f_{k}\in\mathbb{\mathbb{Q}}[x_{1},\ldots,x_{k}]\setminus\mathbb{\mathbb{Q}}[x_{1},\ldots,x_{k-1}]$, $SPT$ and $RTS$ as in the output of Algorithm \[SPT\].\
Output: $ADP$ such that for $1\leq k\leq n$ and for each pair of adjacent cells $(C,C^{\prime})$ of $D(\Pi_{k}(F))$ with $\dim C^{\prime}=\dim C-1$, $ADP(C,C^{\prime})$ is a point in $C$ satisfying Condition \[ADPCond\].
1. Let $r$ be the number of real roots of $f_{1}$. For $1\leq i\leq r+1$:
1. Let $C$ be the $i-th$ $\{f_{1}\}$-sector.
2. If $i>1$ let $C^{\prime}$ be the $i-1$-st $\{f_{1}\}$-section. We have $SPT(C^{\prime})=PT(v,(),\{1\},\{\})$, $v=(\alpha)=RAV(\Omega_{1})$, and $\Omega_{1}=((),f_{1},W,\Theta)$. Let $q\in dbl(W)\cap\mathbb{Q}$ and $q>\alpha$. Set $ADP(C,C^{\prime})=(q)$.
3. If $i\leq r$ let $C^{\prime}$ be the $i$-th $\{f_{1}\}$-section. We have $$SPT(C^{\prime})=PT(v,(),\{1\},\{\})$$ $v=(\alpha)=RAV(\Omega_{1})$, and $$\Omega_{1}=((),f_{1},W,\Theta)$$ Let $q\in dbl(W)\cap\mathbb{Q}$ and $q<\alpha$. Set $ADP(C,C^{\prime})=(q)$.
2. For $1\leq k<n$ and for each non-zero-dimensional cell $C$ of $D(\Pi_{k}(F))$:
1. Let $SPT(C)=PT(v,w,I,J)$ and let $r$ be the number of real roots of $f_{k+1}$ over $C$. For $1\leq i\leq r+1$:
1. Let $S$ be the $i-th$ $\{f_{k+1}\}$-sector over $C$.
2. If $i>1$ let $S^{\prime}$ be the $i-1$-st $\{f_{k+1}\}$-section over $C$. We have $SPT(S^{\prime})=PT(v',w,I\cup\{k+1\},J)$, $v^{\prime}=v\times\alpha=RAV(\Omega_{l+1})$, and $\Omega_{l+1}=(v,g,W,\Theta)$. Let $q\in dbl(W)\cap\mathbb{Q}$ and $q>\alpha$. Set $ADP(S,S^{\prime})=PT(v,w^{\prime},I,J\cup\{k+1\})$, where $w^{\prime}=w\times q$.
3. If $i\leq r$ let $S^{\prime}$ be the $i$-th $\{f_{k+1}\}$-section over $C$. We have $SPT(S^{\prime})=PT(v',w,I\cup\{k+1\},J)$, $v^{\prime}=v\times\alpha=RAV(\Omega_{l+1})$, and $\Omega_{l+1}=(v,g,W,\Theta)$. Let $q\in dbl(W)\cap\mathbb{Q}$ and $q<\alpha$. Set $$ADP(S,S^{\prime})=PT(v,w^{\prime},I,J\cup\{k+1\})$$ where $w^{\prime}=w\times q$.
2. For each cell $C^{\prime}$ of $D(\Pi_{k}(F))$ adjacent to $C$ and such that $\dim C^{\prime}=\dim C-1$:
1. Let $(a_{1},\ldots,a_{k})=PT(v,w,I,J)=ADP(C,C^{\prime})$ and let $$RTS(C^{\prime})=((D_{1},\ldots,D_{m}),(k_{1},\ldots,k_{m}),R)$$
2. Let $S_{1}^{\prime},\ldots,S_{s}^{\prime}$ be the$\{f_{k+1}\}$-sections over $C^{\prime}$, and let $W_{j}^{\prime}$ be the isolating disc of $\Lambda(SPT(S_{j}^{\prime}))$ for $1\leq j\leq s$.
3. Let $I=\{i_{1},\ldots,i_{l}\}$, $J=\{j_{1},\ldots,j_{k-l}\}$, and let $g\in\mathbb{Q}[x_{i_{1}},\ldots,x_{i_{l}},x_{k+1}]$ be $f_{k+1}$ with $x_{j_{t}}$ replaced by $a_{j_{t}}$ for $1\leq t\leq k-l$.
4. Compute $(\Theta;\Omega_{l}^{\prime};v_{1},\ldots,v_{r})=AlgRoots_{l}(v,g,p)$.
5. For $1\leq i\leq r$ refine the isolating disc $W_{i}$ of $\Lambda(v_{i})$ until it is contained in one of $dbl(W_{1}^{\prime}),\ldots,dbl(W_{s}^{\prime})$ or $W_{i}\cap\Delta(0,R/2)=\emptyset$. Let $S$ be the $i$-th $\{f_{k+1}\}$-section over $C$. If $W_{i}\subseteq dbl(W_{j}^{\prime})$, set $ADP(S,S_{j}^{\prime})=PT(v_{i},w,I\cup\{k+1\},J)$, and set $L(i)=S_{j}^{\prime}$. Otherwise if $\Lambda(v_{i})<0$ set $L(i)=-\infty$ else set $L(i)=\infty$.
6. Set $L(0)=-\infty$ and $L(r+1)=\infty$.
7. For $1\leq i\leq r+1$, let $S$ be the $i$-th $\{f_{k+1}\}$-sector over $C$. For each $\{f_{k+1}\}$-sector $S^{\prime}$ over $C^{\prime}$ that lies between $L(i-1)$ and $L(i)$ put $u=w\times\Lambda(SPT(S^{\prime}))$ and set $ADP(S,S^{\prime})=PT(v,u,I,J\cup\{k+1\})$.
3. Return $ADP$.
Let us now prove correctness of Algorithm \[CADAdjacency\]. The working precision $p$ set in step $(1)$ of *SamplePoints* is used in calls to *AlgRoots*. Since *AlgRoots* raises precision as needed to reach its goals, $p$ is just an initial value and can be set arbitrarily e.g. to the number of bits in a double precision number. Steps $(2)$-$(6)$ construct sample points $SPT(C)$ is all cells of $D(F)$, starting with sample points in cells of $D(\Pi_{1}(F))$, and then extending them to sample points in $D(\Pi_{k}(F))$ one coordinate at a time. An important fact to note is that isolating discs in the representations of already constructed sample points may change during the execution of step $(6)$. Namely, in step $(6d)$ the isolating discs of the coordinates of the sample points $SPT(\Pi_{i}(C))$ for all projections of the cell $C$ are replaced with their refinements that were computed in the process of isolating the roots of $f_{k+1}^{C}$. In particular, for any cell $C\in D(F)$ if $SPT(C)=(a_{1},\ldots,a_{n})$ then for any $1\leq k\leq n$ $SPT(\Pi_{k}(C))=(a_{1},\ldots,a_{k})$ and the isolating discs that appear in the representations of any algebraic coordinate $a_{i}$ in $SPT(C)$ and in $SPT(\Pi_{k}(C))$ are equal. Note however, that after *SamplePoints* is finished the representations of $SPT(C)$ are fixed.
In step $(1)$ of *AdjacencyPoints* for each pair of adjacent cells $(C,C^{\prime})$ of $D(\Pi_{1}(F))$ with $\dim C^{\prime}=\dim C-1$ the algorithm constructs a point $$ADP(C,C^{\prime})=(q)\in C$$ such that if $SPT(C^{\prime})=(\alpha)$ and $W$ is the isolating disc of $\alpha$ then $q\in dbl(W)$. At the start of each iteration of the loop in step $(2)$ the algorithm has already constructed a point $ADP(C,C^{\prime})$ for each pair of adjacent cells $(C,C^{\prime})$ of $D(\Pi_{k}(F))$ with $\dim C^{\prime}=\dim C-1$. The points satisfy Condition \[ADPCond\]. Steps $(2a)$ and $(2b)$ construct points $ADP(S,S^{\prime})$ for each pair of adjacent cells $(S,S^{\prime})$ of $D(\Pi_{k+1}(F))$ with $\dim S^{\prime}=\dim S-1$. It is clear that the constructed points satisfy Condition \[ADPCond\]. What we need to show is that the construction will always succeed, pairs of cells $(S,S^{\prime})$ for which $ADP(S,S^{\prime})$ is constructed are adjacent, and $ADP(S,S^{\prime})$ is constructed for every pair of adjacent cells $(S,S^{\prime})$ of $D(\Pi_{k+1}(F))$ with $\dim S^{\prime}=\dim S-1$. Step $(2a)$ constructs $ADP(S,S^{\prime})$ for every pair of adjacent cells from a stack over the same cell $C$. Note that in step $(2a)$ we have $\alpha\in W$ and $(dbl(W)\setminus W)\cap\mathbb{R}$ consists of two intervals, one on each side of $\alpha$, hence we can pick rational numbers $q\in dbl(W)$ with $q>\alpha$ or $q<\alpha$. If cells $S$ and $S^{\prime}$ from stacks over different cells $C$ and $C^{\prime}$ are adjacent and $\dim S^{\prime}=\dim S-1$ then $C$ and $C^{\prime}$ must be adjacent and, by Proposition \[WellBasedProp\], $\dim C^{\prime}=\dim C-1$ and $S$ is a section iff $S^{\prime}$ is a section. This shows that step $(2)$ constructs $ADP(S,S^{\prime})$ for every pair of adjacent cells $(S,S^{\prime})$ of $D(\Pi_{k+1}(F))$ with $\dim S^{\prime}=\dim S-1$.
Let us prove that the construction in step $(2b)$ will always succeed and pairs of cells $(S,S^{\prime})$ for which $ADP(S,S^{\prime})$ is constructed are adjacent. With notation of step $(2b)$, let $$(a_{1},\ldots,a_{k})=PT(v,w,I,J)=ADP(C,C^{\prime})$$ and $$(a_{1}^{\prime},\ldots,a_{k}^{\prime})=PT(v^{\prime},w^{\prime},I^{\prime},J^{\prime})=SPT(C^{\prime})$$ Note that $J^{\prime}\subseteq J$ and $a_{j}^{\prime}=a_{j}$ for $j\in J^{\prime}$. Let $g^{\prime}=f_{k+1}^{C^{\prime}}$ be $f_{k+1}$ with $x_{j}$ replaced by $a_{j}^{\prime}=a_{j}$ for $j\in J^{\prime}$. Then $g$ is equal to $g^{\prime}$ with $x_{j}$ replaced by $a_{j}$ for $j\in J\cap I^{\prime}$. For $i\in I^{\prime}$ let $U_{i}^{\prime}$ be the isolating disk of $a_{i}^{\prime}$ in $v^{\prime}$ and let $U_{i}$ be the isolating disk of $a_{i}^{\prime}$ in the representation of $\Pi(v^{\prime})$ with which $RTS(C^{\prime})$ was computed. Note that, by Remark \[RefRem\], the current representation of $v^{\prime}$ is a refinement of the representation with which $RTS(C^{\prime})$ was computed, hence $U_{i}^{\prime}\subseteq dbl(U_{i})$. Hence, $a_{i}\in dbl(U_{i}^{\prime})$ and $a_{i}\in quad(U_{i})$. By the condition $(5)$ of Definition \[RAV\], for each $1\leq i\leq r$ either $\Lambda(v_{i})$ belongs to one of $W_{1}^{\prime},\ldots,W_{s}^{\prime}$ or $\Lambda(v_{i})\notin\Delta(0,R)$. Therefore we can refine the isolating disc $W_{i}$ of $\Lambda(v_{i})$ so that it is contained in one of $dbl(W_{1}^{\prime}),\ldots,dbl(W_{s}^{\prime})$ or $W_{i}\cap\Delta(0,R/2)=\emptyset$. In the former case the $i$-th $\{f_{k+1}\}$-section over $C$ is adjacent to the $j$-th $\{f_{k+1}\}$-section over $C^{\prime}$, in the latter case the $i$-th $\{f_{k+1}\}$-section over $C$ tends to infinity whose sign is determined by the sign of $\Lambda(v_{i})$. This shows that sections $(S,S^{\prime})$ for which $ADP(S,S^{\prime})$ is constructed are adjacent. Finally, let $S$ and $S^{\prime}$ be sectors over $C$ and $C^{\prime}$ defined in step $(2b(vii))$. Then, by construction in step $(6e)$ of *SamplePoints*, $q=\Lambda(SPT(S^{\prime}))\notin\bigcup_{j=1}^{m}dbl(D_{j})$, and since $W_{j}^{\prime}\subseteq dbl(D_{j})$ (possibly after reordering of indices), $q\notin\bigcup_{j=1}^{s}W_{j}^{\prime}$. Moreover, $-R<q<R$. Since for each $1\leq i\leq r$ either $\Lambda(v_{i})$ belongs to one of $W_{1}^{\prime},\ldots,W_{s}^{\prime}$ or $\Lambda(v_{i})\notin\Delta(0,R)$, the point $PT(v,u,I,J\cup\{k+1\})$ defined in step $(2b(vii))$ belongs to $S$. If $ADP(S,S^{\prime})$ is constructed in step $(2b(vii))$ then $S^{\prime}$ is a sector that lies between sections adjacent to the sections bounding $S$, hence $S$ and $S^{\prime}$ are adjacent.
Empirical Results
=================
An algorithm computing CAD cell adjacencies has been implemented in C, as a part of the kernel of *Mathematica*. The implementation takes a quantified system of polynomial equations and inequalities $S$ and uses *Mathematica* multi-algorithm implementation of CAD to compute a CAF $F$ such that $D(F)$ is a CAD of $\mathbb{R}^{n}$ consistent with the solution set $A$ of $S$. If $F$ is well-based the implementation uses Algorithm \[ADP\] to find the cell adjacencies. The implementation is geared towards solving a specified topological problem, e.g. finding the boundary or the connected components of $A$, hence it avoids computing cell adjacencies for cells that are known not to belong to the closure of $A$. The current implementation also works for non-well-based problems in $\mathbb{R}^{3}$ using ideas from [@ACM2] to extend Algorithm \[ADP\].
The experiments have been conducted on a Linux laptop with a $4$-core $2.7$ GHz Intel Core i7 processor and $16$ GB of RAM. The reported CPU time is a total from all cores used. For each example we give three timings. $t_{CAD}$ is the computation time of constructing a CAF consistent with the solution set the input system. $t_{SP}$ is the time of refining the CAD to a $BP(F)$-invariant CAD of$\mathbb{R}^{n}$ and of constructing sample points in the CAD cells (steps $(1)$-$(6)$ of Algorithm \[ADP\]). Our implementation refines the CAD while constructing sample points, which is why we cannot give separate timings. The third timing, $t_{ADJ}$ is the time of computing cell adjacency information (steps $(7)$-$(9)$ of Algorithm \[ADP\]). We also report the dimension $\dim$ of the embedding space, the number $N_{CELL}$ of cells in the CAD of $A$, the number $N_{ADJ}$ of computed pairs of adjacent cells whose dimensions differ by one, and the number $N_{CC}$ of connected components of $A$.
Find cell adjacencies for a CAD of the union of two unit balls in $\mathbb{R}^{n}$$$x_{1}^{2}+\ldots+x_{n}^{2}\leq1\vee(x_{1}-1)^{2}+\ldots+(x_{n}-1)^{2}\leq1$$ Note that for $n\leq3$ the balls have full-dimensional intersection, for $n=4$ they touch at one point, and for $n>4$ they are disjoint.
$\dim$ $t_{CAD}$ $t_{SP}$ $t_{ADJ}$ $N_{CELL}$ $N_{ADJ}$ $N_{CC}$
-------- ----------- ---------- ----------- ------------ ----------- ----------
$2$ $0.018$ $0.004$ $0.001$ $21$ $42$ $1$
$3$ $0.100$ $0.033$ $0.024$ $179$ $718$ $1$
$4$ $0.489$ $0.175$ $0.112$ $521$ $3898$ $1$
$5$ $1.42$ $0.773$ $0.352$ $954$ $11910$ $2$
$6$ $44.6$ $24.8$ $8.92$ $14050$ $251758$ $2$
: Union of two balls in $\mathbb{R}^{n}$
Here we used modified versions of examples from Wilson’s benchmark set [@W2] (version 4). Of the $77$ examples we selected $63$ that involved at least $3$ variables and we used quantifier-free versions of the examples. In $21$ of the examples the system was not well-based and involved more than $3$ variables, hence our algorithm did not apply. $7$ examples did not finish in $600$ seconds. Of the $35$ examples for which our implementation succeeded, $29$ were well-based and $6$ were not well-based and in $\mathbb{R}^{3}$. On average, $t_{CAD}$ took $55\%$ of the total time, $t_{SP}$ took $34\%$, and $t_{ADJ}$ took $11\%$. Five examples with the largest number of cells are given in Table \[wilson\]. All but the third example are well-based.
Ex \# $\dim$ $t_{CAD}$ $t_{SP}$ $t_{ADJ}$ $N_{CELL}$ $N_{ADJ}$ $N_{CC}$
-------- -------- ----------- ---------- ----------- ------------ ----------- ----------
$2.13$ $4$ $0.109$ $0.263$ $0.210$ $3104$ $10576$ $1$
$2.16$ $3$ $3.06$ $2.65$ $1.27$ $2811$ $37416$ $1$
$6.1$ $3$ $0.768$ $0.794$ $0.312$ $2774$ $8926$ $2$
$5.10$ $4$ $14.9$ $11.2$ $4.01$ $2256$ $63190$ $1$
$6.6$ $6$ $14.6$ $7.85$ $3.52$ $2128$ $76360$ $1$
: \[wilson\]Wilson’s benchmark
Here we took the $32$ 3D solids that appear in *Mathematica* [<span style="font-variant:small-caps;">SolidData</span>]{} and intersected each of them with the solution set of $9(x+y+z)^{2}>z^{2}+1$. All $32$ examples were well-based and in all our implementation succeeded. On average, $t_{CAD}$ took $37\%$ of the total time, $t_{SP}$ took $47\%$, and $t_{ADJ}$ took $16\%$. The five solids which resulted in the largest number of cells are:
1. Steinmetz 6-solid$$\begin{aligned}
& 2x^{2}+(y-z)^{2}\leq2\wedge2x^{2}+(y+z)^{2}\leq2\wedge\\
& 2y^{2}+(x-z)^{2}\leq2\wedge2y^{2}+(x+z)^{2}\leq2\wedge\\
& (x-y)^{2}+2z^{2}\leq2\wedge(x+y)^{2}+2z^{2}\leq2\end{aligned}$$
2. Sphericon$$\begin{aligned}
& (x^{2}+y^{2}\leq(|z|-1)^{2}\wedge x\geq0\wedge-1\leq z\leq1)\vee\\
& (x^{2}+z^{2}\leq(|y|-1)^{2}\wedge x\leq0\wedge-1\leq y\leq1)\end{aligned}$$
3. Steinmetz 4-solid$$\begin{aligned}
& x^{2}+y^{2}\leq1\wedge9x^{2}+y^{2}+8z^{2}\leq9+164/29yz\wedge\\
& 3x^{2}+284/41xy+7y^{2}+82/29yz+8z^{2}\leq9+49/10xz\wedge\\
& 3x^{2}+7y^{2}+49/10xz+82/29yz+8z^{2}\leq9+284/41xy\end{aligned}$$
4. Solid capsule$$\begin{aligned}
& x^{2}+y^{2}+(z-1/2)^{2}\leq1\vee x^{2}+y^{2}+(z+1/2)^{2}\leq1\vee\\
& -1/2\leq z\leq1/2\wedge x^{2}+y^{2}\leq1\end{aligned}$$
5. Reuleaux tetrahedron$$\begin{aligned}
& x^{2}+y^{2}+(19/31+z)^{2}\leq1\wedge\\
& (x-15/26)^{2}+y^{2}+(z-9/44)^{2}\leq1\wedge\\
& (x+11/38)^{2}+(y-1/2)^{2}+(z-9/44)^{2}\leq1\wedge\\
& (x+11/38)^{2}+(y+1/2)^{2}+(z-9/44)^{2}\leq1\end{aligned}$$
The details are given in Table \[solids\].
Ex \# $\dim$ $t_{CAD}$ $t_{SP}$ $t_{ADJ}$ $N_{CELL}$ $N_{ADJ}$ $N_{CC}$
------- -------- ----------- ---------- ----------- ------------ ----------- ----------
$1$ $3$ $254$ $614$ $189$ $156688$ $4320078$ $2$
$2$ $3$ $59.5$ $82.6$ $27.0$ $54256$ $767462$ $2$
$3$ $3$ $52.1$ $78.2$ $20.1$ $24476$ $461614$ $2$
$4$ $3$ $8.11$ $5.42$ $3.06$ $17152$ $84162$ $2$
$5$ $3$ $47.9$ $53.5$ $15.1$ $11756$ $349976$ $2$
: \[solids\]Intersections of solids with $9(x+y+z)^{2}>z^{2}+1$.
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---
abstract: 'We study the ground state and low-energy excitations of fractional quantum Hall systems on a disk at filling fraction $\nu = 5/2$, with Coulomb interaction and background confining potential. We find the Moore-Read ground state is stable within a finite but narrow window in parameter space. The corresponding low-energy excitations contain a fermionic branch and a bosonic branch, with widely different velocities. A short-range repulsive potential can stabilize a charge $+e/4$ quasihole at the center, leading to a different edge excitation spectrum due to the change of boundary conditions for Majorana fermions, clearly indicating the non-Abelian nature of the quasihole.'
author:
- Xin Wan$^1$
- 'Kun Yang$^{2,1}$'
- 'E. H. Rezayi$^3$'
title: 'Edge Excitations and Non-Abelian Statistics in the Moore-Read State: A Numerical Study in the Presence of Coulomb Interaction and Edge Confinement'
---
Fractional quantum Hall (FQH) liquids represent novel states of matter with non-trivial topological order [@wen95], whose consequences include chiral edge excitations and fractionally charged bulk quasiparticles that obey Abelian or non-Abelian fractional statistics. It has been proposed that the non-Abelian quasiparticles can be used for quantum information storage and processing in an intrinsically fault-tolerant fashion [@kitaev03; @freedman02], in which information is stored by the degenerate ground states in the presence of these non-Abelian quasiparticles, and unitary transformations in this Hilbert space can be performed by braiding the quasiparticles [@dassarma05; @bonesteel05]. While many Abelian FQH states have been observed and studied in detail[@wen95], thus far there have been relatively few candidates for the non-Abelian ones. The most promising candidate is the FQH state at Landau level filling fraction $\nu = 5/2$[@willett87]. The leading candidate for the ground state of this system is the Moore-Read (MR) paired state [@moore91], which has been shown [@moore91; @nayak96] to support fractionally charged, non-Abelian quasiparticles. The MR state received strong support from numerical studies using sphere or torus geometries [@morf]. It has been proposed that the non-Abelian nature of the quasiparticles in the MR state may be detected through interference experiments in edge transport [@fradkin98; @stern06]. In order to have a quantitative understanding of the edge physics however, one needs to study the interplay between electron-electron interaction and confining potential, which may lead to edge structures that are more complicated than those predicted by the simplest theory [@chklovskii]. This was found to be the case rather generically in the FQH regime [@wan03].
In anticipation of experimental studies, in this work we perform detailed numerical studies of edge excitations in the 5/2 FQH state in finite-size systems with disc geometry, taking into account the inter-electron Coulomb interaction and a semi-realistic model of the confining potential due to neutralizing background charge. For a limited parameter space, we find the ground state has substantial overlap with the MR state. Within this parameter space we identify the existence of chiral fermionic and bosonic edge modes, in agreement with previous prediction. We find the fermionic mode velocity is much lower than that of the bosonic mode. With suitable short-range repulsive potential at the center, we show that a charge $+e/4$ quasihole can be localized at the center of the system, and its presence changes the spectrum of the fermionic edge mode. This confirms the existence and non-Abelian nature of such fractionally charged quasiparticles.
[*The microscopic model.*]{} We consider a microscopic model of a two-dimensional electron gas (2DEG) confined to a two-dimensional disk, with neutralizing background charge distributed uniformly on a parallel disk of radius $a$ at a distance $d$ above the 2DEG. This distance parameterizes the strength of the confining potential, which decreases with increasing $d$. For $\nu = 5/2$, we explicitly keep the electronic states in the first Landau level (1LL) only, while neglecting the spin up and down electrons in the lowest Landau level (0LL), assuming they are inert. The amount of positive background charge is chosen to be equal to that of the half-filled 1LL, so the system is neutral. The choice of $a = \sqrt{4N}$, in units of $l_{\rm B}$ (magnetic length), guarantees that the disk encloses exactly $2N$ magnetic flux quanta for $N=2P$, corresponding to $\nu = 1/2$ in the 1LL [@note]. The rotationally invariant confining potential comes from the Coulomb attraction between the background charge and the electrons. Using the symmetric gauge, we can write down the following Hamiltonian for the electrons confined to the 1LL: $$\label{eqn:chamiltonian} H_{\rm C} = {1\over 2}\sum_{mnl}V_{mn}^l
c_{m+l}^\dagger c_n^\dagger c_{n+l}c_m +\sum_m U_mc_m^\dagger c_m,$$ where $c_m^\dagger$ is the electron creation operator for the 1LL single electron state with angular momentum $m$, $V_{mn}^l$’s are the corresponding matrix elements of Coulomb interaction for the symmetric gauge, and $U_m$’s are the matrix elements of the confining potential. [*MR ground state.*]{} We diagonalize the Hamiltonian \[Eq. (\[eqn:chamiltonian\])\] for each Hilbert subspace with total angular momentum $M$, and obtain the ground state energy $E(M)$. Figure \[fig:M\_gs\](a) shows $E(M)$ vs $M$ for $N = 12$ electrons in the 1LL in 22 orbitals, which is the minimum to accommodate the corresponding MR state. As illustrated in Fig. \[fig:M\_gs\](b), $M_{\rm gs}$ increases with increasing $d$. This is very similar to what happens in the 0LL [@wan03], and reflects the interplay between electron-electron Coulomb repulsion and confining potential; as the confining potential weakens with increasing $d$, electrons tend to move outward, resulting in bigger $M_{\rm gs}$. $M_{\rm gs}$ coincides with the total angular momentum of the $N = 12$ MR state $M_{\rm MR} = N(2N-3)/2 = 126$ only within a small window: $0.51 \le
d/l_{\rm B} \le 0.76$. This contrasts with the situation for a Laughlin filling fraction $\nu = 1/3$, where $M_{\rm
gs}=N(N-1)/2\nu$ (same as the corresponding Laughlin state) for a substantially bigger window $d < d_c \approx 1.5 l_{\rm
B}$ [@wan03]. When the global ground state has the same total angular momentum as the MR state, the overlap $|\langle \Psi_{\rm
gs}|\Psi_{\rm MR}\rangle|^2$ between the two is about 0.47. We note that, in the absence of confining potential, the corresponding overlaps are 0.46 and 0.45 for $N = 12$ and $14$, respectively. These values are quite substantial, given that the size of Hilbert subspaces are 16,660 and 194,668. But they are well below the overlap ($> 0.95$) in the case of the Laughlin filling $\nu =
1/3$ [@wan03]. The reduced overlap and window for the MR state reflect the fact that the paired state is much weaker compared to the Laughlin state. The overlap is, however, quite sensitive to small changes of system parameters; for example, we can increase the overlap to above 0.7 for $N = 12$, by choosing $a = \sqrt{4N - 4}$, and a change in the $V_1$ pseudopotential, $\delta V_1 = 0.03$. Such sensitivity suggests that the MR state is rather “fragile", consistent with experiments at $\nu=5/2$.
![ \[fig:M\_gs\] (Color online) (a) Ground state energy $E(M)$ in each angular momentum $M$ subspace for $N = 12$ electrons with 22 orbitals in the 1LL (corresponding to $\nu = 1/2$). The positive background charge is at a distance $d = 0.4$, 0.6, and 0.8, in units of $l_{\rm B}$, above the electron plane. The total angular momentum of the global ground state $M_{\rm gs}$ (indicated by arrows) increases from 121, 126, to 136, respectively. The global ground state at $d = 0.6$ has the same total angular momentum $M_{\rm MR} = N(2N-3)/2 = 126$ as the corresponding $N = 12$ MR state. The overlap between the two states is 0.47. The curves for $d
= 0.4$ and 0.8 have been shifted verically by 0.6 and -0.6, respectively. (b) The total angular momentum of the global ground state $M_{\rm gs}$ as a function of $d$. The arrow indicates the plateau at which $M_{\rm gs} = M_{\rm MR}$. ](fig1.eps){width="45.00000%"}
[*Edge excitations.*]{} The MR state has non-trivial topological order. While our numerical results indicate that the ground state has substantial overlap with the MR state for properly chosen system parameters, it does not directly reflect the topological order of the system. One way to probe the topological order is to study edge excitations, which is also of vital experimental importance. For comparison, the Laughlin state supports one bosonic branch of chiral edge excitations, whose properties have been studied in tunneling experiments [@chang03]. For $\nu = 5/2$, a neutral fermionic branch of excitations has been predicted in addition to a bosonic branch [@wen95; @milovanovic96]. The existence of both branches makes the low-energy excitation spectrum of a microscopic model at $\nu = 5/2$ richer, and their experimental consequences more interesting [@fendley].
Figure \[fig:modes\](a) shows the low-energy excitations for pure Coulomb interaction and the confining potential with $d = 0.6$ for 12 electrons in 26 orbitals. Apparently, there is no clear distinction between fermionic and bosonic edge modes as well as bulk modes, due to the relatively small bulk gap, and system size. The situation here is similar to a related study on a rotating Bose gas [@cazalilla05], and will be analyzed in detail elsewhere. Here we focus instead on a model with mixed Coulomb interaction and 3-body interaction for clarity. The 3-body interaction alone generates the MR state as its exact ground state with the smallest total angular momentum. The mixed Hamiltonian is $$\begin{aligned}
\label{eqn:mixedhamiltonian} H &=& (1 - \lambda) H_{\rm C} + \lambda
H_{3B}, \\
H_{3B} &=& - \sum_{i < j <
k}S_{ijk}[\nabla^2_i\nabla^4_j \delta({\bf r}_i - {\bf r}_j)
\delta({\bf r}_i -{\bf r}_k)],\end{aligned}$$ where $S$ is a symmetrizer: $S_{123}[f_{123}]=f_{123}+f_{231}+f_{312}$. We measure energies in units of $e^2/\epsilon l_B$. As we will see, the mixed interaction, which enhances the bulk excitation gap with respect to that of edge excitations, allows for a clear separation between the two in a finite system, effectively increasing the system size.
![ \[fig:modes\] (Color online) Low-energy excitations $\Delta E(\Delta M)$ from exact diagonalization (solid lines) for $N
= 12$ electrons in 26 orbitals in the 1LL (corresponding to $\nu =
1/2$) for (a) Coulomb Hamiltonian \[Eq. (\[eqn:chamiltonian\])\] and (b) mixed Hamiltonian \[Eq. (\[eqn:mixedhamiltonian\]) with $\lambda = 0.5$\]. The neutralizing background charge for the Coulomb part is deposited at $d = 0.6 l_{\rm B}$ above the electron plane. (c) Dispersion curves of bosonic ($E_{\rm b}$) and fermionic modes ($E_{\rm f}$) of the system. These energies can be used to construct the complete edge spectrum for the 12-electron system up to $\Delta
M = 4$ \[dashed bars in (b)\]. ](fig2.eps){width="45.00000%"}
Figure \[fig:modes\](b) shows the low-energy excitations $\Delta
E(\Delta M)$ for 12 electrons in 26 orbitals in the 1LL for the mixed Hamiltonian with $\lambda = 0.5$ and $d = 0.6 l_{\rm B}$. There is a clear separation of the spectrum around $\Delta E = 0.1$, below which we identify as edge modes. The total numbers of these states are 1, 1, 3, 5, and 10 for $\Delta M = 0$-4, which agree with the numbers of edge states expected for the MR state [@milovanovic96]. Notably, the lowest two levels for $\Delta M = 4$ lie very close to each other.
In this case, we can further separate the fermionic and bosonic branches of the edge states. The procedure is similar to but more complicated than the one we used [@wan03] to identify edge modes in the Laughlin case at $\nu = 1/3$, where there is only one bosonic branch of edge modes. The basic idea is to label the low-lying states by two sets of occupation numbers $\{n_{\rm b}(l_{\rm b})\}$ and $\{n_{\rm f}(l_{\rm f})\}$ for bosonic and fermionic modes with angular momentum $l_{\rm b,f}$, respectively. Since the fermionic edge excitations are Majorana fermions that obey antiperiodic boundary conditions [@milovanovic96], $l_{\rm f}$ must be positive half integers and $\sum_{l_{\rm f}} n_{\rm f}(l_{\rm f})$ an even integer because fermion modes are occupied in pairs. The angular momentum and energy of the state, measured from those of the ground state, are $\Delta M = \sum_{l_{\rm b}} n_{\rm b}(l_{\rm b})
l_{\rm b} + \sum_{l_{\rm f}} n_{\rm f}(l_{\rm f}) l_{\rm f}$ and $
\Delta E = \sum_{l_{\rm b}} n_{\rm b}(l_{\rm b}) E_{\rm b}(l_{\rm
b}) + \sum_{l_{\rm f}} n_{\rm f}(l_{\rm f}) E_{\rm f}(l_{\rm f})$. The difficulty here is, in addition to the bosonic modes, we also have fermionic modes, and thus the convolution of fermionic and bosonic modes. Fortunately, we note that the number of states with nearly zero energy coincides with the number of fermionic edge states expected by theory. Accordingly in our construction we will assume these energies to have been evolved from combining two Majorana fermions. Through careful analysis of the low-energy excitations up to $\Delta M = 4$, we obtain the results of bosonic and fermionic mode energies up to $l_{\rm b} = 4$ and $l_{\rm f} =
7/2$, respectively, plotted in Fig. \[fig:modes\](c). The detailed analysis will be published elsewhere. Using these 8 energies \[excluding the trivial $E_{\rm b}(0) = 0$\], we can construct the whole low-energy spectrum of the system up to $\Delta M = 4$, a total of 20 states. The excellent agreement \[see Fig. \[fig:modes\](b)\] justifies our analysis, and thus our result, that, energetically, fermionic modes are well separated from bosonic modes. In contrast to the roughly linear dispersion of the fermionic branch, the energy of the bosonic branch bends down (despite a much bigger initial slope or higher velocity), suggesting a potential vulnerability to edge reconstruction in the bosonic branch [@wan03]. These are not surprising since the bosonic modes are charged; as a result its velocity is dominated by the long-range nature of the Coulomb interaction in the long-wavelength limit, but in the meantime it is also more sensitive to the competition between Coulomb interaction and confining potential which can lead to instability at shorter wavelength.
[*Charge $+e/4$ and $+e/2$ quasiholes.*]{} One of the most important properties of the MR state is that it supports charge $\pm e/4$ quasihole/particle excitations. To demonstrate the unusual fractional charge, we add, to the mixed Hamiltonian with $\lambda =
0.5$ and $d = 0.5 l_{\rm B}$, a short-range potential: $H_W = W
c_0^{\dagger} c_0$, which tends to create quasiparticles or quasiholes at the origin. For small enough repulsive $W$, the ground state of the system should remain MR-like. As $W$ is increased, a quasihole of charge $+e/4$ can appear at the origin, reflected by a change of ground state angular momentum from $M_{\rm gs} = N(2N-3)/2$ to $N(2N-3)/2 + N/2$, and depletion of $1/4$ in the total occupation number of electrons at orbitals with small angular momenta. If $W$ is increased further, a $+e/2$ quasihole, much like a quasihole for the Laughlin state, appears near the origin in the global ground state, whose total angular momentum further increases to $N(2N-3)/2+N$. This is observed for a system of 12 electrons in 24 orbitals (as well as a smaller system of 10 electrons in 20 orbitals). Figure \[fig:quasihole\](a) shows the increase of $M_{\rm gs}$ from 126 to 132 and then to 138 with increasing $W$. Fig. \[fig:quasihole\](b) compares the electron occupation number $n(m)$ in each orbital for $W = 0.0$ ($M_{\rm gs} = 126$, MR-like) and $W = 0.1$ ($M_{\rm gs} = 132$). The accumulated difference in the occupation numbers of the two states, $\sum_{i=0}^m \Delta n(i)$, oscillates around $-0.25$ for $m$ up to about 19, indicating the existence of a $+e/4$ quasihole at the origin. The same comparison for $W = 0.1$ ($M_{\rm gs} = 132$) and $W
= 0.25$ ($M_{\rm gs} = 138$) is plotted in Fig. \[fig:quasihole\](c). Their difference ($\sim -0.25$) indicates the emergence of another $+e/4$ quasihole at the origin, or a $+e/2$ quasihole compared to the MR-like state for $W = 0.0$.
![ \[fig:quasihole\] (Color online) Generation of quasiholes using a short-range repulsion $W c_0^{\dagger} c_0$ in a system of 12 electrons in 24 orbitals, for the mixed Hamiltonian Eq. (\[eqn:mixedhamiltonian\]) with $\lambda=0.5$ and $d = 0.5 l_{\rm B}$. (a) Ground state angular momentum $M_{\rm gs}$ as a function of $W$. (b) Electron occupation number $n(m)$ of the ground state for $W = 0.0$ ($M_{\rm gs} = 126$) and $W = 0.1$ ($M_{\rm gs} = 132$), as well as the accumulated difference in $n(m)$ between the two states, $\sum_{i=0}^m \Delta
n(i)$, which oscillates around -0.25 (dotted line) for $m$ up to about 19, indicating the emergence of a charge $+e/4$ quasihole at $m = 0$. (c) $n(m)$ for $W = 0.1$ and $W = 0.25$ ($M_{\rm gs} = 138$), and their accumulated difference, indicating the appearance of another $+e/4$ quasihole. Low-energy edge excitations of the systems are plotted for (d) $W = 0.0$, (e) $W = 0.1$, and (f) $W = 0.25$. The fermionic mode supports 0, 1, 1, 2 states (solid bars) for $\Delta M
= 1$-4 in (d) ($M_{\rm gs} = 126$), but 1, 1, 2, 2 for $\Delta M = 1$-4 in (e) ($M_{\rm gs} = 132$). The numbers change back to 0, 1, 1, 2 for $\Delta M = 1$-4 in (f) ($M_{\rm gs} = 138$). This suggests that a single +e/4 quasihole (or an odd number of quasiholes) changes the fermionic mode spectrum while a +e/2 quasihole (or, in general, an even number of +e/4 quasiholes) does not.](fig3.eps){width="45.00000%"}
The $+e/4$ quasihole supports a zero energy Majorana fermion mode which is responsible for its non-Abelian nature. This zero mode pairs with the edge excitations and changes their spectra. With quasiholes in the bulk, the fermionic edge excitations are Majorana fermions that obey either periodic (integer $l_{\rm f}$, twisted sector) or antiperiodic (half integer $l_{\rm f}$, untwisted sector) boundary conditions [@milovanovic96]. In the presence of an odd number of $+e/4$ quasiholes (twisted sector), two-fermion edge excitations are shifted by $\delta(\Delta M) = -1$, relative to those in the presence of an even number of such quasiholes (untwisted sector). This is demonstrated in Fig. \[fig:quasihole\](d)-(f) for 12 electrons in 24 orbitals. From $M_{\rm gs} = 126$ in Fig. \[fig:quasihole\](d) ($W = 0.0$, no quasihole present), we count the numbers of fermionic edge states as 0, 1, 1, 2 for $\Delta M = 1$-4. From $M_{\rm gs} = 132$ in Fig. \[fig:quasihole\](e) ($W = 0.1$, one $+e/4$ quasihole present), the numbers change to 1, 1, 2, 2 for $\Delta M = 1$-4. In particular, the existence of a fermionic state at $\Delta M = 1$ clearly indicates the change. From $M_{\rm gs} = 138$ in Fig. \[fig:quasihole\](f) ($W = 0.25$, two $+e/4$ quasiholes present), the numbers change back to 0, 1, 1, 2 for $\Delta M =
1$-4.
[*Summary.*]{} Our results suggest that the Moore-Read (MR) state properly describes a half-filled first Landau level, for properly chosen confinement potential. In this case the system supports chiral edge excitations as well as fractionally charged quaisholes, and their properties agree with theory predictions. We also find that the window of stability of the MR state is rather narrow, and the edge modes may suffer from reconstruction or other instabilities as the confinement potential varies. The nature and consequences of such instabilities are currently under investigation, which will be presented elsewhere along with further details of the present work.
We thank Bert Halperin, Chetan Nayak, Nick Read, Zhenghan Wang and Xiao-Gang Wen for very helpful discussions. This work is supported by NSFC Project 10504028 (X.W.), and NSF grants No. DMR-0225698 (K.Y.) and No. DMR-0606566 (E.H.R.).
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This is a simplification from the real system. In reality the background charge equals the [*total*]{} electron charge that includes the filled 0LL electrons; the latter are also separated from the background charge, and the location of their edge is different from that of the 1LL electrons. The consequences of these will be discussed elsewhere.
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---
abstract: 'The neural ordinary differential equation (Neural ODE) is a novel machine learning architecture whose weights are smooth functions of the continuous depth. We apply the Neural ODE to holographic QCD by regarding the weight functions as a bulk metric, and train the machine with lattice QCD data of chiral condensate at finite temperature. The machine finds consistent bulk geometry at various values of temperature and discovers the emergent black hole horizon in the holographic bulk automatically. The holographic Wilson loops calculated with the emergent machine-learned bulk spacetime have consistent temperature dependence of confinement and Debye-screening behavior. In machine learning models with physically interpretable weights, the Neural ODE frees us from discretization artifact leading to difficult ingenuity of hyperparameters, and improves numerical accuracy to make the model more trustworthy.'
author:
- Koji Hashimoto
- 'Hong-Ye Hu'
- 'Yi-Zhuang You'
bibliography:
- 'holographic\_QCD.bib'
title: Neural ODE and Holographic QCD
---
[UTF8]{}[gbsn]{}
Introduction
============
Applying machine learning to solve physics problems[@Carleo:2019ptp; @Ruehle:2020jrk] has generated a growing research interest in recent years. Machine learning holography is an emerging direction in this field, which introduces artificial intelligence to discover the holographic bulk theory behind generic quantum systems on the holographic boundary. Multiple approaches have been developed to capture different aspects of the holographic duality[@Gan:2017xy; @Hashimoto:2018bnb; @Hashimoto:2018ftp; @You:2017guh; @2019arXiv190300804H; @Hashimoto:2019bih; @Han2020Deep; @Akutagawa:2020yeo]. For example, the entanglement feature learning (EFL)[@You:2017guh] can establish the emergent holographic spacial geometry simply from the entanglement entropy data on the holographic boundary. The anti-de Sitter / deep learning (AdS/DL) correspondence takes a different approach[@Hashimoto:2018bnb; @Hashimoto:2018ftp; @Hashimoto:2019bih; @Akutagawa:2020yeo] by implementing the holographic principle[@Maldacena:1997re; @Gubser:1998bc; @Witten:1998qj] in a deep neural network, where the neural network is regarded as the classical equation of motion for propagating fields on a discretized curved spacetime. Further progress has been made by the neural network renormalization group (Neural RG)[@2019arXiv190300804H], which learns to construct the exact holographic mapping between the boundary and the bulk field theories at the partition function level. All these approaches share a common theme that the emergent dimension of the holographic bulk corresponds to the depth dimension of the deep neural network, and the neural network itself is regarded as the bulk spacetime. As the neural network learns to interpret the holographic boundary data serving from its input layer, the network weights in deeper layers get optimized, which then leads to the optimal holographic bulk description for the boundary data.
However, the development so far has been based on the discretization of the holographic bulk dimension, because the neural network layers are intrinsically discrete in typical deep learning architectures. It is desired to make this dimension continuous, as a smooth holographic spacetime is physically required in the classical limit. In this work, we explore this possibility, based on the recent development of the neural ordinary differential equation (Neural ODE)[@chen2018neural] approach. The Neural ODE is a generalization of the deep residual network[@He2015Deep] to a continuous-depth network with the network weights replaced by a continuous function. It provides a trainable model of differential equations that can evolve the initial input to the final output continuously. The Neural ODE is particularly suitable for the AdS/DL approach because the goal here is precisely to infer the differential equation that describes the propagation of the bulk field in a continuous space-time with smooth geometry. In this context, the continuous network weights of the Neural ODE have a physical interpretation related to the metric function that characterizes the curved spacetime in the holographic bulk. An interpretable spacetime geometry emerges as the neural network is trained, which demonstrate a scenario of machine-assisted discovery in theoretical physics, where the artificial intelligence plays a more active role in the scientific process other than a tool for data processing.
The AdS/DL applied to holographic QCD would be a nice ground to test the effectiveness of the Neural ODE in physics applications. The Neural ODE brings to us two advances: the removal of artificial regularizations and the improvement of accuracy. In previous works [@Hashimoto:2018ftp; @Hashimoto:2018bnb; @Yan:2020wcd; @Akutagawa:2020yeo], due to the discrete nature of the neural network, technical regularization terms are introduced to remove the discretization artifacts and to ensure the smoothness of the network weights.[^1] Such regularization is no longer needed in the Neural ODE approach. Furthermore, for the network to be identified with a field equation in the curved spacetime, the Euler method for the ordinary differential equation was introduced for simplicity, though the Euler integration generically suffers from large numerical errors. Replacing the discrete neural network with the Neural ODE provides a natural interpretation of the metric function in the smooth spacetime, and at the same time, would greatly enhance the accuracy. The improved accuracy of the Neural ODE is simply due to the advanced ODE solver equipped in the Neural ODE framework. The discretization along the integrated coordinate is optimized adaptively, rather than given ad hoc as hyperparameters. This is especially useful when the metric function contains coordinate singularity at the black hole horizon. The required accuracy depends on the purpose and the method of how machine learning is applied.[^2] In our present case of the AdS/DL, as is explicitly shown, the accuracy improvement is sufficient for exploring emergent geometries at various values of temperature.
In this paper, following the holographic QCD framework of Ref. [@Hashimoto:2018bnb], we use the Neural ODE to find bulk spacetimes emergent out of the given data of chiral condensate of lattice QCD. The Neural ODE not only discovers a spacetime which is consistent with that of Ref. [@Hashimoto:2018bnb], but also greatly enhances the power of machine learning method. The emergent geometry turns out to incorporate automatically the presence of the black hole horizon, and the Neural ODE enables us to further explore geometries for different values of temperature, with improved accuracy. The temperature dependence of holographic Wilson loops, calculated by the emergent geometry trained with the Neural ODE, turns out to coincide qualitatively with the known lattice QCD results of the Wilson loops. Interestingly, we find that the radial derivative of the volume factor of the emergent geometry does not depend on the temperature, and the temperature dependence of the chiral condensate solely stems from that of the bulk scalar coupling constant.
The organization of this paper is as follows. In Sec. \[sec:2\], we briefly review the holographic QCD framework adopted in Ref. [@Hashimoto:2018bnb] and the Neural ODE [@chen2018neural]. In Sec. \[sec:3\], we apply the Neural ODE to train the machine (which is equivalent to the holographic QCD system) and find emergent geometry for various values of the temperature. In Sec. \[sec:4\], we introduce a way to calculate consistent full components of the metric from the emergent volume factor, with which we calculate holographic Wilson loops. They qualitatively agree with Wilson loops evaluated in lattice QCD. Sec. \[sec:5\] is for a summary and discussions. Appendix \[sec:A\] is about details of the Neural ODE.
Review: AdS/CFT model and Neural ODE {#sec:2}
====================================
Bulk field theory
-----------------
The holographic principle [@Maldacena:1997re; @Gubser:1998bc; @Witten:1998qj], also known as AdS/CFT correspondence, is a profound relation between a $d$-dimensional quantum field theory (QFT) and a $(d+1)$-dimensional gravity theory. It has been successfully applied to a large class of strongly coupled QFTs in high energy theory and condensed matter theory. Despite its success, a constructive way of finding the holographic gravity dual theory for a given QFT is lacking. If we have the experimental response data of a quantum system under external probing fields, can we model it holographically by a classical field theory in some curved geometry? The entanglement feature learning [@You:2017guh; @Vasseur:2018gfy] and the AdS/DL correspondence [@Hashimoto:2018ftp; @Hashimoto:2018bnb] can answer that question in a concrete setup. Here we briefly review the setup of Ref. [@Hashimoto:2018bnb], for which we apply the Neural ODE method in later sections.
We assume the $d+1$-dimensional bulk spacetime coordinated by $(t,\eta, x_1, \cdots, x_{d-1})$ including the time dimension $t$, the space dimensions $x_i$ and the holographic bulk dimension $\eta$. We assume the translation symmetry except for the $\eta$ direction, and the spacial homogeneity in $(x_1,\cdots,x_{d-1})$, then in the gauge $g_{\eta\eta}=1$, the holographic bulk spacetime can be described by the following metric (we will consider $d=4$ specifically) $$\mathrm{d}s^{2} = -f(\eta)\mathrm{d}t^{2}+\mathrm{d}\eta^{2}+g(\eta)(\mathrm{d}x^{2}_{1}+\cdots+\mathrm{d}x^{2}_{d-1}) \, .
\label{metricd+1}$$ The dual quantum field theory lives in a $d$-dimensional flat spacetime spanned by $(t, x_1, \cdots,x_{d-1})$ on the holographic boundary. We call $\eta$ the radial coordinate and the others are angular directions. The spacetime volume factor is $$\sqrt{|g|}=\sqrt{-\det g}=\sqrt{f(\eta)g(\eta)^{d-1}} \, .$$ A scalar field $\phi$ in this curved spacetime is described by the action: $$\begin{split}
S[\phi] = \dfrac{1}{2}\int \sqrt{|g|}\left( g^{\mu \nu}\partial_{\mu}\phi \partial_{\nu}\phi+m^{2}\phi^2+\frac{\lambda}{2}\phi^4\right) \, .
\end{split}$$ The saddle point equation (the classical equation of motion) $\delta S/\delta \phi=0$ reads, $$\label{eq: EoM}
-\dfrac{1}{\sqrt{|g|}}\partial_{\mu}\left(\sqrt{|g|}g^{\mu\nu}\partial_{\nu}\phi \right)+m^{2}\phi+\lambda \phi^3=0 \, .$$ Since we are interested in homogeneous static condensate in the dual quantum field theory, we assume that $\phi$ is only a function of $\eta$. Then Eq. becomes, $$\begin{aligned}
-\partial_{\eta}^{2}\phi-(\partial_{\eta}\ln \sqrt{|g|})\partial_{\eta}\phi+ m^{2}\phi+\lambda \phi^3=0
\end{aligned}$$ or equivalently, we could write it as $$\begin{aligned}
&\pi = \partial_{\eta}\phi \, ,\\
&\partial_{\eta}\pi+h(\eta)\pi -m^{2}\phi -\lambda \phi^3 = 0 \, ,\label{EOM}
\end{aligned}$$ where the metric function is (with $d=4$) $$h(\eta) \equiv \partial_{\eta}\ln\sqrt{f(\eta)g(\eta)^{d-1}} \, .
\label{hdef}$$ The input data is the pair $\phi(\eta \sim \infty), \pi(\eta\sim \infty)$ near the AdS horizon. And the field will propagate following the classical equation of motion Eq. . On the other hand, there is black hole horizon at $\eta\sim 0$. The on-shell static scalar field satisfies the black hole boundary condition $$\left[\dfrac{2}{\eta}\pi -m^{2}\phi -\lambda \phi^3\right]_{\eta\sim 0}=0 \, ,$$ or equivalently, we could require $$\begin{aligned}
\pi(\eta\sim 0)=0 \, .
\label{blackhole_condition}\end{aligned}$$
The mapping between the asymptotic value of the scalar field $\phi(\eta \sim\infty)$ and the data of the dual quantum field theory is given by the AdS/CFT dictionary with the asymptotically AdS spacetime with the AdS radius $L$ [@Hashimoto:2018bnb], $$\label{eq: asymptotic}
L^{3/2}\phi\sim \alpha e^{-\eta/L}
+ \beta e^{-3\eta/L}-\frac{\lambda \alpha^3}{2L^2}\eta e^{-3\eta/L}$$ for an operator ${\cal O}$ whose dimension is three, corresponding to the bulk scalar field $\phi$ with the mass $m^2 = -3/L^2$. The coefficients are related to the condensate as $$\begin{aligned}
\alpha = \frac{\sqrt{N_c}}{2\pi}m_{\cal O} \, , \quad
\beta = \frac{\pi}{\sqrt{N_c}}\langle {\cal O}\rangle L^3 \, .\end{aligned}$$ Here, $m_{\cal O}$ is the source for the operator ${\cal O}$ of the quantum field theory, and $N_c$ denotes the color number in QFT and hence we set $N_c=3$ as we focus on QCD later. Therefore, the data of one-point function of the quantum field theory $\{m_{\cal O},{\langle\cal O\rangle}\}$ is given, it is mapped to on-shell configuration of $\phi(\eta)$ and $\pi(\eta)$ (by taking derivative on both sides of Eq. ) near the holographic boundary $\eta\sim\infty$.
The experimental data pairs ($\phi(\eta\sim \infty), \pi(\eta\sim \infty)$) can be viewed as the positive data. And they will satisfy the black hole boundary condition Eq. ) after following the classical equation of motion Eq. . We could also view pairs of data $\phi(\eta\sim \infty), \pi(\eta\sim \infty)$ that does not lie on the experimental curve as negative data. We expect those negative data will not satisfy the black hole boundary condition. Therefore, this becomes a binary classification problem, with the propagation equation Eq. . Here, for a given data of the condensate, the parameters in the differential equation to be learned are: the continuous metric function $h(\eta)$, the AdS radius $L$ and interaction coupling $\lambda$ are in general unknown.
We regard Eq. as a neural network, and the network weights are the metric function and other parameters. For that purpose, the numerical method known as the Neural ODE is a perfect framework to find the optimal estimation for those unknown parameters. In the following, we will briefly review the Neural ODE method.
Neural ODE
----------
The Neural ODE [@chen2018neural] is a novel framework of deep learning. Instead of mapping the input to the output by a set of discrete layers, the Neural ODE evolves the input to the output by a differential equation, which is trainable. The general form of the differential equation reads $$\dfrac{\mathrm{d}z(t)}{\mathrm{d}t} = f_{\theta}(z(t), t) \, ,
\label{NODE}$$ where the vector $z$ denotes the collection of hidden variables and $\theta$ denotes all the trainable parameters (which could also be $t$-dependent) in the neural network. Without loss of generality, suppose we have observations at the beginning and end of the trajectory: $\{(z_{0}, t_{0}), (z_{1}, t_{1})\}$. One starts the evolution of the system from $(z_{0}, t_{0})$ for time $t_{1}-t_{0}$ with parameterized velocity function $f_{\theta}(z(t), t)$ using any ODE solver. Then the system will end up at a new state $(z_{1}, t_{1})$. Formally, we could consider optimizing the general loss function $\mathcal{L}$, which explicitly depends on the output $z_1$ as $$\mathcal{L}(z_{1})=\mathcal{L}\left(\int_{t_{0}}^{t_{1}}\mathrm{d}t~f_{\theta}(z(t),t)\right) \, .$$ To back-propagate the gradient with respect to the parameters $\theta$, one introduces the adjoint parameters $a(t) = \frac{\partial \mathcal{L}}{\partial {z}(t)}$ and their corresponding backward dynamics, $$\label{NODE back}
\dfrac{\mathrm{d}{a}(t)}{\mathrm{d}t} = -{a}(t)\cdot\dfrac{\partial {f}_\theta}{\partial {z}} \, .$$ After solving Eqs. and jointly, the parameter gradient can be evaluated from $$\dfrac{\partial \mathcal{L}}{\partial \theta} = \int^{t_{0}}_{t_{1}}a(t)\cdot\dfrac{\partial f_\theta}{\partial \theta}\mathrm{d}t \, .
\label{NODE grad}$$ The derivation Eq. can be found in the appendix.
Emergent spacetime from Neural ODE {#sec:3}
==================================
Learning architecture
---------------------
### Neural ODE and bulk equation
In the form of the first order differential equation, the equations of motion for the bulk field Eq. can be translated to the Neural ODE Eq. by the following identifications: $$\begin{aligned}
(\pi, \phi) \leftrightarrow {z} \, ,
\quad \eta \leftrightarrow t \, .\end{aligned}$$ The bulk metric function $h(\eta)$ corresponds to the neural network weights $\theta$. To make the network depth finite, we introduce the UV and IR cutoffs for the metric as $\eta_{\rm ini}=1$, and $\eta_{\rm fin}=0.1$ in units of the AdS radius $L$.
There are two big advantages of using Neural ODE. First, the metric function is smooth and we do not need to add penalty terms for smoothness. Therefore, we can largely reduce the number of hyper-parameters needed in the network. Second, our Neural ODE uses an adaptive ode solver, called “dopri5.” This gives us much more accuracy in the integration, and it turns out that the equation of motion in the curved geometry is sensitive to the discretization in some region of $\eta$. This adaptive method provides accuracy and efficiency simultaneously.
### Bulk metric parameterization
To make the integration variable monotonically increase from the AdS boundary to the black hole horizon, we made a change of variable $\widetilde{\eta}=1-\eta$ for the metric function, and we model the metric function $h(\widetilde{\eta}$ using the following two ansatz: $$\begin{aligned}
\text{ansatz 1: } & h(\widetilde{\eta}) = \sum_{n=0}^{8}a_{n}\widetilde{\eta}^{n} \, ,
\label{firstc}\\
\text{ansatz 2: }& h(\widetilde{\eta}) = \sum_{n=0}^{8}b_{n}\widetilde{\eta}^{n}+\dfrac{1}{1-\widetilde{\eta}} \, .
\label{secondc}\end{aligned}$$ The first one is the Taylor series around the AdS boundary. The second choice explicitly encodes the divergent behavior of the metric function near the black hole horizon at $\eta=0$. Any black hole horizon with a nonzero temperature has $f(\eta) \propto \eta^2$ with $g(\eta)$ being nonzero constant. Hence, Eq. leads to $h(\eta) \sim 1/\eta$ as the generic behavior of $h(\eta)$ near the horizon $\eta =0$. The second ansatz Eq. explicitly encodes this prior knowledge.
### Lattice QCD data as input
We use the lattice QCD data of RBC-Bielefeld collaboration [@phdthesisUnger] as our input data. The data is the chiral condensate $\mathcal{O}=\bar{q}q$, as a function of its source, the quark mass $m_{q}$. A plot is given in Fig. \[fig:data\] Left. We take the $T=0.208$ \[GeV\] temperature data (the black line in Fig. \[fig:data\] Left), and the detail of the data is listed in Tab. \[table:QCD\_data\].[^3]
We generate positive data and negative data in such a way that if the data’s vertical distance to the experimental curve is less than 0.005, then it is labeled as positive (the label is 0). Otherwise, it is labeled as negative (the label is 1). We collected 10000 positive data and 10000 negative data used for training, as shown in Fig. \[fig:data\] Right. Our goal is to obtain a holographic description of our QCD data using the Neural ODE method. The variation parameters are $\lambda$, $L$ and $h(\eta)$.
------------------- -------------------------------------------------
$m_{q}$\[GeV\] $\langle\bar{\phi}\phi\rangle[({\rm GeV})^{3}]$
\[0.5ex\] 0.00067 0.0063
0.0013 0.012
0.0027 0.021
0.0054 0.038
0.011 0.068
0.022 0.10
\[1ex\]
------------------- -------------------------------------------------
: Chiral condensate as a function of quark mass [@phdthesisUnger], at the temperature $T=0.208$ \[GeV\], converted to physical units [@Hashimoto:2018bnb].
\[table:QCD\_data\]
### Loss function
As for the loss function $\mathcal{L}$, we use $$\begin{split}
\mathcal{L}=\frac{1}{N_{\text{data}}}
\sum_{\rm data} &
\left[
\bigm|
T(\pi(\eta_{\text{fin}});\epsilon,\sigma)-l\bigm|^2
\right.
\\&
\left.+\beta\left(h(\eta_{\text{int}})-4\right)^{2}
\right]
\end{split}
\label{loss}$$ where the first term is the mean square error of the classifier loss function for the output data to approach the true result, Eq. . The function $T(x;\epsilon,\sigma)$ is a specific differentiable nonlinear activation function that maps region $[-\epsilon,\epsilon]$ to 0, and otherwise to 1, in a fuzzy manner, $$T(x;\epsilon,\sigma)=1+0.5\left(\tanh\left(\dfrac{x-\epsilon}{\sigma}\right)-\tanh\left(\dfrac{x+\epsilon}{\sigma}\right)\right) \, .
\label{Tx}$$ The parameter $\sigma$ controls the slope of the boundary as shown in Fig. \[fig:Tx\]. In the mean square error, $l$ is the label of the data ($l=0$ for positive data and $l=1$ for negative data). The second term in Eq. , the $\beta$ penalty term, is to impose the condition that the emergent metric needs to be asymptotically AdS near the boundary $\eta=\eta_{\rm ini}$. Due to nonlinear nature of the ODE function and sensitivity of Neural ODE, one may need to modify the hyperparameters $(\epsilon, \sigma)$ to ensure nonzero value of the gradient during the training.
Emergent metric
---------------
With the architecture described above, we perform the training. We first choose Eq. for the ansatz of the metric function $h(\eta)$. We randomly initialize the training parameters. The initial configuration of the metric function is given in the subplot (c) of Fig.\[fig:res\]. As shown in the subplot (a) of Fig. \[fig:res\], the machine with the initial metric judges all the orange+green data as positive data.
After training with 13000 epochs, the loss is reduced to 0.02. The result is shown in subplot (b) $\&$ (d) of Fig. \[fig:res\]. As we can see the predicted data agrees well with original positive data. We also observe that the emergent metric is a smooth function. The trained metric function reads: $$\begin{aligned}
\begin{aligned}
h(\eta) = & \, 8.2352\widetilde{\eta}^{8} +8.0109\widetilde{\eta}^{7}+ 7.6072\widetilde{\eta}^{6} \\
& +6.9469\widetilde{\eta}^{5} + 150.89\widetilde{\eta}^{4} -130.81\widetilde{\eta}^{3} \\
& + 55.539\widetilde{\eta}^{2}-22.223\widetilde{\eta}^{1}+ 3.7720 \, .
\end{aligned}\end{aligned}$$
The machine also finds the optimal values of the coupling constant and the AdS radius, $$\begin{aligned}
& \lambda = 0.0004 \, , \\
& L = 5.1640 [{\rm GeV}^{-1}] \, .\end{aligned}$$ As we can see in subplot (d) of Fig. \[fig:res\], the metric function $h(\eta)$ which the Neural ODE found has tendency to grow significantly near $\eta \sim 0$. This is indeed the black hole horizon behavior. It is quite intriguing that the machine automatically captures the divergence behavior of the metric function $h(\eta)$ near the black hole horizon.
As a check, we also perform the training with the second ansatz for the metric function $h(\eta)$, [*i.e.*]{} Eq. , which encodes the prior knowledge about the black hole horizon. As shown in Fig. \[fig:emergent\_metric\], the result looks almost the same as that of the first ansatz that does not use the prior knowledge. Therefore, the regularization to implement the black hole horizon in $h(\eta)$ is not necessary. This result indicates that Neural ODE can automatically discover the black hole geometry in the holographic bulk and recover the near-horizon metric behavior without prior knowledge. For convenience, we use the training results of the second ansatz to calculate a physical observable (Wilson loop) in the next section.
![Two emergent metrics for $T=0.208$ \[GeV\] data with different metric ansatz. The solid (dashed) line is the trained result with Eq. \[firstc\] (Eq. \[secondc\]). The two lines are found to overlap with each other, thus the divergence behavior of metric function near the black hole horizon is emergent during the training.[]{data-label="fig:emergent_metric"}](metric_plot.pdf){width="0.8\linewidth"}
Multi-temperature result
------------------------
We also applied the above method to the multi-temperature QCD data given in Tab. \[table:QCD\_data\_variousT\]. During the training, we require different neural networks to share the same value of AdS radius $L$, and the training results are summarized in Tab. \[table:multi-T\]. The model discovers the optimal emergent metric as well as the coupling constant $\lambda$ at each temperature.
------------------- -------------------------------- --------- -------------------------------- --------- -------------------------------- --------- -------------------------------- --------- -------------------------------- --------- --------------------------------
\[0.5ex\] $m_{q}$ $\langle\bar{\psi}\psi\rangle$ $m_{q}$ $\langle\bar{\psi}\psi\rangle$ $m_{q}$ $\langle\bar{\psi}\psi\rangle$ $m_{q}$ $\langle\bar{\psi}\psi\rangle$ $m_{q}$ $\langle\bar{\psi}\phi\rangle$ $m_{q}$ $\langle\bar{\psi}\psi\rangle$
\[0.5ex\] 0.00061 0.056 0.00062 0.049 0.00064 0.034 0.00065 0.019 0.00066 0.011 0.00068 0.0064
0.0012 0.058 0.0012 0.053 0.0013 0.042 0.0013 0.027 0.0013 0.018 0.0014 0.012
0.0024 0.064 0.0025 0.059 0.0025 0.052 0.0026 0.040 0.0026 0.029 0.0027 0.022
0.0049 0.07 0.005 0.068 0.0051 0.065 0.0052 0.058 0.0053 0.048 0.0054 0.038
0.0098 0.08 0.010 0.081 0.010 0.081 0.010 0.079 0.011 0.075 0.011 0.068
0.020 0.095 0.020 0.098 0.020 0.10 0.021 0.10 0.021 0.10 0.022 0.10
\[1ex\]
------------------- -------------------------------- --------- -------------------------------- --------- -------------------------------- --------- -------------------------------- --------- -------------------------------- --------- --------------------------------
\[table:QCD\_data\_variousT\]
[|c||c|c|c|c|c|c|]{}
$T$ & 0.188 & 0.192 & 0.196 & 0.200 & 0.204 & 0.208\
$L$ &5.164 &5.164&5.164&5.164&5.164&5.164\
$\lambda$ & 0.0014 & 0.0011 & 0.0009 & 0.0007 & 0.0005 & 0.0003\
$a_0$ & 3.7671 & 3.7678 & 3.7688 &3.7698 & 3.7709 & 3.7720\
$a_1$ & -22.229 &-22.228& -22.227&-22.226 & -22.225 & -22.223\
$a_2$ &55.533 & 55.534 & 55.535 & 55.536 & 55.537 & 55.539\
$a_3$ &-130.82 & -130.82 & -130.81 & -130.81 &-130.81 & -130.81\
$a_4$ & 150.88 &150.88 & 150.88 &150.88& 150.88 & 150.89\
$a_5$ & 6.939 & 6.9424 & 6.9434 & 6.9443 & 6.9457 & 6.9469\
$a_6$ &7.5981 & 7.6026 & 7.6036 & 7.6044 & 7.6061 & 7.6072\
$a_7$ & 8.0004 & 8.0062 & 8.0071 & 8.0079 & 8.0098 & 8.0109\
$a_8$ & 8.2230 &8.2304 & 8.2313 & 8.2320 & 8.2341 & 8.2352\
$T$ 0.188 0.192 0.196 0.200 0.204 0.208
----------- --------- --------- --------- --------- --------- ---------
$L$ 5.164 5.164 5.164 5.164 5.164 5.164
$\lambda$ 0.0014 0.0011 0.0009 0.0007 0.0005 0.0003
$b_0$ 2.8430 2.8438 2.8447 2.8456 2.8467 2.8474
$b_1$ -24.140 -24.139 -24.138 -24.137 -24.136 -24.135
$b_2$ 55.627 55.628 55.629 55.630 55.631 55.632
$b_3$ -130.22 -130.22 -130.22 -130.22 -130.22 -130.22
$b_4$ 150.79 150.79 150.79 150.79 150.79 150.80
$b_5$ 5.5746 5.5774 5.5790 5.5802 5.5813 5.5820
$b_6$ 4.6816 4.6849 4.6867 4.6880 4.6891 4.6898
$b_7$ 3.5672 3.5710 3.5730 3.5744 3.5756 3.5763
$b_8$ 2.5329 2.5371 2.5394 2.5409 2.5421 2.5428
\[table:multi-T\]
We have two observations of the trained results shown in Tab. \[table:multi-T\]. First, the obtained metric $h(\eta)$ and the AdS radius $L$ do not depend on the temperature $T$. Second, the only dependence on the temperature is encoded solely in the coupling constant $\lambda$ of the scalar field theory.
The former sounds counter-intuitive, since normally the metric itself should be highly dependent on the temperature, and the change in the metric will modify the gravitational fluctuation, which corresponds to the gluon physics. It is easy to resolve this issue. The obtained function is $h(\eta)$ and not the full metric components $f(\eta)$ and $g(\eta)$. Even for the case of the AdS Schwarzschild geometry in which the metric is temperature-dependent, we find $h(\eta)=\frac{4}{L} \coth \frac{4\eta}{L}$ which is temperature independent. In the next section, to compute physical quantities from the emergent $h(\eta)$, we assume some functional form of $g(\eta)$ and discuss the temperature dependence of the metric components.
What the machine found is that the reproduction of the input data mainly relies on the temperature dependence of the coupling constant $\lambda$ in the holographic bulk theory. For lower temperature, we find a strong nonlinear interaction, [*i.e.*]{} larger $\lambda$. The value of $\lambda$ is directly related to the self-coupling of sigma meson. Although we cannot compare our trained results with experiments since the self-coupling has never been precisely measured due to the broad width of the sigma meson, our result provides a unique view of the QCD phase transition, in particular about the mysterious relation between the chiral transition and the deconfinement transition.
Physical interpretation of the emergent spacetime {#sec:4}
=================================================
Reconstruction of the metric
----------------------------
Since in our case the machine learns only $h(\eta)$, to compute physical quantities such as Wilson loop, we need to assume the form of $g(\eta)$ to get $f(\eta)$. Here we assume the functional form of the AdS Schwarzschild configuration, $$\begin{aligned}
g(\eta) = A \left(\cosh\frac{2\eta}{La}\right)^a \, ,
\label{gansatz}\end{aligned}$$ where $A$ and $a$ are temperature-dependent constant. In particular the constant $a$ encodes the dimensionality of the AdS${}_{d+1}$-Schwarzschild as $a = d/4$, and here we just set it as a free parameter. The ansatz Eq. also satisfies the criterion that $g$ is a monotonic function of $\eta$, which is normally required for spacetimes without a bottle neck. The Hawking temperature $T$ constrains the function $f(\eta)$ as $$\begin{aligned}
f(\eta) \sim (2\pi T)^2 \eta^2\end{aligned}$$ so, for our calculation we define a new function $F(\eta)$ as $$\begin{aligned}
f(\eta) = (2\pi TL)^2 \left(\tanh \eta/L\right)^2 F(\eta)
\label{Fdef}
\end{aligned}$$ which satisfies the boundary condition $$\begin{aligned}
\lim_{\eta \to 0} F(\eta) = 1 \, .
\label{F0}\end{aligned}$$ Substituting Eqs. and to Eq. , and perform the integration over $\eta$ with the integration constant fixed by Eq. , we obtain $$\begin{aligned}
F(\eta) = \exp \int_0^\eta \left(
2h(\eta) - \frac{4}{L\sinh (2\eta/L)}-\frac{6}{L} \tanh\frac{2\eta}{La}
\right) d\eta \, .
\label{Fres} \end{aligned}$$ The overall factor $A$ in $g(\eta)$ in Eq. can be fixed by the following asymptotically AdS${}_5$ constraint at $\eta \gg L$, $$\begin{aligned}
f(\eta) \simeq g(\eta) \simeq e^{2\eta/L + \mbox{const.}},
\label{AsAdS}\end{aligned}$$ which implies $h(\eta)\simeq 4/L$ according to Eq. . To determine this constant which we require temperature independent, we expand Eq. around $\eta \gg L$ as $$\begin{aligned}
&\int_0^\eta \left(
2h(\eta) - \frac{4}{L\sinh (2\eta/L)}-\frac{6}{L} \tanh\frac{2\eta}{La}
\right) d\eta
\nonumber \\
& = \frac{2\eta}{L} + c(a) + {\cal O}(1/\eta) \, .\end{aligned}$$ Using this constant $c(a)$, the constraint Eq. determines the normalization of $g(\eta)$ as $$\begin{aligned}
g(\eta) = (2\pi T L)^2 e^{c(a)} \left(2\cosh\frac{2\eta}{La}\right)^a \, .
\label{gdet}\end{aligned}$$ Now, since we require that the constant in Eq. is temperature independent, we have a condition $$\begin{aligned}
\frac{\partial}{\partial T} \left[ T^2 e^{c(a(T))} \right] = 0 \, .\end{aligned}$$ Up to an integration constant, we can numerically solve this equation. Assuming that at $T=0.208$ \[GeV\] we have $a=1$, we find numerically $c(a=1) = 11.1952$. Then the equation above leads to $c(a(T=0.188 \, {\rm [GeV]})) = 11.3984$ and $a(T=0.208 \, {\rm [GeV]})=1.098$. We are going to use $g(\eta)$ given by Eq. and $f(\eta)$ given by Eq. with Eq. for the calculation of physical quantities below.
Wilson loop
-----------
Following the standard method [@Maldacena:1998im; @Rey:1998ik; @Rey:1998bq] for calculating the expectation value of the Wilson loop holographically, we evaluate the Wilson loop for a quark and an antiquark separated by the distance $d$, using our emergent spacetime. The logarithm of the Wilson loop $\langle W \rangle$, which is proportional to the quark potential $V$, is the area of the Euclidean worldsheet of a string hanging down from the AdS boundary. The string reaches $\eta=\eta_0$ at the deepest, and both the quark potential $V(d)$ and the quark distance $d$ are functions of $\eta_0$, as $$\begin{aligned}
d = 2\int_{\eta_0}^\infty
\frac{1}{\sqrt{g(\eta)}}\sqrt{\frac{f(\eta_0)g(\eta_0)}{f(\eta)g(\eta)-f(\eta_0)g(\eta_0)}}d\eta \, ,
\label{d}
\\
2\pi \alpha' V = 2\int_{\eta_0}^\infty \!\!\!\!
\sqrt{f(\eta)}
\sqrt{\frac{f(\eta_0)g(\eta_0)}{f(\eta)g(\eta)-f(\eta_0)g(\eta_0)}}d\eta \, .
\label{Vd}\end{aligned}$$ Here $1/(2\pi \alpha')$ is the string tension which is undetermined in this work. Eliminating $\eta_0$ from these expressions implicitly defines $V(d)$. Note that the integration in $V(d)$ diverges at $\eta = \infty$, and we need to introduce a cut-off for the asymptotic AdS boundary for the calculation.
The quark potential $V(d)$ has another saddle, which is just two straight strings connecting the black hole horizon and the asymptotic boundary, $$\begin{aligned}
2\pi \alpha' V_{\rm Debye}
=2\int^{\eta_0}_0 \!\!\!\!
\sqrt{f(\eta)}d\eta\, .
\label{V}\end{aligned}$$ We need to adopt $V(d)$ in Eq. or $V_{\rm Debye}$, whichever is smaller.
Using the metric obtained in the previous subsection, we calculate the quark potential for each temperature. In Fig. \[fig:Wilson\], we present the quark potential for $T=0.188$ \[GeV\] data and $T=0.208$ \[GeV\] data. They exhibit three phases: at short $d$, the potential is Coulombic, while at large $d$, the potential is flat and Debye-screened, and in the middle range of $d$, the potential is linear, signifying the quark confinement. The set of these features is well-known in lattice QCD simulations (see Fig. \[fig:lattice\]), and, interestingly, our holographic results reproduce these features.[^4]
This reproduction was reported in Ref. [@Hashimoto:2018bnb], and here we further investigate the temperature dependence. As we see in Fig. \[fig:Wilson\], the two plots are identical with each other except for the height of the Debye screening parts. The higher temperature corresponds to the lower height of the flat potential, which is qualitatively consistent with the lattice QCD result, as shown in Fig. \[fig:lattice\].
Summary and discussion {#sec:5}
======================
In this paper, we applied the Neural ODE to the AdS/DL correspondence, where the emergent spacetime in the gravity side of the AdS/CFT correspondence is regarded as a deep neural network. Since the classical spacetime is continuous and smooth, the weights of the network need to be interpreted as a smooth function of the depth, thus the Neural ODE provides a very natural scheme for training the bulk geometry. We followed the setup of Ref. [@Hashimoto:2018bnb] of using the lattice QCD data of QCD chiral condensate to train the neural network. We demonstrated that the Neural ODE indeed worked well to discover a bulk geometry which is holographically consistent with the lattice QCD data. Even without including the black hole boundary condition for the ansatz function of the Neural ODE, the machine found automatically the black hole horizon behavior. This proves the ability of the Neural ODE to automate the proposal of the holographic bulk theory from the holographic boundary data in the AdS/CFT setup.
We performed the training with the training data of lattice QCD at various temperatures and found that the optimal volume factor of the emergent geometries shares the same radial dependence except for the overall normalization. The temperature dependence in the behavior of the QCD chiral condensate simply comes from the bulk scalar coupling constant, which corresponds to the meson couplings. The Wilson loops holographically calculated with the machine-trained emergent geometries appeared to have a correct temperature dependence, as in Fig. \[fig:lattice\].
For a more quantitative evaluation of the emergent spacetime, here we argue that the slope of the linear part of the plots of the quark-antiquark potential, given in Fig. \[fig:Wilson\], corresponds to the QCD string tension $\sigma$. Since in our formulation, the overall normalization $2\pi \alpha'$ is not given, we only look at the ratio of the slope at $T=0.188$ \[GeV\] and the slope at $T=0.208$ \[GeV\]. A numerical fitting of Fig. \[fig:Wilson\] gives $\sigma_{T=0.208{\rm GeV}}/\sigma_{T=0.188{\rm GeV}} \simeq 1.0$. In lattice QCD simulation, this number is expected to be smaller than $1$, because the deconfinement transition (which is not the first-order phase transition) occurs when the QCD string tension goes to zero. So our value $1.0$ still keeps the tendency of the large $N$ gauge theories where the deconfinement transition is expected to be the first order.
In addition, we notice that the string breaking distance, the value of $d$ at the kink in Fig. \[fig:Wilson\], is around $d \sim 10^{-3}$ in the unit of $L \sim 5$ \[GeV${}^{-1}$\], which is too small compared to the expected QCD value $d\sim {\cal O}(1)$\[fm\]. This quantitative discrepancy would be largely due to our assumed functional form of the metric component $g(\eta)$ in Eq. . In this paper we have seen the qualitative feature of the temperature dependence of the Wilson loops to be consistent with lattice QCD results[^5], and further quantitative match will need some different observable data to train $f(\eta)$ and $g(\eta)$ independently.
The Neural ODE is quite effective for physical applications of the machine learning method in which neural network weights have physical meanings. Any physical observable, if looked minutely enough, should be a continuous function of space and time. To identify weights of standard deep neural networks with physical quantities, regularizations to make them a smooth function on the discrete network are necessary, which are rather artificial and often still can not remove discretization artifacts fully. In Neural ODEs, the weights are continuous functions in the first place, which hence reduces unnecessary ingenuity of the regularizations. One of the main improvements from Ref. [@Hashimoto:2018bnb], although the physical setup is the same, is that we could remove the artificial regularizations used in [@Hashimoto:2018bnb], and largely improve the prediction accuracy of the emergent bulk metric at the same time.
Since we obtained the emergent volume factor for each temperature, it is possible to ask what kind of bulk action can allow such a metric as a solution of its equation of motion. There is a lot of work that elaborated possible bulk systems dual to QCD, and the major example would be the Einstein-dilaton system [@Gursoy:2007cb; @Gursoy:2007er]. We want to visit this question in future publications.
We would like to thank T. Akutagawa and T. Sumimoto for valuable discussions. We thank Microsoft Research for the kind hospitality during the workshop “Physics $\cap$ ML.” H.-Y. H. would like to thank Lei Wang for the discussion on Neural ODE. K. H. was supported in part by JSPS KAKENHI Grant Number JP17H06462. H.-Y. H. and Y.-Z. Y. were supported by a startup fund from UCSD.
Neural ODE {#sec:A}
==========
In this appendix, we briefly introduce Neural ODE [@chen2018neural], and how to backpropagate the errors to train parameters. We assume the dynamics of a set of variables $\vec{x}(t)=\{x_{i}(t)\}$ can be described by the ODE specified by a velocity function $\vec{v}=\{v_{i}(\vec{x}(t),t;\theta)\}$, where $\theta$ are training parameters. We call the following equation the forward ODE, $$\dfrac{dx_{i}(t)}{dt}=v_{i}(\vec{x}(t),t;\theta) \, .$$ Given the initial condition $x_i(0)$, the ODE can be integrated from $t=0$ to $t=1$. The loss function $\mathcal{L}$ is a function of the final state, $$\mathcal{L}=\mathcal{L}(\vec{x}(1)).$$ To calculate the gradient with respect to the parameter $\theta$, we first need to calculate the gradient with respect to $\vec{x}(t)$ at each time t. Define the adjoint variable $\vec{a}(t)=\{a_{i}(t)\}$ $$a_{i}(t)=\dfrac{\partial \mathcal{L}}{\partial x_{i}(t)} \, .$$ To derive the dynamics of adjoint variables, we consider the dependence chain $\vec{x}(t)\rightarrow \vec{x}(t+dt)\rightarrow \cdots \rightarrow \mathcal{L}$, $$\dfrac{\partial \mathcal{L}}{\partial x_i(t)}=\dfrac{\partial \mathcal{L}}{\partial x_{j}(t+dt)}\dfrac{\partial x_j(t+dt)}{\partial x_{i}(t)} \, ,$$ where Einstein summation is assumed. Then we find $$\begin{split}
a_{i}(t)&=a_j(t+dt)\dfrac{\partial [x_{j}(t)+v_{j}(\vec{x}(t),t;\theta)dt]}{\partial x_i(t)}\\
&=(\delta_{ij}+\partial_{x_i(t)}v_{j}(\vec{x}(t),t;\theta)dt)a_j(t+dt) \, .
\end{split}$$ Therefore, the adjoint variable follows backward ODE equation, $$\begin{split}
\dfrac{da_i(t)}{dt}=-a_{j}(t)\partial_{x_i(t)}v_{j}(\vec{x}(t),t;\theta)\, ,\label{eq:backprop}
\end{split}$$ $$a_i(0)=\int^{0}_{1}a_j(t)\partial_{x_i(t)}v_{j}(\vec{x},t;\theta)dt \, .$$ To calculate the gradient with respect to the parameter $\theta$, we can collect the gradient for each time step backward, $$\begin{split}
\dfrac{\partial \mathcal{L}}{\partial \theta}&=\int^{0}_{1}\dfrac{\partial \mathcal{L}}{\partial x_i(t)}\dfrac{\partial(x_i(t)-x_i(t-dt))}{\partial \theta}\\
&=\int^{0}_{1}\dfrac{\partial \mathcal{L}}{\partial x_i(t)}\dfrac{\partial v_i(\vec{x},t;\theta)}{\partial \theta}dt \, .
\end{split}$$
[^1]: See Ref. [@Hashimoto:2019bih] for the physical meaning of the regularization as an Einstein action.
[^2]: For example, a hybrid version [@nagai2019self] of the self-learning Monte Carlo [@liu2017self] is a novel way to train effective Hamiltonian while keeping the desired accuracy.
[^3]: See [@Hashimoto:2018bnb] for the conversion method from the lattice QCD unit to the physical unit.
[^4]: In Ref. [@Andreev:2006nw], phenomenological ansatz for the bulk spacetime which is similar to ours and that of Ref. [@Hashimoto:2018bnb] was made. Surprisingly, the machine learns a metric that was proposed independently by humans. K.H. would like to thank Oleg Andreev for bringing the paper to his attention.
[^5]: In fact, we required that the constant in Eq.\[AsAdS\] is independent of the temperature, and if we loosen this condition, the resultant Wilson loops do not match the lattice QCD results.
|
---
abstract: 'The scaling of entanglement entropy for the nearest neighbor antiferromagnetic Heisenberg spin model is studied computationally for clusters joined by a single bond. Bisecting the balanced three legged Bethe Cluster, gives a second Renyi entropy and the valence bond entropy which scales as the number of sites in the cluster. For the analogous situation with square clusters, i.e. two $L \times L$ clusters joined by a single bond, numerical results suggest that the second Renyi entropy and the valence bond entropy scales as $L$. For both systems, the environment and the system are connected by the single bond and interaction is short range. The entropy is not constant with system size as suggested by the area law.'
author:
- 'B. A. Friedman'
- 'G. C. Levine'
date: 'Received: date / Accepted: date'
subtitle: Possible Violation of the Area Law in Dimensions Greater than One
title: Scaling of Entanglement Entropy for the Heisenberg Model on Clusters Joined by Point Contacts
---
[example.eps]{} gsave newpath 20 20 moveto 20 220 lineto 220 220 lineto 220 20 lineto closepath 2 setlinewidth gsave .4 setgray fill grestore stroke grestore
Introduction
============
This paper is a numerical investigation of entanglement entropy of the nearest neighbor isotropic Heisenberg spin 1/2 model on clusters, in particular, a Bethe cluster and two $L \times L$ square clusters joined by a single bond. We are studying the ground state quantum mechanical properties of these models and the coupling $J$ is taken to be the same on every bond (including the single bond joining the clusters), that is the interaction is antiferromagnetic. Recall that for a large Bethe cluster and for the square lattice there is very good numerical evidence that there is long range antiferromagnetic order (suitably defined for a Bethe cluster), though to the best of our knowledge there is no proof. The numerical methods used are spin wave theory [@song], direct diagonalization and valence bond Monte Carlo [@sandvik; @hastings]. There is no sign problem associated with the models so Monte Carlo is an effective numerical method. The quantities calculated by valence bond Monte Carlo are the valence bond entropy [@alet; @chhajlany] and the $n=2$ Renyi generalized entropy $S_2$[@hastings]. Both these quantities are straightforward (given the techniques in ref. [@hastings] ) to calculate by valence bond Monte Carlo. Note that $S_1$, the Von Neumann is not so easy to calculate, however, it is believed that the same essential physics is contained in $S_2$ (but see ref. [@chandran]). Generically, we will refer to all these entropies as entanglement entropy. We only use “balanced” clusters, where the number of even sites is equal to the number of odd sites, thus the ground state has spin 0 [@lieb] and the ground state can be represented by a superposition of valence bond states [@liang; @sandvik].
![14 site three branch Bethe cluster. []{data-label="fig.1"}](figure1n.eps){width="48.00000%"}
In particular, for the present study one of the clusters we shall consider is the three branched bond centered Bethe clusters, see figure 1 for an illustration of the 14 site 3 branched bond centered cluster [@caravan]. As discussed in [@caravan], if you bisect such a cluster, a simple argument shows, the valence bond entropy must scale as the number of sites $N$. In contrast, as we will later numerically demonstrate, the spin-wave technique gives an entropy proportional to $\log{N}$. Because of this apparent contradiction it is important to use an unbiased numerical method, valence bond Monte Carlo, to calculate the Renyi generalized entropy $S_2$. Due to recent advances in numerical techniques[@hastings; @kallin] such calculations can be done accurately for large clusters. A priori, however, one does not know if the clusters one can treat are large enough to see the asymptotic scaling law. One of the objects of the current investigation is to determine if one can realize the asymptotic regime with existing numerical techniques and hardware.
Why are we interested in the scaling law for entanglement entropy? Firstly, if the entanglement entropy scales as $\ln{N}$, DMRG (Density Matrix Renormalization Group) is an effective, unbiased method to calculate the ground state properties. That is, one expects the number of states needed to describe the DMRG blocks to go as $e^{S_1} \approx O(N)$ not say $e^{N}$. A still outstanding issue in condensed matter physics is how Neel order is destroyed by the addition of holes and ultimately is transformed into superconductivity. Since the Bethe cluster has Neel order at 1/2 filling [@kumar; @changlani] , the Hubbard or t-J model on the Bethe cluster would be an effective way to study this issue assuming the DMRG blocks scale with the number of sites, not the number of states, in the blocks.
Secondly, unlike in one dimension, the status of the area law for the entanglement entropy is not as clear [@eisert]. What are the conditions on the Hamiltonian and the cluster for the validity of the area law for a non one dimensional system? Naively, since a single bond connects the two halves of the Bethe cluster, one would expect either a constant or a logarithmic dependence of entanglement entropy on system size. However, for non interacting fermions, one sees $N$ dependent entropy for the Bethe cluster and $\sqrt{N}$ or $\log{N}$ (depending on boundary conditions) for square clusters separated by a single bond [@levine; @caravan]. Is this a pathological feature of non interacting systems?
There have been a number of very interesting papers [@bravyi; @movassagh; @aharonov; @hastings2] where examples of models exhibiting large entanglement entropies or violating the area law, are investigated. To best of our knowledge, the work developed in these papers does not apply directly to the particular systems we consider. Broadly speaking, it seems in the above papers, the lattice or cluster is straightforward to experimentally realize, while the Hamiltonian is difficult to realize in a practical situation. For our models, the Hamiltonian is physically realistic, however, a very large Bethe cluster is hard to realize in an experiment [@degennes]. However, there is no such difficulty in realizing the two $ L \times L$ cluster system.
Bethe-Cluster
=============
Spin-Wave Approach
------------------
To calculate the Von Neumann entropy $S_1$ and the second generalized entropy $S_2$ from a spin-wave approximation we use the approach of [@song] which is easily applied to the Bethe Cluster. Let us consider initially a different situation from that considered in DMRG, namely, we take the subsystem inside the system. As an example, consider in Figure 1, a three site subsystem, consisting of the sites labelled 1,2,3 inside the 14 site bond centered cluster. Figure 2 is the Von Neumann entropy for subsystems of size 3 to 63 for the 254 and 510 site clusters.
![$S_1$ vs. sites in the interior subsystem. The red circles are for the 254 site cluster and the blue diamonds are for 510 sites. The curve is a linear fit to the points for the 510 site cluster.[]{data-label="fig.2"}](figure2n.eps){width="48.00000%"}
We see the Von Neumann entropy scales with the number of sites in the interior cluster assuming one is sufficiently far from the boundary of the cluster. Note for the 254 site cluster the 63 site interior cluster is quite far from the linear fit in figure 2 while for the 510 site cluster the 63 site interior cluster is on the linear fit. The linear scaling is consistent with what one expect from the area law, since for an interior cluster, the number of boundary points scales with the number of sites in the cluster.
Let us now examine a situation of greater similarity to that encountered in the blocking procedure in DMRG. Take a bond centered cluster (figure 1) and pick the subsystem to be the left half of the cluster.
![Entanglement Entropy vs. ln(cluster size) for subsystems bisecting the system. The red circles are for $S_1$ while the blue diamonds are for $S_2$. The green squares are exact diagonalization results for $S_2$.[]{data-label="fig.3"}](figure3n.eps){width="48.00000%"}
This is done in figure 3, the Von Neumann entropy and $S_2$ are plotted vs. the logarithm of the system size. We see both quantities scale as the logarithm of the system size, similar to a one dimensional system. However, as previously mentioned, by the argument of ref. [@caravan], the valence bond entropy scales as the number of sites. Naively, one would expect, say the valence bond entropy and $S_1$ (or $S_2$) to scale the same way with system size ( at least up to logarithms [@kallin2]). Thus either the valence bond entropy scales differently from other entropies or the spin wave calculation gives an incorrect result. In figure 3, the green squares refer to exact diagonalization results for $S_2$ for system sizes 6, 14, and 30. The system sizes accessible to direct diagonalization (the 30 site cluster has state space of dimension approximately $1.6 \times 10^8$ ) are too small to infer the scaling of $S_2$ with system size. Nonetheless, some insight can be gained from plotting, in figure 4, the entanglement entropy for the 30 site cluster vs. the number of sites in the interior subsystem. Perhaps not surprisingly, the entanglement entropy increases as the number of bonds connecting the subsystem to the system increases (up to 7 sites), after which the number of connections decreases and the entropy decreases.
![Entanglement Entropy vs. number of sites in the interior subsystem for a 30 site cluster. The red circles are for $S_1$ while the green squares are for $S_2$. The points were calculated by exact diagonalization.[]{data-label="fig.4"}](figure4n.eps){width="48.00000%"}
Valence Bond Monte Carlo
------------------------
We thus turn to valence bond Monte Carlo [@sandvik] as a method to compute, essentially exactly, the properties of the Heisenberg model on a Bethe Cluster. In figure 5, the valence bond entropy is plotted vs. system size for a bisected system. The valence bond entropy [@alet; @chhajlany] is a natural quantity to compute with valence bond Monte Carlo, as the basis consists of valence bonds and the valence bond entropy counts the number of valence bonds leaving the subsystem. From the figure, one sees that the valence bond entropy scales with system size. Of course, given the argument in [@caravan], this is no surprise. Further insight can be obtained by looking at the value of the valence bond entropy. By the argument of [@caravan], at least 1/3 of the valence bonds (for 1/2 the cluster, i.e. for a 1022 site cluster, roughly 170 bonds) must connect the two halves of the cluster; from figure 3, we see for large clusters, very close to 1/3 of the bonds connect the two halves.
![Valence Bond Entropy vs. system size for a bisected system.[]{data-label="fig.5"}](figure5n.eps){width="48.00000%"}
Let us now consider the $n=2$ Renyi generalized entropy $S_2$. Due to recent advances in computational technique this quantity can be calculated with valence bond Monte Carlo. We apply the methods developed in ref. [@hastings], see also [@kallin]; $S_2(\rho_A)$ is calculated as $-\ln{(\langle \rm{Swap}_A\rangle )}$ where A is the subsystem and $\rm{Swap}_A$ is a swap operator. $\langle \rm{Swap}_A\rangle $ is calculated in the simplest formulation by a double projection Monte Carlo algorithm. In a more sophisticated approach the ratios
$$\frac{\langle \rm{Swap}_{A^{i+r}}\rangle }{\langle \rm{Swap}_{A^{i}}\rangle }$$
are computed from Monte Carlo; from these ratios $\langle \rm{Swap}_A\rangle $ is then calculated. Here $r+i$ is a symbolic notation for a subsystem bigger than the subsystem $i$. For system sizes 6,14, 30 and 62 sites we have used three “different” approaches: the naive (no ratio) approach, the sophisticated approach where $i+r$ consists of the next shell in the Bethe cluster and the brute force “sophisticated” approach where “r” is only one site. Recall there is a shell or layer structure for Bethe clusters, i.e. for the 30 site cluster (take 1/2 the cluster) the first layer has 1 site, layer 2 has 2 sites, layer 3 has 4 sites, layer 4 has 8 sites. All three approaches agree, to within the statistical errors. Exact diagonalization results for 6,14 and 30 site clusters are also in agreement with the Monte Carlo calculations.
![$S_2$ vs. sites in the subsystem for a 254 site cluster. The red crosses are calculated from the $r=1$ ratio method, while the blue diamonds use the shell technique. Statistical errors are smaller than the symbols in the figure. []{data-label="fig.6"}](figure6n.eps){width="48.00000%"}
In figure 6, we plot $S_2$ vs sites in the subsystem for a 254 site cluster. The subsystem is taken to be an interior subsystem as in figure 4. The red crosses are calculated from the r=1 ratio method, while the blue diamonds use the shell technique. Statistical errors are smaller than the symbols in the figure. We see that $S_2$ appears to be a continuous piece wise linear function of the number of sites in the subsystem. It is linear within the shell with a kink in going from one shell to another. The kink (discontinuity in the derivative) is smaller in the shells near the center of the cluster. The decrease in entropy for the final shell is presumably caused by the spins on the boundary of the cluster that are no longer connected by bonds to spins outside the subsystem.
![$S_2$ vs. sites in the subsystem. Green squares are for 126 site system, red circles are for 254 site system and blue diamonds are for 510 site system. Statistical errors are smaller than the symbols in the figure. []{data-label="fig.7"}](figure7n.eps){width="48.00000%"}
We next consider a situation similar to figure 2. In figure 7, we take an interior subsystem and calculate $S_2$ via valence bond Monte Carlo for systems of size 126 sites , green squares, 254 sites, red circles and 510 sites, blue diamonds. Error bars would be smaller than the symbols in the figure. It appears that $S_2$ scales with number of sites in the interior subsystem for a sufficiently small subsystem relative to the system as expected from the area law.
![$S_2$ vs. sites in the system for subsystems bisecting the system. The blue diamonds are the spin wave results, while the red circles are calculated by valence bond Monte Carlo, $r=1$. The green cross is for the shell method but breaking up the large shells to 32 sites.[]{data-label="fig.8"}](figure8n.eps){width="48.00000%"}
Finally in figure 8, we plot $S_2$ vs. system sizes for subsystems that consist of half the system. The blue diamonds are the spin wave results, while the red circles are calculated by valence bond Monte Carlo, r=1. The green cross is for the shell valence Monte Carlo method but breaking up the large shells to 32 sites. For system sizes less than or equal to 62, at least qualitatively, Monte Carlo and spin wave agrees; however as the system sizes grow past this point there is an increasing discrepancy. It appears the spin wave result increases logarithmically with the number of sites while the Monte Carlo results give linear dependence on the number of sites in the system. Does this crossover with system size make sense?
One can present a simple minded argument that rationalizes this behavior. The relevant “aspect” ratio $\alpha$ is the ratio of the number of generations to the number of boundary points, for example for the 6 site cluster $\alpha =\frac{2}{2} $, for 14 sites $\alpha=\frac{3}{4}$ for 30 sites $\alpha=\frac{4}{8}$ etc. . One would expect $\alpha$ must be small to see Bethe cluster rather than 1 dimensional behavior. Hence one would anticipate one needs to study system sizes substantially larger than 30 sites.
A heuristic argument can be made for the magnitude of the slope of $S_2$ vs. system size. Recall that $S_2 = -\ln \langle \phi | \rm{Swap}_A|\phi\rangle $ where $\phi=|\psi_0\rangle |\psi_0\rangle $ i.e. the replicated ground state. Note that $|\psi_0\rangle =\sum_{\alpha} f_{\alpha} |\alpha\rangle $ where $|\alpha\rangle $ is a valence bond state and $f_{\alpha} \ge 0$. Then the normalization $$1= \langle \phi | \phi\rangle = \sum_{\alpha,\beta,\alpha',\beta'} f_{\alpha}f_{\beta}f_{\alpha'}f_{\beta'} \langle \beta'|\langle \alpha'|\alpha\rangle |\beta\rangle$$ and $$\begin{aligned}
\langle \phi|\rm{Swap}_{A}|\phi\rangle =
\\ \sum_{\alpha,\beta,\alpha',\beta'} f_{\alpha}f_{\beta}f_{\alpha'}f_{\beta'} \langle \beta'|\langle \alpha'|\rm{Swap}_{A}|\alpha\rangle |\beta\rangle \end{aligned}$$
All the valence bond states have at least $N/6$ bonds connecting the two halves of the Bethe cluster. Typically, $\frac{N}{6}\frac{1}{3}\frac{1}{3}$ sites in A and an equal number in B are connected by valence bonds in both $|\alpha\rangle $ and $|\beta\rangle $. Assume states $\langle \beta'|$ and $\langle \alpha'|$ with the same valence bonds dominate the normalization and the expectation value of $\rm{Swap}_{A}$. The term $\langle \beta'|\langle \alpha'|\rm{Swap}_{A}|\alpha\rangle |\beta\rangle $ is reduced in comparison to $\langle \beta'|\langle \alpha'|\alpha\rangle |\beta\rangle $ by a factor of $2^{-\frac{N}{6}\frac{1}{3}\frac{1}{3}}$ due to these bonds. Hence $$S_{2}\approx -\ln 2^{-\frac{N}{54}}=\frac{N}{54} \ln 2$$
i.e. $S_{2}$ is proportional to $N$ with a slope of about 1/78. This is comparison to the numerical results in figure 8 which give a slope of roughly $\frac{1}{100}$.
Two Square Clusters Joined by a Single Bond
===========================================
![4X4-4X4 cluster[]{data-label="fig.9"}](figure9.eps){width="24.00000%"}
As in [@caravan] , consider two $L \times L$ clusters linked by a single bond. Here the bond is a spin-spin interaction of strength $J$, the same strength as all the other spin-spin interactions. The single bond is chosen to be as close to the middle of the side as possible. For $L$ even (odd) this bond joins the $L/2$ ( $(L+1)/2$) site on the side facing the other cluster with the corresponding site on the other cluster (figure 9). There are several interesting aspects of this model. Firstly, this model can be realized in a physical system, either solid state or cold atom. Secondly, for $L$ even, there exists valence bond patterns where no valence bond need to go from one $L \times L$ cluster to the other. Lastly, for non interacting fermions, at 1/2 filling, the entanglement entropy scales as $L$, not as a constant or $\ln L$. This is consistent with the Bethe cluster results, both for free fermions and the Heisenberg model, in that the “perimeter” of the model (the sites with only one bond) scales as the number of sites in the cluster.
![$S_2$ vs $L$ for $L \times L$ - $L \times L$ clusters, $L$ even[]{data-label="fig.10"}](figure10n.eps){width="48.00000%"}
![Valence Bond Entropy vs. system size for a bisected system, $L$ even. Error bars are smaller than the data symbols.[]{data-label="fig.11a"}](figure11a.eps){width="48.00000%"}
![Valence Bond Entropy vs. system size for a bisected system, $L$ odd.[]{data-label="fig.11a"}](figure11b.eps){width="48.00000%"}
Figure 10 is a plot of the second Renyi entropy $S_2$ vs. $L$ for $L$ even taking the subsystem as half the cluster calculated using valence bond Monte Carlo. The values of $S_2$ are noticeably smaller than for comparably sized Bethe Clusters. However, $S_2$ does not appear to be a constant or depend on $\ln L$. The calculation appears, to within the large statistical errors and the limited system sizes, to be consistent again with $S_2$ scaling as $L$. To make the case for $L$ more convincing, figure 11 is a plot of valence bond entropy vs. $L$ for various system sizes and $L$ even. One sees a rather clear indication of the valence bond entropy scaling with $L$. The same quantities are plotted for $L$ odd in figure 12. Again one sees the valence bond entropy scaling as $L$. The structure in the graph can be rationalized by noting that for $L = 2n+1$ $n$ even, there is a valence bond pattern with only one valence bond joining the two $ L \times L$ squares and that bond can be chosen to be the middle bond (where the interaction joining the squares is located).
Conclusions
===========
Numerical evidence has been presented that for a Heisenberg model on clusters joined by a single interaction, the entanglement entropy scales as the perimeter, not as a constant, as suggested by the area law. This is consistent with work on non interacting fermions and demonstrates these earlier results are not an artifact of non interacting particles. As discussed in the introduction, this has ramifications for DMRG calculations on Bethe clusters. More generally, the scaling with perimeter places fundamental limitations on DMRG algorithms involving partitioning into clusters. This is analogous to the restrictions placed on heat engines by the thermodynamic entropy where “heat engine” in this case is a quantum calculation (done on a classical computer) and “entropy” is the entanglement entropy. In addition, two $L \times L$ clusters joined by a link is an experimentally realizable system. To directly measure entanglement entropy is difficult (though not impossible [@islam] ). However, recent work on the area law suggests the obstruction to the area law in two dimensions ( see [@marien] p34-39 ) is the existence of edge states. Such (hypothetical) edge states may be easier to measure (and analyze) than directly measuring the entanglement entropy. As well as telling us what we cannot do, insights from the second law of thermodynamics can be used for more rational engine design. An analogous situation could conceivably hold for the “area” law, where now the device analogous to a heat engine is a quantum computer. Proposed modular quantum computer architectures[@monroe1; @monroe2] are quite similar to the clusters joined by links considered in the present work.
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[On One Uniqueness Theorem for M. Rietz Potentials]{}
[We prove that there exists a nonzero holderian function $f:\mathbb{R}\rightarrow\mathbb{R}$ vanishing together with its M. Rietz potential $f\ast
\frac{1}{|x|^{1-\alpha}}$ in all points of some set of positive length. This result improves the one of D. Beliaev and V. Havin [@BH]]{}
[**0. Introduction.**]{} Let $\alpha$ be a real number, $0<\alpha<1$. Let $f:{{\mathbb R}\rightarrow {\mathbb C}}$ be a locally summable function satisfying the following condition: $$\label{condition1} \int_{ {\mathbb R}}{\frac{|f(x)|dx}{1+|x|^{1-
\alpha}}}< +\infty.$$ Let $$(U_\alpha f)(t):=\int_{{{\mathbb
R}}}{\frac{f(x)dx}{|t-x|^{1- \alpha}}},\;t\in\mathbb{R}.$$ In case (\[condition1\]) the function $U_\alpha f$ is defined a.e. on $\mathbb{R}$. We call it *the M. Rietz Potential*, and we call $f$ *the density* of this potential. We write $dom U_\alpha$ (the domain of $U_\alpha$) for the set of all locally summable functions satisying (\[condition1\]).
Let $V\in\mathbb{R}$ be a measurable set; we denote its length as $|V|$. The uniqueness theorem mentioned in the title states that *if $f$ satisfies (\[condition1\]) and Hölder’s condition with an exponent more than $1-\alpha$ in some neighboarhood of $V$, while $|V|>0$ and* $$\label{allnull} f|_V=U_\alpha|_V=0,$$ *then $f=0$ a.e. on $\mathbb{R}$.*
This theorem follows from a slightly more general “uncertainty principle” proven in [@Havin]. It concerns M. Rietz’s potentials of *charges* (not necessarily absolutely continuous with respect to Lebesgue measure) and $\alpha$’s not necessarily from $(0,1)$; for the history of the problem and its connections with the uniqueness problem for Laplase’s equation see also [@HJ] and [@BH]. Havin [@Havin] posted the following question: is it possible to omit Hölder’s condition on $f$ near $V$ in Theorem 1? Moreover, it was still unknown if there exist a nonzero *continuous* function $f\in dom\, U_\alpha$ and a set $V$ of positive length satisfying (\[allnull\]).
In [@BH], it was shown that the answer to the last question is affirmative. However, the function $f$ constructed in [@BH], being continuous, does not satisfy any Hölder’s condition.
In this paper, we build a nonzero *hölderian* function $f\in dom U_\alpha$ vanishing with its M. Rietz potential $U_\alpha f$ on some set of positive length.
As well as in [@BH], we build the desired function using the techniques of ”correction“ which were proposed by D. Menshov and applied to the problems of potential theory in [@AK], [@Binder], [@BW], [@wolff]. Our progress (as compared to [@BH]) is based on improving the correction process using elementary probabilistic techniques (see Lemma 5[^1]). We also mention that we deal with *complex* (not only real-valued!) densities $f$. This detail seems unessential, but it surprisingly simplifies the construction of a desired (*real-valued!*) density $f$, even in case [@BH], when the goal was just to find *continuous* (not necessarily hölderian) $f$.
The main result of this paper is the following theorem.
There exist a nonzero function $f\in
dom\:U_\alpha$, a set $V\subset {\mathbb R}$ of positive length and a positive number $r$, such that $f$ and $U_\alpha f$ are identically zeroes on $V$ and $f$ satisfies Hölder’s condition with exponent $r$.
[**Remark.**]{} The function $f$ we build is complex-valued. In order to get a real-valued density, one should just take its real or imaginary part; at least one of them will not be an identical zero.
I am grateful to V. Havin for introducing me to the problem and for useful discussions.
[**1. Operator $W_\alpha$.**]{} We will need an operator that is in some sense inverse to $U_\alpha$. Let for $g \in C^\infty_0({\mathbb R})$ $$(V_\alpha g)(t):=\frac1\alpha\int_{{{\mathbb
R}}}{g(x)\frac{sgn(t-x)}{|t-x|^ \alpha}dx},$$
$$(W_\alpha g)(t)=((V_\alpha g)(t))',\quad t\in \mathbb{R}.$$
Let $g$ be a function defined on $\mathbb{R}$, $\lambda>0$, $\varepsilon>0$. Denote $(C_{\lambda}g)(x)=g(\lambda x)$, $g_\varepsilon=\frac1\varepsilon C_{1/\varepsilon} g$. The next lemma states some properties of $W$.
1. $W_\alpha(C_0^\infty( {\mathbb R}))\subset {\rm dom}
U_\alpha$;
2. $U_\alpha W_\alpha g = cg,g \in (C_0^\infty(
{\mathbb R}))$;
3. $W_\alpha C_\lambda=\lambda^\alpha C_\lambda
W_\alpha$;
4. $\alpha W_\alpha g = -g'\ast\frac{sgn
x}{|x|^\alpha}$ (we use $\ast$ for the convolution on $\mathbb{R}$);
5. $(W_\alpha g)(t)=(g\ast |x|^{-\beta)}(t), g \in {\rm dom}
W_\alpha, t \notin {\rm supp}\, g.$
Hereinafter $\beta=\alpha+1$. All statements of this lemma are well-known (2) or obvious (1,3,4,5). See, for example, [@BH page 226].
We use the following notation: $I=(-\frac12,\frac12)$; if $Q$ is a bounded interval, then $c_Q$ is its center.
We write $\phi (t)$ for “the finitizator”: $\phi \in C^\infty$, ${\rm supp} \: \phi \subset I$, $\int_I \phi = 1$, $\phi \geq 0$.
In the proof we shall fix positive numbers $p$ and $\lambda$. For a function $h:\mathbb{R}\rightarrow\mathbb{C}$ we introduce its “embedding to the interval $Q$” $h_Q(t):=h(\frac{t-c_Q}{|Q|\lambda})$, $t\in\mathbb{R}$. Finally, let $M_Q(h):=(\frac1{|Q|}\int_Q{|h|^p})^{1/p}.$
[**2. Main Lemma.**]{} The mail tool for the proof of Theorem 1 will be Lemma 4. First, we prove auxiliary lemmas 2 and 3. We state the existence of functions with certain concrete numerical properties. These functions will serve as “building blocks” for our construction. The following proposition is principal for us: for some constant $B<0$ and some $p>0$ there exists a function $h\in C_0^\infty$ with arbitrarily small support satisfying $\int_{\mathbb R}(|1-W_\alpha h|^p-1)<B$. The meaning of this fact is that it allows us to control both the length of support of $h$ (future correcting term) and its influence on the “amount” of potential $W_\alpha$ of the function we are correcting. We shall rearrange this statement for it to be convenient for our purposes.
The function $h$ will be made from “the finitizator” $\phi$ by means of an appropriate scaling.
Let for $\varepsilon>0$, $t\in {\mathbb R}$ $$F^{[\varepsilon]}(t):=(W_\alpha \phi_\varepsilon)(t), \quad
F^{[0]}(t):=|t|^{-\beta}.$$
If a positive number $p$ is sufficiently small, then $$J(p) := \int_{\mathbb R}(|1-F^{[0]}|^p-1)dx < 0$$
[**Proof.**]{} Let $L:=\int_{\mathbb R}\log|1-F^{[0]}|$. Then, as $(a^p-1)/p$ is monotone in $p$ for any $a>0$ and converges to $\log a$ as $p\rightarrow 0$, we have $\lim_{p\searrow
0}\frac{J(p)}{p} = L$ (note that $||1-F^{[0]}(t)|^p-1| \leq
c|t|^{\beta}$, when $|t|$ is large, and if $p<1/\beta$, then the integral $J(p)$ converges in zero as well). But $L$ can be computed exactly: $L=2\pi\cot\frac\pi\beta<0$. One can find the computation, for example, in [@BH page 233].
From now on the number $p$ found in the previous lemma will be fixed.
In the next lemma we pass from $F^{[0]}$ (the potenial $W_\alpha$ of the delta-function) to the potential of some concrete function $\phi_\varepsilon$. We also introduce a “small complex rotation”: multiply $\phi_\varepsilon$ by $e^{i\theta}$ with small $\theta$. This leads to some technical simplifications in the future.
There exist numbers $B<0$, $\theta_0>0$ and $\varepsilon_0>0$, such that if $0\leq\theta\leq\theta_0$ and $0\leq\varepsilon<\varepsilon_0$, then $$J(\varepsilon,\theta):=\int_{\mathbb
R}(|1-e^{i\theta}F^{[\varepsilon]}|^p-1)<B.$$
[**Proof.**]{} Using the homogeneity property of $W_\alpha$ (point 3 of Lemma 1), we get $$\label{estimF} |F^{[\varepsilon]}(t)| \leq
C(\alpha)\min(\frac1{\varepsilon^\beta},\frac1{|t|^\beta}),\quad
t\in {\mathbb R}.$$ It is clear that $F^{[\varepsilon]}$ converges to $F^{[0]}$ pointwise as $\varepsilon\rightarrow 0$. It follows from Lebesgue dominated convergence theorem that $$\lim_{\varepsilon\searrow 0}J(\varepsilon,0)=J(p).$$ Choose $B<0$ to satisfy $J(\varepsilon,0)<2B$ for $\varepsilon\in(0,\varepsilon_0)$. We are to prove that $J(\varepsilon,\theta)\stackrel{\theta\rightarrow
0}{\longrightarrow} J(\varepsilon,0)$ uniformly in $\varepsilon$. Indeed, $$|J(\varepsilon,\theta)-J(\varepsilon,0)| = \left|\int_{\mathbb
R}|1-e^{i\theta}F^{[\varepsilon]}|^p-|1-F^{[\varepsilon]}|^p\right|
\leq
\left|\int_{|x|>\theta^{-p/2}}\right|+\left|\int_{|x|<\theta^{-p/2}}\right|=:J_1+J_2.$$ We have $$J_1=\left|\int_{|x|>\theta^{-p/2}}|1-e^{i\theta}F^{[\varepsilon]}|^p-|1-F^{[\varepsilon]}|^p\right|
\leq 2C \left|\int_{|x|>\theta^{-p/2}}|t|^{-\beta}\right|
\stackrel{\theta\rightarrow 0}{\longrightarrow} 0,$$ as the integral converges in infinity (and does not depend on $\varepsilon$). Then, using the inequality $|a^p-b^p|\leq|a-b|^p$ for $a>0$, $b>0$ and $p\in(0,1)$, we get $$J_2=\left|\int_{|x|<\theta^{-p/2}}|1-e^{i\theta}F^{[\varepsilon]}|^p-|1-F^{[\varepsilon]}|^p\right|=$$ $$=\left|\int_{|x|<\theta^{-p/2}}|e^{-i\theta}-F^{[\varepsilon]}|^p-|1-F^{[\varepsilon]}|^p\right|
\leq$$ $$\leq \int_{|x|<\theta^{-p/2}}|e^{-i\theta}-1|^p \leq
\int_{|x|<\theta^{-p/2}}\theta^p\stackrel{\theta\rightarrow
0}{\longrightarrow} 0.$$ So, $J_1+J_2\stackrel{\theta\rightarrow 0}{\longrightarrow} 0$ uniformly in $\varepsilon\in(0,\varepsilon_0)$, and the lemma is proved.
We are now ready to formulate Lemma 4 – the main tool of further construction. We replace the whole line (from Lemma 3) by a bounded interval, the constant 1 by an arbitrary function with a small oscillation, and finally we bring in a (small) positive parameter $\lambda$. This one is responsible for smallness of the potential $W_\alpha$ of the correcting term far away from the interval we correct on. Denote $\gamma(\lambda):=(1+B\lambda/2)^{1/p}$, where $B$ is a constant from Lemma 3. Notice that $0<\gamma(\lambda)<1$ for any sufficiently small positive $\lambda$.
If $f$ is a function defined on some interval $Q$, we define $osc_Q f:=\sup\limits_{x,y\in Q}(|f(x)-f(y)|)$ (*the oscillation* of $f$ on $Q$).
There exist numbers $\theta>0$, $\lambda_0>0$, $\varepsilon_0>0$, such that for any positive $\lambda< \lambda_0$ one can find a number $\kappa>0$ with the following property: if $0<\varepsilon<
\varepsilon_0$, $Q$ is any bounded interval and a continuous complex-valued function $h$ satisfies $$\label{osc} osc_Q h \leq \kappa |h(c_Q)|,$$ then
1. $M_Q(h-h(c_Q)e^{i\theta}F_Q^{[\varepsilon]}) \leq
\gamma(\lambda)|h(c_Q)|$;
2. $\frac\theta2 |h(t)| \leq
|h(t)-h(c_Q)e^{i\theta}F_Q^{[\varepsilon]}(t)| \leq
\frac{C}{\varepsilon^\beta}|h(t)|,\quad t\in Q$.
Recall that $F^{[\varepsilon]}_Q(t)=F^{[\varepsilon]}(\frac{t-c_Q}{|Q|\lambda})$.
[**Proof.**]{} Take $\varepsilon_0$ and $\theta$ from Lemma 3. First we get an estimate in point 2: $$|h(t)-h(c_Q)e^{i\theta}F_Q^{[\varepsilon]}|\geq
|h(c_Q)||1-e^{i\theta}F_Q^{[\varepsilon]}(t)|-|h(t)-h(c_Q)|\geq
|h(c_Q)|\frac{3\theta}{4} - \kappa|h(c_Q)|=$$ $$=|h(c_Q)|(\frac{3\theta}{4} - \kappa)\geq
|h(t)|\frac1{1+\kappa}(\frac{3\theta}{4} - \kappa)\geq
|h(t)|\frac\theta{2}$$ for any sufficiently small $\kappa$. We have used an elementary inequality $dist(1,\{re^{i\theta}: r \in {\mathbb
R}\})=\sin\theta\geq\frac{3\theta}{4}$, if $\theta>0$ is small.
The right-hand inequality in the point 2 follows clearly from (\[estimF\]).
Now we prove point 1: $$(M_Q (h-h(c_Q)e^{i\theta}F_Q^{[\varepsilon]}))^p \leq (M_Q
(h-h(c_Q)))^p+|h(c_Q)|^p(M_Q(1-e^{i\theta}F_Q^{[\varepsilon]}))^p
\leq$$ $$\leq (osc_Q h)^p+|h(c_Q)|^p
(M_{\lambda^{-1}I}(1-e^{i\theta}F^{[\varepsilon]}))^p.$$
Notice that $$(M_{\lambda^{-1}I}(1-e^{i\theta}F^{[\varepsilon]}))^p=
1+\lambda\int_{\lambda^{-1}I}(|1-e^{i\theta}F^{[\varepsilon]}|^p-1)=$$ $$=1+\lambda\int_{{\mathbb
R}}(|1-e^{i\theta}F^{[\varepsilon]}|^p-1)- \lambda\int_{{\mathbb
R} \backslash
\lambda^{-1}I}(|1-e^{i\theta}F^{[\varepsilon]}|^p-1).$$ Taking (\[estimF\]) into account, we get $$\left|\lambda\int_{{\mathbb R} \backslash
\lambda^{-1}I}(|1-e^{i\theta}F^{[\varepsilon]}|^p-1)\right| \leq
C\lambda^{\alpha+1}=o(\lambda),\lambda\longrightarrow 0,$$ so, if $\lambda$ is sufficiently small, we have $M_{\lambda^{-1}I}(1-e^{i\theta}F^{[\varepsilon]})\leq
1+B\lambda/2<1$. Therefore $$M_Q(h-h(c_Q)e^{i\theta}F^{[\varepsilon]})^p \leq
|h(c_Q)|^p(1+\left(\frac{osc_Q(h)}{h(c_Q)}\right)^p+2B\lambda/3)\leq
|h(c_Q)|^p(1+\kappa^p+2B\lambda/3)).$$ If $\kappa$ is sufficiently small, then $(1+\kappa^p+2B\lambda/3))<1+B\lambda/2$. The lemma is proved.
[**Remark 1.**]{} Careful examination of the proof shows that one can take $\kappa$ equal to $\min((\frac{|B|\lambda}2)^{1/p},\frac\theta{8})$, if $\theta$ is not too large.
[**Remark 2.** ]{} It is the left-hand inequality in the first point for what we pass to complex-valued functions. One cannot obtain such an estimate for real-valued functions.
[**Remark 3.**]{} By this moment we have fixed parameters $p$ and $\theta$. In what follows, constants that we regard as depending on $\alpha$ may also depend on these parameters. Later we shall fix an appropriate $\lambda$ and, thus, $\kappa$ and $\gamma$.
[**3. General idea of the construction.**]{} Now we describe the plan of construction of $f$ and $V$ (see statement of Theorem 1). We shall build a sequence of functions $g_n$, $g_n=g_{n-1}-r_{n-1}$, and a decreasing sequence of sets $V_n\subset
I$ with the following properties:
1. A nonzero function $g_1$ belongs to $C_0^\infty$, and $supp\;g_1\subset\mathbb{R}\backslash I$;
2. $r_k\in
C_0^\infty$, and $supp\;r_k\subset I$ for all $k\in\mathbb{N}$;
3. $\sum\limits_{k=1}^\infty|supp\;r_k|<\frac14$;
4. $|\bigcap\limits_{k=1}^\infty V_n|>\frac34$;
5. $\int_{V_n}|f_n|^p\stackrel{n\rightarrow\infty}{\longrightarrow}0$; here $f_n:=W_\alpha g_n$ and $p$ is the positive number fixed in the previous section
6. Sequences $g_n$ and $f_n$ converge uniformly on $\mathbb{R}$ to some continuous functions $g$ and $f$ correspondingly, and $g=U_\alpha f$.
Let $V:=\bigcap\limits_{k=1}^\infty V_n,$ $V':=\{x\in
I:\;g(x)=0\}.$ It follows from 1, 3 and 4 that $|V'|>\frac34$ and $|V|>\frac34.$ Therefore $|V\cap V'|>\frac12$. It follows from 5 and 6 that $f|_V=0$. Finally, using 2, we conclude that $g|_{\mathbb{R}\backslash I}=g_1|_{\mathbb{R}\backslash I}$, and, thus, the function $g$ is not identically 0. Hence the set $V\cap
V'$ and the function $f$ satisfy all conditions of Theorem 1, except (may be) Hölder’s condition.
Now we describe more precisely the structure of sets $V_n$ and correcting terms $r_k$. Bring in a [*sequence of positive numbers $\{\delta_n\}_1^\infty$*]{}. It will have the following properties: $\delta_1=1, \frac{\delta_n}{\delta_{n+1}} \in
{\mathbb N}$. We denote the partition of the interval $I$ to intervals of length $\delta_n$ as $H_n$. The set $V_n$ will be obtained as the union $\bigcup_{Q\in G_n} Q$, where $G_n$ is some subset of $H_n$. Roughly speaking, the set $G_n$ consists of all intervals on which we haven’t finish correction yet, in particular, for all $k>n$ there holds $supp\;r_k\subset V_n$.
Let us fix a sequence of positive numbers $\{\varepsilon_n\}_{n=1}^\infty$, such that $\sum\limits_{n=1}^\infty\varepsilon_n<\frac14$, and, in addition, $\varepsilon_n$ decay not very fast: $\varepsilon_n^{-1}=O(n^m)$ for some $m>0$. It will be responsible for the length of supports of $r_n$: there will hold an estimate $|supp\;r_n|\leq\varepsilon_n$ for all $n$.
We also demand $supp\;g_1\in(\frac12,\frac32)$, and, moreover, $\forall t\in I f_1(t)\neq 0$. One can take, for example, $g_1:=\phi(x-1)$.
Then, we choose some subset $G_n^g\subset G_n$ and let $$\label{defr_n} r_n:=\sum\limits_{Q\in
G^g_{n+1}}(\lambda\delta_{n+1})^\alpha
f_n(c_Q)(\phi_{\varepsilon_n})_Q e^{i\theta}.$$ So, $$W_\alpha r_n=\sum\limits_{Q\in G^g_{n+1}}
f_n(c_Q)F^{[\varepsilon_n]}_Q e^{i\theta}.$$ Note that in such a definition condition 3 will be satisfied by choosing the sequence $\varepsilon_n$ as described abowe.
The idea is that if $\delta_{n+1}$ is sufficiently small, then on each interval $Q\in G^g_{n+1}$ the oscillation of $f_n$ is small (there holds estimate (\[osc\])), and one can apply Lemma 4 with $f_n$ as $h$. Its result, together with an observation that functions $F^{[\varepsilon]}_Q$ decay sufficiently fast far away from $Q$, allow us, using the notation $V_n^g:=\bigcup\limits_{Q\in G_n^g} Q$, to prove an estimate $$\label{lessEta}
\int_{V^g_{n+1}}|f_{n+1}|^p \leq \eta \int_{V^g_n}|f_n|^p$$ with some $\eta\in(0,1)$. If one chooses $G^g_n$ appropriately (if on each step they occupy a large part of $G_n$), this leads to an estimate of integral over the whole set $V_n$: $$\label{lesseta2} \int_{V_{n}}|f_{n}|^p = O(\eta^{\frac{n}{2}}).$$ Hence condition 5 will be obtained.
[**Remark 1.**]{} The choice of $G_n$ (decreasing $V_n$ on each step) allows us to make functions $f_n$ converge not only in the sense of $L^p(I)$, but uniformly, in partiular, we get an estimate $|f_n(c_Q)|=O(\eta'^n),\;Q\in G_n$, where $\eta'\in(0,1)$.
[**Remark 2.**]{} If we did not worry about the control over modulus of continuity, we could take $G_n^g:=G_n$. Then (\[lesseta2\]) automatically follows from (\[lessEta\]), and the whole construction becomes more simple. Unfortunately, in order to get Hölder’s condition, one should pick out the set $G_n^g$ on each step – this is a set of intervals where the oscillation of $f_n$ is especially small – and make the correction only there.
[**4. Remarks on the estimate of modulus of continuity.**]{} In this section, we explain (not quite rigorously) what does estimates of modulus of continuity of $f$ depend on.
We use the following simple fact: *if a sequence of functions $h_n$ converges on $\mathbb{R}$ to the function $h$, whereas $|h_n-h|\leq C_1\eta_1^n$, and $|h_n'|\leq C_2 R^n$ (here $\eta_1\in(0,1),\; R>1$), then $h$ satisfies Hölder’s condition with an exponent $\log\eta_1/\log\frac{\eta_1}{R} $*.
The condition $|f_n-f|\leq C_1\eta_1^n$, will follow from Remark 1 at the end of the previous section (and, in fact, from estimate (\[lesseta2\])). When one estimates the derivative $f_n'(t)$ of the function $f_n$, the main role is played by the last added term $W_\alpha r_{n-1}$, or, more precisely, the building block $f_{n-1}(c_{Q_t})F^{[\varepsilon_{n-1}]}_{Q_t}
e^{i\theta}$, where the interval $Q_t\in H_n$ is defined by the statement $t\in Q_t$. From the homogeneity properties of $W_\alpha$ one can get (for $Q\in H_n$) an estimate
$$\label{estimFP}
|(F^{[\varepsilon_{n-1}]}_{Q})'(t)|\leq
\frac{c}{\varepsilon^{-\beta-1}\lambda\delta_n}\;t\in\mathbb{R}$$
Thus, one can get (say, on $V_n$) an estimate
$$\label{ocPrime1}|f_n'|\leq
\frac{Cf_n(c_{Q_t})}{\delta_n\varepsilon_{n-1}^{\beta+1}},$$
and so, everywhere,
$$\label{ocPrime}|f_n'|\leq
\frac{C\eta'^n}{\delta_n\varepsilon_{n-1}^{\beta+1}}.$$
This means that the obtained function $f$ would be hölderian if numbers $\delta_n^{-1}$ did not grow faster than some geometric series, in other words, if every time we divided the interval $Q\in
H_n$ into the same number of parts. On the other hand, the exponent $\log\eta_1/\log\frac{\eta_1}{R}$ tends to zero as $R\rightarrow
\infty$, hence it is clear that if $\frac{\delta_n}{\delta_{n+1}}\rightarrow \infty$ as $n\rightarrow\infty$, then we are unable to prove Hölder’s condition with any exponent.
It is clear, however, that if in order to define $\delta_{n+1}$ we use a natural estimate (\[ocPrime1\]) (recall what the necessity to choose small $\delta_{n+1}$ is due to: we need to estimate the oscillation of $f_n$ in order to use Lemma 4 – condition (\[osc\])), then because of the increasing multiplier $\varepsilon_{n-1}^{-\beta-1}$ we should take $\delta_{n+1}/\delta_n$ tending to zero to make (\[osc\]) hold. Therefore we need finer esimates of modulus of continuity of $f_n$, holding, however, not on the whole set $V_{n+1}$, but on some ”good“ part $V_{n+1}^g$ of it.
Note that the building block $F^{[\varepsilon_n]}_Q$ and its derivative $(F^{[\varepsilon_n]}_Q)'$ are large in modulus (as $\varepsilon_n^{-\beta}$ and $\delta_{n+1}^{-1}\varepsilon_n^{-\beta-1}$ correspondingly) only near the center of $Q$; if we consider them outside the interval of the length $\tau |Q|$ and with the same center with $Q$, where $0<\tau<1$, then we get estimates $|F^{[\varepsilon_n]}_Q|\leq C$ and $|(F^{[\varepsilon_n]}_Q)'|\leq C\delta_{n+1}^{-1}$.
One has an idea - exclude this “bad” central part of $Q$ and correct on the remaining part only. Unfortunately, if we drop it forever, this will mean that on each step one eliminates from $V_n$ a subset of length $\tau|V_n|$, and this makes $\cap_{n\in
{\mathbb N}}V_n$ have zero length.
Therefore, for each interval from $H_n$ we bring in a system of its “bad” subsets, and on each of them we shall “make a pause” - not correct during the next few steps, until the partition $H_{n+k}$ becomes so fine that the estimates of $f_n$ and its derivative become satisfactory (requirements to this estimates becomes weaker if $n$ grows). The the pause duration depends on the distance of the corresponding subset from the center of the interval, i.e. on how “bad” $f_n$ is on this subset. According to this, $G_{n+1}$ is divided into two parts: $G_{n+1}^g$, where the correction is made on this step, and $G_{n+1}^d$, where we do not do anything for the time being.[^2]
After that, we shall estimate an amount of intervals from $G_n$ such that have more than one half of their “ancestors” from $G_k,\;
k=0,1,\dots n-1,$, belong to $G_k^d$. It turns out that there are only a few of them (if $G_k^d$ is a small part of $G_k$ for each $k$), and we drop them out. For the rest, we prove an estimate like (\[lesseta2\]), using Lemma 4 and fast decay of $F^{[\varepsilon_{n}]}_{Q}$ away from $Q$.
[**5. Definition of sets $G_n^g$.**]{} To complete the construction, we should determine the sequence $\delta_n$ and sets $G_n$ and $G_n^g$. We need a number of estimates depending on how one picks out “good” subsets $G_n^g$ from $G_n$, but not on the way to choose $G_n$ themselves. We prove these estimates in sections 6, 7 and 8. Later, in section 9, we will define the way to choose $G_n$.
Bring in a positive parameter $\delta$, such that $\delta^{-1}\in
{\mathbb N}$, and let $\delta_n:=\delta^n$.
Then, bring in a parameter $\tau\in (0,1)$. Let $\tau^{-1}\in{\mathbb N}$, and, moreover, $\tau/\delta\in{\mathbb
N}$. We use the following notation: if $a>0$ and $Q$ is a bounded interval, then $Q[a]:=Q\backslash Q'$, where $Q'$ is an interval of the length $a|Q|$ and with the same center as $Q$.
We are now ready to define the set $G_{n+1}^g$. An interval $Q\in
G_{n+1}$ belongs to $G_{n+1}^g$, iff for any $k=0,1,\dots, n-1$ there holds an implication $Q\subset Q'\in G_{n-k}^g\Rightarrow
Q\subset Q'[\tau^{k+1}]$. In other words, if in the $(n-k)$-th step we have made a correction[^3] on the interval $Q'$, then on the next step the correction is forbidden on the set $Q'\backslash Q'[\tau]$, on the $(n-k+2)$-th step it is forbidden on the set $Q'\backslash Q'[\tau^2]$, and so on. It follows from the condition $\tau/\delta\in{\mathbb N}$ that the interval $Q\in G_{n+1},\;Q\subset Q'\in G_{n-k}^g$ either lies in $Q'[\tau^{k+1}]$ or does not intersect it.
In fact, $Q'[\tau^{k+1}]\backslash Q'[\tau^k]$ are the very “bad” subsets of $Q'$; on the $k$-th of them the “length of pause” is $k$ steps.
Let us make some simple, but important observations. It is easy to see that if $Q$ is a bounded interval and $dist(t,c_Q)>3\lambda\varepsilon_n|Q|$, then for any $n\in
{\mathbb N}$ $$\label{Fdaleko} |F^{[\varepsilon_n]}_Q(t)|\leq
C(\alpha)\frac{(\lambda|Q|)^\beta}{|t-c_Q|^\beta},$$ and, besides, $$\label{FPdaleko} |(F^{[\varepsilon_n]}_Q)'(t)|\leq
C(\alpha)\frac{(\lambda|Q|)^\beta}{|t-c_Q|^{\beta+1}}.$$ This means that for all $k\in {\mathbb N}$ and for $t\in
Q[\tau^k]$ $$|F^{[\varepsilon_n]}_Q(t)|\leq
C_1(\alpha)\frac{\lambda^\beta}{\tau^{k\beta}}, \label{goodF}$$ $$|(F^{[\varepsilon_n]}_Q)'(t)|\leq
C_1(\alpha)\frac{\lambda^\beta}{|Q|\tau^{k(\beta+1)}}.
\label{goodPrime}$$ Indeed, for $\tau^k/2>3\lambda\varepsilon_n$ these estimates coincide with the previous ones, and for $\tau^k/2>3\lambda\varepsilon_n$ we can use estimates $$|F^{[\varepsilon_n]}_Q(t)|\leq
\frac{C(\alpha)}{\varepsilon_n^\beta}$$ and $$|(F^{[\varepsilon_n]}_Q)'(t)|\leq
\frac{C(\alpha)}{|Q|\lambda\varepsilon_n^\beta},\quad
t\in\mathbb{R}.$$ We can improve the right-hand inequality in point 2 of Lemma 4 for $t\in Q[\tau^k]$. Indeed, applying (\[goodF\]) instead of (\[estimF\]), we get $$|h(t)-h(c_Q)e^{i\theta}F_Q^{[\varepsilon]}(t)| \leq
\frac{C(\alpha)\lambda^\beta}{\tau^{k\beta}}|h(t)|,\quad t\in
Q[\tau^k].$$ Let $t\in I$. Denote as $Q^n_t$ an element of $H_n$ defined by the condition $t\in Q^n_t$. Let $D^k_n(t):=\#\{l=1,\dots,n: t\in
Q^l_t\backslash Q^l_t[\tau^k]\}$. For $t\in V_n$ let $\widetilde{D}_n(t):=\#\{l=1,\dots,n: Q^l_t\in G^d_l\}$. We know that if $Q^l_t\in G^d_l$, then for some $k_l\in {\mathbb N}$ there holds an inclusion $Q^l_t\subset Q^{l-k_l}_t\backslash
Q^{l-k_l}_t[\tau^{k_l}]$. Moreover, for $l_1\neq l_2$ either $k_{l_1}\neq k_{l_2}$, or $l_1-k_{l_1}\neq l_2-k_{l_2}$. So, with each natural $l\leq n$, such that $Q^l_t\in G^d_l$, we can associate a pair $(l-k_l,k_l)$, such that $Q^l_t\subset
Q^{l-k_l}_t\backslash Q^{l-k_l}_t[\tau^{k_l}]$, and such a mapping will be injective. Hence $\widetilde{D}_n(t)\leq \sum\limits_{k\in
{\mathbb N}}D_n^k(t)=:D_n(t)$. The next section is devoted to estimates of lengths of sets $E_n:=\{t\in I:\: D_n(t)\geq n/2\}$.
[**6. Estimates of lengths of sets $E_n$.**]{}
. For $\tau$ sufficiently small there holds an inequality $$|E_n|\leq \frac{C(\tau)}{n^2},$$ where $C(\tau)$ tends to zero as $\tau\rightarrow 0$.
[**Proof.**]{} Let $${\xi'}_i^{(k)}:=\sum\limits_{Q\in H_i}\chi_{Q\backslash
Q[\tau^k]};$$ $$\xi_i^{(k)}:={\xi'}_i^{(k)}-\tau^k.$$ Then, considering $\xi_i^{(k)}$ as random variables on the probabilistic space $I$ with the measure $dx$, we have $E\xi_i^{(k)}=0$. We can describe sets $E_n$ in terms of functions $\xi_i^{(k)}$ as follows: $$x \in E_n \Leftrightarrow \sum_{k=1}^{n-1} \sum_{i=1}^{n-k}
{\xi'}_i^{(k)}(x) \geq \frac{n}{2}$$ Hence we are to estimate probabilities of the event that sums of random variables $\sum_{i=1}^{n-k} {\xi'}_i^{(k)}$ are large. It is easy to see that random variables ${\xi'}_i^{(1)},\quad
i=1,2,\dots$ are independent: it follows from the fact that $\tau/\delta$ is an integer. Unfortunately, for $k>1$ one cannot say the same thing about ${\xi'}_i^{(k)},\quad i=1,2,\dots$, because $\tau^k/\delta$ is not necessarily an integer. But, for $k>1$ variables ${\xi'}_i^{(k)},\quad i=1,2,\dots$ are still in some sense “almost independent”, and we use it.
Note that if $j\geq i+k$, then $\xi_i^{(k)}$ is a constant on each interval from $H_j$ (it follows from inclusion $\tau^k/\delta^k\in\mathbb{N}$). Hence we can made the following observation:
1. If $i_1+k\leq i_2\leq i_3\leq i_4$, then $E(\xi^{(k)}_{i_1}\xi^{(k)}_{i_2}\xi^{(k)}_{i_3}\xi^{(k)}_{i_4})=0$ (as the function $\xi^{(k)}_{i_2}\xi^{(k)}_{i_3}\xi^{(k)}_{i_4}$ is periodic with a period equal to $\delta^{i_1+k}$, and $\xi^{(k)}_{i_1}$ is constant on each interval from $H_{i_1+k}$).
2. If $i_1\leq i_2\leq i_3\leq i_4-k$, then $E(\xi^{(k)}_{i_1}\xi^{(k)}_{i_2}\xi^{(k)}_{i_3}\xi^{(k)}_{i_4})=0$ (as the function $\xi^{(k)}_{i_4}$ is periodic with a period equal to $\delta^{i_3+k}$, and $\xi^{(k)}_{i_1}\xi^{(k)}_{i_2}\xi^{(k)}_{i_3}$ is constant on each interval from $H_{i_3+k}$).
Now we write $$P(|\sum_{i=1}^n\xi_i^{(k)}|> \varepsilon)\leq
\frac{E(\xi_{1}^{(k)}+\dots+\xi_{n}^{(k)})^4}{\varepsilon^4}=$$ $$=
\frac{\sum\limits_{(i_1,i_2,i_3,i_4)\in\{1,\dots,n\}^4}
E(\xi_{i_1}^{(k)}\xi_{i_2}^{(k)}\xi_{i_3}^{(k)}\xi_{i_4}^{(k)})}{\varepsilon^4}.
\label{sum}$$ First note that $E(\xi_{i_1}^{(k)}\xi_{i_2}^{(k)}\xi_{i_3}^{(k)}\xi_{i_4}^{(k)})\leq
E|\xi_{i_1}^{(k)}|=2\tau^k(1-\tau^k)\leq 2\tau^k$ (the first inequality follows from the inequality $|\xi_{i}^{(k)}|<1$ for all $i$ and $k$). Second, if $(j_1,j_2,j_3,j_4)$ is a non-decreasing permutation of the numbers $(i_1,i_2,i_3,i_4)$, then a term $E(\xi_{i_1}^{(k)}\xi_{i_2}^{(k)}\xi_{i_3}^{(k)}\xi_{i_4}^{(k)})$ may differ from zero only if $j_2-j_1<k$ and $j_4-j_3<k$ (by the above observation). But the number of such fours $(j_1,j_2,j_3,j_4)$ does not exceed $\frac{n(n-1)}2k^2$. Hence, the number of nonzero terms in the numerator in (\[sum\]) does not exceed $4!\frac{n(n-1)}2k^2$. Therefore
$$P(|\sum_{i=1}^n\xi_i^{(k)}|>\varepsilon)\leq
\frac{Cn^2k^2\tau^k}{\varepsilon^4}. \label{cheb}$$
Let $E_n^k:=\{t\in I: D^k_n(t)\geq
n(\sqrt[8]{(4\tau)^k}+\tau^k)\}$. If $\tau$ is so small that $\sum_1^\infty\sqrt[8]{(4\tau)^k}+\tau^k<\frac12$, then $|E_n|\leq\sum_{k=0}^\infty|E_n^k|$. Finally, note that $D_n^k(t)=\sum_{i=1}^n{\xi'}^{(k)}_i(t)=\sum_{i=1}^n{\xi}^{(k)}_i(t)+n\tau^k$, so, using (\[cheb\]), we get $$|E_n|\leq\sum\limits_{k=1}^\infty|E_n^k|\leq
\sum\limits_{k=1}^\infty\frac{Ck^2\tau^\frac{k}{2}}{n^2 2^k}.$$ The lemma follows from this estimate.
[**7. Estimates of $f_n$.**]{} In this section, we prove some estimates for $f_n$’s, main of them are estimate (\[kappa\]), which allows us to apply Lemma 4, and estimate (\[ghvost\]), which shows that for $\lambda$ sufficiently small the terms $f_n(c_{Q'}) F^{[\varepsilon_n]}_{Q'} e^{i\theta}$, corresponding to $Q'\neq Q$, do not change the situation on $Q$ essentially. All the estimates we prove do not depend on the choice of $G_n$, but depend on how we pick out subsets $G_n^g$ (namely, we use estimates (\[goodF\]) and (\[goodPrime\])). The sets $G_n$, as mentioned above, will be defined later. Let $$\label{defT}
T_{n+1}(t):=\sum\limits_{Q\in G^g_{n+1},Q\neq Q^{n+1}_t}f_n(c_Q)
F^{[\varepsilon_n]}_Q e^{i\theta}.$$ Recall that an interval $Q_t^{n+1}\in H_{n+1}$ is defined by condition $t\in Q_t^{n+1}$.
There exists a positive number $\rho=\rho(\alpha)$, such that for $\lambda>0$ sufficiently small and for $\delta=\delta(\lambda)>0$ sufficiently small there holds the following:
1. for all $t\in I$$$|T_{n+1}(t)|\leq
\frac{c(\alpha)\lambda^\beta|f_n(t)|}{\rho};\label{ghvost}$$
2. if $k\leq n$, $x\in V_{n+1}^g$, $y\in I$ and $|x-y|\leq \delta^k$, then $$\label{gfxfy} |f_n(x)|\leq\frac{|f_n(y)|}{\rho^{n-k+1}};$$
3. for all $t\in I$, there holds an estimate $$|T'_{n+1}(t)|\leq
\frac{c(\alpha)\lambda^\beta|f_n(t)|}{\delta^{n+1}\rho};
\label{ghprime}$$
4. for all $t\in V_{n+1}^g$, there holds an estimate $$|f_n'(t)|\leq \frac{c_1(\alpha)\lambda^\beta|f_n(t)|}{\delta^n};
\label{fprime}$$
5. for all $Q\in G_{n+1}^g$, there holds an estimate $$osc_Qf_n\leq \kappa|f_n(c_Q)|\label{kappa}.$$
[**Proof.**]{} Inequality (\[kappa\]) clearly follows from (\[fprime\]), if we take $\delta$ sufficiently small (an interval $Q$ in (\[kappa\]) is of the length $\delta^{n+1}$). We derive (\[ghvost\]) and (\[ghprime\]) from (\[gfxfy\]), which, in turn, follows from (\[ghvost\]) and (\[kappa\]) for preceding n. Finally, (\[fprime\]) follows from (\[kappa\]) and (\[ghprime\]) for preceding n.
The base of induction – (\[kappa\]) and (\[gfxfy\]) for $n=1$ – is provided by the condition $f_1(t)\neq 0,\;t\in I$ (see Section 3) and the choice of $\rho$ and $\delta$ sufficiently small.
Derive (\[ghvost\]) and (\[ghprime\]) from (\[gfxfy\]). Fix $t\in I$. Denote as $G_\epsilon$ the set of all intervals $Q'\in
H_{n+1}$ satisfying the property $dist\,(c_{Q'},Q_t^{n+1})\geq
\epsilon$. Denote $$\sigma_\epsilon(t):=\sum\limits_{Q'\in
G_\epsilon}|F^{[\varepsilon_n]}_{Q'}(t)|,$$ $$\sigma^*_\epsilon(t):=\sum\limits_{Q'\in
G_\epsilon}|(F^{[\varepsilon_n]}_{Q'})'(t)|.$$
We need estimates $$\label{sigma} \sigma_\epsilon(t) \leq
c(\alpha)\lambda^\beta\left(
\frac{\delta^{n+1}}\epsilon\right)^\alpha$$ and $$\label{sigmaP} \sigma^*_\epsilon(t)\leq
c(\alpha)\lambda^\beta\frac{\delta^{(n+1)\beta}}{\epsilon^{\beta+1}}.$$ One can obtain them by estimating each term by (\[Fdaleko\]) and (\[FPdaleko\]) correspondingly, and then estimating the sum by an integral. The detailed proof of the first one can be found in [@BH page 234], the second one can be proved in the same way.
In order to get the above estimate of $|T|$ and $|T'|$, we shall divide terms in the right-hand side of (\[defT\]) into several groups according to their distance from the point $t$. For each group, we estimate $|f(c_Q)|$ by means of (\[gfxfy\]) (the closer to $t$ the interval $Q$ is, the better is this estimate) and then apply (\[sigma\]) (correspondingly, (\[sigmaP\]) for $|T'|$), which, by contrast, becomes better when $\epsilon$ grows.
So, let $G_{n+1}^g:=\bigsqcup_{k\leq n+1}G^{[k]}$, where $G^{[n+1]}:=G^g_{n+1}\backslash G_{\delta^n}$, $G^{[k]}:=(G^g_{n+1}\cup G_{\delta^k})\backslash
G_{\delta^{k-1}}$. For $y\in G^{[k]}$ we have $|f_n(y)|\leq
|f_n(t)|/\rho^{n-k+2}$ and $dist(G^{[k]},t)\geq \delta^k/2$, therefore $$|T_{n+1}(t)|\leq \sum_k\sum\limits_{Q'\in
G^{[k]}}|F^{[\varepsilon_n]}_{Q'}(t)||f_n(t)|/\rho^{n-k+2}\leq$$ $$\leq
c(\alpha)\lambda^\beta|f_n(t)|\sum_k\frac{\delta^{(n+1)\beta}}{\delta^{k\beta}\rho^{n-k+2}}
\leq
c(\alpha)\lambda^\beta|f_n(t)|\rho^{-1}\sum_k\left(\frac{\delta^\beta}{\rho}\right)^{n-k+1};$$ similarly $$|T'_{n+1}(t)|\leq \sum_k\sum\limits_{Q'\in
G^{[k]}}|(F^{[\varepsilon_n]}_{Q'})'(t)||f_n(t)|/\rho^{n-k+2}\leq$$ $$\leq c(\alpha)\frac{\lambda^\beta|f_n(t)|}{\delta^{n+1}}
\sum_k\frac{\delta^{(n+1)(\beta+1)}}{\delta^{k{\beta+1}}\rho^{n-k+2}}
\leq c(\alpha)\frac{\lambda^\beta|f_n(t)|}{\delta^{n+1}\rho}\sum_k
\left(\frac{\delta^{\beta+1}}{\rho}\right)^{n-k+1}.$$ Taking $\delta$ according to the condition $\frac{\delta^\beta}\rho<\frac12$, we get (\[ghvost\]) and (\[ghprime\]).
Now let us prove that (\[gfxfy\]) and (\[fprime\]) follow from (\[ghvost\]), (\[ghprime\]) and (\[kappa\]) for the previous $n$. We need an estimate $$|f_n(x)|\leq
\frac{4}{\theta}|f_{n+1}(x)|\leq\dots\leq
\left(\frac4\theta\right)^k|f_{n+k}(x)|,\quad x\in I,\;n,k\in
{\mathbb N}, \label{snizu}$$ which holds for $\lambda$ sufficiently small. Let us prove it. Let $Q_x^{n+1}\in G_{n+1}^g$. Then we have $$|f_n(x)|\leq\frac2\theta|f_n(x)-f_n(c_{Q_x})e^{i\theta}F^{[\varepsilon_n]}_{Q_x}(x)|\leq$$ $$\leq
\frac2\theta(|f_{n+1}(x)|+|T_{n+1}(x)|)\leq\frac2\theta(|f_{n+1}(x)|+c(\alpha)\lambda^\beta\rho^{-1}|f_n(x)|).$$ The first inequality follows from point 2 of Lemma 4 (which is applicable because of (\[kappa\])), the last one – from (\[ghvost\]). Now, the inequality $|f_n(x)|\leq
\frac{4}{\theta}|f_{n+1}(x)|$ follows from the last estimate, if $2c(\alpha)\lambda^\beta\theta^{-1}\rho^{-1}<\frac12$. If $Q_x^{n+1}\notin G_{n+1}^g$, then it follows from (\[ghvost\]) even easier. So, (\[snizu\]) is proved.
Now, let $k\leq n+1$, $x\in V_{n+2}^g$, $y\in I$ and $|x-y|\leq
\delta^k$. Let $k':=\max\{l\leq n: x\in V_{l+1}^g\}$. From the fact that $x$ is again in a “good” set $V_{n+2}^g$, it follows that $$x\in
Q^{k'+1}_x[\tau^{n-k'+1}] \label{include}.$$ First assume $k\leq k'$. Then $$|f_{n+1}(x)|=|f_n{x}|+|T_{n+1}(x)|\leq
|f_n(x)|(1+c(\alpha)\frac{\lambda^\beta}\rho)\leq\dots\leq$$ $$\leq |f_{k'+1}(x)|(1+c(\alpha)\frac{\lambda^\beta}\rho)^{n-k'}\leq$$ $$\leq (|f_{k'}(x)|+|T_{k'+1}(x)|+|f_{k'}(c_{Q_x^{k'}})|
|F^{[\varepsilon_n]}_{Q_x^{k'+1}}|)(1+c(\alpha)\frac{\lambda^\beta}\rho)^{n-k'}\leq$$ $$\leq |f_{k'}(x)|(1+c(\alpha)\frac{\lambda^\beta}\rho+(1+\kappa)
|F^{[\varepsilon_n]}_{Q_x^{k'+1}}|)(1+c(\alpha)\frac{\lambda^\beta}\rho)^{n-k'}.$$ Applying (\[goodF\]) and taking into account (\[include\]), we write on: $$|f_{n+1}(x)|\leq
|f_{k'}(x)|(1+C(\alpha)\lambda^\beta(\frac{1}\rho+\frac
{(1+\kappa)}{\tau^{(n-k'+1)\beta}})
)(1+c(\alpha)\frac{\lambda^\beta}\rho)^{n-k'}\leq$$ $$\leq
\frac{|f_{k'}(y)|}{\rho^{k'-k+1}}(1+C(\alpha)\lambda^\beta(\frac{1}\rho+\frac
{(1+\kappa)}{\tau^{(n-k'+1)\beta}})
)(1+c(\alpha)\frac{\lambda^\beta}\rho)^{n-k'}\leq$$ $$\leq |f_{n+1}(y)|\frac{\left(
\frac4\theta\right)^{n-k'+1}}{\rho^{k'-k+1}}(1+C(\alpha)\lambda^\beta(\frac{1}\rho+\frac
{(1+\kappa)}{\tau^{(n-k'+1)\beta}}))(1+c(\alpha)\frac{\lambda^\beta}\rho)^{n-k'}.$$ We have used the induction assumption (\[gfxfy\]) for $n=k'$ and (\[snizu\]). Now, if we choose $\rho$ so that $\sqrt{\rho}<\theta/50$ and $\sqrt{\rho}<\tau^\beta/50(C(\alpha))$, we get $$|f_{n+1}(x)|\leq
|f_{n+1}(y)|\frac1{\rho^{k'-k+1}}5^{-n-k'+1}\rho^{-n+k'-1}\cdot$$ $$\cdot
((1+c(\alpha)\frac{\lambda^\beta}\rho)^{n-k'+1}+5^{-n-k'+1}\rho^{-n+k'-1}(1+c(\alpha)\frac{\lambda^\beta}\rho)^{n-k'}).$$ Then we choose $\lambda=\lambda(\rho)$ so that $(1+c(\alpha)\frac{\lambda^\beta}\rho)<2$, and get $$|f_{n+1}(x)|\leq \frac{|f_{n+1}(y)|}{\rho^{n-k'+1+k'-k+1}},$$ and we are done.
Now we prove (\[fprime\]). Let $t\in V^g_{n+2}$. Let, as in the proof of (\[gfxfy\]), $k':=\max\{l\leq n: x\in V_{l+1}^g\}$. Again there holds (\[include\]). We have: $$|(f_{n+1})'(t)|\leq|f'_n(t)|+|T'_{n+1}(t)|\leq\dots\leq$$$$\leq|f'_{k'}(t)|+
|f_{k'}(c_{Q_t^{k'+1}})||(F^{[\varepsilon_{k'}]}_{Q_t^{k'+1}})'(t)|+
\sum\limits_{l=k'+1}^{n+1}|T'_{l}(t)|. \label{prproof}$$ Applying the induction assumption ((\[fprime\]) for $n=k'$) and (\[snizu\]), we have $$|f'_{k'}(t)|\leq
c_1(\alpha)\lambda^\beta\frac{|f_{k'}(t)|}{\delta^{k'}}\leq
c_1(\alpha)\lambda^\beta(4/\theta)^{n-k'+1}\frac{|f_{n+1}(t)|}{\delta^{k'}}.$$ We estimate the second term in (\[prproof\]) using (\[kappa\]), then (\[include\]) and (\[goodPrime\]) and finally (\[snizu\]), in the following way: $$|f_{k'}(c_{Q_t^{k'+1}})||(F^{[\varepsilon_{k'}]}_{Q_t^{k'+1}})'(t)|\leq
(1+\kappa)|f_{k'}(t)|\frac{c(\alpha)\lambda^\beta}{\delta^{k'+1}\tau^{\beta(n-k'+1)}}\leq$$ $$\leq
(1+\kappa)(4/\theta)^{n-k'+1}|f_{n+1}(t)|\frac{c(\alpha)\lambda^\beta}{\delta^{k'+1}
\tau^{\beta(n-k'+1)}}.$$ Finally, using (\[ghprime\]) and then again (\[snizu\]), we get $$|T'_{l+1}(t)|\leq
\frac{c(\alpha)\lambda^\beta|f_l(t)|}{\rho\delta^{l+1}}
\leq(4/\theta)^{n+1-l}\frac{c(\alpha)\lambda^\beta|f_{n+1}(t)|}{\rho\delta^{l+1}}$$ Taking $\delta<\theta\tau^\beta/(4A)$ and substituting all the estimates into (\[prproof\]), we get $$|f'_{n+1}(t)|\leq
\frac{\lambda^\beta|f_{n+1}(t)|}{\delta^{n+1}}(c_1(\alpha)/A^{n-k'+1}+c(\alpha)
\frac{4(1+\kappa)}{\theta\tau^\beta}+\frac4{\rho\theta}\sum_{k=0}^{n-k'+1}A^{-k}).$$ Taking $A>2$, we get (\[fprime\]), if $c_1(\alpha)$ is sufficiently large.
[**8. The end of the proof.**]{} In order to complete the construction, we should define the sets $G_n$. Fix a parameter $\tau$ in order to satisfy inequality $C(\tau)\sum_{n=1}^\infty\frac{1}{n^2}<\frac18$, where $C(\tau)$ is a constant from Lemma 5.
We define the sets $G_{n+1}$ as follows: $G_1:=H_1=\{I\}$; an interval $Q\in H_{n+1}$ belongs to $G_{n+1}$, if $Q\subset V_n$, $$M_Q(f_n)\leq K_n\eta^n,\label{G1}$$ where $K_n$ and $\eta$ will be defined later and, besides, $$\widetilde{D_n}(c_Q)\leq n/2.\label{G2}$$ (in fact, of course, $\widetilde{D_n}$ is a constant on $Q$). Note that the choice of $\tau$ and Lemma 5 guarantee that the total (for all $n$’s) length of $Q\in H_{n+1}$, $Q\subset V_n$, not included in $G_{n+1}$ because of violation of the condition (\[G2\]), does not exceed $\frac18$.
There exists a constant $C'(\alpha)$ such that for all sufficiently small $\lambda$ the following inequalities hold:
1. If $Q\in G^g_{n+1}$, then $$\int_Q|f_{n+1}|^p\leq X\int_Q |f_n|^p.$$
2. If $Q\in G^d_{n+1}$, then $$\int_Q|f_{n+1}|^p\leq Y\int_Q |f_n|^p,$$
where $X:=\gamma(\lambda)^p(1+C'(\alpha)\lambda^\beta)^p(=(1+B\lambda)(1+C'(\alpha)\lambda^\beta)^p)$, $Y:=(1+C'(\alpha)\lambda^\beta)^p$
[**Proof.**]{} Letting $P_{n+1}(t):=f_n(t)-f_n(c_{Q_t})e^{i\theta}F^{[\varepsilon]}_{Q_t}(t)$, we get for $Q\in G_{n+1}^g$: $$\int_{Q}|f_{n+1}|^p = \int_{Q}|P_{n+1}+T_{n+1}|^p \leq
\int_{Q}|P_{n+1}|^p(1+\frac{|T_{n+1}|}{|P_{n+1}|})^p.$$ Applying point 2 of Lemma 4 and the estimate (\[ghvost\]), we get $$\int_{Q}|f_{n+1}|^p\leq \int_{Q}
|P_{n+1}|^p(1+\frac{2c(\alpha)\lambda^\beta}{\rho\theta})^p,$$ and then, estimating the integral using point 1 of Lemma 4, we have $$\int_{Q}|f_{n+1}|^p \leq
(1+B\lambda)(1+C'(\alpha)\lambda^\beta)^p\int_{Q}|f_n|^p.$$ Recall that applicability of Lemma 4 is provided by (\[kappa\]).
The second case is even easier.
If $\lambda$ is sufficiently small, then for all $n$ there holds an inequality $$\int_{V_n}|f_n|^p\leq \eta^n\int_{V_1}
|f_1|^p\label{neqeta}$$ with some $\eta\in (0,1)$.
[**Proof.**]{} Denote as $\Theta_n$ the set $\{0,1\}^n$ (the set of all ordered sets $v=(v_1,\dots,v_n)$ of the length $n$ of zeroes and unities). Define for $v\in\Theta_n$ the set $G^{(v)}\subset
G_{n}$ as the set of all $Q$’s such that for all $k$ $v_k=1$ if and only if $ Q\subset V_k^g$. Let for $v\in\Theta_n$ $Z(v):=\prod\limits_{k=1}^n X^{v_k} Y^{(1-v_k)}$. Let by definition $G_1^g:=I$ and prove by induction in $n$ the estimate $$\int_I|f_1|^p\geq
X\sum_{v\in\Theta_n}Z(v)^{-1} \int_{G^{(v)}}|f_n|^p.
\label{derevo}$$ The base is obvious. Suppose (\[derevo\]) holds for some n. Then $$Z(v)^{-1} \int_{G^{(v)}}|f_n|^p=Z(v)^{-1}
(\int_{G^{(v,1)}}|f_n|^p+\int_{G^{(v,0)}}|f_n|^p) \geq$$ $$\geq
Z(v)^{-1}(X^{-1}\int_{G^{(v,1)}}|f_{n+1}|^p+Y^{-1}\int_{G^{(v,0)}}|f_{n+1}|^p)=$$ $$=Z(v,1)^{-1}\int_{G^{(v,1)}}|f_{n+1}|^p+Z(v,0)^{-1}\int_{G^{(v,0)}}|f_{n+1}|^p.$$ (we used Lemma 7). Thus we have proved the inductive step. Then, for $\lambda$ sufficiently small we have $X<1$, $Y>1$. If $n>N_0$, $v\in \Theta_n$, then $\sum_k v_k < n/2$ implies $G^{(v)}=\varnothing$ (it follows from (\[G2\])), therefore the right-hand side in (\[derevo\]) is not less than $X^{-\frac{n}2+1}Y^{-\frac{n}2}\int_{V_n}|f_n|^p$, and ($\ref{neqeta}$) follows provided $\lambda$ is sufficiently small.
Now we are ready to finish the setup of the construction. What we should do is to make precise the condition (\[G1\]). We take $\eta$ from Lemma 8. As, by Lemma 8, $\int_{V_n}|f_n|^p\leq
C\eta^n$, the total length of all intervals $Q\in H_{n+1}$, such that $Q\subset V_n$ and on $Q$ the condition (\[G1\]) fails, does not exceed $\frac{C}{K_n}$. For $K_n$ we take a sequence growing as some power of $n$ and satisfying the condition $\sum_{n=1}^\infty \frac{C}{K_n}<\frac18$. We get that total length of all intervals dropped out of $V_n$ on all steps because of violation of (\[G1\]) is less than $\frac18$. But one can tell the same thing about the length of all intervals dropped out because of violation of (\[G2\]). Hence $|\bigcap_{i=1}^\infty
V_n|>\frac34$.
It follows from (\[G1\]) and an estimate (\[kappa\]) of the oscillation of $f_n$ that if $Q\in G_{n+1}$, then $|f_n(c_Q)|\leq
C'\eta'^n$. Hence, taking into account (\[estimF\]) and (\[sigma\]) we get $$\label{prirost} |f_n(t)-f_{n+1}(t)|
\leq
\frac{C_1\eta'^n}{\varepsilon_n^\beta}+C_2\lambda^\beta\eta'^n,
t\in\mathbb{R}$$ The first term in the right-hand side corresponds to the building block $F^{[\varepsilon_n]}_{Q^{n+1}_t}$ (if there is any), the second – to all the others. It follows from this estimate that, $f_n$ converges uniformly on ${\mathbb R}$ to some function $f$. We also need an estimate $$\label{major} |f_n(t)-f_{n+1}(t)|\leq
c\frac{\lambda^\beta\delta_{n+1}^\alpha
C'\eta'^n}{|t|^\beta},\qquad t\notin 3I,$$ which follows from (\[Fdaleko\]). Now, (\[major\]) implies that $$\label{maj} |f_n(t)|\leq c|t|^{-\beta},\quad |t|\geq 3/2,$$ so we can, fixing $t$, write $$\int\limits_{\mathbb R}
\frac{f_n(s)ds}{|s-t|^{1-\alpha}}=\int\limits_{|s|\leq
\max(2|t|,3/2)}\frac{f_n(s)ds}{|s-t|^{1-\alpha}}+\int\limits_{|s|\geq
\max(2|t|,3/2)}\frac{f_n(s)ds}{|s-t|^{1-\alpha}}.$$ In the first term, the passage to the limit in the integral is provided by uniform convergence of $f_n$ as $n\rightarrow\infty$, in the second one, the function under integral is majorized by $c|s|^{-2}$. Hence $$\lim\limits_{n\rightarrow\infty}g_n =
\lim\limits_{n\rightarrow\infty}\int\limits_{\mathbb R}
\frac{f_n(s)}{|s-t|^{1-\alpha}}=\int\limits_{\mathbb R}
\frac{f(s)}{|s-t|^{1-\alpha}}$$ (The first inequality follows from Lemma 1). Finally note that the functions $g_n$ converge uniformly to $g$: $$|r_n|\leq \frac{C''\eta'^n}{\varepsilon_n}.$$ So, the program announced in the beginning of Section 3 is completed. Let us now show that the function $f$ satisfies Hölder’s condition.
In this argument, constants may depend on all parameters except $n$. Of course, we use the proposition from the beginning of Section 4. The condition $|f_n-f|\leq C_1\eta_1^n$ follows easily from (\[prirost\]).
Let us estimate the derivative of the correcting term on the $n$-th step:
$$|(W_\alpha r_n)'(t)|=|\frac{d}{dt}(\sum\limits_{Q\in
G^g_{n+1}}f_n(c_Q)F^{[\varepsilon_n]}_Q(t)e^{i\theta})|\leq
\max\limits_{Q\in G^g_{n+1}}|f_n(c_Q)|\sum\limits_{Q\in
G^g_{n+1}}|(F^{[\varepsilon_n]}_Q(t))'|\leq$$ $$\leq\max\limits_{Q\in
G^g_{n+1}}|f_n(c_Q)|\left(|(F^{[\varepsilon_n]}_{Q_t^{n+1}}(t))'|+\sum\limits_{Q\in
H_{n+1},\:Q\neq Q_t^{n+1}}|(F^{[\varepsilon_n]}_Q(t))'|\right).$$
We know that $\max\limits_{Q\in G^g_{n+1}}|f_n(c_Q)|\leq
C'\eta'^n$. Then, we use (\[estimFP\]) for the first term in the brackets, and for the sum – (\[sigmaP\]) with $\epsilon:=\frac{\delta^{n+1}}2$. Thus we have $$|(W_\alpha
r_n)'(t)|\leq C'\eta'^n(\frac
{C_1}{\delta^{n+1}\varepsilon_n}+\frac{C_2}{\delta^{n+1}})\leq
\frac {C_3}{\delta'^{n+1}}$$
Hence $|f_n'(t)|\leq
\frac{C_3}{1-\delta'}\delta'^{-n-1}\leq\frac{C_4}{\delta'^{n+1}}$, and Hölder’s property for $f$ is proved.
[**9. Remark on the order of choice of the parameters.**]{} Recall the order we chose our parameters in. Given $\alpha\in
(0,1)$, we fix $p$ and $\theta$ (Lemmas 3 and 4). Here Lemma 4 holds for all sufficiently small $\lambda$ and $\varepsilon$. Then we fix a sequence $\varepsilon_n$. Independently of other parameters we fix $\tau$. Lemma 6 holds for all sufficiently small $\lambda$ and $\delta$, independently of the choice of $G_n$ (here how small $\lambda$ and $\delta$ should depend on $\rho$ from this lemma, and $\rho$ itself only depends on $\alpha$, $\theta$, $p$, $\tau$, and the setup function $g_1$). Now we choose $\lambda$ such that the multiplier $\eta$ in front of the integral in the right-hand side of (\[neqeta\]) is less than 1 (decreasing $\lambda$ killing the influence of “tails” $T_n$ in comparison with the correction effect provided by point 1 of Lemma 2. Note that the last effect decays when $\lambda$ decreases as well ($\gamma(\lambda) \stackrel{\lambda\rightarrow
0}{\longrightarrow}1$!), but the “tails” dies faster). Finally, we fix $\delta$ in order to provide (\[kappa\]).
So far our considerations did not depend on $G_n$, therefore we did not need to define them. Now we fix a sequence $K_n$ and this finishes the definition of our construction.
[**10. The case of negative $\alpha$.**]{} One can ask whether there is an analogue of Theorem 1 for other M.Rietz’s kernels. In the case of the kernel $|x|^{-\beta},\quad1<\beta<2$, the answer is affirmative; moreover, as the convolution with such a kernel is, at least formally, the inverse operator to $U_{\beta-1}$ (see Lemma 1, points 2 and 5), the example in essential coincides with the one built above.
There exist a nonzero continuous function $g:{\mathbb
R}\rightarrow {\mathbb C},$ $supp\:g\subseteq 3I$ and a set $E$ of positive measure, such that $t\in E \Rightarrow g(t)=0,$ $\int_{\mathbb R} g(x)|t-x|^{-\beta}dx=0$. The last integral converges absolutely for every $t\in E$. The function $g$ satisfies Hölder’s condition with an exponent $\beta-1$.
[**Proof.**]{} We take for $g$ the function built in Theorem 1 (with $\beta-1$ for $\alpha$). Let $$E:=V\cap\left(I\backslash S\right),\quad\mbox{where
}S:=\bigcup\limits_{n\in {\mathbb N}}\bigcup\limits_{Q\in
G^g_{n+1}}3\varepsilon_n(Q-c_Q)+c_Q$$ The idea is that $|S|< \frac34$, but now $supp\; g_n$ is contained “rather deep” inside $S$: $$\label{supp} dist(supp\;g_n,E)\geq\varepsilon_n\delta^{n+1}.$$
We know (point 5 of Lemma 1), that for $t\in E$ $$f_n(t)=\int_{\mathbb R}
g_n(x)|t-x|^{-\beta}dx\stackrel{n\rightarrow
\infty}{\longrightarrow}0.$$ To prove Theorem 2, it is sufficient to justify the passage to the limit in the integral. For the summable majorant we take a function $$\widetilde{g}(x):=|t-x|^{-\beta}\sum\limits_{n=1}^\infty|g_{n+1}(x)-g_n(x)|.$$ Let us estimate the $n$-th term: $$|g_{n+1}-g_n|\leq\sum_{Q\in
G^g_{n+1}}|f_n(c_Q)|(\delta^{n+1}\lambda)^\alpha(\phi_{\varepsilon_n
Q})\leq C'\eta'^n (\delta^{n+1}\lambda)^\alpha\sum_{Q\in
G^g_{n+1}}(\phi_{\varepsilon_n Q}).$$ Check summability of the majorant: $$\int_{\mathbb R} |t-x|^{-\beta}|g_{n+1}(x)-g_n(x)|\leq C'\eta'^n
(\delta^{n+1}\lambda)^\alpha\sum_{Q\in G^g_n}\int_{\mathbb
R}|t-x|^{-\beta}\phi_{\varepsilon_n Q}(x)dx\leq$$ $$\leq C'\eta'^n (\delta^{n+1}\lambda)^\alpha\sum_{Q\in
G^g_n}|t-x^*_Q|^{-\beta}\int_{\mathbb R}\phi_{\varepsilon_n
Q}(x)dx\leq$$ $$\leq C'\eta'^n
(\delta^{n+1}\lambda)^\beta\sum_{Q\in G^g_n}|t-x^*_Q|^{-\beta}.
\label{ocmajor}$$ Here $x^*_Q$ denotes a point of the support of $\phi_{\varepsilon_n Q}(x)$, closest to $t$. Using (\[supp\]) and the fact that the distance between two different points $x^*_Q$ and $x^*_{Q'}$ is no less than $\delta^{n+1}$, we get $$\sum\limits_{Q\in G^g_n}|t-x^*_Q|^{-\beta}\leq
2\sum\limits_{k=0}^\infty(\varepsilon_n\delta^{n+1}+k\delta^{n+1})^{-\beta}\leq
2\delta^{(n+1)(-\beta)}\sum\limits_{k=0}^\infty|\varepsilon_n+k|^{-\beta}\leq$$ $$\leq
2\delta^{(n+1)(-\beta)}(\varepsilon_n^{-\beta}+\sum\limits_{k=1}^\infty
k^{-\beta})\leq C\delta^{(n+1)(-\beta)}\varepsilon_n^{-\beta}
\label{ocsum}$$ Substituting (\[ocsum\]) to (\[ocmajor\]) and summing over all $n$, we get $$\int_{\mathbb R}\widetilde{g}(t)dt\leq
C''\lambda^{-\beta}\sum\limits_{n=1}^\infty\eta'^n\varepsilon_n^{-\beta}<+\infty$$ We should now only prove that $g$ satisfies Hölder’s condition with an exponent $\alpha=\beta-1$. In fact this is a property of the potential $U_{\alpha}$ of any bounded function for which it is defined. To prove it, take $t>0$ and write the following estimate: $$\int_{\mathbb R}||x|^{\alpha-1}-|x-t|^{\alpha-1}|dx =
\int_{(-t;2t)}+\int_{{\mathbb R}\backslash(-t;2t)}=:J_1+J_2.$$ Estimate each term: $$J_1 \leq \int_{(-t;2t)}|x|^{\alpha-1} +
\int_{(-t;2t)}|x-t|^{\alpha-1} = \frac2\alpha(1+2^\alpha)t^\alpha;$$ $$J_2 \leq 2\int_{(t;+\infty)}x^{\alpha-1} -(x+t)^{\alpha-1} \leq
2(\alpha-1)\int_{(t;+\infty)}tx^{\alpha-2} \leq 2t^\alpha.$$ When estimating $J_2$ we have used the inequality $|h(x+t)-h(x)|\leq t\sup\limits_{s\in (x,x+t)}|h'(s)|$ for a smooth function $h$. From this estimates we get $$|(U_\alpha f)(t+\delta)-(U_\alpha f)(t)| \leq \sup\limits_{\mathbb
R}|f|\int_{\mathbb
R}(|t+\delta-x|^{\alpha-1}-||t-x|^{\alpha-1}|)dx \leq
C(\alpha)(\sup\limits_{\mathbb R}|f|)\delta^\alpha.$$ The theorem is proven.
[**Remark 1.**]{} Proving the smoothness of $g$ we didn’t use all the information we had about $f$. In fact $f$, besides it is bounded and belongs to the domain of $U_\alpha$, satisfies Hölder’s condition with some exponent $r>0$. Using this and well-known techniques of estimating operators similar to M. Rietz potential (see, for example, [@Zyg]), one can prove that $g$ satisfies Hölder’s condition with an exponent $\beta-1+r$.
[**Remark 2.**]{} The theorem proven in [@Havin] states that for the potentials $U_\alpha$, $-1<\alpha<0$, uniqueness holds if the density $g$ belongs to $C^{1+\varepsilon}$ with some $\varepsilon>0$. Theorem 2 shows that the last condition cannot be replaced by Hölder’s condition with an exponent $-\alpha$. So, there is a gap between the two results, which decreases when $\alpha\rightarrow -1$. If $\alpha\leq-1$, one does not need any supplementary smoothness condition: the mere existence of the potential is sufficient (see, for example, [@HJ]).
For the cases $\alpha=0$ and $\alpha>1$ (except odd integers for which the uniqueness does not hold in any sense), the question whether one can omit the smoothness conditions imposed in [@Havin] remains open.
[**11. Extension of the results to the multidimensional case.**]{} The result of Theorem 1 can be extended to the case of M. Rietz potentials in spaces $\mathbb{R}^d$ for $d>1$. In this case for $\alpha\in(0,d)$ we consider a set of all measurable functions $f$, satisfying the condition $$\label{Cond1Rn}
\int_{\mathbb{R}^d}\frac{|f(x)|}{1+|x|^{d-\alpha}}dx < + \infty$$ (as above, we denote this set $dom\,U_\alpha$). We let $$U_\alpha f := f\ast |x|^{d-\alpha},\; f\in dom\, U_\alpha,$$ where $\ast$ denotes the convolution in $\mathbb{R}^d$. The case of major interest is $d=2$; $\alpha=1$ (Newton’s potential of the charge concentrated in the plane). Note, however, that in case of $d>1$ there is not any analogue of the uniqueness theorem mentioned in the introduction.
There holds the following generalization of Theorem 1:
For all $d\in\mathbb{N}$ and for all $\alpha\in(0,d)$ there exist a nonzero function $f:\mathbb{R}^d\rightarrow \mathbb{R}$, $f\in
dom\,U_\alpha$ and a set $E\subset \mathbb{R}^n$ of positive Lebesgue measure satisfying the condition $f|_E=0$, $U_\alpha
f|_E=0$, and Hölder’s condition with some positive exponent.
The proof of this theorem is quite similar to the one of Theorem 1. We highlight some details differing in the multidimensional case.
We need an operator $W_\alpha$, “the inverse operator” to $U_\alpha$. The precise expression of this operator (see, for example, [@BH page 241]) does not matter for us, the only thing we need is that if we now denote as $\beta$ the number $d+\alpha$, then points 1,2,3,5 of Lemma 1 still hold.
The role of $I$ will be played by the cube $I^d$, and we shall consequently divide it to congruent cubes with the side equal to $\delta^n$. Instead of “the finitizator” $\phi(x)$ we take the function $\phi(|x|)$
The computations made in [@BH page 241] show that lemma 2 still holds in the multidimensional case. Lemmas 3 and 4 can be derived from it quite similarly to the above.
Taking into account that now $\beta=d+\alpha$, the most of computations in the multidimensional case will repeat one-dimensional literally, if we also replace derivative by gradient everywhere. So, because of point 3 for lemma 1 there still hold (\[estimF\]), (\[Fdaleko\]), (\[FPdaleko\]), (\[goodF\]), (\[goodPrime\]) (now for the cube $Q$ with sides parallel to the coordinate axes the symbol $Q[a]$ denotes $Q\backslash
Q'$, where $Q'$ is a cube obtained from $Q$ by homothety with the center $c_Q$ and the dilation factor $a$).
The remaining part of the construction is the same. Note that estimates (\[sigma\]) and (\[sigmaP\]), playing the key role in the proof of lemma 6, and seeming to depend on the dimension, in fact hold in the literally same form.
[**Remark.**]{} Theorem 2 can be extended to the multidimensional case as well: *for $d<\beta<2d$ there exist a nonzero continuous function $g:{\mathbb{R}^d}\rightarrow {\mathbb C}$ and a set $E$ of positive measure, such that if $t\in E$, then $g(t)=0$, $\int_{\mathbb{R}^d} g(x)|t-x|^{-\beta}dx=0$. The last integral converges absolutely for all $t\in E$. The function $g$ satisfies Hölder’s condition with an exponent $\min\{\beta-d;1\}$*[^4]. The only difference in the proof is the estimate (\[ocsum\]), where sums becomes multiple (of order $d$). They still will converge, because now $\beta>d$.
[99]{} Aleksandrov, A. A., Kargaev, P. P. Hardy classes of functions harmonic in half-space, Algebra i Analiz 5:2 (1993), page 1-73(Russian). Belyaev, D. B. and Havin, V. P. On the uncertainty principle for M.Rietz potentials., Arkiv för Mathematik, 39(2001), 229-233 Binder, I., Theorem on correction by harmonic gradients, Algebra i Analiz 5:2 (1993), page 91-107(Russian). Bourgain, J., Wolff, T., A remark on gradients of harmonic functions in dimention $\geq3$, Colloq. Math. 40/41 (1990), no. 1, 25-260. Havin, V. P., Uncertainty principle for one-dimensional M. Rietz potentials Dokl. AN SSSR 264, $N^o$3 (1982), 559-563.(Russian) Havin, V., Jöricke, B., The Uncertainty Principle in Harmonic Analysis, Springer-Verlag, Berlin,1994. Wolff, T., Counterexamples with harmonic gradients in $\mathbb{R}^3$, Essays on Fourier Analysis in Honor of Elias M. Stein (Princeton, N. J., 1991) (Fefferman, C., Fefferman, R., and Wainger, S., eds.) 321-384, Princeton Univ. Press, Princeton, 1995. Zygmund A., Trigonometric Series, Cambridge University press, 2002.
[^1]: I am grateful to S. Smirnov for useful discussions concerning this lemma.
[^2]: Top indices $g$ and $d$ are the first letters of “go” and “delay”
[^3]: Recall that it means that $Q'$ belongs to the set of indices of summation in the definition (\[defr\_n\]) of the corresponding correcting term $r_{n-k-1}$
[^4]: This smoothness estimate can be obtained by a simple method similar to the one used in the proof of Theorem 2. Using the techniques mentioned in the remark after Theorem 2, one can prove the inclusion $g\in C^{\beta-d+r}$ (in case when $\beta-d+r$ is an integer, we understand it as the corresponding Zygmund class)
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---
abstract: 'To assess the strength of nematic fluctuations with a finite wave vector in a two-dimensional metal, we compute the static $d$-wave polarization function for tight-binding electrons on a square lattice. At Van Hove filling and zero temperature the function diverges logarithmically at $\bq = 0$. Away from Van Hove filling the ground state polarization function exhibits finite peaks at finite wave vectors. A nematic instability driven by a sufficiently strong attraction in the $d$-wave charge channel thus leads naturally to a spatially modulated nematic state, with a modulation vector that increases in length with the distance from Van Hove filling. Above Van Hove filling, for a Fermi surface crossing the magnetic Brillouin zone boundary, the modulation vector connects antiferromagnetic hot spots with collinear Fermi velocities.'
author:
- Tobias Holder
- Walter Metzner
title: Incommensurate nematic fluctuations in two dimensional metals
---
Introduction
============
Metallic two-dimensional electron systems frequently exhibit multiple enhanced fluctuations in the charge, magnetic and pairing channels, which lead to a variety of competing instabilities. In the last decade, nematic order and nematic fluctuations in two-dimensional metals have attracted increasing interest. [@fradkin10] In a homogeneous nematic state an orientational symmetry of the system is spontaneously broken, without breaking the translation invariance. A nematic state can be formed by partial melting of stripe order in a doped antiferromagnetic Mott insulator.[@kivelson98] Alternatively, a nematic state can be obtained from a Pomeranchuk [@pomeranchuk59] instability generated by forward scattering interactions in a normal metal.[@yamase00; @halboth00] On a square lattice, the most natural candidate for a Pomeranchuk instability has a $d_{x^2-y^2}$ symmetry, which breaks the equivalence between $x$- and $y$-directions.
A direct experimental signature of nematic order is a pronounced in-plane anisotropy in transport or spectroscopic measurements, in the absence of additional Bragg peaks indicating a broken translation invariance.[@kivelson03] Evidence for nematic order with a $d_{x^2-y^2}$ symmetry has been observed in several strongly interacting electron compounds. A nematic phase with a sharp phase boundary has been established in a series of experiments on ultrapure $\rm Sr_3 Ru_2 O_7$ crystals in a strong magnetic field.[@ruthenate] Nematic order has also been observed in the high temperature superconductor $\rm Y Ba_2 Cu_3 O_y$ in transport experiments [@ando02; @daou10] and neutron scattering.[@hinkov] Due to the slight orthorhombicity of the $\rm CuO_2$ planes one cannot expect a sharp nematic phase transition in $\rm Y Ba_2 Cu_3 O_y$. However, the strong temperature dependence of the observed in-plane anisotropy indicates that the system develops an intrinsic electronic nematicity, which enhances the in-plane anisotropy imposed by the structure. [@yamase06; @yamase09]
Nematic fluctuations close to a continuous nematic quantum phase transition induce non-Fermi liquid behavior with a strongly momentum dependent decay rate of electronic excitations. [@oganesyan01; @metzner03; @dellanna06; @dellanna07] Mean-field theories for nematic transitions driven by forward scattering interactions in two dimensions indicate that the transition is typically first order at low temperatures, [@kee03; @khavkine04; @yamase05] such that a quantum critical point remains elusive. However, order parameter fluctuations can change the order of the transition, in favor of a continuous transition even at zero temperature.[@jakubczyk09; @yamase11]
So far, only homogeneous nematic states have been considered. However, Metlitski and Sachdev [@metlitski10a; @metlitski10b] recently found a tendency toward formation of a [*modulated*]{} nematic state in a two-dimensional metal with strong antiferromagnetic spin-density wave fluctuations. In that state the nematic order oscillates spatially across the crystal, with a small and generally incommensurate wave vector that points along the Brillouin zone diagonal and increases in length with the distance of the Fermi surface from the Van Hove points. The modulated nematic order occurs as a secondary instability generated by the antiferromagnetic spin fluctuations, and is related by an emergent pseudospin symmetry to the familiar $d$-wave pairing instability.[@metlitski10b]
In this paper, we investigate the possibility of a modulated nematic state from a different starting point. We consider a model of tight-binding electrons on a square lattice with an interaction that has an attractive $d$-wave component for forward scattering in the charge channel. Within a random phase approximation (RPA), we analyze at which wave vector the modulated nematic fluctuations are maximal. For electron densities above Van Hove filling we find the same diagonal wave vector as Metlitski and Sachdev in their spin-density wave model.[@metlitski10a; @metlitski10b] Below Van Hove filling a modulation parallel to the $x$- or $y$-axis can be favored.
In realistic models nematic fluctuations compete with other potential instabilities such as magnetism and superconductivity. Here we do not address this competition. Our model interaction is restricted to a specific $d$-wave charge channel. The competition of modulated nematic fluctuations with antiferromagnetism, superconductivity and charge density wave order in the two-dimensional (extended) Hubbard model has been the subject of a recent renormalization group analysis.[@husemann12]
The paper is structured as follows. In Sec. II we define our model and introduce the $d$-wave charge susceptibility. In Sec. III we analyze the momentum dependence of the static $d$-wave charge susceptibility. We identify the peaks which determine the wave vector of the leading instability in the ground state, and we compute the singular momentum dependence around the peaks. We also determine the shift of the peaks at low finite temperature. We finally conclude in Sec. IV.
Model and susceptibility
========================
We consider a one-band model of electrons on a square lattice with a dispersion $\eps_{\bk}$ and an interaction of the form [@metzner03] $$\label{H_I}
H_I =
\frac{1}{2L} \sum_{\bq} g(\bq) \, n_d(\bq) \, n_d(-\bq) \; ,$$ where $L$ is the number of lattice sites, and $$\label{n_d}
n_d(\bq) = \sum_{\bk} \sum_{\sg=\up,\down} d_{\bk} \,
c_{\bk-\bq/2,\sg}^{\dag} c_{\bk+\bq/2,\sg}$$ is a $d$-wave density fluctuation operator; $d_{\bk}$ is a form factor with $d_{x^2-y^2}$ symmetry such as $d_{\bk} = \cos k_x - \cos k_y$. The coupling function $g(\bq)$ is negative and may depend on the momentum transfer $\bq$. An interaction of the form $H_I$ appears in the $d$-wave charge channel of the two-particle vertex in microscopic models such as the Hubbard or $t$-$J$ model.[@yamase00; @halboth00] Note that the fermionic operators in Eq. (\[n\_d\]) are taken at two momenta $\bk \pm \bq/2$ situated symmetrically around the momentum $\bk$ appearing in the form factor $d_{\bk}$. Hence, for $\bq = \bQ = (\pi,\pi)$, the operator $n_d(\bq)$ differs significantly from the socalled $d$-density wave order parameter $\sum_{\bk,\sg} d_{\bk} c_{\bk + \bQ,\sg}^{\dag} c_{\bk,\sg}$. [@chakravarty01]
For the kinetic energy we use a tight-binding dispersion of the form $$\begin{aligned}
\label{eps_k}
\eps_{\bk} &=& -2t (\cos k_x + \cos k_y) - 4t' \cos k_x \cos k_y\nonumber\\
&&- 2t'' (\cos 2k_x + \cos 2k_y) \; ,\end{aligned}$$ where $t$, $t'$, and $t''$ are the hopping amplitudes between nearest, next-to-nearest, and third-nearest neighbors on the square lattice, respectively. We assume $t > 0$, $t' < 0$ and $t'' \geq 0$ as is adequate for cuprate superconductors. The dispersion has saddle points at $(\pi,0)$ and $(0,\pi)$ for $t'' \leq {t''}^* = (t + 2t')/4$. For $t'' \geq {t''}^*$ the saddle points are shifted to $\pm \big(\pi - \arccos\frac{{t''}^*}{t''},0 \big)$ and $\pm \big(0,\pi - \arccos\frac{{t''}^*}{t''} \big)$. A special situation (Van Hove filling) arises when the Fermi surface touches the saddle points, so that the density of states diverges logarithmically at the Fermi level. The chemical potential corresponding to Van Hove filling is given by $$\label{mu_vh}
\mu_{\rm vh} = \left\{ \begin{array}{cc}
4t' - 4t'' &
\quad \mbox{for} \quad t'' \leq {t''}^* \\
\left( \frac{1}{4} t^2 + tt' + {t'}^2 - 2tt'' \right)/t'' &
\quad \mbox{for} \quad t'' \geq {t''}^*
\end{array} \right. \; .$$ The amount of $d$-wave charge fluctuations can be quantified by the dynamical $d$-wave charge (density) susceptibility [@dellanna06; @yamase04] $$\label{N_d}
N_d(\bq,\om) = - i \int_0^{\infty} dt \, e^{i\om t} \,
\bra [ n_d(\bq,t), n_d(-\bq,0) ] \ket \; ,$$ where $n_d(\bq,t)$ is the time dependent operator corresponding to $n_d(\bq)$ in the Heisenberg picture. Within RPA, the $d$-wave charge susceptibility in the model defined above is given by $$\label{RPA}
N_d(\bq,\om) =
\frac{2\Pi_d^0(\bq,\om)}{1 - 2 g(\bq) \Pi_d^0(\bq,\om)} \; ,$$ with the bare $d$-wave polarization function (particle-hole bubble) $$\label{Pi_d^0}
\Pi_d^0(\bq,\om) = - \int \frac{d^2p}{(2\pi)^2} \,
\frac{f(\xi_{\bp+\bq/2}) - f(\xi_{\bp-\bq/2})}
{\om + i0^+ - (\eps_{\bp+\bq/2} - \eps_{\bp-\bq/2})} \,
d_{\bp}^2 \; .$$ Here and in the following $f$ is the Fermi function and $\xi_{\bk} = \eps_{\bk} - \mu$. The factor 2 in Eq. (\[RPA\]) is due to the spin sum.
An instability toward an ordered state with an order parameter $\bra n_d(\bq^*) \ket \neq 0$ is signalled by a divergence of the static susceptibility $N_d(\bq,0)$ for a certain wave vector $\bq^*$. Within RPA the instability is reached once $2g(\bq^*) \Pi_d^0(\bq^*,0) = 1$ while $2g(\bq) \Pi_d^0(\bq,0) < 1$ for $\bq \neq \bq^*$. For $\bq^* = \b0$ the ordered state is a homogeneous nematic with unbroken translation invariance. The case $\bq^* \neq \b0$ leads to a modulated nematic state with a modulation vector $\bq^*$. In previous studies of the above model it was always assumed that the coupling function $g(\bq)$ is sufficiently strongly peaked at $\bq = \b0$ such that the leading instability is at $\bq^* = \b0$. In the present paper we consider the case where $g(\bq)$ exhibits no or only a weak dependence on $\bq$ in some region around $\bq = \b0$. The leading instability in the $d$-wave charge channel then occurs at wave vectors at which $\Pi_d^0(\bq,0)$ has a peak. In the following we therefore study the structure of the $d$-wave particle-hole bubble, paying particular attention to its extrema.
Static particle-hole bubble
===========================
In this section we analyze the momentum dependence of the $d$-wave particle-hole bubble $\Pi_d^0$ at zero frequency. We will also consider the usual ($s$-wave) particle-hole bubble $\Pi^0$ for comparison. The latter is given by Eq. (\[Pi\_d\^0\]) without the $d$-wave form factor. The structure depends significantly on the electron density which is determined by the chemical potential. There are three qualitatively different cases: below, at, and above Van Hove filling. We will first analyze $\Pi_d^0(\bq,0)$ in the ground state, and treat finite temperatures subsequently.
Global structure and peaks
--------------------------
In Fig. 1 we show $\Pi_d^0(\bq,0)$ in the ground state as a function of $\bq$, for hopping parameters $t = 1$, $t' = - 1/4$, and $t'' = 0$, with three different choices for the electron density below, at, and above Van Hove filling, corresponding to $\mu = \mu_{\rm vh} - 0.01$, $\mu = \mu_{\rm vh}$, and $\mu = \mu_{\rm vh} + 0.05$, respectively. In all cases there are enhanced (negative) values along the lines in the Brillouin zone given by the condition
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$$\label{lines}
\xi_{(\bq + \bG)/2} = 0 \; ,$$
where $\bG$ is a reciprocal lattice vector. Geometrically these lines can be constructed by expanding the Fermi surface by a factor two and then backfolding pieces outside the first Brillouin zone into the first zone, as illustrated in Fig. 2. The momentum transfers $\bq$ satisfying the condition (\[lines\]) are special in that they connect Fermi points with parallel tangents. Eq. (\[lines\]) is the lattice generalization of the condition $|\bq| = 2k_F$ in a continuum system with a circular Fermi surface. With this in mind, one may call such momentum transfers [*$2k_F$-momenta*]{} on the lattice, too, although the Fermi momenta in a lattice system do not have a common modulus $k_F$.[@altshuler95] In the following we will refer to the lines defined by Eq. (\[lines\]) as [*$2k_F$-lines*]{}.
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In Fig. 3 we show the $s$-wave particle-hole bubble $\Pi^0(\bq,0)$ for comparison. There is again some structure tracing the lines given by $\xi_{(\bq + \bG)/2}$. However, the strongest weight now lies in a region near $(\pi,\pi)$ in the Brillouin zone, which is generated by particle-hole excitations connecting the saddle point regions near $(\pi,0)$ and $(0,\pi)$. In the $d$-wave bubble these contributions are suppressed by the $d$-wave form factor, since $d_{\bp} = 0$ for $\bp = (\pm\frac{\pi}{2},\pm\frac{\pi}{2})$. Note that this suppression hinges upon the symmetric choice of the momentum in $d_{\bp}$ at the center of the fermionic momenta $\bp \pm \bq/2$. To obtain the polarization bubble describing $d$-density wave fluctuations as defined by Chakravarty et al.[@chakravarty01] one would have to replace $\bp - \bq/2$ by $\bp$ and $\bp + \bq/2$ by $\bp + \bq$ in Eq. (\[Pi\_d\^0\]). For $\bq = \bQ = (\pi,\pi)$ one then picks up large contributions from the saddle point region.
Incommensurate charge density waves with conventional $s$-wave symmetry have also been discussed for two-dimensional electron systems, especially in the context of cuprate superconductors.[@castellani95] In that scenario, the incommensurate modulation vector is however determined by a competition between Coulomb energy and phase separation tendencies, not by peaks in the polarization function $\Pi^0(\bq,0)$. Peaks in $\Pi^0(\bq,0)$ at small incommensurate wave vectors occurring for large doping (far from half-filling) can be associated with modulated [*ferromagnetic*]{} fluctuations, for example, in a two-dimensional Hubbard model. [@igoshev10; @igoshev11]
At Van Hove filling, the $d$-wave particle-hole bubble is strongly peaked at $\bq = \b0$, which is a crossing point of eight lines satisfying $\xi_{(\bq + \bG)/2} = 0$. The leading $d$-wave charge instability is therefore a homogeneous nematic one in this case.
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Below but close to Van Hove filling, the highest weight is obtained at those points in $\bq$-space where the lines given by Eq. (\[lines\]) are closest to the origin, that is, on the $q_x$- and $q_y$-axes. The leading instability is thus a modulated nematic with a small modulation vector along one of the crystal axes, or a superposition of such modulations. Let us denote the distance of the points of highest weight from the origin by $q_a$. Solving $\xi_{(\bq + \bG)/2} = 0$ for $\bq = (q_a,0)$ and $\bG = - (2\pi,0)$, one obtains $$\begin{aligned}
\label{q_a}
q_a &=& 2 \arccos \left(
\frac{t + 2t' - \sqrt{(t + 2t')^2 - 4t''(2t + \mu)}}{4t''}
\right)
\nonumber \\
&=& 2 \arccos \left( \frac{2t + \mu}{2t + 4t'} \right)
\quad \mbox{for} \quad t'' = 0
\; .\end{aligned}$$ Further away from Van Hove filling, the global maxima of $|\Pi_d^0(\bq,0)|$ can be situated at inflection points of the $2k_F$-lines, which are inherited from inflection points of the Fermi surface. Explicit expressions for these points are rather lengthy so that we refrain from reporting them. They are not linked to any symmetry axis of the lattice.
Above Van Hove filling, the distance of the lines Eq. (\[lines\]) from the origin is given by $$\begin{aligned}
\label{q'_a}
q_a &=& 2 \arccos \left(
\frac{-t + 2t' + \sqrt{(t - 2t')^2 - 4t''(2t - \mu)}}{4t''}
\right)
\nonumber \\
&=& 2 \arccos \left( \frac{2t - \mu}{2t - 4t'} \right)
\quad \mbox{for} \quad t'' = 0
\; .\end{aligned}$$ However, the points $(q_a,0)$ etc. are not the global extrema. Above Van Hove filling, the lines Eq. (\[lines\]) intersect on the diagonals in the Brillouin zone, and the heighest weight is reached at these intersection points. The leading instability is thus a modulated nematic with a diagonal modulation vector $\bq^* = (\pm q_d, \pm q_d)$, or a superposition of such modulations. Solving $\xi_{(\bq + \bG)/2} = 0$ for $\bq = (q_d,q_d)$ and $\bG = - (2\pi,0)$ or $\bG = - (0,2\pi)$, one obtains $$\begin{aligned}
\label{q_d}
q_d &=& 2 \arccos \sqrt{\frac{\mu - 4t''}{4t' - 8t''}}
\nonumber \\
&=& 2 \arccos \sqrt{\frac{\mu}{4t'}}
\quad \mbox{for} \quad t'' = 0
\; .\end{aligned}$$ Close to Van Hove filling, $q_d$ is small. Note that $q_d$ is defined only for $\mu_{\rm vh} \leq \mu \leq 4t''$. For $\mu > 4t''$, no intersection points exist.
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The wave vector $\bq_d = (q_d,q_d)$ connects antiferromagnetic [*hot spots*]{} on the Fermi surface (see Fig. 4). This is because these hot spots lie on the magnetic Brillouin zone boundary and have parallel Fermi surface tangents, and are therefore connected by a diagonal “$2k_F$-vector”. The length of $\bq_d$ is thus twice the distance between the hot spots and the nearest $M$-point $(\pi,0)$ or $(0,\pi)$. An instability toward a modulated nematic state with the same modulation vector $\bq_d$ has been obtained in a recent study of secondary instabilities generated by antiferromagnetic fluctuations.[@metlitski10a; @metlitski10b] In that analysis, $\bq_d$ arises naturally as a vector connecting hot spots since these are the points where (commensurate) antiferromagnetic fluctuations couple most strongly to electronic excitations at the Fermi surface. It is remarkable that the same modulation vector emerges as a peak in the $d$-wave particle-hole bubble, where it is selected by the Fermi surface geometry without any relation to antiferromagnetism.
Expansion around $\bq = \b0$
----------------------------
Away from Van Hove filling, there is a region around $\bq = \b0$ where $\Pi_d^0(\bq,0)$ is relatively flat and isotropic (see Fig. 1). This region is clearly limited by the $2k_F$-lines $\xi_{(\bq + \bG)/2} = 0$. For $|\bq| \ll q_a$ one may expand around $\bq = 0$. The leading terms are [@dellanna06] $$\label{small_q}
\Pi_d^0(\bq,0) = a + c |\bq|^2 + \cO(|\bq|^4)
\; .$$ The first coefficient is given by a weighted density of states at the Fermi level, $$\label{a}
a = - N_{d^2}(\mu) = - \int \frac{d^2k}{(2\pi)^2} \,
d_{\bk}^2 \delta(\mu - \eps_{\bk}) \; ,$$ with a minus sign, so that $a$ is always negative. The second coefficient can be written in the form [@dellanna06] $$\label{c}
c = \frac{1}{16} N'_{d^2\Delta\eps}(\mu) -
\frac{1}{48} N''_{d^2v^2}(\mu) \; .$$ Here $$N_{d^2\Delta\eps}(\eps) = \int \frac{d^2k}{(2\pi)^2} \,
d_{\bk}^2 \Delta\eps_{\bk} \, \delta(\eps - \eps_{\bk})
\; ,$$ with $\Delta = \partial_{k_x}^2 + \partial_{k_y}^2$, and $$N_{d^2 v^2}(\eps) = \int \frac{d^2k}{(2\pi)^2} \,
d_{\bk}^2 v_{\bk}^2 \, \delta(\eps - \eps_{\bk})
\; ,$$ with $v_{\bk} = |\nabla\eps_{\bk}|$, and the primes denote derivatives. Near Van Hove filling, the coefficient $c$ is dominated by the first term in Eq. (\[c\]), since $N_{d^2\Delta\eps}(\mu)$ diverges for $\mu \to \mu_{\rm vh}$, while $N_{d^2 v^2}(\mu)$ remains finite. The sign of $c$ is typically positive for $\mu < \mu_{\rm vh}$, and negative for $\mu > \mu_{\rm vh}$.
Hence, away from Van Hove filling, the interaction term $H_I$ from Eq. (\[H\_I\]) generates a homogeneous nematic instability only if $g(\bq)$ decays sufficiently rapidly at finite $\bq$. That is, below Van Hove filling, $g(\bq_a,0) \Pi_d^0(\bq_a,0)$ with $\bq_a = (q_a,0)$ has to be smaller than $g(\b0) \Pi_d^0(\b0,0)$. Furthermore, above Van Hove filling, $g(\bq_d,0) \Pi_d^0(\bq_d,0)$ with $\bq_d = (q_d,q_d)$ needs to be smaller than $g(\b0) \Pi_d^0(\b0,0)$, and the curvature of $g(\bq)$ around the origin has to compensate for the negative curvature of $\Pi_d^0(\bq,0)$. The RPA analysis therefore indicates that an instability toward a modulated nematic state is more natural. On the other hand, fluctuations beyond RPA may smear out the peaks resulting from the Fermi surface geometry, which could favor a homogeneous nematic state over a modulated one. In principle, such fluctuations may also wipe out a nematic instability completely.[@yamase11]
Shape of ridges and expansion around $\bq^*$
--------------------------------------------
The maxima of $|\Pi_d^0(\bq,0)|$ lie on ridges following the $2k_F$-lines given by $\xi_{(\bq+\bG)/2} = 0$ in the Brillouin zone. The height of the ridges evolves regularly along these lines, with few exceptions. The shape of the ridge determined by the $\bq$-dependence of $\Pi_d^0(\bq,0)$ perpendicular to the lines is generically singular. To discuss the form of this singularity, we introduce a coordinate $q_r$ describing the oriented distance from the closest $2k_F$-line, see Fig. 5. We denote points on the $2k_F$-line by $\bq_{2k_F}$, and the radius of curvature at $\bq_{2k_F}$ by $2m_F v_F$, where $v_F$ is the electron velocity at the corresponding point $\bk_F$ on the Fermi surface (the radius of curvature of the Fermi surface at $\bk_F$ is $m_F v_F$). We consider the generic case where the curvature at $\bq_{2k_F}$ is finite. Exceptional cases with vanishing curvature exist at Van Hove points and at inflection points of non-convex Fermi surfaces. The coordinate $q_r$ is defined positively on the “outer” side of the $2k_F$-line, that is, the side containing the tangent to the line at $\bq_{2k_F}$, and negatively on the “inner” side.
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Before describing the $2k_F$-singularity of $\Pi_d^0(\bq,0)$, it is instructive to recall the behavior of the $s$-wave bubble $\Pi^0(\bq,0)$ for a quadratic dispersion $\eps_{\bk} = |\bk|^2/2m$, which can be expressed in term of elementary functions: [@stern67] $$\label{stern}
\Pi^0(\bq,0) = - \frac{m}{2\pi} +
\Theta(|\bq| - 2k_F) \frac{m k_F}{\pi|\bq|}
\sqrt{\left( \frac{|\bq|}{2k_F} \right)^2 - 1}
\; .$$ Note that $\Pi^0(\bq,0)$ is constant for all momenta satisfying $|\bq| \leq 2k_F$. At the $2k_F$-line, here given by $|\bq| = 2k_F$, the $s$-wave bubble has a square-root singularity with infinite slope on the outer side. For small $q_r = |\bq| - 2k_F$, the bubble has the form $$\label{stern_exp}
\Pi^0(\bq,0) = - \frac{m}{2\pi} +
\Theta(q_r) \frac{\sqrt{m q_r/v_F}}{2\pi} + \cO(q_r^{3/2}) \; .$$ The momentum dependence of the $d$-wave particle-hole bubble near a $2k_F$-line for two-dimensional lattice electrons has a similar form, $$\label{near2k_F}
\Pi_d^0(\bq,0) = \Pi_d^0(\bq_{2k_F},0) +
\Theta(q_r) \frac{d_{\bk_F}^2}{2\pi} \sqrt{m_F q_r/v_F} +
b_F q_r \; ,$$ where $\bk_F$ is the Fermi momentum corresponding to the point $\bq_{2k_F}$ on the $2k_F$-line, $v_F$ and $m_F v_F$ are the the Fermi velocity and the Fermi surface curvature at $\bk_F$, respectively, and $b_F$ is another constant. The square-root singularity in this expression is essentially the same as in Eq. (\[stern\_exp\]). The absence of a term linear in $q_r$ in Eq. (\[stern\_exp\]) is a pecularity of $\Pi^0$ for a quadratic dispersion relation in two dimensions. Usually $b_F$ is a negative number, such that $|\Pi_d^0(\bq,0)|$ decreases with increasing $|q_r|$ in both directions.
The momentum dependence perpendicular to the ridge in Eq. (\[near2k\_F\]) is closely analogous to the corresponding behavior of the s-wave bubble $\Pi^0$ in a two-dimensional lattice system,[@altshuler95] the only difference being the factor $d_{\bk_F}^2$ for the d-wave case. Altshuler et al.[@altshuler95] also specified the momentum dependence for small tangential shifts, in analogy to the isotropic case. That dependence, however, is generally modified by a non-universal variation of the bubble along the ridge.
At points of zero curvature, in particular inflection points, $m_F$ diverges and the expression (\[near2k\_F\]) is not applicable. For $t'/t < 0$ such points exist typically below Van Hove filling, and they can host the global extrema of $\Pi_d^0(\bq,0)$. Above Van Hove filling the global extrema are situated at the crossing points $\bq^* = (\pm q_d, \pm q_d)$ of two $2k_F$-lines on the Brillouin zone diagonal (see Sec. III.A). The momentum dependence of $\Pi_d^0(\bq,0)$ near $\bq^*$ is then given by a superposition of two ridges of the form (\[near2k\_F\]), which are mirror symmetric with respect to the diagonal. The momentum dependence is linear in the edge bounded by the inner side of both ridges, while it is dominated by a square-root singularity with infinite slope in the other three edges formed by the crossing ridges.
Finite temperature
------------------
At finite temperatures the singularities of the particle-hole bubble are smoothed and the peaks are generally shifted with respect to their ground state position. In this section we analyze these effects at low finite $T$.
The particle-hole bubble at $T>0$ can be written as a convolution of the bubble at $T=0$ with the energy-derivative of the Fermi function: $$\label{convol}
\Pi_d^0(\bq;T,\mu) = \int_{-\infty}^{\infty} d\mu' \,
h(\mu - \mu') \, \Pi_d^0(\bq;0,\mu') \; ,$$ where $$h(\xi) = -f'(\xi) =
\frac{1}{4T \cosh^2\left(\frac{\xi}{2T} \right)} \, .$$ Note that we have suppressed the frequency variable ($\om = 0$) in the argument of $\Pi_d^0$ while making the dependences on $T$ and $\mu$ explicit. Note also that $\int d\xi \, h(\xi) = 1$.
We now determine how the ridges along the $2k_F$-lines are shifted and smoothed at $T>0$. To this end, we parametrize the momentum dependence by the oriented distance $q_r$ from the $2k_F$-line defined at fixed $\mu$ and $T=0$ as in the preceding section. We denote the shift of the $2k_F$-line near $\bq_{2k_F}$ at $T=0$ corresponding to a chemical potential $\mu' \neq \mu$ by $q_r^0(\mu')$. Eq. (\[near2k\_F\]) then yields $$\begin{aligned}
\label{Pimu'}
\Pi_d^0(\bq;0,\mu') &=& \Pi_d^0(\bq'_{2k_F};0,\mu') +
b_F [q_r - q_r^0(\mu')] \nonumber\\&&
+a_F \Theta[q_r - q_r^0(\mu')] \sqrt{q_r - q_r^0(\mu')} \; ,\end{aligned}$$ where $a_F = d_{\bk_F}^2 m_F^{1/2}/(2\pi \, v_F^{1/2})$. Neglecting the weak and regular $\mu$-dependence of the height of the ridge one can approximate $\Pi_d^0(\bq'_{2k_F};0,\mu')$ on the right hand side by $\Pi_d^0(\bq_{2k_F};0,\mu)$. Inserting Eq. (\[Pimu’\]) into Eq. (\[convol\]), and using the antisymmetry of $q_r^0(\mu')$ around $\mu$, that is,$q_r^0(\mu + \delta\mu') = - q_r^0(\mu - \delta\mu')$ for small $\delta\mu'$, one obtains $$\begin{aligned}
\label{PiT}
\Pi_d^0(\bq;T,\mu) &=& \Pi_d^0(\bq_{2k_F};0,\mu) + b_F q_r \nonumber\\&&
+a_F \int_{-\infty}^{\infty} d\mu' \, h(\mu - \mu') \,\nonumber\\&&\times
\Theta[q_r - q_r^0(\mu')] \sqrt{q_r - q_r^0(\mu')} \; .\end{aligned}$$ The shift of the ridge at $T > 0$ is given by the solution $q_r^p$ of the equation $\partial_{q_r} \Pi_d^0(\bq;T,\mu) = 0$, that is, $$\label{shift1}
b_F + \frac{a_F}{2}
\int_{-\infty}^{\infty} d\mu' \, h(\mu - \mu') \,
\frac{\Theta[q_r^p - q_r^0(\mu')]}{\sqrt{q_r^p - q_r^0(\mu')}}
= 0 \; .$$ A qualitative contemplation of Eq. (\[PiT\]) reveals that $q_r^p$ is negative. For $\mu'$ near $\mu$, $q_r^0(\mu')$ is a monotonic function of $\mu'$. We denote its inverse function by $\mu'(q_r)$ and linearize $q_r^p - q_r^0(\mu') \approx D [\mu'(q_r^p) - \mu']$, where $D = \tfrac{\partial q_r^0}{\partial\mu'}|_{\mu' = \mu}$. In case that $q_r^0(\mu')$ increases with $\mu'$, Eq. (\[shift1\]) can then be written as $$\label{shift2}
b_F + \frac{a_F}{2 D^{1/2}}
\int_{-\infty}^{\mu'(q_r^p)} d\mu' \,
\frac{h(\mu - \mu')}{\sqrt{\mu'(q_r^p) - \mu'}} = 0 \; .$$ Introducing the variable $\delta\mu' = \mu' - \mu$, and substituting $\delta\mu' = \delta\mu'(q_r^p) u$, where $\delta\mu'(q_r^p) < 0$, the integral in Eq. (\[shift2\]) can be written as $$\begin{aligned}
\label{int1}
&\int_{-\infty}^{\mu'(q_r^p)} d\mu' \,
\frac{h(\mu - \mu')}{\sqrt{\mu'(q_r^p) - \mu'}} \nonumber\\
&\quad = \sqrt{-\delta\mu'(q_r^p)} \int_1^{\infty} \frac {du}{\sqrt{u-1}} \,
\frac{1}{4T \cosh^2 \frac{\delta\mu'(q_r^p) u}{2T}} \; .\end{aligned}$$ One can see that a solution of Eq. (\[shift2\]) requires $|\delta\mu'(q_r^p)| \gg T$ for small $T$. Therefore, we can approximate $\cosh^2 \tfrac{\delta\mu'(q_r^p) u}{2T} \approx
\frac{1}{4} e^{|\delta\mu'(q_r^p)| \, u/T}$. The remaining integral is elementary and yields $$\label{int2}
\int_{-\infty}^{\mu'(q_r^p)} d\mu' \,
\frac{h(\mu - \mu')}{\sqrt{\mu'(q_r^p) - \mu'}} =
\sqrt{\frac{\pi}{T}} e^{-|\delta\mu'(q_r^p)|/T} \; .$$ Inserting this in Eq. (\[shift2\]) yields $$\label{deltamu'}
|\delta\mu'(q_r^p)| =
\frac{1}{2} \, T \ln \frac{T_0}{T} \; .$$ where $T_0 = \pi a_F^2/(4 b_F^2 D)$. Note that indeed $|\delta\mu'(q_r^p)| \gg T$ for small $T$, that is, for $T \ll T_0$. For the shift $q_r^p$ one thus obtains $$q_r^p = - \frac{D}{2} \, T \ln \frac{T_0}{T} \; .$$ Replacing $D$ by $|D|$, the last two equations are valid also in the case where $q_r^0(\mu')$ decreases with $\mu'$. In summary, the ridge is shifted toward the inner side of the $2k_F$-line by an amount of order $T |\log T|$. Hence, the peaks of $\Pi_d^0(\bq,0)$ at $\bq^* = (\pm q_a,0)$ and $(0,\pm q_a)$ below Van Hove filling, and at $\bq^* = (\pm q_d,\pm q_d)$ above Van Hove filling, are also subject to a shift of order $T |\log T|$.
To quantify the smoothing of the peak at the $2k_F$-line at finite temperature, we evaluate $\partial_{q_r}^2 \Pi_d^0$ at $q_r^p$. Substituting $q_r - q_r^0(\mu') = D[\mu'(q_r) - \mu']$ and performing a partial integration, the second derivative of $\Pi_d^0$ with respect to $q_r$ can be written as $$\partial_{q_r}^2 \Pi_d^0 = \frac{a_F}{2D^{3/2}}
\int_{-\infty}^{\infty} d\mu' \, h'(\mu'-\mu) \,
\frac{\Theta[\mu'(q_r) - \mu']}{\sqrt{\mu'(q_r) - \mu'}}
\; .$$ For large $|\delta\mu'(q_r)|/T$ one can approximate $h'(\mu'-\mu) \approx \sgn(D) \, T^{-2} e^{|\delta\mu'|/T}$ and perform the integral explicitly, to obtain $$\partial_{q_r}^2 \Pi_d^0 =
\frac{a_F \sqrt{\pi}}{2|D|^{3/2} T^{3/2}} \,
e^{-|\delta\mu'(q_r)|/T} \; .$$ Inserting $\delta\mu'(q_r^p)$ from Eq. (\[deltamu’\]), one obtains the curvature at the shifted peak position $$\left . \partial_{q_r}^2 \Pi_d^0 \right|_{q_r = q_r^p} =
\frac{a_F \sqrt{\pi}}{2|D|^{3/2} \sqrt{T_0}} \, \frac{1}{T} =
\frac{|b_F|}{|D|} \, \frac{1}{T} \; .$$ The last two equations are valid for any sign of $D$. Hence, the radius of curvature of the peak is proportional to $T$ at small temperatures, with a remarkably simple prefactor.
Conclusion
==========
We have analyzed the strength of nematic fluctuations with a finite wave vector in a two-dimensional metal. To this end we have computed the bare static $d$-wave polarization function $\Pi_d^0(\bq,0)$ as a function of the wave vector $\bq$ for electrons with a tight-binding dispersion on a square lattice. Peaks in $\Pi_d^0(\bq,0)$ indicate at which wave vectors a (modulated) nematic instability occurs in presence of a sufficiently strong attraction in the $d$-wave charge channel.
At Van Hove filling, $\Pi_d^0(\bq,0)$ is strongly peaked at $\bq = 0$, so that the leading nematic instability is homogeneous in this case. Below and close to Van Hove filling, the largest peaks are on the $q_x$- and $q_y$-axes, leading to a modulated nematic state with a small modulation vector along one of the crystal axes. Above Van Hove filling, the largest peaks of $\Pi_d^0(\bq,0)$ are situated at diagonal wave vectors $\bq^* = (\pm q_d,\pm q_d)$, so that the dominant instability leads to a spatially modulated nematic state with a diagonal modulation vector. The same modulated nematic state has been found by Metlitski and Sachdev [@metlitski10a; @metlitski10b] in a recent study of secondary instabilities generated by antiferromagnetic fluctuations in a two-dimensional metal. In that context the wave vector $\bq^*$ is favored because it connects intersections of the Fermi surface with the antiferromagnetic Brillouin zone boundary (hot spots). Remarkably, the peak at the same $\bq^*$ in the $d$-wave polarization function is determined purely by the Fermi surface geometry, without any influence from antiferromagnetic fluctuations.
In all cases the peaks of $\Pi_d^0(\bq,0)$ lie on lines defined by the condition $\eps_{(\bq + \bG)/2} = \mu$, where $\bG$ is a reciprocal lattice vector, which is the lattice analogue of the condition $|\bq| = 2k_F$ in a continuum system. Generically, the momentum dependence of $\Pi_d^0(\bq,0)$ exhibits a square root singularity at these lines, which therefore characterizes also the behavior around the peaks of $\Pi_d^0(\bq,0)$. At low finite temperatures the peaks in the polarazation function are smoothed and shifted by an amount of order $T |\log T|$.
In view of the above results it seems worthwhile to search for modulated nematic instabilities in two-dimensional Hubbard-type models. In a recent functional renormalization group study of the one-band Hubbard model, a modulated nematic instability was found to be typically favorable compared to a homogeneous one, but in any case weaker than antiferromagnetism or $d$-wave superconductivity. Adding a nearest-neighbor repulsion strengthens the nematic fluctuations.[@husemann12] There is more room for nematic instabilities in multi-band systems.[@fischer11] A search for modulated nematic states in such systems would therefore be particularly promising.
We are very grateful to C. Husemann, M. Metlitski, and H. Yamase for valuable discussions.
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abstract: 'Motivated by recent photoemission and pump-probe experiments, we report determinant Quantum Monte Carlo simulations of hybridization fluctuations in the half-filled periodic Anderson model. A tentative phase diagram is constructed based solely on hybridization fluctuation spectra and reveals a crossover regime between an unhybridized selective Mott state and a fully hybridized Kondo insulating state. This intermediate phase exhibits nonlocal hybridization fluctuations and consequentially the so-called “band bending" and a direct hybridization gap as observed in angle-resolved photoemission spectroscopy and optical conductivity. This connects the band bending with the nonlocal hybridization fluctuations as proposed in latest ultrafast optical pump-probe experiment. The Kondo insulating state is only established at lower temperatures with the development of sufficiently strong inter-site hybridization correlations. Our work suggests a unified picture for interpreting recent photoemission, pump-probe, and optical observations and provides numerical evidences for the importance of hybridization fluctuations in heavy fermion physics.'
author:
- Danqing Hu
- 'Jian-Jun Dong'
- 'Yi-feng Yang'
title: 'Hybridization fluctuations in the half-filled periodic Anderson model'
---
Heavy fermion materials, mostly rare earth or actinide intermetallics, provide a model system for studying the localized-to-itinerant transition of strongly correlated electrons [@Stewart1984RMP; @Hewson1997; @Coleman2015; @Onuki2018]. Theoretically, this transition is attributed to collective hybridizations between localized and conduction electrons [@Mott1974; @Doniach1977; @Coleman1983PRB; @Yang2008nature]. A mean-field approximation has often been assumed with a static and uniform hybridization [@Read1983JPC; @Coleman1984PRB; @Millis1987PRB; @Newns1987AdvPhys], leading to many interesting predictions [@Zhang2000PRB; @Burdin2009PRB; @Dzero2010PRL; @Dubi2011PRL; @Ramires2012PRL] and the identification of a characteristic coherence temperature separating the hybridized and unhybridized states [@Iglesias1997PRB; @Burdin2000PRL; @Assaad2002PRB]. The hybridization is manifested by a bending of the conduction bands that also marks the emergence of heavy electrons. However, this simple understanding was questioned by recent angled-resolved photoemission spectroscopy (ARPES) [@Chen2017PRB], which revealed a “band bending" well above the coherence temperature. Although transport and band properties may not have an exact microscopic correspondence, this separation between coherence and hybridization still caused some confusion on the conventional picture.
Fortunately, some light was shed on this issue lately by ultrafast optical pump-probe experiment [@Liu2019arXiv], in which a two-stage hybridization scenario was proposed based on the analysis of anomalous quasiparticle relaxation. While the low-temperature stage starts at the coherence temperature and results in a fluent-dependent relaxation associated with an indirect hybridization gap on the density of states as predicted by the mean-field theory, a precursor ungapped stage was also revealed to exhibit hybridization fluctuations whose onset temperature coincides with that of the “band bending" in ARPES. Such a precursor stage is beyond the mean-field description and has not been sufficiently explored. Although it has been argued that hybridization fluctuations might play an important role in heavy fermion physics [@Pepin2007PRL; @Pepin2008PRB; @Yang2017], further studies have been largely hindered by difficulties in analytical treatment. This is unfortunate because hybridization fluctuations might be the basis of many important heavy fermion phenomena [@Andres1975PRL; @Steglich1976PRL; @Sigrist1991RMP; @Stewart2001RMP; @Gegenwart2008NatPhys; @White2015].
To avoid the analytical difficulties, we propose in this work to study hybridization fluctuations numerically using determinant Quantum Monte Carlo (DQMC) [@Blankenbecler1981PRD; @Assaad2008; @Tomas2012IEEE]. DQMC has led to many useful insights on heavy fermion physics [@Vekic1995PRL; @Capponi2001PRB; @Euverte2013PRB; @Jiang2014PRB; @Wei2017SciRep; @Costa2019PRB; @LufengZhang2019PRB], but this issue has not been well discussed. Although the calculations are often limited at half filling to avoid the sign problem [@Loh1990PRB], the exact numerical results will still allow us to extract some generic properties beyond the mean-field approximation. In particular, one may want to know if there are indeed multiple stages of hybridization as proposed in pump-probe experiment and how they might be connected with the “band bending" in ARPES and the lattice coherence (here referring to the Kondo insulating state with a fully opened indirect hybridization gap as predicted in the mean-field theory). To this end, we constructed a tentative phase diagram based solely on hybridization fluctuation spectra. A partially hybridized precursor state was then revealed that exhibits low-energy hybridization fluctuations with the so-called “band bending" in the dispersion, while the Kondo insulating state is only established at lower temperatures with sufficiently strong inter-site hybridization correlations. This confirms the two-stage hybridization scenario and suggested a consistent interpretation for the photoemission, pump-probe, and optical spectroscopies.
We start with the periodic Anderson model on a two-dimensional square lattice, $$\begin{aligned}
H & =-t\sum_{\langle ij\rangle\sigma}\left(c_{i\sigma}^{\dag
}c_{j\sigma}+c_{j\sigma}^{\dag}c_{i\sigma}\right)+V\sum_{i\sigma}( f_{i\sigma}^{\dag}c_{i\sigma
}+h.c.) \nonumber\\
& \quad+E_{f}\sum_{i\sigma}f_{i\sigma}^{\dag}f_{i\sigma}+U\sum_{i}(n_{i\uparrow}^{f}-\frac{1}{2}) ( n_{i\downarrow}^{f}-\frac{1}{2}),\end{aligned}$$ where $c_{i\sigma}^{\dag}( c_{i\sigma}) $ and $f_{i\sigma}^{\dag}( f_{i\sigma}) $ are the creation (annihilation) operators of conduction and localized $f$ electrons, respectively. $t$ is the hopping integral of conduction electrons between nearest-neighbor sites, and $V$ is the bare hybridization. We set $t=1$ for the energy unit, $U=6$ for the Coulomb interaction of $f$ electrons, and $E_f=0$ for the particle-hole symmetry to avoid the sign problem in the Monte Carlo simulations.
To study hybridization fluctuations, we first introduce the hybridization field, $O_i=\sum_{\sigma}(c^\dagger_{i\sigma}f_{i\sigma}+f^\dagger_{i\sigma}c_{i\sigma})$, and define its correlation function, $$L_{ij}(\tau)=-\left\langle \mathcal{T}_\tau \left[O_i(\tau)-\langle O_i\rangle\right]\left[O_j(0)-\langle O_j\rangle\right]\right\rangle,$$ where $\mathcal{T}_\tau$ is the ordering operator for the imaginary time $\tau$. Unlike the Kondo lattice model, where a static hybridization is nothing but a mean-field artefact, the thermodynamic average $\langle O_i\rangle$ here is always finite and thus not a good quantity to distinguish the physically unhybridized and hybridized states [@Bernhard2000PRB]. It is therefore subtracted to highlight the dynamical hybridization fluctuations. The model is then evaluated numerically with DQMC [@Blankenbecler1981PRD; @Assaad2008; @Tomas2012IEEE]. The imaginary time is discretized into $M$ slices with the inverse temperature $\beta=M\Delta \tau$. At each site and time slice, the interaction is decoupled using the Hubbard-Stratonovich transformation by introducing an auxiliary Ising field. The resulting bilinear Hamiltonian can be treated exactly and the correlation function can be calculated with the help of Wick’s theorem before averaging over all sampled field configurations. Our simulations were performed on an 8$\times$8 square lattice with $M=80$ and examined with larger lattice size and time slices. The hybridization spectral function, $A_{\mathbf{q}}( \omega) =-\frac{1}{\pi}\operatorname{Im}L_{\mathbf{q}}( \omega)$, was solved using the maximum entropy method for $$L_{\mathbf{q}}( \tau) =\int_{-\infty}^{\infty}{\text d}\omega\frac {e^{-\tau\omega}}{e^{-\beta\omega}-1}A_{\mathbf{q}}( \omega),$$ where $L_{\mathbf{q}}( \tau) =\frac{1}{N}\sum_{ij}e^{-i\mathbf{q}\cdot( \mathbf{r}_{i}-\mathbf{r}_{j}) }L_{ij}( \tau)$. The real part of $L_{\mathbf{q}}( \omega) $ was then calculated using the Kramers-Kronig relation. To the best of our knowledge, these quantities have not been well explored in previous studies. The fermionic spectral functions were also calculated following similar standard procedures for comparison.
![(Color online) The real and imaginary parts of the hybridization correlation function, $L_{\mathbf{0}}(\omega)$, with the hybridization parameter $V$ and temperature $T$. The insets are enlarged plots of the imaginary part around $\omega=0$, showing the variation of the low-energy slope in $\operatorname{Im} L_{\mathbf{0}}(\omega)$ with different parameters.[]{data-label="fig1"}](fig1.eps){width="48.00000%"}
Figure \[fig1\] plots the real and imaginary parts of $L_{\mathbf{q=0}}( \omega)$ for varying $V$ at different temperatures. We first consider the high temperature regime. For $T=2.0$, $L_{\mathbf{0}}(\omega)$ changes only slightly with $V$ and shows two peaks at $\omega\approx\pm U$ due to excitations between two $f$ electron Hubbard bands. The finite slope in $\operatorname{Im} L_{\mathbf{0}}(\omega)$ around $\omega=0$ persists for $V=0$ (not shown) and must result from thermal excitations of unhybridized $f$ and conduction electrons. For $T=1.0$ and small $V$, the single valley in $\operatorname{Re} L_{\mathbf{0}}(\omega)$ evolves into a small hump with two valleys at $\omega\approx \pm U/2$, indicating the suppression of thermal excitations with lowering temperature. The two-valley features can be understood from the $V=0$ limit, where the correlation function has an analytical form, $$L_{\mathbf{0}}( \omega)=\frac{2}{N}\sum_{\mathbf{k},\alpha=\pm}\frac{[ f(\alpha U/2) -f( \epsilon_\mathbf{k}) ] (\epsilon_\mathbf{k}-\alpha U/2) }{( \omega+\operatorname*{i}\eta) ^{2}-( \epsilon_\mathbf{k}-\alpha U/2) ^{2}},
\label{eqV0}$$ where $f(x)$ is the Fermi distribution function and $\eta=0^+$ is an infinitesimal cutoff. For a flat band with a half bandwidth $D$, the summation over $\mathbf{k}$ can be evaluated exactly at zero temperature and yield, $L_{\mathbf{0}}\left( \omega\right) =D^{-1}\ln\frac{\left( \omega+\operatorname*{i}\eta\right) ^{2}-\left( U/2\right) ^{2}}{\left( \omega+\operatorname*{i}\eta\right) ^{2}-\left( D+U/2\right) ^{2} }$, which explains the calculated minima and maxima around $\omega=\pm U/2$ and $\pm (D+U/2)$. For large $V$, however, the single-valley shape is recovered.
The two-valley feature can be seen more clearly at $T=0.2$. Correspondingly, the low-energy slope in $\operatorname{Im} L_{\mathbf{0}}(\omega)$ becomes almost zero, indicating diminishing thermal excitations. However, for larger $V$, a small dip appears around $\omega=0$ on top of the hump in $\operatorname{Re} L_{\mathbf{0}}(\omega)$. Accordingly, the imaginary part exhibits a large slope in a small low-energy window followed by a sharp kink before turning to a high-energy plateau above $|\omega|\approx 0.2$. The finite slope must be a quantum effect and indicates a regime with low-energy hybridization fluctuations due to the coupling between conduction and $f$ electrons. Similar features can be found at $T=0.05$ for $V=0.5$, but are suppressed at larger $V$, where the dip in $\operatorname{Re} L_{\mathbf{0}}(\omega)$ is filled in and turns into a smooth maximum, and the slope in $\operatorname{Im} L_{\mathbf{0}}(\omega)$ is also suppressed.
![(Color online) (a, b) Comparison of the different features of the real and imaginary parts of $L_{\mathbf{0}}( \omega)$ in four distinct regimes; (c) The corresponding $f$ electron local density of states. The parameters are: $T=2.0$, $V=0.5$ for regime I; $T= 0.6$, $V=0.25$ for regime II; $T= 0.2$, $V=1.2$ for regime III; and $T = 0.05$, $V=2.0$ for regime IV. (d) Intensity plots of the fermionic spectral function at the Fermi energy in the Brillouin zone evolving with temperature for a fixed $V = 1.0$; (e) The corresponding plots of the dispersion along a chosen path in the Brillouin zone, showing its evolution across the intermediate regime III.[]{data-label="fig2"}](fig2.eps){width="48.00000%"}
The above distinct features of $L_{\mathbf{0}}(\omega)$ suggest four different regimes of the periodic Anderson model. The results are summarized in Figs. \[fig2\](a) and \[fig2\](b). Regimes I and II are governed by background contributions of decoupled conduction and $f$ electrons. For regime I, thermal excitations are large such that the real part of $L_{\mathbf{0}}(\omega)$ has only one valley and the imaginary part has a finite slope; while for regime II, thermal effects are suppressed, revealing two valleys at $\omega=\pm U/2$ in $\operatorname{Re} L_{\mathbf{0}}(\omega)$ due to the Hubbard bands, and the slope in $\operatorname{Im} L_{\mathbf{0}}(\omega)$ is consequentially reduced. The latter corresponds to a selective Mott regime of $f$ electrons that are effectively decoupled from conduction electrons. To see this, we plot the $f$ electron local density of states (DOS) in Fig. \[fig2\](c). The spectra are governed by two broad Hubbard peaks at $\omega=\pm U/2$. In regime I, the valley in between is partially filled by thermal excitations, but in regime II, it is depleted and reveals the Mott gap [@Held2000PRL; @Logan2016JPCM].
.[]{data-label="fig3"}](fig3.eps){width="48.00000%"}
Deviation from the above Mott features defines two hybridized regimes. The small dip on the hump of $\operatorname{Re} L_{\mathbf{0}}(\omega)$ and the large low-energy slope of $\operatorname{Im} L_{\mathbf{0}}(\omega)$ in regime III mark a genuine quantum effect due to low-energy hybridization fluctuations. In regime IV, these features are again suppressed, indicating the crossover into a different phase. This is the Kondo insulating regime, where the slave-boson mean-field theory predicts an artificial boson condensation. The hybridization correlation function can also be evaluated analytically, $$L_{\mathbf{0}}( \omega) =\frac{4}{N}\sum_{\mathbf{k}}\frac{f( E_{\mathbf{k}-}) -f( E_{\mathbf{k}+}) }{(\omega+\operatorname*{i}\eta)^2-\Delta_{\mathbf{k}} ^{2}}\frac{\epsilon_{\mathbf{k}}^2}{\Delta_{\mathbf{k}}},
\label{eqU0}$$ where $E_{\mathbf{k}\pm}=(\epsilon_{\mathbf{k}} \pm \Delta_{\mathbf{k}})/2$ denote two hybridization bands and $\Delta_{\mathbf{k}}=\sqrt{ \epsilon_{\mathbf{k}} ^{2}+\Delta_0^{2}}$ is the direct hybridization gap at each $\mathbf{k}$ with an effective hybridization strength $\Delta_0$ whose magnitude separates the hybridized and unhybridized phases. Thus $\operatorname{Re} L_{\mathbf{0}}(\omega)$ has two minima around $\omega=\pm\Delta_0$. Since $\Delta_{\mathbf k}\ge\Delta_0$ for all $\mathbf{k}$, we have the imaginary part, $\operatorname{Im} L_{\mathbf{0}}(\omega)\propto\sum_{{\mathbf k},\alpha=\pm} \alpha\delta(\omega+\alpha\Delta_{\mathbf k})\epsilon_{\mathbf k}^2/\Delta_{\mathbf k}^2 $, which is gapped for $|\omega|<\Delta_0$. Obviously, the above formula fails in regime III, where we have a large low-energy slope in $\operatorname{Im} L_{\mathbf{0}}(\omega)$ due to the presence of hybridization fluctuations. This has an immediate consequence on the $f$ electron spectra. As shown in Fig. \[fig2\](c), instead of a Kondo insulating gap in the local DOS as in regime IV, we find a broad peak around $\omega=0$, making regime III a precursor ungapped state beyond the mean-field approximation; while in regime IV, the two sharp peaks at lower energyies can be roughly understood from the band hybridization in $E_{\mathbf{k}\pm}$.
To gain further insight, we plot in Fig. \[fig2\](d) the momentum distribution of the total fermionic ($f$ and conduction electrons) spectral intensity at the Fermi energy evolving with temperature for $V=1.0$. We see a clear crossover from a selective Mott regime with two Hubbard bands and a small conduction electron Fermi surface to a Kondo insulating regime where the hybridization gap is fully opened with no discernible spectral weight at the Fermi energy in the whole Brillouin zone. In between, regime III shows a finite spectral weight (not the Fermi surface), albeit with a very different pattern. For clarity, we plot the dispersion in Fig. \[fig2\](e), where a slight band bending is already seen in regime III, but the gap is only partially opened, leaving a finite spectral weight at the Fermi energy and the broad peak in the local DOS. This agrees with the ARPES observation [@Chen2017PRB] and supports the two-stage scenario proposed by pump-probe experiment [@Liu2019arXiv]. The band bending is also an indication of the direct hybridization gap as probed in optical conductivity [@Chen2016RPP]. This gives a consistent interpretation of the high-temperature features in ARPES, pump-probe and optical measurements.
To understand how hybridization fluctuations can further induce the $f$ electron coherence (here the Kondo insulating state) at lower temperature, we compare in Fig. \[fig3\](a) the local and nonlocal contributions to $\operatorname{Re}L_{\mathbf{0}}(\omega)$. Since $L_{\mathbf{0}}(\omega)=N^{-1}\sum_{ij}L_{ij}(\omega)$, the nonlocal part is a sum of all inter-site correlations. We see for regimes I and II, the nonlocal contribution is indiscernible. It only starts in regime III but, quite surprisingly, becomes comparable with the local one in regime IV. Its very existence is an indication of quantum effect. Clearly, while the band bending already appears in regime III, the lattice coherence can only be established later with sufficiently strong inter-site hybridization correlations. It should be noted that the nonlocal correlation is dominantly contributed by the nearest-neighbor term in our calculations. Hence, the Kondo insulator should be viewed more like a short-range-correlated insulator rather than a simple band insulator described by the mean-field picture. This short-range correlation is consistent with previous calculations as well as nuclear magnetic resonance (NMR) observations on doped Kondo lattice [@Wei2017SciRep; @Lawson2019]. We further remark that the development of nonlocal correlations is also manifested in the momentum space. Figure \[fig3\](b) plots the normalized hybridization spectral function, $\tilde{A}_{\mathbf{q}}( \omega)$, along the path $(0,0)-(0,\pi)-(\pi,\pi)-(0,0)$ in the Brillouin zone. The spectra are basically featureless besides the Hubbard bands in regimes I and II. A slight change appears at $(\pi,\pi)$ in regime III, which grows rapidly in regime IV and eventually intrudes into the Mott feature. This reflects a competition between nonlocal hybridization correlations and the local Mott physics. The fact that the former emerges dominantly near $(\pi,\pi)$ seems to also indicate an interplay between hybridization and magnetic fluctuations [@Yang2017].
![(Color online) A tentative phase diagram constructed based solely on $L_{\mathbf{0}}( \omega)$. The points are estimated from the different features of its real part and the lines are a guide to the eye. The background colors reflect the low-energy slope $K$ of its imaginary part. The right panel plots the values of $K$ for $V = 1.0$, whose nonmonotonic temperature dependence clearly demonstrates the separation of four regimes.[]{data-label="fig4"}](fig4.eps){width="47.00000%"}
Putting together, we find it possible to construct a tentative phase diagram of the periodic Anderson model based solely on hybridization fluctuation spectra. The result is shown in Fig. \[fig4\], where the points and dashed lines mark the phase (crossover) boundaries extracted roughly from the features of $\operatorname{Re} L_{\mathbf{0}}(\omega)$, and the background colors reflect the magnitude of the slope, $K=\left.\mathrm{d}\operatorname{Im} L_{\mathbf{0}}( \omega) /\mathrm{d}\omega\right|_{\omega=0}$. We see a rough agreement between the two methods. The phase diagram reveals clearly the four distinct regimes and their overall relationship. This is best demonstrated in the right panel of Fig. \[fig4\] for $V=1.0$, where $K$ undergoes a nonmonotonic variation that separates the different regimes. It is now evident that regime III (at small $V$) bridges the unhybridized selective Mott state (II) and the fully hybridized Kondo insulating state (IV) and marks a crossover from localized to itinerant $f$ electrons. In previous analytical calculations, it has been proposed that the localized-to-itinerant transition at zero temperature may be viewed as a selective Mott transition [@Pepin2007PRLmott; @Pepin2008PRBmott]. This seems to be consistent with our results if regime III could in some way be associated with the crossover regime above the Mott critical end point. Unfortunately, at the moment our calculations are limited at relatively higher temperatures and it is not clear if a straightforward connection can be made. We should note that the presence of a precursor regime above the Kondo insulating phase can also be seen in previous calculations [@Jarrell1993PRL; @Medici2005PRL], but it has not been well discussed in the context of hybridization fluctuations. It will be important if our study can be extended to extremely low temperatures to provide numerical evidences for previous analytical treatment. Recently, it has also been proposed that non-Hermitian physics might lead to exotic properties in a Kondo insulator [@Shen2018PRL; @Yoshida2018PRB; @Michishita2019arXiv]. The so-called exceptional points were argued to be around the high-temperature boundary of the Kondo insulating phase [@Michishita2019arXiv]. In our case, if we make the replacement $\eta\rightarrow \Gamma_k$ in Eq. (\[eqU0\]), we will be able to get a finite slope, $K\propto -\sum_{\mathbf{k}} \Gamma_\mathbf{k}\epsilon_{\mathbf{k}}^2/\Delta_{\mathbf{k}}(\Gamma_\mathbf{k}^2+\Delta_\mathbf{k}^2)^2$, which approaches zero when $\Gamma_{\bf k}\rightarrow 0$ or $\infty$. Thus the finite $K$ in regime III might be associated with the finite dissipation (or lifetime) of hybridization or fermionic excitations in the crossover phase. It would certainly be more intriguing if regime III is a state that could potentially host some exotic non-Hermitian physics.
To summarize, we studied hybridization fluctuations with DQMC for the half-filled periodic Anderson model. This allows us to extract some useful information beyond the mean-field approximation and construct a tentative phase diagram based solely on hybridization fluctuation spectra. We found a crossover from an unhybridized selective Mott state to a fully hybridized Kondo insulating state. In between, there exists an intermediate phase with low-energy hybridization fluctuations and evident band bending. The $f$ electron coherence is only established at lower temperatures with the development of sufficiently strong inter-site hybridization correlations. This confirms the proposed two-stage hybridization scenario based on recent ARPES and pump-probe experiments. The band bending occurs first near the Fermi wave vector of conduction electrons and gives rise to a direct hybridization gap as probed in optical conductivity well above the coherence temperature. We have thus a consistent picture for the high-temperature features of photoemission, pump-probe, and optical spectroscopies. Possible connections with Mott and non-Hermitian physics were also discussed briefly. Our work provides a promising start for numerical studies of hybridization dynamics in causing exotic correlated properties of heavy fermion systems. In the future, we expect to see more insights if our study could be extended to the quantum critical regime or the metallic phase away from the half filling to make a full comparison with previous analytical or experimental conclusions.
This work was supported by the National Natural Science Foundation of China (NSFC Grant No. 11974397, No. 11522435), the National Key R&D Program of China (Grant No. 2017YFA0303103), the State Key Development Program for Basic Research of China (Grant No. 2015CB921303), the National Youth Top-notch Talent Support Program of China, and the Youth Innovation Promotion Association of CAS.
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---
abstract: 'This paper proposes a multi-level feature learning framework for human action recognition using a single body-worn inertial sensor. The framework consists of three phases, respectively designed to analyze signal-based (low-level), components (mid-level) and semantic (high-level) information. Low-level features capture the time and frequency domain property while mid-level representations learn the composition of the action. The Max-margin Latent Pattern Learning (MLPL) method is proposed to learn high-level semantic descriptions of latent action patterns as the output of our framework. The proposed method achieves the state-of-the-art performances, 88.7%, 98.8% and 72.6% (weighted $F_1$ score) respectively, on Skoda, WISDM and OPP datasets.'
address:
- 'State Key Laboratory of Software Development Environment and Key Laboratory of Biomechanics and Mechanobiology of Ministry of Education and Research Institute of Beihang University in Shenzhen, Beihang University, Beijing 100191, China'
- 'Microsoft Research Asia, Beijing 100080, China'
- 'School of Electronic and Information Engineering, Beihang University, Beijing 100191, China'
author:
- Yan Xu
- Zhengyang Shen
- Xin Zhang
- Yifan Gao
- Shujian Deng
- Yipei Wang
- Yubo Fan
- 'Eric I-Chao Chang'
bibliography:
- 'bibfile.bib'
title: 'Learning Multi-level Features For Sensor-based Human Action Recognition'
---
Multi-level ,Human action recognition,Latent pattern ,High-level ,Semantic
Introduction
============
In recent years, sensor-based human action recognition (HAR) plays a key role in the area of ubiquitous computing due to its wide application in daily life [@chennuru2010mobile; @wu2013mobisens; @wu2011senscare; @forster2009unsupervised; @ngo2015similar; @santos2015trajectory]. In most cases, utilization of raw data without special process is impractical since raw recordings confound noise and meaningless components. Therefore, the process of feature extraction and feature learning is supposed to be conducted [@lara2013survey], which is commonly based on the sliding window technique through sampling overlapping frames from signal streams [@bulling2014tutorial]. In general, studies on sensor-based HAR have mainly focused on low-level [@bao2004activity; @ravi2005activity; @huynh2005analyzing; @kang1995application] and mid-level [@bhattacharya2014using; @vollmer2013learning; @huỳnh2007scalable] features.
Low-level features include statistical features [@bao2004activity; @ravi2005activity], frequency-domain features [@huynh2005analyzing; @kang1995application] and some hand-crafted methods [@hammerla2013preserving]. Low-level features are popular owing to their simplicity as well as their acceptable performances across a variety of action recognition problems. However, their simplicity is often accompanied by less discrimination in representation, considering that actions are often highly complex and diverse in daily-life cases.
Compared with low-level features, mid-level features which are mainly obtained through dictionary learning methods [@vollmer2013learning; @huỳnh2007scalable] have proven to be more robust and discriminative [@bhattacharya2014using]. The representations include the sparse coding method [@grosse2012shift] and the bag-of-words (BOW) algorithm [@huỳnh2007scalable]. These methods analyze components (motion primitives) and explore inherent structures of signals. However, when the dictionary size is relatively large, mid-level features would suffer high computation, redundancy in representation and further burden the following classification. In this paper, we reduce the redundancy of the mid-level representation by introducing high-level features, which can achieve better overall performances and be robust to dictionary size.
Many studies have applied probabilistic graphical models [@zhang2013modeling; @bui2008hidden] and pattern-mining methods [@liu2015action; @chikhaoui2011frequent; @gu2010unsupervised; @kim2010human; @palmes2010object] in semantic (high-level) recognition tasks. However, few works focus on feature learning methods in sensor-based HAR. In this paper, a semantic feature learning algorithm is proposed on two motivations. First, it can remove the redundancy in mid-level representation. A compact and discriminative description can be achieved by applying a latent pattern learning method on mid-level features. Second, implementing semantic analysis on sensor signals is an efficient and intuitive way to discover and deal with the variety of action patterns, and thus it can remove ambiguities in descriptions of the action. In particular, the ambiguities mainly come from two aspects, physical level and annotation level, both of which make it difficult to generate generalized good features in HAR tasks [@bulling2014tutorial]. At the physical level, one challenge is that different or even the same people who conduct the same action may produce completely different signals due to changes in environment, which is called the intra-class variability [@zinnen2009multi]. For example, walking style in morning after a good sleep commonly differs from the one after day’s hard work. The other challenge is the inter-class similarity [@amft2007probabilistic], which refers to the similarity among different actions such as ‘drinking milk’ and ‘drinking coffee’. As the inter-class similarity is generally resolved through obtaining additional data from different sensors [@stikic2008adl] or analyzing co-occurring actions [@huynh2008discovery], our work focuses on dealing with the intra-class variability. At the annotation level, a specific action performed in many ways can be cognitively identified as the same one [@aggarwal2011human]. Take ‘cleaning the table’ as an example. It is rational to consider cleaning table from right to left and from up to down as the same action, though they behave absolutely differently in signal records. For those reasons, generalized features that are directly learned from action perspectives would compromise to the common characteristic shared by different action patterns, resulting in ambiguous feature representations [@malisiewicz2011ensemble]. Inspired by Multiple-Instance Learning (MIL) [@dietterich1997solving; @wang2013max], our solution is to mine discriminative latent patterns from actions and construct features based on descriptions of those patterns, which can eliminate ambiguities from both physical and annotation levels. We name this method the Max-margin Latent Pattern Learning (MLPL) method. Instead of being constrained by generic property, the MIL method is a natural solution given that the diversity inside the class can be learnt. Although MIL methods are widely performed in computer vision [@ziaeefard2015semantic; @yi2015human], as for sensor-based HAR problems, relevant works are mainly designed to cope with sparse annotation [@stikic2011weakly; @stikic2009activity]. Instead of dealing with sparsely labeled cases, MLPL proposed in this paper implements MIL to learn discriminative latent patterns of actions, by which high-level features would be acquired.
In this paper, we integrate the advantages of low-, mid- and high-level features and propose the framework known as Multi-Level Complementary Feature Learning (MLCFL). To avoid being confused with the high-level feature learned by the latent pattern learning process, the output feature of our framework is denoted as Multi-Level Complementary Feature (MLCF). In particular, this framework learns multi-level features through three phases, which are respectively designed to analyze signal-based (low-level), components (mid-level) and semantic (high-level) information. In the first phase, the low-level feature (statistical values, FFT coefficients, etc.) is extracted from raw signals. In the second phase, from the component perspective, the mid-level representation can be attained through hard coding processes and occurrence statistics. In the third phase, the MLPL method, from the semantic perspective, is implemented on the Compl feature (the concatenation of low- and mid-level features) to obtain MLCF as the output of this framework. Various experiments on Opp [@chavarriaga2013opportunity; @roggen2010collecting], Skoda [@zappi2008activity] and WISDM [@kwapisz2011activity] datasets show that MLCF possesses higher feature representation ability than low- and mid-level features. Moreover, compared with existing methods, the method we proposed achieves state-of-the-art performances. Our contributions in this paper are as follows:
1. A multi-level feature learning framework MLCFL is constructed, which consists of three phases including low-level feature extraction, mid-level components learning and high-level semantic understanding. The output feature is learned level by level, possessing higher representation ability than low- and mid-level features.
2. An efficient and effective latent pattern learning method MLPL is proposed to learn high-level features, each dimension of which refers to the confidence score of corresponding latent action pattern.
3. Our framework is evaluated on three popular datasets and achieves state-of-the-art performances.
The rest of this paper is organized as follows: Section 2 presents related work; Section 3 describes the MLCFL framework for action recognition; Section 4 presents and analyzes experimental results; finally, we conclude the study in Section 5.
Related work
============
Researches in the area of sensor-based HAR have been ever increasing in the past few years [@hammerla2013preserving; @field2015recognizing; @zhang2013modeling; @bulling2014tutorial]. Numerous methods have been proposed in designing, implementing and evaluating action recognition systems. In this section, we review typically practical methods in terms of feature representation, specifically from low- [@bao2004activity; @ravi2005activity; @huynh2005analyzing; @kang1995application; @hammerla2013preserving; @plotz2011feature] as well as mid- [@bhattacharya2014using; @vollmer2013learning; @zhang2012motion] perspectives and demonstrate how each method is applied to the specific action recognition task. As few works focus on designing high-level features, an additional review is presented on prevalent approaches involving semantic understanding of actions [@huynh2008discovery; @field2015recognizing; @zhang2013modeling; @bui2008hidden; @hospedales2011identifying; @chikhaoui2011frequent; @gu2010unsupervised; @kim2010human; @palmes2010object]. Besides, a brief overview about other representative methods [@stikic2011weakly; @zeng2014convolutional] is summarized.
Low-level features are designed to capture signal-based information. Statistical metrics are the most common approaches, which include mean, variance, standard deviation, energy, entropy and correlation coefficients [@bao2004activity; @ravi2005activity]. Fourier Transform (FT), Wavelet Transform (WT) [@tamura1997classification], Discrete Cosine Transform (DCT) [@he2009activity] as well as auto-regressive (AR) coefficients [@he2008activity] are also commonly applied in HAR tasks for their promising performances. Kang et al. [@kang1995application] analyzed electromyography (EMG) signals by extracting conventional auto-regressive coefficients and cepstral coefficients as features. Hammerla et al. [@hammerla2013preserving] designed the hand-crafted feature based on the Empirical Cumulative Distribution Function (ECDF) to preserve characteristics of inertial signal distribution. Pl[ö]{}tz et al. [@plotz2011feature] improved on that work and proposed the ECDF-PCA feature. They implemented the Principal Component Analysis (PCA) method on signals normalized by ECDF and significantly improved performance. In this paper, statistical values, FFT coefficients and ECDF-PCA are calculated as low-level features to demonstrate the generalization ability of the proposed framework.
Mid-level features are generally extracted from the low-level ones to explore the components and structural information of signals. They are prevalent in HAR tasks for robustness against noise and discrimination in representations [@bhattacharya2014using; @vollmer2013learning]. Huỳnh et al. [@huỳnh2007scalable] and Zhang et al. [@zhang2012motion] implemented the bag-of-words (BOW) model to obtain statistical descriptions of motion primitives. Their works showed the effectiveness of the BOW model in sensor-based action recognition tasks. Blanke et al. [@blanke2009daily] extracted the occurrence statistics feature from low-level actions in a way that is similar to Huỳnh et al. [@huỳnh2007scalable] and then implemented the JointBoosting-framework. One characteristic of their method was to adopt a top-down perspective, using a feature selection algorithm to learn the distinctive motion primitives from the labeled high-level action. Sourav et al. [@bhattacharya2014using] and Christian et al. [@vollmer2013learning] both utilized sparse coding and adopted the convolution basis, which could resist shifts in time and thus reduce redundancy of basis. Our work is similar to [@zhang2012motion], in which the mid-level representation is achieved through hard coding and occurrence statistics.
High-level recognition tasks mainly focus on obtaining intuitive and semantic descriptions of actions. Pattern-mining methods [@huynh2008discovery; @chikhaoui2011frequent; @gu2010unsupervised; @kim2010human; @palmes2010object; @liu2015action] and probabilistic graphical models [@zhang2013modeling; @bui2008hidden] are the most prevalent approaches. Pattern-mining methods explore the diversity in human actions through learning discriminative action patterns and motion primitives. Huynh et al. [@huynh2008discovery] applied probabilistic topic models which stemmed from the text processing community to automatically extract action patterns from sensor data. They described the recognition of daily routines as a probabilistic combination of learned patterns. Liu et al. [@liu2015action] presented an algorithm capable of identifying temporal patterns from low-level actions and utilized these temporal patterns to further represent high-level human actions. Various methods have also been proposed based on probabilistic graphical models [@zhang2013modeling; @bui2008hidden] to capture the temporal, sequential, interleaved or concurrent relationship among motion primitives. However, graphical models are limited to capturing rich temporal relationships in complex actions and also suffer from the exponential increase in calculation when the number of involved actions grows [@liu2015action]. Instead of modeling temporal relationships, we propose an efficient pattern-mining approach, which takes advantage of being compact in representation, intuitive in understanding and efficient in calculation.
Multiple-Instance Learning (MIL) methods in HAR have been widely applied to cope with scarcity of annotation. Maja et al. [@stikic2011weakly] proposed a framework involving MIL as a weakly supervised recognition method to deal with scarcity of labels and proved its robustness to erroneous labels. Similarly, MIL methods with several novel extensions were introduced to handle different annotation strategies in action recognition [@stikic2009activity]. Instead of dealing with annotation scarcity, MLPL proposed in this paper implements MIL to explore latent patterns in human actions, by which the high-level feature would be acquired.
Other representative works in HAR include template matching methods, heuristic methods and deep learning ones. Template matching methods often derived from DTW and LCSS algorithm. Hartmann et al. [@hartmann2010gesture] proposed Segmented DTW to recognize and bound the gesture in each frame through finding the best match among the object and all templates of different classes. Nonsegmented DTW proposed by Stiefmeier et al. [@stiefmeier2008wearable] was a more efficient variation through reusing the previous computation. Nguyen-Din et al. [@nguyen2012improving; @nguyen2014robust] improved LCSS algorithm and proposed WarpingLCSS and SegmentedLCSS, which were more efficient than DTW-based methods and robust to noisy annotation. Besides, heuristic methods are often related to specific tasks [@huynh2005analyzing]and depend on domain knowledge. Reyes-Ortiz et al. [@reyes2014human] designed temporal filters to recognize actions as well as postural transitions. In addition, various studies on deep learning methods have been conducted recently, mainly derived from Convolutional Neural Networks (CNN) [@ha2015multi; @yang2015deep; @ordonez2016deep] and Recurrent Neural Network (RNN) [@palumbo2016human]. These methods explored relationships between the temporal and spatial dependency of recordings and sensors. Zeng et al. [@zeng2014convolutional] applied CNN with partial weight sharing method. The framework they proposed achieved outstanding performance on three popular datasets.
In this paper, contrary to traditional feature extraction and learning methods restricted in low- and mid-level descriptions of signals, we achieve semantic understanding of sensor-based human actions through learning latent action patterns. To the best of our knowledge, our method proposed is the first attempt to explicitly apply a feature learning method to high-level representations in sensor-based HAR tasks. Furthermore, we present a brand new framework to synthesize multi-level features, integrating signal-based (low-level), components (mid-level) and semantic (high-level) information together.
![The flowchart of Multi-Level Complementary Feature Learning (MLCFL). In the first phase, low-level features are extracted from frames and sub-frames. In the second phase, mid-level representations can be obtained through hard coding processes and occurrence statistics. In the third phase, MLPL would be implemented on the Compl feature (the concatenation of low- and mid-level features), where MLCF can be obtained as the output of the framework.[]{data-label="fig:overview"}](fig1_overview.png){width="100.00000%"}
Method
======
Our framework consists of three phases in interpreting the signal: i) low-level feature extraction; ii) mid-level components learning; iii) high-level semantic understanding. The low-level description analyzes the temporal and frequency property of signals while the mid-level representation is a statistical description of either shared or distinctive components (motion primitives). The high-level feature describes the action by distinguishing the specific action pattern it belongs to. The flowchart of the framework is illustrated in [Fig.]{} \[fig:overview\].
Low-level feature extraction {#section:3.1}
----------------------------
In this phase, low-level features are extracted from raw signals to learn properties in the time and frequency domain. Three popular features, namely statistical values, FFT coefficients and ECDF-PCA, are involved in this work and a brief description is presented as follows.
### Statistical values
Statistical metrics analysis is one of the most common approaches to obtaining feature representations of raw signals. Statistical values in this work refer to time-domain features [@bao2004activity; @ravi2005activity], including mean, standard deviation, energy, entropy and correlation coefficients.
### FFT coefficients
Frequency domain techniques [@huynh2005analyzing; @kang1995application] have been extensively applied to capture the repetitive nature of sensor signals. This repetition often correlates to the periodicity property of a specific action such as walking or running. The transformation technique used in this paper is Fourier Transform (FT) through which dominant frequency components in the frequency domain can be procured.
### ECDF-PCA
ECDF-PCA [@plotz2011feature] was brought up on the expertise analysis of inertial signals recorded by the triaxial accelerator sensor. It applies the Empirical Cumulative Distribution Function (ECDF) on signals. Then ECDF feature [@hammerla2013preserving] is procured by inverse equal probability interpolation, through which data are normalized and preserve its inherent structure at the same time. A PCA method is then implemented on this normalized data.
Mid-level components learning
-----------------------------
The mid-level learning method is a general approach in pattern recognition. Dictionary learning methods such as bag-of-words (BOW) [@zhang2012motion] and sparse coding [@bhattacharya2014using] are the most popular approaches for obtaining mid-level representations. Compared with low-level feature extraction which involves analyzing properties in the time and frequency domain, mid-level learning focuses on the structural composition. In this paper, we implement BOW in signal processing. The obtained mid-level feature is the statistical description of components of the signal.
The dictionary is first formed from training data through the K-means algorithm [@huỳnh2007scalable]. In particular, frames are broken into overlapping sub-frames which are smaller in length. Low-level features are extracted from sub-frames and then the K-means method is used to construct the motion-primitive dictionary. $K$ clusters are generated and the dictionary is formed by cluster centers. We define a set of samples as $\{{\bm{\mathrm{x}}}_1,\dots,{\bm{\mathrm{x}}}_n\}, {\bm{\mathrm{x}}}_i\in \bm{\mathrm{R}}^{d\times1}$, $i\in\{1,\dots,n\}$ and each sample ${\bm{\mathrm{x}}}_i$ is associated with an index $z_i\in\{1,\dots,K\}$. If $z_i=j\in\{1,\dots,K\}$, ${\bm{\mathrm{x}}}_i$ is in the $j$-th cluster. The center of the $j$-th cluster is denoted as ${\bm{\mathrm{m}}}_j={\sum_{i=1}^{n}1\{z_i=j\}{\bm{\mathrm{x}}}_i}/{\sum_{i=1}^{n}1\{z_i=j\}}$, ${\bm{\mathrm{m}}}_j\in \bm{\mathrm{R}}^{d\times1}$. ${\bm{\mathrm{m}}}_j$ refers to a word while j refers to the corresponding index in the dictionary. When a new sample ${\bm{\mathrm{x}}}_i$ comes in, the corresponding index $z_i$ can be determined by $z_i=\mathop{\operatorname*{argmin}}_{j\in\{1,\cdots,K\}}({\bm{\mathrm{x}}}_i-{\bm{\mathrm{m}}}_j)({\bm{\mathrm{x}}}_i-{\bm{\mathrm{m}}}_j)^\mathrm{T}$.
![Illustration of symbolic sequences generated by different dictionary sizes in mid-level learning. From top to bottom: raw data, cluster assignments, of which the dictionary size is 100 and 300 respectively.[]{data-label="fig:fig2_symbolic_sequence"}](fig2_symbolic_sequence.jpg){width="90.00000%"}
With the learned dictionary, a new symbolic sequence shown in [Fig.]{} \[fig:fig2\_symbolic\_sequence\] can be derived from raw signals by densely extracting low-level features from sub-frames and retrieving them from the dictionary. In the end, the mid-level feature is represented by occurrence statistics of motion primitives in the frame.
High-level semantic understanding
---------------------------------
Generally speaking, one specific action would be highly identifiable if each pattern of the action had been learnt. Similarly, descriptions capturing properties of each distinctive distribution in the feature space would be more discriminative than the ones that only capture the generic but ambiguous characters of the whole distribution. Inspired by Wang’s method [@wang2013max] of learning a weakly-supervised dictionary for discriminative subjects in images, we propose Max-margin Latent Pattern Learning (MLPL) in sensor-based signal processing. In general, the objective of this algorithm is to identify each specific class by learning a set of latent classifiers, each of which can be a discriminative description of a certain action pattern. The high-level feature is represented by the combination of confidence scores belonging to each latent class. Specifically, the latent pattern learning problem in this work, as shown in [Fig.]{} \[fig:MLPL\], can be divided into two aspects: i) maximizing the inter-class difference, namely the differences between one specific action and other actions; ii) maximizing the intra-class difference, namely the differences among different patterns of one specific action.
![Illustration of Max-margin Latent Pattern Learning (MLPL). Circle and star represent two specific classes. Two different colors of each class are related to two latent classes which can be distinctive descriptions of action patterns. The confidence score is obtained by calculating the distance between the latent class margin and the instance in the feature space. The combination of the confidence scores of all the latent classes forms the semantic feature representation.[]{data-label="fig:MLPL"}](fig3_MLPL.png){width=".5\textwidth"}
We first present a brief notation of Multiple-Instance Learning (MIL). In MIL, a set of bags are defined as [**[X]{}**]{}=$\{{\bm{X}}_1,\dots,{\bm{X}}_m\}$, and each bag contains a set of instances ${\bm{X}}_i=\{{\bm{\mathrm{x}}}_{i1},\dots,{\bm{\mathrm{x}}}_{in}\}$, where ${\bm{\mathrm{x}}}_{ij}\in{{\bm{\mathrm{R}}}^{d\times1}}$. Only one label can be assigned to a bag and instances inside it. The bag would be labeled as positive if there exists at least one positive instance, while being labeled negative only when all of the instances in it are negative. In computer vision, the bag model is a natural description of the image because the image commonly consists of a set of subjects and the label of the image can only be determined by subjects of interest. Compared with the traditional MIL problem, the concept ‘bag’ is simplified in our problem as labels of the whole signal are provided. Therefore, learning the ‘interest’ from the background would be evaded. The goal of our method is to learn various patterns of actions.
To simplify the notation, we define instances from the $i$-th class, $i\in\{1,\dots,m\}$, as a set ${\bm{X}}_i=\{{\bm{\mathrm{x}}}_{i1},\dots,{\bm{\mathrm{x}}}_{in}\}$, where ${\bm{\mathrm{x}}}_{ij}\in{{\bm{\mathrm{R}}}^{d\times1}}$, $j\in\{1,\dots,n\}$. In addition, each set ${\bm{X}}_i$ is associated with the label $Y_i\in\{0,1\}$. When modeling latent patterns in the $i$-th class, $Y_i=1$, otherwise $Y_i=0$. For each class, we assume it contains $K$ latent classes, each of which corresponds to a cluster in the feature space. Intuitively, we assume there are 5 different patterns in class ‘check gaps on the front door’ and ‘close both left door’ respectively. [Fig.]{} \[fig:check steering wheel\] shows examples of patterns (latent classes) learned by MLPL. Multi-scale policy [@park2008hierarchical] is introduced in this paper to learn latent patterns from various semantic levels as well as mitigate uncertainty in determining concrete number of latent classes.
![Illustrations of latent class. Two classes in Skoda, ‘check gaps on the front door’ (a) and ‘close both left door’ (b), with their three latent classes are shown in raw signal. Each row refers to a latent class, where five samples are selected. Rows compare differences among latent classes and columns show the similarity in one specific latent class.[]{data-label="fig:check steering wheel"}](fig4_check_gaps_on_the_front_door.png){width=".9\textwidth"}
(a)
![Illustrations of latent class. Two classes in Skoda, ‘check gaps on the front door’ (a) and ‘close both left door’ (b), with their three latent classes are shown in raw signal. Each row refers to a latent class, where five samples are selected. Rows compare differences among latent classes and columns show the similarity in one specific latent class.[]{data-label="fig:check steering wheel"}](fig4_close_both_left_door.png){width=".9\textwidth"}
(b)
During the learning phase, each instance ${\bm{\mathrm{x}}}_{ij}$ is associated with a latent variable $z_{ij}\in\{0,1,\dots,K\}$. Instance ${\bm{\mathrm{x}}}_{ij}$ is in the $k$-th positive cluster if $z_{ij}=k$ or in negative cluster if $z_{ij}=0$. Considering that MIL is applied in our method to find distinctive latent classes, maximizing differences between one latent class and others is required. Besides, latent classes that belong to the $i$-th class need to be distinguished from the $j$-th class, where $i\neq{j}$. Consequently, from the feature distribution perspective, a natural idea is to maximize the margin among intra- and inter-class. To meet that demand, multi-class SVM is an ideal solution. In particular, SVM with linear kernel is used for its generality and efficiency. Each latent class is associated with a linear classifier, in which $f({\bm{\mathrm{x}}})={\bm{\mathrm{w}}}^\mathrm{T}{\bm{\mathrm{x}}}$ and weighting matrix is defined as $${\bm{\mathrm{W}}}_i=[{\bm{\mathrm{w}}}_0,{\bm{\mathrm{w}}}_1,\dots,{\bm{\mathrm{w}}}_K],{\bm{\mathrm{w}}}_k\in {{{\bm{\mathrm{R}}}^{d\times1}}},k\in{[0,1,\dots,K]}$$ where ${\bm{\mathrm{w}}}_k$ represents the model of the $k$-th latent class if k is positive; ${\bm{\mathrm{w}}}_0$ denotes the negative cluster model. Therefore, we learned $K+1$ linear classifiers for each class and $m*(K+1)$ for total.
Intuitively, the latent label of the instance ${\bm{\mathrm{x}}}_{ij}$ is determined as the most ‘positive’ one. $$z_{ij} = \mathop{\operatorname*{argmax}}_{k\in\{0,\cdots,K\}}{\bm{\mathrm{w}}}_k^\mathrm{T}{\bm{\mathrm{x}}}_{ij}$$
The multi-class hinge loss forces latent classifiers to be distinct from each other. It can be defined as: $$L({\bm{\mathrm{W}}};({\bm{\mathrm{x}}}_{ij},z_{ij}))=\sum_{ij} max(0,1+{\bm{\mathrm{w}}}_{r_{ij}}^\mathrm{T}{\bm{\mathrm{x}}}_{ij}-{\bm{\mathrm{w}}}_{z_{ij}}^\mathrm{T}{\bm{\mathrm{x}}}_{ij})$$ where $r_{ij}=\mathop{\operatorname*{argmax}}_{k\in\{0 \cdots K\},k\neq z_{ij}}{\bm{\mathrm{w}}}_{k}^\mathrm{T}{\bm{\mathrm{x}}}_{ij}$.
The objective function can then be defined as: $$\label{obj_eq}
\begin{split}
\min_{{\bm{\mathrm{W}}},z_{ij}}\sum_{k=0}^{K}{\lVert{{\bm{\mathrm{w}}}_k}\rVert}^2+\alpha\sum_{ij}max(0,1+{\bm{\mathrm{w}}}_{r_{ij}}^\mathrm{T}{\bm{\mathrm{x}}}_{ij}-{\bm{\mathrm{w}}}_{z_{ij}}^\mathrm{T}{\bm{\mathrm{x}}}_{ij}),\\
s.t.\; if\; Y_i=1, z_{ij}>0,and\; if\; Y_i=0,z_{ij}=0,\\
\end{split}$$ $where\; r_{ij}=\mathop{\operatorname*{argmax}}_{k\in\{0,\cdots,K\},k\neq z_{ij}}{\bm{\mathrm{w}}}_k^\mathrm{T}{\bm{\mathrm{x}}}_{ij}.$
The first term in Eq. (\[obj\_eq\]) is for margin regularization and the second term is the multi-class hinge loss maximizing both inter- and intra-class margins. $\alpha$ balances the weight between two terms. In MLPL, all negative instances are utilized in the optimization step. Latent labels in each class are initialized by K-means and updated according to their ‘positiveness’ to each latent class. Constraint in Eq. (\[obj\_eq\]) forces the function to learn latent patterns in the $i$-th class. Though the optimization solution problem is a non-convex one, a local optimization can be guaranteed once the latent information is given [@felzenszwalb2010object]. We take ${\bm{\mathrm{w}}}_{k}^\mathrm{T}{\bm{\mathrm{x}}}_{ij}$ as confidence scores of ${\bm{\mathrm{x}}}_{ij}$ belonging to the $k$-th latent class. Output descriptors are thus represented by the combination of confidence scores belonging to each latent class. Our method is different from [@wang2013max] in three aspects. First, our work focuses on mining latent patterns rather than differentiating subjects of interests from the background. Second, latent classes are learned from all positive instances. The instance selection is removed as there are no background subjects in our problem. Third, instead of using the fixed number of latent classes, multi-scale strategy is adopted so that latent classes can be learnt at different semantic scales. We concatenate features learned by each scale.
: Training Instances [**[X]{}**]{}=$\{{\bm{X}}_1,\dots,{\bm{X}}_m\}$, number of latent patterns per class K : Positive instances ${\bm{X}}_\ell$, negative instances [**[X]{}**]{}$-{\bm{X}}_\ell$
[0.5cm]{}[0cm]{} : For negative instances, $z_{ij}$=0. For positive instances, $z_{ij}$ is initialized by K-means. : $N$ times
[1.2cm]{}[0cm]{} : Solve the multi-class SVM optimization problem $$\begin{aligned}
\min_{{\bm{\mathrm{W}}}_\ell}\sum_{k=0}^{K}{\lVert{{\bm{\mathrm{w}}}_k}\rVert}^2+\alpha\sum_{ij}max(0,1+{\bm{\mathrm{w}}}_{r_{ij}}^\mathrm{T}{\bm{\mathrm{x}}}_{ij}-{\bm{\mathrm{w}}}_{z_{ij}}^\mathrm{T}{\bm{\mathrm{x}}}_{ij}),
\end{aligned}$$ $where\; r_{ij}=\mathop{\operatorname*{argmax}}_{k\in\{0 \cdots K\},k\neq z_{ij}}{\bm{\mathrm{w}}}_{k}^\mathrm{T}{\bm{\mathrm{x}}}_{ij}$.
: For positive instances: $$\begin{aligned}
z_{ij}=\mathop{\operatorname*{argmax}}_{k\in\{1,\cdots,K\}}({\bm{\mathrm{w}}}_k^\mathrm{T}{\bm{\mathrm{x}}}_{ij}-{\bm{\mathrm{w}}}_0^\mathrm{T}{\bm{\mathrm{x}}}_{ij}),
\end{aligned}$$
: The learned classifiers =\[${\bm{W}}_1$,…,${\bm{W}}_m$\]. \
: Instances [$\bf{X}$]{} : $\bf{F(X)}=\bf{W}^\mathrm{T}\bf{X}$
In our MLCFL framework, the MLPL process is implemented on the concatenation of low- and mid-level features, where we obtain MLCF as the output of the feature learning stage and input of the classification stage.
Experiments and results
=======================
In this section, we first describe the datasets, evaluation method and experimental settings for the framework. Moreover, we compare our framework, MLCFL, with other closely related methods and test the framework with three classifiers. Then we demonstrate the effectiveness of Max-margin Latent Pattern Learning (MLPL), the complementary property of low-level and mid-level features. We further conduct intra-personal (tasks performed by one person) and inter-personal (tasks performed by different persons) experiments. In the end of this section, we explore the sensitivity of parameters.
Dataset
-------
We evaluate the proposed Multi-Level Complementary Feature Learning (MLCFL) framework on three popular datasets, namely Skoda, WISDM and Opp. Experimental settings on the three datasets are listed as follows.
[0.5cm]{}[0cm]{} **Skoda** [@zappi2008activity] Skoda Mini Checkpoint contains 16 manipulative gestures, which are performed in a car maintenance scenario and collected by 3D acceleration sensors. Only one worker’s data are recorded, including actions such as ‘open hood’, ‘close left hand door’, etc. The sampling rate is 96Hz. In our experiments, 10 actions and a null class from one right arm-sensor are taken into consideration. 4-fold cross validation is conducted.
[0.5cm]{}[0cm]{} **WISDM** [@kwapisz2011activity] The dataset is collected by Android-based smart phones in 36 users’ pockets. Records comprise 6 daily actions such as ‘jogging’, ‘walking’, etc. Data are collected through controlled laboratory conditions and the sampling rate is 20Hz. We conduct 10-fold cross validation on this dataset.
[0.5cm]{}[0cm]{} **Opp** [@chavarriaga2013opportunity; @roggen2010collecting] Opportunity Activity Recognition is collected by 242 attributes in the scene that simulates a studio flat. The sampling rate of triaxial accelerometers is 64Hz. In this paper, only data collected by the single inertial sensor on the right low arm are utilized, including 13 low-level actions (‘clean’, ‘open’ and ‘close’, etc.) and a null class. 5-fold cross validation is conducted on this dataset.
Evaluation method
-----------------
In HAR, data are severely unbalanced. Some classes are overrepresented while others are scarce. To adapt this characteristic, we apply the weighted $F_1$ score for evaluation.
**Weighted $F_1$ Score** We denote TP and FP as the number of true positives and false positives respectively, and FN as the number of false negatives. Thus, weighted $F_1$ score can be formulated as follows: $$precision_i=\frac{TP_i}{TP_i+FP_i}$$ $$recall_i=\frac{TP_i}{TP_i+FN_i}$$ $$F_1=\sum_i 2\cdot w_i\frac{precision_i\cdot{recall_i}}{precision_i+recall_i}$$ in which $i$ refers to the class index and $w_i$ is the proportion of the $i$-th class in all the classes. Unless mentioned otherwise, we adopt weighted $F_1$ score throughout experiments.
Experiment setup
----------------
To compare with other methods, we follow the dataset settings in Zeng’s work [@zeng2014convolutional]. Specifically, in all three datasets, we utilize data from a single triaxial inertial sensor. Sensor data are segmented into frames using a sliding window with the size of 64 continuous samples and with 50% overlap. We also test window size of 48, 80 and get F1-value 86.8%, 89.5% on Skoda respectively. Since a frame may contain different labels, only frames with a single label are taken into consideration in Skoda and WISDM. But when it comes to Opp, to obtain enough samples, the label of the frame is determined according to the dominant label. Consequently, Skoda, WISDM and Opp contain around 22,000, 33,000 and 21,000 frames, respectively. In cross-validation, folds are created by randomly choosing samples from the dataset.
To demonstrate the generalization ability of the proposed framework, we test three prevalent low-level features, namely statistical values, FFT coefficients and ECDF-PCA. We calculate statistical values (mean, standard deviation, energy, entropy and correlation coefficients) and FFT coefficients the same way as Pl[ö]{}tz et al. [@plotz2011feature] and obtain 23-dimension and 30-dimension feature vectors for each frame respectively. In our practice, statistical values and FFT coefficients with Z-score normalization gain significant improvement in performances than original ones. For ECDF-PCA, we normalize the raw signal through 60 points inverse equal probability interpolation along each channel. Then the PCA process is conducted, where 30 principal components (30 dimensions) are taken as output features. In the mid-level learning phase, the length of the sub-frame for encoding process is set to 20, and the size of the dictionary is set to 300. We concatenate features from three channels of a single triaxial inertial sensor and yield 900-dimension sparse representations. In the high-level learning phase, the input feature is the concatenation of a low-level feature (statistical values with 23 dimensions, FFT coefficients with 30 dimensions, ECDF-PCA with 30 dimensions) and its corresponding mid-level feature (900 dimensions). The weight parameter $\alpha$ in Eq. (\[obj\_eq\]) is set to 1. Besides, the latent-class number $K$ for each class is set to 5 and 10 in two scales. In the step of optimizing ${\bm{\mathrm{W}}}$, we apply linear SVM [@fan2008liblinear] to handle multi-class classification problem. The number of iterations $N$ is set to 3. Without loss of generality, settings for features are the same throughout experiments.
We test the performance of the MLCFL framework with three representative classifiers, namely K-Nearest Neighbors (KNN), Support Vector Machine (SVM) and Nearest Centroid Classifier (NCC). In the K-Nearest Neighbors algorithm (KNN), samples are classified by a majority vote of their neighbors and assigned to the most common label among its $K$ nearest neighbors. $K$ is set to 5 throughout our experiments. Besides, SVM with linear kernel is a prevalent and efficient solution in classification. Specifically, Liblinear [@fan2008liblinear] is used in our experiment[^1]. Nearest Centroid Classifier (NCC) [@chavarriaga2013opportunity] calculates Euclidean Distance between test samples and each centroid (mean) of training classes. The predicted label is allocated according to the nearest class centroid.
Experiments
-----------
In the following sections, we conduct experiments in five respects: (1) comparing with existing methods; (2) conducting classifiers experiments on MLCF; (3) evaluating the effectiveness of MLPL; (4) exploring the complementary property of the low-level feature and mid-level feature; (5) performing intra- and inter-personal experiments.
### Comparing with existing methods
In this section, comparisons of our framework with several published works are shown in Table \[table:compare with existing methods\], including representative methods: statistical values, FFT coefficients, ECDF-PCA [@plotz2011feature] and W-CNN [@zeng2014convolutional]. W-CNN applies CNN with the partial weight sharing technique to perform HAR tasks. This method achieves high performances in single body-sensor based classification tasks on three popular datasets. Statistical values, FFT coefficients and ECDF-PCA are respectively utilized as low-level features in our framework. Their corresponding output features are abbreviated to Stat-MLCF, FFT-MLCF and ECDF-PCA-MLCF. We follow the metrics in the W-CNN work and present results using KNN in the form of classification accuracy.
Table \[table:compare with existing methods\] illustrates that the general performance of MLCF is the highest. Since our algorithm can be applied as the refining process of these low-level features, improvements on statistical values, FFT coefficients and ECDF-PCA are obvious. Despite parameters being generalized on all three datasets, results show that our method still achieves better results than the state-of-the-art method, W-CNN [@zeng2014convolutional], on available Skoda and WISDM. This shows the effectiveness of MLCFL which integrates advantages of low-, mid- and high-level analyzes.
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### Results from different classifiers
In this section, we demonstrate the ‘good’ feature property of MLCF by presenting results from three classifiers. Results from all three datasets are shown in Table \[table:MLCF\].
It can be observed from Table \[table:MLCF\] that MLCF performs better than low- and mid-level features in all three classifiers and achieves the best performance using KNN. Though the sparse representation is preferred by linear SVM, MLCF of 187 (Skoda), 102 (WISDM) and 238 (Opp) dimensions still show obvious enhancement compared with mid-level features of 900 dimensions. Besides, the decision process of NCC is based on the distribution of the feature, which suggests that the distribution of MLCF in the feature space is more regular and discriminative.
### The effectiveness of the MLPL process
In this section, we demonstrate the effectiveness of the Max-margin Latent Pattern Learning (MLPL) method. In particular, we evaluate the MLPL process on three low-level features, their mid-level features and Compl features (the concatenation of low- and mid-level features). Classification results based on KNN are shown in [Fig.]{} \[fig:fvalue\_MLPL\_graph\].
[Fig.]{} \[fig:fvalue\_MLPL\_graph\] shows that MLPL process achieves remarkable improvements on both mid-level and Compl features. The improvement attributes to their sparse representation which is preferred by the linear SVM solution adopted in MLPL [@li2015data]. Compl feature also takes advantage of the complementary property which is further discussed in Section \[sec:lcp\]. Results also show that MLCF, which is generated by implementing MLPL on the Compl feature, obtains the highest performance within group comparison on all three datasets. By contrast, performances of low-level features through MLPL degrade, which is partly due to the low-feature being poorly linearly separable. MLPL is designed to explore distinctive distributions in the feature space. Directly implementing MLPL on the low-level feature is not efficient considering that its low dimensional description is not preferred by the linear classifier inside MLPL. In this condition, MLPL would fail to yield strict boundaries among latent classes.
![Illustrations of the effectiveness of the MLPL process. In each figure, from left to right, black bars show results of low-level feature, mid-level feature and Compl feature; white bars show corresponding results after the MLPL process.[]{data-label="fig:fvalue_MLPL_graph"}](fig5_MLPL_skoda_statistics.png){width="100.00000%"}
![Illustrations of the effectiveness of the MLPL process. In each figure, from left to right, black bars show results of low-level feature, mid-level feature and Compl feature; white bars show corresponding results after the MLPL process.[]{data-label="fig:fvalue_MLPL_graph"}](fig5_MLPL_skoda_fft.png){width="100.00000%"}
![Illustrations of the effectiveness of the MLPL process. In each figure, from left to right, black bars show results of low-level feature, mid-level feature and Compl feature; white bars show corresponding results after the MLPL process.[]{data-label="fig:fvalue_MLPL_graph"}](fig5_MLPL_skoda_pca.png){width="100.00000%"}
![Illustrations of the effectiveness of the MLPL process. In each figure, from left to right, black bars show results of low-level feature, mid-level feature and Compl feature; white bars show corresponding results after the MLPL process.[]{data-label="fig:fvalue_MLPL_graph"}](fig5_MLPL_wisdm_statistics.png){width="100.00000%"}
![Illustrations of the effectiveness of the MLPL process. In each figure, from left to right, black bars show results of low-level feature, mid-level feature and Compl feature; white bars show corresponding results after the MLPL process.[]{data-label="fig:fvalue_MLPL_graph"}](fig5_MLPL_wisdm_fft.png){width="100.00000%"}
![Illustrations of the effectiveness of the MLPL process. In each figure, from left to right, black bars show results of low-level feature, mid-level feature and Compl feature; white bars show corresponding results after the MLPL process.[]{data-label="fig:fvalue_MLPL_graph"}](fig5_MLPL_wisdm_pca.png){width="100.00000%"}
![Illustrations of the effectiveness of the MLPL process. In each figure, from left to right, black bars show results of low-level feature, mid-level feature and Compl feature; white bars show corresponding results after the MLPL process.[]{data-label="fig:fvalue_MLPL_graph"}](fig5_MLPL_opp_statistics.png){width="100.00000%"}
![Illustrations of the effectiveness of the MLPL process. In each figure, from left to right, black bars show results of low-level feature, mid-level feature and Compl feature; white bars show corresponding results after the MLPL process.[]{data-label="fig:fvalue_MLPL_graph"}](fig5_MLPL_opp_fft.png){width="100.00000%"}
![Illustrations of the effectiveness of the MLPL process. In each figure, from left to right, black bars show results of low-level feature, mid-level feature and Compl feature; white bars show corresponding results after the MLPL process.[]{data-label="fig:fvalue_MLPL_graph"}](fig5_MLPL_opp_pca.png){width="100.00000%"}
### Linear complementary property {#sec:lcp}
In this section, we demonstrate how complementary property of low-level and mid-level features improves the performance. In particular, experiments are conducted to present intuitive comparisons among low-, mid- and Compl features using linear SVM. Results are shown in Fig. \[fig:fvalue\_complementarity\_linear\].
![Comparison of Compl feature towards low- and mid-level features using linear SVM.[]{data-label="fig:fvalue_complementarity_linear"}](fig6_compl_skoda_linear.png "fig:"){width=".32\textwidth"} ![Comparison of Compl feature towards low- and mid-level features using linear SVM.[]{data-label="fig:fvalue_complementarity_linear"}](fig6_compl_wisdm_linear.png "fig:"){width=".32\textwidth"} ![Comparison of Compl feature towards low- and mid-level features using linear SVM.[]{data-label="fig:fvalue_complementarity_linear"}](fig6_compl_opp_linear.png "fig:"){width=".32\textwidth"}
[Fig.]{} \[fig:fvalue\_complementarity\_linear\] shows that Compl feature performs relatively better than low-level and mid-level features separately with linear classifier. As the mid-level feature is based on statistical description of motion primitives, it contains information from a structural perspective. By contrast, the low-level feature extracted from the raw data can capture detailed information like statistics property (Statistics values), frequency property (FFT coefficients) and amplitude distribution (ECDF-PCA). Therefore, low- and mid-level features are designed to describe the signal from different perspectives [@huỳnh2007scalable]. Besides, SVM with linear kernel shows advantages in this condition. This is because in the modeling process, feature weights have inherent characters for feature selection. This property is especially exploited and enhanced in MLPL considering that clusters of linear classifiers are involved. Furthermore, latent classes are initialized by K-means in MLPL, thus clusters are likely to be discriminative in distribution. All those factors contribute to gaining better performances when implementing MLPL on Compl feature.
### Intra- and inter-personal experiments
Intra-personal and inter-personal experiments are also conducted to evaluate our framework in daily life scenes. We perform intra-personal tests on Skoda and inter-personal tests on WISDM. In intra-personal experiments, since data in Skoda is recorded by one subject, we divide each class into 6 parts in time sequences and perform 6-fold cross validation. Results obtained by KNN and SVM are shown in Table \[table:intra\]. In inter-personal experiments, we randomly divide 36 subjects into 10 groups. 10-fold cross validation is conducted on this dataset. The results are shown in Table \[table:inter\].
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MLCF with SVM and KNN generally gain the best performance in both intra-personal and inter-personal test, which can be observed from Table \[table:intra\] and Table \[table:inter\], confirming the robustness of our feature learning framework. In intra-personal tests, MLCF yields better performance in KNN than in SVM, suggesting that similarity strategy adopted by KNN is more suitable for MLCF in inter-personal tasks. Besides, high performance MLCF obtained can be contributed to the fact that action patterns of specific subject have been learned during training and thus can be decently represented in testing. By contrast, in inter-personal experiments, MLCF performs better in SVM than in KNN, owing to SVM’s flexibility [@belousov2002flexible] in dealing with varieties among subjects.
### Complexity
We also estimate the overall complexity. Approximately, given fixed-length sampling series, in the first (low-level) stage, the complexity depends on types of low-level features extracted. In the second (mid-level) stage, the complexity is proportional to the product of the size of dictionary and frame. In the third (high-level) stage, the complexity is proportional to the size of latent classes. A typical training and testing time in our experiment at Sokda using FFT as the low feature and Liblinear as classifier is 16 min in training 16,500 samples and 1 minutes in testing 5,500 samples (i7-7700HQ, 2.80 GHZ, 8GB).
Sensitivity of parameters
-------------------------
This section elucidates the evaluation of the variable sensitivity in our framework, including the number of latent classes in the MLPL phase and the dictionary size in the mid-level feature learning phase.
![The effect of the latent class size on MLCF. From left to right, classification results of MLCF are obtained by SVM, KNN and NCC, respectively. The ‘multi-scale’ refers to two scales of 5 and 10.[]{data-label="fig:latent classes on Opp"}](fig7_latent_opp_linear.png "fig:"){width=".32\textwidth"} ![The effect of the latent class size on MLCF. From left to right, classification results of MLCF are obtained by SVM, KNN and NCC, respectively. The ‘multi-scale’ refers to two scales of 5 and 10.[]{data-label="fig:latent classes on Opp"}](fig7_latent_opp_knn.png "fig:"){width=".32\textwidth"} ![The effect of the latent class size on MLCF. From left to right, classification results of MLCF are obtained by SVM, KNN and NCC, respectively. The ‘multi-scale’ refers to two scales of 5 and 10.[]{data-label="fig:latent classes on Opp"}](fig7_latent_opp_ncc.png "fig:"){width=".32\textwidth"}
### Size of latent classes
We evaluate the sensitivity of the latent class size in MLPL on the Opp dataset, the results of which are shown in [Fig.]{} \[fig:latent classes on Opp\]. Latent class sizes are ranging from 5 to 40 and the multi-scale is of 5 and 10. Compared with the single scale, the multi-scale achieves generally the best performance, showing the robustness and effectiveness of learning latent patterns from various semantic levels. From [Fig.]{} \[fig:latent classes on Opp\], it can also be observed that the tendency of the weighted $F_1$ score is inconsistent according to different classifiers. With SVM, the increasing number of latent classes has positive influence before the size reaches around 30 and after that, the performance slightly degrades. The increase occurs under the circumstance that the specific class is clustered into proper number of groups which are linearly separable in feature space while the degradation can be caused by the case that latent classes are too particular and thus over-represented. Specifically, during MLPL, original properly divided latent classes would be further clustered into small clusters, which are not linearly separable. As MLPL outputs the description of the confidence score of each latent class, over clustering would lead to confusion. The max-margin strategy SVM adopts would be sensitive in this confused description. In terms of KNN, the performance is positively related with increment of the latent class’s size, suggesting K-nearest similarity strategy is resistant to over-clustered cases.
### Size of dictionary
We also change the dictionary size in mid-level feature learning phase to evaluate its influences on Compl feature and MLCF. The experiment is conducted on the Skoda dataset with the size of the motion-primitive dictionary ranging from 100 to 800.
![The influence of the motion-primitive dictionary size on Compl feature and MLCF. From left to right, Compl feature and MLCF are derived from statistical values, FFT coefficients and ECDF-PCA, respectively.[]{data-label="fig:size_of_dictionary_skoda"}](fig8_size_of_dictionary_skoda_statistics.png "fig:"){width=".32\textwidth"} ![The influence of the motion-primitive dictionary size on Compl feature and MLCF. From left to right, Compl feature and MLCF are derived from statistical values, FFT coefficients and ECDF-PCA, respectively.[]{data-label="fig:size_of_dictionary_skoda"}](fig8_size_of_dictionary_skoda_fft.png "fig:"){width=".32\textwidth"} ![The influence of the motion-primitive dictionary size on Compl feature and MLCF. From left to right, Compl feature and MLCF are derived from statistical values, FFT coefficients and ECDF-PCA, respectively.[]{data-label="fig:size_of_dictionary_skoda"}](fig8_size_of_dictionary_skoda_pca.png "fig:"){width=".32\textwidth"}
As shown in [Fig.]{} \[fig:size\_of\_dictionary\_skoda\], performances of MLCF positively correlate with the dictionary size in both SVM and KNN while performances of Compl feature increase and then slightly degrade in SVM but continuously degrade in KNN. Although the increase in dictionary size involves more detailed motion-primitive description and the statistical process makes feature robust to noise, larger size may lead to excessively redundant representation, especially when features are concatenated in three channels. Under this condition, when Compl features are directly fed into classifiers, data around 22,000 frames are not sufficient in training models. In contrast, MLCF yields better results when the performance of the Compl feature turns to degrade.
Conclusion
==========
In this paper, we present the MLCFL framework for signal processing in HAR. The framework consists of three parts, obtaining low-level, mid-level and high-level features separately. The low-level feature captures property of raw signals. The mid-level feature achieves component-based representation through hard coding process and occurrence statistics. At high level, the latent semantic learning method MLPL is proposed to mine latent action patterns from concatenation of low- and mid-level features, during which the semantic representation can be achieved. Our framework achieves the state-of-the-art performances, 88.7%, 98.8% and 72.6% (weighted $F_1$ score) respectively, on Skoda, WISDM and OPP datasets. Given that the MLPL method has the ability of discovering various patterns inside the specific class, it is possible to apply this framework in more challengeable scenarios, like tasks without full annoatations. So a potential improvement of our future work is to merge instance selection processing into current framework in order to deal with wealky learning problems.
Author contributions {#author-contributions .unnumbered}
====================
Designed the program and wrote the code: YX ZS XZ SD YG. Provided the data: YX YG. Performed the experiments: ZS XZ SD. Analyzed the data: YG SD YW. Wrote the paper:YX ZS YW YG XZ SD YF EC.
Acknowledgements {#acknowledgements .unnumbered}
================
This work is supported by Microsoft Research under eHealth program, Beijing National Science Foundation in China under Grant 4152033, Beijing Young Talent Project in China, the Fundamental Research Funds for the Central Universities of China under Grant SKLSDE-2015ZX-27 from the State Key Laboratory of Software Development Environment in Beihang University in China.
[1]{}
[^1]: Different from the Liblinear in high-level feature learning, here we use it as a classifier
|
---
author:
- 'Lei Ming,'
- 'Taifan Zheng,'
- 'Yeuk-Kwan E. Cheung'
title: 'Following the density perturbations through a bounce with AdS/CFT Correspondence'
---
Introduction
============
A universe with a bounce process (see for example [@Battefeld:2014uga; @Brandenberger:2016vhg] for two most updated reviews) is a possible solution of the cosmic singularity problem [@Borde:1993xh; @Borde:1996pt] in the standard cosmology within the inflation paradigm [@Guth:2013sya; @Linde:2014nna]. The bounce universe postulates that a phase of matter-dominated contraction precedes the big bang during which the scale factor of the universe reaches a non-zero minimal value. There have been many attempts to extend the standard cosmology beyond the Big Bang, the most notable first effort being the Pre-Big-Bang cosmology [@Gasperini:1992em; @Buonanno:1997zk], and then the Ekpyrotic cosmology [@Khoury:2001wf]. A breakthrough was due to the key observation by D. Wands [@Wands:1998yp] in which he pointed out a scale invariant spectrum of primordial density perturbations can be generated during a matter dominated contraction. Although the spectrum generated in his naive model was later proved to be unstable, it opens a new chapter in cosmological modeling of the early universe.
Building on many pioneering works to utilize AdS/CFT correspondence [@Maldacena:1997re] in cosmological studies [@Kumar:2015gln; @Bzowski:2015clm; @Kumar:2015jxa; @Barbon:2015ria; @Engelhardt:2015gla; @Heidenreich:2015wga; @Engelhardt:2015gta; @Enciso:2015qva; @Banerjee:2015fua; @Battarra:2014tga; @Engelhardt:2014mea; @Morrison:2014jha; @Brandenberger:2013zea; @Enciso:2013lza; @Smolkin:2012er; @Enciso:2012wu; @Barbon:2011ta; @Awad:2009bh; @Awad:2008jf; @Craps:2008cj; @Awad:2007fj; @Turok:2007ry; @Chu:2007um; @Das:2006dz; @Chu:2006pa; @Hamilton:2005ju; @Hertog:2005hu; @Durrer:2002jn], in this paper, we use the correspondence to study how a spectrum generated during the contraction phase can evolve through the bounce in a particular bounce universe model. We are going to conduct our investigations on the coupled scalar tachyon bounce (CSTB) model [@Li:2011nj] constructed earlier, which is based on the D-brane and anti-D-brane dynamics in Type IIB string theory.
The CSTB model has been shown to solve the singularity, horizon and flatness problem [@Cheung:2016oab]; it can produce a scale invariant as well as stable spectrum of primordial density perturbations [@Li:2012vi; @Li:2013bha]. Furthermore predictions testable using dark matter direct detections have been extracted (for a wide class of bounce models) [@Li:2014era; @Cheung:2014nxi; @Cheung:2014pea; @Vergados:2016niz]. An out-of-thermal-equilibrium dynamics of matter production in the the bounce universe makes the bounce scenario very distinct [@Li:2014era] from the standard model of cosmology in which thermal equilibrium dynamics washes out early universe information. A short review of the key ideas can be found [@Cheung:2016wik; @Cheung:2014nxi]. We would like to further corroborate our model by investigating the fluctuations across the bounce.
The fact that CSTB is a string-inspired model and the bounce point may be strongly gravitationally-coupled prompts us to use the AdS/CFT correspondence [@Maldacena:1997re] to study the evolution of the primordial density fluctuations in a Type IIB string background. We take the bulk spacetime metric to be a time dependent $AdS_5 \times S^5$ with its four dimensional part being a FLRW (Friedmann-Lematre-Robertson-Walker) spacetime. In [@Brandenberger:2016egn] a recipe is provided to map the bulk dynamic, fro and back, to the boundary. In this work we improve on their recipe by finding a solution to the dilaton dynamic equation of motion with more realistic Type IIB fields configurations.
According to the AdS/CFT correspondence, which is a strong/weak duality, i.e. when the bulk fields are strongly coupled the boundary is described by a weakly coupled field theory, and vice versa, the bulk fields have dual-operators prescribed by the boundary theory. The dilaton field is related to the square of the gauge field strength, and the gauge coupling of the boundary theory is determined by the vev of the dilaton $\phi$. Therefore the first step is to find a time dependent solution of dilaton equation which, in turn, determines the dynamics of gauge fields on the boundary. Consequently when the boundary gauge field theory becomes weakly coupled during the contraction, we can map the bulk fluctuations onto the boundary and observe its evolution through the bounce.
The bounce process in the bulk could be potentially violent or highly singular in nature – although this is not the case for the CSTB model which enjoys a string theoretical completion at high energy and has a minimum radius – the gauge fields on the boundary, however, evolves most smoothly.
After the bounce, we map the evolved fluctuations – using again the AdS/CFT dictionary – back to the bulk as the gravitational dynamics return to a weakly coupled state. The operation described above hence allows comparing the post-bounce spectrum with the pre-bounce spectrum and checking whether the scale invariance of the spectrum is respected by the bounce process.
The paper is organized as follows. In section \[sec:dilaton\] we present a time dependent dilaton solution with nonzero Ramond-Ramond charges in Type IIB string theory. We describe the cosmic background in which CSTB model can be constructed. In section \[sec:gauge-fields\], we use the results of the previous section to solve the equation of motion of the boundary gauge fields near bounce point, and match the solutions at different evolutionary phases; and finally check whether the spectrum is altered during the bounce. In section \[sec:disc\], we summarize our findings, discuss a potential caveat and remedies. We conclude with outlook on further studies with alternative solutions.
A time dependent dilaton solution to Type IIB supergravity {#sec:dilaton}
==========================================================
First of all we would like to find a solution of the dilaton in Type IIB supergravity with nonzero Ramond-Ramond potentials [@Bergshoeff:2001pv]. The CSTB cosmos is a string cosmological model that can be embedded into an exact string background with appropriate time dependence. The time dependence is necessary for cosmological studies. Altogether we need to generalize the AdS/CFT correspondence to incorporate time dependence in order to study how the spectrum of primordial density perturbations, generated before the bounce, is affected by the bounce dynamics.
The low energy effective theory of Type IIB string is given by [@Polchinski:1998rr]: $$\begin{aligned}
\label{eq:IIBaction}
\begin{split}S_{IIB}&=S_{NS}+S_R+S_{CS}\\
S_{NS}&=\frac1{2\kappa_{10}^2}\int d^{10}x\sqrt{-g}e^{-2\phi}\left(R+4\partial_\mu\phi\partial^\mu\phi-\frac1{12}\left|H_3\right|^2\right)\\
S_R&=-\frac1{4\kappa_{10}^2}\int d^{10}x\sqrt{-g}\left(\left|F_1\right|^2+\frac1{3!}\left|{\widetilde F}_3\right|^2+\frac1{2\times5!}\left|{\widetilde F}_5\right|^2\right)\\
S_{CS}&=-\frac1{4\kappa_{10}^2}\int C_4\wedge H_3\wedge F_3\end{split}\end{aligned}$$ where the field strengths are defined as ${\widetilde F}_3=F_3-C_0\wedge H_3$, ${\widetilde F}_5=F_5-\frac12C_2\wedge H_3+\frac12B_2\wedge F_3$, and $F_3=dC_2$, $F_5=dC_4$, $H_3=dB_2$. The p-forms fields arise from the Ramond-Ramond sector and couple to D-branes of various dimensions; whereas $\phi$ is the dilaton field we are interested in. Note that there is an added constraint which should be imposed on the solution that the 5-form field strength: ${\widetilde F}_5$ is self-dual, ${\widetilde F}_5=\ast{\widetilde F}_5$. The field equations derived from the action (\[eq:IIBaction\]) should be consistent with, but do not imply, it.
The deformed $AdS_5\times S^5$ spacetime metric we will be working on is, $$\label{eq:ads5s5}
ds^2=\frac{L^2}{z^2}\left[- dt^2+a^2(t)\delta_{ij}dx^idx^j+dz^2\right]
+ L^2d\Omega_5^2$$ where $d\Omega_5^2$ being the metric of the unit $S^{5}$ and $a(t)$ being the scale factor of the 4-dimensional FLRW universe and $L$ the AdS radius.
The equation of motions are [@Sfetsos:2010uq]: $$\label{2.3}
\begin{split}
R_{\mu\nu}+2\partial_\mu\partial_\nu\phi
-\frac14{\left(H_3^2\right)}_{\mu\nu}
=&e^{2\phi}\left[\frac12{\left(F_1^2\right)}_{\mu\nu}
+\frac14{\left({\widetilde F}_3^2\right)}_{\mu\nu}
+\frac1{96}{\left({\widetilde F}_5^2\right)}_{\mu\nu}\right]\\
&-\frac14g_{\mu\nu}\left(F_1^2+\frac16{\widetilde F}_3^2
+\frac1{240}{\widetilde F}_5^2\right)\end{split}$$ $$\label{2.4}
R-4\partial_\mu\phi\partial^\mu\phi+4\partial_\mu\partial^\mu\phi
-\frac1{12}H^2=0$$ $$\label{2.5}
\ast{\widetilde F}_3\wedge H_3+d\ast dC_0=0$$ $$\label{2.6}
2d\ast{\widetilde F}_3+H_3\wedge{\widetilde F}_5+\frac12B_2\wedge d {\widetilde F}_5-d C_4\wedge H_3=0$$ $$d \label{2.7}\ast{\widetilde F}_5=H_3\wedge F_3$$ $$\label{2.8}
-2d(e^{-2\phi}\ast H) + 2d(C_0\ast{\widetilde F}_3)
+ dC_2\wedge{\widetilde F}_5 + \frac12C_2\wedge
d{\widetilde F}_5-dC_4\wedge dC_2=0$$ In the above $\mu,\nu=0,1...10$; and the subscripts, $p$, denote the ranks of p-form fields.
We need to make some sensible assumptions to solve this formidable array of equations. A common formula for the self-dual ${\widetilde F}_5$ is [@Macpherson:2014eza]: $$\begin{array}{l}
\begin{aligned}
{\widetilde F}_5
=& r(\sqrt{-g_{00}g_{11}g_{22}g_{33}g_{44}}
dx^0\wedge dx^1\wedge dx^2\wedge dx^3\wedge dx^4 \\
&-\sqrt{g_{55}g_{66}g_{77}g_{88}g_{99}}
dx^5\wedge dx^6\wedge dx^7\wedge dx^8\wedge dx^9)~,
\end{aligned}\\
\end{array}$$ as we would like $r$ to be a constant. Note that ${\widetilde F}_5=dC_4-\frac12C_2\wedge dB_2+\frac12B_2\wedge dC_2$, we can assume that $B_2$ and $C_2$ live on the $AdS_5$ part and $dC_4$ lives on the $S^5$ part.
In the orthonormal basis, we can express them as: $$B_2=f_1dy^0\wedge dy^i+f_2dy^i\wedge dy^j
+f_3dy^i\wedge dy^4+f_4dy^0\wedge dy^4~,$$ $$C_2 = g_1dy^0\wedge dy^i+g_2dy^i\wedge dy^j
+g_3dy^i\wedge dy^4+g_4dy^0\wedge dy^4~,$$ where $i=1,2,3$, $\{dy^\mu\}$ are the orthonormal basis, i.e. $dy^\mu=\sqrt{g_{\mu\mu}}dx^\mu$. To lessen the influence and difficulty caused by forms we assume that the coefficients $f_1 \cdots$, $g_1 \cdots$ are at most linear in $y^0$ and $y^4$, then we can get the expression for the $AdS_5$ part of ${\widetilde F}_5$: $$\begin{array}{l}
\begin{aligned}
\frac12(B_2\wedge dC_2-C_2\wedge dB_2)
= &\frac32\lbrack f_1\frac{\partial g_2}{\partial y^4}
+f_3\frac{\partial g_2}{\partial y^0}
+f_2(\frac{\partial g_1}{\partial y^4}
+\frac{\partial g_3}{\partial y^0})\\
&-g_1\frac{\partial f_2}{\partial y^4}
-g_3\frac{\partial f_2}{\partial y^0}
-g_2(\frac{\partial f_1}{\partial y^4}
+\frac{\partial f_3}{\partial y^0})\rbrack~.
\end{aligned}\\
\end{array}$$
We will take these $f_{i}$ to be constant and $g_{j}$ to be linear in $y^0$ and $y^4$, then the constant, $r$, mentioned above becomes: $$r=\frac32( f_1h_3+f_3h_2+f_2h_1)$$ where $h_1=\frac{\partial g_1}{\partial y^4}+\frac{\partial g_3}{\partial y^0}$, $h_2=\frac{\partial g_2}{\partial y^0}$ and $h_3=\frac{\partial g_2}{\partial y^4}$. Since $f_4$ and $g_4$ won’t appear in the equations of forms, we’ll take them to be zero. Therefore $$H_3=dB_2=0$$ $$dC_2=h_1dy^0 \wedge dy^i \wedge dy^4
+ h_2dy^0 \wedge dy^i\wedge dy^j
+ h_3dy^i\wedge dy^j\wedge dy^4$$ $$dC_4=-rdy^5\wedge dy^6\wedge dy^7\wedge dy^8\wedge dy^9~.$$ Putting these expressions of forms into equations (\[2.5\]) to (\[2.8\]) we arrives at $$\label{2.17}
\frac{\partial C_0}{\partial y^4}=-r\frac{h_3}{h_1}$$ $$\label{2.18}
\frac{\partial C_0}{\partial y^0}=-r\frac{h_2}{h_1}$$ $$\label{2.19}
h_1^2=h_2^2-h_3^2$$ Note here we consider the axion field $C_0$ to be linear in time, $y^0$, and in, $y^4$, the spatial direction transverse to our 4-dimensional universe inside the $AdS_{5}$. So far what we do is to represent the forms by the coeffcients $f_{i}$ and $h_{j}$. In addition, we solve for (\[2.4\]) which is the Euler-Lagrange equation of $\phi$: $$\label{2.20}
2\partial_\mu\partial_\nu\phi =4\partial_\mu\phi\partial_\nu\phi
-\frac12g_{\mu\nu}(R+4\partial_\rho \phi \partial^\rho \phi)~.$$
Putting Equations (\[2.17\]) to (\[2.20\]) into (\[2.3\]), we get the equations of $\phi$ when $\mu\nu=00,ii,44$ (with the metric (\[eq:ads5s5\])): $$\frac{3{\displaystyle\dot a}^2}{a^2}-\frac6{z^2}+2\dot\phi^2
+2\phi_{,z}^2+\frac6{a^2}\phi_{,i}^2
=e^{2\phi}\frac{L^2}{z^2}(\frac{r^2h_2^2}{2h_1^2}
-3h_1^2-3h_2^2+\frac92h_3^2)$$ $$\frac{6a^2}{z^2}-2a\ddot a-\dot a^2 + 2a^2\dot\phi^2
-2a^2\phi_{,z}^2-2\phi_{,i}^2
= e^{2\phi}\frac{a^2L^2}{z^2}\frac{15h_1^2}2$$ $$\frac6{z^2}-\frac{3{\displaystyle\ddot a}}a
-\frac{3{\displaystyle\dot a}^2}{a^2}
+ 2\dot\phi^2+2\phi_{,z}^2-\frac6{a^2}\phi_{,i}^2
=e^{2\phi}\frac{L^2}{z^2} (\frac{r^2h_3^2}{2h_1^2}+3h_1^2+\frac92h_2^2-3h_3^2)$$ These are quadratic first-order partial differential equations of $\phi$. Normally they are hard to solve, however, if we view them as linear equations of $\dot\phi^2$, $\phi_{,z}^2$ and $\phi_{,i}^2$, life becomes much easier: $$\label{2.24}
\dot\phi^2 = \frac14e^{2\phi}\frac{L^2}{z^2}\left(\frac{r^2h_2^2}{3h_1^2}
+\frac{r^2h_3^2}{6h_1^2}+\frac{13}2h_1^2-\frac12h_2^2+2h_3^2\right)
+\frac{3\ddot a}{4a}-\frac1{z^2}$$ $$\label{2.25}
\frac{2\phi_{,i}^2}{a^2}
=\frac16\left[e^{2\phi}\frac{L^2}{z^2}\left(\frac{r^2}2-\frac{27}2h_1^2\right)+\frac{12}{z^2}-\frac{6{\displaystyle\dot a}^2}{a^2} -\frac{3\ddot a}a\right]$$ $$\label{2.26}
\phi_{,z}^2=\frac14e^{2\phi}\frac{L^2}{z^2}\left(\frac{r^2h_2^2}{6h_1^2}+\frac{r^2h_3^2}{3h_1^2}-\frac{13}2h_1^2+2h_2^2-\frac12h_3^2\right)
+\frac1{z^2}~.$$
We would like $\phi$ to be spatially homogeneous, i.e. $\phi_{,i}^2=0$; we can take such an approximation of $e^{2\phi}$ that the right side of equation (\[2.25\]) equals to zero, then $$\label{2.27}
e^{2\phi}\frac{L^2}{z^2}
=(\frac{6\dot a^2}{a^2}+
\frac{3\ddot a}a-\frac{12}{z^2})(\frac{r^2}2-\frac{27}2h_1^2)^{-1}$$ Substituting it into (\[2.24\]) we obtain $$\label{2.28}
\dot\phi=\frac12\sqrt{\frac{6m{\displaystyle\dot a}^2}{a^2}+\frac{3(m+1){\displaystyle\ddot a}}a-\frac{12m+4}{z^2}}$$ with $m\, =\, (\frac{r^2}3+\frac{r^2h_3^2}{2h_1^2}+\frac32h_1^2+\frac92h_2^2-3h_3^2)(\frac{r^2}2-\frac{27}2h_1^2)^{-1}$. The constant captures the effects of form fields $C_2$, $B_2$, $C_0$ on the dilaton, $\phi$. In the next section we will see that it is $\dot\phi$ that matters. Note that we should not solve (\[2.27\]) directly since it is actually a result of an approximation instead of an exact solution. If we want exact solutions to Equations (\[2.24\]) to (\[2.26\]) then the second partial derivatives of $\phi$ should satisfy a constraint equation, $\frac{\partial\dot\phi}{\partial z}=\frac{\partial\phi_{,z}}{\partial t}$.
The evolution of the gauge-field fluctuations {#sec:gauge-fields}
==============================================
The boundary gauge theory is described by $\mathcal{N} =4$ SYM theory, we will follow the notations in [@Brandenberger:2016egn]. The Yang-Mills coupling is determined by the dilaton by $g_{\rm YM} ^2=e^{\phi}$. The boundary theory is strongly coupled in the far past. As the universe contracts, the bulk gravity theory becomes more and more strongly coupled. Before we approach the bounce point, we map the fluctuations onto the boundary as it becomes weakly coupled at this point. We let the gauge field evolve well after the bounce ends and the bulk returns to a weakly coupled state.
After rescaling and gauge fixing, the equations of motion for the Fourier modes of the gauge fields ${\widetilde A}$ becomes [@Awad:2008jf]: $$\label{3.1}
{\displaystyle\ddot {\widetilde A}}_k+(k^2+M^2_{\rm YM}){\widetilde A}_k=0$$ where $$\label{3.2}
M^2_{\rm YM}=\frac{\displaystyle\ddot \phi}{2}
-\frac{\displaystyle\dot \phi^2}{4}~.$$
Let us now zoom into the cosmic dynamics near the bounce point and consider the three phases of universe evolution in the CSTB model [@Li:2011nj]: $$\label{3.3}
{\rm Deflation}: a=e^{-Ht},t_1<t<-t_f$$ $${\rm Smooth\ bounce}: a={\rm cosh}{(Ht)}, -t_1\le t\le t_1$$ $${\rm Inflation}: a=e^{Ht},t_1<t<t_f;$$ where $t_1$ is the time when inflation starts and $t_f$ is when it ends. The bounce process is symmetric about $t=0$. The mapping happens at deflation and inflation phases while the bulk becomes strongly coupled. We solve the equations of motion in each phases:
1. [Deflation:]{}\
Putting (\[3.3\]) and (\[2.28\]) into (\[3.2\]) we arrive at $$\label{eq:Mym}
M^2_{\rm YM}=-\frac{3}{16}(3m+1)H^2+\frac{3m+1}{4z^2}\equiv M~.$$ In (\[eq:Mym\]) all the terms are effectively constant, we denote it as $M$. Putting it into (\[3.1\]) yields $${\widetilde A}_k=D_1(k)e^{\beta t}+D_2(k)e^{-\beta t}$$ where $\beta\equiv\sqrt{-k^2-M}$.
2. [Smooth bounce:]{}\
Taking the first order of $t$ we obtain $$\label{eq:}
M^2_{\rm YM}=\frac{3mH^4t}{\sqrt{-\frac{12m+4}{z^2}+3(m+1)H^2}}
-\frac{1}{4}\left(-\frac{3m+1}{z^2}+\frac{3}{4}(m+1)H^2\right)\equiv Pt+Q$$ which yields $${\widetilde A}_k
=E_1(k){\rm Ai}\left[\frac{-k^2-Q-Pt}{(-P)^{\frac{2}{3}}}\right]
+E_2(k){\rm Bi}\left[\frac{-k^2-Q-Pt}{(-P)^{\frac{2}{3}}}\right]~.$$
3. [Inflation:]{} In this case everything is same as deflation except the value of $t$. Therefore $$\label{eq:3.10}
{\widetilde A}_k=F_1(k)e^{\beta t}+F_2(k)e^{-\beta t}~.$$ We denote $\pm t_0$ as the time of mapping and for the sake of convenience, we set the two modes of ${\widetilde A}_k$ to have the same amplitudes after the first mapping, i.e. $$\label{3.11}
D_1(k)e^{-\beta t_0}=D_2(k)e^{\beta t_0}$$
We assume the arguments of both Airy functions to be small and that the $(-P)^{\frac{1}{3}}t$ term dominates. Then we can asymptotically expand the Airy functions to first power in $q\equiv\frac{-k^2-Q-Pt}{(-P)^{\frac{2}{3}}}$: $$E_1(k){\rm Ai}\left(q\right)=\frac{\left(\frac{1}{3}\right)^{\frac{2}{3}}}{\Gamma \left(\frac{2}{3}\right)}E_1(k)$$ $$E_2(k){\rm Bi}\left(q\right)
=\left[\frac{\left(\frac{1}{3}\right)^{\frac{1}{6}}}{\Gamma \left(\frac{2}{3}\right)}+\frac{\left(\frac{1}{3}\right)^{\frac{5}{6}}}{\Gamma \left(\frac{4}{3}\right)}q\right]E_2(k)$$
Now we can match ${\widetilde A}_k$ and its derivative at the end of deflation and at the beginning of inflation, which we denote as $-t_1$ and $t_1$ respectively. Matching ${\widetilde A}_k$ yields: $$\label{3.14}
D_1(k)e^{-\beta t_1}+D_2(k)e^{\beta t_1}
=\frac{\left(\frac{1}{3}\right)^{\frac{2}{3}}}
{\Gamma \left(\frac{2}{3}\right)}E_1(k)
+\left[\frac{\left(\frac{1}{3}\right)^{\frac{1}{6}}}
{\Gamma \left(\frac{2}{3}\right)}
+\frac{\left(\frac{1}{3}\right)^{\frac{5}{6}}}
{\Gamma \left(\frac{4}{3}\right)} q_1\right]E_2(k)$$ $$F_1(k)e^{\beta t_1}+F_2(k)e^{-\beta t_1}
=\left[\frac{\left(\frac{1}{3}\right)^{\frac{2}{3}}}
{\Gamma \left(\frac{2}{3}\right)}
+\frac{\left(\frac{1}{3}\right)^{\frac{4}{3}}}
{\Gamma \left(\frac{4}{3}\right)}q_2\right]E_1(k)
+\left[\frac{\left(\frac{1}{3}\right)^{\frac{1}{6}}}
{\Gamma \left(\frac{2}{3}\right)}
-\frac{\left(\frac{1}{3}\right)^{\frac{5}{6}}}
{\Gamma \left(\frac{4}{3}\right)}q_2\right]E_2(k)$$ where $q_1\equiv q(-t_1)$ and $q_2\equiv q(t_1)$.
Matching ${\displaystyle \dot{\widetilde A}}_k$ yields: $$D_1(k)\beta e^{-\beta t_1}-D_2(k)\beta e^{\beta t_1}
=\frac{\left(\frac{1}{3}\right)^{\frac{5}{6}}}
{\Gamma \left(\frac{4}{3}\right)}(-P)^{\frac{1}{3}}E_2(k)$$ $$\label{3.17}
F_1(k)\beta e^{\beta t_1}-F_2(k)e^{-\beta t_1}
= \frac{\left(\frac{1}{3}\right)^{\frac{4}{3}}}
{\Gamma \left(\frac{4}{3}\right)}P^{\frac{1}{3}}E_1(k)
+\frac{\left(\frac{1}{3}\right)^{\frac{5}{6}}}
{\Gamma \left(\frac{4}{3}\right)}(-P)^{\frac{1}{3}}~.$$
Solving Equations (\[3.14\]) to (\[3.17\]), we get
\[3.18\] F\_1(k) =
where $$\begin{split}
I_1=&-3^{\frac{1}{3}}\Gamma\left(\frac{2}{3}\right)(-P)^{\frac{2}{3}}
+\beta(-P)^{\frac{1}{3}}\left[3^{\frac{1}{3}}\Gamma\left(\frac{2}{3}\right) (q_1+q_2)+9\Gamma\left(\frac{4}{3}\right)\right]\\
& -\beta^2\left[3^{\frac{1}{3}}\Gamma\left(\frac{2}{3}\right)
q_1q_2+3\Gamma\left(\frac{4}{3}\right)(q_1+2q_2)\right]
\end{split}$$ $$\begin{split}
I_2=&-3^{\frac{1}{3}}\Gamma\left(\frac{2}{3}\right)(-P)^{\frac{2}{3}}
+\beta(-P)^{\frac{1}{3}}\left[-3^{\frac{1}{3}}\Gamma\left(\frac{2}{3}\right)(q_1+q_2)-3\Gamma\left(\frac{4}{3}\right)\right]\\
&-\beta^2\left[3^{\frac{1}{3}}\Gamma\left(\frac{2}{3}\right)q_1q_2
+3\Gamma\left(\frac{4}{3}\right)(q_1+2q_2)\right]~,
\end{split}$$ and
\[3.19\] F\_2(k)=
where $$\begin{split}J_1
=&-3^{\frac{1}{3}}\Gamma\left(\frac{2}{3}\right)(-P)^{\frac{2}{3}}
+\beta(-P)^{\frac{1}{3}}\left[3^{\frac{1}{3}}
\Gamma\left(\frac{2}{3}\right) (q_1-q_2)
+3\Gamma\left(\frac{4}{3}\right)\right]\\
&+\beta^2\left[3^{\frac{1}{3}}\Gamma\left(\frac{2}{3}\right)q_1q_2
+3\Gamma\left(\frac{4}{3}\right)(q_1+2q_2)\right]
\end{split}$$ $$\begin{split}
J_2=&3^{\frac{1}{3}}\Gamma\left(\frac{2}{3}\right)(-P)^{\frac{2}{3}}
+\beta(-P)^{\frac{1}{3}}\left[3^{\frac{1}{3}}
\Gamma\left(\frac{2}{3}\right)(q_1+q_2)
+9\Gamma\left(\frac{4}{3}\right)\right]\\
&-\beta^2\left[3^{\frac{1}{3}}\Gamma\left(\frac{2}{3}\right)q_1q_2
+3\Gamma\left(\frac{4}{3}\right)(q_1+2q_2)\right]
\end{split}$$
We are interested in small wave number limit compared to time scales above, i.e. $kt_1\ll 1$. In the typical inflationary process $Ht_f\sim10^2$ and $Ht_1\sim10^{-2}$. Combining these two facts, we can assume $$\label{3.20}
\beta = \sqrt{-k^2-M} =
\sqrt{-k^2+\frac{3}{16}(3m+1)H^2+\frac{3m+1}{4z^2}}\approx \sqrt{-M}$$ In addition, $Q\sim M$, we have $$\label{3.21}
q_1=\frac{-k^2-Q+Pt_1}{(-P)^{\frac{2}{3}}}
\approx\frac{-Q+Pt_1}{(-P)^{\frac{2}{3}}}~.$$ Similar argument goes for $q_2$ as well. From (\[3.20\]) and (\[3.21\]) we can see that $I_{1}$, $I_{2}$ and $J_{1}$, $J_{2}$ are independent of $k$ when $kt_1\ll1$. From (\[3.11\]) we know $$D_1(k)=\frac{1}{2}{\widetilde A}_k(-t_0)e^{\beta t_0},$$ $$D_2(k)=\frac{1}{2}{\widetilde A}_k(-t_0)e^{-\beta t_0}~.$$ Putting these two into (\[3.18\]) and (\[3.19\]), we obtain, after the second matching, ${\widetilde A}_k$, has the form $${\widetilde A}_k = (G_1e^{\beta t}+G_2e^{-\beta t})
{\widetilde A}_k(-t_0)~,$$ both $G_1$ and $G_2$ being independent of $k$. All in all we can conclude that after the bounce the spectral index is not altered.
The reconstruction of the bulk data from boundary data is elucidated in [@Brandenberger:2016egn], we do not reproduce the arguments here. The punch line is that the $k-$dependence of the bulk fluctuations is completely determined by the $k-$dependence of the gauge field fluctuations, $A_k(t)$, which implies, in turn, that the evolution of the gauge fluctuations will preserve scale invariance across the bounce.
Conclusion and discussion {#sec:disc}
=========================
In this paper we used the AdS/CFT correspondence to show that, when $kt_1\ll 1$ or in the long-wave limits of the dual gauge fields on the boundary, the spectral index of the dilaton fluctuations is not altered as the universe described by the CSTB model undergoes a contraction prior to an expansion. The first step was to find a time-dependent solution of the dialton in a type IIB supergravity on a time-dependent $AdS_5\times S^5$. We generalize the previous proposal of [@Brandenberger:2016egn] in which a certain behavior of the dilaton $\phi$ was assumed. We then utilize the dilaton solution to solve for the dynamics of gauge fields living on the boundary of the $AdS_{5}$. We study the gauge fields near the bounce point and matched their behavior at the transitional points in the different phases of cosmic evolution. The combined profile of gauge field evolution is smooth across the bounce point. The bounce process merely alters the amplitudes of the modes in the density perturbations, and it affects them in the same manner. Therefore it cannot alter the intrinsic scale dependence in the spectrum of matter perturbations generated during the phase of cosmic contraction prior to the bounce. Nevertheless as we can see from (\[3.18\]) and (\[3.19\]), as $k$ becomes larger and larger, i.e. if we do not take long wavelength approximation, the dependence in $k$ begins to show up in the spectrum, the implications of which are under investigation.
A clarifying remark is perhaps needed here to distinguish the two kinds of $k$-modes, and their time dependence, involved in the above discussion. The CST bounce universe undergoes a deflation, before the bounce point, accompanied by horizon crossings with modes with different $k$’s crossing at different times. This makes each $k$-mode in the primordial density perturbations pick up an implicit time dependence: only after this implicit time dependence is carefully taken into account can the spectrum has no overall time dependence. This is the key to the stability analysis on the spectrum generated from the CSTB model [@Li:2013bha; @Li:2012vi]. But this commonly concerned k-dependence in the primordial spectrum is not what we have discussed so far in this work. The $k$-modes in (\[3.1\]) are the $k$-modes of the gauge fields living on the boundary of the $AdS$. They are involved in the mapping procedure and merely encode the bulk dynamics holographically at some particular points on the boundary. Therefore they cannot inject or remove any time dependence in the primordial spectrum. Once the dynamics are mapped onto the boundary, there is no more horizon crossing, the gauge fields evolve under their own equations of motion.
We have made several assumptions and approximations throughout the analysis. Different solutions of the dilaton could be obtained with different ansatz of the Ramond-Ramond field configurations. We have simply chosen the most manageable configuration yet retaining interesting physics. With higher orders of time dependence in the dilaton field we have to expand $M^2_{\rm YM}$ to the higher order in $t$ when solving (\[3.1\]). A systematic study of the field configurations and the corresponding effects on the dilaton field is beyond the scope of this paper. These are nevertheless interesting effects together with higher $\prime\alpha$ effects to the whole analysis, which we hope to address in a future publication.
Another line of research would be to properly set up and study the D-brane and anti-D-brane annihilation process for cosmological modeling. This is the basis for building early universe model from string theory. Going beyond effective field theory approach and beyond kinematic analysis or symmetry arguments can give a more realistic touch to string cosmology. What kind of string compactifications can give rise to a nonsingular universe matching up to the array of precision cosmological observations should be the ultimate question to answer for string cosmologists.
This research project has been supported in parts by the NSF China under Contract 11405084. We also acknowledge the European Union’s Horizon 2020 research and innovation programme under the Marie Skĺodowska-Curie grant agreement No 644121, and the Priority Academic Program Development for Jiangsu Higher Education Institutions (PAPD).
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|
---
abstract: 'The process $e^+e^- \to \pi^+\pi^-2\pi^0\gamma$ is investigated by means of the initial-state radiation technique, where a photon is emitted from the incoming electron or positron. Using of data collected around a center-of-mass energy of $\sqrt{s} = \SI{10.58}{\GeV}$ by the experiment at SLAC, approximately signal events are obtained. The corresponding non-radiative cross section is measured with a relative uncertainty of in the energy region around , surpassing all existing measurements in precision. Using this new result, the channel’s contribution to the leading order hadronic vacuum polarization contribution to the anomalous magnetic moment of the muon is calculated as $(g_\mu^{\pi^+\pi^-2\pi^0}-2)/2 = ({\num{17.9}}\pm {\num{0.1}}_\mathrm{stat} \pm {\num{0.6}}_\mathrm{syst}) \times 10^{-10}$ in the energy range $\SI{0.85}{\GeV} < E_\mathrm{CM} < \SI{1.8}{\GeV}$. In the same energy range, the impact on the running of the fine structure constant at the $Z^0$-pole is determined as $\Delta\alpha^{\pi^+\pi^-2\pi^0}(M^2_\mathrm{Z}) = ({\num{4.44}}\pm {\num{0.02}}_\mathrm{stat} \pm {\num{0.14}}_\mathrm{syst}) \times 10^{-4}$. Furthermore, intermediate resonances are studied and especially the cross section of the process $e^+e^- \to \omega\pi^0 \to \pi^+\pi^-2\pi^0$ is measured.'
author:
- 'J. P. Lees'
- 'V. Poireau'
- 'V. Tisserand'
- 'E. Grauges'
- 'A. Palano'
- 'G. Eigen'
- 'D. N. Brown'
- 'Yu. G. Kolomensky'
- 'M. Fritsch'
- 'H. Koch'
- 'T. Schroeder'
- 'C. Hearty$^{ab}$'
- 'T. S. Mattison$^{b}$'
- 'J. A. McKenna$^{b}$'
- 'R. Y. So$^{b}$'
- 'V. E. Blinov$^{abc}$'
- 'A. R. Buzykaev$^{a}$'
- 'V. P. Druzhinin$^{ab}$'
- 'V. B. Golubev$^{ab}$'
- 'E. A. Kravchenko$^{ab}$'
- 'A. P. Onuchin$^{abc}$'
- 'S. I. Serednyakov$^{ab}$'
- 'Yu. I. Skovpen$^{ab}$'
- 'E. P. Solodov$^{ab}$'
- 'K. Yu. Todyshev$^{ab}$'
- 'A. J. Lankford'
- 'J. W. Gary'
- 'O. Long'
- 'A. M. Eisner'
- 'W. S. Lockman'
- 'W. Panduro Vazquez'
- 'D. S. Chao'
- 'C. H. Cheng'
- 'B. Echenard'
- 'K. T. Flood'
- 'D. G. Hitlin'
- 'J. Kim'
- 'T. S. Miyashita'
- 'P. Ongmongkolkul'
- 'F. C. Porter'
- 'M. Röhrken'
- 'Z. Huard'
- 'B. T. Meadows'
- 'B. G. Pushpawela'
- 'M. D. Sokoloff'
- 'L. Sun'
- 'J. G. Smith'
- 'S. R. Wagner'
- 'D. Bernard'
- 'M. Verderi'
- 'D. Bettoni$^{a}$'
- 'C. Bozzi$^{a}$'
- 'R. Calabrese$^{ab}$'
- 'G. Cibinetto$^{ab}$'
- 'E. Fioravanti$^{ab}$'
- 'I. Garzia$^{ab}$'
- 'E. Luppi$^{ab}$'
- 'V. Santoro$^{a}$'
- 'A. Calcaterra'
- 'R. de Sangro'
- 'G. Finocchiaro'
- 'S. Martellotti'
- 'P. Patteri'
- 'I. M. Peruzzi'
- 'M. Piccolo'
- 'M. Rotondo'
- 'A. Zallo'
- 'S. Passaggio'
- 'C. Patrignani'
- 'H. M. Lacker'
- 'B. Bhuyan'
- 'U. Mallik'
- 'C. Chen'
- 'J. Cochran'
- 'S. Prell'
- 'H. Ahmed'
- 'A. V. Gritsan'
- 'N. Arnaud'
- 'M. Davier'
- 'F. Le Diberder'
- 'A. M. Lutz'
- 'G. Wormser'
- 'D. J. Lange'
- 'D. M. Wright'
- 'J. P. Coleman'
- 'E. Gabathuler'
- 'D. E. Hutchcroft'
- 'D. J. Payne'
- 'C. Touramanis'
- 'A. J. Bevan'
- 'F. Di Lodovico'
- 'R. Sacco'
- 'G. Cowan'
- 'Sw. Banerjee'
- 'D. N. Brown'
- 'C. L. Davis'
- 'A. G. Denig'
- 'W. Gradl'
- 'K. Griessinger'
- 'A. Hafner'
- 'K. R. Schubert'
- 'R. J. Barlow'
- 'G. D. Lafferty'
- 'R. Cenci'
- 'A. Jawahery'
- 'D. A. Roberts'
- 'R. Cowan'
- 'S. H. Robertson'
- 'B. Dey$^{a}$'
- 'N. Neri$^{a}$'
- 'F. Palombo$^{ab}$'
- 'R. Cheaib'
- 'L. Cremaldi'
- 'R. Godang'
- 'D. J. Summers'
- 'P. Taras'
- 'G. De Nardo'
- 'C. Sciacca'
- 'G. Raven'
- 'C. P. Jessop'
- 'J. M. LoSecco'
- 'K. Honscheid'
- 'R. Kass'
- 'A. Gaz$^{a}$'
- 'M. Margoni$^{ab}$'
- 'M. Posocco$^{a}$'
- 'G. Simi$^{ab}$'
- 'F. Simonetto$^{ab}$'
- 'R. Stroili$^{ab}$'
- 'S. Akar'
- 'E. Ben-Haim'
- 'M. Bomben'
- 'G. R. Bonneaud'
- 'G. Calderini'
- 'J. Chauveau'
- 'G. Marchiori'
- 'J. Ocariz'
- 'M. Biasini$^{ab}$'
- 'E. Manoni$^a$'
- 'A. Rossi$^a$'
- 'G. Batignani$^{ab}$'
- 'S. Bettarini$^{ab}$'
- 'M. Carpinelli$^{ab}$'
- 'G. Casarosa$^{ab}$'
- 'M. Chrzaszcz$^{a}$'
- 'F. Forti$^{ab}$'
- 'M. A. Giorgi$^{ab}$'
- 'A. Lusiani$^{ac}$'
- 'B. Oberhof$^{ab}$'
- 'E. Paoloni$^{ab}$'
- 'M. Rama$^{a}$'
- 'G. Rizzo$^{ab}$'
- 'J. J. Walsh$^{a}$'
- 'A. J. S. Smith'
- 'F. Anulli$^{a}$'
- 'R. Faccini$^{ab}$'
- 'F. Ferrarotto$^{a}$'
- 'F. Ferroni$^{ab}$'
- 'A. Pilloni$^{ab}$'
- 'G. Piredda$^{a}$'
- 'C. Bünger'
- 'S. Dittrich'
- 'O. Grünberg'
- 'M. He[ß]{}'
- 'T. Leddig'
- 'C. Voß'
- 'R. Waldi'
- 'T. Adye'
- 'F. F. Wilson'
- 'S. Emery'
- 'G. Vasseur'
- 'D. Aston'
- 'C. Cartaro'
- 'M. R. Convery'
- 'J. Dorfan'
- 'W. Dunwoodie'
- 'M. Ebert'
- 'R. C. Field'
- 'B. G. Fulsom'
- 'M. T. Graham'
- 'C. Hast'
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- 'D. W. G. S. Leith'
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- 'D. B. MacFarlane'
- 'D. R. Muller'
- 'H. Neal'
- 'B. N. Ratcliff'
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- 'M. K. Sullivan'
- 'J. Va’vra'
- 'W. J. Wisniewski'
- 'M. V. Purohit'
- 'J. R. Wilson'
- 'A. Randle-Conde'
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- 'P. F. Harrison'
- 'T. E. Latham'
- 'R. Prepost'
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bibliography:
- '2pi2pi0-pd.bib'
title: ' **Measurement of the $e^+e^-\to\pi^+\pi^-\pi^0\pi^0$ cross section using initial-state radiation at** '
---
-[PUB-17/002]{}\
[SLAC-PUB-17147]{}\
[^1]
[^2]
Introduction {#sec:Introduction}
============
The anomalous magnetic moment of the muon, $g_\mu-2$, exhibits a discrepancy of more than three standard deviations [@Davier:2010nc] between experiment and theory, making it one of the most interesting puzzles in contemporary particle physics. New experiments to improve the measurement of $g_\mu-2$ are starting operation at Fermilab [@Chapelain:2017syu] and J-PARC [@Kitamura:2017xyx]. On the theoretical side [@Passera:2004bj], the QED and weak contributions account for the largest contribution to $g_\mu-2$ and have been calculated with precision significantly exceeding the experiment. The theoretical prediction is limited by the hadronic contributions, which cannot be calculated perturbatively at low energies. Therefore, measured cross sections are used in combination with the optical theorem to compute the hadronic part of $g_\mu-2$. This leads to the dominant uncertainty in the standard model prediction of $g_\mu-2$, which is comparable to the experimental precision. Hence, in order to improve the theoretical prediction, accurate measurements of all hadronic final states are needed. In this paper, we present a new measurement of one of the least known cross sections, $e^+e^- \to \pi^+\pi^-2\pi^0$. This measurement supersedes a preliminary analysis [@Druzhinin:2007cs] from on the same final state. The earlier measurement was performed on approximately half of the data set. Additionally, the new analysis improves the systematic uncertainties of the detection efficiency and of the background subtraction.
The limited precision of this cross section also limits the precision of the running of the fine structure constant $\Delta\alpha$.
The experiment is operated at fixed center-of-mass (CM) energies in the vicinity of . Therefore, the method of initial-state radiation (ISR) is used to determine a cross section over a wide energy range. This method uses events where one of the initial particles radiates a photon, thus lowering the effective CM-energy available for hadron production in the electron-positron annihilation process. Events where the photon is emitted as final-state radiation (FSR) can be neglected since their produced number is extremely low and the FSR photon rarely is sufficiently energetic. Hence, the resulting *radiative* cross section is then converted back into the *non-radiative* cross section using the relation [@Bonneau:1971mk] $$\frac{\mathrm{d}\sigma_{\pi^+\pi^-2\pi^0\gamma}(M)}{\mathrm{d}M} = \frac{2M}{s} \cdot W(s,x,C) \cdot \sigma_{\pi^+\pi^-2\pi^0}(M) \text{.}$$ The radiative cross section of the final state $\pi^+\pi^-2\pi^0$ is denoted by $\sigma_{\pi^+\pi^-2\pi^0\gamma}$, while $\sigma_{\pi^+\pi^-2\pi^0}$ is the non-radiative equivalent. The variable $s$ is the square of the CM energy of the experiment, $x = \frac{2 E^\ast_\gamma}{\sqrt{s}}$, $E^\ast_\gamma$ is the CM energy of the ISR photon, and $M = \sqrt{(1-x)s}$ the invariant mass of the hadronic final state, equivalent to the effective CM energy $E_\mathrm{CM}$ of the hadronic system. The radiator function $W(s,x,C)$ describes the probability at the squared CM energy $s$ for an ISR photon of energy $E^\ast_\gamma$ to be emitted in the polar angle range $| \cos\theta^\ast_\gamma | < C$. It is calculated to leading order in a closed form expression [@Czyz:2000wh], while next-to-leading order effects are accounted for by simulation using PHOKHARA [@Czyz:2008kw; @Czyz:2010hj].
This paper is structured as follows: in Sec. \[sec:babar\], the detector and the analyzed data set are described. Section \[sec:sel\] outlines the basic event selection and the kinematic fit, while Sec. \[sec:bkg\] illustrates the background removal procedure. Acceptance and efficiency determination are explained in Sec. \[sec:eff\]. The main results – cross section and contributions to $a_\mu {\mathrel{\mathop:}=}(g_\mu - 2)/2$ as well as $\Delta\alpha$ – are presented in Sec. \[sec:cs\], followed by the investigation of intermediate resonances in Sec. \[sec:intstr\].
The detector and data set {#sec:babar}
=========================
The experiment was operated at the 2 storage ring at the SLAC National Accelerator Laboratory. Its CM energy was mainly set to the resonance at , while smaller samples were taken at other energies. In this analysis, the full data set around the is used, amounting to an integrated luminosity of [@Lees:2013rw]. The detector is described in detail elsewhere [@Aubert:2001tu; @TheBABAR:2013jta]. The innermost part of the detector is a silicon vertex tracker (SVT), surrounded by the Drift Chamber (DCH), both operating in a magnetic field. Together, the SVT and DCH provide tracking information for charged particles. Neutral particles and electrons are detected in the electromagnetic calorimeter (EMC), which also measures their energy. Particle identification (PID) is provided by the information from the EMC, SVT, and DCH combined with measurements from the internally reflecting ring-imaging Cherenkov detector (DIRC). Muons are identified using information from the instrumented flux return (IFR) of the solenoid magnet, consisting of iron plates interleaved with resistive plate chambers and, in the later runs, limited streamer tubes.
The detector response to a given final state is determined by a detector simulation based on GEANT4 [@Agostinelli:2002hh], which accounts for changes in the experimental setup over time.
Using the AfkQed [@Bevan:2014iga] event generator, based on EVA [@Binner:1999bt; @Czyz:2000wh], simulation samples of ISR channels are produced. These include the signal process (for efficiency calculation) as well as the background channels $\pi^+\pi^-\pi^0\gamma$, $2(\pi^+\pi^-\pi^0)\gamma$, $K^+K^-2\pi^0\gamma$, and $K_\mathrm{s}K^\pm\pi^\mp\gamma$. For the reaction $e^+e^- \to \pi^+\pi^-3\pi^0\gamma$ two simulations exist within AfkQed, which differ by the presence of the intermediate resonances. The simulated processes are $e^+e^- \to \omega2\pi^0\gamma$ (with $\omega \to \pi^+\pi^-\pi^0$) and $e^+e^- \to \eta\pi^+\pi^-\gamma$ (with $\eta \to 3\pi^0$). An $e^+e^- \to \tau^+\tau^-$ sample was generated with KK2f [@Jadach:1999vf]. In addition, the JETSET [@Sjostrand:1993yb] generator is used to obtain a sample of continuum $e^+e^- \to q\bar{q}$ events (*uds*-sample) to investigate non-ISR-background contributions in data.
PHOKHARA [@Czyz:2008kw; @Czyz:2010hj], an event generator for ISR processes that includes the full NLO matrix elements, is used to cross-check the signal simulation and account for next-to-leading order ISR. Final-state radiation is simulated using PHOTOS [@Barberio:1993qi].
Event Selection and Kinematic Fit {#sec:sel}
=================================
For the final state $\pi^+\pi^-2\pi^0\gamma$ two charged tracks and at least five photons must be detected, since only the decay $\pi^0 \to 2\gamma$ is considered. The photon of highest CM energy is chosen as the ISR photon and is required to have an energy of at least . Furthermore, it must lie in the laboratory frame polar angle range $\SI{0.35}{\rad} < \theta_\mathrm{ISR} < \SI{2.4}{\rad}$, in which detection efficiencies have been extensively studied [@Bevan:2014iga]. The distance of closest approach of a charged track to the beam axis in the transverse plane is required to be less than . The distance of the point closest to the beam axis is required to be less than along the beam-axis from the event vertex. Additionally, the tracks are restricted to the polar angle range $\SI{0.4}{\rad} < \theta_\mathrm{tr} < \SI{2.45}{\rad}$ in the laboratory frame and must have a transverse momentum of at least . In order to select the back-to-back topology typical for ISR events with a hard photon, the minimum laboratory frame angle between the ISR photon and a charged track has to exceed .
Photons with an energy in the laboratory frame $E_{\gamma\mathrm{lab}} > \SI{50}{\MeV}$ and with a polar angle within the same range as the ISR photon are considered to build the $\pi^0$ candidates (the charged track vertex is assumed as their point of origin). The invariant mass of each two-photon combination is required to be within of the nominal $\pi^0$ mass [@PDG], while the resolution is about . An event candidate is then built with the two selected tracks, the ISR photon, and any pair of $\pi^0$ candidates with no photons in common, with the further requirement that at least one of the four photons has to have an energy $E_{\gamma\mathrm{lab}} > \SI{100}{\MeV}$.
Candidate events are subjected to a kinematic fit in the hypothesis $e^+e^- \to \pi^+\pi^-2\pi^0\gamma$ with six constraints (four from energy-momentum conservation and two from the $\pi^0$ mass). The photon combination achieving the best fit result is subsequently used in the reconstructed event. The distribution of $\chi^2_{\pi^+\pi^-2\pi^0\gamma}$, the $\chi^2$ of the kinematic fit, is shown in Fig. \[fig:chi2add\] for data and simulation after full selection (also including the selection criteria described in the following paragraphs). The latter distribution is normalized to data in the region $\chi^2_{\pi^+\pi^-2\pi^0\gamma} < 10$, where a lower background level is expected. The $\chi^2_{\pi^+\pi^-2\pi^0\gamma}$ distributions in data and in the AfkQed simulation sample are similar in shape, but the tail of the data distribution shows the presence of background processes, which are discussed in Sec. \[sec:bkg\]. Only events with $\chi^2_{\pi^+\pi^-2\pi^0\gamma} < 30$ are selected.
{width="\linewidth"}
\[fig:chi2add\]
Besides the kinematic fit to the signal hypothesis, the events are subjected to kinematic fits of the background hypotheses $e^+e^- \to \pi^+\pi^-3\pi^0\gamma$, $e^+e^- \to \pi^+\pi^-\pi^0\gamma$, $e^+e^- \to \pi^+\pi^-\pi^0\eta\gamma$, and $e^+e^- \to \pi^+\pi^-2\eta\gamma$ if the detected number of photons is sufficient for the respective hypothesis. As in the signal hypothesis, the photon pairs are constrained to the mass of the $\pi^0$ or $\eta$ meson in the kinematic fit. The same criteria are applied to the photons as well as to the mass of each two-photon combination as in the kinematic fit to the signal hypothesis (replacing the nominal $\pi^0$ mass by the $\eta$ mass where applicable), and the best combination is selected. In the latter three hypotheses above, the resulting $\chi^2$ values are used to reject the corresponding background channels. The contribution from $\pi^+\pi^-\pi^0\gamma$ is suppressed by imposing the requirement $\chi^2_{\pi^+\pi^-\pi^0\gamma} \geq 25$. The possible background channels $\pi^+\pi^-\pi^0\eta\gamma$ and $\pi^+\pi^-2\eta\gamma$ (with $\eta \to 2\gamma$ in both cases) are rejected through the requirements $\chi^2_{\pi^+\pi^-\pi^0\eta\gamma} > \chi^2_{\pi^+\pi^-2\pi^0\gamma}$ and $\chi^2_{\pi^+\pi^-2\eta\gamma} > \chi^2_{\pi^+\pi^-2\pi^0\gamma}$. The background from $e^+e^- \to \pi^+\pi^-3\pi^0\gamma$ is removed as outlined in Sec. \[ssec:bkg5pi\].
Events containing kaons or muons are suppressed by using the PID-algorithms as outlined in Sec. \[ssec:bkgkaon\] and in Sec. \[ssec:bkgmuon\], respectively.
Background {#sec:bkg}
==========
Background events originate from continuum hadron production, hadron production via ISR, and the leptonic channel $e^+e^- \to \tau^+\tau^-$, all shown in Fig. \[fig:allbkg\]. Most events from such processes are removed by the selection outlined above, but specific vetoes are needed for particular channels containing kaons or muons, the latter predominantly produced in the decay $e^+e^- \to \jpsi2\pi^0\gamma \to \mu^+\mu^-2\pi^0\gamma$. Furthermore, remaining background events are subtracted using simulation and sideband subtraction. The channel $e^+e^- \to \pi^+\pi^-3\pi^0\gamma$ was determined to be the largest ISR background contribution. Since this process has not been measured with sufficient precision before, it is treated separately in a dedicated measurement reported below.
{width="\linewidth"}
\[fig:allbkg\]
Continuum Processes
-------------------
The largest background contribution originates from continuum hadron production. In order to subtract this contribution, a simulation based on the JETSET generator [@Sjostrand:1993yb] is used after modifications discussed below to make it more precise. The *uds*-MC events including a true photon (e.g., ISR or FSR photon, but not a photon from, e.g., a $\pi^0$-decay) with $E_\gamma > \SI{3}{\GeV}$ at generator level are discarded. As the remaining continuum MC-sample does not contain ISR events, a photon from a $\pi^0$ decay must be misidentified as an ISR photon for the event to pass the selection criteria. Since the relative fraction of low-multiplicity events in the continuum simulation is rather unreliable, the continuum sample is normalized by comparing the $\pi^0$ peak in the invariant $\gamma_\mathrm{ISR}\gamma$ mass to data (considering all $\gamma_\mathrm{ISR}\gamma$ combinations, where $\gamma_\mathrm{ISR}$ is the selected ISR photon and $\gamma$ corresponds to any photon not already assigned to a $\pi^0$). The normalization scales the number of continuum events down by approximately a factor of three compared to the prediction by the generator (with a relative uncertainty of the normalization of roughly ) and is applied as a function of the invariant mass $M(\pi^+\pi^-2\pi^0)$ to give a precise result over the full energy range. As is visible in Fig. \[fig:allbkg\], continuum processes, which are subtracted using simulation, amount to approximately of data in the peak region.
$e^+e^- \to \pi^+\pi^-3\pi^0\gamma$ {#ssec:bkg5pi}
-----------------------------------
Since this channel has so far only been measured with large uncertainties [@Cosme:1978qe], a dedicated study was performed. For this purpose, candidate events are subjected to the kinematic fit under the hypothesis $e^+e^- \to \pi^+\pi^-3\pi^0\gamma$. In this study continuum background is subtracted using the sample generated by JETSET, while ISR background is subtracted employing the method outlined in Sec. \[ssec:novobkg\]. The resulting measured event spectrum is shown in Fig. \[fig:5pimass-raw\]. The detection efficiency of $\pi^+\pi^-3\pi^0\gamma$ events is calculated using simulated samples of the intermediate states $\omega2\pi^0\gamma$ and $\eta\pi^+\pi^-\gamma$. Due to their distinct kinematics, the $\chi^2_{\pi^+\pi^-3\pi^0\gamma}$ distributions differ and hence the detection efficiencies determined from either $\omega2\pi^0\gamma$ or $\eta\pi^+\pi^-\gamma$ differ by up to from each other, depending on the invariant mass $M_{\pi^+\pi^-3\pi^0}$.
![Measured $M(\pi^+\pi^-3\pi^0)$ distribution of the $e^+e^- \to \pi^+\pi^-3\pi^0\gamma$ background channel in data.[]{data-label="fig:5pimass-raw"}](img/sigma_tot_5pi-34.pdf){width="\linewidth"}
Studying the $3\pi^0$ and $\pi^+\pi^-\pi^0$ invariant mass distributions in data, it was found that – neglecting interference – about of the $\pi^+\pi^-3\pi^0\gamma$ events are produced via $\omega2\pi^0\gamma$ and about via $\eta\pi^+\pi^-\gamma$, both for $M_{\pi^+\pi^-3\pi^0} < \SI{2.9}{\GeVpercsq}$. Hence, less than of the $\pi^+\pi^-3\pi^0\gamma$ events are produced through other channels or phase space. Since there is no simulation of this fraction of events, a mixture according to the measured production fractions of $\omega2\pi^0\gamma$ and of $\eta\pi^+\pi^-\gamma$ is used to estimate the detection efficiency. It has been checked in the almost background-free data sample around the resonance that the efficiency of the $\chi^2_{\pi^+\pi^-3\pi^0}$ requirement is in excellent agreement between data and the simulation mixture, showing relative differences of less than . The difference between the $\omega2\pi^0\gamma$ and $\eta\pi^+\pi^-\gamma$ efficiencies is taken as the uncertainty for the event fraction not simulated by the $\omega2\pi^0\gamma$ or $\eta\pi^+\pi^-\gamma$ samples. This results in a total relative uncertainty of for the $e^+e^- \to \pi^+\pi^-3\pi^0\gamma$ production rate. Other uncertainties are found to be smaller.
The $M_{\pi^+\pi^-3\pi^0}$ invariant mass distributions in the $\omega2\pi^0\gamma$ and $\eta\pi^+\pi^-\gamma$ simulations differ significantly from the measured $\pi^+\pi^-3\pi^0\gamma$ mass distribution. In order to make the simulation samples as realistic as possible and to use them to estimate the background due to $\pi^+\pi^-3\pi^0\gamma$ events in the $\pi^+\pi^-2\pi^0\gamma$ event sample, their $M_{\pi^+\pi^-3\pi^0}$ distributions are adjusted to reproduce the measured event distribution. For this purpose each MC event is weighted with the factor $N_\mathrm{measured}/N_\text{MC true}$ depending on the event mass $M_{\pi^+\pi^-3\pi^0}$, where $N_\mathrm{measured}$ is the number of events measured in data after efficiency correction and $N_\mathrm{MC true}$ is the number of events produced in simulation. The $\pi^+\pi^-2\pi^0\gamma$ selection has different rejection rates for each simulation sample, since the $\pi^+\pi^-2\pi^0\gamma$ selection is sensitive to the kinematics of the production process. Therefore, the efficiencies of the $\eta\pi^+\pi^-\gamma$ and $\omega2\pi^0\gamma$ simulation samples differ by up to from the mixture of both samples. This number is taken as the uncertainty of the events not produced via $\eta\pi^+\pi^-\gamma$ or $\omega2\pi^0\gamma$, where the efficiency of the mixture is assumed.
This study shows that the $e^+e^- \to \pi^+\pi^-3\pi^0\gamma$ background channel is responsible for less than of the events in the peak region $\SI{1}{\GeVpercsq} \leq M(\pi^+\pi^-2\pi^0) < \SI{1.8}{\GeVpercsq}$, less than for $\SI{1.8}{\GeVpercsq} \leq M(\pi^+\pi^-2\pi^0) < \SI{2.7}{\GeVpercsq}$, and less than of the events for higher masses. It is the dominant ISR background contribution, as seen from the result in Fig. \[fig:allbkg\].
Both uncertainties outlined above need to be considered, namely the uncertainty of the $\pi^+\pi^-3\pi^0\gamma$ yield () and the uncertainty of the rejection rate of $\pi^+\pi^-3\pi^0\gamma$ events in the $\pi^+\pi^-2\pi^0\gamma$ selection (). Although both uncertainties have a common source they are conservatively assumed to be independent and added in quadrature. This results in a total relative uncertainty of of the $\pi^+\pi^-3\pi^0\gamma$ background level.
Hence for $\SI{1}{\GeVpercsq} < M(\pi^+\pi^-2\pi^0) < \SI{1.8}{\GeVpercsq}$ the $e^+e^- \to \pi^+\pi^-3\pi^0\gamma$ background yields an uncertainty of less than , for $M(\pi^+\pi^-2\pi^0) < \SI{2.7}{\GeVpercsq}$, and for higher masses, relative to the measured number of $\pi^+\pi^-2\pi^0$ events. As will be shown in Sec. \[ssec:bkgerr\], this is consistent with the independent final estimate for the background systematics.
Kaonic Final States {#ssec:bkgkaon}
-------------------
Two sizable background channels including kaons exist: $e^+e^- \to K^+K^-2\pi^0\gamma$ and $e^+e^- \to K_\mathrm{s}K^\pm\pi^\mp\gamma$. These final states are suppressed by requiring none of the charged tracks to be selected as a kaon by the particle identification algorithm. This algorithm uses a likelihood-based method outlined in Ref. [@Bevan:2014iga] and introduces a systematic uncertainty of $\SI{0.5}{\percent}$. As shown in Fig. \[fig:allbkg\], the remaining background contributions amount to and for $K_\mathrm{s}K^\pm\pi^\mp\gamma$ and $K^+K^-2\pi^0\gamma$, respectively, and are subtracted via simulation.
Muonic Final States {#ssec:bkgmuon}
-------------------
The only sizable muon contribution is produced by the channel $e^+e^- \to \jpsi2\pi^0\gamma \to \mu^+\mu^-2\pi^0\gamma$. Therefore a combined veto is applied. If the invariant mass of the two charged tracks is compatible with the mass and at least one of the charged tracks is identified as a muon, the event is rejected. Tracks are identified as muons using a cut-based approach combining information from the electromagnetic calorimeter and the instrumented flux return [@Aubert:2001tu; @TheBABAR:2013jta]. It is observed that this combined veto rejects up to of the data sample around the $\psitwos$ mass, while its effect is negligible in the remaining mass range. Due to the uncertainty of the selector, a systematic uncertainty of is introduced in the $\psitwos$ region.
Despite the dedicated veto, a number of muon events still survives the selection due to inefficiency and misidentification of the PID algorithm. Since the muon identification efficiency and $\pi^\pm \to \mu^\pm$ misidentification probability are well known for the PID procedures, the remaining muon contribution is calculated from the data and subsequently removed. This yields a remaining muon background at the peak of approximately $\SI{4}{\percent}$ of the data, while the rest of the mass spectrum is negligibly affected.
After removing the muonic backgrounds, no $\psitwos$ peak is observed in data.
Additional Background Contributions
-----------------------------------
Besides the background contributions listed above, the channels $\pi^+\pi^-\pi^0\gamma$ (after selection $<\SI{0.2}{\percent}$ compared to signal) and $\pi^+\pi^-4\pi^0\gamma$ (after selection $<\SI{0.1}{\percent}$ compared to signal) are subtracted using the generator AfkQed.
The generator KK2f [@Jadach:1999vf] is used for the final state $\tau^+\tau^-$ but after the event selection less than 10 events remain to be subtracted, shown in Fig. \[fig:allbkg\]. Other background contributions are negligible.
Alternative Method: Sideband Subtraction {#ssec:novobkg}
----------------------------------------
The sideband subtraction method is a statistical procedure based on the $\chi^2_{\pi^+\pi^-2\pi^0\gamma}$ distribution of the kinematic fit to determine the appropriate number of events to subtract in each mass bin. The number of signal events is calculated as $$N_\mathrm{1s} = \frac{\beta}{\beta - \alpha} N_1 - \frac{1}{\beta - \alpha} N_2 \text{,}$$ where $N_1$ and $N_2$ are the measured event numbers in the signal ($\chi^2 \le 30$) and sideband ($30 < \chi^2 < 60$) regions, respectively, such that $\alpha = N_{2\text{s}}/N_{1\text{s}}$ with events purely from the signal channel and $\beta = N_{2\text{b}}/N_{1\text{b}}$ with events purely from background. The signal $\chi^2$-distribution is taken from simulation, while the background is modeled by the difference between data and signal simulation (normalized at very low $\chi^2$), hence no background simulation is used. The background contribution from continuum processes is subtracted beforehand. The resulting background level compared to data is shown in Fig. \[fig:sbbkg\] as a function of $M(\pi^+\pi^-2\pi^0)$.
{width="\linewidth"}
\[fig:sbbkg\]
Comparison and Systematic Uncertainties {#ssec:bkgerr}
---------------------------------------
The two independent methods of subtracting the remaining background outlined above are compared in order to estimate the corresponding systematic uncertainty. In the calculation of the $\pi^+\pi^-2\pi^0$ cross section the background subtraction procedure based on simulation is used. The relative difference of the result from the sideband method is shown in Fig. \[fig:2pi2pi0csabreldiff\]. From this distribution, systematic uncertainties of in the region ${\SI{1.2}{\GeVpercsq}}< M(\pi^+\pi^-2\pi^0) < \SI{2.7}{\GeVpercsq}$, and for $M(\pi^+\pi^-2\pi^0) > \SI{2.7}{\GeVpercsq}$ are determined. For $\SI{0.85}{\GeVpercsq} \le M(\pi^+\pi^-2\pi^0) < {\SI{1.2}{\GeVpercsq}}$ the systematic uncertainty due to background subtraction is determined for each bin individually from the difference between the two subtraction methods.
![Ratio of the cross sections measured by adopting a background removal procedure based on simulation and on the sideband subtraction method. The horizontal lines indicate the systematic uncertainties for the subtraction of ISR backgrounds.[]{data-label="fig:2pi2pi0csabreldiff"}](img/sigma_tot_chi2.lt.30_200_full_ab-9){width="\linewidth"}
Acceptance and Efficiencies {#sec:eff}
===========================
In order to calculate the efficiency of detecting a $\pi^+\pi^-2\pi^0\gamma$ event with the ISR photon generated in the angular range $| \cos{(\theta_\gamma^\ast)} | < C = \num{0.94}$ as a function of $M(\pi^+\pi^-2\pi^0)$, the detector simulation and event selection are applied to signal simulation. The result is subsequently divided by the number of events before selection, yielding the global efficiency shown in Fig. \[fig:acc\]. The sharp drop observed at low invariant masses is due to the kinematics of the ISR process. Low invariant masses correspond to a very high energetic ISR photon. Momentum conservation then dictates that the hadronic system must be emitted in a relatively small cone in the opposite direction of the ISR photon. Therefore, at small hadronic invariant masses the inefficiency due to overlapping tracks or photons is increased. Because ISR photons are radiated mostly at small polar angles, the probability of losing part of the hadronic system to the non-fiducial volume of the detector is significantly enhanced at small invariant masses.
![The simulated efficiency as a function of the $\pi^+\pi^-2\pi^0$ invariant mass.[]{data-label="fig:acc"}](img/accep_test-5.pdf){width="\linewidth"}
Photon efficiency
-----------------
In order to correct for inactive material, nonfunctioning crystals, and other sources of inefficiency in the photon detection, which may not be included in simulation, a detailed study is performed [@Lees:2012cr]. For this purpose, the photon in $\mu^+\mu^-\gamma$ events is predicted based on the kinematic information from the charged tracks. The probability to detect the predicted photon is then compared between data and simulation. The result is used to correct the detection efficiency of every event as a function of the polar angle of the ISR photon. As a function of $M(\pi^+\pi^-2\pi^0)$, a uniform inefficiency difference of $\Delta\varepsilon_\gamma(\mathrm{MC} - \mathrm{data}) = \SI[parse-numbers=false]{(1.2 \pm 0.4)}{\percent}$ is observed and the total detection efficiency calculated in simulation is reduced accordingly.
Tracking efficiency
-------------------
Efficiency differences between data and MC are also observed in track reconstruction. This is investigated using $e^+e^- \to \pi^+\pi^-\pi^+\pi^-\gamma$ events with one missing track [@Lees:2012cr]. The missing track is predicted using a kinematic fit and the detection efficiency for the missing track is obtained in data and MC. Due to imperfect description of track overlap, small differences uniform in polar angle and transverse momentum exist. These yield a tracking efficiency correction of $\Delta \varepsilon_\mathrm{tr}(\mathrm{MC} - \mathrm{data}) = \SI[parse-numbers=false]{(0.9 \pm 0.8)}{\percent}$ for both tracks combined, slightly reducing the total detection efficiency calculated in simulation.
$\pi^0$ efficiency
------------------
The probability of detecting a $\pi^0$ is studied extensively to uncover possible discrepancies between data and simulation which would need to be corrected. In the ISR process $e^+e^- \to \omega\pi^0\gamma$, the unmeasured $\pi^0$ from the decay $\omega \to \pi^+\pi^-\pi^0$ can be inferred by a kinematic fit. The $\pi^0$ reconstruction efficiency is then determined as the fraction of events in the $\omega$ peak of the $M(\pi^+\pi^-\pi^0_\text{fit})$ distribution in which the $\pi^0$ has been detected. This method is applied to data and simulation to determine differences between them. The resulting $\pi^0$ detection efficiencies yield an efficiency correction of $\Delta\varepsilon_{\pi^0}(\mathrm{MC} - \mathrm{data}) = \SI[parse-numbers=false]{(3.0 \pm 1.0)}{\percent}$ per $\pi^0$ [@Lees:2011zi], which reduces the total detection efficiency calculated in simulation and has been studied to be valid in the full angular and momentum range.
$\chi^2_{\pi^+\pi^-2\pi^0\gamma}$ selection efficiency
------------------------------------------------------
The choice of $\chi^2_{\pi^+\pi^-2\pi^0\gamma} < 30$ is studied by varying this requirement between and , yielding relative differences up to , which is consequently used as the associated uncertainty. This uncertainty is confirmed in a study over a wider range up to $\chi^2_{\pi^+\pi^-2\pi^0\gamma} = 100$, which uses a clean event sample requiring exactly five photons in the final state in addition to the usual selection. The result is shown in Fig. \[fig:chi2test\], where very good agreement between the $\chi^2_{\pi^+\pi^-2\pi^0\gamma}$ distributions in data and simulation is observed.
![The $\chi^2_{\pi^+\pi^-2\pi^0\gamma}$ distributions in the clean sample for data (black points) and the AfkQed generator (red crosses, normalized to the same area as data in the range $\chi^2_{\pi^+\pi^-2\pi^0\gamma} < 10$).[]{data-label="fig:chi2test"}](img/chi2add_nofph5-20.pdf){width="\linewidth"}
Cross Section {#sec:cs}
=============
The main purpose of this analysis is to determine the non-radiative cross section from the measured event rate: $$\sigma_{\pi^+\pi^-2\pi^0}(M) = \frac{\mathrm{d}N_{\pi^+\pi^-2\pi^0\gamma}(M)}{\mathrm{d}\mathcal{L}(M) \cdot \epsilon(M) (1+\delta(M))} \text{.}
\label{eq:nonradcs}$$ Here, $M \equiv M(\pi^+\pi^-2\pi^0)$, $\mathrm{d}N_{\pi^+\pi^-2\pi^0\gamma}$ is the number of events after selection and background subtraction in the interval $\mathrm{d}M$, $\mathrm{d}\mathcal{L}$ the differential ISR-luminosity, $\epsilon(M)$ the combined acceptance and efficiency, and $\delta$ the correction for radiative effects including FSR. The AfkQed generator used in combination with the detector simulation contains corrections for NLO-ISR collinear to the beam as well as FSR corrections implemented by PHOTOS [@Barberio:1993qi]. The NLO-ISR correction is calculated by comparing the generator with PHOKHARA [@Czyz:2008kw], which includes the full ISR contributions up to NLO. An effect of , constant in $M(\pi^+\pi^-2\pi^0)$, is observed and subsequently corrected for. Final-state radiation shifts events towards smaller invariant masses. Therefore, a mass-dependent correction is applied corresponding to the relative change in the content of each mass bin. This is calculated by dividing the simulated event rate with FSR by the event rate without FSR, as shown in Fig. \[fig:FSRcorr\]. The measured event distribution is then divided by the phenomenological fit function to reverse the effect of FSR.
{width="\linewidth"}
\[fig:FSRcorr\]
Besides radiative effects, the mass resolution is considered in the cross section measurement. The invariant mass $M(\pi^+\pi^-2\pi^0)$ has a resolution of in the range of interest. Since the cross section is given in bins of , events with nominal bin-center mass are distributed such that will lie in the central bin, in each neighboring bin, and in the next bins. The effect of the mass resolution has been studied by performing unfolding procedures based on singular value decomposition [@Hocker:1995kb] and Tikhonov regularized $\chi^2$ minimization with L-curve optimization [@Schmitt:2012kp]. It is observed that the effect of the mass resolution is consistent with zero with a systematic uncertainty of .
Once all corrections are applied and the efficiency is determined (including data-MC differences from photon, track and $\pi^0$ detection), Eq. (\[eq:nonradcs\]) is employed to calculate the non-radiative cross section $\sigma$, displayed in Fig. \[fig:cs\] and listed in Table \[tab:csall\].
Removing the effect of vacuum polarization (VP) leads to the *undressed* cross section $\sigma^\mathrm{(0)}$, which is related to its originally *dressed* equivalent $\sigma$ through the transformation [@Eidelman:1995ny] $$\sigma^\mathrm{(0)}_{\pi^+\pi^-2\pi^0}(E_\mathrm{CM}) = \sigma_{\pi^+\pi^-2\pi^0}(E_\mathrm{CM}) \cdot \left( \frac{\alpha(0)}{\alpha(E_\mathrm{CM})} \right)^2 \text{,}
\label{eq:undressedcs}$$ where $\alpha$ is the QED coupling at the center-of-mass energy $E_\mathrm{CM}$, with $\alpha(0) = \num[separate-uncertainty=false]{7.2973525664(17)e-3}$ [@PDG]. The undressed cross section is also listed in Table \[tab:csall\].
Systematic Uncertainties
------------------------
Table \[tab:syst\] shows the systematic uncertainties in this analysis.
The efficiency predicted by the Monte Carlo generator AfkQed is affected by the relative weight of the resonances included in the simulation. The model used in AfkQed includes the $\rho$, $\rho^\prime$, and $\rho^{\prime\prime}$ resonances as well as the intermediate states $\omega\pi^0$, $a_1(1260)\pi$, and a small contribution from $\rho^0f_0$. The corresponding uncertainty due to their relative weight was determined to be less than .
The normalization of the continuum simulation introduces an uncertainty which translates to in the mass range above and below. The PID algorithms in this analysis generate uncertainty from the kaon identification and uncertainty from the combined muon veto above .
Assuming these effects to be uncorrelated, the total systematic uncertainties listed in Table \[tab:syst\] are found in different mass regions. For $M(\pi^+\pi^-2\pi^0) \leq {\SI{1.2}{\GeVpercsq}}$, the systematic uncertainty due to ISR background subtraction is determined bin by bin and ranges from to . In this region the absolute systematic uncertainty due to ISR background subtraction is calculated as $\SI[parse-numbers=false]{(0.455 \cdot E_\text{CM}/\si{GeV} - 0.296)}{\nano\barn}$. In the region below the measurement is compatible with zero.
Comparison to theory and other experiments
------------------------------------------
The measured cross section is compared to existing data in Fig. \[fig:2pi2pi0compilation\]. Our new measurement covers the energy range from to . The previously existing data was collected by the experiments ACO [@Cosme:1972wt; @Cosme:1976tf], ADONE MEA [@Esposito:1977ct; @Esposito:1979dc; @Esposito:1981dv], ADONE $\gamma\gamma2$ [@Bacci:1980zs], DCI-M3N [@Cosme:1978qe], ND [@Dolinsky:1991vq], OLYA [@Kurdadze:1986tc], and SND [@Achasov:2003bv; @Achasov:2009zz]. The new measurement is in reasonable agreement with the previous experiments except for ND, which lies significantly above all others.
![The measured dressed $\pi^+\pi^-2\pi^0$ cross section (statistical uncertainties only).[]{data-label="fig:cs"}](img/sigma_tot_chi2.lt.30_200_full_ab-16){width="\linewidth"}
{width="\linewidth"}
\[fig:2pi2pi0compilation\]
This cross section measurement is an important benchmark for existing theoretical calculations. In Fig. \[fig:EUpred\], the prediction from chiral perturbation theory including $\omega$, $a_1$ and double $\rho$ exchange [@Ecker:2002cw] is shown in comparison to data. The prediction exhibits similar behavior as the measured cross section, underestimating it slightly but especially at low energies this discrepancy is covered by the systematic uncertainties.
![The low-energy part of the vacuum polarization corrected measured undressed cross section (points with statistical uncertainties) compared to the theoretical prediction (line) from Ref. [@Ecker:2002cw].[]{data-label="fig:EUpred"}](img/sigma_tot_chi2.lt.30_200_full_ab-27){width="\linewidth"}
Contribution to $a_\mu$ and $\Delta\alpha$
------------------------------------------
The result of this analysis is of major importance for the theoretical prediction of the muon gyromagnetic anomaly $a_\mu$. Before , the channel $e^+e^- \to \pi^+\pi^-2\pi^0$ was estimated to contribute approximately of the leading order hadronic part of $a_\mu$, but the size of its uncertainty was more than one fifth of the uncertainty of all hadronic contributions combined [@Davier:2003pw].
The theoretical prediction of $a_\mu$ relates the undressed $e^+e^-$ cross section of a given final state $X$ to the corresponding contribution to $a_\mu$ at leading order via [@Jegerlehner] $$a_\mu^X = \frac{1}{4\pi^3} \int_{s^X_\mathrm{min}}^\infty K_\mu(s) \cdot \frac{\sqrt{1 - \frac{4m_e^2c^4}{s}}}{1 + \frac{2m_e^2c^4}{s}} \cdot \sigma^{(0)}_{e^+e^- \to X}(s) \mathrm{d}s \text{,}$$ where $K_\mu(s)$ is the muon kernel function and $m_e$ the electron mass [@PDG]. Integrating over the energy region $\SI{0.85}{\GeV} \le E_\mathrm{CM} \le \SI{1.8}{\GeV}$ we find $$a_\mu^{\pi^+\pi^-2\pi^0} = ({\num{17.9}}\pm {\num{0.1}}_\mathrm{stat} \pm {\num{0.6}}_\mathrm{syst}) \times 10^{-10} \text{,}$$ where the first uncertainty is statistical and the second systematic, giving a total relative precision of .
Before , the world average covered the energy range $\SI{1.02}{\GeV} \le E_\mathrm{CM} \le \SI{1.8}{\GeV}$ and yielded the result[^3] $(\num{16.76} \pm \num{1.31} \pm \num{0.20}_\mathrm{rad}) \times 10^{-10}$ [@Davier:2003pw], implying a total relative precision of . In this region we measure $a_\mu^{\pi^+\pi^-2\pi^0} = ({\num{17.4}}\pm {\num{0.1}}_\mathrm{stat} \pm {\num{0.6}}_\mathrm{syst}) \times 10^{-10}$ in agreement with the previous value. The uncertainties correspond to a total relative precision of . Hence, the relative precision of the measurement alone is a factor higher than the precision of the world data set without .
For comparison with theory predictions it is worthwhile extending the energy range to higher values. Hence, in the energy range $\SI{0.85}{\GeV} \le E_\mathrm{CM} \le \SI{3.0}{\GeV}$ we obtain $a_\mu^{\pi^+\pi^-2\pi^0} = ({\num{21.8}}\pm {\num{0.1}}_\mathrm{stat} \pm {\num{0.7}}_\mathrm{syst}) \times 10^{-10}$.
Similar to $a_\mu$, the measured undressed cross section can be used to determine this channel’s contribution to the running of the fine-structure constant $\alpha$ [@Eidelman:1995ny]: $$\alpha(q^2) = \frac{\alpha(0)}{1 - \Delta\alpha(q^2)} \text{,}$$ where $\Delta\alpha$ is the sum of all higher order corrections and $q^2$ is the squared momentum transfer. The running of $\alpha$ is often evaluated at the $Z^0$ pole ($q^2=M^2_\mathrm{Z}c^2$). In the energy range $\SI{0.85}{\GeV} \le E_\mathrm{CM} \le \SI{1.8}{\GeV}$ the value $$\Delta\alpha^{\pi^+\pi^-2\pi^0}(M^2_\mathrm{Z}c^2) = ({\num{4.44}}\pm {\num{0.02}}_\mathrm{stat} \pm {\num{0.14}}_\mathrm{syst}) \times 10^{-4}$$ is calculated from this measurement. For higher energies, $\SI{0.85}{\GeV} \le E_\mathrm{CM} \le \SI{3.0}{\GeV}$, we find $\Delta\alpha^{\pi^+\pi^-2\pi^0}(M^2_\mathrm{Z}c^2) = ({\num{6.58}}\pm {\num{0.02}}_\mathrm{stat} \pm {\num{0.22}}_\mathrm{syst}) \times 10^{-4}$.
Intermediate Resonances {#sec:intstr}
=======================
The channel $e^+e^- \to \pi^+\pi^-2\pi^0$ is also of interest due to its internal structures. These shed light on the production process of hadrons and can probe theoretical models or provide input for the latter [@Gudino:2015kra]. In Ref. [@Achasov:2003bv] it is suggested that the channel $e^+e^- \to \pi^+\pi^-2\pi^0$ is described completely by the intermediate states $a_1\pi$ and $\omega\pi^0$ in the energy range $\SI{0.98}{\GeV} < E_\mathrm{CM} < \SI{1.38}{\GeV}$. Furthermore, the authors do not observe a $\rho^0$ signal in their data, consistent with earlier measurements [@Akhmetshin:1998df]. In this work, a study of the $a_1\pi$ intermediate state is undertaken but due to the large width of the $a_1$ resonance it is not possible to quantify the $a_1\pi$ contribution. The role of the $\omega\pi^0$ substructure and a possible $\rho^0$ contribution are investigated in this work over a wider energy range than in previous measurements. A complete study of the dynamics of this process would require a partial wave analysis, preferably in combination with the channel $e^+e^- \to \pi^+\pi^-\pi^+\pi^-$. Since this is beyond the scope of this analysis, only selected intermediate states are presented here.
The efficiency as function of the mass of the sub-system is calculated using AfkQed by dividing the mass distribution after $\pi^+\pi^-2\pi^0\gamma$ selection and detector simulation by the distribution of the generated mass. Furthermore, unless stated otherwise no background subtraction is applied to data when graphing the mass distribution of a subsystem.
One important intermediate state is given by the channel $e^+e^- \to \omega\pi^0\gamma \to \pi^+\pi^-2\pi^0\gamma$ with $\mathcal{B}(\omega \to \pi^+\pi^-\pi^0) = \num{0.892} \pm \num{0.007}$ [@PDG]. Fitting a Voigt profile plus a normal distribution (for the radiative tail) to the efficiency corrected $M(\pi^+\pi^-\pi^0)$ distribution, as shown in Fig. \[fig:omegaglobal\], results in an $\omega\pi^0$ production fraction of over the full invariant mass range. The systematic uncertainty is determined as the difference from an alternative fit function. The same fitting procedure is applied in narrow slices of the invariant mass $M(\pi^+\pi^-2\pi^0)$. The resulting number of events is divided by the ISR-luminosity in each mass region, yielding the cross section $\sigma(e^+e^- \to \omega\pi^0\gamma \to \pi^+\pi^-2\pi^0\gamma)$ as a function of the CM-energy of the hadronic system listed in Table \[tab:csomega\] and shown in Fig. \[fig:omerate\] in comparison to existing data [@Bisello:1990du; @Achasov:2000wy; @Akhmetshin:2003ag; @Achasov:2016zvn]. In this case, possible background processes are removed by the fit function. The $\omega\pi^0$ production fraction dominates at low masses, then decreases rapidly, such that it is on the level of already at $M(\pi^+\pi^-2\pi^0) \approx \SI{1.8}{\GeVpercsq}$, decreasing further towards higher masses.
![The measured $\omega$ data peak in the complete $M(\pi^+\pi^-2\pi^0)$ range after selection and efficiency correction.[]{data-label="fig:omegaglobal"}](img/omegafits-3.pdf){width="\linewidth"}
![The measured $e^+e^- \to \omega\pi^0 \to \pi^+\pi^-2\pi^0$ cross sections from different experiments [@Bisello:1990du; @Achasov:2000wy; @Akhmetshin:2003ag; @Achasov:2016zvn] as a function of $E_\mathrm{CM}$ with statistical uncertainties. Data measured in other decays than $\omega \to \pi^+\pi^-\pi^0$ is scaled by the appropriate branching ratio.](img/plotomega-2.pdf){width="\linewidth"}
\[fig:omerate\]
=
------- ------------------
0.924 0.48 $\pm$ 0.08
0.965 2.96 $\pm$ 0.23
1.005 6.26 $\pm$ 0.30
1.045 9.87 $\pm$ 0.37
1.086 10.82 $\pm$ 0.37
1.126 12.45 $\pm$ 0.38
1.167 12.30 $\pm$ 0.36
1.207 14.75 $\pm$ 0.38
1.247 13.95 $\pm$ 0.36
1.288 15.30 $\pm$ 0.37
1.328 14.85 $\pm$ 0.35
1.369 15.37 $\pm$ 0.35
1.409 15.19 $\pm$ 0.34
1.449 15.57 $\pm$ 0.34
1.490 14.22 $\pm$ 0.30
1.530 11.52 $\pm$ 0.26
1.571 9.05 $\pm$ 0.25
1.611 6.66 $\pm$ 0.20
1.652 4.94 $\pm$ 0.20
1.692 3.52 $\pm$ 0.14
1.732 2.21 $\pm$ 0.11
1.773 1.68 $\pm$ 0.09
1.813 1.19 $\pm$ 0.08
1.854 1.30 $\pm$ 0.08
1.894 0.80 $\pm$ 0.07
1.934 0.63 $\pm$ 0.06
1.975 0.65 $\pm$ 0.06
2.015 0.85 $\pm$ 0.06
2.056 0.94 $\pm$ 0.07
2.096 0.95 $\pm$ 0.07
2.136 0.77 $\pm$ 0.06
2.177 0.73 $\pm$ 0.05
2.217 0.58 $\pm$ 0.05
2.258 0.40 $\pm$ 0.04
2.298 0.34 $\pm$ 0.04
2.338 0.35 $\pm$ 0.04
2.379 0.31 $\pm$ 0.03
2.419 0.25 $\pm$ 0.03
2.460 0.20 $\pm$ 0.03
2.500 0.20 $\pm$ 0.03
------- ------------------
: The measured $e^+e^- \to \omega \pi^0 \to \pi^+\pi^-\pi^0\pi^0$ cross section with statistical uncertainties. The relative systematic uncertainty amounts to .
\[tab:csomega\]
Figure \[fig:m2pim2pi0scat\] shows the 2D plot of the $\pi^+\pi^-$ mass vs. the $\pi^0\pi^0$ mass in the range $\SI{1.7}{\GeVpercsq} < M(\pi^+\pi^-2\pi^0) < \SI{2.3}{\GeVpercsq}$, which is chosen to achieve the best prominence of observed structures. In this mass region, the distribution exhibits an excess of events around $M(\pi^+\pi^-) \approx \SI{0.77}{\GeVpercsq}$ and $M(\pi^0\pi^0) \approx \SI{1.0}{\GeVpercsq}$. Investigating this structure in the efficiency corrected one-dimensional distribution in $M(\pi^+\pi^-)$, Fig. \[fig:mpippim\], shows a substantial peak near the $\rho^0$ mass. Figure \[fig:mpi0pi0\] shows that the peak in the $M(\pi^0\pi^0)$ distribution is around the $f_0(980)$ mass with a sharp edge just above the peak. Moreover, this peak vanishes when rejecting events from the $\rho^0$ region in $M(\pi^+\pi^-)$ as observed in Fig. \[fig:mpi0pi0\_norho\], implying production exclusively in combination with a $\rho^0$.
\[fig:m2pim2pi0scat\]
![The $M(\pi^+\pi^-)$ distribution in the invariant mass interval $\SI{1.7}{\GeVpercsq} < M(\pi^+\pi^-2\pi^0) < \SI{2.3}{\GeVpercsq}$ for data after selection and efficiency correction.[]{data-label="fig:mpippim"}](img/intres_17-23-24.pdf){width="\linewidth"}
![The $M(\pi^0\pi^0)$ distribution in the invariant mass interval $\SI{1.7}{\GeVpercsq} < M(\pi^+\pi^-2\pi^0) < \SI{2.3}{\GeVpercsq}$ for data after selection and efficiency correction.[]{data-label="fig:mpi0pi0"}](img/intres_17-23-25.pdf){width="\linewidth"}
![The $M(\pi^0\pi^0)$ distribution excluding the $\rho^0$ mass range in $M(\pi^+\pi^-)$ in the invariant mass interval $\SI{1.7}{\GeVpercsq} < M(\pi^+\pi^-2\pi^0) < \SI{2.3}{\GeVpercsq}$ for data after selection and efficiency correction.[]{data-label="fig:mpi0pi0_norho"}](img/intres_norho_17-23-25.pdf){width="\linewidth"}
In the other two-pion combination, the masses $M(\pi^\pm\pi^0)$ are studied, whose 2D plot is shown in Fig.\[fig:mpimpi0mpippi0scat\]. Correlated $\rho^+\rho^-$ production is visible as a peak around the $\rho^+\rho^-$ mass-crossing and has not been observed before. In the one-dimensional $M(\pi^\pm\pi^0)$ distribution, Fig. \[fig:mpipmpi0\], a large $\rho^\pm$ peak is observed in data.
\[fig:mpimpi0mpippi0scat\]
![The $M(\pi^\pm\pi^0)$ distribution in data after selection and efficiency correction.[]{data-label="fig:mpipmpi0"}](img/intres-32.pdf){width="\linewidth"}
If background processes are subtracted using simulation for continuum and ISR processes (as outlined in Sec. \[sec:bkg\]) and normalization to efficiency is applied, the $e^+e^- \to \pi^+\pi^-2\pi^0$ mass spectrum can be obtained specifically for resonance regions. Restricting the two-$\pi^0$ mass to the $f_0$ region $\SI{0.89}{\GeVpercsq} < M(\pi^0\pi^0) < \SI{1.09}{\GeVpercsq}$ and the $\pi^+\pi^-$ mass to the $\rho^0$ region $\SI{0.63}{\GeVpercsq} < M(\pi^+\pi^-) < \SI{0.92}{\GeVpercsq}$, as indicated by the black ellipse in Fig. \[fig:m2pim2pi0scat\], results in the mass spectrum shown as the blue circles in Fig. \[fig:cs\_intres\]. Similarly, restricting the $\pi^\pm\pi^0$ masses to the $\rho^\pm$ region $\SI{0.63}{\GeVpercsq} < M(\pi^\pm\pi^0) < \SI{0.92}{\GeVpercsq}$, as indicated by the black circle in Fig. \[fig:mpimpi0mpippi0scat\], results in the mass spectrum shown as the red squares in Fig. \[fig:cs\_intres\]. Although backgrounds from processes besides the signal $e^+e^- \to \pi^+\pi^-2\pi^0$ are subtracted, the mass spectra in both resonance regions still include a sizable fraction of events not produced via the intermediate states $\rho^0f_0$ or $\rho^+\rho^-$, respectively. Nonetheless, a peaking structure is visible especially in the $\rho^0f_0$ distribution.
{width="\linewidth"}
\[fig:cs\_intres\]
Branching Fraction {#sec:jpsi}
==================
The peak in the $\pi^+\pi^-2\pi^0$ cross section is used to determine the branching ratio of $\jpsi \to \pi^+\pi^-2\pi^0$. For this purpose, the number of events in the channel $e^+e^- \to \pi^+\pi^-2\pi^0$ normalized to luminosity is obtained from data using the Gaussian fit shown in Fig. \[fig:jpsi\] and is corrected for non-normality of the mass resolution. A linear parametrization is employed for the background, which is dominated by non-resonant $e^+e^- \to \pi^+\pi^-2\pi^0$ production.
From the fit, the product of the integrated cross section and the branching fraction $\jpsi \to \pi^+\pi^-2\pi^0$ is determined: $$\mathcal{B}_{J/\psi \rightarrow \pi^+\pi^-2\pi^{0}}\sigma^{\jpsi}_\text{int} = \SI[parse-numbers=false]{(68 \pm 4_\mathrm{stat} \pm 5_\mathrm{fit} )}{\nano\barn \MeVpercsq} \text{.}$$ From the integrated cross section of a resonance the following relation for calculating the branching fraction is derived [@nagashima1] (with $M_{J/\psi} = \SI[parse-numbers=false]{(3096.900 \pm 0.006)}{\MeVpercsq}$ [@PDG]): $$\begin{aligned}
\mathcal{B}_{J/\psi \rightarrow \pi^+\pi^-2\pi^{0}} \Gamma^{J/\psi}_{ee} &= \frac{N(J/\psi \rightarrow \pi^+\pi^-2\pi^{0}) \cdot M^2_{J/\psi}c^4}{6 \pi^2 \hbar^2 c^2 \cdot \mathrm{d}\mathcal{L}/\mathrm{d}E \cdot \varepsilon} \\
&= \SI[parse-numbers=false]{(28.3 \pm 1.7_\mathrm{stat} \pm 2.1_\mathrm{syst})}{\electronvolt} \text{,}
\end{aligned}$$ where $\varepsilon$ is the detection efficiency and the input uncertainty is negligible. If this value is divided by $\Gamma^{J/\psi}_{ee} = \SI[parse-numbers=false]{(5.55 \pm 0.14)}{\keV}$ [@PDG], the branching fraction follows: $$\mathcal{B}_{J/\psi \rightarrow \pi^+\pi^-2\pi^{0}} = (\num{5.1} \pm \num{0.3}_\mathrm{stat} \pm \num{0.4}_\mathrm{syst} \pm \num{0.1}_\mathrm{input}) \times 10^{-3} \text{,}$$ where the input uncertainty is the propagation of the uncertainties of $M^2_{J/\psi}$, $\Gamma^{J/\psi}_{ee}$, and $\hbar c$. The systematic uncertainty is determined by the systematic uncertainty of the general analysis with the exception of the background subtraction. In this study, the background is subtracted via the fit function and hence its systematic uncertainty is included in the model error, which is determined by fitting several peak and background shapes to data.
Summary and Conclusions {#sec:summary}
=======================
In this study, the cross section $e^+e^- \to \pi^+\pi^-2\pi^0$ is measured with unprecedented precision. At large invariant masses $M(\pi^+\pi^-2\pi^0) > \SI{3.2}{\GeVpercsq}$, a systematic precision of is reached, while in the region $\SI{2.7}{\GeVpercsq} < M(\pi^+\pi^-2\pi^0) < \SI{3.2}{\GeVpercsq}$ it is . In the peak region ${\SI{1.2}{\GeVpercsq}}< M(\pi^+\pi^-2\pi^0) < \SI{2.7}{\GeVpercsq}$ a relative systematic uncertainty of is achieved.
This measurement is subsequently used to calculate the channel’s contribution to $a_\mu$ in the energy range $\SI{0.85}{\GeV} \le E_\mathrm{CM} \le \SI{1.8}{\GeV}$: $$a_\mu(\pi^+\pi^-2\pi^0) = ({\num{17.9}}\pm {\num{0.1}}_\mathrm{stat} \pm {\num{0.6}}_\mathrm{syst}) \times 10^{-10} \text{.}$$ For $\SI{0.85}{\GeV} \le E_\mathrm{CM} \le \SI{3.0}{\GeV}$ we obtain $$a_\mu^{\pi^+\pi^-2\pi^0} = ({\num{21.8}}\pm {\num{0.1}}_\mathrm{stat} \pm {\num{0.7}}_\mathrm{syst}) \times 10^{-10} \text{.}$$
Furthermore, intermediate structures from the channels $\rho^0f_0$ and $\rho^+\rho^-$ are seen. The contribution produced via $\omega\pi^0$ is studied and the cross section measured. The branching fraction $\jpsi \to \pi^+\pi^-2\pi^0$ is determined. For a deeper understanding of the production mechanism, a partial wave analysis in combination with the process $e^+e^- \to \pi^+\pi^-\pi^+\pi^-$ [@Lees:2012cr] is necessary.
Acknowledgments {#sec:Acknowledgments}
===============
We are grateful for the extraordinary contributions of our 2 colleagues in achieving the excellent luminosity and machine conditions that have made this work possible. The success of this project also relies critically on the expertise and dedication of the computing organizations that support . The collaborating institutions wish to thank SLAC for its support and the kind hospitality extended to them. This work is supported by the US Department of Energy and National Science Foundation, the Natural Sciences and Engineering Research Council (Canada), the Commissariat à l’Energie Atomique and Institut National de Physique Nucléaire et de Physique des Particules (France), the Bundesministerium für Bildung und Forschung and Deutsche Forschungsgemeinschaft (Germany), the Istituto Nazionale di Fisica Nucleare (Italy), the Foundation for Fundamental Research on Matter (The Netherlands), the Research Council of Norway, the Ministry of Education and Science of the Russian Federation, Ministerio de Economía y Competitividad (Spain), the Science and Technology Facilities Council (United Kingdom), and the Binational Science Foundation (U.S.-Israel). Individuals have received support from the Marie-Curie IEF program (European Union) and the A. P. Sloan Foundation (USA). The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for United States Government purposes.
[^1]: Deceased
[^2]: Deceased
[^3]: The second uncertainty corresponds to a correction of radiative effects, while the first is the combined statistical and systematic uncertainty.
|
---
author:
- 'Jörg Herbel,'
- 'Tomasz Kacprzak,'
- 'Adam Amara,'
- 'Alexandre Refregier,'
- Aurelien Lucchi
bibliography:
- 'psf\_model\_cnn.bib'
title: Fast Point Spread Function Modeling with Deep Learning
---
Introduction
============
The Point Spread Function (PSF) of an observation describes the mapping of a point source in the sky onto the two-dimensional image plane (e.g. [@Bradt2003]). Diffraction effects in the optical components of the telescope as well as turbulences in the Earth’s atmosphere (for ground-based facilities) cause a point source to be mapped into an extended object with a finite size. Other effects such as imperfections in optical elements along the light path and charge diffusion in digital detectors also contribute to the PSF, which affects all observed objects by smearing their images. Mathematically, this can be described as a convolution of the intrinsic light distribution with a kernel characterizing the PSF. Point sources, such as stars, are effectively noisy samples of the PSF at the position of the star. Extended objects like galaxies appear larger due to the smearing caused by the PSF.
The PSF impacts any measurement of galaxy properties from astronomical images. Therefore, estimating and modeling the PSF is important in many branches of astrophysics and cosmology. For example, the PSF needs to be taken into account when studying galaxy morphology by fitting observed objects with models of the light distribution (e.g. [@Haeussler2007; @Gabor2009]). Another example is the analysis of strong lensing systems [@Birrer2016; @Wong2017]. When modeling the measured light distribution, the PSF has to be included in the model to avoid biased results. Also, cosmic shear measurements (reviewed in [@Refregier2003b; @Hoekstra2008; @Kilbinger2015]) require a precise treatment of the PSF, which induces distortions that can contaminate the measured lensing signal (e.g. [@Heymans2012]).
The impact of incorrectly modeling the PSF on weak lensing measurements has been studied extensively (for example [@Paulin-Henriksson2008; @Paulin-Henriksson2009; @Amara2010; @Massey2013]). This is because the PSF is one of the dominant sources of systematic errors for cosmic shear and thus needs to be taken into account carefully to avoid biases in the cosmological parameter inference. Various methods have been developed to model the PSF for weak lensing analyses. For example, the cosmic shear pipeline of the Kilo-Degree Survey (KiDS) [@Hildebrandt2017] uses shapelets [@Refregier2003a] as basis functions to model the PSF. Another option, pursued by the weak lensing pipeline of the Dark Energy Survey (DES) [@Zuntz2017], is a pixel-based approach [@Bertin2011]. In this case, a basis for PSF modeling is obtained directly from the data by performing a principal component analysis (reviewed in [@Jollife2016]). Further methods to estimate and model the PSF are given in [@Kitching2013].
In recent years, forward modeling has become an increasingly popular method to calibrate shear measurements. A consistent framework for this is the Monte-Carlo Control Loops (*MCCL*) scheme [@Refregier2014; @Bruderer2016; @Herbel2017; @Bruderer2017], originally developed for wide-field imaging data. This method typically requires simulating a large amount of synthetic images with realistic properties. One of the most important features that the simulations need to capture is the PSF pattern in the data, since an incorrectly modeled PSF will lead to model biases in the measured lensing signal. To do this, the PSF model needs to be flexible so as to capture possible complexities in the data while being fast enough to run on large datasets.
In this paper, we construct a PSF model that is informed by data. We do this using a principal component analysis of the stars in our sample, which informs us about the main features our model has to reproduce. The model consists of a base profile that is distorted through a pertubative expansion. This model has several parameters that we fit using a machine-learning method known as Convolutional Neural Network (CNN). Deep-learning methods such as CNNs have been shown to perform especially well on imaging data (e.g. [@LeCun2015]). An additional reason for choosing a CNN is that these networks are very fast once trained, which is crucial in the context of forward modeling and *MCCL*. Deep Learning has gained increasing attention in astrophysics and cosmology recently and it has been applied successfully to various problems in the field (see e.g. [@Ravanbakhsh2016; @Charnock2017; @Hezaveh2017; @Schmelzle2017; @Lanusse2018]). Specifically, we use a deep network to solve a regression problem, namely to map the noisy image of a star to the parameter space of our model, allowing us to simulate the PSF at that position. We test and demonstrate our method using publicly available data from the Sloan Digital Sky Survey (SDSS).
The paper is structured as follows: In section \[sec:sdss-star-sample\], we introduce the data used to demonstrate our method. Section \[sec:psf-model\] describes our model and the data-based approach we pursue to construct it. In section \[sec:cnn\], we present our deep-learning approach to estimate the parameters of our model from images of stars. Our results are shown in section \[sec:results\] and we conclude with section \[sec:conclusion\].
\[sec:sdss-star-sample\]SDSS star sample
========================================
We use publicly available SDSS $r$-band imaging and catalog data to obtain a set of stars on which we demonstrate the applicability of our method. We query the SDSS database[^1] for all $r$-band images from the SDSS data release 14 [@Abolfathi2017] located between and right ascension and and declination. From this set of images, we select the ones with at least one object that fulfills the following criteria: (i) the object was classified as a star in the $r$-band by the SDSS pipeline, (ii) the object has clean photometry, (iii) the object has an $r$-band signal-to-noise ratio of at least 50. The corresponding query submitted to the SDSS database is detailed in appendix \[app:casjobs-sql-query\]. This leads to $248\,866$ images and corresponding catalogs.
To obtain a high-purity sample of stars from this data, we additionally match the SDSS catalogs to the stars found in the *Gaia* data release 1[^2] [@GaiaCollaboration2016a; @GaiaCollaboration2016b]. We cross-match sources in *Gaia* and SDSS based on positions. To be considered a match, a source in SDSS must be closer than $2''$ to a source in the *Gaia* catalog. Additionally, we ensure that there are no other sources in SDSS within a radius of $8.5''$. This criterion guarantees that we only use stars without close neighbors, which would contaminate the images of the objects in our sample.
After defining our sample of stars, we produce $15 \times 15$ pixel cutouts centered on the pixel with the maximum flux. Additionally, at this step, we remove objects that are too close to the border of the SDSS image so that we cannot make a complete stamp, i.e. the peak flux is within 7 pixels of the border. This results in $1\,215\,818$ cutouts. We normalize each cutout to have a maximum pixel intensity of 1, so that all stamps cover similar numerical ranges. This step removes the flux information from the cutouts, which will allow our CNN to process a wide range of stars with fluxes that vary by several orders of magnitude. A random selection of six normalized cutouts is shown in figure \[fig:sdss-stars–vs–cnn-stars\].
\[sec:psf-model\]PSF model
==========================
Model construction
------------------
To construct a model of the PSF images in our sample, we need to know about the features this model has to capture. We therefore need to find the main modes of variation in our data, which will allow us to incorporate these characteristics into our model. One approach to this problem is a principal component analysis (PCA, see [@Jollife2016] for a review), which yields a set of basis vectors our data decomposes into. These basis vectors, called principal components, are ordered according to the amount of variation they introduce in our sample. Therefore, a reasonable reconstruction of our data can be achieved using the first few principal components only, since they account for the most prominent features. For this reason, a PCA is well-suited for building a model that captures the most important characteristics of our data.
To find the principal components of our sample, we perform a singular-value decomposition (SVD, [@Press2007]). We reshape our data into a two-dimensional array $\bm{S}$, whereby each row contains a flattened version of one cutout. Thus, $\bm{S}$ has dimensions $n_\text{star} \times n_\text{pix}$, where $n_\text{star} = 1\,215\,818$ is the number of stars in our sample and $n_\text{pix} = 15^2$ is the number of pixels in one cutout. We then apply the SVD to $\bm{S}$ and reshape the resulting principal components to the original shape of $15 \times 15$ pixels. Note that we do not center our data before performing the SVD, thus we actually compute uncentered principal components. This is intended, since we aim at extracting a basis set that describes the data in terms of perturbations of the mean SDSS PSF image. The reason for this will become clear further below, where we describe our approach to constructing a model of the SDSS PSF based on the results of the SVD. Figure \[fig:svd-sdss\] shows the first 12 uncentered principal components of our sample. The corresponding singular values drop by two orders of magnitude over this range.
![\[fig:svd-sdss\]The first 12 (uncentered) principal components of our SDSS sample, which have been obtained by performing a SVD. The first component is the pixel-wise mean SDSS star. The second and third component indicate shifts of the positions of the centers of the stars. The fourth component arises due to variations in the size of the PSF and the quadrupoles seen in the components 5 and 9 indicate variations in ellipticity. The components 6, 7, 11 and 12 indicate the presence of flexion in the light distributions of the SDSS stars, while the 10th component represents kurtosis. A more detailed description of the distortion effects seen in the principal components is given in section \[subsec:model-description\]. The black reference bar denotes an angular distance of $1''$ (average value computed from all used SDSS images). Note that the color scale is the same for all panels.](svd_sdss.pdf){width="\textwidth"}
It is clear from figure \[fig:svd-sdss\] that our sample decomposition resembles noisy, pixelated versions of the polar shapelets introduced in [@Refregier2003a; @Massey2005]. These basis functions can be used to analyze and decompose localized images and allow for an intuitive interpretation of the decomposition. We make use of the fact that our sample decomposes into shapelets by constructing our PSF model in the following way. First, we choose a base profile that we keep constant for all stars. We then introduce a set of distortion operations and a set of corresponding distortion parameters that allow for deviations from the base profile. These parameters vary from star to star to capture variations of the PSF. Due to our sample decomposing into shapelets, we know which kind of distortion operations we have to apply to our base profile in order to reproduce the characteristics seen in figure \[fig:svd-sdss\]. In the following section, we elaborate on the details of our model. Note that our model is different from shapelets, since we use a non-gaussian base profile.
\[subsec:model-description\]Model description
---------------------------------------------
Our PSF model is fully described by two components: a base profile that is kept constant for all stars and a set of distortion parameters that captures deviations from the base profile. These parameters vary from star to star to allow for variations in the simulated PSF images. The base profile accounts for the first component seen in figure \[fig:svd-sdss\], while the perturbations account for the other components. Simulating a star with this model is done in two steps. First, we draw photons from the base profile. In the second step, we distort the photon positions, inducing the characteristics seen in figure \[fig:svd-sdss\].
We adopt the mixture of two Moffat profiles [@Moffat1969] as our radially symmetric base profile $I$: $$\begin{aligned}
I(r) &= I_1(r) + I_2(r), \\
I_i(r) &= I_{0, i} \left[ 1 + \left( \frac{r}{\alpha_i} \right)^2 \right]^{-\beta_i},\end{aligned}$$ where $r$ is the distance from the center of the profile. The parameters $\beta_i$ set the concentrations of the profiles, while the $\alpha_i$ set the spatial spreads. The parameters $I_{0, i}$ determine the numbers of photons sampled from $I_i$. While the total number of photons for a given star, i.e. the flux, is allowed to vary, we keep the ratio $\gamma = I_{0, 1} / I_{0, 2}$ constant for all stars. Furthermore, we also set $\beta_i$ and $\alpha_i$ to the same values for all stars, which means that initially, all simulated objects are circularly symmetric and have the same size, namely a half-light radius $r_{50}$ of one pixel. This can be used to write $\alpha$ as $$\alpha = \frac{r_{50}}{\sqrt{2^{1 / (\beta - 1)} - 1}} = \frac{1}{\sqrt{2^{1 / (\beta - 1)} - 1}}.$$ To allow for deviations from this base profile, we introduce a set of distortion operations that are applied to the photons after they were sampled from the base profile. In the following, we detail these operations and explain the perturbations they introduce into our base profile. This connects our model to the principal components displayed in figure \[fig:svd-sdss\].
### Size
Variations in the size of the PSF are represented by the fourth principal component, which shows the type of residual that occurs in the case where the size is modeled incorrectly. We parameterize the size in terms of the full width at half maximum (FWHM), whereby we choose to use the same FWHM for $I_1$ and $I_2$. To simulate a star with a given FWHM $F$, we scale the positions of all photons by the corresponding half-light radius, which can be obtained via $$r_{50, i} = \frac{\sqrt{2^{1 / (\beta_i - 1)} - 1}}{2 \sqrt{2^{1 / \beta_i} - 1}} \, F.$$ Let $\vec{\theta} = (\theta_1, \theta_2)$ be the position of a photon sampled from the base profile. The distorted position $\vec{\theta}'$ resulting from a change in size is given by $$\label{eq:size-transformation}
\theta'_i = r_{50} \, \theta_i.$$
### Ellipticity, flexion and kurtosis
The principal components 5 to 12 indicate the presence of ellipticity, flexion and kurtosis in the light distributions of the stars in our sample. We implement these distortions by applying a second transformation following after the first one given by eq. : $$\theta''_i = A_{ij} \theta'_j + D_{ijk} \theta'_j \theta'_k + E_{ijkl} \theta'_j \theta'_k \theta'_l.$$ This transformation is an expansion describing the distorted coordinates $\vec{\theta}''$ in terms of the coordinates $\vec{\theta}'$, which only include changes in the size. The transformation tensors $\bm{A}, \bm{D}, \bm{E}$ can be expressed in terms of derivatives of $\vec{\theta}''$ with respect to $\vec{\theta}'$: $$\label{eq:transformation-derivatives}
A_{ij} = \frac{\partial \theta''_i}{\partial \theta'_j}, \qquad D_{ijk} = \frac{1}{2} \frac{\partial^2 \theta''_i}{\partial \theta'_j \partial \theta'_k}, \qquad E_{ijkl} = \frac{1}{6} \frac{\partial^3 \theta''_i}{\partial \theta'_j \partial \theta'_k \partial \theta'_l}.$$ The same formalism is used to describe the effects of weak gravitational lensing, see [@Bacon2006]. In the following, we describe $\bm{A}$, $\bm{D}$ and $ \bm{E}$ in detail:
- The first-order term in the expansion of $\theta'_i$, $\bm{A}$, accounts for the ellipticity of the PSF, which is represented by the quadrupoles seen in the 5th and the 9th principal component. $\bm{A}$ can be parameterized by two parameters $e_1$ and $e_2$ (see e.g. [@Rhodes2000; @Berge2013]) via $$\begin{aligned}
A &= \begin{cases} \frac{1}{\sqrt{2}} \begin{pmatrix} \text{sgn}(e_2) \, \sqrt{(1 + |e|)(1 + e_1 / |e|)} & - \sqrt{(1 - |e|)(1 - e_1 / |e|)} \\
\sqrt{(1 + |e|) (1 - e_1 / |e|)} & \text{sgn}(e_2) \, \sqrt{(1 - |e|) (1 + e_1 / |e|)} \end{pmatrix}, & |e| > 0, \\
\mathds{1}, & |e| = 0,
\end{cases}\end{aligned}$$ where $|e| = \sqrt{e_1^2 + e_2^2}$.
- The second-order term $\bm{D}$ induces skewness (6th and 7th principal component) and triangularity (11th and 12th principal component) into the simulated light distribution. These two types of distortions are collectively referred to as flexion [@Bacon2006]. They can be parameterized by four variables $f_1$, $f_2$, $g_1$, $g_2$ according to: $$\begin{aligned}
D_{ijk} &= \mathcal{F}_{ijk} + \mathcal{G}_{ijk}, \\
\mathcal{F}_{ij1} &= -\frac{1}{2} \, \begin{pmatrix} 3 f_1 & f_2 \\ f_2 & f_1 \end{pmatrix}, \quad
\mathcal{F}_{ij2} = -\frac{1}{2} \, \begin{pmatrix} f_2 & f_1 \\ f_1 & 3 f_2 \end{pmatrix}, \\
\mathcal{G}_{ij1} &= -\frac{1}{2} \, \begin{pmatrix} g_1 & g_2 \\ g_2 & -g_1 \end{pmatrix}, \quad
\mathcal{G}_{ij2} = -\frac{1}{2} \, \begin{pmatrix} g_2 & -g_1 \\ -g_1 & -g_2 \end{pmatrix}, \end{aligned}$$ where $f_1$ and $f_2$ account for the skewness of the PSF, while $g_1$ and $g_2$ induce triangularity into our simulated light distributions.
- We include one third-order term in our model, namely the one that accounts for kurtosis (symmetrically in all directions). This is motivated by the 10th principal component in figure \[fig:svd-sdss\], which displays the type of residual that arises from moving photons from the bulk of the distribution to the tail or vice-versa. The corresponding transformation that applies a kurtosis $k$ is given by $$\theta''_i = k \, \theta'_i \, \left| \vec{\theta}' \right|^2,$$ such that in our case, $E_{ijmn} = \delta_{ij} \, \delta_{mn} \, k$ for a given kurtosis $k$.
### Centroid position
Finally, we account for shifts of the position of the center of the profile, given by the second and third component in figure \[fig:svd-sdss\]. This is achieved by offsetting all photon positions by a constant amount: $$\theta'''_i = \theta''_i + \Delta \theta_i,$$ whereby $\Delta \theta_i$ denotes a constant offset.
### Summary
To summarize, we construct our PSF model in a data-driven way, which is motivated by the principal components our star sample decomposes into. To simulate a star, we first sample photons from a fixed base profile given by a weighted sum of two Moffat profiles. To perform the random draws, we adopt the sampling procedure detailed in [@Berge2013]. This results in the photon position $\vec{\theta}$. We then apply three successive transformations. The first transformation accounts for the size of the PSF, yielding the photon position $\vec{\theta}'$. The second transformation, which maps $\vec{\theta}'$ to $\vec{\theta}''$, induces ellipticity, skewness, triangularity and kurtosis into the simulated light distribution. The last transformation from $\vec{\theta}''$ to $\vec{\theta}'''$ accounts for shifts of the center of the profile. The coordinates $\vec{\theta}'''$ are the final position of the photon on the simulated image.
Base profile selection
----------------------
In this section, we explain how we choose the base profile of our model. As detailed above, we use a fixed mixture of two Moffat profiles parameterized by three parameters $\beta_1, \beta_2, \gamma$. Since our base profile is intended to account for the first component in figure \[fig:svd-sdss\], we fit it to this mean PSF image from SDSS. To obtain a good fit, we also vary the FWHM of the base profile during the fitting procedure, but we do not use the resulting value later on. We perform the fit with the Particle Swarm Optimizer included in the `CosmoHammer` package [@Akeret2013] and find the best-fit values of $\beta_1 = 1.233$, $\beta_2 = 3.419$, $\gamma = 0.387$.
In the remainder of the paper, we use the base profile defined by these parameter values. Since the mean light distribution we compute from our sample is not perfectly symmetric, we do not obtain a perfect fit. However, this is not an issue here, since the distortion operations we apply after sampling from the base profile account for deviations from circular symmetry.
Parameter estimation with a Convolutional Neural Network {#sec:cnn}
========================================================
In this section, we describe how we estimate the free parameters of our model to reproduce a given star. This problem is the inverse of the forward-modeling process introduced in section \[sec:psf-model\], namely how can we determine the best possible parameter values in order to model a given star? To solve this issue, we use a supervised machine learning technique, specifically a CNN. In the following, we first give a short introduction to Deep Learning and neural networks. We then go into detail about the specific network we use in this paper and explain the training strategy used to learn the parameters of the neural network.
Deep Learning and Convolutional Neural Networks
-----------------------------------------------
In this paper, we focus on supervised machine learning, which requires a set of input examples – images in our case – as well as the desired outputs, such as regression parameters. Given these input/output pairs, a machine learning model is trained to reproduce the output corresponding to each specific input in our dataset. In practice, many choices are left to the user such as the choice of model or the metric function used to evaluate its performance. We will here give a brief overview and refer the reader to [@Murphy2012; @LeCun2015; @Goodfellow2016] for a detailed survey of the relevant literature.
One specific type of model that we consider in this work is a deep neural network, which is part of a broader family of machine learning methods based on learning feature representations from data. Unlike other models, neural networks do not require explicitly constructing features from the input data but can directly learn patterns from the raw input signal. In a nutshell, this is accomplished by using a complex combination of neurons organized in nested layers that can learn a non-linear function of the input data.
In this work, we use a special type of neural network known as Convolutional Neural Network (CNN), see e.g. [@Rawat2017; @Liu2017; @McCann2017], which has recently been achieving state-of-the-art performance on a variety of pattern-recognition tasks. One task where CNNs have particularly excelled is image classification [@Krizhevsky2012], achieving results close to human accuracies [@Simonyan2014]. Three main types of layers are used to build a CNN architecture: convolutional layers, pooling layers and fully connected layers. Additional non-linear activation layers are used to construct non-linear models. Figure \[fig:cnn\] shows a schematic representation of the CNN we use in this paper, which has a standard architecture. In the following, we give a brief description of the different layers of a CNN and their functions. We refer the reader to [@Goodfellow2016] for further details.
![Sketch of the CNN architecture used in this paper. The network consists of multiple convolutional and pooling layers that produce a rich set of features. These are passed to a fully-connected layer, which computes the final output of the CNN. See section \[subsec:implementation-architecture\] for a detailed description of the architecture and section \[subsec:training-strategy\] for details on the output of the CNN.[]{data-label="fig:cnn"}](cnn_schematic.pdf){width="\textwidth"}
Convolutional layers perform discrete convolution operations on their input. These layers are parametrized by a set of learnable filters whose width and height are tunable parameters (we use $3 \times 3$ filters as detailed in the next section) and whose depth extends to the same depth as the input. When feeding the image to the first convolutional layer in the neural network, each filter is slid across the image and for each spatial position, an output value is produced by computing the correlation between the filter and the image content. This process yields one output map for each filter, each having the same size as the input image (assuming that we do not crop the boundaries which might require duplicating boundary pixels). During training, the network learns filters that will activate the most when seeing certain image properties that can discriminate the signal. As each layer has access to a set of several filters, the network will specialize each filter to recognize different image characteristics.
The set of outputs maps produced by each layer is then passed on to the next layer. The layer following a convolutional layer is typically a pooling layer that performs a downsampling operation on the input by merging local input patches, e.g. using an average or maximum operator. For example, a standard pooling layer would subdivide the input map into blocks and take the maximum value in each block, therefore reducing the size of the map as well as the amount of parameters (and computation) in the network. As shown in figure \[fig:cnn\], convolutional and pooling layers are repeated in a sequential fashion multiples times in order to construct a rich set of features from the input signal. Non-linear activation functions are used in between layers to enable the model to learn a non-linear function of the input. The resulting set of output maps (aka features) extracted by this sequence of layers is then passed on to a fully connected layer, which performs the high-level reasoning in order to produce the final output. This operation can typically be computed as a matrix multiplication. If used as the last layer in the network, it outputs a set of scores, one for each output parameter. In the next sections, we will detail the specific architecture used for our experiments as well as the training procedure used to learn the parameters of the neural network.
\[subsec:implementation-architecture\]Implementation and network architecture
-----------------------------------------------------------------------------
The architecture used for our experiments is presented in figure \[fig:cnn\]. It includes three consecutive convolutional layers, whereby each layer uses square filters with a side length of three pixels. The first convolutional layer directly processes the input image using 32 filters. Each $3 \times 3$ filter is applied to the input image using a convolution operation as explained in the previous section. We then repeat this operation in a sequential fashion, whereby we double the number of filters for each following layer, thus we use 64 and 128 filters in the second and third layer. Furthermore, each convolutional layer has adjustable additive bias variables, one per filter. The output of each convolutional layer is passed to a rectified linear unit (ReLU, [@Glorot2011]) followed by a pooling layer. As mentioned earlier, each pooling layer performs a max operation over sub-regions of the extracted feature maps resulting in downsampling by a factor of two. After the three stages of convolutions, activations and pooling, we place a fully connected layer with 2048 ReLU neurons. This step maps 128 feature maps with dimensions of $2 \times 2$ pixels (the outputs of the third convolutional and pooling layer) to 2048 features. The fully connected layer also includes an additive bias term with 2048 adjustable variables. Finally, these 2048 features are mapped to our 10-dimensional output space using a final matrix multiplication and bias addition. The CNN described in this section has approximately $1.16 \cdot 10^6$ trainable parameters. We use the <span style="font-variant:small-caps;">TensorFlow</span> package[^3] [@Abadi2016] to implement our CNN.
\[subsec:training-strategy\]Training strategy
---------------------------------------------
To learn the parameters $\theta$ of the neural network, we measure its performance using a loss function $L(\theta)$. The optimal network parameters $\theta_\text{opt}$ are found by minimizing $L$ with respect to $\theta$, i.e. $\theta_\text{opt} = \min_\theta L(\theta)$. The loss function we use is the Huber loss [@Huber1964], defined as $$\begin{aligned}
L_i(\theta) &= \begin{cases} \frac{1}{2} \left| y^\text{e}_i(\theta) - y^\text{t}_i\right|^2, & \left| y^\text{e}_i(\theta) - y^\text{t}_i \right| \leq \delta, \\
\delta \left| y^\text{e}_i(\theta) - y^\text{t}_i \right| - \frac{1}{2} \delta^2, & \left| y^\text{e}_i(\theta) - y^\text{t}_i \right| > \delta,
\end{cases} \\
L(\theta) &= \frac{1}{N_\text{dim}} \sum^{N_\text{dim}}_{i=1} L_i(\theta). \label{eq:huber-loss}\end{aligned}$$ $y^\text{e}(\theta)$ is a vector of input parameters for our PSF model estimated by the network and $y^\text{t}$ is the corresponding vector of true values. $i$ indexes the dimensions $N_\text{dim} = 10$ of the parameter space of our PSF model and $|\cdot|$ denotes the absolute value. We use this loss function because it is more robust to outliers than the mean squared error (MSE). It is also differentiable everywhere due to the transition to the MSE at residual values smaller than $\delta = 10^{-5}$. When we started experimenting with the CNN, we first used the MSE as loss function. However, we found that this choice led to biases in the parameters estimates produced by the network. We therefore switched to a more robust loss function, which reduced the biases, confirming that they were caused by outliers.
In order to optimize the Huber loss defined in eq. , we use a variant of stochastic gradient descent (e.g. [@Ruder2016]) called <span style="font-variant:small-caps;">Adam</span> [@Kingma2014]. We set the learning rate $\alpha = 0.001$ and the other hyper-parameters of the <span style="font-variant:small-caps;">Adam</span> optimizer to $\beta_1 = 0.9$, $\beta_2 = 0.999$ and $\epsilon = 10^{-8}$, as recommended by [@Kingma2014]. The gradients of the loss function are computed using the back-propagation algorithm implemented in <span style="font-variant:small-caps;">TensorFlow</span> [@Abadi2016].
Furthermore, we train our network to learn a rescaled version of the parameters we aim at estimating to ensure that each dimension of the parameter space contributes approximately equally to the loss function. This is accomplished by rescaling all parameters to the same numerical range, i.e. $y^\text{t}_i \in [-1, 1]$.
To simplify the mapping our CNN has to learn, we apply certain transformations to parts of the parameter space. As can be seen from eq. , $f_1$, $f_2$, $g_1$, $g_2$ have units of $\text{pixel}^{-1}$ and $k$ has units of $\text{pixel}^{-2}$. This means that the change in shape induced by the corresponding transformations depends on the size of the simulated star. In other words, the parameters $f_1$, $f_2$, $g_1$, $g_2$, $k$ are correlated with the size of the simulated PSF. In order to remove this correlation as much as possible, we train our network to learn dimensionless versions of the parameters, which we denote with $\tilde{f}_1, \tilde{f}_2, \tilde{g}_1, \tilde{g}_2, \tilde{k}$. They are obtained via the following transformations: $$\begin{aligned}
\tilde{f}_1, \tilde{f}_2, \tilde{g}_1, \tilde{g}_2 &= \left\{ f_1, f_2, g_1, g_2 \right\} \cdot \text{FWHM}, \\
\tilde{k}&= k \cdot \text{FWHM}^2.\end{aligned}$$
\[subsec:training-data\]Training data
-------------------------------------
The data used to train the network is obtained by uniformly sampling a hypercube in our parameter space and simulating a star for each sample. The ranges of the parameters defining the hypercube are denoted in table \[tab:ranges-training-cube\]. The training data we generate has the same format as our sample of stars from SDSS, meaning that we simulate our training stars using a $15 \times 15$ pixel grid. Initially, each star is placed in the center of the grid (before applying the offsets $\Delta \theta_i$).
Range
------------------- -------------------
$\Delta \theta_i$ $-0.5, 0.5$ pixel
FWHM $1.5, 6.0$ pixel
$e_i$ $-0.25, 0.25$
$\tilde{f}_i$ $-0.25, 0.25$
$\tilde{g}_i$ $-0.1, 0.1$
$\tilde{k}$ $-0.4, 0.4$
: \[tab:ranges-training-cube\]Range of each parameter defining the volume in parameter space that we sample uniformly to create the training data for our CNN.
Given a sample of parameter values, we additionally need to specify the number of photons used to simulate the star. To this end, we also draw random magnitudes between 15 and 20.5, random gain values between 4.5 and 5$\, e^- / \text{ADU}$ and random magnitude zeropoints between 28 and 28.5. We adopt these ranges because they include the majority of stars in our SDSS sample, such that our training data is representative of the SDSS data. When calculating the number of photons from the magnitude, the gain and the magnitude zeropoint, we include Poisson noise.
We draw $10^8$ samples from the hypercube defined above and evaluate our PSF model for each sample in order to obtain a training dataset. Since real images contain background noise, we have to account for this component in order to obtain a network that can be reliably applied to survey data. We add Gaussian noise with mean zero to our simulated stars on the fly when feeding them to network. We sample the corresponding standard deviations uniformly between 3.5 and 5.5$\,$ADUs, since the background level of most of the SDSS images we use is located in this interval. After adding noise, we normalize the simulated images to have a maximum intensity of 1, as was done for the real data (see section \[sec:sdss-star-sample\]).
As mentioned in section \[subsec:training-strategy\], we rescale the parameters the network is supposed to estimate before training the CNN. We re-center the samples drawn from the hypercube to have zero mean and adjust the numerical range to extend from -1 to 1 along each dimension. We then train our CNN by feeding it with batches of simulated stars and the corresponding rescaled true parameters, whereby we set the batch size to 2000 stars. We iterate through three training epochs, meaning that each training star is given to the network three times, each time with a different background noise level. Thus, the network is trained on $3 \cdot 10^8$ stars in total, which takes around $1.5$ days on a machine with 24 CPU cores (no GPUs were used in this work).
\[subsec:cnn-output-corrections\]CNN output corrections
-------------------------------------------------------
We use the <span style="font-variant:small-caps;">Adam</span> optimizer to minimize the Huber loss for each batch of training stars given to the network. This is accomplished by adjusting the weights of the CNN to reduce both the variance with which the estimated values scatter around the true values as well as the biases in the estimated values. These two objectives might not be completely compatible, i.e. there can be a bias-variance trade-off. For example, the optimizer might reduce the spread in one parameter, but the corresponding adjustments in the weights of the CNN lead to a small increment in the bias of this parameter. To correct for this effect, we use linear fits that map the estimated, potentially slightly biased parameter values closer to the true values. These fits are performed after the training is complete. When applying our CNN to estimate the parameters for our PSF model, we always correct the estimates using these linear fits. To obtain the linear corrections, we use $50\,000$ randomly selected stars from the training sample. We perform the fits with the random sample consensus (RANSAC, [@Fischler1981]) method, which is robust to outliers. RANSAC fitting is implemented for example within the <span style="font-variant:small-caps;">scikit-learn</span> library [@Pedregosa2011], which we use for this purpose.
We originally introduced these linear corrections to compensate for biases in the parameter values estimated by the network caused by outliers when training with the MSE instead of the Huber loss (see section \[subsec:training-strategy\]). After switching to the Huber loss, these biases were reduced. Accordingly, the slopes and intercepts found by the RANSAC algorithm are now very close to one and zero, respectively. Thus, the linear corrections change the parameters predicted by the CNN on the percent to sub-percent level only.
\[sec:results\]Results
======================
Here, we present the results from training our CNN introduced in section \[subsec:implementation-architecture\] and using it in combination with the PSF model detailed in section \[subsec:model-description\] to render a model image of each SDSS star in our sample. To train the network on our training data (see section \[subsec:training-data\]), we follow the strategy explained in section \[subsec:training-strategy\]. After the training is complete, we apply the CNN to our SDSS dataset. The resulting parameter estimates are fed into the PSF model to render a model image of each SDSS star. To compute the number of photons used to render a SDSS star, we use the measured magnitude of the object as well as the magnitude zeropoint and the gain of the corresponding exposure.
We first present the performance of the network in conjunction with the PSF model on the SDSS data. We then give results concerning the validation of our network and compare our method to a direct fit of the PSF model to SDSS stars. Finally, we compare our PSF modeling approach to alternative approaches and conclude with an analysis of the impact of our modeling errors on cosmic shear measurements.
\[subsec:perfomance-sdss\]Performance on SDSS
---------------------------------------------
In figure \[fig:sdss-stars–vs–cnn-stars\], we compare six randomly selected stars from our SDSS sample to their model images. We see that our model is able to describe a wide range of PSF images with diverse sizes and shapes. This can also be seen from the residual images, which mainly consist of photon and background noise.
![\[fig:sdss-stars–vs–cnn-stars\]Six randomly selected stars from the SDSS sample compared to their model images. The left columns show the image cutouts of the objects extracted from real SDSS images. The model images are displayed in the middle columns. They are obtained by applying the trained network to the SDSS data and evaluating our PSF model using the resulting parameter estimates. Both the SDSS as well as the modeled images have been normalized to a maximum pixel intensity of 1. The right columns show the residuals that remain after subtracting the SDSS images from the model images.](sdss_stars__vs__cnn_stars.pdf){width="\textwidth"}
To give a global overview of the performance of our model in conjunction with the network, we show the average residual image in figure \[fig:mean-residual\]. As can be seen, there is structure beyond photon and pixel noise left in the mean residual star. However, the average residuals peak at the percent level compared to the peak of the actual image. Furthermore, the projected profiles in the $\theta_1$- and $\theta_2$-direction (see the right panels of figure \[fig:mean-residual\]) show that the SDSS stars are well reproduced by our model. Additionally, we find that the principal components constructed from our modeled stars are consistent with those from the SDSS data (see appendix \[app:svd-cnn\]), further validating our method.
![\[fig:mean-residual\]Mean residual fluxes after subtracting the SDSS stars from their simulated counterparts. The left panel shows the mean residual star obtained from averaging the pixel values of the individual residuals. The middle and the right panel show the mean profiles of the SDSS and the simulated data, which are obtained by averaging the pixel values in $\theta_1$- and $\theta_2$-direction.](mean_residual.pdf){width="\textwidth"}
\[subsec:validation\]Validation of the network
----------------------------------------------
To test further the performance of the CNN, we apply it to simulated data for which we know the ground truth model parameters. Importantly, the network has never seen this validation data during the training, such that it did not have the chance to accidentally overfit and adjust specifically to this dataset. We use two types of validation sets: one training-like validation set and one SDSS-like validation set. The training-like validation sample is statistically identical to the training data and we use it to show that our network did not overfit during the training phase. The SDSS-like validation set on the other hand is statistically close to real survey data. Thus, this validation sample allows us to probe the performance of the CNN in the regime we are interested in.
To generate the training-like validation set, we draw $200\,000$ samples from the hypercube defined in table \[tab:ranges-training-cube\] and evaluate our PSF model for these samples. Also the photon and background noise properties of this validation sample are the same as for the training data. The SDSS-like validation sample is created from the modeled SDSS stars described above in section \[subsec:perfomance-sdss\]. We add Gaussian background noise that corresponds to standard deviations uniformly distributed between 3.5 and 5.5$\,$ADUs (the same interval was used for training the network, see section \[subsec:training-data\]). We then apply the network to both validation samples and obtain two sets of estimated model parameters than can be compared to the input parameters used to generate the validation data.
In figure \[fig:true-vs-pred\], we compare the model parameters estimated from the SDSS-like validation data to the input parameters used to generate this validation set. For most of the objects, the estimates scatter tightly around the blue diagonal lines that represent the ideal one-to-one relation. This means that the network is able to measure reliable parameter values for the majority of stars in our SDSS-like validation sample. Furthermore, we note that the CNN performs very well for the lowest-order distortions, which are size (FWHM) and ellipticity ($\tilde{e}_i$). This is especially important in the context of weak lensing measurements, since the shear signal is very prone to biases in these parameters. Finally, it can be seen from figure \[fig:true-vs-pred\] that the parameter on which the CNN performs less well is the kurtosis $\tilde{k}$. This can be directly linked to the mean residual shown in figure \[fig:mean-residual\], which displays the typical signature caused by small errors in the steepness of the light distribution.
![\[fig:true-vs-pred\]Validation of the performance of the trained CNN. To produce this figure, we apply the network to simulated SDSS-like stars, which are described above in the text. For each object, we then have two sets of parameters available: the ground truth input values used to simulate the star and the values estimated by the CNN. We plot the input values on the $x$- and the estimated values on the $y$-axis. The blue diagonal lines correspond to an ideal one-to-one relation between input and measured values. The displayed color scale applies to all panels. We furthermore report the MSE of our network for each parameter. The given uncertainties correspond to one standard deviation.](cnn_true_vs_pred.pdf){width="\textwidth"}
In figure \[fig:loss\], we show the training error of the network (green line). Each data point corresponds to the average loss of one batch of $2\,000$ training stars (see section \[subsec:training-data\]). As expected, the value of the loss function decreases as the number of processed training examples increases. The figure also shows the loss function evaluated on the two validation sets described above (black lines). To compute the loss of the SDSS-like validation set, we directly average the losses computed for the individual stars in this sample. To compute the loss of the training-like validation set, we first split this sample into 100 batches of $2\,000$ stars, which corresponds to the training batch size. We then compute the average loss of each individual batch and average these 100 loss values to obtain the value displayed in figure \[fig:loss\]. This allows us to also compute an uncertainty on the loss of the training-like validation set, which is given by the grey band in the figure. It corresponds to one standard deviation of the 100 per-batch loss values computed from the split training-like validation set.
![\[fig:loss\]Loss function $L$ used to train the CNN (see eq. ) as a function of the number of training steps (green line). Each data point corresponds to the average loss of one batch of $2\,000$ training stars. We furthermore show the loss function evaluated on the two validation samples described in this section (black lines). To compute the loss of the training-like validation set, we first split this sample into 100 batches of $2\,000$ stars, compute the per-batch losses and then average these values. This allows us to also compute an uncertainty on the loss of the training-like validation set in terms of one standard deviation of the 100 per-batch loss values, which is shown as the grey band. Note that for this figure, we did not apply the linear corrections explained in section \[subsec:cnn-output-corrections\] to the parameter values estimated by the CNN from the two validation sets. We avoided the corrections for this particular figure in order to compare the training and the validation losses on an equal footing, since the corrections were not applied during the training phase either, see section \[subsec:cnn-output-corrections\].](loss.pdf){width="60.00000%"}
As can be seen from figure \[fig:loss\], the final training loss of the network is on the same level as the loss of the training-like validation sample. This means that our network did not overfit. We attribute this to the fact that we only iterated through four training epochs, such that the CNN saw each training star only four times. Furthermore, even though the network is trained on the same objects multiple times, the background noise is different each time (since it is added on the fly during the training phase). This effectively acts as a further regularization which prevents overfitting. Figure \[fig:loss\] also shows that the loss of the SDSS-like validation sample is smaller than the training loss. This is because the SDSS-like validation set is statistically different from the training data. As can be seen from figure \[fig:true-vs-pred\], the SDSS-like validation stars cluster around zero for the ellipticity, flexion and kurtosis parameters. This region contains the highest density of training examples, since the training data is also centered on zero for the parameters $\tilde{e}_i$, $\tilde{f}_i$, $\tilde{g}_i$ and $\tilde{k}$. Thus, most of the SDSS-like validation stars lie in the regions where the network has been trained most extensively and only few objects are close to the boundaries of the hypercube the training data was sampled from. Contrary to this, the fraction of stars close the boundaries of the hypercube is much higher for the uniformly distributed training data. This results in the smaller loss value of the SDSS-like validation set seen in figure \[fig:loss\].
\[subsec:comparison-direct-fitting\]Comparison with direct fitting
------------------------------------------------------------------
In this section, we briefly compare the performance of our CNN to a direct fitting approach, which could be an alternative to using a neural network. We use the same fixed base profile as for the CNN, but instead of using the network to estimate the ten free parameters of our PSF model, we now use an optimization algorithm. The cost function we minimize is the sum of residual pixel values (MSE). To perform the fits, we use the function `scipy.optimize.minimize`, which is part of the SciPy library [@Jones2001]. We try out all applicable optimization algorithms that are available for `scipy.optimize.minimize` and report the results obtained with the algorithm that yields the best average cost function. Our findings are detailed in appendix \[app:direct-fitting\]. In a nutshell, we find that direct fitting yields results similar to the ones obtained with the CNN in terms of residual pixel values. However, the big disadvantage of direct fitting is the runtime. We found that it takes on average several seconds to fit one star, whereby the runtime varies heavily from star to star and can also be significantly larger. Thus, applying this method to large wide-field datasets with hundreds of thousands to millions of stars would be quite intense in terms of computing time. Furthermore, this runtime issue renders direct fitting practically intractable in the context of forward modelling and $\textit{MCCL}$, where one typically has to simulate and analyze data volumes even larger than the actual survey data volume. The CNN on the other hand can process thousands of stars in one second, making this method ideal for forward modeling applications.
Comparison with other PSF modeling techniques
---------------------------------------------
To place our results in a global context, we compare them to results obtained with other PSF modeling techniques. [@Annis2014] present coadded SDSS images and model their PSF. In this case, the PSF modeling is based on a Karhunen–Lo[é]{}ve expansion (cf. PCA). Figure 6 in [@Annis2014] shows the square root of the ratio of the PSF size measured on the SDSS images to the size measured on the modeled PSF. Using the validation sample presented in section \[subsec:validation\], we compute the mean of the square root of the ratio of the true PSF FWHM to the FWHM estimated by the network and find 0.996. This compares well to the ratio given in [@Annis2014], which varies approximately between 0.990 and 1.005 for the $r$-band. We conclude that in terms of the PSF size, our method performs similarly well as the Karhunen–Lo[é]{}ve expansion used in [@Annis2014].
We also compare our method to the PSF modeling in latest DES weak lensing analysis [@Zuntz2017], which relies on a pixel-based approach with <span style="font-variant:small-caps;">PSFEx</span> [@Bertin2011]. Figure 7 in [@Zuntz2017] gives an impression of the corresponding size and ellipticity residuals. Even though [@Zuntz2017] uses a different definition of size and ellipticity than this work, we can still compare at least the orders of magnitude of our results. Using the validation sample described in section \[subsec:validation\], we find a mean squared size error of $\langle \delta \, \text{FWHM}^2 \rangle = 0.0179''$ and mean ellipticity residuals of $\langle \delta \, e_1 \rangle = 0.0005$ and $\langle \delta \, e_2 \rangle = 0.0001$. Since the residuals in the DES analysis are of the same order of magnitude, we conclude that in terms of these numbers, our method performs similarly well as <span style="font-variant:small-caps;">PSFEx</span> on DES data.
Approximate cosmic shear forecast
---------------------------------
Here, we examine the impact of our modeling inaccuracies onto weak lensing measurements. To this end, we calculate the systematic uncertainty $\sigma_\text{sys}^2$ on a shear signal resulting from our PSF modeling errors. To compute this quantity, we use equation (15) from [@Paulin-Henriksson2008]. There are two simplifications related to using this equation: (i) [@Paulin-Henriksson2008] assume that an unbiased estimator of size and ellipticity is available, which is of course not exactly true for our CNN, (ii) the size and ellipticity in [@Paulin-Henriksson2008] correspond to the second-order moments of the PSF light distribution. Due to the presence of flexion and kurtosis, the size and the ellipticity estimated by our network correspond only approximately to second-order moments. However, since we are only aiming at an order-of-magnitude forecast, we are confident that equation (15) from [@Paulin-Henriksson2008] can still give us an idea of the performance of our estimator.
To compute $\sigma_\text{sys}^2$, we use $P^\gamma = 1.84$ and $\langle \left| e_\text{gal} \right|^2 \rangle = 0.16$, as recommended by [@Paulin-Henriksson2008]. We furthermore set $R_\text{gal} / R_\text{PSF} = 1.5$, since one typically uses only galaxies sufficiently larger than the PSF for shear measurements. For the mean PSF size and ellipticity, we use the mean values estimated on the SDSS data. Finally, we use the SDSS-like validation data described in section \[subsec:validation\] to estimate the variance in the error of our predicted sizes and ellipticities. This yields a systematic uncertainty of $\sigma_\text{sys}^2 = 3.25 \cdot 10^{-5}$. This number can be compared to a requirement on $\sigma_\text{sys}^2$ for a SDSS-like survey to not be systematics-limited. In [@Nicola2016], SDSS coadded data was used in combination with other datasets to constrain cosmology. The corresponding survey area is $A_\text{s} = 275\,\text{deg}^2$, the surface density of lensing galaxies is $n_\text{g} \approx 3.2\,\text{arcmin}^{-2}$, the median redshift of the lensing sample is $z_\text{m} \approx 0.6$ and the maximum multipole used for the power spectra is $\ell_\text{max} = 610$. Following [@Amara2007], we translate these numbers into a requirement onto the systematics and obtain $\sigma_\text{sys}^2 \lesssim 1.12 \cdot 10^{-5}$.
This number is slightly smaller than the systematic uncertainty computed above, however, both numbers are of the same order of magnitude. Furthermore, it is important to notice that we model the PSF of individual SDSS exposures in this work. Contrary to that, the requirements we computed correspond to SDSS coadded data, which generally has a better PSF quality than the individual exposures. Also, one would typically average over several stars in the same region to estimate the PSF at a given position, which would further reduce the systematic uncertainty induced by our modelling inaccuracies. We therefore conclude that the performance of our method is on a level suitable for weak-lensing applications.
The requirements for state-of-the-art weak lensing surveys such as DES and KiDs are significantly lower than the numbers computed above. For a DES Y1-like survey, we have $\sigma_\text{sys}^2 \lesssim 3 \cdot 10^{-6}$, see [@Bruderer2017]. For KiDS-450, we find $\sigma_\text{sys}^2 \lesssim 5 \cdot 10^{-6}$. To compute this number, we use $A_\text{s} = 360.3\,\text{deg}^2$, $n_\text{g} = 8.53\,\text{arcmin}^{-2}$, $z_\text{m} \approx 0.6$ and $\ell_\text{max} = 1000$. We took these values from [@Hildebrandt2017], whereby we compute $z_\text{m}$ as the mean of the median redshifts in the four tomographic bins of the KiDS analysis. For both DES and KiDS, $\sigma_\text{sys}^2$ is one order of magnitude smaller than the value we computed for our model ($\sigma_\text{sys}^2 = 3.25 \cdot 10^{-5}$). However, both surveys have a better resolved PSF than SDSS, such that one would expect the CNN to perform better on data from DES or KiDS. Furthermore, as mentioned above, one typically averages over multiple stars to estimate the PSF. Both DES and KiDS are deeper than SDSS, such that the density of stars used for PSF estimation is higher. Finally, both surveys take multiple exposures per field of view, such that multiple samples of the PSF at each position are available (or alternatively coadded data). This increases the quality of the PSF estimation further. For these reasons, we are confident that our method is suitable for state-of-the-art cosmological applications.
\[sec:conclusion\]Conclusion
============================
We presented a novel approach to model the PSF of wide-field surveys. Our method relies on a data-driven PSF model, which is able to describe a wide range of PSF shapes and on Deep Learning to estimate the free parameters of the model. The combination of these two approaches allows us to render the PSF of wide-field surveys at the pixel level. This can be used, for instance, within the *MCCL* framework to generate realistic wide-field image simulations, which are useful for calibrating shear measurements.
We demonstrated our method with publicly available SDSS data. To construct our model, we performed a PCA on our SDSS sample of stars. This allowed us to incorporate the main modes of variation extracted from the data into our model. To this end, we implemented the model in two steps. First, we chose a fixed base profile that is kept constant for all stars. Then, we implemented a set of perturbative distortion operations that account for variations in the PSF. These distortions were designed to enable our model to reproduce the most important principal components extracted from the SDSS sample.
To estimate the free parameters of our PSF model from survey data, we opted for a CNN. This allows for very fast measurements once training is complete, which is vital for incorporating our method into the *MCCL* scheme. We trained the network on a large number of simulated stars and subsequently applied it to the SDSS data. We then generated a model image of each star in our sample. By comparing the modeled star images to their counterparts from SDSS, we demonstrated that our model in conjunction with the CNN is able to reliably reproduce the SDSS PSF. Furthermore, using the simulated version of the SDSS sample, we validated our network and demonstrated that it is able to produce largely unbiased parameter estimates for the big majority of objects in our sample. This was ensured by applying linear corrections to the estimates made by the network. The correction factors were found using RANSAC fits after training the network.
The results presented in this paper offer encouraging prospects for applying our approach to state-of-the-art imaging surveys. Since currently ongoing survey programs have in general a better resolved PSF than SDSS, we expect our approach to perform at least as well as it did on the dataset used here. Future improvements include training the neural network on larger amounts of training data, possibly using GPUs (in this work we only used CPUs for training). Furthermore, if necessary, our model can be extended to include additional higher-order distortions.
This work has made use of data from the European Space Agency (ESA) mission *Gaia* (<https://www.cosmos.esa.int/gaia>), processed by the *Gaia* Data Processing and Analysis Consortium (DPAC, <https://www.cosmos.esa.int/web/gaia/dpac/consortium>). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the *Gaia* Multilateral Agreement. We acknowledge support by grant number 200021\_169130 from the Swiss National Science Foundation.\
This research made use of the Python packages <span style="font-variant:small-caps;">NumPy</span> [@vanderWalt2011], <span style="font-variant:small-caps;">h5py</span>[^4], <span style="font-variant:small-caps;">Matplotlib</span> [@Hunter2007], <span style="font-variant:small-caps;">seaborn</span> [@Waskom2017], <span style="font-variant:small-caps;">jupyter</span> [@Kluyver2016], <span style="font-variant:small-caps;">ipython</span> [@Perez2007], <span style="font-variant:small-caps;">Astropy</span> [@AstropyCollaboration2018], <span style="font-variant:small-caps;">Numba</span> [@Lam2015] and <span style="font-variant:small-caps;">pandas</span> [@McKinney2011].
SDSS CasJobs SQL query {#app:casjobs-sql-query}
======================
Here, we give the SQL queries used to obtain the list of SDSS images from which we extracted our star sample described in section \[sec:sdss-star-sample\]. We ran these queries on the SDSS data release 14.
1. `SELECT DISTINCT fieldID FROM PhotoObjAll INTO mydb.field_id WHERE PhotoObjAll.clean = 1 AND PhotoObjAll.type_r = 6 AND PhotoObjAll.psfFlux_r * SQRT(PhotoObjAll.psfFluxIvar_r) > 50`
2. `SELECT ALL Field.run, Field.field, Field.camcol, Field.ra, Field.dec, Field.gain_r, Field.nMgyPerCount_r, Field.sky_r, Field.skySig_r, Field.pixScale INTO mydb.fieldlist FROM Field INNER JOIN mydb.field_id ON Field.fieldID = field_id.fieldID WHERE Field.ra > 100 AND Field.ra < 280 AND Field.dec > 0 AND Field.dec < 60 `
SVD of the modeled SDSS sample {#app:svd-cnn}
==============================
In this appendix, we display the uncentered principal components of the modeled SDSS stars (figure \[fig:svd-cnn\]), which are also referenced in the main text. Note that sign flips (visible e.g. in the first component when comparing this figure to figure \[fig:svd-sdss\]) are irrelevant here, because they do not change the space of light distributions spanned by the principal components. When decomposing a PSF image $I$ into principal components $\phi_i$, i.e. $I = \sum_i c_i \phi_i$, the decomposition coefficients $c_i$ are given by the projections of the image onto the principal components: $c_i = \langle I, \phi_i \rangle$, where $\langle \cdot \rangle$ denotes a scalar product. Thus, any sign flips in the principal components cancel out: $c_i \phi_i = \langle I, \phi_i \rangle \phi_i = \langle I, (-\phi_i) \rangle (-\phi_i)$.
![\[fig:svd-cnn\]Uncentered principal components of the model images of the SDSS stars described in section \[sec:results\]. The similarity of this decomposition and the one of the SDSS sample, which is given in figure \[fig:svd-sdss\], further validates our method. The color scale is the same for both figures. As explained in the text above, sign flips are not important here, since they cancel out when decomposing a PSF image into its principal components.](svd_cnn.pdf){width="\textwidth"}
Results from direct fitting {#app:direct-fitting}
===========================
Here, we show the results obtained by directly fitting our PSF model to SDSS stars instead of using a CNN, as discussed in section \[subsec:comparison-direct-fitting\]. We found that the Powell optimization algorithm, implemented within the SciPy library (`scipy.optimize.minimize`), yields good results in terms of residual pixel values. Figure \[fig:sdss-stars–vs–fitting-stars\] displays the results obtained by directly fitting the six randomly selected SDSS stars that were also used in figure \[fig:sdss-stars–vs–cnn-stars\]. As can be seen, direct fitting yields results comparable to the ones obtained using the CNN. However, as discussed in the main text, runtime issues render this method virtually unusable for data volumes produced by currently ongoing and upcoming state-of-the-art wide-field surveys.
![\[fig:sdss-stars–vs–fitting-stars\]The figure shows the same six randomly selected stars from the SDSS sample that are also displayed in figure \[fig:sdss-stars–vs–cnn-stars\]. The difference is that the model images where not generated using parameters estimates produced by the CNN, instead, the model parameters were obtained by a direct fitting approach. When comparing these results to the ones obtained using the CNN, see figure \[fig:sdss-stars–vs–cnn-stars\], it is clear that the direct fitting approach performs similarly well as the CNN in terms of residual pixel values. However, direct fitting has a runtime which is orders of magnitudes larger than the runtime of the network, rendering this method practically unusable for large datasets.](sdss_stars__vs__fitting_stars.pdf){width="\textwidth"}
[^1]: <http://skyserver.sdss.org/CasJobs/>
[^2]: <https://gea.esac.esa.int/archive/>
[^3]: Useful tutorials for <span style="font-variant:small-caps;">TensorFlow</span> can be found here: <https://www.tensorflow.org/tutorials/>. One tutorial particularly useful for this paper can be accessed via <https://www.tensorflow.org/tutorials/layers> (accessed on 22/02/2018).
[^4]: <http://www.h5py.org>
|
---
abstract: 'We make an extensive empirical study of the market impact of large orders (metaorders) executed in the U.S. equity market between 2007 and 2009. We show that the square root market impact formula, which is widely used in the industry and supported by previous published research, provides a good fit only across about two orders of magnitude in order size. A logarithmic functional form fits the data better, providing a good fit across almost five orders of magnitude. We introduce the concept of an “impact surface" to model the impact as a function of both the duration and the participation rate of the metaorder, finding again a logarithmic dependence. We show that during the execution the price trajectory deviates from the market impact, a clear indication of non-VWAP executions. Surprisingly, we find that sometimes the price starts reverting well before the end of the execution. Finally we show that, although on average the impact relaxes to approximately $2/3$ of the peak impact, the precise asymptotic value of the price depends on the participation rate and on the duration of the metaorder. We present evidence that this might be due to a herding phenomenon among metaorders.'
author:
- Elia Zarinelli
- Michele Treccani
- 'J. Doyne Farmer'
- Fabrizio Lillo
bibliography:
- 'bibliography\_mi.bib'
title: 'Beyond the square root: Evidence for logarithmic dependence of market impact on size and participation rate'
---
Introduction
============
The market impact[^1] of trades, i.e. the change in price conditioned on signed trade size, is a key property characterizing market liquidity and is important for understanding price dynamics [@bouchaud2008markets]. As shown theoretically in the seminal work of Kyle [@kyle1985continuous], the optimal strategy for an investor with private information about the future price of an asset is to trade incrementally through time. This strategy allows earlier executions to be made at better prices and minimizes execution cost. As done in recent papers, we will call the full orders [*metaorders*]{} and the individual trades used to complete the execution [*child orders*]{}. Here we study the market impact of metaorders and its dependence on other properties, such as participation rate and execution time.
Kyle’s original model [@kyle1985continuous] predicts that market impact should be a linear function of the metaorder size, but this requires a variety of idealized assumptions that may be violated in real markets. Empirical studies have consistently shown that the market impact of a metaorder is a non-linear concave function of its size. The concave nature of market impact is robust, being observed for several heterogeneous datasets in terms of markets, epochs, and style of execution [@toth2011anomalous]. Most earlier studies have concluded that the market impact of a metaorder is well described by a “square root law” of market impact [@torre1997barra; @almgren2005direct; @engle2008measuring; @toth2011anomalous; @mastromatteo2014agent; @brokmann2014; @Iuga14]. Defining market impact $\mathcal{I}$ as the expected average price return (or difference) between the the beginning and the end of a metaorder of size $Q$, the square-root law states that $$\label{eq_rms}
\mathcal{I}(Q) = \pm Y \sigma_D \left( \frac{Q}{V_D} \right)^{\delta}$$ where $\sigma_D$ is the daily volatility of the asset, $V_D$ is the daily traded volume, and the sign of the metaorder is positive (negative) for buy (sell) trades. The numerical constant $Y$ is of order unity and the exponent $\delta$ is in the range 0.4 to 0.7, but typically very close to $1/2$, i.e. to a square root. Notice that the only conditioning variable is the total volume $Q$. This is surprising because it implies that the time taken to complete the metaorder and the participation rate are not individually important for explaining market impact – the total order size $Q$ is all that matters.
Most empirical studies of market impact make use of proprietary data of funds or brokerage firms, since the empirical analysis of metaorder’s impact cannot be performed with public data. Therefore the vast majority of studies rely on a partial view of the market. Exceptions are Refs. [@moro2009market; @tothlillo10] where the whole market is considered and metaorders are reconstructed statistically from brokerage data.
In this paper we perform an extensive empirical investigation of the market impact of metaorders, relying on a dataset of several million metaorders executed in US equity markets on large, medium and small capitalisation stocks. The dataset is heterogeneous, containing metaorders traded by many financial institutions for different purposes, and it spans several years (in the present analysis we consider the period 2007- 2009). The main strengths of our paper are the large number of metaorders and the heterogeneity of their origin. Market impact is very noisy and larger datasets can significantly help in reducing statistical uncertainty; our dataset has almost seven million metaorders, making it more than a factor of four larger than any previous study. Moreover the heterogeneity of institutions and brokers in this dataset guarantees that our results are not specific to a single execution strategy. For comparison in Table \[tab\_number\_0\] we report the approximate number of metaorders investigated in previous literature. It is clear that our sample is more than an order of magnitude larger than the typical size investigated so far. Moreover, in contrast to other studies, the set of funds and brokers is large and heterogeneous.
The main weakness of the dataset is that we have little knowledge and control on the conditions and characteristics of the execution. We do not know if the metaorders were executed for cash reasons or were informed trades (as in [@waelbroeck2013market]). Similarly, we do not know the execution algorithm used by the brokers (even if, as shown below, we can infer some information from the price dynamics during the metaorder execution). Finally, we do not know if trading size was conditioned on movement of the price during execution of the metaorder and if the daily metaorder was part of a longer execution over multiple days. All these effects can potentially bias the sample and have some role in the observed properties of the impact.
Author \# of metaorders Institution
---------------------------------------------- ------------------ ----------------------
Almgren et al. [@almgren2005direct] 700,000 Citigroup
Engle et al. [@engle2008measuring] 230,000 Morgan Stanley
Tóth et al.[@toth2011anomalous] 500,000 CFM
Mastromatteo et al. [@mastromatteo2014agent] 1,000,000 CFM
Brokmann et al. [@brokmann2014] 1,600,000 CFM
Moro et al. [@moro2009market] 150,000 inferred
Bershova et al. [@bershova2013non] 300,000 AllianceBernstein LP
Waelbroeck et al. [@waelbroeck2013market] 130,000 various
Bacry et al. [@Iuga14] 400,000 one broker
: The approximate number of metaorders considered in previous studies, together with the corresponding trading institution where the orders originated. []{data-label="tab_number_0"}
We do several things in this paper, studying the dependence of impact on the ratio of order size and volume as well as other conditioning variables, the development of impact as a function of time, and the relaxation of price once the order is completed, as detailed below.
First, we test the limits of validity of the square root impact law by conditioning it on variables such as the market capitalization of the stock, the participation rate, and the duration of execution. Because we are able to span more than five orders of magnitude of the ratio $Q/V_D$ in Equation \[eq\_rms\], we are able to investigate deviations from the square root law more thoroughly than in previous studies. Indeed we observe consistent deviations for large and small values of $Q/V_D$, indicating that the power law relation of Eq. \[eq\_rms\] is only approximately valid. Instead we find that a logarithmic function (which is more concave) fits the data significantly better.
Second, as suggested by a general class of market impact models, we study how market impact depends jointly on the duration $F$ and the participation rate $\eta$ of the metaorder. Measuring time in units of traded volume, and letting $V_P$ be the volume exchanged by the whole market during the execution of the metaorder, the duration is the fractional volume $F = V_P/V_D$, and the participation rate is the ratio of the order size $Q$ to the market volume while it is being executed, i.e. $\eta = Q/V_D$. This implies that the conditioning variable $Q/V_D$ in Equation \[eq\_rms\] can be written as $$\pi \equiv \frac{Q}{V_D} = \frac{Q}{V_P} \frac{V_P}{V_D} = F \cdot \eta$$ Thus the square root law of Eq. \[eq\_rms\] implicitly assumes that the impact depends only on the square root of the product of $F$ and $\eta$. We will show that this assumption is only approximate and that a more complex functional shape describes the data better. This functional form is described by an [*impact surface*]{} that takes into account the variation of market impact with both participation rate and execution time, or alternatively, any two of the three variables $\pi$, $\eta$ and $F$. We show that the dependence of impact on these individual variables is better described by logarithms than power laws.
Third, we consider how the price changes [*during*]{} the execution of the metaorder. Recent studies [@moro2009market; @bershova2013non; @waelbroeck2013market] find that impact is a concave function of time, i.e. for a given execution size, earlier transactions of the metaorder change the price more than later transactions. By using a much larger dataset we confirm this observation, but we find that the pattern followed by the price during execution does not mirror the dependence of the metaorder on size. To say this more explicitly, consider two metaorders with the same participation rate, one with double the volume of the other. When the larger metaorder is halfway through its execution, will the impact at that point be equal to that of the smaller one that has just completed? The general answer is no: The impact of the larger metaorder at that point in time will be larger than the impact of the smaller one. Interestingly, in some cases we find that the price starts reverting even before the end of the metaorder. We discuss some possible explanations for these findings.
Finally, the fourth question concerns the price dynamics after the conclusion of the metaorder. This topic (not covered in the original paper of Kyle [@kyle1985continuous]) has been receiving increasing attention recently [@moro2009market; @farmer2013efficiency; @bershova2013non; @waelbroeck2013market; @brokmann2014]. Several studies indicate that once the metaorder is executed the market impact relaxes from its peak value and converges to a plateau [@moro2009market; @bershova2013non; @waelbroeck2013market]. The reversion indicates that not all the impact is permanent. Even stronger, a recent study suggests that, up to a proper deconvolution of the market impact with respect to the impact of subsequent metaorders and of the the price momentum, the impact relaxes to zero [@brokmann2014].
The measurement of permanent market impact is difficult for two reasons. First, the price after the end of the metaorder is very noisy, and a careful determination of the average price dynamics requires a large sample of metaorders. Second, if successive metaorders (whether by the same or different traders) are correlated in sign, it might be difficult to isolate the permanent impact of an individual metaorder [@brokmann2014].
By making use of our large and heterogeneous sample, we perform careful measurements of the permanent impact of metaorders, considering different participation rates and durations. For typical metaorder durations and participation rates we find that after the end of the metaorder the price decays to a value which [*on average*]{} is roughly $2/3$ of the peak impact, as suggested by [@farmer2013efficiency] and found empirically by [@bershova2013non]. However, we show that the measured price decay depends on the participation rate and duration of the metaorder. Based on empirical evidence, we postulate that this dependence can be in part explained by a herding phenomenon accounting for the fact that metaorders executed in the same time period tend to have similar sign (buy or sell). Thus correlation between the sign of nearby metaorders might be partly responsible for the level of the plateau reached by permanent impact.
The paper is organized as follows. In Section \[sec:defdat\] we present the definition of the variables and the averaging procedure. We also discuss the dataset and some descriptive statistics. Section \[sec\_models\] presents some models of the price dynamics during the execution of a metaorder, used later to understand the empirical findings. In Section \[sec:measurement\] we present our empirical results and in Section \[fundamentalModels\] we discuss the implications of our empirical results on fundamental models of market impact. Finally, in Section \[sec:conclusions\] we draw some conclusions.
Definitions and Data {#sec:defdat}
====================
In this section we define the parameters we use to describe metaorder execution and the relative measures we consider to quantify market impact. In a second part we describe the database on which our analysis relies and we present some summary statistics of metaorders.
Definitions
-----------
One of the well known facts of intraday financial data is the presence of very strong periodicities. In particular, the level of trading activity is known to vary substantially and consistently between different periods of the trading day, and this intra-day variation affects both the volume profile and the variance of prices. Therefore one minute at the opening is quite different, in terms of volume, from a minute in the middle of the day. In order to take into account the intraday patterns, in this paper we perform all our computations in *volume time*. This consists in moving forward time according to the volume traded in the market. For a trading day, let $V(t)$ be the total volume traded by the market from the opening until (physical) time $t$. We measure volume time via $v=v(t):= V(t)/V(t_c)$, where $t_c$ is the daily closing time and $V(t_c)$ is the volume traded in that day. The relationship between the physical time $t$ and the volume time $v$ is independent of the total daily volume. In particular, $v=0$ at market open and $v=1$ at market close.\
We introduce three non-local parameters characterising the execution of a metaorder buying/selling ($\epsilon = \pm 1$) $Q$ shares in a physical time interval $[t_s,t_e]$. The **participation rate** $\eta$ is defined as the ratio between the volume $Q$ traded by the metaorder and the volume traded by the whole market during the execution interval $$\eta := \frac{Q}{V(t_e)-V(t_s)}.$$ The **duration** $F$ of a metaorder in volume time is defined by $$F := v(t_e)-v(t_s) = \frac{V(t_e)-V(t_s)}{V(t_c)}.$$ The **daily fraction** $\pi$ is defined as the ratio between the volume $Q$ traded by the metaorder and the volume traded by the market in the whole day, i.e. $\pi := {Q}/{V(t_c)}$. The metaorders we consider are executed within a single trading day, therefore these parameters are between $0$ and $1$. The three variables are clearly not independent, because it is $\pi = \eta \cdot F$.\
To quantify the market impact of the execution of a metaorder we define $s(v)$ as the logarithm of the price $S(v)$ at volume time $v$ rescaled by the daily volatility $\sigma_D$, i.e. $s(v) := {\log S(v)}/{\sigma_D}$. Letting $\epsilon$ be the sign of the metaorder, and $\Omega$ be any set of information upon which the market impact is conditioned, the market impact at time $v$ of a metaorder that started at time $v_s<v$ is
$$\mathcal{I}(v | \Omega) := \mathbb{E}\left[ \left. \epsilon \left( s(v) -s(v_s) \right) \right| \Omega \right].
\label{eq_imp_v}$$
We will consider conditioning sets $\Omega$ involving $\eta$, $F$ and $\pi$ as well as global information like the market capitalisation of the traded stock or the year when the metaorder is executed. With $\mathbb{E}\left[ \cdot \right|\Omega]$ we refer to the sample average over all metaorders belonging to the same set $\Omega$.
We will consider three types of impact. The **immediate** market impact quantifies how market impact builds up during the execution of the metaorder, i.e. $v_s<v< v_e$. After the conclusion of the execution of the metaorder, $v>v_e$, the market impact relaxes toward the **permanent** market impact. The **temporary** market impact is measured at the moment $v=v_e$ when the metaorder is completed, i.e. $$\label{eq_imp_tmp}
\mathcal{I}_{tmp}(\Omega) := \mathbb{E}\left[ \left. \epsilon \left( s(v_e) -s(v_s) \right) \right| \Omega \right].$$ The temporary market impact[^2] conditioned on the daily fraction $\pi$ defines the [*market impact curve*]{} $\mathcal{I}_{tmp}(\Omega=\{ \pi \} )$. This is the quantity that has received the most attention in previous studies of market impact. The temporary market impact conditioned on both the participation rate $\eta$ and the duration $F$ defines the [*market impact surface*]{}, $\mathcal{I}_{tmp}(\Omega=\{ \eta, F \} )$. The immediate market impact conditioned on both the participation rate $\eta$ and the duration $F$ defines the [*market impact trajectory*]{} $\mathcal{I}(v | \Omega=\{ \eta, F\})$, i.e. how the impact reaches the market impact surface $\mathcal{I}_{tmp}(\Omega=\{ \eta, F \} )$ during the execution of the metaorder.
Metaorder execution data {#sec_meta}
------------------------
Our analysis relies on the database made available by Ancerno, a leading transaction-cost analysis provider (*www.ancerno.com*)[^3]. The database contains data gathered by Ancerno on metaorder execution from the main investment funds and brokerage firms in the U.S. For each metaorder we consider the stock symbol, the volume $Q$, the sign $\epsilon$, the starting time $t_s$ of the metaorder and the time $t_e$ when the metaorder is completed. Our analysis has been performed on a subset of the database, containing metaorders traded on the U.S. equity market from January 2007 to December 2009. Before filtering, this subset contains 28,386,564 metaorders. This is more than an order of magnitude larger than any previous measurements of this kind (see table \[tab\_number\_0\]). All metaorders are completed within one trading day. We introduce the following filters:
- **Filter 1**: we select the stocks which belong to the Russell3000 index. This filter is introduced in order to have the time series of the price for each analysed metaorder. In this way we also discard metaorders executed on highly illiquid stocks.
- **Filter 2**: we select metaorders ending before 4:01 PM.
- **Filter 3**: we select metaorders whose duration is longer than 2 minutes.
- **Filter 4**: we select metaorders whose participation rate $\eta$ is smaller than $0.3$
year raw Filter 1 Filter 2 Filter 3 Filter 4
------ ------------ ------------ ------------ ----------- -----------
2007 9,216,333 6,904,656 3,082,767 2,130,045 1,976,382
2008 9,955,238 8,074,103 4,035,043 2,731,572 2,563,674
2009 9,214,993 7,622,703 3,954,355 2,552,092 2,404,827
tot 28,386,564 22,601,462 11,072,165 7,413,709 6,944,883
: Number of metaorders surviving each filter introduced in the analysis.[]{data-label="tab_number"}
JPM XOM MSFT GE PG BAC CSCO AAPL T GS
-------- -------- -------- -------- -------- -------- -------- -------- -------- --------
37,179 36,676 36,112 35,490 34,216 34,163 33,750 32,007 31,652 30,921
QCOM HPQ WMT VZ MRK C PFE GOOG SLB JNJ
30,765 29,915 29,898 27,975 27,106 26,767 26,392 26,202 25,821 25,602
: Ticker symbol of the most traded stocks and corresponding number of metaorders.[]{data-label="tab_top"}
The number of metaorders surviving each filter is reported in table \[tab\_number\]. In table \[tab\_top\] we present the stock symbols with the largest number of metaorders in the dataset and their number. It is interesting to note that for the top 20 stocks the metaorders recorded in the Ancerno database are responsible of around 5% of the daily volume. It is evident that Filter 2 cuts a significant fraction of metaorders. All these metaorders last exactly 410 minutes, starting at 9:30 AM and ending at 4:20 PM. A detailed investigation of these orders strongly suggests that the initial and final time of these orders are not reliable, and we suspect that for these orders the times communicated to Ancerno are not accurate. For example, these orders have systematically lower participation rate than the other orders, suggesting an effective shorter time span of execution. In order to avoid introducing data that might be spurious, we drop these metaorders, at the cost of significantly reducing our sample.\
In conclusion, for each metaorder in the dataset we recover the relative daily fraction $\pi$, the participation rate $\eta$, and the duration $F$. By exploiting the price data, we also recover the time series of the price $s(v)$ during and after the execution of the metaorder.
Market price data
-----------------
In order to augment the information in the metaorder data described in the previous subsection we augment it with market data. The latter are historical data provided by Kibot (*www.kibot.com*),consisting of one-minute time series giving the Date, Time, Open, High, Low, Close, Volume of $3,500$ stocks in the Russell3000 index. We consider as a proxy of the daily volatility $\sigma_D= (S_h-S_l)/S_o$, where $S_{h,l,o}$ are the high-low-open price of the day. Given the time interval $[t_s,t_e]$ and the volume $Q$ of a metaorder, this dataset makes it possible to compute its participation rate $\eta$, daily rate $\pi$, and duration $F$. We also use this database to measure the price dynamics during and after the execution of the metaorder.
Metaorder statistics
--------------------
![Time series of metaorders active on the market for AAPL in the period March-April 2008. Buy (Sell) metaorders are depicted in blue (red). The thickness of the line is proportional to the metaorder participation rate. More metaorders in the same instant of time give rise to darker colours. Each horizontal line is a trading day. We observe very few blanks, meaning that there is almost always an active metaorder from our database, which is of course only a subset of the number of orders that are active in the market.[]{data-label="fig_aapl"}](aapl.pdf){width="100.00000%"}
We now present some descriptive statistics. In Figure \[fig\_aapl\] we show the time series of the metaorders for Apple (AAPL) in the period March-April 2008. There is a significant number of metaorders active every day. In most trading days there is a “mood", i.e. on any given day most metaorders have the same sign, indicating a possible herding effect. Later we will quantify this metaorder overlap and we will discuss its possible role on the shape of market impact.
![Estimation of the probability density function of the participation rate $\eta$ (top left), duration $F$ (top right), and daily fraction $\pi$ (bottom left). All these panels are in log-log scale and the first two shows also the best fit with a power law function in the region bounded by the two vertical dashed lines. The bottom right panel shows the logarithm of the estimated joint probability density function $p(\eta,F)$ in double logarithmic scale of the duration $F$ and the participation rate $\eta$. []{data-label="fig_stat"}](stat_eta.pdf "fig:"){width="40.00000%"} ![Estimation of the probability density function of the participation rate $\eta$ (top left), duration $F$ (top right), and daily fraction $\pi$ (bottom left). All these panels are in log-log scale and the first two shows also the best fit with a power law function in the region bounded by the two vertical dashed lines. The bottom right panel shows the logarithm of the estimated joint probability density function $p(\eta,F)$ in double logarithmic scale of the duration $F$ and the participation rate $\eta$. []{data-label="fig_stat"}](stat_F.pdf "fig:"){width="40.00000%"}\
![Estimation of the probability density function of the participation rate $\eta$ (top left), duration $F$ (top right), and daily fraction $\pi$ (bottom left). All these panels are in log-log scale and the first two shows also the best fit with a power law function in the region bounded by the two vertical dashed lines. The bottom right panel shows the logarithm of the estimated joint probability density function $p(\eta,F)$ in double logarithmic scale of the duration $F$ and the participation rate $\eta$. []{data-label="fig_stat"}](stat_pi.pdf "fig:"){width="40.00000%"} ![Estimation of the probability density function of the participation rate $\eta$ (top left), duration $F$ (top right), and daily fraction $\pi$ (bottom left). All these panels are in log-log scale and the first two shows also the best fit with a power law function in the region bounded by the two vertical dashed lines. The bottom right panel shows the logarithm of the estimated joint probability density function $p(\eta,F)$ in double logarithmic scale of the duration $F$ and the participation rate $\eta$. []{data-label="fig_stat"}](prova.pdf "fig:"){width="40.00000%"}
We then investigate the distributional properties of the parameters characterising metaorders, namely the participation rate $\eta$, the duration $F$, and the daily fraction $\pi$. The statistics are performed by aggregating the $6,944,883$ filtered metaorders.
We find that the participation rate $\eta$ and the duration $F$ are both well approximated by a truncated power-law distribution over several orders of magnitude. The estimated probability density function of the participation rate $\eta$ is shown in log-log scale in the top left panel of Figure \[fig\_stat\]. A power law fit in the region $10^{-4}\le \eta \le 0.1$, i.e. over three orders of magnitude gives a best fit exponent $a=-0.864 \pm 0.001$. The top right panel of Figure \[fig\_stat\] shows the estimated probability density function of the duration $F$ of a metaorder. A power law fit in the intermediate region bounded by the two vertical dashed lines ($0.01\le F \le 0.5$) gives a power-law exponent $a=-0.932 \pm 0.003$. Thus in both cases the power law is very heavy tailed, meaning that there is substantial variability in both the partition rate and duration of the orders over a large range. Note that in both cases the variability is intrinsically bounded (and therefore the power law is automatically truncated) by the fact that by definition $\eta \le 1$ and $F \le 1$. In addition, for $p(F)$, there is a small bump on the right extreme of the distribution corresponding to all-day metaorders. The deviation from a power law for small $F$ is forced automatically by our filter retaining only orders lasting at least $2$ minutes, which in volume time corresponds on average to $2/390\simeq 0.005$.
The bottom left panel shows the probability density function of the daily fraction $\pi$. In this case the distribution is less fat tailed, and in particular it is clearly not a power law. This is potentially an important result, as the predictions of some theories for market impact depend on this, and have generally assumed power law behavior [@gabaix06; @farmer2013efficiency].
Since two of the three variables characterizing a metaorder are sufficient to derive the third one ($\pi = \eta \cdot F$), it is important to study the correlation between them, especially in light of the multivariate regression we perform below. The bottom right panel of Figure \[fig\_stat\] shows the logarithm of the estimated joint probability density function $p(\eta,F)$ in double logarithmic scale as a function of the duration $F$ and the participation rate $\eta$. The linear correlation between the two variables is very low ($-0.022 $). The main contribution coming from the extreme regions, i.e. $\eta$ very large, implies $F$ very small and vice versa. This means, as expected, that very aggressive metaorders are typically short and long metaorders more often have a small participation rate.
Heuristic models of market impact {#sec_models}
=================================
Before presenting our empirical results on market impact, we consider some simple heuristic models of price dynamics. By “heuristic" we mean that these are reduced form models that are chosen because they are intuitively reasonable and they are useful, e.g. for computing optimal trade execution strategies. We distinguish these from more fundamental models that try to explain the form of the market impact function from first principles. These models provide a useful framework to investigate and to interpret our measures of market impact, presented in the next Section. In particular it will provide a context to interpret some of the non-intuitive aspects of the relationship between immediate impact and temporary impact. In Section \[fundamentalModels\] we return to discuss some of the implications of our empirical work for fundamental models.
The Almgren-Chriss model {#sec_ac}
------------------------
We consider first a simplified version of the Almgen-Chriss model [@almgren2001optimal] in continuous time. We assume that a metaorder with participation rate $\eta$ is executed incrementally within $t \in [0,T]$[^4]. The total traded quantity is $Q= \eta T$ and the instantaneous trading rate is $q(t)=-\dot x(t)$, where $x(t)$ is the metaorder quantity that remains to be traded. The price dynamics is $$S(t)=S(0)+a\int_{0}^{t} q(s) ds + \sigma\int_{0}^{t} \mathrm{d}W_s \ ,$$ where $W_t$ is a Wiener process. Due to the linearity of the impact function, the immediate impact as a function of time is $${\cal I}(t|\Omega = \{ \eta,T \})= a (Q-x(t)) = a (\eta T-x(t)) \ .$$ Assuming that during a buy metaorder the trader only buys and never sells, ${\cal I}(t|\Omega = \{ \eta ,T \})$ is a non decreasing function of time converging to the temporary impact $\mathcal{I}_{tmp}(\Omega = \{ \eta, T \})={\cal I}(T|\Omega = \{ \eta,T \})= aQ = a \eta T $, independently of the trading profile followed during the execution. Thus the temporary market impact is a linear function of metaorder duration $T$, for fixed participation rate $\eta$ (see the red solid line in Figure \[fig\_ac\_0\]).
![ Temporary market impact $\mathcal{I}_{tmp}(\Omega=\{ \eta, T \})$ as a function of the metaorder duration $T$ evaluated in the framework of the simplified version of the Almgren-Chriss model with $a=1$, $\sigma =1$ and $\eta =1$ (red line). We show also the immediate market impact trajectory $\mathcal{I}(t|\Omega=\{ \eta, T \})$ for several values of the risk-aversion parameter $\lambda = \{ 0.1,0.5,1\}$ and several metaorder durations $T=\{ 5,10,20 \}$ (blue lines).[]{data-label="fig_ac_0"}](ac_plot.pdf){width="53.00000%"}
What does the immediate impact look like under the Almgen-Chriss model? Even though the impact is linear, if the optimised execution schedule is risk adverse, the immediate impact trajectory [*during*]{} the execution does not necessarily overlap with the curve described by the temporary impact as a function of $T$ (see the blue lines in Figure \[fig\_ac\_0\]). As a concrete example, consider the optimal trading profile of the original Almgren-Chriss model for a generically risk averse investor. In this case the function to be minimized is the expected cost plus $\lambda$ ($>0$) times the variance of the cost. The optimal solution is [@almgren2001optimal] $$x(t)=Q\frac{\sinh k(T-t)}{\sinh kT}$$ where $k=\sqrt{\lambda \sigma^2/a}$, and $\sigma$ is the volatility of the price. For a risk neutral strategy ($\lambda=0$), the solution is a strategy with constant velocity, $q(s)=\eta$. This is the simplest (and probably more widespread) execution strategy, the so called Volume Weighted Average Price (VWAP) scheme. More risk averse strategies correspond to higher $\lambda$ and lead to more front loaded executions. We observe in Figure \[fig\_ac\_0\] that for higher levels of risk aversion the immediate impact deviates more from the temporary impact. In all cases the price reaches the temporary impact from above. This is due to the front loading property of the strategy. For a strategy where the trading rate increases during execution, the price would reach the temporary impact from below. Only in the case of VWAP (i.e. risk neutral strategy) do immediate and temporary impact overlap.
The propagator model {#sec_prop}
--------------------
A more sophisticated model is the propagator model, devised to take into account the non linear and immediate properties of market impact. The propagator model was initially proposed by Bouchaud et al. [@bouchaud2004fluctuations] in (discrete) transaction time and independently introduced by Lillo and Farmer [@lillo2004long] (the latter as a model where price moves in response to the unexpected component of the order flow, see also Taranto et al. [@taranto2014adaptive]). An interesting extension that goes beyond the propagator model was very recently proposed in [@Donier14].
The continuous-time version of the propagator model, discussed by Gatheral [@gatheral2010no], is $$S(t) = S(0) + \int_{0}^{t} f(q(s)) G(t-s) \mathrm{d}s +\int_{0}^{t} \sigma(s)\mathrm{d}W_s,$$ where $G(t)$ is a decaying function describing the temporal dependence of the impact and the function $f(q)$ is an odd function describing the volume dependence of the impact. The Almgren and Chriss model can be recovered by setting $f(q)= aq$ and $G(t)={\mathbb 1}_{t\ge0}$. We consider here a small variation of the Gatheral model where $s(t):= \log S(t)/ \sigma_D$ evolves in volume time $v$ according to $$\label{eq:prop2}
s(v) = s(0) + \int_{0}^{v} f(q(s)) G(v-s) \mathrm{d}s +\int_{0}^{v} \mathrm{d} W_s.$$ In order to deal with nondimensional quantities, we rescale the instantaneous trading rate $q(s)$ by the daily traded volume $V(t_c)$. Specifically, we consider the propagator model with power-law impact function $f(q) = q^{\delta}$ and power-law decay kernel $G(t) = t^{-\gamma}$. For $\delta = 1$ and $\gamma = 0$ one recovers the Almgren-Chriss model. Using this model, Gatheral [@gatheral2010no] shows that the condition $\delta + \gamma \ge 1$ is necessary to exclude price manipulation. In the case of linear market impact, $f(q)=q$, Gatheral et al. [@gatheral2012transient] obtained the optimal condition and derived the explicit form of the optimal strategy in a expected cost minimisation problem. In the general case of non linear impact the problem is more involved [@dang2012optimal; @curato14]. In this paper we are not interested in solving the optimisation problem but rather in calculating the market impact for different classes of trading strategies.\
For the simple VWAP strategy characterised by trading rate $q(s)=\eta$ and duration $F$, the temporary market impact is $$\mathcal{I}_{tmp}(\Omega=\{ VWAP, \eta, F\}) : = \int_{0}^{F} f(q(s)) G(F-s) \mathrm{d}s = f(\eta) \int_0^F G(F-s) ds.$$ The factorization between the temporal and the volume dependence hypothesized in the propagator model immediately leads to a factorization of the temporary impact into a part depending only on $\eta$ and a part depending only on $F$. In the special case $f(q) = q^{\delta}$ and $G(t) = t^{-\gamma}$ this becomes $$\label{eq_imp_vwap_1}
\mathcal{I}_{tmp}(\Omega=\{ VWAP, \eta, F\}) = \frac{1}{1-\gamma} \eta^{\delta} F^{1-\gamma}. $$ This relation shows that within the propagator model with a power-law impact function and a power-law decay kernel, the temporary impact is a factorisable power-law function of both the participation rate $\eta$ and of the duration $F$. The “macroscopic” exponents describing the shape of the temporary market impact surface $\mathcal{I}_{tmp}(\Omega=\{ VWAP, \eta, F\})$ are inherited from the “microscopic” exponents describing the market impact function of individual trades $\delta$ and the decaying kernel $\gamma$. If the model is at the critical condition according to Gatheral, $\delta+\gamma=1$, the temporary impact does not depend on the duration $F$ and the participation rate $\eta$ separately, but only on their product, i.e. the daily fraction $\pi=\eta \cdot F$. Hence the market impact surface is fully characterized by the market impact curve. Notice that with $\gamma=\delta=1/2$, one recovers the square-root impact curve, Eq. \[eq\_rms\], with $Y=2$. Finally, the immediate and the permanent market impact, as defined by Eq. \[eq\_imp\_v\], are
$$\label{eq_imp_vwap_2}
\mathcal{I}(z | \Omega=\{ VWAP, \eta, F\}) =
\begin{cases}
\frac{1}{1-\gamma} \eta^{\delta} F^{1-\gamma} z^{1-\gamma} & \text{if } z < 1,\\
\frac{1}{1-\gamma} \eta^{\delta} F^{1-\gamma} \left( z^{1-\gamma} - (z-1)^{1-\gamma} \right)& \text{if } z>1.
\end{cases}$$
where $z = v/F$. For large values of $z$ the impact decays to zero as a power law function with exponent $\gamma$. On the other hand, immediately after the metaorder completion, the price decay follows a power law with exponent $1-\gamma$.
The calculation of the immediate and temporary market impact becomes more involved if we consider general execution schemes where the trading velocity is not constant. For purely illustrative purposes, we consider a class of execution schemes of $Q$ shares characterised by the instantaneous monotonic trading rate $$\label{example}
q(s)=\frac{Q}{V(t_c)}\frac{(\alpha+1)}{F^{\alpha+1}}(F-s)^\alpha,~~~~~~~~~~\alpha>-1.$$ For positive (negative) $\alpha$ the relative trading profile trades more (less) at the beginning of the period, while $\alpha=0$ corresponds to a VWAP scheme. It is possible to show that for this class of schemes the temporary market impact is $$\label{tempimpprop}
\mathcal{I}_{tmp}(\Omega=\{ \alpha, \eta, F\}) = \eta^\delta F^{1-\gamma} \frac{(1+\alpha)^\delta}{1+\alpha\delta-\gamma},$$ while the immediate and permanent market impact are
$$\begin{aligned}
\mathcal{I}(z | \Omega=\{ \alpha, \eta, F\}) = \eta^{\delta} F^{1-\gamma}(\alpha +1)^{\delta} \int_0^z \mathrm{d}s \frac{(1-s)^{\alpha \delta}}{(z-s)^{\gamma}} {\mathbb 1}_{s\le 1} = \nonumber \\
\begin{cases}
\frac{1}{1-\gamma} \eta^\delta (1+\alpha)^\delta F^{1-\gamma} z^{1-\gamma} ~~ _2\mathcal{F}_1(1,-\delta\alpha;2-\gamma;z) & \text{if } z < 1,\\
\frac{1}{1+\alpha \delta} \eta^\delta (1+\alpha)^\delta F^{1-\gamma} z^{-\gamma} ~~ _2\mathcal{F}_1(1,\gamma;2+\delta\alpha;\frac{1}{z}) & \text{if } z>1,
\end{cases}\end{aligned}$$
where $z=v/F$ is the rescaled time and $_2\mathcal{F}_1$ is the hypergeometric function. In order to avoid the divergence of the temporary impact ($z=1$), one has to impose $1+\alpha\delta-\gamma>0$.\
![ Temporary market impact (red line) $\mathcal{I}_{tmp}(\Omega=\{ \alpha, \eta, F\})$ as a function of the metaorder duration $F$ for fixed participation rate $\eta=1$ and different values of trading rate profiles $\alpha$ for the propagator model with $\gamma=\delta=1/2$. We consider a back loaded profile ($\alpha = -1/2$, top left), a VWAP profile ($\alpha = 0$, top right), and two front loaded profiles ($\alpha = 1$ and $\alpha = 4$, bottom). In each panel we also show the immediate market impact trajectory (blue lines) $\mathcal{I}(v | \Omega=\{ \alpha, \eta, F\})$ for some values of $F=0.25,0.5,0.75,1$ as a function of volume time $v$. []{data-label="fig_ac"}](qwerty_2.pdf){width="100.00000%"}
Figure \[fig\_ac\] shows the temporary market impact $\mathcal{I}_{tmp}(\Omega=\{ \alpha, \eta, F\})$ as a function of the metaorder duration $F$ at fixed participation rate $\eta$ for different values of trading profiles $\alpha$ in a propagator model with $\gamma=\delta=1/2$. For some values of the duration $F$ we also show the immediate market impact trajectory $\mathcal{I}(v | \Omega=\{ \alpha, \eta, F\})$, i.e. the trajectories followed by the price to reach the temporary impact. There are three important comments one can make observing these figures: (i) The temporary market impact depends on the trading profile (see also Eq. \[tempimpprop\]). This is a general consequence of the non-linearity of the impact function $f$. As we have seen before in the Almgren-Chriss model, the temporary impact is independent of the trading profile. (ii) As in the Almgren-Chriss model, the more the trading profile deviates from VWAP, the more the immediate impact trajectories deviate from the temporary impact. For front (back) loaded strategies, $\alpha>0$ ($\alpha<0$), the trajectories reach the temporary impact from above (below); (iii) Even if the trade sign is always the same during execution (e.g. buys for a buy metaorder), the impact trajectories can be non-monotone, since they reach a maximum and decay to the temporary impact, [*before*]{} the end of the metaorder (see the cases $\alpha=1$ and $\alpha=4$ in Fig. \[fig\_ac\]). In other words, the price reversion, well documented after the end of the metaorder, starts during the metaorder’s execution if the trading profile is front loaded enough. As we will see below, this is exactly what we observe for real metaorder executions in some of the data.
Measurements of market impact {#sec:measurement}
=============================
Market impact curve: testing the square-root formula
----------------------------------------------------
![Measured temporary market impact $\mathcal{I}_{tmp}(\Omega=\{ \pi \} )$ of a metaorder as a function of the daily rate $\pi$, defined as the ratio of the traded volume and the daily volume. The scale is double logarithmic; the dashed read line is the best fit to a power-law and the solid blue curve is the best fit to a logarithm.[]{data-label="fig_toth0"}](toth.pdf){width="80.00000%"}
The temporary market impact curve $\mathcal{I}_{tmp}(\Omega=\{ \pi \} )$ is defined as the price change conditioned on the daily fraction $\pi$. The square-root impact formula (Eq. \[eq\_rms\]) states that the temporary market impact curve is described, at least to a first approximation, by a power-law function: $$\label{eq_rms2}
\mathcal{I}_{tmp}(\Omega=\{ \pi \}) =Y \pi^\delta \ . $$ Previous studies find $\delta$ in the range 0.4 to 0.7 [@almgren2005direct; @toth2011anomalous; @mastromatteo2014agent; @brokmann2014]. Figure \[fig\_toth0\] shows the shape of the measured temporary market impact curve for our data in double logarithmic scale. This plot is obtained by dividing the data into evenly populated bins according to the daily rate $\pi$ and computing the conditional expectation of the impact for each bin. Here and in the other figures of this paper, the error bars are standard errors. Note that the range of $\pi$ spans more than five orders of magnitude. We observe that for $\pi$ roughly in the range from $10^{-3}$ to $10^{-1}$ the points, to a first approximation, lie on a straight line. Nonetheless, a clear concavity is evident, since for large and small $\pi$ the impact curve bends down.
Performing a nonlinear regression on the function $f(\pi|Y,\delta) = Y \pi^{\delta}$, the best fitting parameters are $\hat Y = 0.15 \pm 0.01$ and $\hat\delta=0.47\pm0.01$. The value of the exponent is consistent with that found in previous work [@almgren2005direct; @toth2011anomalous; @mastromatteo2014agent; @brokmann2014]. In order to compare different functional forms, we consider the Weighted Root Mean Square Error, $E_{RMS}$, as a measure of the goodness of fit[^5]. For the power law we find $E_{RMS}(f(\hat{Y}, \hat{\delta})) = 6.70$. The concavity of the market impact shape depicted in double-logarithmic scale suggests that a function more concave than a the power-law might better explain the data. As an alternative we fit a function of the form $g(\pi|a,b) = a \log_{10}(1+b\pi)$. The shape of $g(\pi|a,b)$ is linear for values of $b\pi \ll 1$ and logarithmically concave for $b\pi > 1$. The estimated best fitting parameters are $\hat a = 0.028 \pm 0.001$ and $\hat b=465 \pm 33$ and the relative goodness of fit and $E_{RMS}(g(\hat{a}, \hat{b})) = 2.80$, which is quite dramatically better than that for the power law. The light blue line in Figure \[fig\_toth0\] is the best fitting logarithmic curve. From the figure and from the values of $E_{RMS}$, we conclude that the logarithmic functional form describes our data better than the power-law (square root) functional form.\
![Temporary market impact of a metaorder as a function of the daily fraction $\pi$, defined as the ratio of the traded volume and the daily volume. The scale is double logarithmic and the lines are a best fit with a power-law function $f(\pi| Y, \delta)$ (dashed) and a logarithmic function $g(\pi | a,b)$ (solid). The top left panel considers three different years separately, the top right panel considers the market capitalisation of the traded stock separately, the bottom left panel considers different subsets of the metaorder duration $F$, and the bottom right panel considers different subsets of the participation rate $\eta$. The values of the best fitting parameters and the goodness of fit for the power law and the logarithmic function are reported in table \[tab\_fit\] for each year and market capitalisation. []{data-label="fig_toth"}](toth_ye.pdf "fig:"){width="38.00000%"} ![Temporary market impact of a metaorder as a function of the daily fraction $\pi$, defined as the ratio of the traded volume and the daily volume. The scale is double logarithmic and the lines are a best fit with a power-law function $f(\pi| Y, \delta)$ (dashed) and a logarithmic function $g(\pi | a,b)$ (solid). The top left panel considers three different years separately, the top right panel considers the market capitalisation of the traded stock separately, the bottom left panel considers different subsets of the metaorder duration $F$, and the bottom right panel considers different subsets of the participation rate $\eta$. The values of the best fitting parameters and the goodness of fit for the power law and the logarithmic function are reported in table \[tab\_fit\] for each year and market capitalisation. []{data-label="fig_toth"}](toth_cap.pdf "fig:"){width="38.00000%"}\
![Temporary market impact of a metaorder as a function of the daily fraction $\pi$, defined as the ratio of the traded volume and the daily volume. The scale is double logarithmic and the lines are a best fit with a power-law function $f(\pi| Y, \delta)$ (dashed) and a logarithmic function $g(\pi | a,b)$ (solid). The top left panel considers three different years separately, the top right panel considers the market capitalisation of the traded stock separately, the bottom left panel considers different subsets of the metaorder duration $F$, and the bottom right panel considers different subsets of the participation rate $\eta$. The values of the best fitting parameters and the goodness of fit for the power law and the logarithmic function are reported in table \[tab\_fit\] for each year and market capitalisation. []{data-label="fig_toth"}](toth_dur.pdf "fig:"){width="38.00000%"} ![Temporary market impact of a metaorder as a function of the daily fraction $\pi$, defined as the ratio of the traded volume and the daily volume. The scale is double logarithmic and the lines are a best fit with a power-law function $f(\pi| Y, \delta)$ (dashed) and a logarithmic function $g(\pi | a,b)$ (solid). The top left panel considers three different years separately, the top right panel considers the market capitalisation of the traded stock separately, the bottom left panel considers different subsets of the metaorder duration $F$, and the bottom right panel considers different subsets of the participation rate $\eta$. The values of the best fitting parameters and the goodness of fit for the power law and the logarithmic function are reported in table \[tab\_fit\] for each year and market capitalisation. []{data-label="fig_toth"}](toth_eta.pdf "fig:"){width="38.00000%"}
We now consider the problem of how the temporary impact curve depends on other conditioning variables. As suggested in [@toth2011anomalous], the square-root law of market impact seems to be a very robust statistical regularity. It does not appear to depend on the traded instrument (equities, futures, FX, etc.) or time period (from the mid-nineties, when liquidity was provided by market makers, to present day electronic markets). We verify the robustness of the temporary market impact curve by conditioning on different time periods present in the database and on the market capitalisation of the traded instrument. Figure \[fig\_toth\] (top panels) shows the results of our analysis: quite remarkably, the shape of the temporary market impact curve is roughly independent on both the trading period (2007, 2008 and 2009) and the stock capitalisation (small, medium and large, according to the classification provided by the Ancerno database). The estimated best fitting parameters and the relative goodness of fit are reported in table \[tab\_fit\]. We observe that, with the exception of the small capitalisation conditioning, the logarithmic function is always better in explaining the data.
We now shift our focus to measuring the temporary market impact curve while conditioning on the parameters characterising the execution of a metaorder. As pointed out in [@gatheral2010no], the square-root formula depends only on the daily fraction $\pi$, which implies that the temporary market impact is independent of both the duration $F$ and the participation rate $\eta$. We investigate this point by conditioning our measurements on the participation rate and duration of the metaorder. The bottom panels of Figure \[fig\_toth\] show the results of this analysis We notice that in both cases the three curves are locally approximated by a power law function, essentially because the two types of conditioning reduce significantly the span of data on the abscissa. However it is clear that the exponents of the power law are different and, as a consequence, the superposition of the three subsets gives a logarithmically concave function. In particular the power law exponent decreases from $0.58$ to $0.42$ when we condition impact on $F$, and from $0.74$ to $0.47$ when we condition on $\eta$. In all cases the error on the expoenent is $0.02$. Therefore the exponent, and thus the impact function, depends on the conditioning variable ($F$ or $\eta$).
Year 2007 2008 2009
----------- ----------------- ----------------- ----------------- -----------------
Power law $\hat{Y}$ 0.13$\pm$0.01 0.12$\pm$0.01 0.15$\pm$0.01
$\hat{\delta} $ 0.41$\pm$0.02 0.41$\pm$0.02 0.46$\pm$0.01
$E_{RMS} $ 5.47 4.96 3.18
Logarithm $\hat{a} $ 0.029$\pm$0.001 0.025$\pm$0.001 0.032$\pm$0.001
$\hat{b} $ 491$\pm$29 547$\pm$56 316$\pm$25
$E_{RMS} $ [**1.36**]{} [**2.11**]{} [**2.23**]{}
Mkt. Cap Large Medium Small
Power Law $\hat{Y}$ 0.19$\pm$0.01 0.15$\pm$0.01 0.12$\pm$0.01
$\hat{\delta} $ 0.51$\pm$0.02 0.46$\pm$0.02 0.42$\pm$0.02
$E_{RMS} $ 1.38 1.32 [**0.69**]{}
Logarithm $\hat{a} $ 0.030$\pm$0.001 0.030$\pm$0.001 0.027$\pm$0.001
$\hat{b} $ 441$\pm$21 400$\pm$35 428$\pm$62
$E_{RMS} $ [**0.40**]{} [**0.69**]{} 1.03
: Estimated values of the best fitting parameters $\hat{Y}$ and $\hat{\delta}$ for the fitting function $f(\pi | Y, \delta) = Y \pi^{\delta}$ and $\hat{a}$ and $\hat{b}$ for the fitting function $g(\pi | a,b) = a \log_{10}(1+b \pi )$ and the corresponding goodness of fit. Each fit is performed conditioning the sample data on the execution year (top table) and market capitalisation (bottom table). In bold face there is the fits with the smallest $E_{RMS}$.[]{data-label="tab_fit"}
In conclusion, when plotted as a function of the daily rate $\pi$, the temporary market impact curve $\mathcal{I}_{tmp}(\Omega=\{ \pi \})$ is clearly described by a concave function, well fitted by a logarithmic function and only locally approximated by a square root function. Interestingly strong concavity for very large volumes has been quoted for CFM metaorders also in reference [@Donier14]. In the next section we show that the impact does depend on $F$ and $\eta$ separately, and that the collapse seen here is significantly due to a compensation effect from the data aggregation of orders with different conditioning parameters.
### Inferring latent order book from market impact\[virtualOrderBook\]
In a recent paper Toth et al. [@toth2011anomalous] present a theory connecting the shape of the market impact to the one of latent order book. They argue that true order book does not reflect the actual supply and demand that are present in the market, due to the fact that participants do not reveal their true intensions. Latent order book becomes visible when price moves and thus can be inferred from the shape of market impact.
Specifically, let ${\cal V}(x|b,n)$ be the volume available in the latent order book at log price $x$. A metaorder of size $\pi$ will generate a market impact ${\cal I}(\pi)$ solving the equation $$\label{eqlob}
\pi=\int_{x_0}^{x_0+{\cal I}(\pi)} {\cal V}(x|b,n)~~dx$$ where $x_0$ is the log price at the beginning of the metaorder execution. If the profile ${\cal V}$ is a linear function of log price, then market impact is a square root function of the traded volume $\pi$, as suggested by [@toth2011anomalous]. On the other hand, if the profile is constant, the market impact is linear in the traded volume. Here we assume a parametric form, and which for the center of the order book allows us to easily interpolate between a linear vs. a constant order book. We then compute the expected market impact and we fit the parameters of the latent order book profile on real data of market impact. Specifically, we consider the normalized function $${\cal V}(x|b,n):= \frac{1}{Y}\frac{ x^n \exp(bx)}{\int_0^1 \mathrm{d} y \ y^n \exp(by)}
\label{bookProfile}
$$ where $x \in [0,1]$, ${\cal V}(0|b,n)=0$ and ${\cal V}(1|b,n)=1/Y$, where $Y$ is a normalizing constant. For $x \ll 1/b$ the profile grows as a polynomial, while for $x \gg 1/b$ it grows as an exponential. We can invert Eq. \[eqlob\] to derive $\mathcal{I}(\pi |Y,b,n)$ obtaining the following cases. When $n=0$, we can perform the analytical calculation and recover the previously introduced logarithm function: $\mathcal{I}(\pi |Y,b,n=0) :=Y \log(1 + c\pi )/\log(1+c)$ where $c=\exp(b)-1$. The market impact function grows linearly for $\pi \ll 1/c$ and logarithmically for $\pi \gg 1/c$. When $n=1$ we obtain a market impact function $\mathcal{I}(\pi |Y,b,n=1)$ that grows as a square root function for small $x$ and logarithmically for large $x$. By keeping $n$ as a free parameter we can infer the order book shape near the best by fitting the impact function.
Because the data set is so large we divide the data in into $N_{bins}$ evenly populated bins. For each bin $i$ we measure the average daily rate $\pi_i$, the average impact $\mathcal{I}_i$, and the standard error on the sample impact $SE(\mathcal{I}_i)$. Then, via a non-linear weighted optimisation, we obtain the best fitting parameters of the impact models $\mathcal{I}(\pi |Y,b,n=0)$, $\mathcal{I}(\pi |Y,b,n=1)$, and $\mathcal{I}(\pi |Y,b,n)$ and we calculate the Weighted Root Mean Squared Error of each model. The large number of bins in the fitting procedure compared to the number of parameters minimizes the risk of overfitting[^6]. We have also tested the conclusions by varying the number of bins and find that for $N_{bins} > 500$ the results are independent of the number of bins.
The results of this procedure are reported in figure \[fig\_lob\]. We observe that the model $\mathcal{I}(\pi |Y,b,n=0)$ describes the data better than the model $\mathcal{I}(\pi |Y,b,n=1)$. The fitted value of the parameter $1/c$ discriminating the linear from the logarithmic regime in the impact in the case $n=0$ is $\pi^*\simeq 2 \times 10^{-3}$, indicating that when $\pi\ll\pi^*$ market impact is linear, while above this value the impact starts to be logarithmic. The model with three free fitting parameters $\mathcal{I}(\pi |Y,b,n)$ clearly improves the goodness of fit. The value of the inferred exponent, $\hat{n}=0.22$, is close to zero, suggesting an almost flat order book profile near the best bid/ask positions, even if the noise observed in the left part of figure \[fig\_lob\] is quite large. From this we conclude that our data is consistent with an exponential form of the latent order book for large volumes (see also the discussion in Section X of the recent paper [@Donier14]).
![The inferred impact function based on a latent order book profile parameterized with Eq. (\[bookProfile\]) and solved using Eq. (\[eqlob\]). The inferred impact is plotted on double logarithmic scale as a function of the normalized order size $\pi$. See Eq. (\[bookProfile\]) for the interpretation of the parameters. []{data-label="fig_lob"}](lob_xy_500.pdf){width="80.00000%"}
Market impact surface: beyond the square-root formula
-----------------------------------------------------
![Non-parametric estimation of the impact surface ${\cal I}_{tmp}(\Omega = \{ F,\eta \})$ as a function of the duration $F$ and the participation rate $\eta$. The three axes are in logarithmic scale. The orange surface represents the double power-law function $f(\eta,F|\hat{Y},\hat{\delta},\hat{\gamma_1}) = \hat{Y} \eta^{\hat{\delta}} F^{\hat{\gamma_1}}$ with the empirically fit parameters $\hat{Y}$, $\hat{\delta}$ and $\hat{\gamma_1}$. The blue surface represents the double logarithmic function $g(\eta,F|\hat{a},\hat{b},\hat{c}) = \hat{a} \log_{10}(1+\hat{b} \eta) \cdot \log_{10}(1+\hat{c} F)$ with the empirically fit parameters $\hat{a}$, $\hat{b}$ and $\hat{c}$. []{data-label="fig_toth_3d"}](toth_3d_log.pdf){width="95.00000%"}
We now consider the dependence of the temporary market impact $\mathcal{I}_{tmp}(\Omega=\{ \eta,F \})$ on the participation rate $\eta$ and the duration $F$ of metaorder execution. Please recall that in Figure \[fig\_stat\] we demonstrated that the correlation between these two parameters is quite low.
Figure \[fig\_toth\_3d\] shows a non parametric estimation of the market impact surface $\mathcal{I}_{tmp}(\Omega=\{ \eta,F \}) $ for the roughly five million metaorders in our database[^7]. The three axes of the plot are in logarithmic scale to highlight the concave shape of the market impact surface. If the temporary impact is described by a power-law function both in $\eta$ and in $F$, i.e. $$\label{eq_rms3}
\mathcal{I}_{tmp}(\Omega=\{ \eta,F \}) = Y \cdot \eta^{\delta} \cdot F^{\gamma_1}$$ the market impact surface is a plane in logarithmic scale. Figure \[fig\_toth\_3d\] shows that a linear functional form is only an approximate representation of the empirical surface. In fact the surface is clearly concave (in log scale), and almost flattens out in the top left corner. We perform a non-linear regression of the measured temporary market impact surface $\mathcal{I}_{tmp}(\Omega=\{ \eta,F \})$ with a power law function $f(\eta, F|Y,\delta,\gamma_1)= Y \eta^{\delta} F^{\gamma_1}$, according to equation \[eq\_rms3\]. The best fitting parameters are $\hat Y = 0.207 \pm 0.005$, $\hat \delta=0.52 \pm 0.01$ and $\hat \gamma_1 = 0.54 \pm 0.01$. The Root Mean Square Error for this model is $E_{RMS}(f(\hat{Y},\hat{\delta},\hat{\gamma_1}))=2.46$. Interestingly these values are very close to those predicted by the critical propagator model with $\delta=\gamma=1/2$. The orange plane in Figure \[fig\_toth\_3d\] is the functional form $f(\eta,F|\hat{Y},\hat{\delta},\hat{\gamma_1})$ with the best fitting parameters.
![Contour plot of the residuals of the fitting function $f(\eta,F|\hat{Y},\hat{\delta},\hat{\gamma_1}) = \hat{Y} \eta^{\hat{\delta}} F^{\hat{\gamma_1}}$ (left panel) and of the fitting function $g(\eta,F|\hat{a},\hat{b},\hat{c}) = \hat{a} \log_{10}(1+\hat{b} \eta) \cdot \log_{10}(1+\hat{c} F)$ (right panel) as a function of the plane $\eta$ – $F$. Positive (negative) residuals are in blue (red). The power law clearly has macroscopic structure in the residuals; this is much less so for the logarithm.[]{data-label="fig_res"}](map_0.pdf "fig:"){width="50.00000%"} ![Contour plot of the residuals of the fitting function $f(\eta,F|\hat{Y},\hat{\delta},\hat{\gamma_1}) = \hat{Y} \eta^{\hat{\delta}} F^{\hat{\gamma_1}}$ (left panel) and of the fitting function $g(\eta,F|\hat{a},\hat{b},\hat{c}) = \hat{a} \log_{10}(1+\hat{b} \eta) \cdot \log_{10}(1+\hat{c} F)$ (right panel) as a function of the plane $\eta$ – $F$. Positive (negative) residuals are in blue (red). The power law clearly has macroscopic structure in the residuals; this is much less so for the logarithm.[]{data-label="fig_res"}](map_1.pdf "fig:"){width="50.00000%"}
In order to quantify the deviations of the surface from the best fitting power-law function $f(\eta,F|\hat{Y},\hat{\delta},\hat{\gamma_1})$, in the left panel of Figure \[fig\_res\] we show the residuals of the fit as a function of $\eta$ and $F$. Positive residuals are in blue, while negative residuals are in red. A clear non-random pattern emerges, since residuals in the center are typically positive, while those in the periphery are negative. This is an indication of the approximate description provided by Eq. \[eq\_rms3\] and therefore by the square root law: the impact surface is concave even in logarithmic scale. This suggests that an improvement of the parametrisation of the market impact surface could be obtained considering a logarithmic functional form[^8] $g(\eta, F | a,b,c) = a \log_{10}(1+b \eta) \log_{10}(1+c F)
$. By means of a non-linear regression, we obtain the best fitting parameters: $\hat a = 0.035 \pm 0.001$, $\hat b = 60 \pm 3$ and $\hat c = 61 \pm 2$. The Mean Square Error of the fit is $E_{RMS}(g(\hat{a}, \hat{b}, \hat{c})) = 1.44$. This last value is much smaller than the $E_{RMS}$ of the double power law function of Eq. \[eq\_rms3\] and, as in the previous section, indicates that logarithmic functions better describe temporary market impact. The blue surface in figure \[fig\_toth\_3d\] represents the functional form $g(\eta, F | \hat{a},\hat{b},\hat{c})$ evaluated with the best fitting parameters. In the right panel of figure \[fig\_res\] we present the residuals of the regression $g(\eta, F | \hat{a},\hat{b},\hat{c})$: the pattern present on the left panel for the power law fit is very strongly attenuated, indicating once more a better fit.
The analysis of the residuals also allows us to understand why the square root gives a relatively good collapse of the data, apparently independent from internal and external conditioning variables. In fact, consider the structure of the residuals in the left panel of Figure \[fig\_toth\_3d\]. Since $\pi=\eta F$, conditioning on $\pi$ means taking averages over diagonal strips going from the bottom left to the top right part of the plane $(\eta, F)$. This averaging includes positive and negative residuals that partly cancel out, giving the observed data collapse. Nonetheless, the disaggregation of the data done here indicates that there is indeed dependence on $F$ and $\eta$ separately.\
![Contour plot of the best fitting local exponent $\hat{\delta}(\eta, F)$ (left panel) and $\hat{\gamma_1}(\eta, F)$ measured in the plane $\eta$ – $F$ adopting the power law function. Green (brown) regions corresponds to locally fitted exponent $\hat{\delta}(\eta, F)$ and $\hat{\gamma_1}(\eta, F)$ larger (smaller) than 0.5[]{data-label="fig_proj"}](exp_delta_2d.pdf "fig:"){width="50.00000%"} ![Contour plot of the best fitting local exponent $\hat{\delta}(\eta, F)$ (left panel) and $\hat{\gamma_1}(\eta, F)$ measured in the plane $\eta$ – $F$ adopting the power law function. Green (brown) regions corresponds to locally fitted exponent $\hat{\delta}(\eta, F)$ and $\hat{\gamma_1}(\eta, F)$ larger (smaller) than 0.5[]{data-label="fig_proj"}](exp_gamma_2d_n.pdf "fig:"){width="50.00000%"}
To see deviations of the impact surface from a power law from another point of view, we measure the exponent $\delta$ describing the dependence on $\eta$ and the exponent $\gamma_1$ for the dependence on $F$ via a local non-linear fitting of the functional form $f(\eta, F| Y, \delta, \gamma_1)$ [^9]. Figure \[fig\_proj\] shows the local estimation of $\delta$ and $\gamma$ as a function of $F$ and $\eta$. The structure is very clear. The green region on the left indicates that the local exponent $\hat{\delta}(\eta,F)$ is consistently larger for small $\eta$, and consistently smaller than $0.5$ for large $\eta$. The range of variation is significant, with $\hat{\delta}$ varying from roughly $0.1$ to $0.9$.
Similarly, the behavior of the exponent $\hat{\gamma_1}(\eta,F)$ describing the power law scaling on $F$ shows clear structure, though the behavior is a bit more complicated. For small $F$ the exponent $\hat{\gamma_1}$ is close to one. For intermediate values of $F$ the exponent is close to $0.3$. The behavior for larger values of $F$ is more complicated, with high exponents for low participation rates and visa versa for high participation rates.
In conclusion, the non trivial structure appearing in the investigation of the local exponents $\delta$ and $\gamma$ suggests that the logarithmic function $g(\eta, F|a,b,c)$ better describes the temporary market impact surface. The square-root predicted values $\gamma=\delta=0.5$ only works well in the central region of the $\eta-F$ plane.\
Market impact during the execution of the metaorder
---------------------------------------------------
![ Immediate market impact (solid lines) of metaorders of different participation rate $\eta$ (increasing from the top left to the bottom right panel). Each solid line corresponds to the immediate market impact of metaorders with duration $F_i < F < F_{i+1}$. The temporary market impact is marked by a circle.[]{data-label="fig_transient"}](new_tra_1.pdf "fig:"){width="40.00000%"} ![ Immediate market impact (solid lines) of metaorders of different participation rate $\eta$ (increasing from the top left to the bottom right panel). Each solid line corresponds to the immediate market impact of metaorders with duration $F_i < F < F_{i+1}$. The temporary market impact is marked by a circle.[]{data-label="fig_transient"}](new_tra_2.pdf "fig:"){width="40.00000%"}\
![ Immediate market impact (solid lines) of metaorders of different participation rate $\eta$ (increasing from the top left to the bottom right panel). Each solid line corresponds to the immediate market impact of metaorders with duration $F_i < F < F_{i+1}$. The temporary market impact is marked by a circle.[]{data-label="fig_transient"}](new_tra_4.pdf "fig:"){width="40.00000%"} ![ Immediate market impact (solid lines) of metaorders of different participation rate $\eta$ (increasing from the top left to the bottom right panel). Each solid line corresponds to the immediate market impact of metaorders with duration $F_i < F < F_{i+1}$. The temporary market impact is marked by a circle.[]{data-label="fig_transient"}](new_tra_5.pdf "fig:"){width="40.00000%"}
In this section we focus on the immediate market impact, i.e. how the market impact builds up during the execution of a metaorder. We consider the following question: Given two metaorders with the same participation rate $\eta$ and different durations $F_1$ and $F_2$ ($F_1<F_2$), should we expect that the market impact reached at time $F_1$ is the same for the two metaorders? A priori, when the first is at its very end while the second one is still being executed, they should be indistinguishable from the point of view of market impact, since the public information available up to this time is the same for both metaorders (see also the discussion in the very recent paper of [@Iuga14]).
In Figure \[fig\_transient\] we show the result of our analysis[^10]. The four panels refer to four different bins of the participation rate $\eta$. In each panel we consider 10 bins of duration $F_i$. For each of them a line represents the average price path during the execution $\mathcal{I}(v | \Omega=\{ \eta, F\})$ for metaorders with duration $F_i < F < F_{i+1}$. The circles are the temporary impacts for metaorders of volume time duration $F_i$. The figure clearly gives a [*negative*]{} answer to our question. In fact, the price trajectories deviate from the temporary market impact described by the circles. For small participation rates this effect is more evident and price trajectories are well above the immediate impact. Notice also that in some cases the price reverts [*before*]{} the end of the metaorder[^11]. A similar behavior was very recently shown in Ref. [@Iuga14]. For larger participation rates the price trajectories become closer and closer to the circles representing the temporary impact.
The discussion presented in Section \[sec\_models\] helps us understanding this behaviour. We have seen that within the Almgren-Chriss model, the market impact trajectories deviate from the temporary market impact surface if the execution profile deviates from the VWAP trading profile. Front-loaded execution profiles, used for example by risk averse investors, generate market impact trajectories that stay above the market impact surface as shown in figure \[fig\_ac\]. However the Almgren-Chriss model is not able to reproduce some of the main features of market impact since it predicts a linear market impact. The propagator model, on the contrary, better reproduces the concavity of the market impact surface and consistently makes it possible to recover the square-root law describing the market impact curve. As we have seen in section \[sec\_prop\], the model predicts that, also in this case, front-loaded execution schemes have market impact trajectories that depart from the impact surface and reach it from above. Consistently, the presence of a decaying kernel for the impact, makes it possible to reproduce the fact that price starts to revert [*before*]{} the end of the execution. If the trading pressure is softer than the market recovering force, market impact starts reverting.
These results strongly suggest that the typical trading profile used by the brokers in our database is not the widespread VWAP trading profile but a front loaded execution scheme. This might be due to risk aversion or in order to avoid losing a profit opportunity. In fact, if the price is expected to increase, it is better to buy more at the beginning of the metaorder and less at the end. Unfortunately, these alternatives can not be tested within the information contained in the Ancerno database.\
Impact decay and permanent impact
---------------------------------
In this section we consider the temporal dependence of the price after the execution of the metaorder, i.e. how the market impact relaxes. The long term limit of the price, when all the temporary effects have dissipated, is called permanent impact. Recently there has been a debate about the value of the permanent impact and the dynamics of the price after the end of the metaorder. Under the assumption that the metaorder size distribution has a power law tail with exponent $1.5$, the model of Farmer et al. [@farmer2013efficiency] predicts a decay of the impact to a permanent value of roughly $2/3$ of the peak impact. Note however, that this assumption does [*not*]{} appear to be met in our data – as seen in Figure \[fig\_stat\], $\pi$ clearly does not have a power law distribution. This has serious consequences, since in this case their theory does not predict a square root market impact function. See Section \[fundamentalModels\] for a more complete discussion.
Here we consider the market impact trajectory $\mathcal{I}(v|\Omega = \{ \eta,F\})$ after the end of the execution of the metaorder, i.e. for $v>F$. In order to compare metaorders with different durations $F$, we rescale time as $z=v/F$. In this way at metaorder completion it is $z=1$ independently from the metaorder duration $F$. We also rescale the market impact trajectory by dividing by the market impact at the end of the execution of the metaorders $\mathcal{I}(v=F|\Omega = \{ \eta,F\})$, i.e. $$\mathcal{I}_{ren}(z|\Omega = \{ \eta,F\}) := \frac{\mathcal{I}(z|\Omega = \{ \eta,F\})}{\mathcal{I}(z=1|\Omega = \{ \eta,F\})}.$$ The decay of the impact is presumably less dependent on the execution scheme than the immediate impact, and therefore its study could be used, at least in principle, to investigate how well the propagator model describes the price dynamics. As we will see, this is strictly true only if we can neglect the order flow of other metaorders (for example if the participation rate of the conditioning metaorder is large enough).
![Decay of temporary market impact after the execution of the meteorder. We follow the normalised market impact path $\mathcal{I}_{ren}(z)$ as a function of the rescaled variable $z=v/F$, without conditioning on any variable. The market impact path of each metaorder is followed also in the following day. The red horizontal line corresponds to 2/3, as predicted by the model of Farmer et al. [@farmer2013efficiency].[]{data-label="fig_permanent_all"}](imp_all.pdf){width="60.00000%"}
We first consider all the metaorders together and we compute the average rescaled path followed by the price after the end of the metaorder. The result is shown in Figure \[fig\_permanent\_all\]. We observe that the price decays toward a value which is remarkably close to (though slightly higher than) $2/3$ of the peak impact. This is agreement with the results obtained for example in [@moro2009market; @bershova2013non].
![Decay of temporary market impact after the execution of a metaorder. We follow the normalised market impact path $\mathcal{I}_{ren}(z|\Omega = \{ \eta,F\})$ as a function of the rescaled variable $z=v/F$. Within each panel the solid lines correspond to the average market impact trajectory for metaorders with different durations $F$; the four panels correspond to different participation rates $\eta$. We consider the price dynamics up to the end of day when the metaorder was placed. The black line corresponds to the prediction of the propagator model with $\delta = 0.5$. Overnight returns and the price path of subsequent days are not considered in our analysis. []{data-label="fig_permanent"}](dur_per_intra_2.pdf "fig:"){width="38.00000%"} ![Decay of temporary market impact after the execution of a metaorder. We follow the normalised market impact path $\mathcal{I}_{ren}(z|\Omega = \{ \eta,F\})$ as a function of the rescaled variable $z=v/F$. Within each panel the solid lines correspond to the average market impact trajectory for metaorders with different durations $F$; the four panels correspond to different participation rates $\eta$. We consider the price dynamics up to the end of day when the metaorder was placed. The black line corresponds to the prediction of the propagator model with $\delta = 0.5$. Overnight returns and the price path of subsequent days are not considered in our analysis. []{data-label="fig_permanent"}](dur_per_intra_3.pdf "fig:"){width="38.00000%"}\
![Decay of temporary market impact after the execution of a metaorder. We follow the normalised market impact path $\mathcal{I}_{ren}(z|\Omega = \{ \eta,F\})$ as a function of the rescaled variable $z=v/F$. Within each panel the solid lines correspond to the average market impact trajectory for metaorders with different durations $F$; the four panels correspond to different participation rates $\eta$. We consider the price dynamics up to the end of day when the metaorder was placed. The black line corresponds to the prediction of the propagator model with $\delta = 0.5$. Overnight returns and the price path of subsequent days are not considered in our analysis. []{data-label="fig_permanent"}](dur_per_intra_4.pdf "fig:"){width="38.00000%"} ![Decay of temporary market impact after the execution of a metaorder. We follow the normalised market impact path $\mathcal{I}_{ren}(z|\Omega = \{ \eta,F\})$ as a function of the rescaled variable $z=v/F$. Within each panel the solid lines correspond to the average market impact trajectory for metaorders with different durations $F$; the four panels correspond to different participation rates $\eta$. We consider the price dynamics up to the end of day when the metaorder was placed. The black line corresponds to the prediction of the propagator model with $\delta = 0.5$. Overnight returns and the price path of subsequent days are not considered in our analysis. []{data-label="fig_permanent"}](dur_per_intra_5.pdf "fig:"){width="38.00000%"}
The large size of our metaorder database allows us to perform an analysis of the price decay conditioning on the duration and participation rate. Figure \[fig\_permanent\] shows the results. The four panels refer to increasing values of the participation rate. In each panel the market impact path of metaorders with several durations is presented. We follow the relaxation of the market impact trajectories up to three times the duration of the metaorder, but we avoid introducing overnight returns and following the price on subsequent days. In each panel we also show the prediction of the propagator model for the market impact trajectory when $\gamma = 0.5$ (black line).
The figure shows that the price decay and its long term limit depend on $\eta$ and $F$. For small participation rates (top panels) the average permanent impact (across durations) is close to $2/3$ peak impact. However this is also the regime where we observe the strongest dependence of the permanent impact on $F$. Longer metaorders relax more slowly than shorter metaorders, and at the end of the period examined remain at higher price levels. This effect is bigger for smaller participation rates[^12]. On the contrary, for the largest participation rates the renormalised market impact paths of metaorders are all very similar. The market impact relaxes toward zero and we do not observe any flattening of the curve in the considered time window. Quite interestingly, in this regime the market impact decay is well described by the prediction of the propagator model with $\gamma=0.5$, while for small and intermediate participation rates the price is systematically higher than the value predicted by the propagator model.
![Decay of temporary market impact after the execution of the metaorder. We follow the renormalised market impact path $\mathcal{I}_{ren}(z|\Omega = \{ \eta,F\})$ as a function of the rescaled variable $z=v/F$. Each solid line corresponds to the average market impact trajectory computed on metaorders characterised of low (top row) and high (bottom row) participation rate $\eta$ and several durations $F$ (see legend). We consider metaorders with long durations $0.224<F<1$ and we follow the price path also in the following days (in contrast with the analysis of figure \[fig\_transient\]). []{data-label="fig_cnt_2"}](dur_per_cnt_2_2.pdf "fig:"){width="38.00000%"} ![Decay of temporary market impact after the execution of the metaorder. We follow the renormalised market impact path $\mathcal{I}_{ren}(z|\Omega = \{ \eta,F\})$ as a function of the rescaled variable $z=v/F$. Each solid line corresponds to the average market impact trajectory computed on metaorders characterised of low (top row) and high (bottom row) participation rate $\eta$ and several durations $F$ (see legend). We consider metaorders with long durations $0.224<F<1$ and we follow the price path also in the following days (in contrast with the analysis of figure \[fig\_transient\]). []{data-label="fig_cnt_2"}](dur_per_cnt_2_3.pdf "fig:"){width="38.00000%"}\
![Decay of temporary market impact after the execution of the metaorder. We follow the renormalised market impact path $\mathcal{I}_{ren}(z|\Omega = \{ \eta,F\})$ as a function of the rescaled variable $z=v/F$. Each solid line corresponds to the average market impact trajectory computed on metaorders characterised of low (top row) and high (bottom row) participation rate $\eta$ and several durations $F$ (see legend). We consider metaorders with long durations $0.224<F<1$ and we follow the price path also in the following days (in contrast with the analysis of figure \[fig\_transient\]). []{data-label="fig_cnt_2"}](dur_per_cnt_2_4.pdf "fig:"){width="38.00000%"} ![Decay of temporary market impact after the execution of the metaorder. We follow the renormalised market impact path $\mathcal{I}_{ren}(z|\Omega = \{ \eta,F\})$ as a function of the rescaled variable $z=v/F$. Each solid line corresponds to the average market impact trajectory computed on metaorders characterised of low (top row) and high (bottom row) participation rate $\eta$ and several durations $F$ (see legend). We consider metaorders with long durations $0.224<F<1$ and we follow the price path also in the following days (in contrast with the analysis of figure \[fig\_transient\]). []{data-label="fig_cnt_2"}](dur_per_cnt_2_5.pdf "fig:"){width="38.00000%"}
We have also performed the previous analysis following the market impact decay on subsequent days. This allows us to include metaorders with longer duration in the analysis. Considering metaorders with the same duration as before, we observe that the global picture changes slightly only for metaorders with very large participation rate. In the other cases it is approximatively unchanged (data not shown). Considering metaorders with even longer duration, see Figure \[fig\_cnt\_2\], we observe the appearance of clear plateaux with height 0.8 - 1 times the peak impact. It is worth also noticing that here, but also in the top left panel of Figure \[fig\_permanent\], the reversion of the price before the end of the metaorder is much more clearly visible, as explained in footnote \[foot\]. Note that the market impact trajectories of metaorders in this analysis often contain the overnight return (contrary to the previous analysis). As seen by [@brokmann2014], we observe that the price decay essentially stops when the trading day ends. However the presence of overnight returns increases significantly the already large noise in the determination of permanent market impact.
### The role of metaorder autocorrelations
The picture emerging from the previous analysis can be partly clarified by taking into account the autocorrelation of the sign of metaorders. Positive autocorrelations in the signs of metaorders will make the market impact of a single metaorder relaxation artificially high, as it becomes impossible to isolate metaorders from each other. Moreover this effect is stronger for longer metaorders, since the probability of overlapping with other metaorders is larger. It is also larger for lower participation rates, since the market impact is easily overwhelmed by that of metaorders with larger participation rates. On the contrary, we expect that the effect is milder for shorter metaorders, because of the lower probability of overlap, and larger participation rates, because the effect of metaorders with lower participation rates on price becomes negligible.
duration (mins) number \# overlaps same sign opposite sign
----------------- --------- ------------- ----------- ---------------
0 - 10 368,484 1.7 0.548 0.452
10 - 25 117,756 3.0 0.553 0.447
25 - 50 71,031 4.6 0.553 0.447
50 - 100 52,931 6.6 0.547 0.453
100 - 200 43,884 8.0 0.546 0.454
200 - 390 49,411 9.3 0.543 0.457
0 - 390 703,497 3.54 0.548 0.452
: [*An analysis of overlap of metaorders.*]{} We consider the metaorders with participation rate $\eta > 0.005$ traded on the 100 most populated stocks from January 2007 to December 2009. This set has 703,497 metaorders. We consider nonintersecting bins according to the duration of the metaorders (first column) and their relative number (second column). For each metaorder we consider the time interval from the beginning up to $3$ times its duration. We count the metaorders in the whole set overlapping with the selected time interval. For each subset we report the average number of overlapping metaorders ( \# overlaps, third column). As expected, the number of overlaps increases with the duration of the metaorder. We then measure the fraction of the overlapping metaorders which have the same or opposite sign as the selected metaorder (fourth and fifth columns). We observe that a constant average fraction ($\sim 55 \%$) of the overlapping metaorders have the same sign, independent of the duration. This finding quantifies the autocorrelation of the sign in the time series of the metaorders. []{data-label="tab_anom"}
The overlap of the metaorders present in our database is summarized in Table \[tab\_anom\]. We observe that, considering the time interval from the beginning of the metaorder up to three times its duration after the end of the execution, on average, a given metaorder overlaps on average with 3.5 other metaorders. As expected, the average number of overlapping metaorders is larger for longer metaorders (around 2 for the shortest ones and around 10 for the longest ones). On average, $55 \%$ of these metoarders have the same sign. This implies that, on average, a metaorder is surrounded by more metaorders of the same sign than of the opposite signs, and this effect enhances the measured impact.
Very recently Ref. [@brokmann2014] considers trades from the same fund and traded following a signal and show that they present a strong autocorrelation in time. The authors suggest that a positive autocorrelation of sign of the metaorders can keep the impact artificially high. They suggested a method to deconvolve their own trades to remove both their own impact and the information, finding zero permanent impact on the time scale of 15 days. It is important to highlight that, although consistent with [@brokmann2014], our measure of the autocorrelation of metaorders is obtained by using an extensive database covering the trading activity of many different investors, rather than all the metaorders of the same fund. Thus our analysis points out a herding among funds in their trading of metaorders, rather than metaorders by the same institution in the attempt to exploit medium term signals as in Ref. [@brokmann2014].
The positive autocorrelation of metaorder signs qualitatively explains the findings on price decay. Market impact trajectories of metaorders with very large participation rate are negligibly perturbed by the other metaorders and their trajectories are roughly independent of duration (bottom right panel of Figure \[fig\_permanent\]). Moreover, the market impact trajectory is quite well described by the propagator model. On the other hand, we have seen that the market impact trajectory of metaorders with lower participation rate and longer durations deviates from each other and from the prediction of the propagator model (top panels of Figure \[fig\_permanent\]). We speculate that, in this case, the market impact trajectories are kept artificially high by the effect of other metaorders with the same sign and non-negligible participation rate. The fact that this effect is stronger for low participation rates is consistent with our explanation. Although interesting and worthy of investigation, a more in depth analysis of this aspect is beyond the scope of this paper.
Implications for fundamental models \[fundamentalModels\]
=========================================================
One of the motivations for this paper is to test fundamental theories for market impact. In this section we review these theories and discuss their possible implications in relation to the results presented here. We also offer some caveats, discussing possible effects that might distort our results.
The latent order book approach of Toth et al.
---------------------------------------------
Toth et al. [@toth2011anomalous] present a theory for market impact based on the concept of a latent order book. The key idea is that the true order book does not reflect the actual supply and demand that are present in the market, due to the fact that participants do not reveal their true intensions. They show that for prices to be diffusive, i.e. for the variance to grow linearly with time, it is necessary for the latent order book to have a linear profile around the current price, which implies a square root impact function. This is supported by simulations of a simple agent-based model. They make no prediction about how prices should relax after execution is completed, though in subsequent empirical work this group suggests that once the predictive advantage of a trading strategy has been removed the price relaxes slowly to zero [@brokmann2014].
That fact that we observe a logarithm for temporary impact appears to contradict the theory of Toth et al. While we do observe that the square root is an approximation over part of the range, we see substantial deviations. In addition the fact that we observe an impact surface with logarithmic dependence on $\eta$ and $F$ separately is not consistent with their theory. However see the caveats given below, as well as the discussion of the implications for the latent order book in Section \[virtualOrderBook\].
The fair pricing approach of Farmer et al.
------------------------------------------
Farmer et al [@farmer2013efficiency] derive a fair pricing principle that, when combined with the martingale property of prices, predicts that the average execution price should equal the final price when the metaorder has completed and prices have been allowed to relax. This is done by deriving a Nash equilibrium between informed traders and liquidity providers, in a setup that requires much stronger assumptions than the theory of Toth et al. above [@toth2011anomalous]. (This model can be viewed as an extension of Kyle’s original model, but with more realistic assumptions. Farmer et al. assume batch executions and that the beginning and end of metaorders is known by market participants. The functional form of market impact depends on the distribution of metaorder sizes. Under the assumption that the cumulative distribution of metaorder sizes is a power law with exponent $-3/2$ they predict a square root impact and that after execution prices should revert to 2/3 of their peak value.
The analogous quantity to metaorder size studied here is $\pi = Q/V$. From Figure \[fig\_stat\] it is clear that this is not distributed according to a power law[^13]. As a result, it is not clear what this model implies. Further work is needed to fit a functional form to the distribution of $\pi$ and work out the predictions for market impact under the fair pricing principle, but this is beyond the scope of this paper.
Other theories
--------------
Several other theories deserve mention. The theory of Gabaix et al. [@gabaix06] also predicts a square root for market impact. However, this theory requires a very strong assumption, namely that the utility function of investors has absolute risk aversion, i.e. they assume that investors have a utility function of the form $\mu - \sigma^\delta$, where $\mu$ is the mean of returns and $\sigma$ is the standard deviation and $\delta = 1$. If $\delta = 2$, for example, then the impact becomes linear.
In view of our results a theory that is particularly worthy of mention is the PhD thesis of Austin Gerig[^14] [@gerigthesis]. This model was an historical precursor to the theory of Farmer et al. discussed above. As they did, Gerig assumed the prices form a martingale and that the starting and stopping times of metaorders are observable, but made a different auxiliary assumption. This theory deserves special mention because it is the only theory that we are aware of that predicts a logarithmic dependence for market impact.
A few caveats
-------------
Our data has limitations and we should issue some caveats. In our data it is not possible to observe the strategic intentions of the agents originating the metaorders. There may be preferential biases that are invisible to us. In particular, suppose that execution of buy metaorders is sometimes cancelled before completion if the price rises too much (or if selling if the price falls too much). This will systematically bias the sample to make impact appear more concave. Even if the true impact were a square root, this could make the measured impact more concave. Nonetheless, such effects would have to be substantial, and it seems a bit surprising that they would result in such good agreement with a logarithmic functional form.
Another important caveat that should be mentioned is the normalization by daily volume. We make the implicit assumption, which has been almost universally made in prior work, that liquidity is proportional to daily volume. This provides a (time varying) point of reference for market impact. This is an assumption, and is not part of the predictions of any of the fundamental theories discussed above. A failure of the core assumption that daily volume is the correct way to measure liquidity could easily distort the shape of the impact function. The only exception to the above is Kyle’s original 1985 model and the new Kyle-Obizhaeva market invariance model, which predict a more complicated liquidity scaling [@Kyle14]. We have not tested any such alternatives.
Finally we should remind the reader that we truncate all metaorders that are longer than one day in duration (so that a metaorder that persists for $n$ days is treated as $n$ separate metaorders). However, our inspection of the data suggests that this is rare – see Figure \[fig\_aapl\].
Conclusions {#sec:conclusions}
===========
We have presented the most extensive empirical analysis of the market impact of the execution of large trades performed so far, at least in terms of the number of metaorders and heterogeneity of their originators that have been analyzed. The large dataset allows us to reduce the statistical uncertainty in the analysis and thereby make stronger inferences about the functional form of market impact. We have also linked together the raw data on metaorders with minute-by-minute data on prices, so that we can study time dependent effects, such as the immediate impact as a metaorder is executed and the reversion after it is completed. Our results extend but also contrast with what is commonly believed about market impact. Some of our main conclusions are as follows:
- Market impact conditional on the daily fraction $\pi$ (the ratio between the volume and the average daily volume) is remarkably well described by a logarithmic function over more than four orders of magnitude. In contrast, the square root impact law, which is widely used in academia and industry, approximates market impact only for a couple of orders of magnitude in $\pi$. Thus the form of market impact is strongly concave, even more so than suggested by the square root law.
- The [*market impact surface*]{} captures an inherently bivariate dependence of impact on participation rate and duration. As before, this bivariate dependence is much better represented by a logarithm than by a power law. Furthermore, the good “collapse” seen by conditioning on $\pi$ alone is substantially due to a compensation effect between residuals. That is, we show that impact depends on $F$ and $\eta$ separately; however, when one aggregates by conditioning only on $\pi$, the dependences tend to cancel each other.
- During execution the price trajectory deviates from the temporary market impact and sometimes the price starts reverting well before the end of the execution. This strongly suggests that market impact is decaying even as the metaorder is being executed. We believe the lack of correspondence is due to front-loaded execution. (This also reflects a limitation of our analysis; we do not have detailed timestamps for the execution of the metaorder, and so we are forced to assume uniform execution).
- The propagator model is only a good description of metaorder impact for metaorders with high participation rate. This is likely due, at least in part, to the overlap (herding) between metaorders of the same sign, which for moderate to low participation rate modifies the price dynamics considerably.
- Prices clearly show a strong tendency toward reversion after metaorder execution ends. This behavior strongly depends on both participation rate and duration. For high participation rates orders of all duration relax in essentially the same way, consistent with a propagator in which impact relaxes to zero as the square root of time. In contrast for low participation rates orders of different duration behavior quite differently, with orders of longer duration relaxing more slowly than those of short duration.
Our results present several modeling challenges, since none of the available models are fully capable of explaining our results. Indeed, the only model that we are aware of that predicts logarithmic behavior is one due to the work of Farmer, Gerig and Lillo, which is reported in Austin Gerig’s thesis [@gerigthesis]. These results are particularly surprising when compared to the work of [@toth2011anomalous]. In that work they show that square root behavior of impact is a necessary condition for diffusive behavior of prices, and that deviations from this should therefore result in arbitrage. This raises the question of how the observed logarithmic impact can avoid this problem. Please note however the caveats given in the previous section. At the very least our work suggests the need for more large studies of market impact. Unless there are biases in our results as discussed above, our work suggests that current fundamental theories of market impact have serious problems and that models of market impact require further development.
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank E. Bacry, J.-P. Bouchaud, J. Gatheral, P. Kyle, and H. Waelbroeck for useful and inspiring discussions. EZ, MT, and FL wish to thank Enrico Melchioni for the constant encouragement. The opinions expressed here are solely those of the authors and do not represent in any way those of their employers.
[^1]: Also called price impact
[^2]: This quantity is sometimes also called peak impact. Temporary impact should not be confused with the temporary component of impact, used for example in the Almgren-Chriss model [@almgren2001optimal].
[^3]: ANcerno Ltd. (formerly the Abel Noser Corporation) is a widely recognised consulting firm that works with institutional investors to monitor their equity trading costs. Its clients include pension plan sponsors such as the California Public Employees’ Retirement System (CalPERS), the Commonwealth of Virginia, and the YMCA retirement fund, as well as money managers such as MFS (Massachusetts Financial Services), Putman Investments, Lazard Asset Management, and Vanguard. Previous academic studies that use Ancerno data include [@puckett2008short; @goldstein2009brokerage; @chemmanur2009role; @jame2010organizational; @goldstein2011purchasing; @puckett2011interim; @busse2012buy]. In particular, the authors of [@puckett2008short] give evidence regarding the existence of weekly institutional herding, often resulting in intense buying and selling episodes which may affect the efficiency of security prices. The authors investigate the contemporaneous and subsequent abnormal returns of securities that institutional herds sell or buy. They bring evidence that stocks that herds buy outperform the stocks that herds sell prior to and during the week of portfolio formation. Then, intense sell herds are followed by return reversals while the contemporaneous returns associated with intense buy herds are permanent.
[^4]: Here we do not distinguish volume and physical time.
[^5]: The Weighted Root Mean Square Error of a function $g(x|a,b)$ with parameters $a$ and $b$ to reproduce the observations $\{ y_i \}_{1 \le i \le N}$ of the explanatory variables $\{ x_i\}_{1 \le i \le N}$ is defined by $$E_{RMS}(g(a, b)) = \sqrt{ \frac{1}{N}\sum_i^N \left( \frac{y_i-g(x_i| a, b) }{SE(y_i)} \right)^2 }$$ where $SE(y_i)$ is the standard error associated with the observation $y_i$. The smaller $E_{RMS}$, the better the fit.
[^6]: The Akaike Information Criterion leads to the same conclusions on the relative performance of the different functional forms.
[^7]: In the present analysis we subset for $\eta>10^{-3}$, in order to avoid a strongly noisy part of the domain. We subset also for $F<0.5$, since the remaining part the plot shows the most discrepant behaviour compared to the square root.
[^8]: Note that the parameters $a$ are $b$ are not necessarily the same as those used in the parametrisation of the impact curve. The relation between the parameters of the curve and of the surface depends on the joint distribution of $F$ and $\eta$.
[^9]: More in detail, we divide the dataset in $n_1=10$ ($n_2=10$) evenly-populated subsets according to $\eta$ ($F$). Each measure in the dataset is labelled by an integer $i_1\in \{1, \dots, n_1\}$ ($i_2 \in \{1, \dots, n_2\}$) according to the subset the measure belongs to. Each measure in the dataset is then labelled by a couple of integers $b=(i_1,i_2)$ identifying a bin. We measure the sample mean of the temporary market impact $I_b^r$ of the metaorders belonging to the same bin $b$ and the average of the relative sample mean of $\eta^r_b$ and $F^r_b$. We define a square identified by the center $c=(c_1,c_2)$, where $c_1 \in \{ 1, \dots, n_1\}$, by considering the bins in the neighbourhood of the center $S_1=\{ c_1-2,c_1-1, c_1, c_1+1, c_1+2\}$ and $S_2=\{ c_2-2,c_2-1, c_2, c_2+1, c_2+2\}$. The square is defined by $S(c_1,c_2)=S_1 \times S_2$. We select the bins belonging to a square $S(c_1,c_2)$ and we fit the realised temporary market impact $I^r_b$ as a function of the relative realised $\eta^r_b$ and $F^r_b$. We consider a fitting function of the form $f(\eta,F) = C \eta^{\delta}F^{1-\gamma}$. We measure the best fitting parameters $\hat{\delta}(c)$ and $\hat{\gamma}(c)$ as a function of the center of the square $c=(c_1,c_2)$.
[^10]: A methodological comment is in order. The Ancerno database does not provide the number, time, volume, or price of each individual transaction (or child-order) through which the metaorder has been executed. For this reason we follow the price dynamics by using the public price information with one-minute time resolution.
[^11]: Note that, with this method, we are able to investigate in full detail the the market impact path in the early stage of the execution, because for orders with duration $F_i < F < F_{i+1}$ we follow the price up to $v=F_i$, missing the very last part. As a consequence, in figure \[fig\_transient\] the magnitude of the reversion of the market impact path during the execution is underestimated. On the contrary, this feature becomes more evident in the analysis performed in the following section, see Figure \[fig\_permanent\] and \[fig\_cnt\_2\]. In that case, we follow the market impact path with great attention to the late stage of the execution. We will observe that long metaorders with small participation rate present a strong reversion, as clearly visible in Figure \[fig\_cnt\_2\].\[foot\]
[^12]: This observation that metaorders with low participation rate revert more slowly is consistent with the notion that reversion depends on detection by others of the presence or absence of the order. The beginning or end of a low participation rate order is more difficult to detect, and should require more time for a given level of certainty, giving a more sluggish reaction to completion of the order. Of course there may also be other explanations.
[^13]: One complication is that we only study metaorders that are executed within the course of a single day, which truncates the distribution. Nonetheless, based on Figure \[fig\_stat\] it seems unlikely that removing this truncation would restore a power law.
[^14]: This was joint work with J.D. Farmer and F. Lillo.
|
---
abstract: 'In high intensity focused ultrasound (HIFU) systems using non-ionizing methods in cancer treatment, if the device is applied to the body externally, the HIFU beam can damage nearby healthy tissues and burn skin due to lack of knowledge about the viscoelastic properties of patient tissue and failure to consider the physical properties of tissue in treatment planning. Addressing this problem by using various methods, such as MRI or ultrasound, elastography can effectively measure visco-elastic properties of tissue and fits within the pattern of stimulation and total treatment planning. In this paper, in a linear path of HIFU propagation, and by considering the smallest part of the path, including voxel with three mechanical elements of mass, spring and damper, which represents the properties of viscoelasticity of tissue, by creating waves of HIFU in the wire environment of MATLAB mechanics and stimulating these elements, pressure and heat transfer due to stimulation in the hypothetical voxel was obtained. Through the repeatability of these three-dimensional elements, tissue is created. The measurement was performed on three layers. The values of these elements for liver tissue and kidney of sheep in a practical example and outside the body are measured, and pressure and heat for three layers of liver and kidney tissue of an organism were obtained by applying ultrasound signals with a designed model. This action is repeated in three different directions, and the results are then compared with simulation software for ultrasound, as a reference to U.S. Food and Drug Administration (FDA) measures for HIFU, as well as comparisons of results with an operational method for an HIFU cell. The temperature of modeling on the liver for the practical mode in the first and third layers is 17, 16, and 24 percent, and for the software simulator of the HIFU, the measures are 12.9, 17.9, and 15 percent relative absolute changes. The results for kidney tissue for the layers mentioned is 6, 5.7, and 14.5 percent for the simulator of the HIFU, and 4, 5, and 14 percent compared to the practical mode, demonstrating relative absolute changes. The percentage of absolute changes in pressure for liver and kidney tissue in conducted simulation for the simulator of HIFU also gained 9 percent. It was also observed that treatment planning using the properties of visco-elasticity are especially effective based upon experiments conducted as part of this study.'
author:
- |
Saeed Reza Hajian $^{1}$, Ali Abbaspour Tehrani Fard $^{2}$, Majid Pouladian $^{3,*}$\
and Gholam Reza Hemmasi $^{4}$ [^1] [^2] [^3] [^4] [^5] [^6]
title: '**Modeling pressure distribution and heat in the body tissue and extract the relationship between them in order to improve treatment planning in HIFU** '
---
Introduction
============
Cancer treatment methods can be categorized in two ways: noninvasive and invasive. HIFU is a method of treatment that is primarily used for the removal of solid tumors by creating heat. This method is completely non-invasive. As a radiosurgery procedure it does not damage tissue, and treatment can be performed in one or more sessions. This treatment is done in two ways. In the first method, a probe is inserted through the transrectal area into the desired tissue. The second method is extracorporeal. HIFU waves are radiated from outside into the tissue of the tumor. This device follows standards approved by FDA and CE which stands for European Conformity. Today in different countries, HIFU centers are being established with this system, which is widely used in the treatment of prostate, brain and breast cancer [@kremkau1979cancer]. As seen in Figure \[fig1\], the use of this way compared to general hyperthermia has the advantages that including them can refer to less painful time and high heat in the tumor necrosis [@randal2002high].
![Difference in heat in normal hyperthermia and HIFU[]{data-label="fig1"}](fig1.jpg){width="\linewidth"}
\
Today, this method is used as an extracorporeal treatment for cancers of the liver, pancreas, prostate and bone. As seen in Figure 2, the HIFU method, when using the extracorporeal method, can be effective in the treatment of cancers of the soft tissue by focusing on sound waves.
![Block diagram using HIFU in destroying the interior of the tumor[]{data-label="fig2"}](fig2.jpg){width="\linewidth"}
HIFU (High Intensity Focused ultrasound) has been studied by numerous researchers as a non-invasive treatment for many clinical problems that require surgery. HIFU is a modern medical method that is bloodless, and portable, and it requires no general anesthesia. This method has limitations, such as the potential to enlarge the size of the lesions created, the need for a cooling system, the inability to treatment by air or bone, problems associated with monitoring of treatment during the healing process, and the inability to plan before treatment [@jenne1997ct] [@amin2005hifu] [@suri2001two]. Including HIFU applications can be referred for the eradication of cancerous tumors in the prostate, uterus, breast, liver and intestines [@cheng2015contrast] [@zhou2014high]. Eradication of a brain tumor is done through the skull as an aid to surgery or minimally invasive surgery usually by means of HIFU applications [@sarraf2016deepad] [@sarraf2016classification] [@sarraf2014brain] Recently, many researchers have shown interesting for new applications of HIFU which include cardiac disease such as coronary artery stenosis, cardiomyopathy, hypertrophy and other disorders of the atrial septum, open blood-brain barrier, and fetal tissue [@shui2015high] Clement McDonald and his colleagues, by analyzing thermal imaging data based on two magnetic images, predicted the effect of HIFU in the treatment of a number of fibroids in different areas that are visible with imaging after treatment [@imankulov2015feasibility] [@saverino2016associative] [@sarraf2016robust].
Heating Biological Model
========================
A heating biological model is defined by equation \[eq1\], through which tissue can be calculated during treatment. In this model, the geometry of tissue and the heat created by absorption of HIFU should be fully studied; Full details of the heat capacity of the tissue should be revealed. $$\label{eq1}
Q=\frac{-K.A. (\Delta T). (\Delta t)}{(\Delta L)}$$\
The flow of heat passing through tissue obtained using the above equation is as follows: $$\label{eq2}
f=\frac{-K.\Delta T}{\Delta L}$$ In these equations, $f$ is heat flow, $k$ is weakened heat conduction, $\Delta T$ is the greatest changes in temperature; $\Delta L$ is the length of tissue, and $A$ is the tissue level and $\Delta t$ exposure time. In the case of small lengths, the equation wi ll be as follows: $$\label{eq3}
f=-k\Delta T$$ Heating conductivity inside tissue is obtained through biological heating equation \[eq4\]. In this equation, $p$ is tissue density, $c$ is specific heat coefficient, $q_{s}$ is absorption of HIFU energy, qp is the rate of heat flow through the blood, and $q_{m}$ is indicative of metabolic activity. $$\label{eq4}
\rho c \frac{\partial T}{\partial t} = \bigtriangledown (k \bigtriangledown T)+ q_{s} + q_{p} + q_{m}$$ The best way to investigate heating effects is to calculate the ultrasound absorption rate, blood flow and metabolic activity. Factors such as blood density, specific heat, and temperature of blood affect blood flow. In Figure \[fig3\], a general view of the heating of a capillary network and its parameters is displayed. The heating energy of a capillary network using equation 5 is obtained.
![Thermal energy input and output balance of tissue.[]{data-label="fig3"}](fig3.jpg){width="\linewidth"}
$$\label{eq5}
q_{p} = -w_{b} \rho_{b} c_{b} \delta_{t}(T_{f}-T_{i})$$
\
In this regard, $w_{b}$ is blood flow, $\rho_{b}$ is blood density, $c_{b}$ is specific heat of blood, $\delta_{t}$ is specific heat of tissue, $T_{f}$ is final heat, and $T_{i}$ is initial heat. By placement of the relationship in equation \[eq4\], we will arrive at:\
$$\label{eq6}
\rho c. \frac{\partial T}{\partial t}= \bigtriangledown (k \bigtriangledown T) + q_{s} - w_{b} \rho_{b} c_{b} \delta_{t}(T_{f}-T_{i})+q_{m}$$ In the above equation, metabolic activity is negligible in front of the heat generated, and we will arrive at:\
$$\label{eq7}
\rho c. \frac{\partial T}{\partial t}= \bigtriangledown (k \bigtriangledown T) + q_{s} - w_{b} \rho_{b} c_{b} \delta_{t}(T_{f}-T_{i})$$\
According to Equation \[eq7\], the amount HIFU absorbed energy due to the model designed for tissue can be modified. The energy is associated with the similarity of tissue, density, and other biological factors. Usually when treatment is done, treatment points should be shown separately in separate images, and the operator should be specify separately that the same tissue is shown in Figure \[fig4\]. Bright points represent treatment carried out by ultrasound.
![Treatment done in vivo tissue with HIFU and treatment outcomes.[]{data-label="fig4"}](fig4.jpg){width="\linewidth"}
In the capsule area, the temperature in the central and parietal regions is different than that shown in Figure \[fig5\] [@habash2007thermal].
![Treatment done in vivo tissue with HIFU and treatment outcomes.[]{data-label="fig5"}](fig5.jpg){width="\linewidth"}
As previously noted, the removal of unhealthy tissue by two phenomena of increasing the temperature and the cavitation in tissues is done. In the meantime, in some cases, thermal images are taken that have special importance. Below, thermal images before and after the treatment are shown. At the time of treatment, usually images will be thermographic (Figure \[fig6\]) so that we can note the difference before and after treatment.
![Treatment done in vivo tissue with HIFU and treatment outcomes.[]{data-label="fig6"}](fig6.jpg){width="\linewidth"}
Software Modeling
-----------------
In this model, each layer of tissue using three elements of spring, damper and mass, as shown in Figure \[fig7\], is shown. In order to define the direction of modeling used to measure elasticity, viscosity and mass of tissue of elements, spring and damper are placed in parallel with each other, and mass is shown in a series with these two in circuit [@hajian2015new].
![Selected model in simulation[]{data-label="fig7"}](fig7.jpg){width="\linewidth"}
In this model, a three-layer tissue is intended. Therefore, these three elements, as shown in Figure \[fig8\], are repeated, and the last layer is connected to the ground.
![Model for three layers[]{data-label="fig8"}](fig8.jpg){width="\linewidth"}
### Input Signal
At the beginning, an ultrasound wave enters the tissue, where we define an input signal which consists of three pulses with a maximum range of 10 volts (Figure 9).
![Square wave designed to simulate[]{data-label="fig9"}](fig9.jpg){width="\linewidth"}
Simulation results
------------------
In this paper, a linear method is used for simulation, and the mass, spring and damper model is expressed. The HIFU simulator is used to measure HIFU, a phantom, as Figure 10 is considered. By putting the values of the mass, spring and damper in the simulation, pressure of HIFU on a tissue can be achieved, including a layer of the kidney.
![Figure of phantom intended for related software.[]{data-label="fig10"}](fig10.jpg){width="\linewidth"}
Then, using liver values, the appropriate software estimates for heat and pressure are applied in different layers. In the case of radiation from three directions and heat in the target point to liver tissue on the tumor (target), a temperature as high as 81 C for the modeling will be obtained. In this case, paths of radiations are considered, as shown in Figure \[fig11\]. Liver and kidney comparative graphs of temperature as a result of HIFU radiation in the linear part are shown in three layers as Table \[table1\]. Between the practical and simulator test, little difference was observed.
![Test and design done to extract heat and pressure to liver tissue[]{data-label="fig11"}](fig11.jpg){width="\linewidth"}
\
In Figure \[fig11\], through a 50 percent reduction in the amount of viscoelasticity using the same path, the temperature of the first through third layer is obtained (Table 2).
\
In this case, in the target point a 76 $^{\circ}$ C temperature, and for the second path a temperature of the first layer of 41 $^{\circ}$ C, and for the second and third paths 30 and 22 $^{\circ}$ C, respectively, are achieved. This issue proves the effects of viscoelasticity on the temperature and the pressure, and the relative change in the target temperature is 6 $^{\circ}$ C and the first layer is 14%. Performing this issue for pressure causes it to create relative changes of 9 percent as a result of changing viscoelasticity of a path. Measurement in this simulation is done by removing a living tissue from the body. If it is in any path, the value of the physical parameter is measured, and biophysical tissue damage in the same path is prevented as a practical matter. For this purpose, the volume of contour between the HIFU probe to the target in the included K layer and each layer is composed of a voxel matrix, and viscoelastic elements should be extracted in a non-invasive manner (imaging) and placed in the designed model, which is a different method. Additionally, if we want the temperature at the target point to remain at a certain level, mechanical properties can be changed. Heat at a given point or the excitation signal of the HIFU probe is modified, and by this action, inverse treatment planning will occcur.
![Compare the temperatures of the liver (Figure A) and kidney (Figure B) due to wave propagation of the HIFU in depth in various modes[]{data-label="fig12"}](fig12.jpg){width="\linewidth"}
As shown in Figure \[fig12\], the percent of relative change of temperature measured in the first, second and third layers for the designed model to practical mode was obtained, and it was at 4, 5, and 14. Using the simulator HIFU, 6, 5.7, and 14.5 percent were obtained. The values for the liver tissue for the practical mode in the first and third layers, respectively, is 16, 17, and 24%, and relative changes using the simulator HIFU are 12.9, 17.9, and 15 percent. A comparison chart of liver and kidney tissue pressure, described in Figure 13, is displayed. The results of this figure in two modes using the software of simulator HIFU and model simulation have been carried out, and the following numbers were achieved: 14.59 and 13.39 MPa. In calculating for pressure mode, 9% relative changes are observed. It should be noted that in pressure mode between the liver and kidneys, no significant changes were observed.
![Compare the liver and kidney pressures due to wave propagation of HIFU in depth in various positions[]{data-label="fig13"}](fig13.jpg){width="\linewidth"}
Conclusion
----------
HIFU is a method that is used primarily for the removal of solid tumors through the creation of heat. This method requires the deliberate design of appropriate treatment. In this way, designing the appropriate treatment prevents damage to healthy tissues and results in the best treatment for the patient. Hence, it is very important. In this article, we sought to investigate pressure changes in kidney and liver tissue by modeling and stimulating HIFU processes to determine optimal values, to prevent complications, and to provide the optimal treatment method. The model and assumption represent a new method in treatment. Planning and precision in this case are also significant factors when comparing the HIFU simulator. The advantages of this model can be attributed to proper temperature distribution based on medical need and lack of skin burns caused by sound mechanical effects. Additionally using this method causes set temperature to be employed at specific limits, and as a result, normal tissue is not damaged.\
The presence of blood effects and pulse can be considered as pulse; in this case, the mechanical elements will be changed with time. In addition, blood changes the heat distribution in the tissue, to overcome this problem, a correction factor should be considered. In this article, the correction factor of stimulation as an electrical wave in the scale of MV was considered to the input of tissue that in Simulink simulation was not isolated from ultrasound stimulation.\
\
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[^1]: $^{1}$ Medical Radiation Eng. Department, Engineering Faculty, Science and Research Branch, Islamic Azad University, Tehran, Iran
[^2]: $^{2}$ Assistant Professor, Department of Electrical Engineering, Sharif University of Technology, Tehran, Iran
[^3]: $^{3}$ Chief of Biomedical Eng. Faculty, Science and Research Branch, Islamic Azad, University, Tehran, Iran
[^4]: $^{4}$ Assistant professor, Research Centre of Gastroenterology & Hepatology of Firouzgar Hospital, Iran University of Medical Science, Tehran, Iran
[^5]: $^{*}$ Corresponding Author: Majid Pouladian, Head of Biomedical Engineering Faculty, Science and Research Branch, Islamic Azad University,Tehran (Iran)
[^6]: **Keywords: *HIFU, simulator HIFU, pressure and heat, mechanical model, electrical model, viscoelasticity, voxel, ultrasound elastography***
|
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abstract: 'Let $G$ be a directed planar graph on $n$ vertices, with no directed cycle of length less than $g{\geqslant}4$. We prove that $G$ contains a set $X$ of vertices such that $G-X$ has no directed cycle, and $|X|{\leqslant}\tfrac{5n-5}9$ if $g=4$, $|X|{\leqslant}\tfrac{2n-5}4$ if $g=5$, and $|X|{\leqslant}\tfrac{2n-6}{g}$ if $g{\geqslant}6$. This improves recent results of Golowich and Rolnick.'
address: 'Laboratoire G-SCOP (CNRS, Univ. Grenoble-Alpes), Grenoble, France'
author:
- Louis Esperet
- Laetitia Lemoine
- Frédéric Maffray
title: |
Small feedback vertex sets\
in planar digraphs
---
[^1]
A directed graph $G$ (or digraph, in short) is said to be acyclic if it does not contain any directed cycle. The *digirth* of a digraph $G$ is the minimum length of a directed cycle in $G$ (if $G$ is acyclic, we set its digirth to $+\infty$). A *feedback vertex set* in a digraph $G$ is a set $X$ of vertices such that $G-X$ is acyclic, and the minimum size of such a set is denoted by $\tau(G)$. In this short note, we study the maximum $f_g(n)$ of $\tau(G)$ over all planar digraphs $G$ on $n$ vertices with digirth $g$. Harutyunyan [@Har11; @HM15] conjectured that $f_3(n){\leqslant}\tfrac{2n}5$ for all $n$. This conjecture was recently refuted by Knauer, Valicov and Wenger [@KVW16] who showed that $f_g(n){\geqslant}\tfrac{n-1}{g-1}$ for all $g{\geqslant}3$ and infinitely many values of $n$. On the other hand, Golowich and Rolnick [@GR15] recently proved that $f_4(n){\leqslant}\tfrac{7n}{12}$, $f_5(n){\leqslant}\tfrac{8n}{15}$, and $f_g(n){\leqslant}\tfrac{3n-6}{g}$ for all $g{\geqslant}6$ and $n$. Harutyunyan and Mohar [@HM15] proved that the vertex set of every planar digraph of digirth at least 5 can be partitioned into two acyclic subgraphs. This result was very recently extended to planar digraphs of digirth 4 by Li and Mohar [@LM16], and therefore $f_4(n){\leqslant}\tfrac{n}2$.
This short note is devoted to the following result, which improves all the previous upper bounds for $g {\geqslant}5$ (although the improvement for $g=5$ is rather minor). Due to the very recent result of Li and Mohar [@LM16], our result for $g=4$ is not best possible (however its proof is of independent interest and might lead to further improvements).
\[th:main\] For all $n{\geqslant}3$ we have $f_4(n){\leqslant}\tfrac{5n-5}9$, $f_5(n){\leqslant}\tfrac{2n-5}{4}$ and for all $g{\geqslant}6$, $f_g(n){\leqslant}\tfrac{2n-6}{g}$.
In a planar graph, the degree of a face $F$, denoted by $d(F)$, is the sum of the lengths (number of edges) of the boundary walks of $F$. In the proof of Theorem \[th:main\], we will need the following two simple lemmas.
\[lem:1\] Let $H$ be a planar bipartite graph, with bipartition $(U,V)$, such that all faces of $H$ have degree at least 4, and all vertices of $V$ have degree at least 2. Then $H$ contains at most $2|U|-4$ faces of degree at least 6.
Assume that $H$ has $n$ vertices, $m$ edges, $f$ faces, and $f_6$ faces of degree at least 6. Let $N$ be the sum of the degrees of the faces of $H$, plus twice the sum of the degrees of the vertices of $V$. Observe that $N=4m$, so, by Euler’s formula, $N{\leqslant}4n+4f-8$. The sum of degrees of the faces of $H$ is at least $4(f-f_6)+6f_6=4f+2f_6$, and since each vertex of $V$ has degree at least 2, the sum of the degrees of the vertices of $V$ is at least $2|V|$. Therefore, $4f+2f_6+4|V|{\leqslant}4n+4f-8$. It follows that $f_6{\leqslant}2 |U|-4$, as desired.
\[lem:2\] Let $G$ be a connected planar graph, and let $S=\{F_1,\ldots,F_k\}$ be a set of $k$ faces of $G$, such that each $F_i$ is bounded by a cycle, and these cycles are pairwise vertex-disjoint. Then $\sum_{F \not\in S}
(3d(F)-6){\geqslant}\sum_{i=1}^k (3d(F_i)+6)-12$, where the first sum varies over faces $F$ of $G$ not contained in $S$.
Let $n$, $m$, and $f$ denote the number of vertices, edges, and faces of $G$, respectively. It follows from Euler’s formula that the sum of $3d(F)-6$ over all faces of $G$ is equal to $6m-6f=6n-12{\geqslant}6\sum_{i=1}^k
d(F_i)-12$. Therefore, $\sum_{F\not\in S} (3d(F)-6){\geqslant}6\sum_{i=1}^k
d(F_i)-12 - \sum_{i=1}^k (3d(F_i)-6)=\sum_{i=1}^k (3d(F_i)+6)-12$, as desired.
We are now able to prove Theorem \[th:main\].
*Proof of Theorem \[th:main\].* We prove the result by induction on $n{\geqslant}3$. Let $G$ be a planar digraph with $n$ vertices and digirth $g{\geqslant}4$. We can assume without loss of generality that $G$ has no multiple arcs, since $g{\geqslant}4$ and removing one arc from a collection of multiple arcs with the same orientation does not change the value of $\tau(G)$. We can also assume that $G$ is connected, since otherwise we can consider each connected component of $G$ separately and the result clearly follows from the induction (since $g{\geqslant}4$, connected components of at most 2 vertices are acyclic and can thus be left aside). Finally, we can assume that $G$ contains a directed cycle, since otherwise $\tau(G)=0{\leqslant}\min\{\tfrac{5n-5}9,\tfrac{2n-5}{4},\tfrac{2n-6}{g}\}$ (since $n{\geqslant}3$).
Let ${\mathcal{C}}$ be a maximum collection of arc-disjoint directed cycles in $G$. Note that ${\mathcal{C}}$ is non-empty. Fix a planar embedding of $G$. For a given directed cycle $C$ of ${\mathcal{C}}$, we denote by $\overline{C}$ the closed region bounded by $C$, and by $\mathring{C}$ the interior of $\overline{C}$. It follows from classical uncrossing techniques (see [@GW97] for instance), that we can assume without loss of generality that the directed cycles of ${\mathcal{C}}$ are pairwise non-crossing, i.e. for any two elements $C_1,C_2 \in {\mathcal{C}}$, either $\mathring{C_1}$ and $\mathring{C_2}$ are disjoint, or one is contained in the other. We define the partial order $\preceq$ on ${\mathcal{C}}$ as follows: $C_1 \preceq C_2$ if and only if $\mathring{C_1}\subseteq \mathring{C_2}$. Note that $\preceq$ naturally defines a rooted forest ${\mathcal{F}}$ with vertex set ${\mathcal{C}}$: the roots of each of the components of ${\mathcal{F}}$ are the maximal elements of $\preceq$, and the children of any given node $C\in {\mathcal{F}}$ are the maximal elements $C' \preceq C$ distinct from $C$ (the fact that ${\mathcal{F}}$ is indeed a forest follows from the non-crossing property of the elements of ${\mathcal{C}}$).
Consider a node $C$ of ${\mathcal{F}}$, and the children $C_1,\ldots,C_k$ of $C$ in ${\mathcal{F}}$. We define the closed region ${\mathcal{R}}_C=\overline{C}-\bigcup_{1{\leqslant}i {\leqslant}k} \mathring{C_i}$. Let $\phi_C$ be the sum of $3d(F)-6$, over all faces $F$ of $G$ lying in ${\mathcal{R}}_C$.
\[cl:1\] Let $C_0$ be a node of ${\mathcal{F}}$ with children $C_1,\ldots,C_k$. Then $\phi_{C_0}{\geqslant}\tfrac{3}2 (g-2)k+\tfrac32g$. Moreover, if $g{\geqslant}6$, then $\phi_{C_0}{\geqslant}\tfrac{3}2 (g-2)k+\tfrac32g+3$.
Assume first that the cycles $C_0,\ldots,C_k$ are pairwise vertex-disjoint. Then, it follows from Lemma \[lem:2\] that $\phi_{C_0}{\geqslant}(k+1)(3g+6)-12$. Note that since $g{\geqslant}4$, we have $(k+1)(3g+6)-12{\geqslant}\tfrac{3}2 (g-2)k+\tfrac32g$. Moreover, if $g{\geqslant}6$, $(k+1)(3g+6)-12{\geqslant}\tfrac{3}2 (g-2)k+\tfrac32g+3$, as desired. As a consequence, we can assume that two of the cycles $C_0,\ldots,C_k$ intersect, and in particular, $k{\geqslant}1$.
Consider the following planar bipartite graph $H$: the vertices of the first partite set of $H$ are the directed cycles $C_0,C_1,\ldots,C_k$, the vertices of the second partite set of $H$ are the vertices of $G$ lying in at least two cycles among $C_0,C_1,\ldots,C_k$, and there is an edge in $H$ between some cycle $C_i$ and some vertex $v$ if and only if $v\in C_i$ in $G$ (see Figure \[fig:ex\]). Observe that $H$ has a natural planar embedding in which all internal faces have degree at least 4. Since $k{\geqslant}1$ and at least two of the cycles $C_0,\ldots,C_k$ intersect, the outerface also has degree at least 4. Note that the faces $F_1,\ldots,F_t$ of $H$ are in one-to-one correspondence with the maximal subsets ${\mathcal{D}}_1,\ldots,{\mathcal{D}}_t$ of ${\mathcal{R}}_{C_0}$ whose interior is connected. Also note that each face of $G\cap {\mathcal{R}}_{C_0}$ is in precisely one region ${\mathcal{D}}_i$ and each arc of $\bigcup_{i=0}^{k} C_i$ (i.e. each arc on the boundary of ${\mathcal{R}}_{C_0}$) is on the boundary of precisely one region ${\mathcal{D}}_i$. For each region ${\mathcal{D}}_i$, let $\ell_i$ be the number of arcs on the boundary of ${\mathcal{D}}_i$, and observe that $\sum_{i=1}^{t}\ell_i=\sum_{j=0}^{k} |C_j|$. Let $\phi_{{\mathcal{D}}_i}$ be the sum of $3d(F)-6$, over all faces $F$ of $G$ lying in ${\mathcal{D}}_i$. It follows from Lemma \[lem:2\] (applied with $k=1$) that $\phi_{{\mathcal{D}}_i}{\geqslant}3\ell_i-6$, and therefore $\phi_{C_0}=\sum_{i=1}^{t}\phi_{{\mathcal{D}}_i}{\geqslant}\sum_{i=1}^{t}(3\ell_i-6)$.
![The region ${\mathcal{R}}_{C_0}$ (in gray) and the planar bipartite graph $H$. \[fig:ex\]](ex)
A region ${\mathcal{D}}_i$ with $\ell_i{\geqslant}4$ is said to be of *type 1*, and we set $T_1=\{1{\leqslant}i{\leqslant}t \,|\, {\mathcal{D}}_i \mbox{ is of type 1}\}$. Since for any $\ell {\geqslant}4$ we have $3\ell-6{\geqslant}\tfrac{3\ell}2$, it follows from the paragraph above that the regions ${\mathcal{D}}_i$ of type 1 satisfy $\phi_{{\mathcal{D}}_i}{\geqslant}\tfrac{3\ell_i}2$. Let ${\mathcal{D}}_i$ be a region that is not of type 1. Since $G$ is simple, $\ell_i=3$. Assume first that ${\mathcal{D}}_i$ is bounded by (parts of) two directed cycles of ${\mathcal{C}}$ (in other words, ${\mathcal{D}}_i$ corresponds to a face of degree four in the graph $H$). In this case we say that ${\mathcal{D}}_i$ is of *type 2* and we set $T_2=\{1{\leqslant}i{\leqslant}t \,|\, {\mathcal{D}}_i \mbox{ is of type 2}\}$. Then the boundary of ${\mathcal{D}}_i$ consists in two consecutive arcs $e_1,e_2$ of some directed cycle $C^+$ of ${\mathcal{C}}$, and one arc $e_3$ of some directed cycle $C^-$ of ${\mathcal{C}}$. Since $g{\geqslant}4$, these three arcs do not form a directed cycle, and therefore their orientation is transitive. It follows that $|C^+|{\geqslant}g+1$, since otherwise the directed cycle obtained from $C^+$ by replacing $e_1,e_2$ with $e_3$ would have length $g-1$, contradicting that $G$ has digirth at least $g$. Consequently, $\sum_{i=0}^k |C_i|{\geqslant}(k+1)g+|T_2|$. If a region ${\mathcal{D}}_i$ is not of type 1 or 2, then $\ell_i=3$ and each of the 3 arcs on the boundary of ${\mathcal{D}}_i$ belongs to a different directed cycle of ${\mathcal{C}}$. In other words, ${\mathcal{D}}_i$ corresponds to some face of degree 6 in the graph $H$. Such a region ${\mathcal{D}}_i$ is said to be of *type 3*, and we set $T_3=\{1{\leqslant}i{\leqslant}t \,|\, {\mathcal{D}}_i \mbox{ is of type 3}\}$. It follows from Lemma \[lem:1\] that the number of faces of degree at least 6 in $H$ is at most $2(k+1)-4$. Hence, we have $|T_3|{\leqslant}2k-2$.
Using these bounds on $|T_2|$ and $|T_3|$, together with the fact that for any $i\in T_2\cup
T_3$ we have $\phi_{{\mathcal{D}}_i} {\geqslant}3\ell_i-6=3=\tfrac{3\ell_i}2 -\tfrac{3}2$, we obtain:
$$\begin{aligned}
\phi_{C_0} & = & \sum_{i\in T_1} \phi_{{\mathcal{D}}_i} + \sum_{i\in T_2}
\phi_{{\mathcal{D}}_i} + \sum_{i\in T_3} \phi_{{\mathcal{D}}_i} \\
& {\geqslant}& \sum_{i=1}^t \tfrac{3\ell_i}2 - \tfrac32 |T_2| - \tfrac32
|T_3|\\
& {\geqslant}& \tfrac32\, \sum_{i=0}^k |C_i| - \tfrac32 |T_2| - \tfrac32
(2k-2)\\
& {\geqslant}& \tfrac32 (k+1)g-3k+3 \, = \, \tfrac32 (g-2)k+\tfrac32g+3, \end{aligned}$$
as desired. This concludes the proof of Claim \[cl:1\].$\Box$
Let $C_1,\ldots,C_{k_\infty}$ be the $k_\infty$ maximal elements of $\preceq$. We denote by ${\mathcal{R}}_\infty$ the closed region obtained from the plane by removing $\bigcup_{i=1}^{k_\infty} \mathring{C_i}$. Note that each face of $G$ lies in precisely one of the regions ${\mathcal{R}}_C$ ($C \in {\mathcal{C}}$) or ${\mathcal{R}}_\infty$. Let $\phi_\infty$ be the sum of $3d(F)-6$, over all faces $F$ of $G$ lying in $R_\infty$. A proof similar to that of Claim \[cl:1\] shows that $\phi_\infty{\geqslant}\tfrac32 k_\infty (g-2)+3$, and if $g{\geqslant}6$, then $\phi_\infty{\geqslant}\tfrac32 k_\infty (g-2)+6$.
We now compute the sum $\phi$ of $3d(F)-6$ over all faces $F$ of $G$. By Claim \[cl:1\],
$$\begin{aligned}
\phi & = & \phi_\infty + \sum_{C \in {\mathcal{F}}} \phi_C\\
& {\geqslant}&\tfrac32 k_\infty (g-2)+3 + (|{\mathcal{C}}|-k_\infty) \tfrac32(g-2)+ |{\mathcal{C}}|
\cdot \tfrac32g \\
& {\geqslant}& (3g-3) |{\mathcal{C}}|+3.\end{aligned}$$
If $g{\geqslant}6$, a similar computation gives $\phi {\geqslant}3g |{\mathcal{C}}|+6 $. On the other hand, it easily follows from Euler’s formula that $\phi=6n-12$. Therefore, $|{\mathcal{C}}|{\leqslant}\tfrac{2n-5}{g-1}$, and if $g{\geqslant}6$, then $|{\mathcal{C}}|{\leqslant}\tfrac{2n-6}{g}$.
Let $A$ be a set of arcs of $G$ of minimum size such that $G-A$ is acyclic. It follows from the Lucchesi-Younger theorem [@LY78] (see also [@GR15]) that $|A|=|{\mathcal{C}}|$. Let $X$ be a set of vertices covering the arcs of $A$, such that $X$ has minimum size. Then $G-X$ is acyclic. If $g=5$ we have $|X|{\leqslant}|A|=|{\mathcal{C}}|{\leqslant}\tfrac{2n-5}{4}$ and if $g{\geqslant}6$, we have $|X|{\leqslant}|A|=|{\mathcal{C}}|{\leqslant}\tfrac{2n-6}{g}$, as desired. Assume now that $g=4$. In this case $|A|=|{\mathcal{C}}|{\leqslant}\tfrac{2n-5}{3}$. It was observed by Golowich and Rolnick [@GR15] that $|X|{\leqslant}\tfrac13(n+|A|)$ (which easily follows from the fact that any graph on $n$ vertices and $m$ edges contains an independent set of size at least $\tfrac{2n}3-\tfrac{m}3$), and thus, $|X|{\leqslant}\tfrac{5n-5}{9}$. This concludes the proof of Theorem \[th:main\].$\Box$
Final remark {#final-remark .unnumbered}
============
A natural problem is to determine the precise value of $f_g(n)$, or at least its asymptotical value as $g$ tends to infinity. We believe that $f_g(n)$ should be closer to the lower bound of $\tfrac{n-1}g$, than to our upper bound of $\tfrac{2n-6}g$.
For a digraph $G$, let $\tau^*(G)$ denote the the infimum real number $x$ for which there are weights in $[0,1]$ on each vertex of $G$, summing up to $x$, such that for each directed cycle $C$, the sum of the weights of the vertices lying on $C$ is at least $1$. Goemans and Williamson [@GW97] conjectured that for any planar digraph $G$, $\tau(G){\leqslant}\tfrac32 \tau^*(G)$. If a planar digraph $G$ on $n$ vertices has digirth at least $g$, then clearly $\tau^*(G){\leqslant}\tfrac{n}{g}$ (this can be seen by assigning weight $1/g$ to each vertex). Therefore, a direct consequence of the conjecture of Goemans and Williamson would be that $f_g(n){\leqslant}\tfrac{3n}{2g}$.
[99]{}
A. Harutyunyan, *Brooks-type results for coloring of digraphs*, PhD Thesis, Simon Fraser University, 2011.
M.X. Goemans and D.P. Williamson, *Primal-Dual Approximation Algorithms for Feedback Problems in Planar Graphs*, Combinatorica [**17**]{} (1997), 1–23.
N. Golowich and D. Rolnick, *Acyclic Subgraphs of Planar Digraphs*, Electronic J. Combin. [**22(3)**]{} (2015), \#P3.7.
A. Harutyunyan and B. Mohar, *Planar Digraphs of Digirth Five are 2-Colorable*. J. Graph Theory [**84(4)**]{} (2017), 408–427.
K. Knauer, P. Valicov, and P.S. Wenger, *Planar Digraphs without Large Acyclic Sets*, J. Graph Theory [**85(1)**]{} (2017), 288–291.
Z. Li and B. Mohar, *Planar digraphs of digirth four are 2-colourable*, Manuscript, 2016. <http://arxiv.org/abs/1606.06114>
C. Lucchesi and D. H. Younger, *A minimax theorem for directed graphs*, J. London Math. Society [**2**]{} (1978), 369–374.
[^1]: The authors are partially supported by ANR Project Stint (<span style="font-variant:small-caps;">anr-13-bs02-0007</span>), and LabEx PERSYVAL-Lab (<span style="font-variant:small-caps;">anr-11-labx-0025</span>).
|
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abstract: 'We report the results of spectroscopic observations and numerical modelling of the [H[ii]{} ]{}region IRAS18153$-$1651. Our study was motivated by the discovery of an optical arc and two main-sequence stars of spectral type B1 and B3 near the centre of IRAS18153$-$1651. We interpret the arc as the edge of the wind bubble (blown by the B1 star), whose brightness is enhanced by the interaction with a photoevaporation flow from a nearby molecular cloud. This interpretation implies that we deal with a unique case of a young massive star (the most massive member of a recently formed low-mass star cluster) caught just tens of thousands of years after its stellar wind has begun to blow a bubble into the surrounding dense medium. Our two-dimensional, radiation-hydrodynamics simulations of the wind bubble and the [H[ii]{} ]{}region around the B1 star provide a reasonable match to observations, both in terms of morphology and absolute brightness of the optical and mid-infrared emission, and verify the young age of IRAS18153$-$1651. Taken together our results strongly suggest that we have revealed the first example of a wind bubble blown by a main-sequence B star.'
date: 'Accepted 2016 December 12. Received 2016 December 12; in original form 2016 October 24'
title: 'IRAS18153$-$1651: an [H[ii]{} ]{}region with a possible wind bubble blown by a young main-sequence B star[^1] '
---
\[firstpage\]
circumstellar matter – stars: massive – stars: winds, outflows – ISM: bubbles – HII regions – ISM: individual objects: IRAS18153$-$1651.
Introduction {#sec:intro}
============
Hot massive stars are sources of fast line-driven winds (Snow & Morton 1976; Puls, Vink & Najarro 2008), whose interaction with the circum- and interstellar medium results in the origin of bubbles and shells of various shapes and scales (Johnson & Hogg 1965; Lozinskaya & Lomovskij 1982; Chu, Treffers & Kwitter, 1983; Dopita et al. 1994). Since most (if not all) massive stars form in a clustered way (Lada & Lada 2003; Gvaramadze et al. 2012), the wind bubbles produced by individual members of star clusters are unobservable because they merge into a single much more extended structure – a superbubble (McCray & Kafatos 1987). Numerous examples of such superbubbles (with diameters ranging from several tens of pc to kpc scales) were revealed in H$\alpha$ photographic surveys of the Magellanic Clouds and other nearby galaxies (e.g. Davies, Elliott & Meaburn 1976; Meaburn 1980; Courtes et al. 1987; Hunter 1994).
To produce a well-shaped circular bubble or shell a massive star should be isolated from the destructive influence of winds from other massive stars (cf. Nazé et al. 2001). To achieve this, it should leave the parent cluster either because of a few-body dynamical encounter with other massive stars (e.g. Poveda, Ruiz & Allen 1967; Oh & Kroupa 2016) or binary supernova explosion (e.g. Blaauw 1961; Eldridge, Langer & Tout 2011), or it should be the only massive star in the parent cluster. In the first case, the wind bubble around the star running away from its birth place rapidly becomes elongated (Weaver et al. 1977) and transforms into a bow shock (van Buren & McCray 1988) if the star is moving supersonically with respect to the local interstellar medium (ISM). Stellar motion alone is hence the main reason that closed structures around hydrogen-burning massive field stars are not observed[^2]. The only known possible exception is the Bubble Nebula (e.g. Christopoulou et al. 1995; Moore et al. 2002), which is produced by the runaway O6.5(n)fp (Sota et al. 2011) star BD+602522, whose luminosity class, however, is unknown because of the peculiar shape of the He[ii]{} $\lambda$4686 line. The origin of this nebula might be attributed to a situation in which a bow-shock-producing star encounters a density enhancement (cloudlet) on its way, resulting in a temporal formation of a closed bubble with the star located near its leading edge.
{width="14cm"}
In the second case, the wind-blowing star is the only massive star (a B star of mass of $8-10 \, {\rm\,M_\odot}$) in a star cluster of mass of about $100 \, {\rm\,M_\odot}$ (e.g. Kroupa et al. 2013). Such stars with their weak winds produce momentum-driven bubbles (Steigman, Strittmatter & Williams 1975) in the dense material of the parental molecular cloud, which could be detected under favourable conditions, e.g. if the cluster was formed near the surface of the cloud. In this paper, we report the discovery of an optical arc within the circular mid-infrared shell (known as IRAS18153$-$1651) and argue that it represents the edge of a young ($\sim10^4$ yr) wind bubble produced by a main-sequence B star residing in a low-mass star cluster. In Section\[sec:bub\], we present the images of the arc, IRAS18153$-$1651 and two stars associated with them, as well as review the existing data on these objects. In Section\[sec:obs\], we describe our optical spectroscopic observations of the arc and the two stars. In Section\[sec:sta\], we classify the stars and model their spectra. In Section\[sec:arc\], we derive some parameters of the arc and propose a scenario for the origin of the arc and the shell around it. In Section\[sec:num\], we present and discuss results of numerical modelling, which we carried out to support the scenario. We summarize and conclude in Section\[sec:sum\].
IRAS18153$-$1651 and its central stars {#sec:bub}
======================================
{width="14cm"}
The nebula, which is the subject of this paper, was serendipitously discovered in the archival data of the [*Spitzer Space Telescope*]{} during our search for bow shocks generated by OB stars running away from the young massive star clusters NGC6611 and NGC6618 embedded in the giant [H[ii]{} ]{}regions M16 and M17, respectively (for motivation and some results of this search, see Gvaramadze & Bomans 2008; cf. Gvaramadze et al. 2014a). The [*Spitzer*]{} data we used were obtained with the Multiband Imaging Photometer for [*Spitzer*]{} (MIPS; Rieke et al. 2004) within the framework of the 24 and 70 Micron Survey of the Inner Galactic Disk with MIPS (Carey et al. 2009) and with the [*Spitzer*]{} Infrared Array Camera (IRAC; Fazio et al. 2004) within the Galactic Legacy Infrared Mid-Plane Survey Extraordinaire (Benjamin et al. 2003).
In the MIPS 24$\mu$m image the nebula appears as a limb-brightened, almost circular shell of radius $\approx$27 arcsec, with the western edge brighter than the other parts of the shell (see Fig.\[fig:neb\], upper left panel). This image also shows the presence of a point-like source, which is somewhat offset from the geometric centre of the shell in the north-west direction. In the SIMBAD data base the nebula is named IRAS18153$-$1651, and we will use this name hereafter. At the distance of IRAS18153$-$1651 of 2 kpc (see below), 1 arcsec corresponds to $\approx$0.01 pc, so that the linear radius of the shell is $\approx$0.26 pc.
IRAS18153$-$1651 and the point source within it are also visible in all (3.6, 4.5, 5.8 and $8.0\,\mu$m) IRAC images (Fig.\[fig:neb\]). In these images, the nebula lacks the circular shape and has a more complicated appearance. At 8$\mu$m one still can see a more or less circular shell, whose west side overlaps with a bar-like structure stretched in the north-south direction. At shorter wavelengths, both the shell and the “bar" become more diffuse. This morphology and the brightness asymmetry of the shell at 24$\mu$m indicate that the nebula is interacting with a dense medium in the west. This inference is supported by the data of the APEX Telescope Large Area Survey of the Galaxy (ATLASGAL; Schuller et al. 2009). Fig.\[fig:hub\] presents the ATLASGAL 870$\mu$m image of the region centred on IRAS18153$-$1651 (left-hand panel) and the [*Spitzer*]{} 24 and 8$\mu$m images of the same region (middle and right-hand panels, respectively) with the 870$\mu$m image overlayed in black contours. In the ATLASGAL image one can see two filaments of cold dense gas intersecting each other in a nodal point (dubbed “hub–N" in Busquet et al. 2013; see below for more detail), while comparison of this image with the two other ones shows that IRAS18153$-$1651 is apparently interacting with the hub–N and one of the filaments (see also below).
The point source in IRAS18153$-$1651 can also be seen in all ($J,H,K_{\rm s}$) Two-Micron All Sky Survey (2MASS) images (Skrutskie et al., 2006) as well as in the images provided by the Digitized Sky Survey II (DSS-II) (McLean et al. 2000). The IRAC images clearly show that this source is composed of two nearby stars (see also Fig.\[fig:acq\]). The coordinates of these stars, as given in the GLIMPSE Source Catalog (I + II + 3D) (Spitzer Science Center 2009), are: RA(J2000)=$18^{\rm h} 18^{\rm m} 16\fs21$, Dec.(J2000)=$-16\degr 50\arcmin 38\farcs8$ (hereafter star1) and RA(J2000)=$18^{\rm h} 18^{\rm m}
16\fs28$, Dec.(J2000)=$-16\degr 50\arcmin 36\farcs7$ (hereafter star2). This catalogue also provides the IRAC band magnitudes for stars1 and 2 separately. The two stars are also resolved by the the UKIDSS Galactic Plane Survey (Lucas et al. 2008), which gives for them $H$- and $K$-band magnitudes. The total $B$- and $V$-band magnitudes of the two stars are, respectively, 15.84$\pm$0.05 and 14.14$\pm$0.05 (Henden et al. 2016). The details of the stars are summarized in Table\[tab:det\], to which we also added their spectral types, effective temperatures and surface gravities, based on our spectroscopic observations and spectral analysis (presented in Sections\[sec:obs\] and \[sec:sta\], respectively).
star1 star2
-------------------- --------------------------------- ---------------------------------
SpT B1V B3V
$\alpha$ (J2000) $18^{\rm h} 18^{\rm m} 16\fs21$ $18^{\rm h} 18^{\rm m} 16\fs28$
$\delta$ (J2000) $-16\degr 50\arcmin 38\farcs8$ $-16\degr 50\arcmin 36\farcs7$
$H$ (mag) 10.28$\pm$0.02 11.16$\pm$0.02
$K$ (mag) 9.39$\pm$0.02 10.89$\pm$0.02
$[3.6]$ (mag) 9.00$\pm$0.14 —
$[4.5]$ (mag) 8.98$\pm$0.15 —
$[5.8]$ (mag) 8.83$\pm$0.08 10.14$\pm$0.21
$T_{\rm eff}$ (kK) 22$\pm$2 20$\pm$2
$\log g$ 4.2$\pm$0.2 4.4$\pm$0.2
: Details of two stars in the centre of IRAS18153$-$1651. The spectral types, SpT, effective temperatures, $T_{\rm eff}$ and surface gravities, $\log g$, are based on our spectroscopic observations and spectral analysis. The coordinates and IRAC photometry are from the GLIMPSE Source Catalog (I + II + 3D) (Spitzer Science Center 2009). The $H$ and $K$ photometry is from Lucas et al. (2008). []{data-label="tab:det"}
The DSS-II images of IRAS18153$-$1651 show diffuse emission to the west of stars1 and 2, extending to the edge of the 24$\mu$m shell. This emission is also clearly seen in the H$\alpha$+\[N[ii]{}\] image (see Fig.\[fig:neb\]) obtained in the framework of the SuperCOSMOS H-alpha Survey (SHS; Parker et al. 2005), which also reveals a clear arcuate structure located at about 12 arcsec (or 0.11 pc in projection) from the stars (note that these stars are not in the geometric centre of the arc; see Section\[sec:com\] for discussion of this issue). The orientation of the arc suggests that it could be shaped by the winds of the central stars and that is what has motivated us to carry out the research presented in this paper.
A literature search showed that the region containing IRAS18153$-$1651 has been studied quite extensively during the last years (Busquet et a. 2013, 2016; Povich et al. 2016; Santos et al. 2016). These studies, however, only briefly touch IRAS18153$-$1651 and are mostly devoted to environments of this object. Below we review some relevant information about IRAS18153$-$1651.
IRAS18153$-$1651 is a part of the infrared dark cloud G14.225-0.506 (identified with [*Spitzer*]{} by Peretto & Fuller 2009), which, in turn, is a central region of the south-west extension of the massive star-forming region M17 (M17SWex; Povich & Whitney 2010). Using [*Spitzer*]{} and 2MASS data, Povich & Whitney (2010) concluded that M17SWex is a precursor to an OB association and that this cloud as a whole will produce $>200$ B stars and perhaps not (m)any O stars. [*Chandra*]{} observations of G14.225-0.506 confirmed that M17SWex currently hosts no O-type stars (Povich et al. 2016). Observational evidence suggests that M17SWex is located in the Carina-Sagittarius arm at a distance of 2kpc (Povich et al. 2016). In what follows, we adopt this distance for IRAS18153$-$1651 as well.
Radio observations of the dense NH$_3$ gas in G14.225-0.506 by Busquet et al. (2013) showed that this cloud consists of a net of eight molecular filaments, some of which intersect with each other in two regions of enhanced density (named hub–N and hub–S). IRAS18153$-$1651 is located at about 1.5 arcmin south-east from one of these two nodal points (hub–N) and is bounded from the west side by a filament (named F10–E) stretching for $\approx$2.5 arcmin from the hub–N to the south (see fig.2 in Busquet et al. 2013 and Fig.\[fig:hub\]). Busquet et al. (2013) attributed the origin of the filaments to the gravitational instability of a thin layer threaded by a magnetic field. Polarimetric observations of background stars at optical and near-infrared wavelengths showed that the magnetic field lines in G14.225-0.506 are perpendicular to most of the filaments and to the (elongated) cloud as a whole (Santos et al. 2016), which further supports the possibility that the regular magnetic field plays an important role in the formation of parallel filaments observed in M17SWex and other star-forming regions. Interestingly, Santos et al. (2016) found that the magnetic field lines are not perpendicular to the filament F10–E and hub–N (which is also elongated in the north-south direction) and suggested that this discrepant behaviour might be caused by expansion of the [H[ii]{} ]{}region IRAS18153$-$1651. Additional evidence suggesting the interaction between IRAS18153$-$1651 and the filament F10–E and hub–N was presented in Busquet et al. (2013, 2016).
The detection of numerous young stellar objects (Povich & Whitney 2010) and H$_2$O masers (Jaffe, Güsten & Downes 1981; Wang et al. 2006) in M17SWex (including G14.225-0.506) points to ongoing star formation in this region, while the presence of several bright IR sources, of which IRAS18153$-$1651 (with its bolometric luminosity of $\sim10^4 \, {\rm\,L_\odot}$ and number of Lyman continuum photons per second of $<1.5\times10^{46}$; Jaffe, Stier & Fazio 1982) is one of the brightest, implies that a number of massive stars have already formed there. [*Chandra X-ray Observatory*]{} imaging study of M17SWex revealed that IRAS18153$-$1651 contains one of the two richest concentrations of X-ray sources detected in this star-forming region (Povich et al. 2016). Povich et al. (2016) suggested that IRAS18153$-$1651 might represent a site for massive cluster formation, and noted that the infrared and radio continuum luminosities of this [H[ii]{} ]{}region indicate that it is powered by a B1-1.5V star. Similarly, Busquet et al. (2016) mentioned unpublished VLA 6cm observations of IRAS18153$-$1651, which reveal “a cometary [H[ii]{} ]{}region ionized by a B1 star with the head of the cometary arch pointing toward hub–N". Our spectroscopic observations of stars1 and 2 confirm that IRAS18153$-$1651 is powered by a B1V star (see the next section) and suggest that this [H[ii]{} ]{}region indeed contains a recently formed star cluster (see Section\[sec:arc\]).
Spectroscopic observations {#sec:obs}
==========================
To classify the central stars of IRAS18153$-$1651 and clarify the nature of the optical arc, we obtained long-slit spectra with the Gemini Multi-Object Spectrograph South (GMOS-S) and the Cassegrain Twin Spectrograph (TWIN) of the 3.5-m telescope in the Observatory of Calar Alto (Spain).
Gemini-South {#sec:GS}
------------
To obtain spectra of stars1 and 2, we used the Poor Weather time at Gemini-South under the programme ID GS-2011B-Q-92. GMOS-S provided coverage from 3800 to 6750Å with a resolving power of $\approx$3000. The spectra were collected on 2012 May 7 under good seeing conditions ($\approx$0.6 arcsec), but some thick clouds (extinction around 1mag). The slit of width of 0.75 arcsec was aligned along stars1 and 2, i.e. with a position angle (PA) of PA=$31\fdg5$, measured from north to east. The desired signal-to-noise ratio of $\sim$150 was achieved with a total exposure time of 9$\times$150s. The bias subtraction, flat-fielding, wavelength calibration and sky subtraction were executed with the GMOS package in the Gemini library of the [iraf]{}[^3] software. In order to fill the gaps in GMOS-S’s CCD, the observation was divided into three series of exposures obtained with a different central wavelength, i.e. with a 5Å shift between each exposure. The extracted spectra were obtained by averaging the individual exposures, using a sigma clipping algorithm to eliminate the effects of cosmic rays. The average wavelength resolution is $\approx$0.46Å pixel$^{-1}$ (full width at half-maximum (FWHM) $\approx$4.09Å), and the accuracy of the wavelength calibration estimated by measuring the wavelength of 10 lamp emission lines is 0.061Å. A spectrum of the white dwarf H600 was used for flux calibration and for removing the instrument response. Unfortunately, due to the weather conditions, any absolute measurement of the flux is not possible.
![GMOS $g\prime$-band acquisition image of the central stars of IRAS18153$-$1651, separated from each other by $\approx$2.3 arcsec or $\approx$0.02 pc in projection. The brightest of the stars is star1. The relative positions of the GMOS and TWIN slits are shown by a solid (blue) and a dashed (red) rectangle, respectively. The widths of the rectangles of 0.75 and 2.1 arcsec correspond to the widths of the slits. []{data-label="fig:acq"}](fig3.eps){width="8cm"}
Calar Alto {#sec:CA}
----------
An additional spectrum of IRAS18153$-$1651 was obtained with TWIN on 2012 July 12 under the programme ID H12-3.5-013. Three exposures of 600s were taken. The set-up used for TWIN consisted of the gratings T08 in the first order for the blue arm (spectral range 3500–5600 Å) and T04 in the first order for the red arm (spectral range 5300–7600 Å) which provide a reciprocal dispersion of 72 Å${\rm mm}^{-1}$ for both arms. The resulting FWHM spectral resolution measured on strong lines of the night sky and reference spectra was 3.1–3.7 Å. The Calar Alto observation was mostly intended to get a spectrum of the optical arc. Correspondingly, the slit of $240\times2.1$ arcsec$^2$ was oriented at PA=80$\degr$ to cross the brightest part of the arc. The seeing was variable, ranging from $\simeq$1.5 to 2.0 arcsec. Spectra of He–Ar comparison arcs were obtained to calibrate the wavelength scale and the spectrophotometric standard star BD+$33\degr$2642 (Oke 1990) was observed at the beginning of the night for flux calibration.
The primary data reduction was done using [iraf]{}: the data for each CCD detector were trimmed, bias subtracted and flat corrected. The subsequent long-slit data reduction was carried out in the way described in Kniazev et al. (2008). The blue and red parts of the spectra were reduced independently for all three exposures, then aligned along the rows using the IRAF [apall]{} task, and finally summed up.
Stars1 and 2: spectral classification and modelling {#sec:sta}
===================================================
{width="14cm"}
To classify stars1 and 2, we used their GMOS spectra. The spectra are dominated by H and He[i]{} absorption lines (see Fig.\[fig:mod\]). No He[ii]{} lines are visible in the spectra, which implies that both stars are of B type. Using the EW(H$\gamma$)–absolute magnitude calibration by Balona & Crampton (1974) and the measured equivalent widths of EW(H$\gamma$)=4.5$\pm$0.1Å and 6.3$\pm$0.4Å for stars1 and 2, respectively, we estimated their spectral types as B1V and B3V. The apparent luminosity resulting from the B1V classification of star1 is consistent with the location of IRAS18153$-$1651 at the distance of 2 kpc (cf. Povich et al. 2016). We note that asymmetric profiles of the He[i]{} lines in the spectrum of star2 suggest that this star might be a binary system. Nonetheless, the low spectral resolution and signal-to-noise ratio did not allow us to confirm this.
We modelled the spectra using the approach described in Castro et al. (2012; see also Lefever et al. 2010). The technique is rooted in a previously calculated [fastwind]{}[^4] (Santolaya-Rey, Puls & Herrero 1997; Puls et al. 2005) stellar atmosphere grid (e.g. Simon-Diaz et al. 2011; Castro et al. 2012) and a $\chi^{2}$-based algorithm, searching for the best set of parameters that reproduce the main strong optical lines observed between $\approx$4000$-$5000Å (labelled in Fig.\[fig:mod\]). With the best-fitting models for stars1 and 2 (see Fig.\[fig:mod\]), we derived effective temperatures of $T_{\rm eff}$=22000$\pm$2000K and 20000$\pm$2000K, and surface gravities of $\log g$=4.2$\pm$0.2 and 4.4$\pm$0.2, respectively.
Main-sequence stars of these effective temperatures are sources of radiatively driven winds, which are strong enough to appreciably modify the ambient medium. In Table\[tab:krt\] we give mass-loss rates, $\dot{M}$, and terminal wind velocities, $v_\infty$, predicted by Krtička (2014) for main-sequence B stars with $T_{\rm eff}$ in the range of temperatures derived for stars1 and 2, i.e. for $T_{\rm eff}\in[18\,000,24\,000]$K. In this table we also give the rates of Lyman and dissociating Lyman-Werner photons (respectively, $Q_0$ and $Q_{\rm FUV}$) for the same range of effective temperatures (taken from Diaz-Miller, Franco & Shore 1998). The spectral classification of stars1 and 2 implies that $T_{\rm eff}$ of star1 should be at the upper end of the temperature range derived from spectral modelling, while that of star2 – at the lower end (cf. Kenyon & Hartmann 1995). In the following, we adopt for these stars the effective temperatures of 24 and 18 kK, respectively. From Table\[tab:krt\] it then follows that the mechanical wind luminosity, $L_{\rm w}=\dot{M}v_{\infty}^2 /2$, and radiation fluxes of star1 are more than two orders of magnitude higher then those of star2, so that we will consider the former star as the main energy source in IRAS18153$-$1651.
--------------- -------------------------------------- ------------------------- --------------------- ---------------------
$T_{\rm eff}$ $\dot{M}$ $v_\infty$ $Q_0$ $Q_{\rm FUV}$
(kK) (${{\rm\,M_\odot}\, {\rm yr}^{-1}}$) (${{\rm\,km\,s^{-1}}}$) (${\rm s}^{-1}$) (${\rm s}^{-1}$)
18 $9.1\times10^{-12}$ 820 $1.86\times10^{43}$ $2.04\times10^{46}$
20 $3.4\times10^{-11}$ 1290 $1.35\times10^{44}$ $6.03\times10^{46}$
22 $7.9\times10^{-11}$ 1690 $6.31\times10^{44}$ $1.38\times10^{47}$
24 $3.9\times10^{-10}$ 1700 $2.40\times10^{45}$ $2.45\times10^{47}$
--------------- -------------------------------------- ------------------------- --------------------- ---------------------
: Mass-loss rates and terminal wind velocities of main-sequence B stars of solar metallicity and different effective temperatures as predicted by Krtička (2014), and rates of Lyman ($Q_0$) and dissociating Lyman-Werner ($Q_{\rm FUV}$) photons from Diaz-Miller et al. (1998).[]{data-label="tab:krt"}
Optical arc {#sec:arc}
===========
The GMOS slit was oriented along stars1 and 2 and therefore it does not cross the optical arc. Correspondingly, we did not detect any signatures of nebular emission in the 2D spectrum. On the contrary, the slit of the TWIN spectrograph was oriented in such a way that it intersects the region of maximum brightness of the arc. Below we discuss the spectrum of the arc obtained with this spectrograph.
In the TWIN 2D spectrum the arc becomes visible via its emission lines of H$\alpha$, H$\beta$, \[N[ii]{}\] $\lambda\lambda$6548, 6584 and \[S[ii]{}\] $\lambda\lambda$6717, 6731 superimposed on strong continuum emission, which extends to the west of stars1 and 2 for about 20 arcsec (see also Fig.\[fig:Ha\]). A part of this spectrum is presented in Fig.\[fig:2D\]. Such composed spectra are typical of ionized reflection nebulae like the Orion Nebula (see e.g. plate XXIX in Greenstein & Henyey 1939). Correspondingly, we attribute the continuum emission to the starlight scattered by dust in the shell around the [H[ii]{} ]{}region.
We extracted a 1D spectrum over the optical arc by summing up, without any weighting, all rows from the area of an annulus with an outer radius of 20 arcsec centred on stars1 and 2 and the central $\pm$3 arcsec excluded. The resulting spectrum is presented in Fig.\[fig:neb-spec\]. The emission lines detected in the spectrum were measured using the programs described in Kniazev et al. (2004). Table\[tab:int\] lists the observed intensities of these lines normalized to H$\beta$, $F$($\lambda$)/$F$(H$\beta$), the reddening-corrected line intensity ratios, $I$($\lambda$)/$I$(H$\beta$), and the logarithmic extinction coefficient, $C$(H$\beta$), which corresponds to a colour excess of $E(B-V)$=2.08$\pm$0.23 mag. In Table\[tab:int\] we also give the electron number density derived from the intensity ratio of the \[S[ii]{}\] $\lambda\lambda$6716, 6731 lines, $n_{\rm
e}$(\[S[ii]{}\]). Both $C$(H$\beta$) and $n_{\rm e}$(\[S[ii]{}\]) were calculated under the assumption that $T_{\rm e}=6\,500$ K (see Section\[sec:sim\]). All calculations were done in the way described in detail in Kniazev et al. (2008).
The upper and middle panels in Fig.\[fig:Ha\] plot, respectively, the H$\alpha$ line and continuum intensities and the H$\alpha$ heliocentric radial velocity distribution along the slit. A comparison of these panels with the SHS and MIPS images of IRAS18153$-$1651 (presented for convenience in the bottom panel of Fig.\[fig:Ha\]) shows that in the west direction from stars1 and 2 the H$\alpha$ and continuum emission extend to the edge of the mid-infrared nebula, and that the H$\alpha$ intensity peaks at the position of the arc. In the opposite direction, the H$\alpha$ line intensity is prominent up to a distance comparable to the radius of the arc, while the continuum emission fades at a shorter distance. From the lower panel of Fig.\[fig:Ha\], we derived the mean heliocentric radial velocity of the H$\alpha$ emission of $31\pm5 \, {{\rm\,km\,s^{-1}}}$. Using this velocity and assuming the distance to the Galactic Centre of $R_0$=8.0 kpc and the circular rotation speed of the Galaxy of $\Theta _0 =240 \, {{\rm\,km\,s^{-1}}}$ (Reid et al. 2009), and the solar peculiar motion $(U_{\odot},V_{\odot},W_{\odot})=(11.1,12.2,7.3) \, {{\rm\,km\,s^{-1}}}$ (Schönrich, Binney & Dehnen 2010), one finds the local standard of rest velocity of $27\pm5 \, {{\rm\,km\,s^{-1}}}$, which agrees well with that of the cloud G14.225-0.506 (Jaffe et al. 1982; Busquet et al. 2013) and the star-forming region M17SWex as a whole (e.g. Povich et al. 2016).
The electron number density could also be derived from the surface brightness of the arc in the H$\alpha$ line, $S_{{\rm H}\alpha}$, measured on the SHS image (cf. equation 4 in Frew et al. 2014): $$\begin{aligned}
n_{\rm e}\approx 1.7 \, {\rm cm}^{-3} \left({l\over 1 \, {\rm
pc}}\right)^{-0.5}\left({T_{\rm e} \over 10^4 \, {\rm K}}\right)^{0.45} \nonumber \\
\times \left({S_{{\rm H}\alpha}\over 1 \, {\rm
R}}\right)^{0.5}e^{1.1E(B-V)} \, ,
\label{eqn:den}\end{aligned}$$ where $l$ is the line-of-sight thickness of the arc and 1R$\equiv$1Rayleigh=5.66$\times10^{-18}$ erg cm$^{-2}$ s$^{-1}$ arcsec$^{-2}$ at H$\alpha$ (here we assumed that $n_{\rm
e}$ and $T_{\rm e}$ are constant within the arc).
Using equations(1) and (2) in Frew et al. (2014) and the flux calibration factor of 20.4 counts pixel$^{-1}$ R$^{-1}$ from their table1, and adopting the observed \[N[ii]{}\] to H$\alpha$ line intensity ratio of $0.36^{+0.30} _{-0.17}$ from Table\[tab:int\] (here \[N[ii]{}\] corresponds to the sum of the $\lambda$6548 and $\lambda$6584 lines), we obtained the peak surface brightness of the arc (corrected for the contribution from the contaminant \[N[ii]{}\] lines) of $S_{{\rm H}\alpha}\approx$30$^{+3} _{-4}$R. Then, using equation(\[eqn:den\]) with $E(B-V)$=2.08$\pm$0.23 mag, $l$$\approx$0.23 pc (we assumed that the arc is a part of a spherical shell of inner radius of 12 arcsec and thickness of 5 arcsec) and $T_{\rm e}$=6500K (see Section\[sec:sim\]), one finds $n_{\rm e}$$\approx$160$^{+45} _{-35}$ ${\rm cm}^{-3}$, which agrees within the error margins with $n_{\rm e}$(\[S[ii]{}\]) given in Table\[tab:int\].
To find the actual H$\alpha$ surface brightness of the optical arc, $S_{{\rm H}\alpha} ^{\rm act}$, one needs to estimate the attenuation of the H$\alpha$ emission line in magnitudes, $A$(H$\alpha$), in the direction towards IRAS18153$-$1651, which is related to the visual extinction, $A_V=R_VE(B-V)$, through the following relationship: $A$(H$\alpha$)=$0.828A_V$ (e.g. James et al. 2005). Assuming a ratio of total to selective extinction of $R_V=3.1$ and using $E(B-V)$ from Table\[tab:int\], one finds $A$(H$\alpha$)=5.34$\pm$0.59 mag and $S_{{\rm H}\alpha}
^{\rm act}\approx4\,100^{+3\,700} _{-2\,100}$R.
Emission-line objects can be classified by using various diagnostic diagrams (see e.g. Kniazev, Pustilnik & Zucker 2008; Frew & Parker 2010, and references therein), of which the most frequently used one is $\log$(H$\alpha$/(\[S[ii]{}\] $\lambda\lambda$6716, 6731)) versus $\log$(H$\alpha$/(\[N[ii]{}\] $\lambda\lambda$6548, 6584)). For the optical arc one has $0.56^{+0.18} _{-0.20}$ versus 0.45$\pm$0.27 (see Table\[tab:int\]). These values place the arc in the area occupied by [H[ii]{} ]{}regions, although the large error bars allow the possibility that the arc is located within the domain occupied by supernova remnants, which means that the emission of the arc might be due to shock excitation.
Proceeding from this, one can suppose that the arc is either the edge of an asymmetric stellar wind bubble, distorted by a density gradient and/or stellar motion (e.g. Mackey et al. 2015, 2016), or a bow shock if the relative velocity between the wind-blowing star and the ambient medium is higher than the sound speed of the latter. The detection of two early type B stars separated from each other by only $\approx$2.3 arcsec (or 0.02 pc in projection) and the presence of a concentration of X-ray sources around them (see Section\[sec:bub\]) suggest that we deal with a recently formed star cluster with stars1 and 2 being its most massive members (cf. Gvaramadze et al. 2014b). This, in turn, implies that the mass of the cluster is about $100 \, {\rm\,M_\odot}$ (e.g. Kroupa et al. 2013). We hypothesize therefore that the arc is the edge of the wind bubble blown by the wind of star1 and suggest that the one-sided appearance of the bubble is caused by the interaction between the bubble and a photoevaporated flow from the molecular cloud to the west of the star (see Section\[sec:bub\]).
Further, we interpret the more extended (mid-infrared) nebula around the arc as an [H[ii]{} ]{}region, so that the radius of the nebula is equal to the Strömgren radius $R_{\rm S}$. Proceeding from this, one can estimate the electron number density of the ambient medium: $$n_{\rm e}=\sqrt{{3Q_0\over 4\pi\alpha _{\rm B}R_{\rm S}^3}} \, ,
\label{eqn:dens}$$ where $\alpha _{\rm B} =3.4\times10^{-10}T^{-0.78}$ is the CaseB recombination coefficient (Hummer 1994). This assumes that the electron and H$^+$ number densities are the same, which is true for B stars because they cannot ionize helium. If $T$=6500K (see Section\[sec:sim\]) and assuming $Q_0=2.4\times10^{45}
\, {\rm s}^{-1}$ (see Table\[tab:krt\]), one finds from equation(\[eqn:dens\]) that $n_{\rm e}\sim100 \, {\rm cm}^{-3}$.
In such a dense medium the bubble created by the wind of star1 will soon become radiative (i.e. the radiative losses of the shocked wind material at the interface with the ISM become comparable to $L_{\rm w}$; cf. Mackey et al. 2015). This happens at the moment (McCray 1983): $$t_{\rm rad}=1.5\times10^4 L_{33} ^{0.3} n_{100} ^{-0.7} \, {\rm yr} \, ,
\label{eqn:time}$$ when the radius of the bubble is $$R_{\rm rad}=0.2L_{33} ^{0.4} n_{100} ^{-0.6} \, {\rm pc} \, ,
\label{eqn:rad}$$ where $L_{33}=L_{\rm w}/(10^{33} \, {\rm erg} \, {\rm s}^{-1})$ and $n_{100}=n/(100 \,
{\rm cm}^{-3})$. Subsequent evolution of the bubble follows the momentum-driven solution by Steigman et al. (1975) and its radius is given by: $$R(t)=R_{\rm rad}(t/t_{\rm rad})^{1/2} \, .
\label{eqn:ste}$$ With $\dot{M}=3.9\times10^{-10} \, {{\rm\,M_\odot}\, {\rm yr}^{-1}}$ and $v_{\infty}=1700 \, {{\rm\,km\,s^{-1}}}$ (see Table\[tab:krt\]), assuming $n=200 \, {\rm cm}^{-3}$ (see Section\[sec:sim\]), and using equations(\[eqn:time\]) and (\[eqn:rad\]), one finds that $L_{\rm w}\approx3.6\times10^{32}
\, {\rm erg} \, {\rm s}^{-1}$, $t_{\rm rad}\approx7\,000$yr and $R_{\rm rad}\approx0.09$pc. Then, it can be seen from equation(\[eqn:ste\]) that the radius of the bubble would be equal to the observed radius of the optical arc of $\approx$0.11pc if the bubble was formed only $\approx11\,000$yr ago[^5]. This inference is supported by numerical modelling presented in the next section.
Numerical Modelling {#sec:num}
===================
Simulations {#sec:sim}
-----------
To support our scenario for the origin of the optical arc, we ran 2D axisymmetric simulations using the <span style="font-variant:small-caps;">pion</span> code (Mackey & Lim 2010; Mackey 2012). The calculations solve the Euler equations of hydrodynamics and are accurate to second order in space and time. In addition, the non-equilibrium ionization fraction of hydrogen and heating and cooling processes are also calculated, mediated by the ionizing (EUV) and non-ionizing UV (FUV) radiation field that is calculated by a raytracing scheme. A uniform grid was used with cylindrical coordinates $z\in[-0.512,0.512]$pc and $R\in[0,0.512]$pc resolved by 1536 and 768 grid zones, respectively, for a grid resolution of $5\times10^{-4}$pc. Rotational symmetry around $R=0$ is assumed, and the simulations used the same general setup as Mackey et al. (2015).
We consider a single stellar source (a B1 star), with $T_{\rm eff}=24$kK, using the wind and radiation properties in Table\[tab:krt\]: $\dot{M}=4\times10^{-10} {{\rm\,M_\odot}\, {\rm yr}^{-1}}$, $v_\infty=1700 \, {{\rm\,km\,s^{-1}}}$, $Q_0=2.40\times10^{45}$s$^{-1}$ and $Q_{\rm FUV}=2.45\times10^{47}$s$^{-1}$. The FUV radiation heats the neutral gas around the star through photoelectric heating on grains, following the implementation of Henney et al. (2009). A uniform ISM was set up around the star with a H number density of $n_{\rm H}$=100, 200 and 300cm$^{-3}$. In the following, however, we present only the results of simulations with $n_{\rm H}$=200cm$^{-3}$ (or a mass density of $\rho=4.68\times10^{-22}$gcm$^{-3}$) because they better match the observations. At $z>0.1$pc the ISM was set to be 5 times denser, to mimic the nearby molecular cloud (although the real structure of the cloud must be more complicated, with substructure and density gradients; see Section\[sec:bub\] and Fig.\[fig:hub\]). The gas is initially at rest and in pressure equilibrium, with $p/k_{\rm B}=1.1\times10^4\,$Kcm$^{-3}$, where $p$ is the gas pressure and $k_{\rm B}$ is the Boltzmann constant.
The evolution of this simulation is now described. The [H[ii]{} ]{}region expands rapidly in a spherical manner until it reaches the density jump at $z=0.1$pc. At this interface a shock is transmitted into the dense medium and a second shock is reflected back into the [H[ii]{} ]{}region. This reflected flow subsequently develops into a photoevaporation flow from the dense medium. The equilibrium temperature of the [H[ii]{} ]{}region is $T\approx6500$K because of the very soft spectrum of the B1 star. The stellar wind drives a hot, expanding bubble within the [H[ii]{} ]{}region that is initially spherical. Subsequently this bubble is impacted by the trans-sonic flow from the dense medium through the [H[ii]{} ]{}region and is deformed. This creates a compressed and overpressurised layer between the dense molecular cloud and wind bubble. The ISM displaced by the wind bubble also creates an overdense layer in all directions at the interface between wind bubble and [H[ii]{} ]{}region. These two overdense regions are the brightest emitters in H$\alpha$. A snapshot from this simulation is shown in Fig.\[fig:simsDT\], where gas density and temperature are plotted on logarithmic scales.
![Gas density (upper half-plane) and temperature (lower half-plane) for the 2D simulation of the wind bubble and [H[ii]{} ]{}region described in the text, after 0.018Myr of evolution. The units of density are $\log (\rho/{\rm g}\,{\rm cm}^{-3})$ and of temperature are $\log(T/{\rm K})$. This plot shows the simulated 2D plane, i.e., *not* projected onto the plane of the sky.[]{data-label="fig:simsDT"}](fig8.eps){width="45.00000%"}
Synthetic maps and comparison with observations {#sec:com}
-----------------------------------------------
Infrared emission from dust is calculated by postprocessing snapshots from the simulation using the [torus]{} code (Harris 2000, 2015; Kurosawa et al. 2015) following the method described in Mackey et al. (2016). A very similar approach using [torus]{} has also been used to model bow shocks around O stars (Acreman, Stevens & Harries 2016). Briefly, we use [torus]{} as a Monte-Carlo radiative transfer code to calculate the radiative equilibrium temperature of dust grains throughout the simulation, assuming radiative heating by the central star. The dust-to-gas ratio is assumed to be 0.01 in the ISM material (Draine et al. 2007), and we use a Mathis, Rumpl & Nordsieck (1977) grain-size distribution defined by the minimum (maximum) grain size $a_{\rm min}=0.005\,\mu$m ($a_{\rm max}=0.25\,\mu$m) and a power law index $q=3.3$. The calculations here are for spherical silicate grains (Draine & Lee 1984). [torus]{} calculates the dust temperature at all positions and then produces synthetic emission maps at different wavelengths from arbitrary viewing angles. Emission maps at 24$\mu$m are shown in Fig.\[fig:simsIR\] for 5 different viewing angles, and the observational image from [*Spitzer*]{} is shown on the same scale for comparison. The projection where the line-of-sight is at 60$\degr$ to the symmetry axis of the simulation provides a reasonable match to the observations, both in terms of morphology and absolute brightness. The exception is that the observational image seems to have emission from the position of the star itself, absent from our models.
{width="33.00000%"} {width="33.00000%"} {width="33.00000%"} {width="33.00000%"} {width="33.00000%"} {width="33.00000%"}
We also calculated H$\alpha$ emission from the simulation at a projection angle of 60$\degr$, because this angle was the best match to the 24$\mu$m data. We did this using [visit]{} (Childs et al. 2012) and, while the relative brightness of each pixel is correctly calculated, there may be some scaling offset in the overall normalization of the image, so the absolute brightness should be treated with caution. The result is plotted in Fig.\[fig:simsHA\], where the left-hand panel shows the emission map over the full range of brightnesses and the right-hand one shows only the brightest 10 per cent of the predicted emission, to mimic the case where the brightest emission is just above the noise level (see Section\[sec:arc\]). The brightest part of the simulated H$\alpha$ nebula is located within the [H[ii]{} ]{}region, about 10–15 arcsec from the star and well within the infrared bubble. This agrees very well with the observations in Fig.\[fig:neb\]. Furthermore, Fig.\[fig:Ha\] shows that the arc is about a factor of 2 brighter than the rest of the [H[ii]{} ]{}region in H$\alpha$ emission, comparable to what is shown in the left panel of Fig.\[fig:simsHA\].
Taken together, the agreement of the synthetic infrared and optical emission maps with the observational data is very encouraging. In particular, it suggests that H$\alpha$ imaging that is a factor of a few deeper would give a much clearer picture of the structure of this nebula, and strongly test our assertion that we are seeing a wind bubble and [H[ii]{} ]{}region from a main-sequence B star. The simulation setup is very simple, with little fine-tuning to match the observations. If further observations support the interpretation as a young stellar wind bubble, then more detailed 3D simulations including a clumpy or turbulent medium and a more realistic density structure of the molecular cloud would be clearly warranted. Particularly, one can expect that the presence of the denser material to the north-west of IRAS18153$-$1651 (hub–N) would result in a stronger photoevaporation flow from this direction, which would explain the observed offset of stars1 and 2 from the geometric centre of the mid-infrared bubble as well as from that of the optical arc (cf. Ngoumou et al. 2013).
\
\
Summary and conclusion {#sec:sum}
======================
In this paper, we have reported the study of a compact, almost circular mid-infrared nebula associated with the IRAS source 18153$-$1651 and located in the region of ongoing massive star formation M17SWex. The study was motivated by the discovery of an optical arc near the centre of the nebula, leading to the hypothesis that it might represent a bubble blown by the wind of a young massive star. To substantiate this hypothesis, we obtained optical spectra of the arc and two stars near its focus with the Gemini-South and the 3.5-m telescope in the Observatory of Calar Alto (Spain). The stars have been classified as main-sequence stars of spectral type B1 and B3, while the line ratios in the spectrum of the arc indicated that its emission is due to photoionization [**$<...>$**]{}. These findings allowed us to suggest that we deal with a recently formed low-mass ($\sim100 \, {\rm\,M_\odot}$) star cluster (with the two B stars being its most massive members) and that the optical arc and IRAS18153$-$1651 are, respectively, the edge of a wind bubble and the [H[ii]{} ]{}region produced by the B1 star. We also suggested that the one-sided appearance of the wind bubble is the result of interaction between the bubble and a photoevaporation flow from a molecular cloud located to the west of IRAS18153$-$1651 (the presence of such a cloud is evidenced by submillimetre and radio observations). We supported our suggestions by simple analytical estimates, showing that the radii of the bubble and the [H[ii]{} ]{}region would fit the observations if the age of IRAS18153$-$1651 is $\sim10\,000$yr and the number density of the ambient medium is $\sim100 \, {\rm cm}^{-3}$. These estimates were validated by two-dimensional, radiation-hydrodynamics simulations investigating the effect of a nearby dense cloud on the appearance of newly forming wind bubble and [H[ii]{} ]{}region. We found that synthetic H$\alpha$ and 24$\mu$m dust emission maps of our model wind bubble and [H[ii]{} ]{}region show a good match to the observations, both in terms of morphology and surface brightness.
To conclude, taken together our results strongly suggest that we have revealed the first example of a wind bubble blown by a main-sequence B star.
Acknowledgements
================
VVG acknowledges the Russian Science Foundation grant 14-12-01096. JM acknowledges funding from a Royal Society–Science Foundation Ireland University Research Fellowship (No. 14/RS-URF/3219). AYK acknowledges support from the National Research Foundation (NRF) of South Africa and the Russian Foundation for Basic Research grant 16-02-00148. TJH is funded by an Imperial College Junior Research Fellowship. This research was supported by the Gemini Observatory, which is operated by the Association of Universities for Research in Astronomy, Inc., on behalf of the international Gemini partnership of Argentina, Brazil, Canada, Chile, and the United States of America, and has made use of the NASA/IPAC Infrared Science Archive, which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration, the SIMBAD data base and the VizieR catalogue access tool, both operated at CDS, Strasbourg, France.
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[^1]: Based on observations obtained at the Gemini Observatory (processed using the Gemini [iraf]{} package), which is operated by the Association of Universities for Research in Astronomy, Inc., under a cooperative agreement with the NSF on behalf of the Gemini partnership: the National Science Foundation (United States), the National Research Council (Canada), CONICYT (Chile), Ministerio de Ciencia, Tecnología e Innovación Productiva (Argentina), and Ministério da Ciência, Tecnologia e Inovação (Brazil), and observations collected at the Centro Astronómico Hispano Alemán (CAHA), operated jointly by the Max-Planck Institut für Astronomie and the Instituto de Astrofisica de Andalucia (CSIC).
[^2]: Note that a runaway massive star can produce a short-living ($\sim10^4$ yr) circular shell during the advanced stages of evolution if it is surrounded by a dense material comoving with the star, i.e. the dense matter shed during the red supergiant phase, and if it is massive enough to evolve afterwards into a Wolf-Rayet star (Gvaramadze et al. 2009).
[^3]: [iraf]{}: the Image Reduction and Analysis Facility is distributed by the National Optical Astronomy Observatory (NOAO), which is operated by the Association of Universities for Research in Astronomy, Inc. (AURA) under cooperative agreement with the National Science Foundation (NSF).
[^4]: The stellar atmosphere code [fastwind]{} provides reliable synthetic spectra of O- and B-type stars taking into account non-local thermodynamic equilibrium effects in spherical symmetry with an explicit treatment of the stellar wind.
[^5]: We note that these estimates are very approximate and should be considered with caution.
|
---
abstract: 'A graviton of a nonzero mass and decay width propagates five physical polarizations. The question of interactions of these polarizations is crucial for viability of models of massive/metastable gravity. This question is addressed in the context of the DGP model of a metastable graviton. First, I argue that the well-known breakdown of a naive perturbative expansion at a low scale is an artifact of the weak-field expansion itself. Then, I propose a different expansion – the constrained perturbation theory – in which the breakdown does not occur and the theory is perturbatively tractable all the way up to its natural ultraviolet cutoff. In this approach the couplings of the extra polarizations to matter and their selfcouplings appear to be suppressed and should be neglected in measurements at sub-horizon scales. The model reproduces results of General Relativity at observable distances with high accuracy, while predicting deviations from them at the present-day horizon scale.'
---
[ NYU-TH-04/03/22]{}
0.9cm
**Weakly-coupled metastable graviton**
0.7cm
Gregory Gabadadze
0.3cm
*Center for Cosmology and Particle Physics*
*Department of Physics, New York University, New York, NY, 10003, USA*
1.9cm
Introduction
============
An idea that the observable acceleration of the Universe could be a result of large-distance modification of gravity is attractive, and is experimentally testable [@DDG]–[@Nima]. Moreover, large-distance modifications of gravity give rise to a conceptually new approach to the long-standing cosmological constant problem [@DGS; @ADDG]. Hence, development of the models of modified gravity becomes an important task.
The DGP model [@DGP] is a covariant theory of the large-distance modification of gravity (see, e.g., [@Dick]–[@Padila]). Interactions in this model are mediated by a single graviton that lives in infinite-volume five-dimensional space-time. This graviton resembles a massive 4D spin-2 state since it has five polarizations. Experimental constraints on extra light states with gravitational interactions are rather severe, therefore, the question of how these extra polarizations interact with observable matter and with themselves becomes crucial for the consistency of the model.
From a 4D perspective, the graviton behaves as a state of a nonzero decay width[^1]. As such it shares certain properties of a massive graviton (and massive non-Abelian gauge fields). If non-Abelian gauge fields are prescribed mass $m_V$ by hand (i.e., without using the Higgs mechanism), then the nonlinear amplitudes of the theory become strong at a scale $m_V/g$, where $g$ is a gauge-coupling constant. A similar effect exists in a theory of gravity in which mass $m_g$ is introduced by hand as in Ref. [@PF]. This was shown in Ref. [@Arkady], by calculating nonlinear corrections to a spherically symmetric static body of the gravitational radius $r_g$, in which case the weak-field approximation breaks down at a scale $\Lambda_m\sim m_g/ (r_gm_g)^{1/5}$ [@Arkady]. This breakdown is a universal property of the perturbative expansion in the massive theory as it can be understood by looking at tree-level trilinear graviton vertex diagrams [@DDGV]. A simplest trilinear (graviton)$^3$ diagram gives rise to the $1/m_g^4$ singularity [@DDGV]. For small $m_g$ this diagram is enhanced even though it is multiplied by an extra power of the Newton constant. This can also be understood in terms of interactions of the longitudinal polarizations becoming strong [@AGS]. For the pure gravitational sector itself the corresponding scale $\Lambda_m$ reduces to $m_g/ (m_g/\mpl)^{1/5}$, which can be made only as big as $m_g/ (m_g/\mpl)^{1/3}$ by adding higher nonlinear terms [@AGS].
The reason for the breakdown of perturbation theory at a low scale can be traced back to terms in the graviton propagator that contain products of the structure \[sing\] with similar structures or with the flat space metric. These terms do not manifest themselves in physical amplitudes at the linear level since they are multiplied by conserved currents, however, they enter nonlinear diagrams leading to the breakdown of perturbation theory for massive non-Abelian gauge fields or massive gravity [^2].
In the model of Ref. [@DGP] the graviton decay width is introduced by adding to General Relativity a term that is reparametrization invariant (see the action (\[1\])). As a result, the terms similar to (\[sing\]) in the propagator are gauge dependent. In a simple gauge adopted in Ref. [@DGP] they read as follows: , \[singDGP\] where $m_c\sim 10^{-33}~{\rm eV}$ is a counterpart of $m_g$ and $p\equiv \sqrt{p^2}$ is a root of the Euclidean momentum square. Because of the singularity in $m_c$ in (\[singDGP\]), perturbation theory breaks down precociously [@DDGV]. However, this breakdown is an artifact of an ill-defined perturbative expansion – the known exact solutions of the model have no trace of the breakdown scale (see Ref. [@DDGV]). This shows that if one sums up all the tree-level perturbative diagrams, then the breakdown scale should disappear. A more ambitious program is to asks for quantum consistency of the model as an effective theory with a cutoff. Such a consistency can only be established within the perturbation theory. The latter is ill-defined above the scale $m_g/ (m_g/\mpl)^{1/3}$ [@Luty] (see also [@Rubakov]), because of the same nonlinear diagrams that make the weak-field expansion ill-defined already at the classical level [@DDGV]. However, it is not clear whether the new scale that emerges in perturbation theory has any physical significance. The perturbative breakdown at the quantum level could be an artifact of the technical method itself. As was recently argued by Dvali [@Dvali], in a certain toy sigma model a similar breakdown takes place, however, the full resumed solution of the theory is valid well above the naive breaking scale.
In the next section I will discuss in detail the reasons for the breakdown of perturbation theory in the DGP model. Then, in section 3 I will show how one can [*define*]{} a different perturbative procedure such that the singular terms like (\[singDGP\]) are eliminated from the propagator and the UV behavior of the amplitudes is regular all the way up to the cutoff of the theory. The effective one-graviton exchange amplitude between two sources with the stress-tensors $T_{\mu\nu}$ that I will calculate in section 3 takes the form: =[1p\^2+m\_cp]{}( T\_\^2- [12]{}T\^2 [ p\^2+2m\_cp p\^2+3m\_cp]{}). \[Adgp0\] This amplitude interpolates between the four-dimensional behavior at $p\gg m_c$ and the five-dimensional behavior at $p\ll m_c$. It has no poles on a physical Riemann sheet and satisfies requirements of unitarity, analyticity and causality.
Notice that the naive perturbation theory in the DGP model has no strong coupling scale if the localized 5D Einstein-Hilbert term is used on the brane worldvolume [@GS] and in the “dielectric” regularization of the DGP model [@Massimo2]. In higher dimensional generalizations of the DGP model [@DG; @Massimo1], the perturbation theory is well-defined and there is no problem in the first place as was recently shown in [@GS]. The concern of the present work is the five-dimensional model with the four-dimensional induced term (see the action (\[1\]) below).
What’s wrong with perturbative expansion?
=========================================
A brief answer to the above question is as follows. The DGP model has two [*dimensionful*]{} parameters: the Newton constant $G_N$ and the graviton lifetime $m_c$. As a result, the naive perturbative expansion in powers of $G_N$ is contaminated by powers of $1/m_c$. Hence, for small values of $m_c$ perturbation theory breaks down for the unusually low value of the energy scale. The question is whether this breakdown is an artifact of perturbation theory, or it could be that the breaking scale is truly a physical scale at which the model needs certain UV completion. Depending on a concrete model at hand, the either of above two possibilities could be realized.
To study these issues in detail we consider the action of the DGP model [@DGP] S=[\^2 ]{} d\^4xR(g)+ [\^[3]{} ]{}d\^4xdy [R]{}\_[5]{}([|g]{}), \[1\] where $R$ and ${\cal R}_{5}$ are the four-dimensional and five-dimensional Ricci scalars, respectively, and $\m$ stands for the gravitational scale of the bulk theory. The analog of the graviton mass is $m_c= 2\m^3/\mpl^2$. The higher-dimensional and four-dimensional metric tensors are related as (x,y=0)g(x). \[bargg\] There is a boundary (a brane) at $y=0$ and ${\bf Z}_2$ symmetry across the boundary is imposed. The presence of the boundary Gibbons-Hawking term is implied to warrant the correct Einstein equations in the bulk. Matter fields are assumed to be localized on a brane and at low energies, that we observe, they do not escape into the bulk. Hence, the matter action is completely four-dimensional $S_M=\int d^4x L_M$. Our conventions are: $ \eta_{AB} =
{\rm diag}\, [+----]\,; A,B = 0,1,2,3,{\it 5}\,;\mu,\nu = 0,1,2,3.$
The simplest problem is to calculate the Green’s function $D^{\mu\nu;\alpha\beta}$ and the amplitude of interaction of two sources $T_{\mu\nu}$ on the brane \_[1-graviton]{} T\_D\^[;]{} T\_. \[A\] In order to perform perturbative calculations one has to fix a gauge. A simple way adopted in [@DGP] is to use harmonic gauge in the bulk $\partial^A h_{AB}= \partial_B h^C_C/2$. In this case the one-graviton exchange amplitude on the brane (in the momentum space) takes the form: \_[1-graviton]{}(p,y=0)=[T\^2\_[1/3]{} p\^2+m\_cp]{}, \[A13\] where we denote the Euclidean four-momentum square by $p^2$: p\^2-p\^p\_=-p\_0\^2+p\_1\^2+ p\_2\^2+p\_3\^2p\_4\^2+p\_1\^2+ p\_2\^2+p\_3\^2, \[psquare\] and T\^2\_[1/3]{} 8G\_N ( T\^2\_-[13]{}TT ). \[T1third\] We can see that ${\cal A}_{\rm 1-graviton}$ in (\[A13\]) is non-singular in the $m_c\to 0$ limit (moreover, the pole in the amplitude is on a nonphysical Riemann sheet, see discussions in the next section.). However, the gauge dependent part of the propagator $D^{\mu\nu;\alpha\beta}$ contains terms proportional to (\[singDGP\]). These terms do not enter the one-graviton exchange amplitude, but, they do contribute to higher-order tree-level non-linear diagrams which blow up in the $m_c\to 0$ limit [@DDGV].
The same fact is reflected in the expression for the trace of $h_{\mu\nu}\equiv g_{\mu\nu}-\eta_{\mu\nu}$ which in the harmonic gauge takes the form [@DGP]: \^\_(p, y=0) =-[T 3m\_cp]{}. \[trace\] (Hereafter the tilde-sign denotes Fourier-transformed quantities and we put $8\,\pi\,G_N\,=1$.). From this expression we learn that: (i) ${\tilde h}^\mu_\mu $ is a propagating field in this gauge; (ii) ${\tilde h}^\mu_\mu $ propagates as a 5D field, i.e., it does not see the brane kinetic term; (iii) The expression for ${\tilde h}^\mu_\mu $ is singular in the limit $m_c\to 0$. The gauge dependent part of the momentum-space propagator ${\tilde D}(p,y)$ contains the terms $p_\mu \,p_\nu {\tilde h} $, which, due to (\[trace\]), give rise to the singular term (\[singDGP\]). Hence, to understand the origin of the breakdown of perturbation theory, one should look at the origin of the $1/m_c$ scaling in (\[trace\]).
The singular behavior of ${\tilde h}^\mu_\mu $ is a direct consequence of the fact that the four-dimensional Ricci curvature $R(g)$ in the linearized approximation is forced to be zero by the $\{55\}$ and/or $\{\mu 5\}$ equations of motion. This can be seen by direct calculation of $R$ and of those equations, but it is more instructive to see this by using the ADM decomposition. The $\{55\}$ equation reads: R=(K\^\_)\^2-K\_\^2, \[ADM\] where $K_{\mu\nu}$ denotes the extrinsic curvature. Since $K\sim {\cal O} (h)$ the above equation implies that the four-dimensional curvature $R\sim {\cal O} (h^2)$ and in the linearized order $R$ vanishes. Let us now see how this leads to the singular behavior of $h$ in (\[trace\]). The junction condition across the brane contains two types of terms: there are terms proportional to $m_c$ and there are terms that are independent of $m_c$. The former come from the bulk Einstein-Hilbert action while the latter appear due to the worldvolume Einstein-Hilbert term. In the trace of the junction condition the $m_c$–independent term is proportional to the four-dimensional Ricci scalar $R$. On the other hand, as we argued above, $R$ has no linear in $h$ term in the weak-field expansion, simply because these terms cancel out due to the $\{55\}$ and/or $\{\mu 5\}$ equations. Therefore, in the linearized approximation the junction condition contains only the terms that come from the bulk. These terms are proportional to $m_c h$. This inevitably leads to the trace of $h$ (\[trace\]) that is singular in the $m_c\to 0$ limit and triggers the breakdown of the perturbative approach as discussed above.
The above arguments suggest that the two limiting procedures, first truncating the small $h$ expansion and only then taking the $m_c\to 0$ limit, do not commute with each other. Therefore, the right way to perform the calculations is either to look at exact solutions of classical equations of motion, as was argued in [@Arkady; @DDGV], or to retain at least quadratic terms in the equations. The obtained results won’t be singular in the $m_c\to 0$ limit.
However, neither of the above approaches addresses the issue of quantum gravitational loops. Since the loops can only be calculated within a well-defined perturbation theory, one needs to construct a new perturbative expansion that would make diagrams tractable at short distances.
In the next section we will propose to rearrange perturbation theory in such a way that the consistent answers be obtained in the weak-field approximation. This can be achieved if the linearized gauge-fixing terms can play the role similar to the nonlinear terms. We will see that this requires a certain nontrivial procedure of gauge-fixing and choosing of appropriate boundary conditions.
Constrained perturbative expansion
==================================
Below we develop a perturbative approach that allows to perform calculations in the weak-field approximation without breaking the expansion at a low scale.
We recall that in the DGP model the boundary (the brane) breaks explicitly translational invariance in the $y$ direction, as well as the rotational symmetry that involves the $y$ coordinate. However, this fact is not reflected in the linearized approximation – the linearized theory that follow from (\[1\]) is invariant under five-dimensional reparametrizations[^3]. This line of arguments suggests to introduce constraints in the linearized theory that would account for the broken symmetries. It is clear that an arbitrary set of such constraint cannot be consistent with equations of motion with boundary conditions on the brane and at $y\to \infty$. However, by trial and error a consistent set of constraints and gauge conditions can be found. Below we introduce this set of equations step by step. We start by imposing the following condition: B\_ \_h\_[55]{} + \^h\_ =0 . \[bmu\] Furthermore, to make the kinetic term for the $\{\mu 5\}$ component invertible we set a second condition: B\_5\^h\_[5]{}=0. \[b5\] At a first sight, the two conditions (\[bmu\]) and (\[b5\]) fix all the $x$-dependent gauge transformations and make the gauge kinetic terms non-singular and invertible. However, at a closer inspection this does not appear to be satisfactory. One can look at the $\{\mu\nu \}$ component of the equations of motion and integrate this equation w.r.t. $y$ from $-\epsilon $ to $\epsilon$, with $\epsilon \to 0$. After the integration, all the terms with $B_\mu$ and $B_5$ vanish. The resulting equation (which is just the Israel junction condition) taken by its own, is invariant under the following four-dimensional transformations h\^\_(x\^, y)|\_[y=0]{}=h\_(x, y)|\_[y=0]{}+ \_\_|\_[y=0]{}+ \_\_|\_[y=0]{}. \[branetransf\] This suggests that in the $m_c\to 0$ limit the gauge kinetic term on the brane is not invertible. As a result, the problem of a precocious breakdown of perturbation theory discussed in the previous section arises. To avoid this difficulty one can introduce the following term on the brane worldvolume: S - \^2d\^4x dy (y)( \^h\_ - [12]{} \_h\^\_ )\^2. \[gauge\] This makes the graviton kinetic term of the brane invertible even in the $m_c \to 0$ limit. At this stage, the partition function can be [*defined*]{} as follows: Z\_[gf]{}=[lim]{}\_[,0]{} dh\_[AB]{} [exp]{} i ( S + S+ \^3d\^4x dy {[B\_5\^22]{} + [B\_\^2 2]{} } ). \[zgf\] Here $S$ and $\Delta S$ are given in (\[1\]) and (\[gauge\]) respectively, and the limit $\alpha,\gamma\to 0$ enforces (\[bmu\]) and (\[b5\]). Before proceeding further, notice that Eqs. (\[bmu\]) and (\[b5\]) would have been just gauge-fixing conditions if the boundary were absent (e.g., in a pure 5D theory with no brane). However, in the present case, the above equations, when combined with the junction condition across the brane, enforce certain boundary conditions on the brane. Therefore, Eqs. (\[bmu\]) and (\[b5\]) do more than gauge-fixing, and $\gamma$ and $\alpha $ cannot be regarded as gauge fixing parameters. The prescription given by (\[zgf\]) is to calculate first all Green’s functions and then take the limit $\alpha, \gamma \to 0$. Because of this, the results of the present calculations differ from [@DGP] where other boundary conditions were implied.
Using (\[zgf\]) we calculate below the propagator $D$ and the amplitude ${\cal A}$ defined in (\[A\]). We will see that there are no terms in $D$ that blow up as $m_c\to 0$.
We start with the equations of motion that follow from (\[zgf\]). The $\{\mu\nu\}$ equation on the brane reads &[m\_c2]{}& \_[-]{}\^[+]{}dy ( \_D\^2h\_- \_\_D\^2h\^\_ +\_\_5 h\_[5]{}+ \_\_5 h\_[5]{} - 2 \_ \^\_5 h\_[5]{} )+ G\_\^[(4)]{}\
&-& (\_\_h\_ +\_\_h\_ - \_\_h\^\_-\_ \_\_h\^ +[12]{} \_ \_4\^2 h\^\_) = T\_, \[munubrane\] where $\epsilon \to 0$, $\partial_D^2\equiv \partial_A\partial^A$, $\partial_4^2\equiv \partial_\mu\partial^\mu$. In (\[munubrane\]) we retained only terms that are nonzero in the $\epsilon \to 0$ limit. Furthermore, $G_{\mu\nu}^{(4)}$ denotes the 4D Einstein tensor: G\^[(4)]{}\_= \_4\^2h\_- \_ \_ h\^\_- \_\_ h\^\_+ \_\_h\^\_- \_ \_4\^2h\^\_+ \_\_\_h\^. \[G4\] The $\{\mu\nu\}$ equation in the bulk takes the form: & \_D\^2h\_& -\_\_D\^2h\^\_- \_\^h\_ - \_\^h\_ + \_\_h\^\_\
&+& \_\_\_h\^ +\_ \_4\^2h\_[55]{}-\_\_h\_[55]{} +\_\_5 h\_[5]{}\
&+& \_\_5 h\_[5]{} -2 \_ \^\_5 h\_[5]{}-[1]{} ( \_\_h\_[55]{}+ \_\^h\_)= 0. \[munubulk\] As a next step we turn to the $\{\mu 5 \}$ equation which reads as follows: \_4\^2 h\_[5]{} -\_\^h\_[5]{} -\_5 (\^h\_ -\_h\^\_) - [1]{} (\_\^h\_[5]{})=0. \[mu5\] Finally, the $\{5 5 \}$ equation takes the form \^2\_4 h\^\_-\_\_h\^ -[1]{} (\^2\_4h\_[55]{} + \_\_h\^) =0. \[55\] After the calculation is done the limit $\alpha, \gamma \to 0$ should be taken. We turn to the momentum space w.r.t. four worldvolume coordinates: \_[AB]{}(x, y) = d\^4p e\^[ipx]{} [h]{}\_[AB]{}(p, y). \[mom\] From the above equations we calculate the response of gravity to the source $T_{\mu\nu}$. In the limit $\alpha, \gamma \to 0$ the results are as follow: \_(p, y) [1p\^2+m\_cp]{}( T\_- [12]{}\_T [ p\^2+2m\_cp p\^2+ 3m\_cp]{})e\^[-p|y|]{}. \[hmunu\] We note that in this expression there are no terms similar to (\[singDGP\]), unlike to what happens in the harmonic gauge [@DGP] where the singular terms are present.
For the off-diagonal components we find that ${\tilde h}_{\alpha 5}\sim \gamma \,p_\alpha $, and \_[5]{}(p, y) 0. \[hmu5\] Finally, \_[55]{} - [r2]{} [h]{}\^\_ [r2]{}[Tp\^2+3m\_cp]{}e\^[-p|y|]{}, \[h55\] with $r\equiv (p^2+2m_c p)/(p^2+m_cp)$. The amplitude on the brane, as was already stated in (\[Adgp0\]), takes the form \_[1-graviton]{}(p, y=0) =[1p\^2+m\_cp]{}( T\_\^2- [12]{}T\^2 [ p\^2+2m\_cp p\^2+3m\_cp]{}). \[Adgp\] A remarkable property of this amplitude is that it interpolates between the 4D behavior at $p\gg m_c$ \_[4D]{}(p, y=0) [1p\^2 ]{}( T\_\^2- [12]{}T\^2 ), \[A4D\] and the 5D amplitude at $p\ll m_c$ \_[5D]{}(p, y=0) [1m\_cp ]{}( T\_\^2- [13]{}T\^2 ). \[A5D\] This amplitude has no vDVZ discontinuity [@Iwa; @vdv; @Zakharov].
It is instructive to rewrite the amplitude (\[Adgp\]) in the following form: \_[1-graviton]{} =[T\^2\_[1/2]{} p\^2+m\_cp]{} +[16]{}T\^2 [g(p\^2)p\^2+m\_cp]{}, \[Adgpsplit\] where T\^2\_[1/2]{} ( T\^2\_-[12]{}TT ), \[T1half\] and g(p\^2) [3m\_cp p\^2+3m\_cp]{}. \[gp\] The first term on the r.h.s. of (\[Adgpsplit\]) is due to two transverse polarizations of the graviton, while the second term is due to an extra scalar polarization. The scalar acquires a momentum-dependent form-factor. The form-factor is such that at sub-horizon distances, i.e., when $p\gg m_c$, the scalar decouples. At these scales the effects of the extra polarization is suppressed by a factor $m_c/p$ (e.g., in the Solar system this is less than $10^{-13}$). However, the scalar polarization kicks in at super-horizon scales, $p\ll m_c$, where the five dimensional laws or gravity are restored.
Let us discuss the above results in more detail. For this we study the pole structure of the amplitude (\[Adgpsplit\]). There are two nontrivial poles p\^2=-m\_cp, [and]{} p\^2=-3m\_cp. \[poles\] Let us find the positions of these poles on a complex plane of the Minkowskian momentum square $p_\mu^2$, where $p^2=p_\mu^2 {\rm exp}{(-i\pi)}$. For this we note that there is a branch cut from zero to plus infinity on the complex plane (see Fig. 1). The pole at $p^2=0$ is just the origin of the branch cut. Because of the cut the complex plane has many sheets (the propagator is multivalued function due to the square root in it). It is straightforward to show that both of the poles in (\[poles\]) are on a [*non-physical*]{}, second Riemann sheet. Moreover, the positions of these poles are far away from the branch cut (usual particle physics resonances appear on non-physical sheets close to the branch cut, the above poles, however, are located on a negative semi-axis of the second Riemann sheet). Hence, the physical Riemann sheet is pole free[^4]. The poles on a nonphysical sheet correspond to metastable states that do not appear as [*in*]{} and [*out*]{} states in the S-matrix [@Veltman]. Using the contour of Fig. 1 that encloses the plane with no poles, and taking into account the jump across the cut, the four-dimensional Källen-Lehmann representation can be written for the amplitude (\[Adgpsplit\]). The latter warrants four-dimensional analyticity, causality and unitarity of the amplitude (\[Adgp\])[^5]. Although the above interpretation is the only correct one, one could certainly adopt the following provisional picture that might be convenient for intuitive thinking. The second pole in (\[poles\]) can be interpreted as a “metastable ghost” with a momentum-dependent decay width that accompanies the fifth polarization and cancels its contributions at short distances. Remarkably, this state does not give rise to the usual instabilities because it can only appear in [*intermediate states*]{} in Feynman diagrams, but does not appear in the [*in*]{} and [*out*]{} states in the S-matrix elements. In this respect, it is more appropriate to think that the scalar graviton polarization acquires the form-factor $g(p)$ (\[gp\]).
The above results seem somewhat puzzling from the point of view of the Kaluza-Klein (KK) decomposition. Conventional intuition would suggest that the spectrum of the KK modes consists of massive spin-2 states. The Källen-Lehmann representation for the amplitude as a sum w.r.t. these massive states would give rise to the tensorial structure where the first term on the r.h.s. of (\[Adgpsplit\]) is proportional to $T^2_{1/3}$, instead of $T^2_{1/2}$. In this case, the remaining part of the amplitude on the r.h.s. would have a [*negative*]{} sign. This might be thought of as a problem. However, this is not so. The crucial difference of the present approach from the conventional KK theories is that the effective 4D states are mixed states of an infinite number of tensor and scalar modes. What is responsible for the mixing between the different spin states is the brane-induced term and the present procedure of imposing the constraints. In the covariant gauge that we discuss the trace of $h$ propagates and mixes with tensor fields. From the KK point of view this would look as an infinite tower of states with wrong kinetic terms. However, at least in the linearized approximation, the trace is a gauge artifact (similar to the zeroth component of the gauge field in covariantly gauge-fixed QED or QCD). Nevertheless, the effect of the trace part is that the true physical eigenmodes do not carry a definite four-dimensional spin of a local four-dimensional theory (see also [@Massimo2]). Because of this there is no reason to split the amplitude (\[Adgpsplit\]) into the term that is proportional to $T^2_{1/3}$ and the rest.
The question of interactions of these states in the full nonlinear theory is not addressed in the present work. What happens with the diagrams in which the “metastable ghosts” propagate in the loops (the unitarity cuts of which should give production of these multiple states) remains unknown. However, since the theory possesses 4D reparametrization invariance, we expect that these questions will find answers similar to those of non-Abelian gauge fields. Further studies are being conducted to understand these issues.
Conclusions
===========
To summarize briefly, a new, [*constrained perturbative expansion*]{} was proposed. In this approach perturbation theory is well-formulated. The resulting amplitude interpolates between the 4D behavior at observable distances and 5D behavior at super-horizon scales. This is due to the scalar polarization of the graviton that acquires a momentum-dependent form-factor. As a result, the scalar decouples with high accuracy from the observables at sub-horizon distances.
The model can potentially evade the no-go theorem for massive/metastable gravity [@vdv], that states that for the cancellation of the extra scalar polarization one should introduce a ghost that would give rise to instabilities [@vdv; @DGP1; @DGP2]. In the present model, at least in the linearized approximation, such instabilities do not occur. The convenient (although not precise) picture is to think of a “metastable ghost” that exists only as an intermediate state in Feynman diagrams which does not appear in the final states at least in the linearized theory. Since this state cannot be emitted in physical processes, it does not give rise to the usual instability. The latter property is similar to the observation made in the “dielectric” regularization of the DGP model in [@Massimo2].
The questions that remain open concern the gauge-fixing and interactions in the full non-linear theory where the Faddeev-Popov ghosts are expected to play a crucial role. These issues will be addressed elsewhere.
I would like to thank Gia Dvali and Massimo Porrati for valuable comments, and Max Libanov for useful critical remarks.
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[^1]: In 6D or higher [@DG], it behaves as a massive graviton with a decay width [@Wagner; @Kiritsis; @DHGS].
[^2]: Unfortunately, the massive gravity in 4D [@PF] is an unstable theory [@Deser] with an instability time scale that can be rather short [@GG].
[^3]: If instead of the boundary we consider a dynamical brane of a nonzero tension, then the five-dimensional Poincare symmetry is nonlinearly realized and one has to include a Nambu-Goldstone mode on the brane.
[^4]: Because of the $\sqrt{p^2}$ dependence of the propagator there are two choices of the sign of the square root. We choose the sign as above. For this choice the poles do not appear on the physical sheet and Euclidean Green’s functions decay for large $y$. However, an opposite choice of the sign of $\sqrt{p^2}$ can also be adopted. This would correspond to a different branch of the theory. On that branch, if we insist on flat brane, we find tachyonic poles on a physical Riemann sheet. This indicates that Minkowski space on that branch is unstable. The unstable classical solutions found in Ref. [@Luty] do precisely correspond to this choice of the sign of the square root. On that branch one can also obtain the selfaccelerated solution without introducing the cosmological constant [@D; @DDG]. This branch is decoupled (at least classically) from the branch that we are discussing in this work.
[^5]: [*A priori*]{} it is not clear why the theory that is truly higher-dimensional at all scales should have respected 4D analyticity and causality.
|
---
abstract: |
While greedy algorithms have long been observed to perform well on a wide variety of problems, up to now approximation ratios have only been known for their application to problems having *submodular* objective functions $f$. Since many practical problems have non-submodular $f$, there is a critical need to devise new techniques to bound the performance of greedy algorithms in the case of non-submodularity.
Our primary contribution is the introduction of a novel technique for estimating the approximation ratio of the greedy algorithm for maximization of monotone non-decreasing functions based on the curvature of $f$ without relying on the submodularity constraint. We show that this technique reduces to the classical $(1 - 1/e)$ ratio for submodular functions. Furthermore, we develop an extension of this ratio to the adaptive greedy algorithm, which allows applications to non-submodular stochastic maximization problems. This notably extends support to applications modeling incomplete data with uncertainty.
author:
- 'J. David Smith'
- 'My T. Thai'
bibliography:
- 'kdd17.bib'
title: |
Deterministic & Adaptive Non-Submodular Maximization\
via the Primal Curvature
---
Introduction
============
It is well-known that greedy approximation algorithms perform remarkably well, especially when the traditional ratio of $(1 - 1/e) \approx 0.63$ [@nemhauser_analysis_1978] for maximization of [*submodular*]{} objective functions is considered. Over the four decades since the proof of this ratio, the use of greedy approximations has become widespread due to several factors. First, many interesting problems satisfy the property of *submodularity*, which states that the marginal gain of an element never increases. If this condition is satisfied, and the set of possible solutions can be phrased as a uniform matroid, then one of the highest general-purpose approximation ratios is available “for free” with the use of the greedy algorithm. Second, the greedy algorithm is exceptionally simple both to understand and to implement.
A concrete example of this is the *Influence Maximization* problem, to which the greedy algorithm was applied with great success – ultimately leading to an empirical demonstration that it performed near-optimally on real-world data [@li_why_2017]. Kempe et al. showed this problem to be submodular under a broad class of influence diffusion models known as *Triggering Models* [@kempe_maximizing_2003]. This led to a number of techniques being developed to improve the efficiency of the sampling needed to construct the problem instance (see e.g. [@borgs_maximizing_2014; @tang_influence_2015; @nguyen_stopandstare_2016] and references therein) while maintaining a $(1 - 1/e - \epsilon)$ ratio as a result of the greedy algorithm. This line of work ultimately led to a $(1 - \epsilon)$-approximation by taking advantage the dramatic advances in sampling efficiency to construct an IP that can be solved in reasonable time [@li_why_2017]. In testing this method, it was found that greedy solutions performed near-optimally – an unexpected result given the $1 - 1/e$ worst-case.
For non-submodular problems, no general approximation ratio for greedy algorithms is known. However, due to their simplicity they frequently see use as simple baselines for comparison. On the Robust Influence Maximization problem proposed by He & Kempe, the simple greedy method was used in this manner [@he_robust_2016]. This problem consists of a non-submodular combination of Influence Maximization sub-problems and aims to address uncertainty in the diffusion model. Yet despite the non-submodularity of the problem, the greedy algorithm performed no worse than the bi-criteria approximation [@he_robust_2016].
Another recent example of this phenomena is the socialbot reconnaissance attack studied by Li et al. [@li_privacy_2016]. They consider a minimization problem that seeks to answer how long a bot must operate to extract a certain level of sensitive information, and find that the objective function is (adaptive) submodular only in a scenario where users disregard network topology. In this scenario, the corresponding maximization problem, Max-Crawling, has a $1 - 1/e$ ratio due to the work of Golovin & Krause [@golovin_adaptive_2011]. However, this constraint does not align with observed user behaviors. They give a model based on the work of Boshmaf et al. [@boshmaf_socialbot_2011], who observed that the number of mutual friends with the bot strongly correlates with friending acceptance rate. Although this model is no longer adaptive submodular, the greedy algorithm still exhibited excellent performance. Thus we see that while submodularity is *sufficient* to imply good performance, it is is not *necessary* for the greedy algorithm to perform well.
This, in turn, leads us to ask: is there any tool to theoretically bound the performance of greedy maximization with non-submodularity? Unfortunately, this condition has seen little study. Wang et al. give a ratio for it in terms of the worst-case rate of change in marginal gain (the *elemental curvature* $\alpha$) [@wang_approximation_2014]. This suffices to construct bounds for non-submodular greedy maximization, though for non-trivial problem sizes they quickly approach 0. We note, however, that the $\alpha$ ratio still encodes strong assumptions about the worst case: that the global maximum rate of change can occur an arbitrary number of times.
Motivated by the unlikeliness of this scenario, our proposed bound instead works with an estimate of how much change can occur during the $k$ steps taken by the greedy algorithm.
The remainder of this paper is arranged as follows: First, we briefly cover the preliminary material needed for the proofs and define the class of problems to which they apply (Sec. \[sec:prelim\]). We next define the notion of curvature used and develop a proof of the ratio based on it, with an extension to adaptive greedy algorithms, and show it is equivalent to the traditional $1 - 1/e$ ratio for submodular objectives (Sec. \[sec:ratio-theory\]), and conclude with a reflection on the contributions and a discussion of future work (Sec. \[sec:conclusion\]).
**Contributions.**
- A technique for estimating the approximation ratio of greedy maximization of non-submodular monotone non-decreasing objectives on uniform matroids.
- An extension of this technique to adaptive greedy optimization, where future greedy steps depend on the success or failure of prior steps.
|
---
abstract: 'We determine all values of the parameters for which the cell modules form a standard system, for a class of cellular diagram algebras including partition, Brauer, walled Brauer, Temperley-Lieb and Jones algebras. For this, we develop and apply a general theory of finite dimensional algebras with Borelic pairs. The theory is also applied to give new uniform proofs of the cellular and quasi-hereditary properties of the diagram algebras and to construct quasi-hereditary 1-covers, in the sense of Rouquier, with exact Borel subalgebras, in the sense of König. Another application of the theory leads to a proof that Auslander-Dlab-Ringel algebras admit exact Borel subalgebras.'
author:
- Kevin Coulembier
- Ruibin Zhang
title: Borelic pairs for stratified algebras
---
Introduction
============
The theory of quasi-hereditary algebras was initiated by Cline, Parshall and Scott in [@CPS; @Scott] and provided a unified framework for studying modular representation theory of semisimple algebraic groups and the BGG category ${\mathcal O}$. A central role in this theory is played by the [*standard modules*]{} of quasi-hereditary algebras. Cellular algebras were introduced by Graham and Lehrer in [@CellAlg] and include many [*diagram algebras*]{} such as Brauer, Iwahori-Hecke and BMW algebras. These algebras are of central importance in representation theory and low dimensional topology. Cellular algebras have a class of natural modules, known as the [*cell modules*]{}. For many values of their parameters, Brauer and BMW algebras are also quasi-hereditary and then the standard modules coincide with the cell modules. This remains true for a range of cellular algebras.
In [@Nakano1; @Nakano2], it is proved that, in most cases, the cell modules of Iwahori-Hecke algebras of type $A$ behave as the standard modules of some quasi-hereditary algebra, even though the Iwahori-Hecke algebra is itself not quasi-hereditary. This is formulated into the statement that “the cell modules form a [*standard system*]{}”, see [@DR], and is equivalent to the condition that the algebra admits a [*cover-Schur algebra*]{}, in the sense of [@HHKP; @Rou]. The result in [@Nakano1] extends to Brauer algebras, by [@Paget], to partition and BMW algebras, by [@HHKP], and to Iwahori-Hecke algebras of type $B$, by [@Rou]. An important consequence is that multiplicities in cell filtrations are well-defined.
In the present paper, we determine when the cell modules form a standard system for a variety of diagram algebras: partition, Brauer, walled Brauer, Temperley-Lieb and Jones algebras. In the cases of the Brauer and partition algebras, this reproduces (and completes) the corresponding results of [@Paget; @HHKP]. We will achieve this by developing a construction which is in part inspired by the theory of exact Borel subalgebras.
In [@Koenig], König introduced the notion of an [*exact Borel subalgebra*]{} of a quasi-hereditary algebra, inspired by the Borel subalgebra of a semisimple Lie algebra. When an exact Borel subalgebra exists, the standard modules are given explicitly as the modules induced from the simple modules of that subalgebra. Not every quasi-hereditary algebra admits an exact Borel subalgebra. However, it was proved in [@KKO] that every quasi-hereditary algebra over an algebraically closed field is Morita equivalent to a (quasi-hereditary) algebra admitting an exact Borel subalgebra. Although the proof is constructive, it would not be trivial to apply it to construct the Morita equivalent algebra and its exact Borel subalgebra for a given quasi-hereditary algebra. In fact, few explicit examples of Morita equivalent algebras with exact Borel subalgebras for prevalent quasi-hereditary algebras are known, see [*e.g.*]{} [@Koenig; @PSW].
We will define a generalisation of the concept of exact Borel subalgebras, [*viz.*]{} [*Borelic pairs $(B,H)$ of arbitrary algebras*]{} $A$, and develop the theory of algebras with such pairs. We prove that key structural properties of $A$ are determined by those of $H$. For instance, if $H$ is semisimple then $A$ is quasi-hereditary, and, roughly speaking, cellularity of $H$ implies cellularity of $A$. Furthermore, in the latter case, the cell modules of $A$ form a standard system if and only if this is the case for $H$. These and further applications are more precisely discussed in Section \[IntroBP\].
We use this general theory to study the diagram algebras mentioned above, in particular for (i) determining when the cell modules form standard systems and (ii) constructing Morita equivalent algebras with exact Borel subalgebras. For any of the diagram algebras $A$, we construct an algebra $C$, satisfying a double centraliser property with $A$. The algebra $C$ also admits a Borelic pair, which allows to prove that it is cellular, determine when the cell modules form a standard system, and find out when it is quasi-hereditary. In the latter case, $C$ admits an exact Borel subalgebra. In most cases, $C$ is Morita equivalent to $A$ and in the remaining cases, it is a $1$-faithful cover in the sense of [@Rou]. In some cases, $C$ constitutes a previously unknown ‘Schur algebra’. To the best of our knowledge, this is the first purely diagrammatic description of a Schur algebra. Aside from the new results on the algebras $A$, our constructions also yields a new proof of their cellularity and quasi-heredity.
Standardly based and base stratified algebras
---------------------------------------------
In [@JieDu], Du and Rui introduced a generalisation of cellular algebras, called [*standardly based algebras*]{}. We will prove that any algebra over an algebraically closed field can be given at least one standardly based structure. Standardly based algebras also come with cell modules. When the standardly based algebra is cellular, the two types of cell modules coincide.
In [@HHKP], Hartman [*et al.*]{} defined the concept of [*cellularly stratified algebras*]{}, giving a powerful tool to determine when cell modules form a standard system. We introduce the corresponding weaker notion of [*base stratified algebras*]{}, which is a significant simplification that is better compatible with the notion of Borelic pairs, and actually suffices for studying when the cell modules of a cellular algebra form a standard system. This generalisation will also allow the periplectic Brauer algebra, which is a non-cellular diagram algebra, to be studied using our techniques, see [@peri].
\[tabb\]
$$\xymatrix{
&& *+[F]\txt{stratified} \\
*+[F]\txt{standardly\\ based} &*+[F]\txt{exactly stratified}\ar[ur]&&*+[F]\txt{standardly stratified}\ar[ul]\\
&&*+[F]\txt{ exactly\\ standardly stratified}\ar[ul]\ar[ur]&*+[F]\txt{strongly\\ standardly stratified}\ar[u]\\
*+[F]\txt{base\\stratified}\ar[uu]\ar[urr]&&*+[F]\txt{properly stratified}\ar[u]\ar[ur]\\
&&*+[F]\txt{quasi-hereditary}\ar[u]\ar[ull]
}$$
Overview of the structures on (finite dimensional) algebras
-----------------------------------------------------------
The different structures on algebras we study are summarised in Figure 1. [*Properly stratified algebras*]{} were introduced in [@Dlab; @Kluc]. The notion of [*(standardly) stratified algebras*]{} were introduced in [@CPSbook Chaper 2]. The notion of [*strongly standardly stratified algebras*]{} was studied in [@ADL], although the term“standardly stratified” was used, leading to inconsistency with [@CPSbook]. The term “strongly standardly stratified algebras” was coined in [@Frisk], which also introduced [*exactly standardly stratified algebras*]{}, as “weakly properly stratified algebras”.
An arrow means that one structure implies the other one. For arrows between different versions of standardly stratified algebras, the (proper) standard modules remain the same. The arrow from quasi-hereditary to base stratified only holds when the field is algebraically closed and in this case the standard modules are mapped to the cell modules. A general base stratified algebra has standard modules, cell modules and proper standard modules. The first type of modules are different from the other two, unless the algebra is quasi-hereditary.
Borelic pairs {#IntroBP}
-------------
We generalise the notion in [@Koenig] of exact Borel subalgebras from quasi-hereditary algebras to exactly standardly stratified algebras, where the generalisation to properly stratified algebras was already introduced by Klucznik and Mazorchuk in [@Kluc].
We further generalise this theory by introducing the notion of [*Borelic pairs*]{} $(B,H)$ of an arbitrary finite dimensional algebra $A$. We prove that, when $A$ admits a Borelic pair $(B,H)$, it is standardly stratified. Moreover, if $H$ is quasi-local, $A$ is strongly standardly stratified. We also consider a stronger notion of [*exact Borelic pairs*]{}, leading to exactly standardly stratified algebras. Then we find that, if $H$ is quasi-local (resp. semisimple) $A$ is properly stratified (resp. quasi-hereditary) and in each case $B$ is an exact Borel subalgebra. If $H$ is standardly based, $A$ is base stratified, so in particular standardly based. Furthermore, the cell modules of $H$ form a standard system if and only the standard modules of $A$ form a standard system.
Overview of main applications
-----------------------------
Since we will almost exclusively deal with [*finite dimensional, unital, associative*]{} algebras, we don’t mention these characteristics in the following statements.
\[ThmA\] Over a perfect field ${\Bbbk}$, the Auslander-Dlab-Ringel algebra of a ${\Bbbk}$-algebra $R$ is quasi-hereditary with exact Borel subalgebra.
This will be proved in Theorem \[ThmAus\]. The weak assumption that ${\Bbbk}$ is perfect is not required for the quasi-heredity of ADR algebras, see [@DRAus], but our construction of the exact Borel subalgebra fails without it. This exact Borel subalgebra and the ones in Theorem \[ThmC\] seem unrelated to the construction in [@KKO], see e.g. Remark \[NotKKO\]. The quasi-heredity of ADR algebras has the following consequence, see Theorem \[AllBased\].
\[CorB\] Any algebra over an algebraically closed field has a standardly based structure.
The following theorem, which uses a total quasi-order $\preccurlyeq$ and a partial order $\le$ on the set of simple modules, will be proved in Theorems \[ThmDiagram1\], \[ThmMor\] and \[ThmCov\] and Corollary \[CorCell\].
\[ThmC\] Consider an arbitrary field ${\Bbbk}$ and a fixed $\delta\in{\Bbbk}$. Let $A$ be the partition algebra ${\mathcal{P}}_n(\delta)$, the Brauer algebra ${\mathcal B}_n(\delta)$, the walled Brauer algebra ${\mathcal B}_{r,s}(\delta)$, the Jones algebra $J_n(\delta)$ or the Temperley-Lieb algebra ${{\mathrm{TL}}}_n(\delta)$, with $n,r,s\in{\mathbb{N}}$, $n>1$, $r\ge 1$ and $s\ge 1$. Then
1. $A$ is cellular if $\mathscr{C}^3_A$;
2. $(A,\preccurlyeq)$ is exactly standardly stratified(\*) if $\mathscr{C}^1_A$;
3. $(A,\le)$ is quasi-hereditary if $\mathscr{C}^1_A$ and $\mathscr{C}^2_A$;
4. $A$ admits a cover $C$ which is:
- Morita equivalent to $A$ if $\mathscr{C}^1_A$;
- quasi-hereditary with exact Borel subalgebra if $\mathscr{C}^2_A$;
- always exactly standardly stratified(\*) with exact Borel subalgebra;
- cellular and base stratified if $\mathscr{C}^3_A$.
[ | l | l | l |l |]{}\
algebra $A$& $\mathscr{C}^1_A$& $\mathscr{C}^2_A$ & $\mathscr{C}^3_A$\
${\mathcal{P}}_n(\delta)$ & $\delta\not=0$ & ${{\rm{char}}}({\Bbbk})\not\in [2,n]$&$\emptyset$\
${\mathcal B}_n(\delta)$ & $\delta\not=0$ or $n$ odd & ${{\rm{char}}}({\Bbbk})\not\in [2,n]$&$\emptyset$\
$J_n(\delta)$ & $\delta\not=0$ or $n$ odd & $n$ odd and ${{\rm{char}}}({\Bbbk})\not\in [3,n]$ or & $x^i-1$ splits over ${\Bbbk}$,\
& &$n$ even and ${{\rm{char}}}({\Bbbk})\not\in \{2\}\cup[3,n/2]$& for $i\in\{n,n-2,\ldots\}$\
${{\mathrm{TL}}}_n(\delta)$ & $\delta\not=0$ or $n$ odd & $\quad\emptyset$&$\emptyset$\
${\mathcal B}_{r,s}(\delta)$ & $\delta\not=0$ or $r\not=s$ & ${{\rm{char}}}({\Bbbk})\not\in [2,\max(r,s)]$&$\emptyset$\
When $\mathscr{C}^1_A$ is satisfied, but not $\mathscr{C}^2_A$, the algebra $A$ is not quasi-hereditary, for any order.
(\*) In case $A=J_n(\delta)$, they are furthermore properly stratified.
Here and throughout the paper, $\emptyset$ represents the ‘empty’ condition. So condition $\emptyset$ is always satisfied.
When ${\mathcal{C}}^1_A$ is not satisfied, it follows from [@CellQua] that $A$ is not quasi-hereditary, for any order. Hence, we obtained a [*Morita equivalent algebra with exact Borel subalgebra for all cases when $A$ is quasi-hereditary*]{}. The quasi-heredity and cellularity of $A$ in Theorem \[ThmC\] were previously obtained by a variety of methods in [@AST1; @CDDM; @CellAlg; @CellQua; @partition; @Xi]. Here we find a new unified proof, which also constructs the exact Borel subalgebras. The quasi-heredity of $J_n(\delta)$ has previously only been stated, in [@CellQua Proposition 4.2], without the explicit conditions. Next we determine when the cell modules of $A$ form a standard system.
\[ThmD\] If the field ${\Bbbk}$ is algebraically closed, all algebras $A$ in Theorem \[ThmC\] are cellular. The cell modules form a standard system if and only if the following condition is satisfied:
[ | l | l | ]{}\
algebra $A$& condition\
${\mathcal{P}}_n(\delta)$ & $\delta\not=0$ if $n=2$; and $\begin{cases}{{\rm{char}}}({\Bbbk})\not\in\{2,3\}\qquad\mbox{or}\\{{\rm{char}}}({\Bbbk})=3\mbox{ and }n=2\end{cases}$\
${\mathcal B}_n(\delta)$ & $\delta\not=0$ if $n\in \{2,4\}$; and $\begin{cases}{{\rm{char}}}({\Bbbk})\not\in\{2,3\}\qquad\mbox{or}\\{{\rm{char}}}({\Bbbk})=3\mbox{ and }n=2\end{cases}$\
$J_n(\delta)$ & $\delta\not=0$ if $n\in \{2,4\}$; and $\begin{cases} {{\rm{char}}}({\Bbbk})\not\in [3,n] & \mbox{if~$n$ is odd} \\
{{\rm{char}}}({\Bbbk})\not\in\{2\}\cup[3,n/2]&\mbox{if~$n$ is even}\end{cases}$\
${{\mathrm{TL}}}_n(\delta)$ & $\delta\not=0$ if $ n\in \{2,4\}$\
${\mathcal B}_{r,s}(\delta)$ &$\delta\not=0$ if $(r,s)\in\{(1,1),(2,2)\}$; and $\begin{cases}{{\rm{char}}}({\Bbbk})\not\in\{2,3\}\qquad\mbox{or}\\{{\rm{char}}}({\Bbbk})=3\mbox{ and }\max(r,s)\le 2\end{cases}$\
This follows from Theorems \[ThmDiagram2\], \[ThmMor\] and \[ThmCellStan\]. The results for ${\mathcal B}_n(\delta)$ were previously obtained in [@Paget] and most of the claim for ${\mathcal{P}}_n(\delta)$ was proved in [@HHKP]. This theorem, together with [@DR Theorem 2] and Lemma \[LemSchur\], yields the following consequence.
\[CorE\] The cellular algebra $A$, under the condition in Theorem \[ThmD\], admits a Schur algebra in the sense of [@HHKP Definition 12.1] or Definition \[DefSchur\]. Moreover, the cell multiplicities in modules which admit cell filtrations are independent of the specific filtration.
When the condition ${\mathcal{C}}^1_A$ in Theorem \[ThmC\] is not satisfied and the cover $C$ is hence not Morita equivalent to $A$, in most cases it is still a $1$-faithful cover. This yields a quasi-hereditary $1$-cover in the sense of [@Rou] when $C$ is quasi-hereditary ([*i.e.*]{} when ${\mathcal{C}}^2_A$ is satisfied).
\[ThmF\] Consider an arbitrary field ${\Bbbk}$ and the cover $C$ of $A$ in Theorem \[ThmC\], then $C$ is a 1-faithful quasi-hereditary cover if $\mathscr{C}^2_A$ and the condition in the table is satisfied.
[ | l | l | ]{}\
algebra $A$& condition (along with ${\mathcal{C}}_A^2$)\
${\mathcal{P}}_n(\delta)$ & $n\not=2$ or $\delta\not=0$\
${\mathcal B}_n(\delta)$ & $n\not\in \{2,4\}$ or $\delta\not=0$\
$J_n(\delta)$ & $n\not\in \{2,4\}$ or $\delta\not=0$\
${{\mathrm{TL}}}_n(\delta)$ & $n\not\in \{2,4\}$ or $\delta\not=0$\
${\mathcal B}_{r,s}(\delta)$ & $(r,s)\not\in\{(1,1),(2,2)\}$ or $\delta\not=0$\
Under these conditions, the algebra $C$ is the Schur algebra predicted in Corollary \[CorE\].
This is proved in Theorems \[Thm0Cover\] and \[ThmMor\]. If, for $A={\mathcal B}_n(\delta)$, we take the stronger condition ‘$n$ even or $\delta\not=0$’, but relax condition ${\mathcal{C}}^2_A$ to ‘${{\rm{char}}}({\Bbbk})\not\in\{2,3\}$’, the Schur algebra of Corollary \[CorE\] is constructed in a more combinatorial way in [@Henke], see also [@Bowman] for the walled Brauer algebra. The combination of our methods with the ones in [@Henke] can be used to construct the Schur algebras of Corollary \[CorE\] in full generality, see Remark \[SchurGen\]. The result in Theorem \[ThmF\] also leads to the following question. The answer is likely to grow with $n,r,s$.
What is the maximal $k\in{\mathbb{N}}$ for which the quasi-hereditary covers in Theorem \[ThmF\] are $k$-faithful?
Finally, we remark that our treatment of Theorems \[ThmA\], \[ThmC\] and \[ThmD\] provide alternative proofs for several results in [@AST1; @CDDM; @DRAus; @CellAlg; @HHKP; @Paget; @CellQua; @partition; @Xi]. Our approach does not rely on any of the results in these papers.
Structure of the paper
----------------------
In Section \[SecPrel\], we recall some useful results and introduce some notation. Sections \[SecPre\] to \[SecBase\] constitute Part \[GenThe\]. Here, the general theory of borelic pairs is developed. In Part \[Examp\] we apply these results to a variety of examples. Concretely, in Section \[SecADR\] we consider Auslander-Dlab-Ringel algebras, in Section \[SecBGG\] a thick version of the BGG category ${\mathcal O}$ and in Section \[SecDia\] the class of diagram algebras. In Part \[Faith\] (Section \[SecCov\]) we determine the precise homological connection between the diagram algebras and their covers constructed in Section \[SecDia\]. In Appendix \[TheApp\], we recall some known technical properties of stratified algebras, which are not easy to find with proof in the literature.
Preliminaries {#SecPrel}
=============
We work over an arbitrary ground field ${\Bbbk}$. Unless specified otherwise, we make no assumptions on its characteristic and do not require it to be algebraically closed. A field is called [*perfect*]{} if every algebraic field extension of ${\Bbbk}$ is separable. Examples are algebraically closed fields, fields of characteristic zero and finite fields. We set ${\mathbb{N}}=\{0,1,2,\cdots\}$.
Algebras and modules
--------------------
By an algebra $A$ over ${\Bbbk}$, we will always mean an [*associative, unital, finite dimensional*]{} algebra. All subalgebras of an algebra $A$ are assumed to contain the identity $1_A$. Modules over $A$ are supposed to be unital, in the sense that $1_A$ acts as the identity. Furthermore, they are always assumed to be left modules and finitely generated, unless explicitly specified otherwise. The corresponding module category is denoted by $A$-mod. The Jordan-Hölder multiplicity of a simple module $L$ in $M\in A$-mod will be denoted by $[M:L]$.
Two algebras $A$ and $B$ are [*Morita equivalent*]{} if there is an equivalence of categories $$A\mbox{-mod }\cong\; B\mbox{-mod}.$$ We write this as $A\;\stackrel{M}{=}\; B$. A [*local*]{} algebra is an algebra with unique maximal left ideal. We say that an algebra is [*quasi-local*]{} if it is Morita equivalent to the direct sum of local algebras, so if there are no extensions between non-isomorphic simple modules.
Jacobson radical
----------------
The Jacobson radical ${{\rm rad}}A$ of an algebra $A$ is the intersection of all maximal left ideals, see [@Lam Chapter 4]. An algebra $A$ is called [*semisimple*]{} if its Jacobson radical is zero, which is equivalent to the condition that $A$ is the direct sum of simple algebras by [@Lam Theorem 4.14]. All finite dimensional unital algebras are [*semiprimary*]{}, meaning that $A/{{\rm rad}}A$ is semisimple and ${{\rm rad}}A$ is nilpotent, see [@Lam Theorem 4.15].
A ${\Bbbk}$-algebra $A$ is called [*separable*]{} if for any field extension $\mathbb{K}$ of ${\Bbbk}$, $A\otimes_{{\Bbbk}}\mathbb{K}$ is a semisimple $\mathbb{K}$-algebra. Every simple algebra over a perfect field is separable. If $A$ has a semisimple subalgebra $S$, such that $A=S\,\oplus \,{{\rm rad}}A$ as vector spaces, we say that $A$ is a [*Wedderburn algebra.*]{} Wedderburn’s principal theorem states that any algebra such that $A/{{\rm rad}}A$ is separable is Wedderburn, see [@Weibel Exercise 9.3.1].
For a subspace $I\subset A$ (not containing $1_A$) and an $A$-module $M$, we denote by $$\label{eqInvariants}M^I=\{v\in M\,|\, xv=0,\quad\mbox{ for all $x\in I$}\},$$ the $I$-invariants of $M$. In case $I$ is a two-sided ideal, the space $M^I$ comes with a natural $A/I$-module structure.
Idempotents {#SecIdem}
-----------
An [*idempotent*]{} $e\in A$ is [*primitive*]{} if the left $A$-module $Ae$ is indecomposable. Two idempotents $e$ and $f$ in $A$ are [*equivalent*]{} if and only if $Ae\cong Af$ as left $A$-modules, or equivalently if there are $a,b\in A$ such that $e=ab$ and $f=ba$.
For an algebra $A$, there is a one-to-one correspondence between the isomorphism classes of simple unital modules, the equivalence classes of primitive idempotents and the isomorphism classes of projective covers of the simple modules. So there is a (finite) set $\Lambda=\Lambda_A$ with corresponding simple module $L(\lambda)$, primitive idempotent $e_\lambda$ and projective module $P(\lambda):=A e_\lambda$ for each $\lambda\in \Lambda_A$, exhausting the classes irredundantly, such that $$e_{\lambda}L(\lambda')\not=0\quad\Leftrightarrow\quad\lambda'=\lambda\quad\Leftrightarrow\quad{\mathrm{Hom}}_A(P(\lambda), L(\lambda'))\not=0.$$
Centraliser algebras {#SecCover}
--------------------
For an algebra $C$ with idempotent $e$, the centraliser algebra of $e$ is $C_0=eCe$. We consider the pair of adjoint functors $(F,G)$: $$\label{eq630}
\xymatrix{
C\text{-}\mathrm{mod}\ar@/^/[rrrrrr]^{F=eC\otimes_C-\,\cong\, e-\,\cong\,\mathrm{Hom}_C(Ce,-)}&&&&&&
C_0\text{-}\mathrm{mod}.\ar@/^/[llllll]^{G={\mathrm{Hom}}_{C_0}(eC,-)}
}$$ We have $F\circ G\cong {\mathrm{Id}}$ on $C_0$-mod, and $F$ is exact while $G$ is left exact.
### {#SecSecCover}
We say that $C$ is a [*cover*]{} of $C_0$ if the restriction of $F$ to the category of projective $C$-modules is fully faithful. In other words, the canonical morphism $$C\,\stackrel{\sim}{\to}\, {\mathrm{End}}_{C}(C)^{{{\rm op}}}\,\stackrel{F}{\to}\, {\mathrm{End}}_{C_0}(e C)^{{{\rm op}}}; \qquad c\mapsto\alpha_c,\quad\mbox{with}\quad \alpha_c(x)=xc\;\;\mbox{ for all $x\in eC$,}$$ is an isomorphism if and only if $C$ is a cover.
### {#SecMor}
Assume that $C=CeC$, then every primitive idempotent of $C$ is equivalent to one contained in $eCe$. Hence $Ce$ is a projective generator for $C$-mod, so it follows that $F$ is faithful. Since we already had $F\circ G\cong {\mathrm{Id}}$, it follows that $F$ is an equivalence, so $C$ and $C_0$ are Morita equivalent.
Orders and partitions
---------------------
A [*partial quasi-order*]{} (also known as pre-order) $\preccurlyeq$ is a binary relation which is reflexive and transitive. When a partial quasi-order is also anti-symmetric, it is a [*partial order*]{}, and will usually be denoted by $\le$. When $\preccurlyeq$ satisfies the condition that, for all elements $s,t$, at least one of $s\preccurlyeq t$ or $t\preccurlyeq s$ holds, it is a [*total quasi-order*]{}, or simply a [*quasi-order*]{}.
For a partial quasi-order $\preccurlyeq$, we use the notation $s\prec t$ when $s\preccurlyeq t$ but $t\not\preccurlyeq s$. We also write $s\sim t$ when $s\preccurlyeq t$ and $t\preccurlyeq s$. An [*extension*]{} $\preccurlyeq_e$ of a partial quasi-order $\preccurlyeq$ is a partial quasi-order such that $s\preccurlyeq t$ implies $s\preccurlyeq_et$, and such that $s\sim t$ if and only if $s\sim_e t$. In particular, an extension of a partial order remains a partial order.
An [*$n$-decomposition ${\mathcal{Q}}$ of a set*]{} $S$ is an ordered disjoint union $S=\sqcup_{i=0}^n S_i$ into $n+1$ subsets. In case the set $S$ is finite, decompositions are in natural bijection with total quasi-orders. For the decomposition ${\mathcal{Q}}$ the corresponding total quasi-order $\preccurlyeq_{{\mathcal{Q}}}$ on $S$ is given by $s\preccurlyeq_{{\mathcal{Q}}} t$ if and only if $s\in S_i$ and $t\in S_j$ with $i\ge j$. To each decomposition ${\mathcal{Q}}$, we also associate a partial order $\le_{{\mathcal{Q}}}$, defined by $s<_{{\mathcal{Q}}} t$ if and only if $s\in S_i$ and $t\in S_j$ with $i> j$. Note that we cannot obtain every partial order in this way.
We will use the term [*partition*]{} in two different situations. A [*partition of $n\in{\mathbb{N}}$*]{} is a (non-strictly) decreasing sequence of natural numbers adding up to $n$. When $\lambda$ is a partition of $n$ we write $\lambda\vdash n$. For $p>0$, we say that $\lambda\vdash n$ is $p$-regular if the sequence of the partition does not contain $p$ times the same non-zero number. For a field ${\Bbbk}$, we say that $\lambda$ is ${\Bbbk}$-regular if either ${{\rm{char}}}({\Bbbk})=0$ or $\lambda$ is ${{\rm{char}}}({\Bbbk})$-regular, and write this as $\lambda\vdash_{{\Bbbk}} n$. Secondly, we will use the notion of a [*partition of a set*]{}, which is a grouping of the elements into non-empty subsets (or equivalence classes).
Stratifying ideals {#SecStratId}
------------------
An idempotent ideal $J$ in an algebra $A$ is a two-sided ideal which satisfies $J^2=J$. This means $J=AeA$ for some idempotent $e$, see [*e.g.*]{} [@APT p673]. This idempotent $e$ is not uniquely determined by $J$. However, it follows easily, see [*e.g.*]{} [@APT p673], that $AeA=A\tilde{e}A$, for two idempotents $e$ and $\tilde e$, implies $$\label{MoreAe0}eAe\;\stackrel{M}{=}\; \tilde{e}A\tilde{e}.$$
### {#SecStratId2}
We introduce some specific types of stratifying ideals, using in particular terminology from [@CPS; @CPSbook; @Frisk; @Kluc]. An idempotent ideal $J=AeA$ in $A$ is
1. an [*exactly stratifying ideal*]{} if:
- the right $A$-module $J_A$ is projective.
2. a [*standardly stratifying ideal*]{} if:
- the $A$-module ${}_AJ$ is projective.
3. a [*exactly standardly stratifying ideal*]{} if:
- the left $A$-module ${}_AJ$ and right $A$-module $J_A$ are projective.
4. a [*strongly standardly stratifying ideal*]{} if:
- the $A$-module ${}_AJ$ is projective and
- the algebra $eAe$ is quasi-local.
5. a [*properly stratifying ideal*]{} if:
- the left $A$-module ${}_AJ$ and right $A$-module $J_A$ are projective and
- the algebra $eAe$ is quasi-local.
6. a [*heredity ideal*]{} if:
- the $A$-module ${}_AJ$ is projective and
- the algebra $eAe$ is semisimple.
All these ideals are stratifying in the sense of [@CPSbook Definition 2.1.1], by [@CPSbook Remark 2.1.2(b)]. A heredity ideal is properly stratifying by [@APT Corollary 5.3], the other relations in Figure 1 are by definition.
\[MoreAe\] By equation , the definitions depend only on $J$ and not on $e$.
### {#SecChain}
We say that a chain of idempotent ideals $$\label{stratchain}
0=J_0\subsetneq J_1\subsetneq \cdots \subsetneq J_{m-1}\subsetneq J_m =A,$$ has length $m-1$. With this convention, the trivial chain $0=J_0\subset J_1=A$ has length $0$. Chains of length $k$ are in one-to-one correspondence with $k$-decompositions of $\Lambda$. The decomposition corresponding to a chain of idempotent ideals is defined by setting $\lambda\in\Lambda_i$ for the minimal $i$ for which $J_{i+1} L(\lambda)\not=0$. Consequently, the chains of idempotent ideals of $A$ are also in one-to-one correspondence with the total quasi-orders of $\Lambda$.
### {#section}
If for a chain , each ideal $J_{i}/J_{i-1}$ is a standardly, exactly standardly, strongly standardly or properly stratifying ideal in $A/J_{i-1}$, the chain is also called standardly, exactly standardly, strongly standardly or properly stratifying. If each ideal $J_{i}/J_{i-1}$ is a heredity ideal, the chain is called heredity.
We will take the convention of denoting the image of an idempotent $e\in A$, in the quotient $A/J$, for some idempotent ideal $J$ again by $e$.
\[DefAi\] For a chain , we can choose idempotents $f_i$ for $i\in\{1,\cdots,m\}$ such that $J_i=Af_iA$, with $f_if_j=f_i=f_jf_i$ if $i\le j$ and $f_m=1$. We define the algebras $$A^{(j)}=f_{j+1}(A/J_{j})f_{j+1},\quad\mbox{ for }\quad j\in\{0,1,\ldots, m-1\},$$ which are only uniquely associated to the chain up to Morita equivalence, by .
Standardly stratified algebras {#SecStrat}
------------------------------
An algebra $A$ with some partial quasi-order $\preccurlyeq$ on $\Lambda$ will be denoted as $(A,\preccurlyeq)$.
\[DefA1\]${}$ Consider an algebra $(A,\preccurlyeq)$, for $\preccurlyeq$ a total quasi-order, and the chain of idempotent ideals corresponding to $\preccurlyeq$. We say that $(A,\preccurlyeq)$ is
1. [*standardly stratified*]{}, if the chain is standardly stratifying;
2. [*exactly standardly stratified*]{}, if the chain is exactly standardly stratifying;
3. [*strongly standardly stratified*]{}, if $\preccurlyeq$ is a total order and the chain is strongly standardly stratifying;
4. [*properly stratified*]{}, if $\preccurlyeq$ is a total order and the chain is properly stratifying;
5. [*quasi-hereditary*]{}, if $\preccurlyeq$ is a total order and the chain is heredity.
\[RemTriv\] The trivial chain $0=J_0\subset J_1=A$ is an exactly standardly stratifying chain for any algebra $A$. It is thus essential to specify the quasi-order $\preccurlyeq$ when speaking about types of standardly stratified algebras.
\[RemSSS\] If $\preccurlyeq$ is a total order and the chain corresponding to $\preccurlyeq$ is standardly (resp. exactly standardly) stratifying, it is automatically strongly standardly (resp. properly) stratifying, as the algebra $A^{(i)}$ will only have one simple module up to isomorphism.
There is also a module theoretic approach to standardly stratified algebras, where we no longer demand the orders to be total. The equivalence between both definitions (when the order is total) is well-known by e.g. [@CPS; @CPSbook; @Dlab; @Frisk], see also Appendix \[AppEq\].
\[DefA2\] Consider an algebra $(A,\preccurlyeq)$ for some partial quasi-order $\preccurlyeq$ on $\Lambda$. For any $\lambda\in\Lambda$, let $L(\lambda)$ denote the corresponding simple $A$-module and $P(\lambda)$ its projective cover.
1. If there is a set of $A$-modules $\{S(\lambda),\lambda\in \Lambda\}$ such that for any $\lambda,\mu\in\Lambda$:
- we have $[S(\lambda):L(\mu)]=0$ unless $\mu\preccurlyeq \lambda$, and
- there is a surjection $P(\lambda){\twoheadrightarrow}S(\lambda)$ such that the kernel has a filtration where the section are isomorphic to modules $S(\nu)$ for $\lambda \prec\nu$,
we say that $(A,\preccurlyeq)$ is [*standardly stratified.*]{}
2. If $(A,\preccurlyeq)$ is standardly stratified and there are also modules $\{\overline{S}(\lambda),\lambda\in \Lambda\}$ with $[\overline{S}(\lambda):L(\lambda)]=1$ and $[\overline{S}(\lambda):L(\mu)]=0$ unless $\mu=\lambda$ or $\mu\prec\lambda$, such that each module $S(\lambda)$ has a filtration where the sections are isomorphic to modules $\overline{S}(\mu)$ with $\mu\sim\lambda$, we say that $(A,\preccurlyeq)$ is [*exactly standardly stratified.*]{}
3. If $\preccurlyeq$ is a partial order and $(A,\preccurlyeq)$ is standardly stratified as in (1), we say that $(A,\preccurlyeq)$ is [*strongly standardly stratified.*]{}
4. If $\preccurlyeq$ is a partial order and $(A,\preccurlyeq)$ is exactly standardly stratified as in (2), we say that $(A,\preccurlyeq)$ is [*properly stratified.*]{}
5. If $\preccurlyeq$ is a partial order, $(A,\preccurlyeq)$ is standardly stratified as in (1) and in addition $[S(\lambda):L(\lambda)]$=1 for all $\lambda\in \Lambda$, we say that $(A,\preccurlyeq)$ is [*quasi-hereditary.*]{}
Observe that if $(A,\preccurlyeq)$ is standardly stratified, $(A,\preccurlyeq')$ is also standardly stratified, for any extension $\preccurlyeq'$ of $\preccurlyeq$.
### {#SAj}
The modules $S(\lambda)$ in Definition \[DefA2\] have simple top $L(\lambda)$ and are known as [*standard modules*]{}. They satisfy $S(\lambda)\cong Ae_\lambda/J_{j-1}e_\lambda$ with $j$ the largest such that $J_{j-1}L(\lambda)=0$. The modules $\overline{S}(\lambda)$ are [*proper standard modules*]{} and are given by $S(\lambda)/(Ae_\lambda{{\rm rad}}S(\lambda))$.
Faithful covers and standard systems {#IntroFaithCov}
------------------------------------
In [@Rou Definition 4.37], Rouquier introduced notions of [*faithfulness*]{} of [*quasi-hereditary covers*]{}. We extend this to standardly stratified algebras. Consider a cover $C$ as in \[SecSecCover\], which is standardly stratified. Such a cover is called [*$j$-faithful*]{} if the functor $F={\mathrm{Hom}}_C(Ce,-)$ induces isomorphisms $${\mathrm{Ext}}^i_C(M,N)\;\,\tilde\to\;\, {\mathrm{Ext}}^i_{C_0}(FM,FN),\qquad\forall \;0\le i\le j,\label{ifaith}$$ for all modules $M,N$ admitting a filtration with sections given by proper standard modules. Important examples of quasi-hereditary $1$-covers are the $q$-Schur algebras of the Hecke algebra, by [@Nakano2] and the analogue of [@qSchAndrew; @qSchJie] in type $B$, by [@Rou Theorem 6.6].
Consider an abelian category ${\mathcal{C}}$ and a partially ordered set $(S,\le)$. As in [@DR Section 3] or [@HHKP Definition 10.1], a [*standard system in ${\mathcal{C}}$ for $S$*]{} (also known as an exceptional sequence) is a set of objects $\{\Theta(p)\,|\, p\in S\}$ in ${\mathcal{C}}$, such that for all $p,q\in S$:
1. ${\mathrm{End}}_{{\mathcal{C}}}(\Theta(p))$ is a division ring;
2. ${\mathrm{Hom}}_{{\mathcal{C}}}(\Theta(p),\Theta(q))=0$ unless $p\le q$;
3. ${\mathrm{Ext}}^1_{{\mathcal{C}}}(\Theta(p),\Theta(q))=0$ unless $p< q$.
For a quasi-hereditary algebra $(A,\le)$, the standard modules $\{S(\lambda),\,\lambda\in\Lambda\}$ form a standard system in $A$-mod for $(\Lambda,\le)$, see [*e.g.*]{} [@DR Lemmata 1.2, 1.3 and 1.6].
Standardly based algebras and Schur algebras {#SecSBalg}
--------------------------------------------
There is a close connection, and large overlap, between quasi-hereditary algebras and [*cellular algebras*]{}, see [@CellAlg Remark 3.10], [@StructureCell Proposition 4.1 and Corollary 4.2], [@CellQua Theorem 1.1] and [@Cao Theorem 1.1]. Moreover, an interesting concept of “cellularly stratified” algebras was recently introduced in [@HHKP], which combines properties of cellular and stratified algebras. Comparing or combining properties of cellular and stratified algebras is often complicated by the involutive anti-automorphism $\imath$ in the definition of a cell datum in [@CellAlg Definition 1.1]. Omitting $\imath$ in the definition leads to the concept of standardly based algebras.
We use a reformulation of [@JieDu Definition 1.2.1]. A [*standardly based structure*]{} of an algebra $A$ is a poset $L$, with two-sided ideals $A^{\ge p}$ for each $p\in L$, such that for all $p,q\in L$
- $A^{\ge p}\supseteq A^{\ge q}$ if $p<q$, hence $A^{>p}:=\cup_{q>p}A^{\ge q}$ in $A^{\ge p}$ is an ideal in $A^{\ge p}$
- we can take complements $A^{(p)}$ of $A^{>p}$ in $A^{\ge p}$, such that $\bigoplus_{p\in L}A^{(p)}=A$;
- $A^{\ge p}/A^{>p}\cong W(p)\otimes_{{\Bbbk}} W'(p)$ as $A$-bimodules for a left module $W(p)$ and right module $W'(q)$.
The modules $W(p)$ will be referred to as the [*cell modules*]{} of $A$.
\[remMorSB\] A standardly based structure of an algebra is preserved under Morita equivalences, see [@Yang Section 3], meaning that the ideals are naturally linked. In particular, the cell modules are mapped to the corresponding cell modules.
By [@JieDu Theorem 2.4.1], $\Lambda$ can be naturally identified with a subset of $L$. We consider $\Lambda$ then as a poset for the inherited partial order from $L$. For $\lambda\in\Lambda\subset L$, we have furthermore that ${\mathrm{Top}}W(\lambda)=L(\lambda)$ and $$[W(\lambda):L(\lambda)]=1\qquad\mbox{and}\qquad [W(\lambda):L(\mu)]=0\quad\mbox{unless } \mu\le \lambda.$$ By [@JieDu Proposition 2.4.4], every indecomposable projective module $P(\lambda)$ with $\lambda\in\Lambda$ has a filtration with sections given by modules $W(p)$ such that $$(P(\lambda):W(\lambda))=1\qquad\mbox{and}\qquad (P(\lambda):W(p))=0\quad\mbox{unless } p\ge \lambda.$$ Hence if $L=\Lambda$, the standardly based algebra is quasi-hereditary with standard modules $W(\lambda)$. Conversely, over an algebraically closed field, every quasi-hereditary algebra is standardly based for $L=\Lambda$ (as posets) and $W(\lambda)=S(\lambda)$, by [@JieDu Theorem 4.2.3].
\[ExS1\] Consider $A:={\Bbbk}[x]/(x^{t}-1)$ for a field ${\Bbbk}$ such that $x^t-1$ splits as $$x^t-1=\prod_{i=1}^t(x-\omega_i),$$ for not necessarily distinct $\omega_i\in{\Bbbk}$. Then consider $L:=\{1,2,\cdots,t\}$ with usual order and $$a_k=\prod_{i=1}^{k-1}(x-\omega_i)\;\;\mbox{ for }\; 2\le k\le t\quad\mbox{ and } a_1=1.$$ The ideals $A^{\ge i}={{\rm{Span}}}\{a_i, a_{i+1},\ldots,a_t\}$ give a standardly based structure of $A$. $A$ is even cellular for involution $\imath={\mathrm{id}}_A$ the identity, by [@JieDu Lemma 1.2.4], or [@CellAlg Example 1.3].
\[ExS2\] For any field ${\Bbbk}$, the group algebra ${\Bbbk}{\mathbb{S}}_t$ of the symmetric group ${\mathbb{S}}_t$ on $t$ symbols is cellular and hence standardly based for $L=\{\lambda\vdash t\},$ equipped with the partial order obtained by reversing the dominance order on partitions, see [@CellAlg Example 1.2]. The ideals are obtained from the Murphy basis and the cell modules are the Specht modules of [@James Section 4]. The algebra ${\Bbbk}{\mathbb{S}}_t$ is then cellular for the involution $\imath$, which is the linearisation of the inversion on the group ${\mathbb{S}}_t$.
The following definition is essentially [@HHKP Definition 12.1], see also [@Henke; @Rou].
\[DefSchur\]A [*(cover-)Schur algebra*]{} of a standardly based algebra $A$ is a quasi-hereditary $1$-cover $({\mathcal{S}},\le)$, such that $(\Lambda_{{\mathcal{S}}},\le)=(L_A,\le)$ and $F(S(p))\cong W(p)$ for all $p\in L_A$.
By [@Rou Corollary 4.46], a cover-Schur algebra of a standardly based algebra is unique, up to Morita equivalence, if it exists. We use the specification cover-Schur algebra since the generalised Schur algebra representing the orthogonal or symplectic group in the double centraliser property for the Brauer algebra does not act as a Schur algebra in the above sense.
\[GenThe\]
Pre-Borelic pairs {#SecPre}
=================
In this section we introduce the notion of pre-Borelic pairs for arbitrary algebras. These will lead to Borelic pairs in the next section, where we will also prove that these contain König’s notion of exact Borel subalgebras for quasi-hereditary algebras as a special case.
Definition and properties
-------------------------
Fix an algebra $A$.
\[defB\]
A pair $(B,H)$ of subalgebras $H\subset B\subset A$ forms a [*pre-Borelic pair of*]{} $A$ if there exists a two-sided ideal $B_+$ in $B$, with $B=H\oplus B_+$, such that
(I) $A$ is projective as a right $B$-module and $B$ is projective as a left $H$-module;
(II) taking $B_+$-invariants, $({{\rm Res}}^A_B-)^{B_+}$, as in , yields an equivalence of categories between simple $A$-modules and simple $H$-modules;
(III) the ideal $B_+$ is contained in the Jacobson radical ${{\rm rad}}B$ of $B$.
### {#ee0}
Label the set of isomorphism classes of simple $A$-modules by $\Lambda$ as in Section \[SecIdem\]. By (II), we can use the same set for $H$. We use the notation $L(\lambda)$ (resp. $L^0(\lambda)$) for the corresponding simple modules over $A$ (resp. $H$), and hence $$\label{LL0}
L^0(\lambda)\;\cong \;L(\lambda)^{B_+},\qquad\mbox{ for all $\lambda\in\Lambda$}.$$ By assumption (III), we also have a one-to-one correspondence between simple $B$-modules and simple $H$-modules, so $$\label{eqLABH}\Lambda=\Lambda_A=\Lambda_B=\Lambda_H.$$ We use the same notation $L^0(\lambda)$, for the simple $B$-module with trivial $B_+$-action defined as the inflation of the $H$-module $L^0(\lambda)$. For each $\lambda\in\Lambda$, there is an idempotent $e^0_\lambda\in H\subset B\subset A$, primitive in $H$ (but generally not in $A$) such that $e^0_\lambda L^0(\lambda)\not=0$.
### {#DefDelta}
For any $H$-module $N$, interpreted as a $B$-module with trivial $B_+$-action, we define $$\Delta_N\;:=\;{{\rm Ind}}^A_B N=A\otimes_BN.$$ By adjunction, we have $$\label{FroDel}{\mathrm{Hom}}_A(\Delta_N,M)\;\cong \;{\mathrm{Hom}}_H(N,M^{B_+}),\qquad\mbox{for all $M\in A$-mod.}$$
\[TopStan\] Consider a pre-Borelic pair $(B,H)$ of $A$. For any $N\in H$[-mod]{} with simple top $L^0(\lambda)$, for some $\lambda\in \Lambda$, we have $${\mathrm{Top}}\Delta_N\;\cong\; L(\lambda).$$
Equations and imply that, for $\lambda,\lambda'\in\Lambda$, we have $$\dim{\mathrm{Hom}}_A(\Delta_N,L(\lambda'))=\dim{\mathrm{Hom}}_H(N,L^0(\lambda'))=\delta_{\lambda,\lambda'},$$ proving the claim.
\[PPlambda\] Given a pre-Borelic pair $(B,H)$, define $P_\lambda= Ae_\lambda^0$, for any $\lambda\in \Lambda$. Then $$P_\lambda\;\cong\; \bigoplus_{\mu\in\Lambda}P(\mu)^{\oplus c_\mu^\lambda}\quad\mbox{with}\quad c_\mu^\lambda:= [{{\rm Res}}^A_BL(\mu):L^0(\lambda)].$$
This follows from $A\otimes_B Be^0_\lambda\cong Ae^0_\lambda$ and adjunction.
### {#section-1}
For a fixed pre-Borelic pair $(B,H)$, we introduce the modules $$\label{Standards}
\Delta(\lambda)\;=\;\Delta_{He^0_\lambda}\;=\; A\otimes_B He^0_\lambda,\qquad\mbox{and}\qquad \overline{\Delta}(\lambda)\;=\;\Delta_{L^0(\lambda)}\;=\; A\otimes_B L^0(\lambda).$$ We say that an $A$-module $N$ [*has a $\Delta$-flag ${\mathcal{M}}$ of length $d\in{\mathbb{N}}$*]{} if there are submodules $$\label{eqFlag}N= M_0\supset M_1\supset M_2\supset\cdots\supset M_{d-1}\supset M_d=0 ,$$ such that for each $0\le i< d$, we have $M_i/M_{i+1}\cong \Delta(\lambda)$, for some $\lambda\in\Delta$. For such a $\Delta$-flag ${\mathcal{M}}$, we introduce $$\label{DefMult}(N:\Delta(\lambda))_{{\mathcal{M}}}\,:=\; \sum_{i=0}^{d-1}\dim {\mathrm{Hom}}_A\left(M_i/M_{i+1},L(\lambda)\right).$$ By definition and Lemma \[TopStan\], we then have $$\label{flagGro}[N]=\sum_{\mu\in\Lambda}\,(N:\Delta(\mu))_{{\mathcal{M}}}\;[\Delta(\mu)],$$ in the Grothendieck group $G_0(A)$ of $A$-mod.
By left exactness of ${\mathrm{Hom}}_A(-,L(\lambda))$, equation implies the following inequality.
\[BoundMult\] If and $A$-module $M$ has a $\Delta$-flag ${\mathcal{M}}$, then $$\dim{\mathrm{Hom}}_A(M,L(\lambda))\;\le\;(M:\Delta(\lambda))_{{\mathcal{M}}},\qquad\mbox{for all $\lambda\in\Lambda$.}$$
A special case: idempotent graded algebras {#SecPreSpec}
------------------------------------------
For an algebra $A$, we choose a decomposition $$\label{decomp1}1_A\;=\; e^\ast_0+e^\ast_1+\cdots+e^\ast_n,$$ where $e^\ast_i$ are mutually orthogonal, but not necessarily primitive, idempotents. For notational convenience, we allow some of these idempotents to be zero. We have a ${\mathbb{Z}}$-grading on $A$: $$\label{grBr}A_j=\bigoplus_{i=\max(0,-j)}^{\min(n, n-j)} e^\ast_i A e^\ast_{i+j},\qquad\mbox{for all $j\in{\mathbb{Z}}$}.$$ Indeed, we have $A=\bigoplus_j A_j$ as vector spaces and furthermore $$A_jA_k=\bigoplus_i e^\ast_i A e^\ast_{i+j} Ae^\ast_{i+j+k}\subset A_{j+k},\qquad\mbox{for all $j,k\in{\mathbb{Z}}$}.$$ We call this the [*idempotent grading*]{} associated to the (ordered) choice of $e_i^\ast$ in and set $$A_+:=\bigoplus_{j>0}A_j\;\mbox{ and }\;A_{-}:=\bigoplus_{j<0}A_j.$$
\[BHgrad\] An [*${\mathbb{N}}$-graded subalgebra*]{} of an idempotent graded algebra $A$ is a graded subalgebra $B$ such that $B_{j}=0$ for $j<0$. For such $B$, set $$H\,:=\,B_0\,=\,\bigoplus_{i\in{\mathbb{N}}}e^\ast_i B e^\ast_i\quad\mbox{ and}\qquad B_+\,:=\,\bigoplus_{j>0}B_j.$$ An ${\mathbb{N}}$-graded subalgebra $B$ is [*complete*]{} if $$\label{eqComplete}A=B\oplus A_{-}B.$$
Note that any ${\mathbb{Z}}$-graded subalgebra $B$ satisfying is automatically ${\mathbb{N}}$-graded. Furthermore, it is easy to check that equation is equivalent to $$\label{eqComplete2}A\;=\; B\,\oplus\, A_-A.$$
We will use freely the fact that, as follows from the definitions, [*any module $M$, over $A$, $B$ or $H$, is automatically a ${\mathbb{Z}}$-graded module*]{}, by setting $$\label{eqDefZGrad}M_{-j}:=e^\ast_{j}M,\quad\mbox{ for }\;\quad j\in\{0,1,\cdots,n\}.$$
\[LemGradBor\] Consider a complete ${\mathbb{N}}$-graded subalgebra $B$ of an idempotent graded algebra $A$. If $A_B$ and ${}_HB$ are projective, then $(B,H)$ is a pre-Borelic pair.
Condition (I) in Definition \[defB\] is given.
Condition (II) is the requirement that $\{L(\lambda)^{B_+},\lambda\in \Lambda\}$ be a complete set of non-isomorphic simple $H$-modules. Consider the simple $A$-module $L(\lambda)$, which we equip with ${\mathbb{Z}}$-grading as in . Let $k_0$ be the maximal degree for which $L(\lambda)_{k_0}$ is non-zero. We clearly have $L(\lambda)_{k_0}\subset L(\lambda)^{A_+}$. Now take some $w\in L(\lambda)_i$, for $i<k_0$. Since $L(\lambda)$ is simple, we have $L(\lambda)_{k_0}\subset A_+ w$. In particular $w\not\in L(\lambda)^{A_+}$. It then follows quickly that $L(\lambda)_{k_0}=L(\lambda)^{A_+}$. Similarly, simplicity of the $A$-module $L(\lambda)$ implies that each $L(\lambda)_j$, so in particular $L(\lambda)^{A_+}=L(\lambda)_{k_0}$, is simple as an $A_0$-module. Equation implies that $A_0=H\oplus (A_{-}A_+)_0$. Consequently, we have $A_0v=Hv$ for any $v\in L(\lambda)^{A_+}$ and the restriction of $L(\lambda)^{A_+}$ to an $H$-module remains simple. Equation implies $A_+\subset AB_+, $ from which it follows that $L(\lambda)^{B_+}=L(\lambda)^{A_+}$ as $H$-modules. Hence, $$\label{simpleComp}L^0(\lambda):=L(\lambda)^{B_+}$$ is a simple $H$-module.
Now we consider an arbitrary simple $H$-module $L^0$, which we can interpret as a simple $B$-module contained in one degree, say $-i$ for $i\in{\mathbb{N}}$, so $L^0=e_i^\ast L^0$. The $A$-module $M:={{\rm Ind}}^A_B L^0$ satisfies $e_j^\ast M=0$ unless $j\ge i$, with $e_i^\ast M\cong L^0$, by equation . It thus follows that $M\not=0$, and that any proper submodule $S$ of $M$ satisfies $e_i^\ast S=0$. Hence, taking the union of all proper submodules yields the unique maximal submodule. We conclude that ${{\rm Ind}}^A_B L^0$ has simple top. It follows from adjunction that this top is $L(\lambda)$ if and only if $L^0\cong L^0(\lambda)$. This implies that the simple modules in exhaust all simple $H$-modules and that $L^0(\lambda)\cong L^0(\mu)$ implies $L(\lambda)\cong L(\mu)$, which proves condition (II).
Condition (III) is immediate from the ${\mathbb{N}}$-grading on $B$. This concludes the proof.
The lemma motivates the following definition.
\[GrPreBor\] Let $A$ be an idempotent graded algebra with complete ${\mathbb{N}}$-graded subalgebra $B$. If $A_B$ and ${}_HB$ are projective with $H:=B_0$, we call $B$ a [*graded pre-Borelic algebra*]{}.
The reason we just work with the algebra $B$ for graded pre-Borelic subalgebras, instead of a pair, is that $H:=B_0$ is defined through the grading on $B$.
\[RemJchain\] An idempotent grading on an algebra $A$ implies a chain of idempotent ideals. Set $f_k:=\sum_{j<k}e^\ast_j$, for $1\le k\le n+1$ and $J_k:=Af_kA$, leading to the chain $$\label{chainProp}0=J_0\subset J_1\subset \cdots\subset J_{n}\subset J_{n+1}=A.$$ Using the chain , we can rewrite equation as $$\label{eqABJ}e^\ast_j A\;=\; e^\ast_j B\,\oplus\, e^\ast_jJ_{j},\qquad\mbox{for all }\,0\le j\le n.$$ In particular, $e_j^\ast Ae_j^\ast=e_j^\ast He_j^\ast\oplus e_j^\ast J_je_j^\ast$, and for the algebras in Remark \[RemSSS\] we find $$\label{eqAiH}A^{(i)}=f_{i+1} (A/J_i)f_{i+1}=e_i^\ast (A/J_i)e_i^\ast\cong e_i^\ast He_i^\ast=He_i^\ast=e_i^\ast H.$$
\[LemHB\] Let $A$ be an idempotent graded algebra with complete ${\mathbb{N}}$-graded subalgebra $B$. Then ${}_HB$ is projective if and only if the left $A^{(j)}$-module $f_{j+1} A/J_j$ is projective for $0\le j\le n$.
By , we have $ He_j^\ast\cong A^{(j)}$ and by , $e_j^\ast B\cong f_{j+1} (A/J_j)$, which implies the statement.
\[LemABAM\] Consider an idempotent graded algebra $A$ with graded pre-Borelic algebra $B$, and an $He_i^\ast\cong A^{(i)}$-module $M$. We have $$\Delta_M=A\otimes_B M\;\cong\; (A/J_i)f_{i+1}\otimes_{A^{(i)}}M.$$
We will prove that the right adjoints of the functors $A\otimes_B-$ and $(A/J_i)f_{i+1}\otimes_{A^{(i)}}-$ are isomorphic. These right adjoints are given respectively by $$e_i^\ast({{\rm Res}}^A_B-)^{B_+}\qquad\mbox{and}\qquad {\mathrm{Hom}}_A(Af_{i+1}/J_if_{i+1},-).$$ When applied to $N\in A$-mod, the first corresponds to taking all elements in $e_i^\ast N$ which are annihilated by $B_+$ and thus by $A_+$. The second corresponds to taking all elements in $f_{i+1}N$ which are annihilated by elements of $J_if_{i+1}$. These two clearly coincide.
Borelic pairs {#borelic-pairs}
=============
Definition
----------
Consider an algebra $A$ with pre-Borelic pair $(B,H)$.
### {#section-2}
We define an equivalence relation $\sim_H$ on $\Lambda$, where $\lambda\sim_H\mu$ if and only if $L^0(\lambda)$ and $L^0(\mu)$ are in the same indecomposable block of $H$-mod. This equivalence relation is hence trivial if and only if $H$ is quasi-local. In particular, the equivalence relation is trivial when $B_+={{\rm rad}}B$. A partial quasi-order $\preccurlyeq$ on $\Lambda$ is said to be [* $H$-compatible*]{} if $$\lambda\sim_H\mu\; \Rightarrow\; \lambda\sim\mu,$$ where we recall that $\lambda\sim\mu$ stands for $\lambda\preccurlyeq\mu$ and $\mu\preccurlyeq \lambda$.
\[DefBorPair\] Consider an $H$-compatible partial quasi-order $\preccurlyeq$ on $\Lambda$. We say that $(B,H)$ is a [*Borelic pair*]{} of $(A,\preccurlyeq)$ if, for all $\lambda,\mu\in \Lambda$, we have
1. $[B_+e^0_\lambda:L^0(\mu)]=0\mbox{ unless }\mu\succ \lambda; $
2. $[\Delta(\lambda):L(\mu)]=0\mbox{ unless }\; \mu \preccurlyeq\lambda;$
A Borelic pair $(B,H)$ is called [*exact*]{} if $[\overline{\Delta}(\lambda):L(\lambda)]=1$, for all $\lambda\in\Lambda$.
Properties
----------
Now we start exploring the properties of Borelic pairs. Whenever a partial quasi-order $\preccurlyeq$ on $\Lambda$ is considered in relation to a pre-Borelic pair $(B,H)$ of $A$, we assume it to be $H$-compatible.
\[PlambdaFilt\] Consider an algebra $(A,\preccurlyeq)$ with Borelic pair $(B,H)$, then the module $P_\lambda=A\otimes_B Be^0_\lambda$ admits a $\Delta$-flag. In particular, the module $K$ defined through the short exact sequence $$\label{sesPD}0\to K\to P_\lambda\to \Delta(\lambda)\to 0,$$ has a $\Delta$-flag ${\mathcal{M}}$ with $$(K: \Delta(\mu))_{{\mathcal{M}}}=0\quad \mbox{unless} \;\;{\mu}\succ{\lambda}.$$
We start from the short exact sequence of $B$-modules $$0\to B_+ e^0_\lambda\to Be^0_\lambda\to He_\lambda^0\to 0.$$ We set $N:=B_+ e^0_\lambda$, and apply the exact functor $A\otimes_B-$, see Definition \[defB\](I), to the above short. This yields the short exact sequence $$0\to A\otimes_BN\to P_\lambda\to \Delta(\lambda)\to 0.$$
We claim that the $B$-module $N=B_+ e^0_\lambda$ has a filtration with sections given by the $B$-modules $He_\mu^0=Be_\mu^0/B_+e_\mu^0$ with $\mu\succ \lambda$. Firstly, by Definition \[defB\](I), ${}_HBe^0_\lambda$ and $He_\lambda^0$ are projective, so also $N$ is projective as an $H$-module. Since $\sim$ is $H$-compatible, we have a decomposition $${}_HN\;=\;\bigoplus_{[\mu]}N_{[\mu]},$$ where $[\mu]$ runs over the equivalence classes of $\sim$ and $[N_{[\mu]}:L^0(\nu)]=0$ unless $\mu\sim\nu$. Furthermore, $N_{[\mu]}$ is projective as an $H$-module, so a direct sum of modules $He^0_\kappa$ with $\kappa\in [\mu]$. By Definition \[DefBorPair\](1), $N_{[\mu]}=0$ unless $\mu\succ\lambda$. Take $\mu\in\Lambda$ such that $N_{[\mu']}=0$ if $\mu'\succ\mu$. This actually constitutes a $B$-submodule with trivial $B_+$ action, by Definition \[DefBorPair\](1). We can then proceed iteratively with the module $N/N_{[\mu]}$. This yields the desired filtration.
As ${{\rm Ind}}^A_B-$ is exact by Definition \[defB\](I), this filtration of $N$ induces the desired filtration of $K:=A\otimes_BN$.
\[CorPP\] Consider an algebra $(A,\preccurlyeq)$ with Borelic pair $(B,H)$, the projective module $P_\lambda$ is the direct sum of $P(\lambda)$ and certain $P(\mu)$ with $\mu\succ\lambda$.
It follows from Lemmata \[PlambdaFilt\] and \[BoundMult\] that $$\dim{\mathrm{Hom}}_A(P_\lambda,L(\lambda))=1,\qquad\mbox{and}\qquad{\mathrm{Hom}}_A(P_\lambda,L(\mu))=0,\;\mbox{ for~$\mu\not\succ\lambda$.}$$ This proves the requested decomposition of $P_\lambda$ into projective covers.
\[LemBasis\] Consider an algebra $(A,\preccurlyeq)$ with Borelic pair $(B,H)$. The modules $\Delta(\lambda)$ induce a ${\Bbbk}$-basis $\{[\Delta(\lambda)]\,|\, \lambda\in\Lambda\}$ of $G_0(A)$.
The indecomposable projective $A$-modules induce a basis $\{[P(\lambda)]\,|\, \lambda\in\Lambda\}$. Corollary \[CorPP\] implies that another basis is given by $\{[P_\lambda]\,|\, \lambda\in\Lambda\}$. Lemma \[PlambdaFilt\] and equation then imply that $\{[\Delta(\lambda)]\,|\, \lambda\in\Lambda\}$ also forms a basis.
By equation and Lemma \[LemBasis\], the multiplicities $(N:\Delta(\mu))_{{\mathcal{M}}}$ coincide for all possible $\Delta$-flags ${\mathcal{M}}$, if $(B,H)$ is a Borelic pair. Hence, we leave out the reference to ${\mathcal{M}}$.
The following lemma states that, in case $\preceq$ is actually a partial order, the modules $\{\Delta(\lambda)\,|\,\lambda\in\Lambda\}$ form a standard system.
\[LemDmor\]\[LemExt1\] For an algebra $(A,\preccurlyeq)$ with Borelic pair $(B,H)$ and $\lambda,\mu\in\Lambda$, we have
1. ${\mathrm{Hom}}_A(\Delta(\lambda),\Delta(\mu))=0\qquad\mbox{unless }\; \lambda\preccurlyeq \mu;$
2. ${\mathrm{Ext}}^1_A(\Delta(\lambda),\Delta(\mu))=0\quad\mbox{unless}\quad \lambda\prec\mu;$
3. ${\mathrm{Ext}}^1_A(\Delta(\lambda),M)=0$ for any $M\in A\mbox{\rm{-mod}}$ for which $[M:L(\nu)]=0$ if $\nu\succ\lambda$.
Part (1) is a consequence of Definition \[DefBorPair\](2) and Lemma \[TopStan\]. Part (2) is a special case of part (3), by Definition \[DefBorPair\](2).
Now we consider $M$ as in part (3). Consider $K$ as in Lemma \[PlambdaFilt\]. The contravariant left exact functor ${\mathrm{Hom}}_A(-,M)$ applied to yields a surjection $${\mathrm{Hom}}_A(K,M){\twoheadrightarrow}{\mathrm{Ext}}^1_A(\Delta(\lambda),M).$$ By Lemmata \[PlambdaFilt\] and \[TopStan\], the left-hand side (and therefore the right-hand side) is zero.
\[DomDelta\] Consider an algebra $(A,\preccurlyeq)$ with Borelic pair $(B,H)$. If for an $A$-module $M$ with $\Delta$-flag and $\lambda\in\Lambda$, we have $$(M:\Delta(\lambda))\not=0\qquad\mbox{and}\qquad(M:\Delta(\mu))=0\quad\mbox{if $\mu\succ\lambda$,}$$ then there exists a module $M'$ with $\Delta$-flag for which we have a short exact sequence $$0\to \Delta(\lambda)\to M\to M'\to 0.$$
This follows immediately from Lemma \[LemExt1\].
\[LemSurjN\] Consider an algebra $(A,\preccurlyeq)$ with Borelic pair $(B,H)$. If we have a surjection $M_1{\twoheadrightarrow}M_2$ where $M_1,M_2$ have $\Delta$-flags, the kernel $K$ also has a $\Delta$-flag.
We prove this by induction on the length of the $\Delta$-flag of $M_1$. Since all $\Delta(\lambda)$ have different simple top $L(\lambda)$, see Lemma \[TopStan\], it follows that there are no epimorphisms $\Delta(\lambda){\twoheadrightarrow}\Delta(\mu)$, different from identities. It follows easily that the claim in the lemma is thus true for flags of length 1, meaning if $M_1\cong \Delta(\lambda)$, for some $\lambda\in\Lambda$.
First assume that there exists $\mu\in\Lambda$ such that $(M_1:\Delta(\mu))\not=0$, but $(M_2:\Delta(\nu))=0$ for all $\nu \succcurlyeq\mu$. Without restriction, we can take such $\mu$ such that $(M_1:\Delta(\kappa))=0$, for all $\kappa\succ\mu$. By Lemma \[DomDelta\], we have $\Delta(\mu)\hookrightarrow M_1$ with cokernel with $\Delta$-flag and such that the composition with $M_1{\twoheadrightarrow}M_2$ is zero. Thus there exists a morphism $\iota$ making $$\xymatrix{
K\ar@{^{(}->}[rr]&&M_1\ar@{->>}[rr]&&M_2\\
&&\Delta(\mu)\ar@{^{(}->}[u]\ar@{.>}[llu]^{\iota}
}$$ commute. Clearly $\iota$ is a monomorphism and the kernel $K$ of $M_1{\twoheadrightarrow}M_2$ will thus have a $\Delta$-flag if the kernel $K/\Delta(\mu)$ of $M_1/\Delta(\mu){\twoheadrightarrow}M_2$ has one.
Now we assume that there exists no $\mu$ as in the previous paragraph. We take $\lambda\in\Lambda$ for which we can apply Lemma \[DomDelta\] to obtain a monomorphism $\Delta(\lambda)\hookrightarrow M_2$. If we would have $[K:L(\nu)]\not=0$, for some $\nu\succ\lambda$, then by Definition \[DefBorPair\](2), we must have $(M_2:\Delta(\mu))\not=0$, for some $\mu\succcurlyeq\nu\succ \lambda$. This contradicts the assumption in the beginning of this paragraph. By Lemma \[LemExt1\](3), we thus have ${\mathrm{Ext}}^1_{A}(\Delta(\lambda),K)=0$. This means we get a morphism $\iota$, making $$\xymatrix{
K\ar@{^{(}->}[rr]&&M_1\ar@{->>}[rr]&&M_2\\
&&\Delta(\lambda)\ar@{^{(}->}[rru]\ar@{.>}[u]^{\iota}
}$$ commute. Since $\iota$ is a monomorphism, $K$ is the kernel of $M_1/\Delta(\lambda){\twoheadrightarrow}M_2/\Delta(\lambda)$.
In both cases, we have thus reduced the problem to a case where the length of the $\Delta$-flag of $M_1$ is strictly lower.
\[corPFlag\] Consider an algebra $(A,\preccurlyeq)$ with Borelic pair $(B,H)$. Then the kernel $K$ of $P(\lambda){\twoheadrightarrow}\Delta(\lambda)$ has a $\Delta$-flag with $$(K: \Delta(\nu))=0\quad \mbox{unless} \;\;{\nu}\succ{\lambda}.$$
We prove by induction that $P(\lambda)$ has a $\Delta$-flag. If there are no $\nu\succ\lambda$, then $P_\lambda=P(\lambda)$ by Corollary \[CorPP\]. Hence it has a $\Delta$-flag by Lemma \[PlambdaFilt\].
If we already established that $P(\mu)$ has a $\Delta$-flag for all $\mu\succ\lambda$, then it follows from the inclusion $P(\lambda)\hookrightarrow P_\lambda$ of the direct summand $P(\lambda)$ into $P_\lambda$ and Lemma \[LemSurjN\] that $P(\lambda)$ has a $\Delta$-flag.
Lemma \[LemSurjN\] then further implies that the kernel of $P(\lambda){\twoheadrightarrow}\Delta(\lambda)$ has a $\Delta$-flag and the restriction on multiplicities is inherited from Lemma \[PlambdaFilt\].
Exact Borel subalgebras for exactly standardly stratified algebras {#IntroQH}
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Before we present the properties of algebras with Borelic pairs we need to review the notion of exact Borel subalgebras.
### {#DefKoe}
An [*exact Borel subalgebra of a quasi-hereditary algebra*]{} $(A,\le)$, as introduced by König in [@Koenig], is a subalgebra $B$ satisfying the following three conditions from [@KKO Definition 2.2].
(i) The simple modules of $B$ are also labelled by $\Lambda_A$, so $\Lambda_A=\Lambda_B=\Lambda$ and we denote the simple $B$-modules by $\{L^0(\lambda)\,|\, \lambda\in\Lambda\}$. The only simple subquotients in the radical of the projective cover of $L^0(\lambda)$ are $L^0(\mu)$ with $\mu>\lambda$.
(ii) The functor $A\otimes_B-$ is exact.
(iii) There is an isomorphism $A\otimes_B L^0(\lambda)\cong S(\lambda)$ for each $\lambda\in\Lambda$.
For an algebra $B$ with simple modules $L^0(\lambda)$ for $\lambda\in\Lambda$, choose a decomposition $1=\sum_{\lambda\in\Lambda}\overline{e}_\lambda$, where $\{\overline{e}_\lambda\}$ is a set of mutually orthogonal idempotents, satisfying $\overline{e}_\mu L^0(\lambda)=0$ unless $\mu=\lambda$.
\[BorelWed\] An exact Borel subalgebra $B$ of a quasi-hereditary algebra $(A,\le)$ is Wedderburn. We have $$B\;=\; H\,\oplus\, {{\rm rad}}B\qquad\mbox{ with }\; H:=\bigoplus_{\lambda\in\Lambda}\overline{e}_\lambda\, B\, \overline{e}_\lambda.$$
By condition (i) above, we have $\overline{e}_\mu \,B \,\overline{e}_\lambda=0$ unless $\mu \ge \lambda$, showing that $$\oplus_{\mu\not=\lambda} \overline{e}_\lambda \,B\, \overline{e}_\mu=\oplus_{\mu>\lambda} \overline{e}_\lambda\, B \,\overline{e}_\mu\;\subset\; {{\rm rad}}B.$$ That $\overline{e}_\lambda \,B \,\overline{e}_\lambda$ is semisimple follows immediately from the fact that $[B\overline{e}_\lambda:L^0(\lambda)]$ is equal to the number of direct summands in $B\overline{e}_\lambda$ by condition (i). Hence, $H$ is semisimple and the displayed inclusion an equality.
In [@Kluc Definition 1], the notion of exact Borel subalgebras was generalised to properly stratified algebras and it can be further generalised to exactly standardly stratified algebras.
\[DefEB\] An [*exact Borel subalgebra of an exactly standardly stratified algebra*]{} $(A,\preccurlyeq)$ is a subalgebra $B$ such that the following four conditions are satisfied.
1. The simple modules of $B$ are also labelled by $\Lambda_A$, so $\Lambda_A=\Lambda_B=\Lambda$ and we denote the simple $B$-modules by $\{L^0(\lambda)\,|\, \lambda\in\Lambda\}$. We have $\overline{e}_\mu \, B\, \overline{e}_\lambda=0$ unless $\lambda\preccurlyeq \mu$.
2. The functor $A\otimes_B-$ is exact.
3. The algebra $$H:=\bigoplus_{\lambda\sim\mu} \overline{e}_\lambda\, B\,\overline{e}_\mu\;\cong\; B/\left(\bigoplus_{\mu \prec\lambda}\overline{e}_\lambda \, B\, \overline{e}_\mu\right)$$ is such that the left $H$-module ${}_HB$ is projective. The interpretation of $H$ as a quotient algebra of $B$ allows to interpret $H$-modules as $B$-modules by inflation.
4. There are isomorphisms $A\otimes_B L^0(\lambda)\cong \overline{S}(\lambda)$ and $A\otimes_B P^0(\lambda)\cong {S}(\lambda)$, with $P^0(\lambda)$ the $H$-projective cover of $L^0(\lambda)$, for each $\lambda\in\Lambda$.
\[CompareEB\] There is no conflict between Definition \[DefEB\] and the special case of quasi-hereditary algebras in \[DefKoe\]:
- An exact Borel subalgebra $B$ of a quasi-hereditary algebra $(A,\le)$ is an exact Borel subalgebra of $(A,\le)$, interpreted as an exactly standardly stratified algebra. This follows from Lemma \[BorelWed\].
- Consider an exact Borel subalgebra $B$ of an exactly standardly stratified algebra $(A,\preccurlyeq)$ which happens to be quasi-hereditary. As $S(\lambda)\cong \overline{S}(\lambda)$, Definition \[DefEB\](4) implies that $H$ is semisimple. It then follows that the multiplicity of $L^0(\lambda)$ in its projective cover over $B$ is 1. Hence $B$ is an exact Borel subalgebra of $(A,\preccurlyeq)$ as a quasi-hereditary algebra.
If $(A,\preccurlyeq)$ is properly stratified, Definition \[DefEB\] is almost identical to [@Kluc Definition 1]. The latter additionally requires projectivity of $B_H$.
Algebras with (exact) Borelic pairs {#Sec4}
-----------------------------------
\[ThminfQH\] If an algebra $A$ admits a Borelic pair $(B,H)$ for an $H$-compatible partial quasi-order $\preccurlyeq$, then $(A,\preccurlyeq)$ is standardly stratified with $S(\lambda):=\Delta(\lambda)$. If moreover $\preccurlyeq$ is a partial order (implying that $H$ is quasi-local) then $(A,\preccurlyeq)$ is strongly standardly stratified.
Consider $(A,\preccurlyeq)$ with Borelic pair $(B,H)$. That the modules $S(\lambda):=\Delta(\lambda)$ satisfy Definition \[DefA2\](1) follows from Definition \[DefBorPair\](2) and Corollary \[corPFlag\]. The same reasoning for the case where $\preccurlyeq$ is a partial order implies that $(A,\preccurlyeq)$ is strongly standardly stratified as in Definition \[DefA2\](3).
\[PropKo\] Consider an algebra $(A,\preccurlyeq)$ for some partial quasi-order $\preccurlyeq$.
1. Assume that $\preccurlyeq$ is a partial order. The following statements are equivalent, for any subalgebra $B$ of $A$:
- $B$ is Wedderburn and $(B,B/{{\rm rad}}B)$ is an exact Borelic pair.
- $(A,\preccurlyeq)$ is quasi-hereditary and $B$ is an exact Borel subalgebra.
2. Assume that $\preccurlyeq$ is a partial order. The following statements are equivalent, for any subalgebras $H\subset B$ of $A$:
- $(B,H)$ is an exact Borelic pair.
- $(A,\preccurlyeq)$ is properly stratified and $B$ is an exact Borel subalgebra.
3. The following statements are equivalent, for any subalgebras $H\subset B$ of $A$:
- $(B,H)$ is an exact Borelic pair.
- $(A,\preccurlyeq)$ is exactly standardly stratified and $B$ is an exact Borel subalgebra.
We start the proof of this theorem with the following lemma.
\[LempreBor\] If $A$ is exactly standardly stratified with an exact Borel subalgebra $B$ with subalgebra $H$ as in Definition \[DefEB\], then $(B, H)$ is a pre-Borelic pair.
We check the conditions in Definition \[defB\]. Condition \[defB\](I) follows from Definition \[DefEB\](2) and (3). We define the two-sided ideal $B_+:=\oplus_{\lambda\not\sim\mu}\overline{e}_\lambda B\overline{e}_\mu=\oplus_{\lambda\succ\mu}\overline{e}_\lambda B\overline{e}_\mu$, for which we have $B=H\oplus B_+$. Condition \[defB\](III) is clearly satisfied.
As the modules ${S}(\lambda)$ of \[DefA2\](1) have simple top $L(\lambda)$, Definition \[DefEB\](4) leads, using adjunction, to $$\delta_{\lambda,\mu}\;=\;\dim{\mathrm{Hom}}_B(P^0(\lambda),L(\mu))\;=\;\dim{\mathrm{Hom}}_H(P^0(\lambda),L(\mu)^{B_+})\;=\; [L(\mu)^{B_+}: L^0(\lambda)].$$ Condition \[defB\](II) follows from this equation.
It suffices to prove part (3). Part (2) is a special case and so is part (1) by remark \[CompareEB\].
Firstly, we assume that $(B,H)$ is an exact Borelic pair. By Theorem \[ThminfQH\], $A$ is standardly stratified. Furthermore, the standard module $S(\lambda)$ has a filtration where the sections are given by $\overline{S}(\mu):=\overline{\Delta}(\mu)$ by Definition \[defB\](I), with $\mu\sim_H\lambda$. By Definition \[DefBorPair\] we have $[\overline{S}(\mu):L(\mu)]=1$, so $(A,\preccurlyeq)$ is exactly standardly stratified.
Now we continue by proving that $(B,\preccurlyeq)$ is an exact Borel subalgebra, as in Definition \[DefEB\]. Condition \[DefEB\](1) is satisfied by equation and Definition \[DefBorPair\](1). Condition \[DefEB\](2) follows from \[defB\](I). To prove the two remaining conditions \[DefEB\](3) and (4), we just need to show that the subalgebra of $B$ defined in \[DefEB\](3), which we denote by $H'$, is the same as the subalgebra $H$ in the pair $(B,H)$. Definition \[DefBorPair\](1) implies that $H'\subset H$. The other inclusion follows from the fact that $\sim$ is assumed to be $H$-compatible.
Secondly, we assume that $(A,\preccurlyeq)$ is exactly standardly stratified and $B$ is an exact Borel subalgebra. By Lemma \[LempreBor\], $(B,H)$ is a pre-Borelic pair, so it remains to check the conditions in Definition \[DefBorPair\]. Condition \[DefBorPair\](1) follows from \[DefEB\](1) and \[DefEB\](3). Condition \[DefBorPair\](2) follows from Definitions \[DefA2\](1) and \[DefEB\](4). Hence $(B,H)$ is a Borelic pair. The extra condition for exact Borelic pairs follows from \[DefEB\](4).
A special case: graded Borelic subalgebras {#ssspecial}
------------------------------------------
Fix an idempotent graded algebra $A$ as in Section \[SecPreSpec\]. We have the corresponding chain of idempotent ideals and hence an $n$-decomposition ${\mathcal{Q}}$ of $\Lambda=\sqcup_{i=0}^n\Lambda_i$, as in Section \[SecChain\], with corresponding quasi-order and partial-order. We now introduce these directly.
\[lepleq\] Introduce the function $F_A:\Lambda\to \{0,1,\cdots,n\}$, where $F_A(\lambda)$ is the minimal $i\in{\mathbb{N}}$ such that there is an idempotent $f\in e_i^\ast Ae_i^\ast$, equivalent to $e_\lambda$. Then we have $\Lambda_i:=\{\lambda\in\Lambda\,|\, F_A(\lambda)=i\}$.
1. The [*idempotent quasi-order*]{} is the total quasi-order $\preccurlyeq_{{\mathcal{Q}}}$ on $\Lambda$ such that $\mu\preccurlyeq_{{\mathcal{Q}}} \lambda$ if and only if $F_A(\mu)\ge F_A(\lambda)$.
2. The [*idempotent partial order*]{} is the partial order $\le_{{\mathcal{Q}}}$ on $\Lambda$ such that $\mu<_{{\mathcal{Q}}} \lambda$ if and only if $F_A(\mu)> F_A(\lambda)$
\[LemHcomp\] For any $\lambda\in\Lambda$ we have $e_\lambda^0\in He^\ast_i$ with $i=F_A(\lambda)$. In particular, the partial quasi-order $\preccurlyeq_{{\mathcal{Q}}}$ is $H$-compatible.
It follows from equation that $L^0(\lambda)$ is a vector space contained in the highest degree of the graded vector space $L(\lambda)$. This implies that $e^\ast_iL^0(\lambda)\not=0$ for the lowest $i$ for which there is an idempotent $e$ equivalent to $e_\lambda$ such that $ee^\ast_i=e$.
Hence, for that minimal $i$, which is $F_A(\lambda)$, we have $e_\lambda^0e_i^\ast=e^0_\lambda$.
\[grBrBo\] Consider an algebra $A$ with an idempotent grading. Let $B$ be a graded pre-Borelic subalgebra with $H:=B_0$.
1. The pair $(B,H)$ is an exact Borelic pair for the idempotent quasi-order $\preccurlyeq_{{\mathcal{Q}}}$.
2. If $H$ is quasi-local, $(B,H)$ is an exact Borelic pair for the idempotent partial order $\le_{{\mathcal{Q}}}$.
Combining this with Theorem \[PropKo\] yields the following corollary.
\[CorgrBrBo\] Consider an algebra $A$ with an idempotent grading and graded pre-Borelic subalgebra $B$ with $H:=B_0$.
1. Then $(A,\preccurlyeq_{{\mathcal{Q}}})$ is exactly standardly stratified with exact Borel subalgebra $B$.
2. If $H$ is quasi-local, $(A,\le_{{\mathcal{Q}}})$ is properly stratified with exact Borel subalgebra $B$.
3. If $H$ is semisimple, $(A,\le_{{\mathcal{Q}}})$ is quasi-hereditary with exact Borel subalgebra $B$.
The corollary indicates that our methods might be well-suited to extend to the setting of [@AffineK]. With notation introduced in Appendix \[AppKl\], it can be paraphrased as: If an idempotent graded algebra $A$ has graded pre-Borelic subalgebra $B$ with $H\in{\mathcal B}$, then $A$ is ${\mathcal B}$-properly stratified, for ${\mathcal B}$ one of the classes of algebras ${\mathcal{D}}$, ${\mathcal{L}}$ or ${\mathcal{S}}$.
First observe that $\overline{\Delta}(\lambda)$, with grading as in , lives in degrees $\{-j,-j-1,\ldots\}$ with $j=F_A(\lambda)$, by Lemma \[LemHcomp\] and equation . Moreover, the degree $-j$-component is precisely $L^0(\lambda)$. Each simple module $L(\mu)$ lives similarly in degree $\{-k,-k-1,\ldots\}$ with $k=F_A(\mu)$. Therefore we find that $$\label{deltabar}
[\overline{\Delta}(\lambda):L(\lambda)]=1\quad\mbox{ and }\quad [\overline{\Delta}(\lambda):L(\mu)]=0\mbox{ unless $\mu\le_{{\mathcal{Q}}}\lambda$}.$$ Now we focus on the proof of part (1). By we only need to check conditions (1) and (2) in Definition \[DefBorPair\]. We have $$[B_+e^0_\lambda:L^0(\mu)]=\sum_{j>0}\dim e_\mu^0 B_j e_\lambda^0.$$ By Lemma \[LemHcomp\] we find that $e^0_\mu B_j e^0_\lambda\not=0$ for $j>0$ implies that $\mu\succ_{{\mathcal{Q}}}\lambda$, proving condition \[DefBorPair\](1). The module $\Delta(\lambda)=A\otimes_{B}He_\lambda^0$ has a filtration with sections $\overline{\Delta}(\nu)=A\otimes_{B}L^0(\nu)$ with $\nu\sim_H \lambda$, by Definition \[defB\](I). Hence, by equation , $[\Delta(\lambda):L(\mu)]\not=0$ implies there is a $\nu\in\Lambda$ with $$\mu\,\le_{{\mathcal{Q}}}\,\nu\,\sim_H \lambda\qquad\mbox{and hence}\qquad \mu\preccurlyeq_{{\mathcal{Q}}} \lambda.$$ So condition \[DefBorPair\](2) for $\preccurlyeq_{{\mathcal{Q}}}$ is satisfied.
For the proof of part (2) we only need to observe that $<_{{\mathcal{Q}}}$ and $\prec_{{\mathcal{Q}}}$ are identical and that, since $H$ is quasi-local, now $\Delta(\lambda)$ has a filtration where each section is isomorphic to $\overline{\Delta}(\lambda)$.
By the proofs in Sections \[SecPreSpec\] and \[ssspecial\], we find that Corollary \[CorgrBrBo\] remains true for a ${\mathbb{N}}$-graded subalgebra $B$, such that $A_B$ and ${}_HB$ are projective, and for which $$A_+\subset AB_+\qquad\mbox{and}\qquad
A_0=H\oplus (A B_+)_0.$$ However, in all examples of quasi-hereditary algebras with exact Borel subalgebras we will encounter in Part II, the stronger condition is satisfied.
Base stratified algebras {#SecBase}
========================
Definition
----------
The following definition is inspired by [@HHKP Propositions 5.1 and 5.2].
\[BasedId\] A [*base stratifying ideal*]{} in an algebra $A$ is an idempotent ideal $J=AeA$ which is exactly standardly stratifying, [*i.e.*]{} ${}_AJ$ and $J_A$ are projective, see \[SecStratId2\], and such that $eAe$ is standardly based, see Section \[SecSBalg\].
\[ExQHBS\] By Corollary \[CorB\], [*any*]{} exactly standardly stratifying ideal can be given the structure of a base stratifying ideal, when working over an algebraically closed field.
By equation and Remark \[remMorSB\], the definition above does not intrinsically depend on the choice of the idempotent $e$.
We use Definition \[BasedId\] to introduce a generalisation of the concept of “cellularly stratified algebra” of [@HHKP Definition 2.1], see also [@HHKP Section 5].
\[DefBS\] Consider an algebra $A$ with an $n$-decomposition ${\mathcal{Q}}$: $\Lambda=\sqcup_{i=0}^n\Lambda_i$. We consider the corresponding chain of idempotent ideals $\{J_i\,|\, 0\le i\le n+1\}$ of \[SecChain\]. Then $(A,{\mathcal{Q}})$ is [*base stratified*]{} if each $J_i/J_{i-1}$ is a base stratifying ideal in $A/J_{i-1}$.
It follows easily from Remark \[remMorSB\] and the equivalence of Definition \[DefA1\](2) and \[DefA2\](2) that a base stratification is a property of the Morita class of an algebra.
\[RemParTriv\] A trivial example of a base stratified algebra is any standardly based algebra with the $0$-decomposition of $\Lambda$. It is hence essential to keep track of the decomposition ${\mathcal{Q}}$ of a base stratified algebra $(A,{\mathcal{Q}})$ in order to make non-trivial statements.
### {#DefL}
For a base stratified algebra $(A,{\mathcal{Q}})$, we make a choice of idempotents $\{f_l\,|\, 1\le l\le n+1\}$ as in Remark \[DefAi\] and consider the corresponding algebras $A^{(l)}$. By definition $A^{(l)}$ is standardly based for some poset $L_l$. We consider the set $L:=\sqcup_{i=0}^n L_i=\{(i,p)\,|\, 1\le i\le n\,,\; p\in L_i\}$. We make $L$ into a poset by setting $(i,p)\le (j,q)$ if $i>j$, or $i=j$ and $p\le q$, so we take the lexicographic ordering, up to reversal of the order on natural numbers.
\[ThmBaSt\] A base stratified algebra $(A,{\mathcal{Q}})$ is standardly based for the poset $L$. The cell modules of $A$ satisfy $$W(i,p)\;\cong\; A f_{i+1}\otimes _{f_{i+1}Af_{i+1}} W^0_i(p),$$ with $W^0_i(p)$ the cell modules of $A^{(i)}$, regarded as $f_{i+1} Af_{i+1}$-modules with trivial $f_{i+1}J_{i}f_{i+1}$-action.
We use induction on $n$, where ${\mathcal{Q}}$ is an $n$-decomposition. If $n=0$, there is nothing to prove. Assume the statement is true for $n-1$ and consider the case $n$. Clearly the algebra $\tilde{A}:=A/J_1$ is base stratified and hence standardly based for the poset $L'=\sqcup_{i=1}^n L_i$. We fix corresponding ideals $\tilde{A}^{\ge (l,p)}$ with $1\le l\le n$ and $p\in L_l$.
We set $K:=A^{(0)}=f_1Af_1$. By Definition \[BasedId\] and Lemma \[LemStu\] we have $$J_1\;\cong\; Af_1\otimes_{f_1Af_1}f_1A\;\cong\; Af_1\otimes_K K\otimes_K f_1A.$$ By assumption, $K$ is standardly based and we set $$A^{\ge (0,p)}\;:=\;Af_1\otimes_K K^{\ge p}\otimes_K f_1A,$$ for all $p\in L_0$. The fact that the quotients $A^{\ge (0,p)}/A^{> (0,p)}$ satisfy the desired properties then follows from the exactness of the functors $Af_1\otimes_K-$ and $-\otimes_K f_1A$ in Lemma \[LemStu\]. In particular, we have $$W(0,p)\;\cong\; A f_{1}\otimes _{K} W^0_0(p),$$
The set of ideals of $A$ is then completed by taking $A^{\ge(l,p)}$ to be an arbitrary ideal in $A$ containing $J_1$ with $A^{\ge(l,p)}/J_1=\tilde{A}^{\ge (l,p)}$ for $l>0$ and $p\in L_l$.
It follows from the considerations in Remark \[MoreAe\] that the cell module $W(i,p)$ of $A$ does not depend on the actual choice of idempotents $\{f_i\}$.
Our setup gives a very simple proof of a generalisation of [@HHKP Theorem 10.2].
\[corHKKP\] Maintain the notation and assumptions of Theorem \[ThmBaSt\].
1. For $0\le i\le j\le n$, $p\in L_i$ and $q\in L_j$, we have $${\mathrm{Ext}}^k_A(W(i,p), W(j,q))\;\cong\; \delta_{i,j}\,{\mathrm{Ext}}^k_{A^{(i)}}(W^0_i(p),W_i^0(q)),\qquad\mbox{for all $k\in{\mathbb{N}}$}.$$
2. The cell modules of $A$ form a standard system for poset $L$ if and only if, for each $0\le i\le n$, the cell modules of $A^{(i)}$ form a standard system for poset $L_i$.
Part (1) follows from Theorem \[ThmBaSt\] and Lemma \[NewLemStr\], part (2) from part (1).
As, by definition, a base stratified algebra is exactly standardly stratified, it has (proper) standard modules $S(\lambda)$ and $\overline{S}(\lambda)$, for $\lambda\in\Lambda$, see Section \[SecStrat\]. For easy comparison with the cell modules we actually write $S(i,\lambda)$, instead of $S(\lambda)$, for $\lambda\in\Lambda_i\subset\Lambda$, with same convention for proper standard modules.
Consider a base stratified algebra $(A,{\mathcal{Q}})$. For any fixed $i$, $\lambda\in\Lambda_i$ and $p\in L_i$, we have
- $S(i,\lambda)$ has a filtration where the sections are modules $W(i,q)$, $q\in L_i$ with $q\ge \lambda$;
- $W(i,p)$ has a filtration where the sections are modules $\overline{S}(i,\mu)$, $\mu\in\Lambda_i$.
The filtration of $S(i,\lambda)$ is induced from the filtration of the projective left $A^{(i)}$-module using its cell modules. The filtration of $W(i,p)$ is induced from the Jordan-Hölder decomposition series of the $A^{(i)}$-module $W^0_i(\lambda)$.
Base-Borelic pairs {#BBP}
------------------
Given an algebra $A$, we fix a decomposition of unity as in . Assume that we have a graded pre-Borelic algebra $B$ as in Definition \[GrPreBor\]. We have the corresponding decomposition ${\mathcal{Q}}$ as in Section \[SecPreSpec\].
\[ThmIdGrB\] Consider an algebra $A$ with a graded pre-Borelic algebra $B$ such that $H=B_0$ is standardly based. Then $(A,{\mathcal{Q}})$ is base stratified.
Corollary \[CorgrBrBo\](1) and the equivalence of Definition \[DefA1\] and \[DefA2\] implies that the chain is exactly standardly stratifying. Equation then implies the chain is base stratifying.
\[CorStB\] Consider an algebra $A$ with a graded pre-Borelic algebra $B$ such that $H$ is standardly based. For each $0\le i\le n$ consider the poset $L_i$ (containing $\Lambda_i$) of the standardly based algebra $ H e^\ast_i$. Then $A$ is standardly based for the poset $L$ defined in \[DefL\].
Moreover, the cell modules over $A$ form a standard system for $L$, if and only if the cell modules for $H e^\ast_i$ form a standard system for $L_i$, for each $0\le i\le n$.
The first statement follows from the combination of Theorems \[ThmIdGrB\] and \[ThmBaSt\]. The claim about standard systems follows from Proposition \[corHKKP\](2) and Theorem \[chainProp\](1).
\[CellB\] Maintain the notation and assumptions of Corollary \[CorStB\]. The cell modules of the algebra $A$ satisfy $$W(i,p)\cong\Delta_{W^0_i(p)}=A\otimes_B W^0_i(p),$$ with $W^0_i(p)$ the cell module for $He^\ast_i$, with $p\in L_i$.
This follows from Lemma \[LemABAM\] and Theorem \[ThmBaSt\].
Schur algebras
--------------
The following observation is implicit in [@HHKP Sections 11-13].
\[LemSchur\] Consider a standardly based algebra $A$. The cell modules $\{W(p), \,p\in L\}$, form a standard system for $(L_A,\le)$ in $A$[-mod]{} if and only if A admits a cover-Schur algebra as in Definition \[DefSchur\].
The existence of a Schur algebra clearly implies that the cell modules form a standard system. Now consider a standardly based algebra such that $\{W(p)\}$ forms a standard system.
Set $W=\oplus_p W(p)$. We denote by ${{\mathcal{F}}}(W)$ the exact category of $A$-modules with filtrations where each section is a direct sum of summands of $W$. By [@DR Section 3], ${{\mathcal{F}}}(W)$ contains unique indecomposable objects $Y(p)$, with $p\in L$, such that ${\mathrm{Ext}}^1_A(Y(p),W)=0$ and there is a surjection $Y(p){\twoheadrightarrow}W(p)$ with kernel in ${{\mathcal{F}}}(W)$. Since $P(\lambda)\in {{\mathcal{F}}}(W)$, by Section \[SecSBalg\], we have in particular $Y(\lambda)\cong P(\lambda)$ for $\lambda\in\Lambda_A\subset L_A$. Hence, we can take $m_p\in {\mathbb{N}}$ such that $$A\cong {\mathrm{End}}_A(\bigoplus_{\lambda\in\Lambda_A}P(\lambda)^{\oplus m_\lambda})^{{{\rm op}}},\quad\mbox{and we define} \quad {\mathcal{S}}:= {\mathrm{End}}_A(\bigoplus_{p\in L_A}Y(p)^{\oplus m_p})^{{{\rm op}}}.$$ Consider the idempotent $f\in{\mathcal{S}}$ corresponding to the projection of $\bigoplus_{p\in L_A}Y(p)^{\oplus m_p}$ onto the summand $ \bigoplus_{\lambda\in\Lambda_A}P(\lambda)^{\oplus m_\lambda}$. Then we have $f{\mathcal{S}}f\cong A$ and we can interpret $\bigoplus_{p\in L_A}Y(p)^{\oplus m_p}$ as a $(A,{\mathcal{S}})$-bimodule of the form $f{\mathcal{S}}$.
By [@DR Section 3], $({\mathcal{S}},\le)$ is quasi-hereditary and the functor $$G={\mathrm{Hom}}_A(\bigoplus_{p\in L_A}Y(p)^{\oplus m_p},-)\;:\; A\mbox{-mod}\to {\mathcal{S}}\mbox{-mod}.$$ is exact and fully faithful on ${{\mathcal{F}}}(W)$ and maps the cell modules of $A$ to the standard modules of ${\mathcal{S}}$. Since the functor $$F=f-\;:\; {\mathcal{S}}\mbox{-mod}\to A\mbox{-mod}.$$ satisfies $F\circ G={\mathrm{Id}}$, it follows immediately that $F$ induces isomorphisms of the appropriate homomorphism spaces and first extension groups, which implies ${\mathcal{S}}$ is a $1$-faithful quasi-hereditary cover.
\[Examp\]
Auslander-Dlab-Ringel algebras {#SecADR}
==============================
In [@Auslander], Auslander proved that each finite dimensional unital algebra $R$ has a cover $A(R)$ with finite global dimension. In [@DRAus], Dlab and Ringel proved that $A(R)$ is quasi-hereditary. In this section, we prove that $A(R)$ admits an exact Borel subalgebra (while giving an independent proof of the quasi-heredity), under the weak assumption that $R$ is Wedderburn.
The Auslander construction {#introAus}
--------------------------
There are several Morita equivalent versions of the algebra $A(R)$ which appear in the literature. Let $R$ be a ${\Bbbk}$-algebra. Consider the [*right*]{} regular $R$-module and its radical, which is the Jacobson radical ${\mathcal{J}}$ of $R$. For each $i\ge 1$, we let $M_R^i$ be the largest direct summand of the [*right*]{} $R$-module $R/{\mathcal{J}}^i$ such that no direct summand of $M_R^i$ appears as a direct summand of some $M_R^j$ for $j<i$. We set $n$ to be the largest integer for which $M_R^n\not=0$. Alternatively, $n$ is defined as the smallest positive integer for which ${\mathcal{J}}^{n}=0$.
The [*ADR-algebra*]{} $A(R)$ is the cover of $R$ defined as $$A(R):={\mathrm{End}}_R(\oplus_{i=1}^nM_R^i).$$ We label the simple [*right*]{} $R$-modules by $\Lambda^0:=\Lambda_{R^{{{\rm op}}}}$. We have idempotents $\{c^\ast_\lambda\,|\,\lambda\in\Lambda^0\}$ in $R$ such that the top of the right $R$-module $c_\lambda^\ast R$ is of type $\lambda$ and such that $1_R=\sum_\lambda c^\ast_\lambda$.
For each $\lambda\in \Lambda^0$, let $\ell(\lambda)$ be the Loewy length of $c_\lambda^\ast R$. For $l\in{\mathbb{N}}$ we set $$c_{\ge l}^\ast=\sum_{\lambda\in\Lambda^0\,|\, \ell(\lambda)\ge l}c_\lambda^\ast,$$ so in particular $c_{\ge 1}^\ast=1_R$. Then we have $$M_R^i\;\cong\;c_{\ge i}^\ast R/{\mathcal{J}}^i,\quad\mbox{ for~$1\le i\le n$}.$$ In particular, every direct summand of $M_R^i$ has Loewy length $i$.
Construction of the Borelic subalgebra
--------------------------------------
From now on, we assume that $R$ is Wedderburn, meaning $R=S\oplus{\mathcal{J}}$, for a semisimple subalgebra $S$. Then we have a direct sum of simple algebras $$S\;=\;\bigoplus_{\lambda\in\Lambda^0} S c_\lambda^\ast,$$ where the idempotent $c_\lambda^\ast$ is now central in $S$. The space $c_\lambda^\ast S$ is then naturally a subspace of $c_\lambda^\ast R/{\mathcal{J}}^i$, which is bijectively mapped to the top of $c_\lambda^\ast R/{\mathcal{J}}^i$ under the canonical surjection.
We let $e^\ast_i\in A(R)$ be the projection onto the summand $M_R^{i}$ for $1\le i\le n$. This yields a decomposition of unity as in , with $e_0^\ast=0$. We define a subspace $B=\bigoplus_{i,j}e_i^\ast B e_j^\ast$ of $A(R)$, by $$e_i^\ast B e_j^\ast \;:=\;\{\alpha\in{\mathrm{Hom}}_R(M_R^j,M_R^i)\,|\, \alpha(c_{\ge j}^\ast) \in c_{\ge i}^\ast S\subset M^i_R\},\qquad\forall\;1\le i,j\le n.$$ In particular, we find $e_i^\ast B e_j^\ast =0$ if $i>j$, by considering Loewy lengths. If $i\le j$, then $c_{\ge j}^\ast c_{\ge i}^\ast =c_{\ge j}^\ast $, so $\alpha\in e_i^\ast Be_j^\ast$ even implies $\alpha(c_{\ge j}^\ast) \in c_{\ge j}^\ast S$.
\[LemmaAus\] The subspace $B$ is a graded pre-Borelic subalgebra of $A(R)$ regarded as an idempotent graded algebra.
Set $A:=A(R)$. By construction, $B$ is a ${\mathbb{Z}}$-graded subspace of $A$ containing $1_A$. Now we prove that $B$ is in fact an algebra. Take $\alpha\in e^\ast_i B e^\ast_k$ and $\beta\in e^\ast_k Be^\ast_j$ with $i\le k\le j$. It follows easily that $\alpha\beta\in B$ from the fact that $S$ is a subalgebra of $R$.
To prove that $B$ is complete, it suffices to prove equation , [*viz.*]{} that for $i\le j$, $$\label{rewcom}e^\ast_i Ae^\ast_j\;=\; e^\ast_i B e^\ast_j\oplus \sum_{k <i}e^\ast_i Ae^\ast_k Ae^\ast_j.$$ For this we observe that the morphisms in $e_i^\ast Be_j^\ast$ are spanned by the surjections from indecomposable summands in $M_R^j$ onto their factor modules appearing in $M_R^i$. A natural complement of that space is given by morphisms from $M_R^j$ with image in the radical of $M_R^i$. As this image will have Loewy length strictly lower than $i$, the morphism factors through $M_R^k$ (a direct sum of all factor modules of Loewy length $k$ of the projective $R$-modules) for some $k<i$. This complement is hence precisely $\sum_{k <i}e^\ast_i Ae^\ast_k Ae^\ast_j$.
Now we show that the complete ${\mathbb{N}}$-graded subalgebra $B$ is pre-Borelic. The algebra $$\label{eqHADR}H=B_0\cong\bigoplus_{i=1}^n c_{\ge i}^\ast S$$ is semisimple and therefore ${}_HB$ is projective. Now we prove that $A_B$ is projective. Take first $\alpha\in e_k^\ast Ae_1^\ast$, for some $k\ge 1$. By construction, $\alpha$ corresponds to a monomorphism $D\hookrightarrow M_R^k$, with $D$ a direct summand of the semisimple module $M_R^1$. Let $e$ denote the idempotent in $e_1^\ast Ae_1^\ast\cong S$ such that $D=M^1_Re$. In particular, we have $\alpha=\alpha e$. Since $\alpha$, restricted to $D$ is injective, it follows that the canonical epimorphism $eB{\twoheadrightarrow}\alpha B$ given by $x\mapsto \alpha x$ is an isomorphism. It follows that $$Ae_1^\ast A=Ae_1^\ast B$$ is a projective right $B$-module.
Now assume that we have proved that $A(\sum_{k<i}e_i^\ast)A$ is projective as a right $B$-module, for $i>1$. We take $\alpha\in Ae_i^\ast$ and associate a right $R$-module $$N_\alpha=\{v\in M^i_R\,|\,\alpha(v)R\mbox{ has Loewy length strictly less than $i$.}\}$$ The image of $N_\alpha$ under the projection $M^i_R{\twoheadrightarrow}{\mathrm{Top}}M^i_R$, yields a direct summand $X_{\alpha}$ of ${\mathrm{Top}}M^i_R$. Note that $N_\alpha=N_{\alpha'}$ if and only if $X_\alpha=X_{\alpha'}$, since the radical of $M^i_R$ has Loewy length $i-1$. We have $X_\alpha={\mathrm{Top}}(M^i_R) f$ for some idempotent $f\in Sc^\ast_{\ge i}\subset e_i^\ast Ae_i^\ast$. It follows that $N_\alpha=N_e$, for the idempotent $e:=e_2^\ast-f$. Observe that $A(\sum_{k<i}e_i^\ast)A$ is the ideal of morphisms whose image has Loewy length strictly less than $i$. For $\beta\in e_i^\ast B$, we thus have $\alpha\beta\in A(\sum_{k<i}e_i^\ast)A$ if and only if ${{\rm{im}}}\beta\subset N_\alpha=N_e$. Working inside the right $B$-module $A/(A(\sum_{k<i}e_i^\ast)A)$, it thus follows that $\alpha B\cong eB$.
Hence we find that $A(\sum_{k\le i}e_i^\ast)A/A(\sum_{k<i}e_i^\ast)A$ is also projective. This means that also $A(\sum_{k\le i}e_i^\ast)A$ is projective. It follows that $A_B$ is projective by iteration, which concludes the proof.
Main theorem on Auslander-Dlab-Ringel algebras
----------------------------------------------
\[ThmAus\] Consider a field ${\Bbbk}$. If the algebra $R$ is Wedderburn (for instance if ${\Bbbk}$ is perfect), the simple modules of the ADR algebra $A(R)$ are labelled by $$\label{LAR}\Lambda_{A(R)}=\{(i,\lambda)\; \mbox{with }\;1\le i\le \ell(\lambda) \mbox{ and }\lambda\in \Lambda^0\}.$$ Moreover, $\left(A(R),\le_{{\mathcal{Q}}}\right)$ is quasi-hereditary and admits an exact Borel subalgebra.
Since $H$ in equation is semisimple, the statement follows from Lemma \[LemmaAus\], by using Corollary \[CorgrBrBo\](3) and equation .
By [@DRAus], the ADR algebra is quasi-hereditary, regardless of wether $R$ is Wedderburn. However, if $R$ is not Wedderburn, Theorem \[ThmAus\] does not yield an exact Borel subalgebra. In this case, the field ${\Bbbk}$ cannot be algebraically closed and the results in [@KKO] also do not ensure the existence of exact Borel subalgebras in Morita equivalence classes.
All algebras are standardly based
---------------------------------
\[AllBased\] Any algebra over an algebraically closed field ${\Bbbk}$ is standardly based.
Consider an arbitrary algebra $R$. As the field is perfect, Theorem \[ThmAus\] implies that $A(R)$ is quasi-hereditary. By [@JieDu Theorem 4.2.7], $A(R)$ is standardly based. As $R\cong e^\ast A(R)e^\ast$ for some idempotent $e^\ast$, it follows that $R$ is also standardly based, by [@Yang Proposition 3.5].
### {#Explanation}
The standardly based structure can be derived explicitly from the above proof. Take a simple module $L$ in the socle of the right regular $R$-module. Acting on this with the left $R$-action leads to a two-sided ideal $W\otimes L$ in $R$ for some left $R$-module $W$. Factoring out this ideal and continuing the procedure yields the structure.
Jie Du informed us that he was aware of this result. In the unpublished manuscript [@preprint], it is proved that any split finite dimensional algebra over any field is standardly based, using the same approach as in \[Explanation\].
Example from Lie theory {#SecBGG}
=======================
We use the general theory to construct a very elementary Lie theoretic example of a properly stratified algebra with exact Borel subalgebra. A more advanced example can be found in [@Kluc Section 7]. In this section we set ${\Bbbk}={\mathbb{C}}$.
Thick category ${\mathcal O}$
-----------------------------
We consider the category ${\mathcal O}^{k}$, for $k\in{\mathbb{Z}}_{\ge 1}$, studied in e.g. [@Soergel]. For a reductive Lie algebra ${\mathfrak{g}}$, with triangular decomposition ${\mathfrak{g}}={\mathfrak{n}}^-\oplus{\mathfrak{h}}\oplus {\mathfrak{n}}^+$ and Borel subalgebra ${\mathfrak{b}}={\mathfrak{h}}\oplus{\mathfrak{n}}^+$, the category ${\mathcal O}^{k}$ is the category of all ${\mathfrak{g}}$-modules which
- are finitely generated;
- have a basis, where each basis element $v$ satisfies $h^kv\in{{\rm{Span}}}\{h^jv\,|\, j<k\}$, if $h\in{\mathfrak{h}}$;
- are locally $U({\mathfrak{n}}^+)$-finite.
If $k=1$, we get the ordinary BGG category ${\mathcal O}$. For a module $M$ in ${\mathcal O}^k$ and $\nu\in{\mathfrak{h}}^\ast$, we consider the generalised weight space $$M_{(\nu)}:=\{v\in M\,|\, (h-\nu(h))^k v=0,\;\mbox{ for all }\,h\in{\mathfrak{h}}\}, \quad\mbox{so }\;M=\bigoplus_{\nu\in{\mathfrak{h}}^\ast}M_{(\nu)}.$$ The category ${\mathcal O}^k$ decomposes into subcategories ${\mathcal O}^k_\chi$ based on the central characters $\chi$ of $U({\mathfrak{g}})$. The simple objects in ${\mathcal O}^k_\chi$ are the simple highest weight modules $L(\mu)$ with $\mu$ in the Weyl group orbit corresponding to $\chi$, see [@Humphreys Section 1.7], which we denote by $\Lambda$. We consider the Bruhat (partial) order $\le$ on $\Lambda$ of [@Humphreys Section 5.2]. The module $\widetilde{M}_{n,k}(\lambda)$ with $\lambda\in \Lambda$, for $n\in{\mathbb{N}}$, introduced in [@SHPO Section 4], is projective and does not depend on $n$, for $n$ large enough, by [@SHPO Proposition 12]. We denote this module by $P_\lambda$. By the construction in [@SHPO Section 4], this module is generated by a vector $v_\lambda\in (P_\lambda)_{(\lambda)}$, and we have an isomorphism $${\mathrm{Hom}}_{{\mathcal O}^k}(P_\lambda,M)\,\;\tilde{\to}\;\, M_{(\lambda)};\quad \alpha\mapsto \alpha(v_\lambda).$$ As $\oplus_\lambda P_\lambda$ is a projective generator of ${\mathcal O}_\chi^k$, see [@SHPO; @KKM; @Soergel], we have $${\mathcal O}_\chi ^k\cong A\mbox{-mod}\qquad\mbox{with}\qquad A:={\mathrm{End}}_{{\mathcal O}_\chi^k}\left(\bigoplus_{\lambda\in\Lambda}P_\lambda\right)^{{{\rm op}}}.$$ We denote the identity morphism of $P_\lambda$ by $e^\ast_\lambda$. The modules $P_\lambda$ are not indecomposable, but they represent precisely the modules with corresponding notation in Lemma \[PPlambda\].
Main theorem on thick ${\mathcal O}$
------------------------------------
The following theorem generalises [@Koenig Theorem D].
The algebra $(A,\le)$ is properly stratified with exact Borel subalgebra.
The fact that $(A,\le)$ is properly stratified was already pointed out in [@KKM Corollary 9(a)], due to the Morita equivalence in [@Soergel Proposition 1].
Let us define subalgebras $B,H,\overline{N}$ of $A$. For arbitrary $\lambda,\mu\in{\mathfrak{h}}^\ast$, we set $$\begin{aligned}
e^\ast_\lambda Be^\ast_\mu&:=&\{\alpha\in {\mathrm{Hom}}_{{\mathcal O}^k}(P_\lambda,P_\mu)\,|\, \alpha(v_\lambda) \in U({\mathfrak{b}})v_\mu\}\\
e^\ast_\lambda He^\ast_\mu&:=&\{\alpha\in {\mathrm{Hom}}_{{\mathcal O}^k}(P_\lambda,P_\mu)\,|\, \alpha(v_\lambda) \in U({\mathfrak{h}})v_\mu\}\\
e^\ast_\lambda \overline{N}e^\ast_\mu&:=&\{\alpha\in {\mathrm{Hom}}_{{\mathcal O}^k}(P_\lambda,P_\mu)\,|\, \alpha(v_\lambda) \in U({\mathfrak{n}}^-)v_\mu\}.\end{aligned}$$ By construction, $e_\lambda^\ast He_\mu^\ast=0$ unless $\mu=\lambda$ and $He_\lambda^\ast\cong {\mathbb{C}}[x]/(x^k)$ is a local algebra. Since $P_\mu$ is generated by $v_\mu$, it follows from the PBW theorem in [@Humphreys Section 0.5] that $P_\mu$ is spanned by vectors of the form $u_1u_2v_\mu$, where $u_2\in U({\mathfrak{b}})$ and $u_1\in U({\mathfrak{n}}^-)$. Moreover, since all simple constituents have highest weight in $\Lambda$, it follows that we can take a basis of such vectors where each $u_2v_\mu\in (P_\mu)_{(\nu)}$ for some $\nu\in\Lambda$. This implies that $A=\overline{N}B.$
Now consider $\alpha\in e^\ast_\lambda\overline{N}e^\ast_\mu$, with $\alpha(v_\lambda)=Yv_\mu$, for some $Y\in U({\mathfrak{n}}^-)$. For any $\beta\in e^\ast_\mu Be^\ast_\nu$, with $\beta(v_\mu)=X_\beta v_\nu$, we have $$\alpha\beta(v_\lambda)=\beta(Y v_\mu)=YX_\beta v_\nu.$$ Since the modules $P_\mu$ are $U({\mathfrak{n}}^-)$-free, we find that $\alpha\beta=0$ if and only if $\beta=0$. It follows that $\alpha B\cong e^\ast_\mu B$ and hence that $A_B$ is projective. Similarly it follows that ${}_HB$ is projective. From the PBW theorem we find $$\label{ConsPBW}U({\mathfrak{g}})\;=\; U({\mathfrak{n}}^-)U({\mathfrak{b}})\,=\, U({\mathfrak{b}})\,\oplus\, {\mathfrak{n}}^-U({\mathfrak{g}})\,=\, U({\mathfrak{b}})\,\oplus\, U({\mathfrak{g}})_{<} U({\mathfrak{g}}),$$ where $U({\mathfrak{g}})_{<}$ is the subspace of $U({\mathfrak{g}})$ of all elements which are negative weight vectors for the adjoint ${\mathfrak{h}}$-action. Now consider an arbitrary extension $\le^e$ of $\le$, which is a total order, meaning we can identify $(\Lambda,\le^e)$ with a subset of ${\mathbb{N}}$. The basis of $P_\mu$ mentioned above and equation imply equation . We thus find that $B$ is a graded exact Borelic subalgebra. The results then follow from Corollary \[CorgrBrBo\](2).
Algebras in diagram categories {#Sec8}
==============================
\[SecDia\] We will obtain many examples of the types of algebras introduced in Part I, as algebras of morphisms in the partition category. This category is a natural generalisation of the Brauer category of [@Deligne; @BrCat] and the partition algebra of [@Jones; @PartM].
Category algebras {#GrAlg}
-----------------
For a category ${\mathcal{C}}$ with finitely many objects and morphisms, the [*category algebra*]{} ${\Bbbk}[{\mathcal{C}}]$ is given, as a vector space, by all formal sums of morphisms in ${\mathcal{C}}$ with coefficients in ${\Bbbk}$. This is an algebra for the natural product of composition. In particular, for a finite group $G$, which is a category with one object where all morphisms are isomorphisms, we denote the group algebra by ${\Bbbk}G$. If ${\mathcal{C}}$ is [*${\Bbbk}$-linear*]{} with finitely many objects and finite dimensional morphism spaces, we define the category algebra as $${\Bbbk}[{\mathcal{C}}]\;:=\;\bigoplus_{x,y\in {{\rm{Ob}}}{\mathcal{C}}}{\mathrm{Hom}}_{{\mathcal{C}}}(x,y)\qquad\mbox{with}\qquad e^\ast_y{\Bbbk}[{\mathcal{C}}]e^\ast_x= {\mathrm{Hom}}_{{\mathcal{C}}}(x,y),$$ where $e^\ast_z$ is the identity morphism of $z\in{{\rm{Ob}}}{\mathcal{C}}$.
Categories of diagrams {#SecDefCat}
----------------------
Consider an arbitrary field ${\Bbbk}$ and a fixed element $\delta\in{\Bbbk}$.
### Partition category
The partition category ${\mathcal{P}}(\delta)$ is ${\Bbbk}$-linear with set of objects ${\mathbb{N}}$. A ${\Bbbk}$-basis of ${\mathrm{Hom}}_{{\mathcal{P}}(\delta)}(i,j)$ is given by all partitions of the set $\{1,2,\cdots,i,1',2',\cdots, j'\}$. We will also view a partition as an equivalence relation on the set.
For a partition $p$ of the set $S$ and a partition $q$ of the set $T$, with $$S=\{1,2,\cdots,i,1',2',\cdots, j'\}\quad\mbox{and}\quad T=\{1',2',\cdots,j',1'',2'',\cdots, k''\},$$ we define the partition $q\ast p$ of the set $$S\cup T=\{1,2,\cdots,i,1',2',\cdots, j',1'',2'',\cdots, k''\},$$ corresponding to the minimal equivalence relation generated by $p$ and $q$.
We derive two properties from $q\ast p$. Firstly, it induces a partition $q\odot p$ of the set $$\{1,2,\cdots,i,1'',2'',\cdots, k''\},$$ where two elements in the latter set are equivalent if and only if they were so in $S\cup T$. The second property is the number $d(p,q)$ of equivalence classes in $q\ast p$ which are contained in $S\cap T=\{1',2',\cdots,j'\}$. Now we identify the partitions $p$, $q$ and $q\odot p$ with basis elements in respectively ${\mathrm{Hom}}_{{\mathcal{P}}(\delta)}(i,j)$, ${\mathrm{Hom}}_{{\mathcal{P}}(\delta)}(j,k)$ and ${\mathrm{Hom}}_{{\mathcal{P}}(\delta)}(i,k)$. The product (composition) $qp=q\circ p$ is defined as $\delta^{d(p,q)}q\odot p$, which extends bilinearly to $${\mathrm{Hom}}_{{\mathcal{P}}(\delta)}(j,k)\,\times\,{\mathrm{Hom}}_{{\mathcal{P}}(\delta)}(i,j)\;\to\;{\mathrm{Hom}}_{{\mathcal{P}}(\delta)}(i,k).$$
It is easily checked that for the above definitions, ${\mathcal{P}}(\delta)$ is indeed a (${\Bbbk}$-linear) category. For any $n\in{\mathbb{Z}}_{>1}$, the partition algebra is $${\mathcal{P}}_n(\delta)={\mathrm{End}}_{{\mathcal{P}}(\delta)}(n).$$
We will think graphically of the sets (and their partitions) by imagining the elements of $\{1,2,\cdots,i\}$ to be $i$ dots on a horizontal line, ordered from left to right and the elements of $\{1',2',\cdots, j'\}$ to be similarly ordered on a horizontal line above the other one.
### Brauer category
The Brauer category ${\mathcal B}(\delta)$, as introduced in [@BrCat Definition 2.4], is a ${\Bbbk}$-linear subcategory of ${\mathcal{P}}(\delta)$, with the same set of objects ${{\rm{Ob}}}{\mathcal B}(\delta)={\mathbb{N}}$. The morphism spaces are spanned by the partitions into subsets containing exactly two elements. Such partitions will be graphically represented as [*Brauer diagrams*]{}. A $(k,l)$-Brauer diagram consists of $k$ points on a horizontal line, $l$ points on a parallel line above the first line and $(k+l)/2$ lines, each connecting two points. Composing a $(k,l)$- and a $(i,k)$-Brauer diagram, by drawing the first on top of the second one and using the procedure of composing general partitions yields a $(i,l)$-Brauer diagram, up to a power of $\delta$. We have for instance $$\label{defOA}
\begin{tikzpicture}[scale=0.9,thick,>=angle 90]
\begin{scope}[xshift=8cm]
\draw (0,0) to [out=90,in=-180] +(.6,.6);
\draw (.6,.6) to [out=0,in=90] +(.6,-.6);
\draw (.6,0) to [out=90,in=-180] +(.6,.6);
\draw (1.2,.6) to [out=0,in=90] +(.6,-.6);
\node at (2.2,0.5) {$\circ$};
\draw (2.6,0) to [out=60,in=-120] +(1.2,1);
\draw (3.2,0) to [out=110,in=-70] +(-0.6,1);
\draw (3.2,1) to [out=-90,in=-180] +(.6,-.6);
\draw (3.8,.4) to [out=0,in=-90] +(.6,.6);
\node at (5,0.5) {=};
\node at (5.6,0.5) {$\delta$};
\draw (5.9,0) to [out=90,in=-180] +(.3,.3) to [out=0,in=90] +(.3,-.3);
\node at (12.5,0.5) {in ${\mathrm{Hom}}_{{\mathcal B}(\delta)}(4,0)\times{\mathrm{Hom}}_{{\mathcal B}(\delta)}(2,4)\to {\mathrm{Hom}}_{{\mathcal B}(\delta)}(2,0)$.};
\end{scope}
\end{tikzpicture}$$ For any $n\in{\mathbb{Z}}_{>1}$, the Brauer algebra is $${\mathcal B}_n(\delta)={\mathrm{End}}_{{\mathcal B}(\delta)}(n).$$
The lines in Brauer diagrams which connect the lower and upper line will be referred to [*propagating lines*]{}. Lines connecting two points on the lower line are called [*caps*]{} and lines connecting two points on the upper are [*cups*]{}.
### Walled Brauer category
The walled Brauer algebra ${\mathcal B}_{r,s}(\delta)$ is a subalgebra of the Brauer algebra ${\mathcal B}_{r+s}(\delta)$ satisfying $${\mathcal B}_{r,s}(\delta)\cong {\mathcal B}_{s,r}(\delta)\quad\mbox{and} \quad{\mathcal B}_{n,0}(\delta)\cong {\Bbbk}{\mathbb{S}}_n.$$
There are several options to define a “walled Brauer category”. The most natural is as the category $\underline{{\rm Rep}}_0({\rm GL}_\delta)$ of [@Comes2 Section 3.2], see also [@Deligne], which has a tensor category structure. However, that category decomposes into blocks, and each such block is equivalent to a subcategory of the Brauer category ${\mathcal B}(\delta)$, which we realise as follows.
For any $p\in {\mathbb{N}}$, the subcategory ${}^p{\mathcal B}(\delta)$ of ${\mathcal B}(\delta)$ has set of objects $p+2{\mathbb{N}}\subset {\mathbb{N}}={{\rm{Ob}}}{\mathcal B}(\delta)$ and the morphisms ${\mathrm{Hom}}_{{}^p{\mathcal B}(\delta)}(p+2i,p+2j)$ are spanned by the Brauer diagrams which are “well-behaved” with respect to a straight vertical line separating the left $p+i$ and $p+j$ dots from the right $i$ and $j$ dots on the two lines. Well-behaved means that propagating lines do not cross the straight line, but cups and caps intersect it precisely once. For $r\ge s>0$, we have $${\mathcal B}_{r,s}(\delta)={\mathrm{End}}_{{}^{r\text{-}s}{\mathcal B}(\delta)}(r+s).$$
### Jones category
A [*$(k,l)$-Jones diagram*]{} is a partition of a set of $k+l$ dots into pairs (so a Brauer diagram), which can be drawn without intersections when the $l$ dots are on the outer boundary of an annulus and the $k$ dots on the inner boundary of the annulus. The Jones category $J(\delta)$ is the ${\Bbbk}$-linear subcategory of ${\mathcal B}(\delta)$, with the same set of objects ${\mathbb{N}}$ and where the morphisms are spanned by the Jones diagrams. For $n\in{\mathbb{Z}}_{>1}$, the Jones algebra is $$J_n(\delta)={\mathrm{End}}_{J(\delta)}(n).$$ For instance, we have $J_2(\delta)\cong{\mathcal B}_2(\delta)$.
### Temperley-Lieb category
The Temperley-Lieb category ${{\mathrm{TL}}}(\delta)$ is a subcategory of the Brauer (or Jones) category with the same set of objects ${\mathbb{N}}$, but morphisms are spanned by the diagrams without intersections. This is the category of [@BFK Section 2.2], specialised at $q$ with $-q-q^{-1}=\delta$. For $n\in{\mathbb{Z}}_{>1}$, the Temperley-Lieb algebra (of type A) is $${{\mathrm{TL}}}_n(\delta)={\mathrm{End}}_{{{\mathrm{TL}}}(\delta)}(n).$$ For instance, we have ${{\mathrm{TL}}}_2(\delta)\cong {\mathcal B}_{1,1}(\delta)$.
### The category ${{\rm{FI}}}$
Usually, ${{\rm{FI}}}$ is introduced as the category of which the objects are all finite sets and morphisms are all injective maps between sets. We take the equivalent full subcategory with ${\mathbb{N}}$ as set of objects, where $n\in{\mathbb{N}}$ is identified with some set of cardinality $n$. This is a subcategory of ${\mathcal{P}}(\delta)$, for arbitrary $\delta$, but is not ${\Bbbk}$-linear.
Triangular structure and truncation {#SecNHN}
-----------------------------------
We will distinguish three types of partitions.
1. Partitions of type $H$: These are partitions into subsets of exactly two elements, one on each line (one primed and one unprimed).
2. Partitions of type $N^+$: These are partitions where
- each element of the upper line is contained in a set with at least one element of the lower line and no other elements of the upper line.
- for $k$, resp. $l$, minimal in the set containing $i'$, resp. $j'$, we have $i<j\Leftrightarrow k<l$.
3. Partitions of type $N^-$: These are partitions where
- each element of the lower line is contained in a set with at least one element of the upper line and no other elements of the lower line.
- for $i'$, resp. $j'$, minimal in the set containing $k$, resp. $l$, we have $k<l\Leftrightarrow i<j$.
By definition, partitions of type $H$ must appear in ${\mathrm{Hom}}_{{\mathcal{P}}(\delta)}(j,j)$, for some $j\in{\mathbb{N}}$, while those of type $N^+$, resp. $N^-$, must appear in ${\mathrm{Hom}}_{{\mathcal{P}}(\delta)}(i,j)$, for $i<j$, resp. $i>j$.
As a special case we have Brauer diagrams of the three corresponding types.
1. Brauer diagrams of type $H$: These diagrams consist solely of propagating lines.
2. Brauer diagrams of type $N^+$: After removing all caps one is left with only non-crossing propagating lines.
3. Brauer diagrams of type $N^-$: After removing all cups one is left with only non-crossing propagating lines.
The subspace of one of the category algebras spanned by all diagrams of type $H$ is also denoted by $H$, with similar convention for $N^+$ and $N^-$. We draw some examples: $$\label{defOA}
\begin{tikzpicture}[scale=0.9,thick,>=angle 90]
\begin{scope}[xshift=8cm]
\draw (0,1) to [out=-90,in=-180] +(.6,-.6);
\draw (.6,.4) to [out=0,in=-90] +(.6,.6);
\draw (.6,1) to [out=-90,in=-180] +(.6,-.6);
\draw (1.2,.4) to [out=0,in=-90] +(.6,.6);
\node at (2.7,0.5) {$\in N^-$,};
\draw (4.6,0) to [out=60,in=-120] +(1.2,1);
\draw (5.2,0) to [out=110,in=-70] +(-0.6,1);
\draw (5.8,0) to [out=110,in=-70] +(-0.6,1);
\draw (6.4,0) to [out=90,in=-90] +(0,1);
\node at (7.1,0.5) {$\in H,$};
\draw (9.6,0) to [out=110,in=-70] +(-0.6,1);
\draw (10.8,0) to [out=130,in=-50] +(-1.2,1);
\draw (10.2,0) to [out=90,in=-180] +(.6,.4);
\draw (10.8,.4) to [out=0,in=90] +(.6,-.4);
\draw (9,0) to [out=75,in=-180] +(1.5,.8);
\draw (10.5,.8) to [out=0,in=105] +(1.5,-.8);
\node at (12.7,0.5) {$\in N^+$.};
\end{scope}
\end{tikzpicture}$$
For all the categories, except ${}^p{\mathcal B}(\delta)$, we denote the identity morphism in ${\mathrm{End}}(i)$ by $e_i^\ast$. For ${}^p{\mathcal B}(\delta)$ we use $e_i^{\ast}$ for the identity morphism of $p+2i\in {{\rm{Ob}}}{}^p{\mathcal B}(\delta)$.
For each $n\in{\mathbb{Z}}_{>1}$, we introduce the full subcategory ${\mathcal{P}}^{\le n}(\delta)$, resp. ${{\rm{FI}}}^{\le n}$, of ${\mathcal{P}}(\delta)$, resp. ${{\rm{FI}}}$, with objects $\{0,1,2\cdots, n\}\subset {\mathbb{N}}$. Their category algebras satisfy .
We observe that ${\mathcal B}(\delta)$ decomposes into two subcategories, one with objects $2{\mathbb{N}}$ and one with objects $2{\mathbb{N}}+1$. Hence, for $n\in{\mathbb{Z}}_{>1}$, we define ${\mathcal B}^{\le n}(\delta)$, $J^{\le n}(\delta)$ and ${{\mathrm{TL}}}^{\le n}(\delta)$ as the full subcategories of resp. ${\mathcal B}(\delta)$, $J(\delta)$ and ${{\mathrm{TL}}}(\delta)$, with objects $${\mathscr{J}}(n)\;:=\; \{n,n-2,\cdots,n-2\lfloor \frac{n}{2}\rfloor\}.$$ The corresponding category algebras satisfy $1=\sum_{i\in{\mathscr{J}}(n)}e^\ast_i$. For $p\in{\mathbb{N}}$ and $n\in{\mathbb{Z}}_{>0}$, the category ${}^p{\mathcal B}^{\le n}(\delta)$ is the full subcategory of ${}^p{\mathcal B}(\delta)$ with objects $$\{p,p+2,\cdots,p+2n\}.$$ The category algebra satisfies equation .
The category algebras
---------------------
We consider the category algebras of the previous section with idempotents $e_i^\ast$. We use the corresponding decomposition ${\mathcal{Q}}$ of $\Lambda$, and the idempotent quasi-order $\preccurlyeq_{{\mathcal{Q}}}$ and idempotent partial order $\le_{{\mathcal{Q}}}$ on $\Lambda$ of Definition \[lepleq\].
\[ThmDiagram1\] Fix an arbitrary field ${\Bbbk}$ and $\delta\in{\Bbbk}$.
1. The algebras ${\Bbbk}[{\mathcal{P}}^{\le n}(\delta)]$, ${\Bbbk}[{\mathcal B}^{\le n}(\delta)]$, ${\Bbbk}[{}^p{\mathcal B}^{\le n-p}(\delta)]$ and ${\Bbbk}[{{\rm{FI}}}^{\le n}(\delta)]$ are
- quasi-hereditary for $\le_{{\mathcal{Q}}}$, with exact Borel subalgebra if ${{\rm{char}}}({\Bbbk})\not\in [2,n]$;
- exactly standardly stratified for $\preccurlyeq_{{\mathcal{Q}}}$, with exact Borel subalgebra;
- base stratified for decomposition ${\mathcal{Q}}$.
2. The algebra ${\Bbbk}[J^{\le n}(\delta)]$ is
- quasi-hereditary for $\le_{{\mathcal{Q}}}$ with exact Borel subalgebra, if ${{\rm{char}}}({\Bbbk})$ does not divide any element of ${\mathscr{J}}(n)$;
- properly stratified for $\le_{{\mathcal{Q}}}$, with exact Borel subalgebra;
- base stratified for decomposition ${\mathcal{Q}}$ if the polynomials $x^{i}-1$, for $i\in{\mathscr{J}}(n)$, split over the field ${\Bbbk}$.
3. The algebra ${\Bbbk}[{{\mathrm{TL}}}^{\le n}(\delta)]$ is
- quasi-hereditary for $\le_{{\mathcal{Q}}}$, with exact Borel subalgebra;
- base stratified for decomposition ${\mathcal{Q}}$.
When the above condition for quasi-heredity is not satisfied, the category algebra is not quasi-hereditary, for any partial order on $\Lambda$ (with $\Lambda$ given in Lemma \[LemExBH\]).
The statements on the exact standard stratification in Theorem \[ThmDiagram1\](1) can be refined by replacing $\preccurlyeq_{{\mathcal{Q}}}$ by a partial quasi-order $\preccurlyeq$, such that $<$ is the same as $\prec_{{\mathcal{Q}}}$, but $\sim$ reflects the block decomposition of ${\Bbbk}{\mathbb{S}}_t$, see [@James Section 21].
If ${\Bbbk}$ is algebraically closed, the algebras in Theorem \[ThmDiagram1\] are all standardly based by Theorem \[ThmBaSt\], with poset $L:=\sqcup_{i}L_i$ and $L_i$ the posets for the standardly based algebra $He_i^\ast$, as can be obtained from Examples \[ExS1\] and \[ExS2\] and Lemma \[LemExBH\].
\[ThmDiagram2\] Assume that ${\Bbbk}$ is algebraically closed. The cell modules of $C$ form a standard system for $(L,\le)$ if and only if the condition below is satisfied.
[ | l | l | l | ]{}\
algebra $C$& condition& Set $L\,=\,\sqcup_{i}L_i$\
${\Bbbk}[{\mathcal{P}}^{\le n}(\delta)]$ & ${{\rm{char}}}({\Bbbk})\not\in\{2,3\}$ or ${{\rm{char}}}({\Bbbk})=3$ and $n=2$&$\{(i,\mu)\,|\, 0\le i\le n,\; \mu\vdash i\}$\
${\Bbbk}[{\mathcal B}^{\le n}(\delta)]$ & ${{\rm{char}}}({\Bbbk})\not\in\{2,3\}$ or ${{\rm{char}}}({\Bbbk})=3$ and $n=2$ &$\{(i,\mu)\,|\, i\in \mathscr{J}(n),\; \mu\vdash i\}$\
${\Bbbk}[J^{\le n}(\delta)]$ & $i$ not divisible by ${{\rm{char}}}({\Bbbk})$, for $i\in{\mathscr{J}}(n)$ &$\{(i,\omega)\,|\, i\in \mathscr{J}(n),\; \omega\in{\Bbbk},\,\, \omega^i=1\}$\
${\Bbbk}[{{\mathrm{TL}}}^{\le n}(\delta)]$ & $\emptyset$&${\mathscr{J}}(n)$\
${\Bbbk}[{{\rm{FI}}}^{\le n}(\delta)]$ & ${{\rm{char}}}({\Bbbk})\not\in\{2,3\}$ or ${{\rm{char}}}({\Bbbk})=3$ and $n=2$& $\{(i,\mu)\,|\, 0\le i\le n,\; \mu\vdash i\}$\
${\Bbbk}[{}^p{\mathcal B}^{\le n}(\delta)]$ & ${{\rm{char}}}({\Bbbk})\not\in\{2,3\}$&$\{(i,\mu,\nu)\,|\, 0\le i\le n,$\
& or ${{\rm{char}}}({\Bbbk})=3$ and $n+p\le 2$&$\mu\vdash p+i,\,\nu\vdash i \}$\
We start the proofs of the theorems with the following proposition.
\[PropExBH\] Let $C$ be one of the category algebras in Theorem \[ThmDiagram2\], equipped with the idempotent grading of . The subspaces $H,N^+$ and $N^-$ of $C$, defined in Section \[SecNHN\] are subalgebras and $B:=HN^+$ is a graded pre-Borelic subalgebra of $C$.
One verifies that the subspaces are subalgebras for ${\mathcal{P}}(\delta)$. The other cases follow by restriction. Further, we can classify partitions into those where each element of the upper line is contained in a set/class with at least one element of the lower line and no other elements of the upper line, and the rest. In each of the cases, the first type of diagrams span $B$, whereas those of the second type span $C_-B$, proving equation . So we find that $B$ is a complete ${\mathbb{N}}$-graded subalgebra of $C$.
It hence remains to prove that $C_B$ and ${}_HB$ are projective modules. We have $C=N^-B$ and by associating to each partition the number of classes which contain elements of both rows (for Brauer diagrams this is just the number of propagating lines) we find a decomposition $$C=\bigoplus_{i}N^- e_i^\ast B.$$ For any partition $q$ in $N^- e_i^\ast$ we have $qB\cong e_i^\ast B$, proving that $C_B$ is projective. Similarly it follows that for any partition $q$ in $e^\ast_i B$ we have $Hq\cong He_i^\ast$, which concludes the proof.
For the cyclic group $C_t$ of order $t$, we have ${\Bbbk}C_t\cong {\Bbbk}[x]/(x^t-1)$. We denote its labelling set of simple modules by $\Lambda_C^{{\Bbbk},t}$. In case the polynomial $x^t-1$ splits over ${\Bbbk}$, for instance when ${\Bbbk}$ is algebraically closed, we can take $\{\omega\in{\Bbbk}\,|\, \omega^t=1\}$ for this set.
\[LemExBH\] The following table summarises the structure of the subalgebra $H\subset C$, its labelling set $\Lambda_C$ of isoclasses of simple modules and the criterion for $H$ to be semisimple.
[ | l | l | l | l |]{}\
algebra $C$& algebra $H$ & set $\Lambda_C=\Lambda_H=\sqcup_i \Lambda_i$ & semisimplicity criterion for $H$\
${\Bbbk}[{\mathcal{P}}^{\le n}(\delta)]$ & $\bigoplus_{i=0}^n {\Bbbk}{\mathbb{S}}_i $ &$\{(i,\mu)\,|\, 0\le i\le n,\; \mu\vdash_{{\Bbbk}} i\}$&${{\rm{char}}}({\Bbbk})\not\in[2,n]$\
${\Bbbk}[{\mathcal B}^{\le n}(\delta)]$ &$\bigoplus_{i\in{\mathscr{J}}(n)} {\Bbbk}{\mathbb{S}}_i $ & $\{(i,\mu)\,|\, i\in \mathscr{J}(n),\; \mu\vdash_{{\Bbbk}} i\}$ &${{\rm{char}}}({\Bbbk})\not\in[2,n]$\
${\Bbbk}[J^{\le n}(\delta)]$ & $\bigoplus_{i\in{\mathscr{J}}(n)} {\Bbbk}C_i $ & $\{(i,\omega)\,|\, i\in \mathscr{J}(n),\; \omega\in \Lambda_C^{{\Bbbk},i}\}$ &${{\rm{char}}}({\Bbbk})\nmid i$, for $i\in{\mathscr{J}}(n)$\
${\Bbbk}[{{\mathrm{TL}}}^{\le n}(\delta)]$ &$\bigoplus_{i\in{\mathscr{J}}(n)} {\Bbbk}$ &${\mathscr{J}}(n)$&$\emptyset$\
${\Bbbk}[{{\rm{FI}}}^{\le n}(\delta)]$ & $\bigoplus_{i=0}^n {\Bbbk}{\mathbb{S}}_i $ &$\{(i,\mu)\,|\, 0\le i\le n,\; \mu\vdash_{{\Bbbk}} i\}$ &${{\rm{char}}}({\Bbbk})\not\in[2,n]$\
${\Bbbk}[{}^p{\mathcal B}^{\le n}(\delta)]$ & $\bigoplus_{i=0}^n {\Bbbk}{\mathbb{S}}_{p+i}\times{\mathbb{S}}_i$ & $\{(i,\mu,\nu)\,|\, 0\le i\le n,$ &${{\rm{char}}}({\Bbbk})\not\in[2,p+n]$\
& & $\mu\vdash_{{\Bbbk}} p+i,\,\nu\vdash_{{\Bbbk}} i \}$ &\
The structure of the algebra $H$ follows from its definition in Section \[SecDefCat\]. We use equation . The simple modules of ${\Bbbk}{\mathbb{S}}_t$ are labelled by ${\Bbbk}$-regular partitions of $t$, see [@James Section 11]. By Maschke’s theorem, for a finite group $G$, the algebra ${\Bbbk}G$ is semisimple if and only if the order $|G|$ is not divisible by ${{\rm{char}}}({\Bbbk})$. Hence, ${\Bbbk}{\mathbb{S}}_t$ is semisimple if and only if ${{\rm{char}}}({\Bbbk})>t$ or ${{\rm{char}}}({\Bbbk})=0$.
We freely use Proposition \[PropExBH\] and Lemma \[LemExBH\].
That the algebras are based-stratified follows from Theorem \[ThmIdGrB\] and Examples \[ExS1\] and \[ExS2\]. The exact standard stratification follows from Corollary \[CorgrBrBo\](1). The quasi-heredity follows from Corollary \[CorgrBrBo\](3). It is easily checked that the group algebra ${\Bbbk}C_i$ is quasi-local, hence $C={\Bbbk}[J^{\le n}(\delta)]$ is properly stratified by Corollary \[CorgrBrBo\](2).
Now we prove that in the remaining cases $C$ is not quasi-hereditary. By equation and Lemma \[NewLemStr\], $C$ will have infinite global dimension as soon as $H$ has. For any finite group $G$, the algebra ${\Bbbk}G$ is Frobenius and hence self-injective. In particular, the global dimension of ${\Bbbk}G$ is finite if and only if it is zero. In conclusion, when the criteria for semisimplicity of $H$ in Lemma \[LemExBH\] are not satisfied, $C$ has infinite global dimension. In particular $C$ is not quasi-hereditary for any order, by [@CPS] or [@APT Corollary 6.6].
The following lemma can be derived from [@Nakano1] with minor additional work.
\[LemHN\] Assume that ${\Bbbk}$ is algebraically closed and $i>1$.
- The Specht modules of ${\Bbbk}{\mathbb{S}}_i$ form a standard system if and only if $$\begin{cases}{{\rm{char}}}({\Bbbk})\not\in \{2,3\}\,\,\,\mbox{ or}\\ {{\rm{char}}}({\Bbbk})=3 \,\mbox{ and }\,i=2.\end{cases}$$
- The cell modules of the standardly based algebra ${\Bbbk}C_i$ of Example \[ExS1\] form a standard system if and only if ${{\rm{char}}}({\Bbbk})$ does not divide $i$.
For ${{\rm{char}}}({\Bbbk})\not\in\{2,3\}$, [@Nakano1 Theorem 6.4(b)(ii)] implies that the Specht modules of ${\Bbbk}{\mathbb{S}}_i$ form a standard system. When ${{\rm{char}}}({\Bbbk})=3$, ${\Bbbk}{\mathbb{S}}_2$ is semisimple and the (simple) Specht modules thus form a standard system.
Now we prove that, in the remaining cases, the Specht modules do not constitute a standard system, for any order. The trivial module $S_1$ and the sign module $S_2$ are both Specht modules. Assume ${{\rm{char}}}({\Bbbk})=2$, then $S_1\cong S_2$, contradicting property (2) in the definition of a standard system in \[IntroFaithCov\]. Now assume ${{\rm{char}}}({\Bbbk})=3$ and $i\ge 3$. We will prove that $${\mathrm{Ext}}^1_{{\Bbbk}{\mathbb{S}}_i}(S_1,S_2)\not=0\not={\mathrm{Ext}}^1_{{\Bbbk}{\mathbb{S}}_i}(S_2,S_1),$$ contradicting property (3) in the definition of a standard system. The modules $M$ and $N$ corresponding to the extensions are given by $$M=\langle v,w\rangle\qquad \mbox{with }\quad s_j v=v+c_j w\;\mbox{ and }\; s_j w=-w,$$ for $s_{j}$, $j\in\{1,\ldots,i-1\}$ the generators of ${\mathbb{S}}_i$ and $c_j\in{\Bbbk}$ arbitrary, and $$N=\langle v,w\rangle\qquad\mbox{with }\quad s_j w=-w+d_j v\;\mbox{ and }\; s_j v=v,$$ for $d_j\in{\Bbbk}$ arbitrary. It can be verified that in characteristic 3, there are no conditions on the coefficients $c_j,d_j$ coming from imposing the braid relations. However, when $c_j\not= c_{j'}$ or $d_j\not= d_{j'}$, for $1\le j\not=j'\le i-1$, the modules do not decompose and the extensions hence do not split.
For ${\Bbbk}C_i\cong{\Bbbk}[x]/(x^i-1)$, the cell modules are two-by-two non-isomorphic if and only if all $i$-th roots of $1$ in the algebraically closed field ${\Bbbk}$ are different. This is equivalent to demanding that ${\Bbbk}C_i$ is semisimple, in which case the cell modules are simple and form a standard system for any order. So a necessary and sufficient condition is that ${{\rm{char}}}({\Bbbk})$ does not divide $i$.
By Corollary \[CorStB\] and Proposition \[PropExBH\], this follows from Lemmata \[LemExBH\] and \[LemHN\].
\[RemnotS\] Consider the case ${{\rm{char}}}({\Bbbk})\not\in[2,n]$. We have proved that ${\Bbbk}[{\mathcal B}^{\le n}(\delta)]$ is quasi-hereditary with exact Borel subalgebra. This is not a [*strong*]{} exact Borel subalgebra as defined in [@Koenig], as $H$ is not a maximal semisimple subalgebra. In fact, generically, ${\Bbbk}[{\mathcal B}^{\le n}(\delta)]$ will be semisimple itself, as follows from [@Rui] and Theorem \[ThmMor\] below.
\[NotKKO\] The exact Borel subalgebra $B$ of $C:={\Bbbk}[{\mathcal B}^{\le n}(\delta)]$ does not satisfy the property $${\mathrm{Ext}}^k_C(C\otimes_B M,C\otimes_B N)\;\cong\;{\mathrm{Ext}}^k_B(M,N),\qquad \forall \,k\ge 2,$$ for arbitrary $M,N\in B$-mod, which is satisfied for the exact Borel subalgebra predicted by the general theory of [@KKO], see [@KKO Theorems 1.1 and 10.5]. Indeed, we observe that $B$, and hence the extension group on the right-hand side, does not depend on $\delta$. The extension group in the left-hand side depends heavily on $\delta$. As mentioned in Remark \[RemnotS\], for generic $\delta$ the left-hand side will vanish. It is easily checked that it does not vanish for [*e.g.*]{} $\delta=0$. This proves that the displayed isomorphism cannot be true in general.
For $C={\Bbbk}[{{\rm{FI}}}^{\le n}]$, we have $B=H$ as $N^+=0$. So for ${{\rm{char}}}({\Bbbk})\not\in[2,n]$, ${\Bbbk}[{{\rm{FI}}}^{\le n}]$ is quasi-hereditary with semisimple exact Borel subalgebra. It can easily be checked that for the reversal of the order $\le_{{\mathcal{Q}}}$, we have that ${\Bbbk}[{{\rm{FI}}}^{\le n}]$ is an exact Borel subalgebra of itself.
Morita equivalences
-------------------
\[ThmMor\] For an arbitrary field ${\Bbbk}$, we have the following Morita equivalences under the respective conditions given in the table:
[ | l | l | ]{}\
Morita equivalence & condition\
${\mathcal{P}}_n(\delta)\;\,\,\stackrel{M}{=}\,\;\,{\Bbbk}[{\mathcal{P}}^{\le n}(\delta)]$ & $\delta\not=0$\
${\mathcal B}_n(\delta)\;\,\,\stackrel{M}{=}\;\,\,{\Bbbk}[{\mathcal B}^{\le n}(\delta)]$ & $\delta\not=0$ or $n$ odd\
$J_n(\delta)\;\,\,\stackrel{M}{=}\;\,\,{\Bbbk}[J^{\le n}(\delta)]$ & $\delta\not=0$ or $n$ odd\
${{\mathrm{TL}}}_n(\delta)\,\,\stackrel{M}{=}\,\,{\Bbbk}[{{\mathrm{TL}}}^{\le n}(\delta)]$ & $\delta\not=0$ or $n$ odd\
${\mathcal B}_{p+n,n}(\delta)\,\stackrel{M}{=}\,{\Bbbk}[{}^p{\mathcal B}^{\le n}(\delta)]$ & $\delta\not=0$ or $p\not=0$\
Theorem \[ThmMor\] and Lemma \[LemExBH\] together determine the labelling set $\Lambda_A$ of the simple modules of the diagram algebras $A$, under the condition in the right column of the table. In almost all cases $\Lambda_A$ is known by [@CDDM; @CellAlg; @partition; @Xi].
We start the proof of Theorem \[ThmMor\] by constructing special elements in the category algebras, which will also be essential for constructions in Part III.
\[Lemab\] Consider algebras $C$, $A=e_n^\ast Ce_n^\ast$ and $i\in I_A(n)$, as in the table below.
[ | l | l | l | ]{}\
Category algebra $C$ & Diagram algebra $A$ & $I_A(n)$\
${\Bbbk}[{\mathcal{P}}^{\le n}(\delta)]$& ${\mathcal{P}}_n(\delta)$ & $\{0,1,\ldots,n\}$\
${\Bbbk}[{\mathcal B}^{\le n}(\delta)]$& ${\mathcal B}_n(\delta)$ & ${\mathscr{J}}(n)$\
${\Bbbk}[J^{\le n}(\delta)]$& $J_n(\delta)$ & ${\mathscr{J}}(n)$\
${\Bbbk}[{{\mathrm{TL}}}^{\le n}(\delta)]$& ${{\mathrm{TL}}}_n(\delta)$ & ${\mathscr{J}}(n)$\
${\Bbbk}[{}^p{\mathcal B}^{\le n}(\delta)]$& ${\mathcal B}_{p+n,n}(\delta)$ & $\{0,1,\ldots,n\}$\
Then there are elements $a_i\in e_n^\ast Ce_i^\ast$ and $b_i\in e_i^\ast Ce_n^\ast$ such that $c^\ast_i:=a_ib_i\in A$ and $e_i^\ast$ satisfy $$\begin{cases} e_i^\ast= b_ia_i\quad \mbox{and}\quad (c^\ast_i)^2=c^\ast_i&\mbox{ if either $\delta\not=0$ or~$i+p\not=0$}\\
(c^\ast_i)^2=0=a_ib_i&\mbox{ if~$\delta=0$ and~$i+p=0$.}
\end{cases}$$ Here, we set $p=0$ in all cases, except for ${}^p{\mathcal B}^{\le n}(\delta)$.
There are many different choices for $a_i$ and $b_i$. The constructions given below for the various cases produce different $a_i$ and $b_i$ in the overlapping situations.
Firstly we consider the partition category (so $p=0$) and arbitrary $1\le i\le n$. We put $$b_i=\{\{1,1'\},\{2,2'\},\cdots,\{i,i'\},\{i+1,i+2,\cdots,n\}\}\qquad\mbox{and}$$ $$a_i=\{\{1,1'\},\{2,2'\},\cdots,\{i-1,(i-1)'\},\{i,i',(i+1)',(i+2)',\cdots,n'\}\}.$$ We have $b_i a_i= e^\ast_i $, which implies that $c^\ast_i:=a_ib_i$ is an idempotent. In case $i=0$ we define $a_0\in {\mathrm{Hom}}_{{\mathcal{P}}(\delta)}(0,n)$ and $\overline{a}_0\in {\mathrm{Hom}}_{{\mathcal{P}}(\delta)}(n,0)$ as the partitions into one set. When $\delta\not=0$, we set $b_0=\overline{a}_0/\delta$, which leads to $b_0a_0=e_0^\ast$ and an idempotent $c^\ast_0=a_0b_0$. When $\delta=0$ we set $b_0=\overline{a}_0$, in which case $c^\ast_0:=a_0b_0$ squares to zero. The above already deals with the partition category completely, although the diagrams that will be constructed below for the other categories can also be used for the partition category when $i\in{\mathscr{J}}(n)$.
Now we consider the cases ${\mathcal B}_n(\delta)$, $J_n(\delta)$ and ${{\mathrm{TL}}}_n(\delta)$ (so $p=0$). For arbitrary $i\in{\mathscr{J}}(n)$ we introduce three Brauer diagrams, which are also Temperley-Lieb and hence Jones diagrams. We consider two diagrams $\overline{a}_i$ and $\widehat{a}_i$ with $(n-i)/2$ caps and $i$ propagating lines and $a_i$ with $(n-i)/2$ cups and $i$ propagating lines. Note that the definition of $\widehat{a}_i$ requires $i>0$.
$$\label{defOA}
\begin{tikzpicture}[scale=0.9,thick,>=angle 90]
\begin{scope}[xshift=8cm]
\node at (0,0.5) {$\overline{a}_i=$};
\draw (.7,0) -- +(0,1);
\draw (1.3,0) -- +(0,1);
\draw [dotted] (1.6,.5) -- +(1,0);
\draw (2.9,0) -- +(0,1);
\draw (3.5,0) -- +(0,1);
\draw (4.1,0) to [out=90,in=-180] +(.3,.3) to [out=0,in=90] +(.3,-.3);
\draw (5.3,0) to [out=90,in=-180] +(.3,.3) to [out=0,in=90] +(.3,-.3);
\draw [dotted] (6.2,.1) -- +(1,0);
\draw (8,0) to [out=90,in=-180] +(.3,.3) to [out=0,in=90] +(.3,-.3);
\end{scope}
\end{tikzpicture}$$
$$\label{defBB}
\begin{tikzpicture}[scale=0.9,thick,>=angle 90]
\begin{scope}[xshift=8cm]
\node at (0,0.5) {$\widehat{a}_i=$};
\draw (.7,0) -- +(0,1);
\draw (1.3,0) -- +(0,1);
\draw [dotted] (1.6,.5) -- +(1,0);
\draw (2.9,0) -- +(0,1);
\draw (3.5,0) to [out=90,in=-180] +(.3,.3) to [out=0,in=90] +(.3,-.3);
\draw (4.7,0) to [out=90,in=-180] +(.3,.3) to [out=0,in=90] +(.3,-.3);
\draw [dotted] (5.7,.1) -- +(1,0);
\draw (7.4,0) to [out=90,in=-180] +(.3,.3) to [out=0,in=90] +(.3,-.3);
\draw (8.6,0) to [out=135,in=-45] +(-5.1,1);
\end{scope}
\end{tikzpicture}$$
$$\label{defA}
\begin{tikzpicture}[scale=0.9,thick,>=angle 90]
\begin{scope}[xshift=8cm]
\node at (0,0.5) {$a_i=$};
\draw (.7,0) -- +(0,1);
\draw (1.3,0) -- +(0,1);
\draw [dotted] (1.6,.5) -- +(1,0);
\draw (2.9,0) -- +(0,1);
\draw (3.5,0) -- +(0,1);
\draw (4.1,1) to [out=-90,in=-180] +(.3,-.3) to [out=0,in=-90] +(.3,.3);
\draw (5.3,1) to [out=-90,in=-180] +(.3,-.3) to [out=0,in=-90] +(.3,.3);
\draw [dotted] (6.2,.9) -- +(1,0);
\draw (8,1) to [out=-90,in=-180] +(.3,-.3) to [out=0,in=-90] +(.3,.3);
\end{scope}
\end{tikzpicture}$$
If $\delta\not=0$, we can set $b_i:=\delta^{(i-n)/2}\overline{a}_i$ and then we have $b_ia_i=e_i^\ast$. If $i\not=0$, we can set $b_i:=\widehat{a}_i$ and then we have $b_ia_i=e_i^\ast$. In either case, we automatically find that $c^\ast_i:=a_ib_i$ is an idempotent. When both $\delta=0$ and $i=0$, we set $b:= \overline{a}_0$ and then $c^\ast_0=a_0b_0$ squares to zero. This completes the cases ${\mathcal B}_n(\delta)$, $J_n(\delta)$ and ${{\mathrm{TL}}}_n(\delta)$.
The case ${\mathcal B}_{p+n,n}(\delta)$ behaves similarly. For instance, if $p>0$ and $\delta=0$, we can take $$\begin{tikzpicture}[scale=0.9,thick,>=angle 90]
\begin{scope}[xshift=4cm]
\node at (0,1) {$c_0^\ast:=$};
\draw (1.2,0) -- +(0,2);
\draw [dotted] (1.4,1) -- +(0.6,0);
\draw (2.2,0) -- +(0,2);
\draw (2.8,0) to [out=30, in=150] +(5.4,0);
\draw (3.4,0) to [out=32, in=148] +(4.2,0);
\draw (3.4,2) to [out=-32, in=-148] +(4.8,0);
\draw (3.9,0) [dotted] -- +(0.7,0);
\draw (3.9,2) [dotted] -- +(1.2,0);
\draw (4.9,0) to [out=80,in=100] +(1.2,0);
\draw (5.5,0) to [out=120, in=-60] +(- 2.7,2);
\draw (5.5,2) to [out=-90,in=180] +(.3,-.3) to [out=0,in=-90] +(.3,.3);
\draw[line width =2] (5.8,0) -- +(0,2);
\draw (6.5,0) [dotted] -- +(0.7,0);
\draw (6.5,2) [dotted] -- +(1.2,0);
\end{scope}
\end{tikzpicture}$$ where there are $p$ propagating lines and $n$ cups and caps.
Any of the category algebras satisfies $$1_C=\sum_{i\in I_A(n)}e^\ast_i,$$ with $I_A(n)$ the set in Lemma \[Lemab\]. By Lemma \[Lemab\], the condition in the right column of the table in the theorem implies that $e_i^\ast\in Ce_n^\ast C$, for all $i\in I_A(n)$. Consequently, we have $1_C\in Ce_n^\ast C$, and hence $C=Ce_n^\ast C$. The conclusion thus follows from \[SecMor\].
Cellularity
-----------
We prove that the category algebras and the diagram algebras are cellular in the sense of [@CellAlg Definition 1.1].
### {#section-3}
For all $i,j\in{\mathbb{N}}$, we define a ${\Bbbk}$-vector space isomorphism $$\imath:{\mathrm{Hom}}_{{\mathcal{P}}(\delta)}(i,j)\to {\mathrm{Hom}}_{{\mathcal{P}}(\delta)}(j,i),$$ which is determined by demanding that any partition is mapped to its “horizontal flip”, which simply identifies the $i$ dots on the lower line of partitions in ${\mathrm{Hom}}_{{\mathcal{P}}(\delta)}(i,j)$ with those on the upper line of partitions in ${\mathrm{Hom}}_{{\mathcal{P}}(\delta)}(j,i)$. This clearly extends to an involutive anti-algebra morphism of ${\Bbbk}[{\mathcal{P}}^{\le n}(\delta)]$, which furthermore restricts to involutive anti-algebra morphisms of ${\Bbbk}[{\mathcal B}^{\le n}(\delta)]$, ${\Bbbk}[{}^{p}{\mathcal B}^{\le n}(\delta)]$ and ${\Bbbk}[{{\mathrm{TL}}}^{\le n}(\delta)]$. On a diagram with only propagating lines, interpreted as an element of a symmetric group, $\imath$ acts as inversion. For ${\Bbbk}[J^{\le n}(\delta)]$ we use a different involution $\imath$, [*viz.*]{} $\imath$ is defined on a Jones diagram as its horizontal and vertical flip. In particular, this stabilises any Jones diagram with only propagating lines.
Consider an arbitrary field ${\Bbbk}$. Let $C$ be ${\Bbbk}[{\mathcal{P}}^{\le n}(\delta)]$, ${\Bbbk}[{\mathcal B}^{\le n}(\delta)]$, ${\Bbbk}[J^{\le n}(\delta)]$, ${\Bbbk}[{{\mathrm{TL}}}^{\le n}(\delta)]$ or ${\Bbbk}[{}^p{\mathcal B}^{\le n}(\delta)]$. Then $C$ is cellular for involution $\imath$ when it satisfies the condition to be base stratified in Theorem \[ThmDiagram1\].
We take an arbitrary extension of the partial order of \[DefL\] on $L=\sqcup_i L_i$ to a total order, again denoted by $\le$. The ideals $C^{\ge (l,p)}$ for $(l,p)\in L$ of the standardly based algebra $C$ then form a chain of ideals, which is a refinement of the chain of ideals $J_i=Cf_i C$. We introduce the space $J_i'$ spanned by all partitions where exactly $i$ subsets contain dots of the upper and lower row, or the intersection of that space with the relevant category algebra. Then we have $$J_i = J_i'\oplus J_{i-1}\qquad\mbox{and}\qquad \imath(J_i')=J'_i.$$ By extending the cellular structures on $He_i^\ast$ in Examples \[ExS1\] and \[ExS2\] to $C$, we construct similar decompositions $$C^{\ge (l,p)}=\left(C^{\ge (l,p)}\right)'\;\oplus\; C^{\ge (l,p_1)}$$ for the refinement, where $p_1$ is the (unique) maximal $q\in L_l$ with $q<p$. Together with the above decompositions, it then follows easily that [@CellQua Definition 2.2] is satisfied for the refined chain and $C$ is hence cellular, see also [@JieDu Lemma 1.2.4].
As $\imath(e_n^\ast)=e_n^\ast$, [@StructureCell Proposition 4.3] implies the following property.
\[CorCell\] The algebras ${\mathcal{P}}_n(\delta)$, ${\mathcal B}_n(\delta)$, ${\mathcal B}_{r,s}(\delta)$ and ${{\mathrm{TL}}}_n(\delta)$ are cellular with respect to the restriction of $\imath$. If the polynomials $x^i-1$, for $i\in{\mathscr{J}}(n)$, split over ${\Bbbk}$, the algebra $J_n(\delta)$ is cellular with respect to the restriction of $\imath$.
\[PropCMod\] Consider any of the cellular algebras $A$ in Corollary \[CorCell\] with the corresponding category algebra $C$ in Lemma \[Lemab\]. The cell modules of $A$ are labelled by $L_A:=L_C$ in Theorem \[ThmDiagram2\], and are given by $$W_A(i,p):=e_n^\ast W_C(i,p)\quad\mbox{with}\quad W_C(i,p)=C\otimes_B W^0_i(p)\quad\mbox{ for }\; i\in I_A\;\mbox{ and }\, p\in L_i.$$
We use Proposition \[CellB\]. Since $$e_n^\ast C\otimes_B W^0_i(p)\not=0,$$ for any $W^0_i(p)$, the proposition follows from the proof of [@StructureCell Proposition 4.3].
\[Faith\]
Covers of the diagram algebras {#SecCov}
==============================
The two main methods to prove that cell modules of a cellular algebra form a standard system are construction of a certain quasi-hereditary 1-cover ([*viz.*]{} a cover-Schur algebra), see [@Nakano1; @Henke]; and cellular/base stratification, see [@HHKP] and Section \[Sec8\].
When the diagram algebra $A$ in Lemma \[Lemab\] is Morita equivalent to $C$, the base stratification of $C$ solves the problem also for $A$. In this section, we use a combination of both approaches above, for when $A$ is not Morita equivalent to $C$. We prove that the base stratified algebra $C$ is almost always a $1$-faithful cover of $A$, in the sense of Section \[IntroFaithCov\]. Even though $C$ might not be quasi-hereditary, this determines when the cell modules of $A$ form a standard system.
Connection with the coarse filtration
-------------------------------------
In Section \[SecDefCat\], we defined the diagram algebras as the endomorphism algebras of the objects of the corresponding category. Remarkably, we can reconstruct the respective categories starting from the diagram algebra, by using the elements $\{a_i,b_i,c_i^\ast\,|\, i\in I_A(n)\}$ of Lemma \[Lemab\].
\[thmcoarseP\] Consider an arbitrary field ${\Bbbk}$, $\delta\in{\Bbbk}$ and let ${\mathcal{A}}$ be ${\mathcal{P}}(\delta)$, ${\mathcal B}(\delta)$, ${}^p{\mathcal B}(\delta)$, $J(\delta)$ or ${{\mathrm{TL}}}(\delta)$. For $n\in{\mathbb{Z}}_{>0}$, set $A:={\mathrm{End}}_{{\mathcal{A}}}(n)$. For any $i,j\in I_A(n)$, we have an isomorphism $${\mathrm{Hom}}_{{\mathcal{A}}}(j,i)\;\tilde{\to}\; {\mathrm{Hom}}_A(Ac_i^\ast,Ac_j^\ast);\quad x\mapsto \alpha_x\;\mbox{ with}\quad \alpha_x(c_i^\ast)=a_i xb_j,\quad\mbox{for~$x\in {\mathrm{Hom}}_{{\mathcal{A}}}(j,i)$}.$$Moreover, the composition of such morphisms on both sides agrees contravariantly.
As a special case we have the following corollary, giving an alternative description of the category algebra $C$ connected to the diagram algebra $A$.
\[CorCatAlgAlt\] Maintain the notation of Theorem \[thmcoarseP\] and consider the category algebra $C:={\Bbbk}[{\mathcal{A}}^{\le n}]$. We have an isomorphism of left $A$-modules $$Ac^\ast_j\;\stackrel{\sim}{\to}\; {\mathrm{Hom}}_{{\mathcal{A}}}(j,n)=e_n^\ast Ce_j^\ast,\quad c_j^\ast\mapsto a_j,\qquad\mbox{for }\; j\in I_A(n),$$ and an isomorphism of algebras $$C\;\cong\; {\mathrm{End}}_A(\,\bigoplus_{j\in I_A(n)} Ac^\ast_j\,)^{{{\rm op}}}.$$
### {#section-4}
Before we prove the theorem, we elaborate on the idempotents $c_i^\ast$. The partition algebra has a filtration by two-sided ideals, known as the [*coarse filtration*]{}: $$0=J_0\subsetneq J_1\subsetneq J_2\subsetneq \cdots\subsetneq J_{n}\subsetneq J_{n+1}={\mathcal{P}}_n(\delta),$$ see [*e.g.*]{} [@Xi Lemma 4.6]. Here, $J_i$ is the ideal spanned by those partitions of $$\label{SSS}
S=S_1\sqcup S_2=\{1,2,\ldots,n\}\sqcup\{1',2',\ldots, n'\},$$ where at most $i-1$ of the subsets contain elements of both $S_1$ and $S_2$, for each $1\le i\le n+1$. Note that we have $J_i=Ac^\ast_{i-1}A$ for $1\le i\le n+1$. Hence, $J_i$ is an idempotent ideal, if either $i\not=1$ or $\delta\not=0$. When $\delta=0$, we have $J_1^2=0$.
When $\delta\not=0$, the coarse filtration is an exactly standardly stratifying chain, even quasi-heredity when ${{\rm{char}}}({\Bbbk})\not\in[2,n]$, which leads to the corresponding properties of ${\mathcal{P}}_n(\delta)$ in Theorem \[ThmC\].
### {#section-5}
By restricting the coarse filtration of ${\mathcal{P}}_n(\delta)$, we obtain coarse filtrations of ${\mathcal B}_n(\delta)$, ${\mathcal B}_{r,s}(\delta)$, $J_n(\delta)$ and ${{\mathrm{TL}}}_n(\delta)$. These coarse filtrations are based on the number of propagating lines. The corresponding idempotents (or the nilpotent element) generating the ideals are also given by $\{c_i^\ast\}$ in Lemma \[Lemab\].
Now we start the proof of Theorem \[thmcoarseP\] with the following lemma.
\[LemAc00\] Let $A$ be one of the diagram algebras in Theorem \[thmcoarseP\], with $n$ even in case $A$ is ${\mathcal B}_n(\delta)$, $J_n(\delta)$ or ${{\mathrm{TL}}}_n(\delta)$. We have an isomorphism $${\mathrm{Hom}}_A(Ac_0^\ast,A)\;\,\tilde{\to}\;\,c_0^\ast A,\quad\alpha\mapsto \alpha(c_0^\ast).$$
We have a monomorphism $$\phi:{\mathrm{Hom}}_A(Ac_0^\ast,A)\;\,\hookrightarrow\;\, A,\quad\alpha\mapsto \alpha(c_0^\ast).$$ First we claim that $c_0^\ast A$ is contained in the image of $\phi$. We have the $A$-bimodule isomorphism $$Ac_0^\ast\otimes_{{\Bbbk}} c_0^\ast A\;\tilde\to \; A c_0^\ast A;\qquad ac_0^\ast \otimes c_0^\ast b\mapsto ac_0^\ast b,$$ which is clear by construction, or follows from the cellular structure. In particular, for any $x\in c_0^\ast A$, the above gives rise to a morphism $\alpha_x$ from $Ac_0^\ast$ to $A$, given by $\alpha_x(ac_0^\ast)=ax$. Hence, $\phi(\alpha_x)=x$, which proves the claim.
Now we will prove that the image of $\phi$ is contained in $c_0^\ast A$. If $c_0^\ast$ is an idempotent, this follows from $\alpha(c_0^\ast)=c_0^\ast\alpha(c_0^\ast)$. Consider therefore $(c_0^\ast)^2=0$ and some element $a\in A$ in the image of $\phi$, so $a=\alpha(c_0^\ast)$ for some $\alpha$. We prove that $a\in c_0^\ast A$ by considering cases separately. We will use the conventions of \[IntroC\] for the elements $c_i^\ast$.
Firstly, consider $A={\mathcal{P}}_n(0)$. Observe that $a$ must satisfy $c_1^\ast a=a$, since $c_1^\ast c_0^\ast=c_0^\ast$. Hence, $a$ must be a linear combination of partitions of where $S_2=\{1',\ldots, n'\}$ is contained in a subset. So $a=a_1+a_2$, where $a_1$ is a linear combination of partitions where $S_2$ is a subset (hence $a_1\in c_0^\ast A$) and $a_2$ is a linear combination of partitions which have a subset which strictly contains $S_2$. We define $u=u_1-u_2$, with $u_1,u_2$ partitions of as: $$\label{equ}u_1:=\{\{1,2,\ldots,n,1'\},\{2',3',\ldots,n'\}\},\quad u_2:=\{\{1,2,\ldots,n,2',3',\ldots,n'\},\{1'\}\}.$$ Since $uc_0^\ast=0=u a_1$, we must have $ua_2=0$. However, $u_1a_2$ is a linear combination of partitions where $\{1'\}$ is strictly contained in a subset, whereas $u_2a_2$ is a linear combination of partitions where $\{1'\}$ is a subset. Hence we must have $u_1a_2=0=u_2a_2$. However, the partition $v$ consisting of one set satisfies $vu_1a_2=a_2$. This shows that $a_2=0$ and hence that $a\in c_0^\ast A$.
Consider $A$ equal to $ {\mathcal B}_n(0)$, $J_n(0)$ or ${{\mathrm{TL}}}_n(0)$ with $n$ even. As the case $n=2$ is straightforward, we assume $n\ge 4$. We now have that $c_2^\ast a=a$. This means $a$ is a linear combination of diagrams with top row of the form $$\label{diagfora}
\begin{tikzpicture}[scale=0.9,thick,>=angle 90]
\begin{scope}[xshift=8cm]
\draw (2.8,0.8) -- +(0,0.2);
\draw (3.4,0.8) -- +(0,0.2);
\draw (4,1) to [out=-90,in=-180] +(.3,-.3) to [out=0,in=-90] +(.3,.3);
\draw (5.2,1) to [out=-90,in=-180] +(.3,-.3) to [out=0,in=-90] +(.3,.3);
\draw [dotted] (6.1,.9) -- +(1,0);
\draw (7.5,1) to [out=-90,in=-180] +(.3,-.3) to [out=0,in=-90] +(.3,.3);
\end{scope}
\end{tikzpicture}$$ where the first two dots can be arbitrarily connected. Consider also the diagram $$\label{Defw}
\begin{tikzpicture}[scale=0.9,thick,>=angle 90]
\begin{scope}[xshift=8cm]
\node at (-3,0.5) {$w:=$};
\draw (-1.7,1) to [out=-90,in=-180] +(.3,-.3) to [out=0,in=-90] +(.3,.3);
\draw (-1.7,0) to [out=50, in=-130] +(1.2,1);
\draw (-1.1,0) to [out=90,in=-180] +(.3,.3) to [out=0,in=90] +(.3,-.3);
\draw (.1,0) -- +(0,1);
\draw (.7,0) -- +(0,1);
\draw (1.3,0) -- +(0,1);
\draw [dotted] (1.6,.5) -- +(1,0);
\draw (2.9,0) -- +(0,1);
\draw (3.5,0) -- +(0,1);
\end{scope}
\end{tikzpicture}$$ Since $wc_0^\ast=c_0^\ast$ we must also have $wa=a$. This implies that the top rows of the diagrams , appearing in $a$, must have only caps. It now follows easily that $a\in c_0^\ast A$.
Now set $A:={\mathcal B}_{r,r}(0)$ and observe that $a=c_1^\ast a$. We focus on the case $r>1$ and define $$\label{Defv}
\begin{tikzpicture}[scale=0.9,thick,>=angle 90]
\begin{scope}[xshift=4cm]
\node at (0,1) {$v:=$};
\draw (2.8,0) to [out=50, in=130] +(3.3,0);
\draw (2.8,2) to [out=-30, in=-150] +(6,0);
\draw (3.4,0) -- +(0,2);
\draw [dotted] (3.9,1) -- +(1,0);
\draw (5.5,0) -- +(0,2);
\draw[line width =2] (5.8,0) -- +(0,2);
\draw (6.7,0) -- +(0,2);
\draw [dotted] (7.1,1) -- +(0.6,0);
\draw (8.2,0) -- +(0,2);
\draw (8.8,0) to [out=120, in=-60] +(-2.7,2);
\end{scope}
\end{tikzpicture}$$ which satisfies $vc_0^\ast=c_0^\ast$. The conclusion follows as for the previous case.
First we prove that ${\mathrm{Hom}}_{{\mathcal{A}}}(j,i)\to {\mathrm{Hom}}_A(Ac_i^\ast,Ac_j^\ast)$ is indeed an isomorphism. We distinguish four different cases.
i\) When $c_i^\ast$ and $c_j^\ast$ are idempotents, the proposed map is obviously well-defined and has inverse $\alpha\mapsto b_i \alpha(c_i^\ast)a_j$.
ii\) Now assume that $i=0$ and $(c_0^\ast)^2=0$, but $j>0$, so that $c_j^\ast$ is an idempotent. By Lemma \[LemAc00\], it suffices to prove that $$\phi:\;{\mathrm{Hom}}_{{\mathcal{A}}}(j,0)\;{\to}\; c_0^\ast Ac_j^\ast;\qquad x\mapsto a_0 xb_j,$$ is an isomorphism. An inverse to $\phi$ is constructed by mapping any diagram $d$ in $c_0^\ast Ac_j^\ast$ to $d'a_j$ where $d'$ is the diagram obtained from $d$ by omitting the top row (essentially forgetting the diagram $a_0$), proving that $\phi$ is an isomorphism.
iii\) The case where $c_i^\ast$ is an idempotent, but $c_j^\ast=c_0^\ast$ is not, follows similarly, by using ${\mathrm{Hom}}_{A}(Ac_i^\ast,Ac_j^\ast)\cong c_i^\ast Ac_j^\ast$.
iv\) Finally, assume $i=j=0$ and $(c_0^\ast)^2=0$. Using the monomorphism $Ac_0^\ast\hookrightarrow A$ and the isomorphism in Lemma \[LemAc00\] shows that the image of the injective composition $${\mathrm{Hom}}_{A}(Ac_0^\ast,Ac_0^\ast)\hookrightarrow {\mathrm{Hom}}_A(Ac_0^\ast, A)\stackrel{\sim}{\to} c_0^\ast A$$ is the one-dimensional space $c_0^\ast A\cap Ac_0^\ast={\Bbbk}c_0^\ast$. So both ${\mathrm{Hom}}_{A}(Ac_0^\ast,Ac_0^\ast)$ and ${\mathrm{Hom}}_{{\mathcal{A}}}(0,0)$ are one-dimensional and it follows that the morphism in the lemma is an isomorphism.
Now take $x\in {\mathrm{Hom}}_{{\mathcal{A}}}(j,i)$ and $y\in {\mathrm{Hom}}_{{\mathcal{A}}}(k,j)$, so $xy\in {\mathrm{Hom}}_{{\mathcal{A}}}(k,i)$. We claim that $$\alpha_y\circ\alpha_x=\alpha_{xy}.$$ We have $\alpha_{y}\circ\alpha_x(c_i^\ast)=\alpha_y(a_i xb_j)$. We can take $x'\in {\mathrm{Hom}}_{{\mathcal{A}}}(n,i)$ such that $x' a_j=x$, then $$\alpha_y(a_i xb_j)=a_ix' \alpha_y(c_j^\ast)=a_ix' a_jyb_k=a_ixyb_k,$$ proving the claim.
Covers {#SecCovers}
------
\[ThmCov\]Let $C$ be any of the category algebras in Lemma \[Lemab\] and $A=e^\ast_nCe^\ast_n$ the corresponding diagram algebra. Then $C$ is a cover of $A$.
The condition in \[SecSecCover\] follows from Corollary \[CorCatAlgAlt\].
In most cases, the cover $C$ is Morita equivalent to $A$, by Theorem \[ThmMor\]. Now we focus on the remaining cases. Consider the full subcategory $\overline{{{\mathcal{F}}}}$ of $C$-mod of modules which admit a filtration with sections $\overline{\Delta}(\mu)$, $\mu\in\Lambda$. Recall from Section \[SecCover\] the exact functor $$F:=e_n^\ast C\otimes_C-\cong e_n^\ast- \;:C\mbox{-mod}\;\to\; A\mbox{-mod}.$$
\[Thm0Cover\] Let $A$ be ${\mathcal{P}}_n(0)$ with $n>2$; ${\mathcal B}_n(0)$, $J_n(0)$ or ${{\mathrm{TL}}}_n(0)$ with $n$ even and $n>4$; or $A={\mathcal B}_{r,r}(0)$ with $r>2$. For all $M,N\in {\overline{\mathcal{F}}}$, the functor $F$ induces isomorphisms $${\mathrm{Hom}}_{C}(M,N)\;\,\tilde\to\;\, {\mathrm{Hom}}_A(FM,FN)\quad\mbox{ and }\,\quad {\mathrm{Ext}}^1_{C}(M,N)\;\,\tilde\to\;\, {\mathrm{Ext}}^1_A(FM,FN).$$ Hence, the cover $C$ is $1$-faithful.
\[SchurGen\] Consider one of the algebras $C$ in Theorem \[ThmDiagram2\] under the condition that its cell modules form a standard system. Due to the base stratification, the Schur algebra of $C$ can be obtained similarly to [@Paget; @HHKP; @Henke]. The corresponding Schur algebra of $C$ is also naturally a Schur algebra of $A$ in case $C$ and $A$ are Morita equivalent, but also under the conditions in Theorem \[Thm0Cover\]. This will be worked out in more detail elsewhere.
We make preparations for the proof of the theorem. The left exact functor $$G:={\mathrm{Hom}}_{A}(e_n^\ast C,-):\; A\mbox{-mod}\;\to\; C\mbox{-mod},$$ is right adjoint to $F$. Hence we have the adjoint (unit) natural transformation $$\eta:{\mathrm{Id}}\to G\circ F.$$
\[LemGFiso\] Let $A$ be ${\mathcal B}_n(0)$, $J_n(0)$ or ${{\mathrm{TL}}}_n(0)$, for $n$ even, ${\mathcal{P}}_n(0)$ or ${\mathcal B}_{r,r}(0)$.
1. For any $M$ in $C$[-mod]{}, $\eta_M$ induces an isomorphism $M\cong G\circ F(M)$ if and only if $$e_0^\ast M\;\to\; {\mathrm{Hom}}_A(Ac^\ast_0,e_n^\ast M):\quad\, v\mapsto \beta_v,\;\;\mbox{ where }\,\beta_v(c_0^\ast)=a_0v,$$ is an isomorphism, with $a_0$ from Lemma \[Lemab\].
2. For any $M$ in $C$[-mod]{}, we have ${\mathcal{R}}_1G\circ F(M)=0$ if and only if $${\mathrm{Ext}}^1_A(Ac_0^\ast,e_n^\ast M)=0.$$
We evaluate $\eta$ on $M$, to get the following morphism of $C$-modules: $$\eta_M:M\to G\circ F(M)\stackrel{\sim}{\to}{\mathrm{Hom}}_A(e_n^\ast C,e_n^\ast M);\;\quad v\mapsto \alpha_v,\;\;\alpha_v(c)=cv.$$ This restricts to vector space morphisms $$\eta^i:\;\,e_i^\ast M\to{\mathrm{Hom}}_A(e_n^\ast Ce_i^\ast,e_n^\ast M)\quad\mbox{for}\quad i\in I_A.$$ We also introduce $$\rho^i:\;\, {\mathrm{Hom}}_A(e_n^\ast Ce_i^\ast,e_n^\ast M)\to e_i^\ast M;\;\quad \alpha\mapsto b_i\alpha(a_i),$$ with $a_i,b_i$ as in Lemma \[Lemab\]. Then $\eta^i$ and $\rho^i$ are mutually inverse if $i\not=0$.
Thus we find that $\eta_M$ is an isomorphism if and only if $\eta^0$ is. Part (1) then follows from applying the isomorphism ${\mathrm{Hom}}_A(e_n^\ast Ce_0^\ast, e_n^\ast M)\cong {\mathrm{Hom}}_A(Ac^\ast_0,e_n^\ast M)$ from Corollary \[CorCatAlgAlt\].
To prove part (2), we observe that we have $${\mathcal{R}}_1G\cong {\mathrm{Ext}}^1_A(e_n^\ast C,-)\cong {\mathrm{Ext}}^1_A(\bigoplus_{j\in I_A}Ac_j^\ast,-)\cong {\mathrm{Ext}}^1_A(Ac_0^\ast,-),$$ by Corollary \[CorCatAlgAlt\], since $Ac_j^\ast$ is projective for $j\not=0$.
The proof of the lemma also gives the following result.
\[Corin\] Let $A$ be as in Lemma \[LemGFiso\]. For $i\not=0$, we have $${\mathrm{Hom}}_A(Ac_i^\ast,e_n^\ast M)\;\cong\; e_i^\ast M,\quad\mbox{for any $M\in A${\rm -mod}.}$$
### {#IntroC}
We fix some of the $c_i^\ast$ in Lemma \[Lemab\]. For ${\mathcal B}_n(0)$, $J_n(0)$ or ${{\mathrm{TL}}}_n(0)$ with $n$ even, take $$\begin{tikzpicture}[scale=0.9,thick,>=angle 90]
\begin{scope}[xshift=4cm]
\node at (0.4,0.5) {$c_0^\ast=$};
\draw (1.9,0) to [out=90,in=-180] +(.3,.3) to [out=0,in=90] +(.3,-.3);
\draw (3.1,0) to [out=90,in=-180] +(.3,.3) to [out=0,in=90] +(.3,-.3);
\draw (4.3,0) to [out=90,in=-180] +(.3,.3) to [out=0,in=90] +(.3,-.3);
\draw [dotted] (5.2,.1) -- +(1,0);
\draw (6.6,0) to [out=90,in=-180] +(.3,.3) to [out=0,in=90] +(.3,-.3);
\draw (1.9,1) to [out=-90,in=-180] +(.3,-.3) to [out=0,in=-90] +(.3,.3);
\draw (3.1,1) to [out=-90,in=-180] +(.3,-.3) to [out=0,in=-90] +(.3,.3);
\draw (4.3,1) to [out=-90,in=-180] +(.3,-.3) to [out=0,in=-90] +(.3,.3);
\draw [dotted] (5.1,.9) -- +(1,0);
\draw (6.6,1) to [out=-90,in=-180] +(.3,-.3) to [out=0,in=-90] +(.3,.3);
\node at (9.4,0.5) {$c_2^\ast=$};
\draw (10.9,0) -- +(0,1);
\draw (11.5,0) to [out=90,in=-180] +(.3,.3) to [out=0,in=90] +(.3,-.3);
\draw (12.7,0) to [out=90,in=-180] +(.3,.3) to [out=0,in=90] +(.3,-.3);
\draw [dotted] (13.6,.1) -- +(1,0);
\draw (15,0) to [out=90,in=-180] +(.3,.3) to [out=0,in=90] +(.3,-.3);
\draw (16.2,0) to [out=140,in=-40] +(-4.7,1);
\draw (12.1,1) to [out=-90,in=-180] +(.3,-.3) to [out=0,in=-90] +(.3,.3);
\draw (13.3,1) to [out=-90,in=-180] +(.3,-.3) to [out=0,in=-90] +(.3,.3);
\draw [dotted] (14.1,.9) -- +(1,0);
\draw (15.6,1) to [out=-90,in=-180] +(.3,-.3) to [out=0,in=-90] +(.3,.3);
\end{scope}
\end{tikzpicture}$$ $$\begin{tikzpicture}[scale=0.9,thick,>=angle 90]
\begin{scope}[xshift=4cm]
\node at (9.4,0.5) {$c_4^\ast=$};
\draw (10.9,0) -- +(0,1);
\draw (11.5,0) -- +(0,1);
\draw (12.1,0) -- +(0,1);
\draw (12.7,0) to [out=90,in=-180] +(.3,.3) to [out=0,in=90] +(.3,-.3);
\draw [dotted] (13.6,.1) -- +(1,0);
\draw (15,0) to [out=90,in=-180] +(.3,.3) to [out=0,in=90] +(.3,-.3);
\draw (16.2,0) to [out=133,in=-47] +(-3.5,1);
\draw (13.3,1) to [out=-90,in=-180] +(.3,-.3) to [out=0,in=-90] +(.3,.3);
\draw [dotted] (14.1,.9) -- +(1,0);
\draw (15.6,1) to [out=-90,in=-180] +(.3,-.3) to [out=0,in=-90] +(.3,.3);
\node at (16.6,0) {.};
\end{scope}
\end{tikzpicture}$$ For $A={\mathcal B}_{r,r}(0)$, take $$\begin{tikzpicture}[scale=0.9,thick,>=angle 90]
\begin{scope}[xshift=4cm]
\node at (1.8,1) {$c_0^\ast:=$};
\draw (2.8,0) to [out=30, in=150] +(6,0);
\draw (2.8,2) to [out=-30, in=-150] +(6,0);
\draw (3.4,0) to [out=32, in=148] +(4.8,0);
\draw (3.4,2) to [out=-32, in=-148] +(4.8,0);
\draw (3.9,0) [dotted] -- +(1.2,0);
\draw (3.9,2) [dotted] -- +(1.2,0);
\draw (5.5,0) to [out=90,in=-180] +(.3,.3) to [out=0,in=90] +(.3,-.3);
\draw (5.5,2) to [out=-90,in=180] +(.3,-.3) to [out=0,in=-90] +(.3,.3);
\draw[line width =2] (5.8,0) -- +(0,2);
\draw (6.5,0) [dotted] -- +(1.2,0);
\draw (6.5,2) [dotted] -- +(1.2,0);
\node at (10.8,1) {$c_1^\ast:=$};
\draw (11.8,0) to [out=30, in=150] +(5.4,0);
\draw (12.4,0) to [out=32, in=148] +(4.2,0);
\draw (12.4,2) to [out=-32, in=-148] +(4.8,0);
\draw (12.9,0) [dotted] -- +(0.7,0);
\draw (12.9,2) [dotted] -- +(1.2,0);
\draw (13.9,0) to [out=80,in=100] +(1.2,0);
\draw (14.5,0) to [out=120, in=-60] +(- 2.7,2);
\draw (14.5,2) to [out=-90,in=180] +(.3,-.3) to [out=0,in=-90] +(.3,.3);
\draw[line width =2] (14.8,0) -- +(0,2);
\draw (15.5,0) [dotted] -- +(0.7,0);
\draw (15.5,2) [dotted] -- +(1.2,0);
\draw (17.8,0) -- +(0,2);
\end{scope}
\end{tikzpicture}$$ $$\begin{tikzpicture}[scale=0.9,thick,>=angle 90]
\begin{scope}[xshift=4cm]
\node at (1.8,1) {$c_2^\ast:=$};
\draw (2.8,0) -- +(0,2);
\draw (3.4,0) to [out=32, in=148] +(4.2,0);
\draw (4,2) to [out=-32, in=-148] +(3.6,0);
\draw (3.9,0) [dotted] -- +(0.7,0);
\draw (4.4,2) [dotted] -- +(0.7,0);
\draw (4.9,0) to [out=80,in=100] +(1.2,0);
\draw (5.5,0) to [out=120, in=-60] +(- 2.1,2);
\draw (5.5,2) to [out=-90,in=180] +(.3,-.3) to [out=0,in=-90] +(.3,.3);
\draw[line width =2] (5.8,0) -- +(0,2);
\draw (6.5,0) [dotted] -- +(0.7,0);
\draw (6.5,2) [dotted] -- +(0.7,0);
\draw (8.2,0) -- +(0,2);
\draw (8.8,0) -- +(0,2);
\node at (9.1,0) {.};
\end{scope}
\end{tikzpicture}$$ For ${\mathcal{P}}_n(0)$, take $$c_0^\ast=\{\{1,2,\ldots,n\},\{1',2',\ldots,n'\}\},\quad c_1^\ast=\{\{1,1',2',\ldots,n'\},\{2,3,\ldots,n\}\},\mbox{ and}$$ $$c_2^\ast=\{\{1,1'\},\{2,2',3',\ldots,n'\},\{3,4,\ldots,n\}\}.$$
We also introduce $\gamma=\gamma_A$, where $\gamma_A=1$ for $A={\mathcal B}_{r,r}(0)$ or ${\mathcal{P}}_n(0)$ and $\gamma_A=2$ otherwise.
\[Ac0A\] Consider $A$ equal to ${\mathcal B}_n(0)$, $J_n(0)$ or ${{\mathrm{TL}}}_n(0)$ for $n$ even and $n\ge 4$, ${\mathcal{P}}_n(0)$ with $n\ge 2$, or ${\mathcal B}_{r,r}(0)$ with $r\ge 2$. We have $${\mathrm{Hom}}_A(Ac_0^\ast, A/Ac_0^\ast A)=0.$$
This is a stronger version of Lemma \[LemAc00\]. It can be proved using the same arguments. Consider $\alpha:Ac_0^\ast\to A/Ac_0^\ast A$ and $a\in A$ such that $a+Ac_0^\ast A=\alpha(c_0^\ast)$.
For ${\mathcal B}_n(0)$, $J_n(0)$ or ${{\mathrm{TL}}}_n(0)$, we have $c_2^\ast c_0^\ast=c_0^\ast$, so we can assume that $a\in c_2^\ast A$. As $w$ in equation is an idempotent satisfying $wc_0^\ast=c_0^\ast$, we can further assume that $wa=a$. These two conditions on $a$ show that it must be a linear combination of diagrams with $n/2$ caps, so $a\in Ac_0^\ast A$ and hence $\alpha=0$.
The proof for ${\mathcal B}_{r,r}(0)$ is identical, by using the idempotent $v$ in equation . Also the proof for ${\mathcal{P}}_n(0)$ works along the same lines, by using $uc_0^\ast=0$ with $u=u_1-u_2$ in .
\[esc0\] Let $A$ be as in Lemma \[Ac0A\]. There exists an exact sequence $$Ac^\ast_{2\gamma}\to Ac_{\gamma}^\ast\to Ac_0^\ast\to 0.$$
First, let $A$ be ${\mathcal B}_n(0)$, $J_n(0)$ or ${{\mathrm{TL}}}_n(0)$. The map $Ac_2^\ast{\twoheadrightarrow}Ac_0^\ast$ is defined as $a\mapsto ac_0^\ast$ for any $a\in Ac_2^\ast$, where surjectivity follows from $c_2^\ast c_0^\ast=c_0^\ast$. Now we determine the kernel $K$ of this epimorphism. From the structure of $c_2^\ast$ and $c_0^\ast$ it follows that $K$ is the spanned by all diagrams without propagating lines and by all elements of the form $d_1-d_2$, where $d_1,d_2\in Ac_2^\ast$ are diagrams satisfying the following conditions. For $k\in \{1,2\}$, the diagram $d_k$ has two propagating lines, connecting $1$ to $i_k$ and $n$ to $j_k$, giving four different dots $\{i_1,i_2,j_1,j_2\}$. There is a cap in $d_k$ which connects dots $i_l$ and $j_l$, with $\{k,l\}=\{1,2\}$. Finally, removing these caps and propagating lines in $d_1$ and $d_2$ yield identical $(n-2,n-4)$-Brauer diagrams.
An example of such a $d_1-d_2$, with $i_1=1'$, $j_1=2'$, $i_2=3'$ and $j_2=4'$, is given by $$\label{Defx}
\begin{tikzpicture}[scale=0.9,thick,>=angle 90]
\begin{scope}[xshift=8cm]
\node at (1.4,0.5) {$x:=$};
\draw (2.9,0) -- +(0,1);
\draw (3.5,0) to [out=90,in=-180] +(.3,.3) to [out=0,in=90] +(.3,-.3);
\draw (4.7,0) to [out=90,in=-180] +(.3,.3) to [out=0,in=90] +(.3,-.3);
\draw [dotted] (5.6,.1) -- +(1,0);
\draw (6.9,0) to [out=90,in=-180] +(.3,.3) to [out=0,in=90] +(.3,-.3);
\draw (8.2,0) to [out=147,in=-33] +(-4.7,1);
\draw (4.1,1) to [out=-90,in=-180] +(.3,-.3) to [out=0,in=-90] +(.3,.3);
\draw (5.2,1) to [out=-90,in=-180] +(.3,-.3) to [out=0,in=-90] +(.3,.3);
\draw [dotted] (6.1,.9) -- +(1,0);
\draw (7.5,1) to [out=-90,in=-180] +(.3,-.3) to [out=0,in=-90] +(.3,.3);
\node at (9.5,0.5) {$-$};
\draw (10.9,0) to [out=70,in=-110] +(1.2,1);
\draw (11.5,0) to [out=90,in=-180] +(.3,.3) to [out=0,in=90] +(.3,-.3);
\draw (12.7,0) to [out=90,in=-180] +(.3,.3) to [out=0,in=90] +(.3,-.3);
\draw [dotted] (13.6,.1) -- +(1,0);
\draw (14.9,0) to [out=90,in=-180] +(.3,.3) to [out=0,in=90] +(.3,-.3);
\draw (16.2,0) to [out=136,in=-44] +(-3.5,1);
\draw (10.9,1) to [out=-90,in=-180] +(.3,-.3) to [out=0,in=-90] +(.3,.3);
\draw (13.2,1) to [out=-90,in=-180] +(.3,-.3) to [out=0,in=-90] +(.3,.3);
\draw [dotted] (14.1,.9) -- +(1,0);
\draw (15.5,1) to [out=-90,in=-180] +(.3,-.3) to [out=0,in=-90] +(.3,.3);
\end{scope}
\end{tikzpicture}$$
Now we claim that $K=Ax$. That the span of all diagrams without propagating lines is in $Ax$ follows easily from multiplying $x$ with the diagram having a cup and cap connecting the first two dots and otherwise only vertical propagating lines. For $d_1-d_2\in K$ as above, we consider the three algebras separately.
For $A={\mathcal B}_n(0)$, we can consider a diagram $a\in{\mathbb{S}}_n$, where $1$ is connected to $i_1$, $2$ to $j_1$, $3$ to $i_2$ and $4$ to $j_2$. It then follows easily that $a$ can be completed such that $ax=d_1-d_2$.
For $A={{\mathrm{TL}}}_n(0)$, take an arbitrary $d_1-d_2$ as above and consider the $n/2-2$ cups which appear in both diagrams $d_1$ and $d_2$. It follows easily that this information determines $d_1-d_2$ uniquely, up to sign. Now we consider the unique diagram $a\in A$ which contains those $n/2-2$ cups, the $n/2-2$ caps which appear in $c_4^\ast$ and four propagating lines. Then we find $ax=\pm (d_1-d_2)$. Finally, the case $A=J_n(0)$ follows similarly.
As $c_4^\ast x=x$, we have a surjection $Ac_4^\ast{\twoheadrightarrow}K$, with $K=Ax$, proving the exact sequence.
For the two remaining algebras one proves, similarly to the above, that the kernel of $Ac_1^\ast {\twoheadrightarrow}Ac_0^\ast$ is generated by $x=c_2^\ast x$, given by $$\begin{tikzpicture}[scale=0.9,thick,>=angle 90]
\begin{scope}[xshift=4cm]
\node at (1.5,1) {$x:=$};
\draw (2.8,0) to [out=30, in=150] +(5.4,0);
\draw (3.4,0) to [out=32, in=148] +(4.2,0);
\draw (3.4,2) to [out=-32, in=-148] +(4.8,0);
\draw (3.9,0) [dotted] -- +(0.7,0);
\draw (3.9,2) [dotted] -- +(1.2,0);
\draw (4.9,0) to [out=80,in=100] +(1.2,0);
\draw (5.5,0) to [out=120, in=-60] +(- 2.7,2);
\draw (5.5,2) to [out=-90,in=180] +(.3,-.3) to [out=0,in=-90] +(.3,.3);
\draw[line width =2] (5.8,0) -- +(0,2);
\draw (6.5,0) [dotted] -- +(0.7,0);
\draw (6.5,2) [dotted] -- +(1.2,0);
\draw (8.8,0) -- +(0,2);
\node at (10.5,1) {$-$};
\draw (11.8,0) to [out=30, in=150] +(5.4,0);
\draw (12.4,0) to [out=32, in=148] +(4.2,0);
\draw (11.8,2) to [out=-30, in=-150] +(6,0);
\draw (13,2) to [out=-35, in=-145] +(3.6,0);
\draw (12.9,0) [dotted] -- +(0.7,0);
\draw (13.6,2) [dotted] -- +(0.7,0);
\draw (13.9,0) to [out=80,in=100] +(1.2,0);
\draw (14.5,0) to [out=120, in=-60] +(- 2.1,2);
\draw (14.5,2) to [out=-90,in=180] +(.3,-.3) to [out=0,in=-90] +(.3,.3);
\draw[line width =2] (14.8,0) -- +(0,2);
\draw (15.5,0) [dotted] -- +(0.7,0);
\draw (15.2,2) [dotted] -- +(0.7,0);
\draw (17.8,0) to [out=100,in=-80] +(-0.6,2);
\end{scope}
\end{tikzpicture}$$ for $A={\mathcal B}_{r,r}(0)$, and by $$x:=\;\;\{\{1,1'\},\{2,3,\ldots,n\},\{2',3',\ldots,n'\}\}\;-\;\{\{1'\},\{2,3,\ldots,n\},\{1,2',3',\ldots,n'\}\},$$ for $A={\mathcal{P}}_n(0)$.
\[LemDi\] For $A$ as in Lemma \[LemGFiso\], let $D_i:=e_n^\ast C\otimes_B He_i^\ast$. Then $$D_i\,\cong\, Ac_i^\ast/Ac_{i-\gamma}^\ast Ac_i^\ast,\;\,\mbox{ for }\quad\;i\in I_A.$$
Set $f'=\sum_{j<i}e_j^\ast$ and $f=f'+e_i^\ast$. Lemma \[LemABAM\] and equation then imply that $$C\otimes_B He_i^\ast\;\cong\; \left(C/Cf'C\right)f.$$ By Corollary \[CorCatAlgAlt\], the $A$-module $e_n^\ast C\otimes_B He_i^\ast$ is a quotient of $Ac_i^\ast$. Furthermore, the submodule $e_n^\ast C\sum_{j<i}e_i^\ast Ce_i^\ast$ corresponds precisely to the submodule in $Ac_i^\ast$ spanned by diagram which have strictly fewer than $i$ propagating lines (subsets which contain dots on both lines). This is precisely $Ac_{i-\gamma}^\ast Ac_i^\ast$.
\[PropHom\] For $A$ in Lemma \[Ac0A\], the unit $\eta:{\mathrm{Id}}\to G\circ F$ induces isomorphisms $$G\circ F(\overline{\Delta}(i,\nu))\;\cong\;\overline{\Delta}(i,\nu),\qquad\mbox{for all $i\in I_A$ and~$\nu\in\Lambda_i$.}$$
First we prove the case $i=0$. We have $\overline{\Delta}(0,\emptyset)\cong Ce_0^\ast$ and $F \overline{\Delta}(0,\emptyset)\cong Ac_0^\ast$ by Lemma \[LemDi\]. Lemma \[LemGFiso\](1) applied to $M=Ce_0^\ast$ shows that $\eta_M$ is indeed an isomorphism, by Theorem \[thmcoarseP\] for $i=j=0$.
For $i>0$, we have $e_0^\ast \overline{\Delta}(i,\nu)=0$ for all $\nu\in \Lambda_i$, so by Lemma \[LemGFiso\](1), we only need to prove $$\label{Hom000}{\mathrm{Hom}}_A(Ac_0^\ast, e_n^\ast \overline{\Delta}(i,\nu))=0.$$ Recall that $He_{\gamma}^\ast$ is a group algebra and hence self-injective. In particular, for any simple module $L^0(\gamma,\nu)$ of $He_\gamma^\ast$ we have $L^0(\gamma,\nu)\hookrightarrow He_\gamma^\ast$ and consequently $e_n^\ast\overline{\Delta}(\gamma,\nu)\hookrightarrow D_\gamma$. By Lemma \[LemDi\], $D_\gamma$ is a direct summand of $A/Ac_0^\ast A$. Hence Lemma \[Ac0A\] implies $${\mathrm{Hom}}_A(Ac_0^\ast, D_\gamma)=0.$$ Thus the above equation implies equation , concluding the case $i=\gamma$. Finally, for $i>\gamma$, we use Lemma \[esc0\], which implies that the left-hand side of equation is a subspace of ${\mathrm{Hom}}_A(Ac^\ast_\gamma,e_n^\ast \overline{\Delta}(i,\nu))$, which is equal to $e_\gamma^\ast \overline{\Delta}(i,\nu)$ by Corollary \[Corin\]. Now $e_\gamma^\ast \overline{\Delta}(i,\nu)=0$ since $i>\gamma$, and equation is again satisfied.
### {#IntroY}
From now on we will assume that for $A$ equal to ${\mathcal B}_n(0)$, $J_n(0)$ or ${{\mathrm{TL}}}_n(0)$ we have $n>4$. This allows to introduce diagrams $y_1,y_2\in A$ as $$\begin{tikzpicture}[scale=1,thick,>=angle 90]
\begin{scope}[xshift=4cm]
\node at (7,0.5) {$y_1:=$};
\draw (7.9,0) -- +(0,1);
\draw (8.5,0) -- +(0,1);
\draw (9.1,0) to [out=60,in=-110] +(1.2,1);
\draw (9.7,0) to [out=90,in=-180] +(.3,.3) to [out=0,in=90] +(.3,-.3);
\draw [dotted] (10.6,.1) -- +(1,0);
\draw (12,0) to [out=90,in=-180] +(.3,.3) to [out=0,in=90] +(.3,-.3);
\draw (13.2,0) to [out=120,in=-60] +(-2.3,1);
\draw (9.1,1) to [out=-90,in=-180] +(.3,-.3) to [out=0,in=-90] +(.3,.3);
\draw [dotted] (12.14,.9) -- +(0.4,0);
\draw (12.6,1) to [out=-90,in=-180] +(.3,-.3) to [out=0,in=-90] +(.3,.3);
\draw (11.5,1) to [out=-90,in=-180] +(.3,-.3) to [out=0,in=-90] +(.3,.3);
\node at (15,0.5) {$y_2:=$};
\draw (15.9,0) to [out=60,in=-110] +(1.2,1);
\draw (16.5,0) to [out=60,in=-110] +(1.2,1);
\draw (17.1,0) to [out=60,in=-110] +(1.2,1);
\draw (17.7,0) to [out=90,in=-180] +(.3,.3) to [out=0,in=90] +(.3,-.3);
\draw [dotted] (18.6,.1) -- +(1,0);
\draw (20,0) to [out=90,in=-180] +(.3,.3) to [out=0,in=90] +(.3,-.3);
\draw (21.2,0) to [out=120,in=-60] +(-2.3,1);
\draw (15.9,1) to [out=-90,in=-180] +(.3,-.3) to [out=0,in=-90] +(.3,.3);
\draw [dotted] (20.14,.9) -- +(0.4,0);
\draw (20.6,1) to [out=-90,in=-180] +(.3,-.3) to [out=0,in=-90] +(.3,.3);
\draw (19.5,1) to [out=-90,in=-180] +(.3,-.3) to [out=0,in=-90] +(.3,.3);
\end{scope}
\end{tikzpicture}$$ Similarly, for $A={\mathcal B}_{r,r}(0)$ we will assume $r>2$, which allows to introduce $$\begin{tikzpicture}[scale=0.9,thick,>=angle 90]
\begin{scope}[xshift=4cm]
\node at (1.6,1) {$y_1:=$};
\draw (2.8,0) -- +(0,2);
\draw (3.4,0) to [out=32, in=148] +(4.2,0);
\draw (3.4,2) to [out=-30, in=-150] +(4.8,0);
\draw (4.6,2) to [out=-40, in=-140] +(2.4,0);
\draw (3.9,0) [dotted] -- +(0.7,0);
\draw (4.9,2) [dotted] -- +(0.4,0);
\draw (4.9,0) to [out=80,in=100] +(1.2,0);
\draw (5.5,0) to [out=120, in=-60] +(- 1.5,2);
\draw (5.5,2) to [out=-90,in=180] +(.3,-.3) to [out=0,in=-90] +(.3,.3);
\draw[line width =2] (5.8,0) -- +(0,2);
\draw (6.5,0) [dotted] -- +(0.7,0);
\draw (6.25,2) [dotted] -- +(0.4,0);
\draw (8.2,0) to [out=100,in=-80] +(-0.6,2);
\draw (8.8,0) -- +(0,2);
\node at (10,1) {$y_2:=$};
\draw (10.8,0) to [out=80, in=-100] +(0.6,2);
\draw (11.4,0) to [out=32, in=148] +(4.2,0);
\draw (10.8,2) to [out=-30, in=-150] +(6,0);
\draw (12.6,2) to [out=-40, in=-140] +(2.4,0);
\draw (11.9,0) [dotted] -- +(0.7,0);
\draw (12.9,2) [dotted] -- +(0.4,0);
\draw (12.9,0) to [out=80,in=100] +(1.2,0);
\draw (13.5,0) to [out=120, in=-60] +(- 1.5,2);
\draw (13.5,2) to [out=-90,in=180] +(.3,-.3) to [out=0,in=-90] +(.3,.3);
\draw[line width =2] (13.8,0) -- +(0,2);
\draw (14.5,0) [dotted] -- +(0.7,0);
\draw (14.25,2) [dotted] -- +(0.4,0);
\draw (16.2,0) to [out=100,in=-80] +(-0.6,2);
\draw (16.8,0) to [out=100,in=-80] +(-0.6,2);
\end{scope}
\end{tikzpicture}$$ For all four algebras a direct computation shows that $$\label{eqyx}
x\;=\; y_1x-y_2x,$$ with $x$ as introduced in the proof of Lemma \[esc0\].
For $A={\mathcal{P}}_n(0)$, we will assume that $n>2$, and we introduce $y\in A$, $$y:=\quad \{\{1,1'\},\{2,2'\},\{3,4,\ldots,n\},\{3',4',\ldots,n'\}\}\,-$$ $$\{\{1,1'\},\{2'\},\{3,4,\ldots,n\},\{2,3',4',\ldots,n'\}\}+\{\{1'\},\{1,2'\},\{3,4,\ldots,n\},\{2,3',4',\ldots,n'\}\}.$$ It follows immediately that $yx=0$.
\[LemHom01\] Let $A$ be as in Theorem \[Thm0Cover\]. Then $${\mathrm{Ext}}^1_{A}(Ac_0^\ast,Ac_0^\ast)=0.$$
First we will prove ${\mathrm{Hom}}_{A}(Ax,Ac_0^\ast)=0.$ Consider $\phi:Ax\to Ac_0^\ast$ and $a:=\phi(c_0^\ast)$.
First, let $A$ be ${\mathcal B}_n(0)$, $J_n(0)$ or ${{\mathrm{TL}}}_n(0)$, the case ${\mathcal B}_{r,r}(0)$ is proved similarly. As $c_4^\ast x=x$, we have $a\in c_4^\ast A c_0^\ast$. By equation , we must have $$a=y_1 a- y_2a.$$ This means that $a$ is a linear combination of diagrams which have the $n/2$ caps of $c_0^\ast$, the $n/2-2$ cups of $c_4^\ast$ and another cup connecting either $1',2'$, or $3',4'$. The only such diagram is $c_0^\ast$. However, $y_1c_0^\ast=y_2c_0^\ast$ and hence $a=0$.
For $A={\mathcal{P}}_n(0)$, we have $a\in c_2^\ast Ac_0^\ast$, with $\dim c_2^\ast Ac_0^\ast=2$. With $y$ from \[IntroY\], we must have $ya=0$. It follows by direct computation that $ya=0$ for $a\in c_2^\ast Ac_0^\ast$ implies $a=0$.
Hence, in every case we have indeed, ${\mathrm{Hom}}_{A}(Ax,Ac_0^\ast)=0$. Recall the short exact sequence $$\label{sesK}0\to Ax\to Ac_\gamma^\ast \to Ac_0^\ast\to 0$$ from the proof of Lemma \[esc0\]. This implies an exact sequence $$\label{ExtHomeq}{\mathrm{Hom}}_A(Ax,M)\to {\mathrm{Ext}}^1_A(Ac_0^\ast,M)\to 0,$$ for any $A$-module $M$, as $Ac_\gamma^\ast$ is projective. As we established that ${\mathrm{Hom}}_A(Ax,Ac_0^\ast)=0$, the statement follows.
\[LemHom02\] Let $A$ be as in Theorem \[Thm0Cover\]. Then $${\mathrm{Ext}}^1_A(Ac_0^\ast, Ac_{2\gamma}^\ast/Ac_{\gamma}^\ast Ac_{2\gamma}^\ast)=0.$$
We will prove ${\mathrm{Hom}}_{A}(Ax,Ac_{2\gamma}^\ast/Ac_{\gamma}^\ast Ac_{2\gamma}^\ast)=0,$ then the statement follows from . Consider $\phi:Ax\to Ac_{2\gamma}^\ast/Ac_{\gamma}^\ast Ac_{2\gamma}^\ast$ and $a:=\phi(c_0^\ast)$.
First, let $A$ be ${\mathcal B}_n(0)$, $J_n(0)$ or ${{\mathrm{TL}}}_n(0)$, the case ${\mathcal B}_{r,r}(0)$ is proved similarly. As $c_4^\ast x=x$, we have $a\in c_4^\ast A/(Ac_2^\ast) c_4^\ast$. By equation , we must have $$a=y_1 a- y_2a.$$ This means that $a$ is represented by a linear combination of diagrams, where each contains $4$ propagating lines, the $n/2-2$ cups and caps of $c_4^\ast$, but also either the cup connecting $1',2'$ or $3',4'$. This is an inconsistency, so $a=0$.
For $A={\mathcal{P}}_n(0)$, the dimension of $c_2^\ast (A/Ac_0^\ast A)c_2^\ast$ is 2. It follows quickly that no non-zero element $a$ satisfies $ya=0$, so $a=0$.
\[ext02\] Let $A$ be as in Theorem \[Thm0Cover\]. Then $${\mathrm{Ext}}^1_A(Ac_0^\ast,Ac_\gamma^\ast/Ac_0^\ast Ac_\gamma^\ast)=0.$$
Set $M:=Ac_\gamma^\ast/Ac_0^\ast Ac_\gamma^\ast$. The short exact sequence and Lemma \[Ac0A\] imply a short exact sequence $$0\to {\mathrm{Hom}}_A(Ac_\gamma^\ast,M)\to{\mathrm{Hom}}_A(Ax,M)\to {\mathrm{Ext}}^1_A(Ac_0^\ast,M)\to 0.$$ We will prove that for each algebra, $$\dim {\mathrm{Hom}}_A(Ax,M)\;\le\; d:=\dim c_\gamma^\ast(A/Ac_0^\ast A)c_\gamma^\ast =\dim H e_\gamma^\ast,$$ proving that the extension group must vanish.
First, let $A$ be ${\mathcal B}_n(0)$, $J_n(0)$ or ${{\mathrm{TL}}}_n(0)$, the case ${\mathcal B}_{r,r}(0)$ is proved similarly. If $a$ is the image of $x$ under a morphism $Ax\to M$, then, by equation , we have $$a= c_4^\ast a\quad\mbox{and}\quad a=y_1a-y_2a.$$ Hence $a$ must be represented by a linear combinations of diagrams, containing the $n/2-1$ caps of $c_2^\ast$, the $n/2-2$ cups of $c_4^\ast$ and a cup which either connects $1',2'$ or $3',4'$. The dimension of the space of such $a$ is $4$ for ${\mathcal B}_n(0)$ and $J_n(0)$ and $2$ for ${{\mathrm{TL}}}_n(0)$. In each case, imposing the actual condition that $a=y_1a-y_2a$, leaves half of the dimensions. For each case, this yields precisely $d$.
For $A={\mathcal{P}}_n(0)$, we have $d=1$ and the dimension of $c_2^\ast (A/Ac_0^\ast A)c_1^\ast$ is $3$. The subspace of elements that are annihilated by left multiplication with $y$ also has dimension $1$.
\[PropExt\] Maintain the notation and assumptions of Theorem \[Thm0Cover\]. We have $${\mathcal{R}}_1G\circ F(\overline{\Delta}(i,\nu))=0,\qquad\mbox{for all $i\in I_A$ and~$\nu\in\Lambda_i$.}$$
By Lemma \[LemGFiso\](2), it suffices to prove that $$\label{eqExtvan}{\mathrm{Ext}}^1_A(Ac_0^\ast,e_n^\ast\overline{\Delta}(i,\nu))=0.$$ By Lemma \[esc0\], this space is a subquotient of $${\mathrm{Hom}}_A(Ac_{2\gamma}^\ast,e_n^\ast\overline{\Delta}(i,\nu)),$$ which is zero when $i>2\gamma$, by Corollary \[Corin\]. Hence we focus on $i \in\{0,\gamma,2\gamma\}$.
By Lemma \[LemDi\], we have $e_n^\ast\overline{\Delta}(0,\emptyset)\cong Ac_0^\ast$, so is satisfied for $i=0$ by Lemma \[LemHom01\].
As $He_i^\ast$ is self-injective, for $D_i$ in Lemma \[LemDi\], we have a short exact sequence $$0\to e_n^\ast \overline{\Delta}(i,\nu)\to D_i\to Q_\nu\to 0,$$ where $Q_\nu$ has a filtration with sections $e_n^\ast\overline{\Delta}(i,\nu')$ with $\nu'\in\Lambda_i$. This gives an exact sequence $${\mathrm{Hom}}_A(Ac_0^\ast,Q_\nu)\to {\mathrm{Ext}}^1_A(Ac_0^\ast,e_n^\ast \overline{\Delta}(i,\nu))\to {\mathrm{Ext}}^1_A(Ac_0^\ast,D_i).$$ If $i\not=0$, the left-hand space is zero by . For $i\in\{\gamma,2\gamma\}$, the right-hand side is zero by Lemmata \[LemHom02\] and \[ext02\]. Hence the middle term is zero and is satisfied.
By Proposition \[PropExt\], we have $$\label{eqRG}
{\mathcal{R}}_1G(FM)={\mathrm{Ext}}^1_{A}(e_n^\ast C,FM)=0,\qquad\mbox{for any $M\in\overline{{{\mathcal{F}}}}$.}$$ Consider a short exact sequence $M_1\hookrightarrow M{\twoheadrightarrow}M_2$, with $M_i$ (and hence also $M$) in $\overline{{{\mathcal{F}}}}$. Using the above vanishing of cohomology, we find a commutative diagram with exact rows $$\label{commdia}\xymatrix{
0\ar[r] & M_1 \ar[r]\ar[d]^{\eta_{M_1}} & M\ar[r]\ar[d]^{\eta_M} &
M_2\ar[r]\ar[d]^{\eta_{M_2}} & 0\\
0\ar[r] & GFM_1 \ar[r] & GFM\ar[r] & GFM_2\ar[r] &0.
}$$ This implies that, if $\eta_{M_1}$ and $\eta_{M_2}$ are isomorphisms, so is $\eta_M$. Proposition \[PropHom\] can then be used to prove, by induction on the length of the filtration that we have $$\label{eqeta}\eta_M:\;\;M\;\,\tilde\to\;\, G\circ F(M),$$ for any $M\in\overline{{{\mathcal{F}}}}$. Hence, we have an isomorphism $${\mathrm{Hom}}_C(M,N)\cong {\mathrm{Hom}}_C(M,G\circ F( N))\cong {\mathrm{Hom}}_A(FM,FN),$$ induced by $F$. Now we consider arbitrary $M_1,M_2\in\overline{{{\mathcal{F}}}}$. The morphism $$F:\;{\mathrm{Ext}}^1_C(M_2,M_1)\to {\mathrm{Ext}}^1_A(FM_2,FM_1)$$ has left inverse induced by $G$, by . As furthermore $F\circ G\cong {\mathrm{Id}}$, the above morphism is an isomorphism. This concludes the proof.
Cell modules and standard systems
---------------------------------
\[ThmCellStan\] Assume that the field ${\Bbbk}$ is algebraically closed. Let $A$ be ${\mathcal B}_n(0)$, $J_n(0)$ or ${{\mathrm{TL}}}_n(0)$ with $n$ even, ${\mathcal{P}}_n(0)$ or ${\mathcal B}_{r,r}(0)$. The cell modules of the cellular algebra $A$ form a standard system if and only if the following condition is satisfied.
[ | l | l | ]{}\
algebra $A$& condition\
${\mathcal{P}}_n(0)$ & $n>2$ and ${{\rm{char}}}({\Bbbk})\not\in\{2,3\}$\
${\mathcal B}_n(0)$ & $n\not\in \{2,4\}$ and ${{\rm{char}}}({\Bbbk})\not\in\{2,3\}$\
$J_n(0)$ & $n\not\in \{2,4\}$ and ${{\rm{char}}}({\Bbbk})\not\in[2,n/2]$\
${{\mathrm{TL}}}_n(0)$ & $n\not\in \{2,4\}$\
${\mathcal B}_{r,r}(0)$ & $r>2$ and ${{\rm{char}}}({\Bbbk})\not\in\{2,3\}$\
First we prove the following lemma.
\[NegPn\] The cell modules of
1. ${\mathcal B}_2(0)$, $J_2(0)$, ${{\mathrm{TL}}}_2(0)$ and ${\mathcal B}_{1,1}(0)$ do not form a standard system, for any partial order;
2. ${\mathcal B}_4(0)$, $J_4(0)$, ${{\mathrm{TL}}}_4(0)$, ${\mathcal{P}}_2(0)$ and ${\mathcal B}_{2,2}(0)$ do not form a standard system.
First we prove part (1), we have $${{\mathrm{TL}}}_2(0)\cong{\mathcal B}_{1,1}(0)\cong {\Bbbk}[x]/(x^2).$$ The only cellular structure on this algebra gives two cell modules which are isomorphic. If ${{\rm{char}}}({\Bbbk})\not=2$, we have $${\mathcal B}_2(0)\cong J_2(0)\cong {\Bbbk}[x]/(x^2)\oplus {\Bbbk},$$ so the result follows as above. If ${{\rm{char}}}({\Bbbk})=2$, the algebra ${\mathcal B}_2(0)\cong J_2(0)$ has two isomorphic cell modules, induced from the sign and trivial ${\Bbbk}{\mathbb{S}}_2$-module, by Proposition \[PropCMod\].
Now we prove part (2). We have $c_{2\gamma}^\ast=1$ and, as $c_\gamma^\ast$ is an idempotent, equation implies $${\mathrm{Hom}}_A(Ax,A/Ac_\gamma^\ast A)\cong {\mathrm{Ext}}^1(Ac_0^\ast, A/Ac_\gamma^\ast A).$$ As $Ac_0^\ast\cong W(0,\emptyset)$ and $A/Ac_\gamma^\ast A$ has a filtration with sections of the form $W(2\gamma,\nu)$ with $\nu\in L_{2\gamma}$, the right-hand side must be zero in order to have a standard system for $(L,\le)$. However, we claim that there exists a non-zero morphism $$\phi: Ax\to A/Ac_\gamma^\ast A.$$ For $A={{\mathrm{TL}}}_4(0)$, we have $A/Ac_2^\ast A\cong {\Bbbk}$ and we can set $\phi(x)= 1$. For $A={\mathcal{P}}_2(0)$ we have $A/Ac_1^\ast A\cong{\Bbbk}{\mathbb{S}}_2$ and we can set $\phi(x)=1-s$, for $s$ the generator of ${\mathbb{S}}_2$. The other cases are left as an exercise.
By Proposition \[PropCMod\], under the conditions in Theorem \[Thm0Cover\], the cell modules of $A$ form a standard system if and only if the cell modules of $C $ form a standard system. The necessary and sufficient condition for the latter is given in Theorem \[ThmDiagram2\].
For the remaining cases, [*i.e.*]{} when Theorem \[Thm0Cover\] is not applicable, the cell modules do not form a standard system by Lemma \[NegPn\].
Stratified algebras {#TheApp}
===================
Homological stratification
--------------------------
We have the following alternative characterisations of standardly and/or exactly stratifying ideals.
\[LemStu\] Consider an idempotent ideal $J=AeA$ in $A$.
1. The ideal $J$ is standardly stratifying (${}_AJ$ is projective) if and only if
- multiplication induces an $A$-bimodule isomorphism $Ae\otimes_{eAe}eA\;\tilde\to\;J$, and
- the left $eAe$-module $eA$ is projective.
2. The ideal $J$ is exactly stratifying ($J_A$ is projective) if and only if
- multiplication induces an $A$-bimodule isomorphism $Ae\otimes_{eAe}eA\;\tilde\to\;J$, and
- the right $eAe$-module $Ae$ is projective.
We prove part (1), as part (2) then follows from considering $A^{{{\rm op}}}$. If $Ae\otimes_{eAe}eA\;\tilde\to\;J$, then the left $A$-module $J$ is induced from the left $eAe$-module $eA$. If the latter is projective then ${}_AJ$ is also projective, so the ideal $J$ is standardly stratifying. If $J$ is standardly stratifying, the conclusion follows from the proof of [@CPSbook Remark 2.1.2(b)].
An [*exact stratification of an algebra*]{} $A$ is a chain of ideals such that $J_{i}/J_{i-1}$ is exactly stratifying in $A/J_{i-1}$, see \[SecStratId2\](0), for $1\le i\le m$. For a chain we consider the algebras $A^{(i)}$ as in Remark \[DefAi\] and we identify $A^{(i)}$-modules with $f_{i+1}Af_{i+1}$-modules having trivial $f_{i+1}J_{i}f_{i+1}$-action. The following principle is well-known.
\[NewLemStr\] Consider an algebra $A$ with an exact stratification . For any $A^{(l)}$-module $M$ and $A^{(j)}$-module $N$, with $0\le l\le j\le m-1$, let $\widetilde{M}:=Af_{l+1}\otimes _{f_{l+1}Af_{l+1}}M$ and $\widetilde{N}:=Af_{j+1}\otimes_{f_{j+1}Af_{j+1}}N$. Then $${\mathrm{Ext}}^k_{A}(\widetilde{M},\widetilde{N})\;\cong\; \delta_{jl}\,{\mathrm{Ext}}^k_{A^{(l)}}(M,N),\qquad\mbox{for any }\; k\in{\mathbb{N}}.$$
First we claim that $J_{l}\widetilde{M}=0=J_{l}\widetilde{N}$. It suffices to show that $f_{l}$ acts as zero. Now $f_{l}$ is included in $f_{j+1}J_{l}f_{j+1}$, since $l\le j$. Action of $f_{j+1}J_{l}f_{j+1}$ on $\widetilde{N}$ gives $$f_{j+1}J_{l}f_{j+1}\otimes _{f_{j+1}Af_{j+1}} N=0.$$ This observation and [@CPSbook equation (2.1.2.1)] imply $${\mathrm{Ext}}^k_A(\widetilde{M}, \widetilde{N})\;\cong\; {\mathrm{Ext}}^k_{A/J_{l}}(\widetilde{M}, \widetilde{N}).$$ Consider the functors $$\Upsilon_i=(A/J_i)f_{i+1}\otimes_{A^{(i)}}-\;:A^{(i)}\mbox{{\rm-mod}}\to A/J_i\mbox{{\rm-mod}}.$$ By Lemma \[LemStu\](2) these are exact. As $A/J_{l}$-modules we have $\widetilde{M}\cong\Upsilon_lM$, and as $A/J_{j}$-modules $\widetilde{N}\cong\Upsilon_jN.$ Since the exact functor $\Upsilon_l$ is left adjoint to the exact functor $f_{l+1}-$, we have $${\mathrm{Ext}}^k_{A/J_{l}}(\widetilde{M}, \widetilde{N})\;\cong\; {\mathrm{Ext}}^k_{A^{(l)}}\left(M,f_{l+1}\Upsilon_j N\right).$$ Since we have $$f_{l+1}(A/J_{l}) f_{j+1}\;=\;\begin{cases} 0 &\mbox{for }\, l<j, \\ A^{(l)} &\mbox{for }\, l=j, \end{cases}$$ we find $f_l\Upsilon_l\cong{\mathrm{Id}}$ and $f_l\Upsilon_j=0$ if $l<j$, which concludes the proof.
Equivalence of ring and module theoretic definitions {#AppEq}
----------------------------------------------------
In case the quasi-order $\preceq$ is total, Definitions \[DefA1\] and \[DefA2\] agree. We give an overview of where this is proved, and use the numbering corresponding to Definitions \[DefA1\] and \[DefA2\].
1. This is [@CPSbook Theorem 2.2.3].
2. By [@Frisk Proposition 7], Definition \[DefA2\](2) is equivalent to demanding that Definition \[DefA2\](1) holds both for $A$ and $A^{{{\rm op}}}$. By definition the same relation holds between Definition \[DefA1\](1) and (2). So this case follows from the above case (1).
3. Both in Definitions \[DefA1\] and \[DefA2\] we find that going from (1) to (3) only corresponds to restricting from quasi-orders to orders, see Remark \[RemSSS\]. This case thus also follows from (1).
4. As above, this case follows from case (2) by going from quasi-orders to orders. Alternatively one can apply results in [@Dlab Theorem 5].
5. This is [@CPS Theorem 3.6].
Comparison with Kleshchev’s terminology {#AppKl}
---------------------------------------
For any class ${\mathcal B}$ of Noetherian, positively graded, connected algebras, Kleshchev introduced in [@AffineK Section 6] the notions of ${\mathcal B}$-standardly stratified, ${\mathcal B}$-properly stratified and ${\mathcal B}$-quasi-hereditary algebras. Even though the aim of [@AffineK] is to study infinite dimensional and graded algebras, the definitions also cover all the cases in Definition \[DefA1\]. Therefore, we assume that every grading is reduced to the zero component and introduce the classes ${\mathcal{S}}\subset {\mathcal{L}}\subset {\mathcal{D}}$, where ${\mathcal{D}}$ contains the finite dimensional unital algebras, ${\mathcal{L}}$ the quasi-local algebras and ${\mathcal{S}}$ the semisimple algebras. Then we have the following identification between our notions and those in [@AffineK]:
[ | l | l| l | l |l |]{}\
&& ${\mathcal B}={\mathcal{D}}$& ${\mathcal B}={\mathcal{L}}$ & ${\mathcal B}={\mathcal{S}}$\
${\mathcal B}$-standardly && standardly & strongly standardly&quasi-\
stratified && stratified & stratified &hereditary\
${\mathcal B}$-properly & &exactly standardly & properly&quasi-\
stratified &&stratified & stratified & hereditary\
${\mathcal B}$-quasi && properly & properly&quasi-\
-hereditary &&stratified & stratified&hereditary\
This follows by comparing [@AffineK Definitions 6.1 and 6.2] with \[SecStratId2\], using Lemma \[LemStu\].
Acknowledgement {#acknowledgement .unnumbered}
---------------
The research was supported by Australian Research Council Discover-Project Grant DP140103239 and an FWO postdoctoral grant.
The authors thank Steffen König, Julian Külshammer and Volodymyr Mazorchuk for very useful comments on the first version of the manuscript.
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K. Coulembier[[email protected]]([email protected])
School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia
R. B. Zhang[[email protected]]([email protected])
School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia
|
---
author:
- |
J.A. Gracey,\
Department of Applied Mathematics and Theoretical Physics,\
University of Liverpool,\
P.O. Box 147,\
Liverpool,\
L69 3BX,\
United Kingdom.
title: 'The Nambu–Jona-Lasinio model at $O(1/N^2)$.'
---
[**Abstract.**]{} We write down the anomalous dimensions of the fields of the Nambu–Jona-Lasinio model or chiral Gross Neveu model with a continuous global chiral symmetry for the two cases $U(1)$ $\times$ $U(1)$ and $SU(M)$ $\times$ $SU(M)$ at $O(1/N^2)$ in a $1/N$ expansion.
The Nambu–Jona-Lasinio, (NJL), model was introduced in [@1] as a theory with a continuous global chiral symmetry which is broken dynamically. It has also been studied in the context of hadronic physics following the early work of [@A] and lately in, for example, [@2], and chiral perturbation theory [@3] since it corresponds to a low energy effective theory of the strong interactions. Further, there has also been significant interest in analysing models with four fermi interactions, like the NJL model, in other contexts such as the standard model, [@B; @4; @5]. For instance, it was initially pointed out in [@B] and subsequently studied in more detail in [@4; @5] that the Higgs boson of the standard model could be regarded as a composite field built out of fermions in much the same way that the $\sigma$ and $\pi$ fields of the two dimensional chiral Gross Neveu, (CGN), model are bound states of the fundamental fermions of the model, [@6].
One of the most widely used techniques to examine the CGN or NJL models is the large $N$ expansion where the number of fundamental fields $N$ is allowed to become large, [@6], and $1/N$ can therefore be used as a dimensionless coupling constant. Consequently one can show, for example, that the models are renormalizable in $2$ $\leq$ $d$ $<$ $4$ dimensions in this approach [@7] and simultaneously deduce that they possess a rich structure, such as dynamical symmetry breaking. Whilst the leading order large $N$ analysis is well understood, it is important to go beyond the leading order to improve our knowledge of the quantum structure. Recently, new techniques to achieve this for theories with fermionic interactions were introduced in [@8] for the $O(N)$ Gross Neveu model based on the critical point self consistency methods to calculate critical exponents in the bosonic $O(N)$ $\sigma$ model, [@9; @10]. In this letter we present the results of the application of the same techniques to the $4$-fermi models with continuous chiral symmetry by writing down the anomalous dimensions of each of the fields in arbitrary dimensions at $O(1/N^2)$. This is one order beyond any previous analysis and it is necessary to have such higher order expressions in order to improve our understanding of the areas mentioned above. A further motivation for such a computation lies in the accurate independent evaluation of these quantities in three dimensions in order to provide precise estimates to compare with recent numerical simulations of $4$-fermi models, [@11].
More concretely, the lagrangians of the models we have considered are, [@6], $$L ~=~ i \bar{\psi}^i{\partial \!\!\! /}\psi^i + \sigma \bar{\psi}^i\psi^i
+ i\pi\bar{\psi}^i\gamma^5\psi^i - \frac{1}{2g^2}(\sigma^2 + \pi^2)$$ for the NJL model with a global $U(1)$ $\times$ $U(1)$ chiral symmetry and $$L ~=~ i \bar{\psi}^{iI}{\partial \!\!\! /}\psi^{iI} + \sigma \bar{\psi}^{iI}
\psi^{iI} + i\pi^a\bar{\psi}^{iI}\gamma^5\lambda^a_{IJ}\psi^{iJ}
- \frac{1}{2g^2}({\sigma^{}}^2 + {\pi^a}^2)$$ for the same model but with an $SU(M)$ $\times$ $SU(M)$ chiral symmetry. The bosonic fields $\sigma$ and $\pi^a$ are auxiliary and $g$ is the perturbative coupling constant. In (2) the generalized Pauli matrices, $\lambda^a_{IJ}$, $1$ $\leq$ $a$ $\leq$ $(M^2 - 1)$, $1$ $\leq$ $I$ $\leq$ $M$ are normalized to $\mbox{Tr}(\lambda^a\lambda^b)$ $=$ $4T(R)\delta^{ab}$ and we used the conventions of [@13] and [@14]. Both (1) and (2) involve $N$-tuplets of fermions $1$ $\leq$ $i$ $\leq$ $N$ and this $N$ will become our expansion parameter. We also use the conventions of [@11] in defining the properties of $\gamma^5$ as $\{\gamma^\mu,\gamma^5\}$ $=$ $0$, $\mbox{tr}(\gamma^5
\gamma^{\mu_1}\ldots\gamma^{\mu_n})$ $=$ $0$ and $(\gamma^5)^2$ $=$ $1$ with $\mbox{tr}1$ $=$ $2$. We remark that our $\gamma^5$ conventions in $d$-dimensions retain the anticommutativity property. This differs from the definition given in [@C] which is one formulation used to perform consistent renormalization calculations using dimensional regularization where the spacetime dimension is changed to provide a way of handling infinities, [@C; @D]. There, [@C], one loses Lorentz invariance in the full $d$-dimensional space where $[\gamma^\mu,\gamma^5]$ $=$ $0$ when $\mu$ is not an index in the physical spacetime. By contrast, the method of [@9] uses propagators defined in arbitrary but [*fixed*]{} dimensions which are consistent with Lorentz and conformal symmetry. Therefore it seems more appropriate here to retain the anticommutativity of the $\gamma^5$ for a Lorentz invariant formulation. (Indeed an alternative to [@C] for treating $\gamma^5$ in dimensionally regularized calculations retains this condition, [@D].)
In [@8], the model with $\pi$ $=$ $0$ was solved at $O(1/N^2)$ by the self consistency approach of obtaining critical exponents and we have used the same methods to treat (1) and (2). Briefly, this involves solving for critical exponents at the $d$-dimensional fixed point of the field theory where the model possesses an extra scaling or conformal symmetry. With this scaling property one writes down the most general form the propagators of the fields can take, consistent with Lorentz and conformal symmetry, where the powers of the scaling form are related to the dimension and therefore the anomalous dimension of the fields, [@11]. To obtain analytic expressions for these critical exponents one represents the skeleton Dyson equations of various Green’s functions by the scaling forms and solves the resulting representation of the Dyson equations for the unknown exponents order by order in $1/N$. The power of the method is illustrated by the fact that the leading order results are deduced algebraically and agree with previous work, whilst the new results at the subsequent order are deduced with a minimal amount of effort, \[10-12\].
For (1), the asymptotic scaling forms of the respective propagators are, in coordinate space, in the critical region, $$\psi(x) ~\sim~ \frac{A{x \!\!\! /}}{(x^2)^\alpha} ~~,~~
\sigma(x) ~\sim~ \frac{B}{(x^2)^\beta} ~~,~~
\pi(x) ~\sim~ \frac{C}{(x^2)^\gamma}$$ where $A$, $B$ and $C$ are amplitudes which are independent of $x$ and the dimensions of the fields are defined as $$\alpha ~=~ \mu + {\mbox{\small{$\frac{1}{2}$}}}\eta ~~,~~ \beta ~=~ 1 - \eta - \chi_\sigma ~~,~~
\gamma ~=~ 1 - \eta - \chi_\pi$$ where $d$ $=$ $2\mu$ is the spacetime dimension, $\eta$ is the fermion anomalous dimension and $\chi_\sigma$ and $\chi_\pi$ are the anomalous dimensions of the respective vertices. The anomalous dimensions are expanded in powers of $1/N$ via, for example, $\eta$ $=$ $\sum_{i=1}^\infty \eta_i/N^i$. By substituting the scaling forms (3) into the skeleton Dyson equations with dressed propagators, which are illustrated in figs. 1-3, we were able to determine $\eta_2$ using results obtained in [@11]. We found $$\begin{aligned}
\eta_1 &=& - \, \frac{2\Gamma(2\mu-1)}{\Gamma(\mu-1)\Gamma(1-\mu)
\Gamma(\mu+1)\Gamma(\mu)} \\
\eta_2 &=& \eta^2_1 \left[ \Psi(\mu) ~+~ \frac{2}{\mu-1}
{}~+~ \frac{1}{2\mu} \right]\end{aligned}$$ where $\Psi(\mu)$ $=$ $\psi(2\mu-1)$ $-$ $\psi(1)$ $+$ $\psi(2-\mu)$ $-$ $\psi(\mu)$ and $\psi(x)$ is the logarithmic derivative of the $\Gamma$-function, as well as $\chi_{\sigma \, 1}$ $=$ $\chi_{\pi \, 1}$ $=$ $0$. Equation (6) represents the first $O(1/N^2)$ quantity to be determined for (1).
The $O(1/N^2)$ corrections to the vertex anomalous dimensions were deduced by following the analogous calculation for the $O(N)$ Gross Neveu model given in [@14]. It involves studying the scaling properties of the $3$-point functions at $O(1/N^2)$ using a method which developed the leading order work of [@15; @16] for the bosonic $\sigma$ model on $CP(N)$. Rather than illustrate the large number of graphs which arise at $O(1/N^2)$, we have given in figure 4 the basic structure of the distinct graphs which arise, though the graphs with vertex counterterms have not been shown. The basic integrals corresponding to each of the graphs have been given in [@14] and it was therefore a straightforward exercise to manipulate the graphs which occur in (1) to be proportional to integrals whose values are already known, [@14]. For (1), at $O(1/N^2)$ we found that the degeneracy in the $(\sigma,\pi)$ sector was not lifted but unlike at leading order the fields now have a non-zero anomalous dimension, which is a new feature, ie $$\chi_{\sigma \, 2} ~=~ \chi_{\pi \, 2} ~=~
- \, \frac{\mu^2(4\mu^2-10\mu+7) \eta^2_1}{2(\mu-1)^3}$$
We have also carried out the same calculation for the non-abelian case (2) and we note the results obtained at leading order are $$\eta_1 ~=~ \frac{\tilde{\eta}_1}{2} \left[{{\frac{1}{M}}}+{\frac{C_2(R)}{T(R)}}\right]$$ where $\tilde{\eta}_1$ $=$ $-\,2\Gamma(2\mu-1)/[\Gamma(\mu+1)\Gamma(\mu)
\Gamma(1-\mu)\Gamma(\mu-1)]$ and $$\begin{aligned}
\chi_{\sigma \, 1} &=& \frac{\mu\tilde{\eta}_1}{2(\mu-1)}
\left[{{\frac{1}{M}}}-{\frac{C_2(R)}{T(R)}}\right] \\
\chi_{\pi \, 1} &=& \frac{\mu\tilde{\eta}_1}{2(\mu-1)}
\left[ {\frac{C_2(R)}{T(R)}}- {{\frac{1}{M}}}- \frac{C_2(G)}{2T(R)} \right]\end{aligned}$$ where $\lambda^a \lambda^a$ $=$ $4C_2(R) I$, $f^{acd} f^{bcd}$ $=$ $C_2(G)
\delta^{ab}$ and $C_2(R)$ $=$ $(M^2-1)/2M$, $C_2(G)$ $=$ $M$ for $SU(M)$, [@12; @13]. Several leading order exponents were calculated in [@11] for $SU(2)$ $\times$ $SU(2)$ and our results are in agreement with them which provides us with a partial check on our calculation. At next to leading order the expressions are more involved compared to (6) and (7). We found $$\begin{aligned}
\eta_2 &=& \frac{\tilde{\eta}^2_1}{4} \left[ \left({{\frac{1}{M}}}+{\frac{C_2(R)}{T(R)}}\right)^2
\left(\Psi(\mu) + \frac{2}{\mu-1} + \frac{1}{2\mu} \right) \right. \\
&+& \left. \frac{\mu}{(\mu-1)} \left( \left( {{\frac{1}{M}}}-{\frac{C_2(R)}{T(R)}}\right)^2
+ \frac{C_2(G)C_2(R)}{2T^2(R)} \right)
\left( \Psi(\mu) + \frac{3}{2(\mu-1)} \right) \right] \nonumber\end{aligned}$$ $$\begin{aligned}
\chi_{\sigma \, 2} &=& \frac{\mu\tilde{\eta}^2_1}{4(\mu-1)^2}
\left[ (2\mu-1)\left( \frac{1}{M^2} - \frac{C^2_2(R)}{T^2(R)} \right)
\left( \Psi(\mu) + \frac{1}{(\mu-1)}\right) \right. \nonumber \\
&+& \left. \frac{\mu C_2(R)C_2(G)}{2T^2(R)} \left( \Psi(\mu)
+ \frac{1}{(\mu-1)} \right) + \frac{3\mu}{2(\mu-1)} \left( \frac{1}{M}
- \frac{C_2(R)}{T(R)}\right)^2 \right. \nonumber \\
&+& \left. \frac{5\mu C_2(R)}{(\mu-1)T(R)} \left({{\frac{1}{M}}}- {\frac{C_2(R)}{T(R)}}\right)
- \frac{2\mu}{(\mu-1)}\left(\frac{1}{M^2} - \frac{C^2_2(R)}{T^2(R)} \right)
\right. \nonumber \\
&+& \left. \frac{\mu}{2(\mu-1)}\left({{\frac{1}{M}}}+{\frac{C_2(R)}{T(R)}}\right)^2
- \frac{\mu(2\mu^2-5\mu+4)}{(\mu-1)M} \left({{\frac{1}{M}}}+ \frac{3C_2(R)}{T(R)} \right)
\right. \nonumber \\
&+& \left. \frac{\mu}{M}\left( 3(\mu-1)\Theta(\mu)
- \frac{(2\mu-3)}{(\mu-1)}\right) \left({{\frac{1}{M}}}- {\frac{C_2(R)}{T(R)}}\right) \right]\end{aligned}$$ and $$\begin{aligned}
\chi_{\pi \, 2} &=& \frac{\mu\tilde{\eta}^2_1}{4(\mu-1)^2}
\left[ \Psi(\mu) + \frac{1}{(\mu-1)} \right] \nonumber \\
&& \times \left[(2\mu-1)\left({{\frac{1}{M}}}+{\frac{C_2(R)}{T(R)}}\right)
\left({\frac{C_2(R)}{T(R)}}-{{\frac{1}{M}}}-\frac{C_2(G)}{2T(R)}\right) \right. \nonumber \\
&&- \left. \frac{\mu C_2(G)}{2T(R)} \left({\frac{C_2(R)}{T(R)}}-\frac{2}{M}-\frac{C_2(G)}{2T(R)}
\right) \right] \nonumber \\
&+& \frac{3\mu^2\tilde{\eta}^2_1}{4(\mu-1)^3}
\left({\frac{C_2(R)}{T(R)}}-{{\frac{1}{M}}}-\frac{C_2(G)}{2T(R)}\right)^2 \nonumber \\
&+& \frac{5\mu^2\tilde{\eta}^2_1}{16(\mu-1)^2M} \left[ 4 \left({\frac{C_2(R)}{T(R)}}-{{\frac{1}{M}}}\right)
- {\frac{C_2(G)}{T(R)}}\right] \nonumber \\
&+& \frac{2\mu^2\tilde{\eta}^2_1}{(\mu-1)^3} \left[ \frac{1}{M^2}
-\frac{C^2_2(R)}{T^2(R)} - \frac{C_2(G)}{2T(R)M} - \frac{C^2_2(G)}{8T^2(R)}
+ \frac{C_2(R)C_2(G)}{T^2(R)} \right. \nonumber \\
&+& \left. \frac{1}{16} \left( \left( {\frac{C_2(R)}{T(R)}}+{{\frac{1}{M}}}\right)^2 - \frac{C_2(G)}
{MT(R)} - \frac{3C_2(G)C_2(R)}{2T^2(R)} + \frac{C^2_2(G)}{2T^2(R)} \right)
\right. \nonumber \\
&-& \left. \frac{3}{16}\left( {{\frac{1}{M}}}- {\frac{C_2(R)}{T(R)}}+ \frac{C_2(G)}{2T(R)} \right)^2
- \frac{(2\mu^2-5\mu+4)}{8} \left( \frac{3}{M^2}
+ \frac{C_2(R)}{T(R)M} \right. \right. \nonumber \\
&-& \left. \left. \frac{C_2(G)}{4T(R)}\left({\frac{C_2(R)}{T(R)}}-\frac{3}{M}\right)
+\left({\frac{C_2(R)}{T(R)}}-{{\frac{1}{M}}}\right)^2 \right) \right. \nonumber \\
&-& \left. \frac{(\mu-1)^2}{8} \left( 3\Theta(\mu) - \frac{(2\mu-3)}{(\mu-1)^2}
\right) \left(\frac{1}{M^2} - \frac{C_2(R)}{MT(R)} + \frac{C_2(G)}{MT(R)}
\right. \right. \nonumber \\
&-& \left. \left. \left({\frac{C_2(R)}{T(R)}}-{{\frac{1}{M}}}- \frac{C_2(G)}{4T(R)} \right)
\left( {\frac{C_2(R)}{T(R)}}-{{\frac{1}{M}}}- \frac{C_2(G)}{2T(R)} \right) \right) \right]\end{aligned}$$ where $\Theta(\mu)$ $=$ $\psi^\prime(\mu)$ $-$ $\psi^\prime(1)$, and we have used the results of [@12] in manipulating the $f^{abc}$ and $d^{abc}$ tensors which arise in the $3$-loop graphs of fig. 4. We have expressed our results in as general a form as possible which allows one to check that each expression does agree with the analogous results of [@8; @14] and the $O(N)$ model.
We conclude with the observation that our results will prove to be extremely useful in establishing which other models lie in the same universality class as (1) and (2) since, for example, we have provided independent analytic expressions which can now be expanded in powers of $\epsilon$ and compared with $\epsilon$-expansions of critical exponents of other models deduced from the corresponding perturbative renormalization group functions.
[**Acknowledgement.**]{} The author thanks Dr S.J. Hands for a brief conversation.
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Fig. 1.
: Dressed skeleton Dyson equation for $\psi$.
Fig. 2.
: Dressed skeleton Dyson equation for $\sigma$.
Fig. 3.
: Dressed skeleton Dyson equation for $\pi$.
Fig. 4.
: Vertex corrections with dressed propagators.
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